{"Bibliographic":{"Title":"Lecture notes for use at workshops on surveying instrumentation and coordinate computation","Authors":"","Publication date":"1971","Publisher":""},"Administrative":{"Date created":"08-16-2023","Language":"English","Rights":"CC 0","Size":"0000263543"},"Pages":["Holdahl\nTA\n562\n.L43\n1971\nU.S. DEPARTMENT OF COMMERCE\nNational Oceanic and Atmospheric Administration\nNational Ocean Survey\nRockville, Maryland\nLECTURE NOTES FOR USE AT WORKSHOPS ON SURVEYING\nINSTRUMENTATION AND COORDINATE COMPUTATION\nBy\nJoseph F. Dracup\nCarl F. Kelley\nGeorge B. Lesley\nRaymond W. Tomlinson\nOffice of National Geodetic Survey","TA\n562\nL43\n1971\nTABLE OF CONTENTS\nPage\n1\nTheodolites\nby Carl F. Kelley\nBasic Principle - Electro-Optical Distance\n32\nMeasuring Instruments\nby George B. Lesley\n60\nElectronic Distance Measuring Instruments\nby Raymond W. Tomlinson\n80\nRectangular Coordinate Systems\nby Joseph F. Dracup\nCorrections to Measured Lengths and\n105\nComputation of Grid Azimuths\nby Joseph F. Dracup\n120\nComputations and Adjustments\nby Joseph F. Dracup\n213\nBibliographies\nNOTE: Sketch and table numbers are not numbered\nconsecutively. Consequently, references to\nsketches and tables in the text of a particular\nreport are limited to that report only.\nJAN 30 1995\nN.O.A.A.\nU.S. DEPT. OF COMMERCE","THEODOLITES\nCarl F. Kelley\nSupervisory Geodesist\nGeodesy Division\nNational Geodetic Survey\nNational Ocean Survey\nNational Oceanic and Atmospheric Administration\nU. S. Department of Commerce\nRockville, Maryland 20852\nIntroduction\nMany surveyors avoid using or even becoming familiar\nwith theodolites because they fear the name, theodolite,\nand its inference to control surveys. This fear is unjus-\ntified as the definitions below will illustrate.\nTHEODOLITE: A precise surveying instrument consisting\nof an alidade with telescope, mounted on\nan accurately graduated circle, equipped\nwith necessary levels and reading de-\nvices. Sometimes the alidade carries\na graduated vertical circle. The two\nclasses of theodolites are direction\ninstruments and repeating instruments.\nA surveying instrument composed of a\nTRANSIT:\nhorizontal circle graduated in circular\nmeasure and an alidade with a telescope\nwhich can be reversed in its supports\nwithout being lifted therefrom.","Based on these definitions it can be stated that a\ntheodolite is nothing more than a glorified transit. How-\never, such instruments provide a system for obtaining\nreliable horizontal and vertical directions with a minimum\nof effort.\nThis text is designed to familiarize the reader with\ntheodolites in general and point out major principles in\ntheir operation. As most theodolites are similar, to avoid\nrepetition small differences will not be explained. However,\nto maintain continuity all aspects of one type theodolite,\nthe Wild T-2, will be discussed with other types being\nreferenced whenever necessary. Specific information on the\ninstrument of your choice may be found in the manual which\nwill accompany your instrument. Read this manual prior to\noperating the instrument.\nUnpacking the Instrument\nTheodolites are normally transmitted in a shipping\ncase which, including instrument, has a total weight of\napproximately 35 pounds. The removal of the instrument,\nwhich is in a carrying case, from the shipping container\nwill present no problem. It is accomplished utilizing the\ncarrying strap or handle on the carrying case. As with all\ninstruments be sure to place the case containing the instru-\nment on a firm level surface when setting it down.\n2","The cover, lid, or door of the carrying case is then\nremoved or opened as the case may be. The cover for the\nT-2 case is connected to the base of the case by two catches.\nBy pulling outward on the top of these catches, the top is\nfreed and may be lifted off the instrument and base.\nThe instrument is connected to the base of the carrying\ncase during transportation to prevent damage. The T-2 is\nloosened from the base by freeing three sliding levers that\nhold it in place and sliding them away from the instrument.\nInstruments in wooden carrying cases are normally held in\nplace by rotating wooden blocks and slid into the case\nrather than set on the base. The T-2 is always placed in\nthe case with the telescope in the vertical position eye\npiece down and clamped although this procedure may differ\nwith other instruments. The T-2 is then grasped by both Y\naxes, near the base, and removed from the case base and\nplaced on the tripod. The instrument should be held in a\nnear vertical position during this operation and care taken\nto assure that it is secured to the tripod thus preventing\nits being knocked off.\nThe reverse procedure is followed when replacing the\ninstrument in the carrying case, which is where it should\nbe when transported from station to station. In no case\nshould the theodolite be left attached to the tripod, and\n3","the two units carried in a horizontal position as the\ninstrument especially the spindle could be damaged.\nTripod Setup, Leveling Instrument\nPrior to unpacking the instrument, the tripod should\nbe set over the point with the plate approximately level.\nThis is accomplished by the use of a plumb bob which is\ninserted in the fastening plug. When performing this\norientation, be sure the plug is in the center of the tri-\npod to allow for final centering when the instrument is\nplaced on the tripod. The legs of the tripod are firmly\ndug into the ground (for stabilizing remove any sod which\nmay be around the legs of the instrument) by stepping on\nthe foot pieces. If good observations are to be obtained,\nyou must have a stable setup. In some cases it is necessary\nto anchor or stake the tripod legs, place a walking plat-\nform around the tripod and so forth to insure stability.\nAfter placing the instrument on the tripod, final\ncentering is performed. This is often performed with a\nplumb bob, but when the instrument is equipped with an op-\ntical plummet, it may be used. Some tripods such as the\nKern are equipped with a centering rod or forced centering\ndevice. The methods for utilizing the forced centering rod\nand checking the optical plummet will be outlined in the\ninstrument's handbook.\nUsing the circular level or bull's-eye bubble, bring\nthe instrument into approximate level. Using the horizontal\n4","or plate level, the instrument is now leveled exactly.\nRotate the instrument SO that the level vial is parallel\nto any two of the three leveling screws. Bring the bubble\nto center, using these two screws as shown in Figure l.\nTurn the instrument 90° and center the bubble, using the\nthird screw. Return to the first position to see that the\nbubble is still centered. Now, turn the instrument 180°.\nIf the bubble is centered, the instrument is now level. If\nnot, (Figure 2) the level is out of adjustment.\nT-3\nT-2\n5\n5\n10\n20\n25\n30\n15\n0\n11\nFigure 1\n111\nFigure 2\nI\nFigure 3\nUnless the bubble is out a large amount, it should\nnot be adjusted as it adds to the time required to level the\ninstrument and has a tendency to wear the threads of the\n5","adjusting screws and nuts. Instead, after the normal level-\ning procedure has been completed, rotate the alidade SO\nthat the bubble axis is in line with one of the foot screws.\nNote, by the graduation, the position of the lower end of the\nbubble (toward the vertical circle), then rotate to 180°.\nNote again the position of the same (lower) end of the bubble\n(Figures 1 & 2). The mean of these two readings is the so-\ncalled \"reversing position\" of the bubble. Assume, for\nexample, that the first reading is 4.6 and the second reading\nafter rotating 180° is 7.2. The reversing point is 5.9. Now\nturn the alidade 90° SO that the bubble axis is parallel to\nthe other two foot screws. Bring the bubble to its reversing\nposition by rotating these foot screws equally in opposite\ndirections (Figure 3). This procedure may be repeated to\nrefine the determination of the reversing position. Once\nthe reversing position is determined, all that needs to be\ndone to level the theodolite is to use the foot screws, only,\nto bring the level to its reversing position in each of the\ntwo directions of the alidade 90° apart.\nNormally, only one end of the bubble need be considered\nsince it is not likely to change significantly in length\nduring the period of observations. By employing this method,\nthe level of the theodolite may be checked and corrected very\nsimply without resorting to adjusting the vial itself.\n6","The key to this method is that once the reversing\nposition of the bubble is well determined and the theodolite\nis leveled as described, then the vertical axis on which the\nalidade rotates is truly vertical and the bubble will remain\nin the same position, whether centered or not, in any direc-\ntion the alidade is rotated. The reversing position of the\nbubble may drift slightly from day to day or even during a\nperiod of observations; thus, it should be checked before\neach observing period and occasionally during the period.\nIt should seldom be necessary to readjust the bubble vial.\nWhen it is necessary to readjust the vial, the instru-\nment should be leveled as indicated above and the bubble\nbrought to center, using the level adjusting screw nuts\nonly. If the means of adjusting the level vial is a screw\nwhich opposes a spring, the last turn in bringing the bubble\nto center should be made against the spring.\n0\n00\nFigure 4\nThe bubble travels in the direction of the\nleft thumb when you are facing the instrument.\nPrior to starting observations a final check should be\nmade with the instrument in level to assure that it has re-\nmained centered over the point.\n7","Parallax and Focusing\nParallax is the apparent movement of the cross-hairs\nover the image of the target when the eye is moved from\nside to side. The following procedures will eliminate\nparallax:\n1. Using the black milled ring on the eye-piece,\nmake the crosswires very sharp and black while\nsighting on a moderately bright background approxi-\nmately 300 meters away or on the sky.\n2. Sight on an object approximately 100 meters away\nand focus it sharply with the focusing ring.\n3. Move the head from side to side to see if parallax\nexists.\n4. If it does, use the milled ring to eliminate the\nparallax. It will be necessary to correct the\nfocus each time the milled ring is moved.\n5. Note the setting on the milled ring so that it\nmay be reset after dark when this test is not\npossible.\nFor focusing of the reading microscope, the milled\noccular tube is rotated while the graduation become sharply\ndefined.\nPointings\nThe definition of the term \"pointing\" is critical as\nthe operation is involved in performing this act. To obtain.\n8","relatively rapid and accurate readings, the observer should\npoint and not aim at his target (s) . Aiming, or excessive\ntime expended while pointing on an object; will cause eye\nstrain and give the object the appearance of moving. The\nfirst effort or when the target first appears to be properly\naligned with the cross hairs is normally the most reliable.\nWhen moving the target into the cross hairs with the\nslow motion screw, it is recommended that the final movement\nbe made in a clockwise direction. This procedure will elimi-\nnate small errors caused by slack in the slow motion screen.\nAll pointing should be made by placing the object near\nthe point where the horizontal and vertical wires cross and\npreferably at their intersection. Small deviations from\nthis point are allowable, but all pointings should be made\nwith the object in the same relationship with the cross hairs\nto eliminate errors due to possible tilt of the hairs.\nThe following figures illustrate typical cross hair\narrangements and indicate the recommended location of the\ntarget with a dot. These locations are for horizontal di-\nrections.\nFigure 5\n9","The horizontal and vertical circles are normally read\nusing a microscope which adjoins the telescope. In some\ninstruments such as the T-2, it is necessary to rotate a\nknob allowing you to view either the horizontal or vertical\ncircle. By turning this knob clockwise to the stop, the\nhorizontal circle is visible, and likewise by turning it\ncounterclockwise to the stop, the vertical circle may be\nviewed. In some instruments both circles are visible at\nthe same time.\nThe illumination of the circles should be as bright\nand equally illuminated as possible. During the measurement\nof an angle, the illumination should not be changed. In\nthe reading microscope appear the images of the diametri-\ncally opposite portions of the circle, separated by a\nfine horizontal line, and below that is the scale image\nof the micrometer drum (this system varies with different\ninstruments). . Circles for the Wild T-2, DKM 2A and THEO 010\nare shown in Figure 6. As this figure illustrates the\nDKM 2A provides a digitized reading double circle system.\nThe 10 minute multiples are bracketed by the index lines\nfor every setting of the micrometer.\n10","28 129 130\nJENA THEO 010\n5\n129° 25' 47.5\nOLE\n60€\n80\n98\n98\nWILD T-2\n265 266\n265° 47' 23.6\n20\n30\n124\n5432\nV\nKERN DKM 2-A\n56° 23' 37\".0\nH\n56\n5\n4\n3\n2\n33\n30 40\nFigure 6\n11","Horizontal Circle\nThe graduation of the circle may be determined by\ndividing the number of divisions between degrees into 60.\nReading of this circle may be accomplished using the index\nmarker or by counting the number of divisions between degree\nreadings which are 180° opposite and multiplying by 1/2 the\nvalue of each graduation.\n59\n98\n266\n265\nFigure 7\nEach division equals 20 minutes.\nThere are four divisions between 265° and 85°.\n4 X 20 = 40 minutes.\n2\nReadings therefore indicate 265° 40'.\nThe additional minutes and seconds are obtained from\nthe micrometer circle.\nVertical Circle\nThe same procedure may be used for reading the vertical\ncircle. . Using the procedure of counting the division is\noften more reliable than using the index marker as it may\nbe in error. The circles must be brought into coincidence\nbefore these readings are made.\n12","Electrical Illumination\nMost modern theodolites are equipped with electrical\nillumination in addition to mirrors which may be used for\nobtaining light from natural sources. It is recommended\nthat electrical illumination be used for all observations\nas it provides a constant and bright illumination for all\nplate readings. You may also vary the intensity of your\nlight by use of a rheostat.\nA diffused light bulb is essential in properly illumi-\nnating the horizontal circle. You may find that frosted\nbulbs do not accomplish this to a desirable degree. For\nthis reason many bulb tubes are equipped with colored lens\nto further diffuse the light. The bulbs may be modified\nby painting them white or placing a piece of paper over the\nend of the light tube when the colored filament is not\navailable.\nHorizontal Plate Circle Settings\nWhen observing horizontal angles, it is essential\nthat different plate settings be used for each position.\nThese positions should be selected in such a manner that\nthe entire range of the circle and micrometer are utilized.\nThis is necessary to minimize any errors that may be present\nin the circle and micrometer. The following tables for 4,\n8, 12, and 16 positions will accomplish this objective and\nare therefore recommended.\n13","CIRCLE SETTINGS\n(1)\nFour Positions of Circle\nWild T-3\n5-Minute\n10-Minute\nMicrometer\nMicrometer Drum\nMicrometer Drum\nCircle\nReadings\n0° 00' 40\"\n0°\n0°\n00'\n10\"\n00'\n10 units\nl\n45\n45\n40\n45\n02\n00\n25 units\n2\n01\n50\n3\n90\n03\n10\n90\n05\n10\n90\n00\n35 units\n4\n04\n40\n135\n20\n135\n07\n135 00\n50 units\n(2)\nEights Positions of Circle\n40\n10\n0\n00\n10 units\n1\n0\n00\n0\n00\n01\n25\n22\n00\n25 units\n2\n22\n01\n50\n22\n45\n45\n40\n45\n3\n03\n10\n02\n00\n35 units\n67\n4\n67\n04\n67\n20\n03\n55\n00\n50 units\n40\n00\n10 units\n5\n90\n00\n90\n05\n10\n90\n06\n6\n112\n01\n50\n112\n25\n112\n00\n25 units\n40\n135\n07\n135\n00\n35 units\n7\n135\n03\n10\n8\n04\n08\n157\n20\n157\n55\n157\n00\n50 units\n14","(3) Twelve Positions of Circle\nWild T-3\n5-Minute\n10-Minute\nMicrometer\nMicrometer Drum\nMicrometer Drum\nCircle\nReadings\n0° 00'\n40\"\n0° 00'\n10\"\n0° 00'\n10 units\nl\n15\n00\n25 units\n2\n15\n01\n50\n15\n01\n50\n3\n30\n03\n10\n30\n03\n30\n30\n00\n35 units\n4\n45\n04\n45\n45\n20\n05\n10\n00\n50 units\n60\n40\n60\n06\n60\n5\n00\n50\n00\n10 units\n6\n08\n75\n01\n50\n75\n30\n75\n00\n25 units\n7\n90\n03\n10\n90\n00\n10\n90\n00\n35 units\n8\n04\n50 units\n105\n20\n105\n01\n50\n105\n00\n40\n120\n120\n00\n10 units\n9\n120\n00\n03\n30\n135\n00\n25 units\n10\n135\n01\n50\n135\n05\n10\n06\n11\n150\n03\n10\n150\n50\n150\n00\n35 units\n165\n165\n04\n165 08 30\n00\n50 units\n12\n20\n15","(4) Sixteen Positions of Circle\nWild T-3\nMicrometer\n10-Minute\n5-Minute\nCircle\nReadings\nMicrometer Drum\nMicrometer Drum\n0° 00\"\n10 units\n0°\n10\"\n40\"\n001\n0° 00'\n1\n25 units\n11\n01\n25\n11\n00\n11\n01\n50\n2\n35 units\n40\n22\n00\n22\n02\n22\n03\n10\n3\n50 units\n33\n00\n04\n03\n55\n4\n20\n33\n33\n45\n10 units\n45\n00\n45\n40\n05\n10\n00\n5\n56\n25 units\n56\n06\n00\n25\n56\n6\n01\n50\n67\n35 units\n67\n40\n00\n07\n67\n03\n10\n7\n78\n50 units\n78\n08\n00\n78\n04\n55\n8\n20\n10 units\n90\n00\n40\n90 00 10\n90\n00\n9\n25 units\n101\n00\n01\n25\n101\n01\n50\n101\n10\n40\n112\n00\n35 units\n112\n02\n10\n112\n03\n11\n50 units\n123\n00\n04\n123\n03\n55\n20\n123\n12\n10 units\n135\n00\n40\n135\n05\n10\n00\n13\n135\n146\n25 units\n146\n06\n00\n25\n146\n14\n01\n50\n40\n35 units\n157\n00\n157\n07\n157\n03\n10\n15\n168\n50 units\n168\n08\n00\n55\n168\n04\n16\n20\nThe Kern DKM-3 theodolite is an example of an instrument where\nthe micrometer drum has a range of 5 minutes.\nThe Wild T-2 and the Kern DKM-2 theodolites are examples of\ninstruments whose micrometer drums have a range of 10 minutes.\n16","Recording\nThe following formats illustrate the recommended form\nfor recording horizontal and vertical observational data:\nHorizontal Directions\nStation: BALDY\nInstrument: Wild T-2 Date: 7/31/68\nMean Direction\nPosition Observed D or R\nO\n\"\n* \"\nMean D & R\n\"\nl\nMARBLE\nD\n0 00 15 16 15.5\n180 00 10 12 11.0 13.2 00.0\nVALE\n79 47 46 48 47.0\nD\n259 47 40 43 41.5 44.2 31.0\nRUSH\n104\nD\n09\n22\n23\n22.5\n284 09 18 19 18.5 20.5 07.3\n2\nMARBLE\nR\n191 01 28 30 29.0\n11 01 33 34 33.5 31.2 00.0\nVALE\nR\n270 49 02 04 03.0\n90 49 06 07 06.5 04.8 33.6\nRUSH\nR\n295 10 35 36 35.5\n115 10 40 40 40.0 37.8 06.6\n*To insure against blunders, the micrometer should be\nbrought into coincidence twice and both the readings re-\ncorded. The spread between the readings should not ex-\nceed 3\" or 3 units for a Wild T-3.\n17","Vertical Directions\nStation: BALDY\nInstrument Wild T-2\nDate 7/31/68\nTel\nMean\nPosition Observed\nD or R\no\nMean\nD & R Direction\n1\nMARBLE\nD\n87 10 07.0 05.0 06.0\nR\n272 50 03.5 04.5 04.0 02.0 87 10 01.0\n87 10 02.5 02.5 02.5\nD\nR\n272 50 10.0 08.0 09,0 53,5 87 09 56.7\nD\n87 10 09.0 07,5 08.2\nR\n272 50 08.5 09.0 08.8 59.4 87 09 59.7\n87\n09\n59.1\n90 19 10.0 12.0 11.0\n2\nVALE\nD\n269 41 08.0 06.0 07.0 04.0 90 19 02.0\nR\n90 19 11.0 12.0 11.5\nD\n269 40 58.0 59.0 58.5 03.0\n01.5\nR\n90 19 08.0 10.0 09.0\nD\n269 41 06.0 08.0 07.0 02.0\nR\n01.0\n90 19 02.2\nObserving Procedures\nObservations should be made following procedures which\nremove the collimation error of the instrument and range\nover the complete circle and micrometers. Recommended\ncircle settings for various instruments and number of positions\nwere given earlier.\nAlthough other observational practices could be de-\nveloped that would satisfy the above requirements, the pro-\ncedures given in US C&GS Special Publication No. 247 are as\nsimple as most and once the pattern is developed become\nalmost automatic.\n18","To illustrate this technique, let us say several signals\nare to be observed from a point. One signal, usually the one\nfurthermost to the left is selected as the initial. The\ncircle and micrometer are set for a particular position. It\nis not necessary to set the seconds (or units for the Wild T-3)\non the micrometer exactly as specified, but every effort\nshould be made to have the setting within 10\" (or 5 units for\nthe T-3). Each signal is then observed in a clockwise order\nand the results recorded. At the last target, the telescope\nis reversed (plunged) and the procedure repeated in a reverse\n(counterclockwise) order. The observed seconds are meaned\nand the initial direction is subtracted from each observation,\nthus referencing the measurements to an initial of 0° 00'\n00.00. This completes one position. To continue the obser-\nvations, it is not necessary to have the telescope in the\ndirect position. The next circle setting as given in Table A\nmay be increased by 180° if the telescope is in reverse.\nIf a first-order theodolite is used, the rejection limit\nis 4\" from the mean for first-order surveys and 5\" from the\nmean for second and third-order surveys for limited surveys\nwhere it would be more likely a Kern DKM-2 or its equivalent\nwould be utilized and for these instruments a rejection limit\nof 5\" from the mean is acceptable for all orders of accuracy.\nAn observation which falls outside the specified tolerance\nshould be rejected and reobserved using the same circle\nsetting.\n19","Observations of Astronomic Azimuths\nIn order to obtain reliable astronomic azimuth values,\nit is necessary that the theodolite be equipped with a good\nlevel vial. Therefore, it is recommended that a striding\nlevel should be used with second-order theodolites such as\nthe Wild T-2. The vial should be graduated from left to\nright with zero at the vertical circle end of the vial. A\nstrip of adhesive tape may be used for this purpose and the\nwhite background will make the numbers easy to read.\nIf the value of the level vial is not available with\nthe instrument, it may be determined in the field following\nthe procedure explained later.\nSpecial care must be employed to maintain the ver-\ntical axis of the instrument as nearly vertical as possible\nthroughout the azimuth observations. This requires keeping\nthe bubble at/or very close to its \"reversing point\", or\nthe point at which the bubble remains stationary when the\nalidade is rotated. This condition is achieved entirely\nby adjustment of the footscrews and is independent of the\nlevel adjustment. The latter adjustment merely controls\nthe position of the bubble reversing point and is not criti-\ncal to the observations as long as the reversing point is\nfairly well centered. If necessary, the bubble can be\nbrought close to its reversing point at the end of each\nposition by a slight \"touching up\" of the footscrews,\n20","especially when the vial is broadside to the direction of\nthe star. Bubble readings will be recorded immediately\nafter each pointing on the star, and also on the mark if\nthe line departs more than 1 degree from the horizontal.\nCareful control of the vertical axis as outlined above\nwill render unnecessary the recording of bubble readings\nof inclined sights of less than 5 degrees.\nLevel Readings When Using Plate Level\nW\nE\nD\n08.0\n21.1\nR\n22.2\n08.9\n14.1\n+1.9\n12.2\nLevel Readings Using Stride Level\nW\nE\n08.0\n21.1)\n) made prior to star\n22.2\n08.9)\n14.1\n+1.9\n12.2\n08.2\n21.3\n22.0\n08.7\n13.8\n+1.2\n12.6\n+3.1 2 = 1.6\nAdditional equipment needed to observe a second-order\nazimuth are a portable short wave battery powered radio\nreceiver with a frequency range of from about l to 30\nmegacycles per second. Time comparisons should be made at\nthe station so that the chronometer will not be moved after\ncomparisons.\n21","A standard chronometer or watch with a good rate is\nnecessary for use in azimuth observations. Chronometers are\nvery delicate instruments and should be handled and packed\nwith care. Information from \"Apparent Places of Fundamental\nStars\" for the current year is needed for azimuth observa-\ntions and computations.\nTime will be recorded in the following manner:\nTelescope\nObject Observed\nPolaris\n7h\n38m\n475.0\nDirect\n40.0\n41\nReverse\n7\n40\n13.5\n7\nMean\n2\n53.0\nDiff.\nDetermination of Level Value\nThe value of plate or striding level vials for Wild T-2,\nWild T-3, and Kern DKM-2 theodolites may be determined by\ndifferential vertical angles using the following method. For\nmaximum accuracy and greater simplicity a second theodolite,\na transit works equally well, is recommended as the target\nfor the level determination. This will require illuminating\nthe target crosswires, making sure the instrument is focused\nnear infinity. This can be accomplished by focusing the\ninstrument on a sufficiently distant physical feature.\n1. Select an area for the instrument support stand which\nwill provide maximum stability for the level deter-\nmination.\n22","2. If a second instrument is not available, select a\nclearly defined target for the vertical observations\nat least 0.25 mile distant. The observations should\nbe made when there is a minimum change in refraction\n(normally between 1200 Noon and 4 P.M.)\n3. When attaching the tribrach to the observing stand,\nmake certain the grooves are oriented such that two\nlegs of the theodolite will be parallel to the direc-\ntion of the target and the instrument is shaded.\n(Figure 8. )\n4. The instrument should be leveled paying close atten-\ntion to the cross leveling; that position which\nplaces the level vial perpendicular to the target\ndirection. No additional cross-leveling is made\nduring a set, but may be performed between sets.\n5. With the theodolite pointed on target, set the\nhorizontal plate to read 0° 00', thus facilitating\nthe placing of the level vial parallel to the target\nline-of-sight by simply rotating the instrument to\nread 90° or 270° on the circle. The altitude level\nslow motion screw is not used for the determination\nand should be tightened to put the level out of\naction.\n6. Placing the instrument at a right angle to the\ntarget line-of-sight, move the plate or striding\n23","bubble near one end of the vial by turning both\nparallel foot screws an equal amount. Read and\nrecord both ends of the bubble, allowing approxi-\nmately one minute for the bubble to come to rest.\n7. Repoint on the target, reading and recording the\nvertical angle. Since we are interested only in\nangular tilt (difference in vertical angles) re-\nquired to move the level bubble a given distance,\ntelescope reversal is not required.\n8. Repeat step six, moving the bubble near the opposite\nend of the vial.\n9. Repeat step seven. This completes one determination\nof the vial.\n10. In order to assess the uniformity of the graduated\nvial and obtain the best mean value, it is necessary\nto determine a value from each graduation. The\nfollowing abstracts exemplify the method used, noting\nthat the mean level differences (column 3) decrease\napproximately one division with each successive\ndetermination.\n11. The number of determinations per set will be governed\nby the length of the bubble and the number of vial\ngraduations, however, these abstracts indicate the\noptimum number obtainable.\n24","12. The value of the level is obtained from a minimum\nof two sets. The range between the determinations\nwithin a set shall not exceed two seconds. The\nsets shall not differ by more than one second.\nWild T-3 Plate Bubble\nWild T-2 Stride Level\nWild T-2 Plate Bubble\n25","Value of One Division of Bubble\nWild T-2 XXXXXXX\nSet 1 of 2\nStriding Level\nLevel Mean\nMean\nDiff.\nSeconds\nL\nR\nDiff.\nVertical Angle\nSeconds\nSeconds\nvalue per\nDivision\n1. 09.7 22.8\n270 04 39-4\n40.0\n04 05-06\n2.\n17.0\n30.1\n34.5/7.30=04.73\n7.30\n05.5\n3. 11.0 24.1 6.00\n04 32-30\n25.5/6.00=04.25\n31.0\n4. 15.9 29.1 4.95\n04 07-06\n06.5\n24.5/4.95=04.95\n04 24-24\n5. 12.0 25.1 3.95\n24.0\n17.5/3.95=04.43\n6. 14.9 28.1 2.95\n04 10-09\n14.5/2.95=04.92\n09.5\n7. 13.0 26.2 1.90\n04 19-20\n10.0/1.90=05.26\n19.5\n8. 13.9 27.1 0.90\n04 14-14\n14.0\n5.5/0.90=06.11\n04.95\nmean of seven\nValue of One Division of Bubble\nWild T-3 XXXXXXX\nSet 1 of 2\nSeconds\nlevel\nSeconds Double\nvalue per\nmean\nL\nR\ndiff. Vertical Angle\nDiff.\ndivision\nsum\n1. 27.2 07.7\n89 20 50.2-50.5 100.7\n2. 22.3 03.2 4.70\n42.0-42.3 84.3 *32.8/4.70=06\"98\n3. 25.9 06.7 3.55\n20 48.8-48.8 97.6 26.6/3.55=07.49\n4. 23.2 04.0 2.70\n20 43.7-43.9 87.6 20.0/2.70=07.41\n5. 25.0 05.8 1.80\n20 47.3-47.3 94.6 14.0/1.80=07.78\n6. 24.0 04.9 0.95\n20 45.3-45.4 90.7 07.8/0.9508.21\n07.57\nmean of five\n* Double difference of sum of seconds\n100.7\ne.g.\n- 84.3\n16.4 X 2 = 32.8\n26","Value of One Division of Bubble\nKern DKM 2 XXXXXX\nStriding Level\nSet 1 of 2\nSeconds\nLevel Mean\nSeconds\nSeconds\nValue per\nL\nR\nDiff.\nVertical Angle\nDivision\n88 02 48-49\n48.5\n04.4\n17.2\n1.\n20.4\n02 06-06\n06.0\n42.5/3.25=13.08\n3.25*\n2.\n07.7\n2.6\n02 40-41\n40.5\n34.5/2.60=12.27\n17.9\n3.\n05.0\n19.6\n02 18-19\n18.5\n22.0/1.75=12.57\n4.\n06.8\n1.75\nmean of three 12.97\n* This value is the mean of the differences between\nlevel readings for this set and the previous set.\nL\nR\ne.g.\n04.4\n17.2\n3.3 + 3.2 = 3.25\n1.\n2\n20.4\n2.\n07.7\n3.2\n3.3\n27","-\nSchematic diagram of the THEO 010 Theodolite\n28","Vertical circle\nIlluminating mirror for\nthe diaphragm\nKnob for coincidence\nsetting\nIlluminating mirror for\nthe vertical circle\nClamping screw for\nvertical circle\nRing for focussing\ntelescope\nInverter knob\nEyèpiece for reading\nmicroscope\nEyepiece of telescope\nHorizontal level\nTangent screw for\naltitude\nTangent screw for\nazimuth\nReflector for collimation\nlevel\nCircular level\nIlluminating mirror for\nhorizontal circle\nEyepiece for optical\ncentering\nOne of the 3 levelling\nscrews\nTightening screw\nUniversal-Theodolite WILD T 2\n29","30","I\nI\n2\n10\n3\n4\n5\n11\n6\n12\n7\n13\n14\n8\n15\n16\n9\n17\n1\nHorizontal axis clamp\n10\nIndex error adjusting screw\n2\nFinder collimator\n11\nCircle reading eyepiece\n3\nIlluminating mirror\n12\nMicrometer knob\n4\nFocusing ring\n13\nVertical axis clamp\n5\nTelescope eyepiece\n14\nHorizontal slow-motion screw\n6\nVertical slow-motion screw\n15\nLeveling knob\n7\nOptical plummet\n16\nTerminal for electric illumination\n8\nCoarse-fine circle orienting drive\n17\nCentering tripod head\n9\nLocking lever\nKERN DKM 2-A\n31","PASIC PRINCIPLE\nELECTRO-OPTICAL DISTANCE MEASURING INSTRUMENTS\nBy George B. Lesley\nNOAA, National Geodetic Survey\nCorbin, Virginia 22446\nThe first precise electro-optical distance measuring\ninstrument was introduced in 1948. It was large\nand cumbersome, observations had to be made at\nnight, and it was limited in range, but simpli-\nfied the hard task of establishing a precise dis-\ntance with calibrated invar tapes.\nToday several types of electro-optical distance\nmeasuring instruments are available. All of these\ninstruments fall into one of two major categories\nbeing long range and short range.\nAs a general rule, the short range instruments\nare more compact and portable, while the long\nrange counterpart is larger and less portable.\nVirtually all the new instruments manufactured in\nthe past five years, both long and short range,\nhave transistorized circuitry and 12 VDC power\nsources. Most of these instruments fall into\none of two categories as far as their light source\nis concerned, that being infrared and red laser\n32","light. The short range instruments generally\nfall in the infrared category and the long range\nin the red laser category. Several of the short\nrange instruments feature direct distance readout\nwhile most of the long range instruments require\nreduction computations for distance determination.\nThe red laser and the infrared has an advantage\nover the previously used tungsten light in that\na filter can be used at the instrument end of\na line being measured which will pass only the\ntransmitted light. This system makes daylight\nmeasurements possible because direct sunlight and\nother reflected stray light, which represents\nnoise, is blocked from entry into the instrument.\nInfrared is often considered to be light. Tech-\nnically, light is that part of the spectrum that\nis visible to the eye and infrared is that part\nof the spectrum that is between visible light and\nradio waves. This article will be on Geodimeter\nmodels 6A and 8 which use visible light.\nGeodimeter Models 6A and 8 use practically the\nsame measuring principle. The primary difference\nbetween the Model 8 and the Model 6 series is that\n33","the Model 8 uses a KDP Cell modulator and a laser\nlight whereas the Model 6 uses a Kerr Cell modulator\nand a tungsten or mercury vapor light. Both models\nuse a phase resolver to resolve the proportional\npart of the quarter-wavelength and the same modu-\nlating frequencies.\nThe Geodimeter uses the following principles:\nVelocity of light, VL = 299,792,500 m/s\nModulation Frequencies, F1 = 29,970,000 Hz\nF2 = 30,044,922 Hz\nF3 = 31,468,500 Hz\nF4 = 31,465,500 Hz\nF1 = 29,968,500 Hz\nReceiver Frequencies,\nF2 = 30,043,422 Hz\nF3 = 31,467,000 Hz\nF4 = 31,467,000 Hz\nRefractive Index, RI = 1.0003086\nThe unit lengths are computed as follows:\n299,792,500\n= 2.500 000\nU1 =\n4(29,970,000)(1.0003086)\n299,792,500\n= 2.493 766\nU2 =\n+(30,044,922)(1.0003086)\n299,792,500\n= 2.380 952\nU3 =\n4(31,468,500)(1.0003086)\n299,792,500\n= 2.381 179\nU4 =\n4(31,465,500)(1.0003086)\n34","The refractive index, RI, is uncorrected for temperature,\npressure, humidity, and color of light. The formulas for\nthe correction are as follows:\n6,328A o Laser Light = 1 + 0.00010 7925 X\nP\n273.2 + t\n1.5026e10-\n= 1 + 0.00010 9129 X P\nO\n5,500A\nLight\n273.2 + t\n273.2 + t\nP = Pressure in millimeters of mercury\nt = Temperature in degrees centigrade\ne = Vapor pressure in millimeters of mercury\nThe last term in the formula, correction for relative\nhumidity, is seldom greater than one part per million\nand is usually taken from a graph.\nMIRROR\nINSTRUMENT\n+\n+\n+\n+\nFigure 1\nMeasuring method, showing unit lengths for one measurement\n2000m\n1000m\n+\nF1,U1\nF2,U2\nF3,U3\nFigure 2\nMeasuring method using three different frequencies\n35","Distances are determined by methods shown in figures\n1 and 2. In figure 1, unit lengths for one measuring\nfrequency are shown; however, the instrument uses\nthree different frequencies, as shown in figure 2.\nThe instrument transmits pieces of light, and the\nlengths of the pieces for each measuring frequency\nare determined by computing U1, U2 and U3. The instru-\nment measures the last part of a unit length (Fig.1),\nwhich is usually less than 2.5 meters. In 1000 meters\nthere are 400 unit lengths for F1, 401 for F2 and\n420 for F3. When F1 has transmitted 400 unit lengths,\nit is lagging F2 by one full unit length. When F1 has\ntransmitted 21 unit lengths, it is lagging F3 by one\nfull unit length.\nAt 1000 meters, all unit lengths will coincide; but\nF1 and F3 will have an even number of unit lengths\nand F2 will have an uneven number of them. At 2000\nmeters, all unit lengths will coincide, and all fre-\nquencies will have an even number of unit lengths.\nThe unit lengths will start through another cycle\nand repeat the same procedure for the next 2000 meters.\nTherefore, any distance measured will have to be\nknown to within 2000 meters or less when using three\nmodulating frequencies.\n36","In order to resolve the distance to 50,000 meters or\nless the fourth frequency is added and only F3 and F4\nare used to determine how many 2000 meter increments\nthere are. The unit lengths will coincide the first\ntime at 25,000 meters but F3 will have an even number\nof unit lengths and F4 will have an uneven number of\nunit lengths. At 50,000 meters F3 and F4 will coincide\nagain and both of them will have an even number of\nunit lengths.\nCOMPUTATIONS\nA recording and computation form is shown in figure 3.\nThe observations are with a Model 8 but the same\nprinciple is used with the Model 6A. The form should\nbe fairly selfexplanatory and every step of the com-\nputation will not be discussed, however, a few steps\nwill be explained.\nOne complete measurement consists of readings on all\nfour frequencies and each frequency consists of four\nphases on interior Calibrate \"C\" and four phases on\nthe Mirror \"R\". The sign of the first phase reading\nis recorded for each set of four phases to show what\npart of the unit length is being measured. It is not\nimportant that the values on phases 1-2 or 3-4 are\n37","in close agreement but the difference should not\nexceed 35 to 40 divisions. The difference between the\nsum of 2-3 and 1-4 should be in close agreement and\nshould not exceed 20 to 25 divisions.\nThe values L1, L2' and L3' are computed by subtracting\nthe calibrate value from the mirror value (R-C), however,\nthe signs of \"R\" and \"C\" must be the same before sub-\ntracting. If one unit length is added, the sign will\nchange and if two unit lengths are added the sign will\nremain the same. If \"C\" is greater than \"R\" and the signs\nare opposite, add 2.500 meters to \"R\" before subtracting\n\"R-C\". If \"C\" is greater than \"R\" and the signs are the\nsame, add 5.000 meters to \"R\" before subtracting \"R-C\".\nIf L2' or L3' is smaller than L1, add 5.000 meters to\nL2 or L3 in order to have a positive value for \"A\"\nand \"B\".\nObservations are made with frequency 4 to determine the\nnumber of 2000 meter increments in the distance measured\nand should not be used in the mean with D1, D2 and D3\nbecause there is no oscillator for F4. An additional\ncapacitor is switched into the F3 circuit in order to\nlower the transmitting frequency 3000 Hz for F4.\nIn order to obtain L4, the reflector value is subtracted\n38","1.30 m\nHT. MIRROR = 13.72 m\n+ .233\nF4: If the signs are different,\n.238\n.541\n.542\n.779\n.775\n1.554\n1052.378\n8000.000\n= 9052.378\n- 0.447\n9051.931\nbefore subtracting \"C-R\".\nadd 2,500 meters to \"C\"\nC\n= 4.058\n= 4.993\n= .935\nFREQUENCY 4\n8300\n= 8000\n13 July 1970\nC. C. Glover\nHT. GEOD. =\n=\n=\n=\n=\nPROJECT NO.\n.475\n.470\n.776\n.775\n1.246\n1.250\n2.496\nn = 10000 G-D'\nMEAN D1, D2, D3,\nOBSERVER\n(nearest 2000)\nL3 (R-Conly)\nL4 = (C-R)\nG = (L3*-L4)\nSUM CORR'NS.\nR\nD (Slope dist.)\nDATE\n+\nMitchell N. Base 1965 ToMitchell S. Base 1965\nn\nD\nSUM = - 0.447 M\n+ .375\n.378\n.684\n.687\n1.062\n1.062\n2.124\n4.755\n1054.755\nD3 = 1052.374\n2.381\n3 Prism AGA Type\n- 0.950\n+ 0.211\n- 0.027\n0.000\n+ 0.319\nR\nFREQUENCY 3\n= 4.993\n= 4.755\n= 2.385\n2.370\n49.8\nF3\n= 1050.\nCulpeper\n+L3 =\n-K3 =\n=\n=\n=\n=\n=\n=\n=\n.377\n.380\n.686\n.688\n1.066\n1.065\n2.131\nF1, F2, F3: If the signs are different, add 2,500 meters to \"R\" before subtracting \"R-C\".\nREFLEX NO.\nATMOS. CORRECTION\nLOCATION\nD'\nC\nSTATION\nL3 = 0.9524 x L3\nREFLEX CONST.\n35.2 ppm\nGEOD. CONST.\nL3' (R-C)\nREFLEX ECC.\n4.992\n1054.992\n2.618\nD2 = 1052.374\n+\nGEOD. ECC.\nF2\n1050.\nB x 21\n-L1\n.323\n.326\n.633\n.632\n.959\n.955\n1.914\nB\nC\n=\n=\n=\n1172ft\nFREQUENCY 2\n23.9°C\nMEAN\n-K2\n+L2\n+\nD'\n68%\n= 1043.\nVirginia\n= 5.005\n= 4.992\n= 2.385\n= 2.608\n2.385\nD1 = 1052.385\n80056\n.326\n.329\n.633\n.960\n.959\n1.919\n.631\nMIRROR\nGEOD. NO.\nR\n1050.\nSTATION\n24.2\n1165\nF1\nSTATE\n+\nL2 = 0,9975 x L2\"\nU.S. DEPARTMENT OF COMMERCE\nENVIRONMENTAI SCIENCE SERVICES\nADMINISTRATION\n=\n=\nL2' (R-C)\nGEODIMETER MODEL 8 OBSERVATIONS\nGEOD.\n.306\n.305\n.611\n.607\n.916\n.913\n1.829\n+L1\nA x 400\nD'\n2024\n23.4\n1180\nR\n-L1\nFREQUENCY 1\nA\nMIRROR\n1000.\n50.\n1050.\n24.2\n1165\n+ .333\n.334\n.639\n.638\n.973\n1.944\n2.385\n.971\n69.5/77.0\nC\n=\n=\n=\nGEOD.\nA x 400 MINUS B X 21\n2020\n23.9\n1180\nESSA FORM 76-33 (9-69)\n(nearest hundreds)\n(B X 21 nearest 5)\nSUM\nLI=(R-C) =\nESSA FORM 76-33\nTOTAL SUM\nSUM 2 & 3\nSUM1 &4\nPHASE\n2\n3\n4\n1\n=\nPRESS.\nRI.HU.\nTEMP.\nTIME\n(9-69)\nD'","from the calibrate value (C-R). This procedure differs\nfrom the first three frequencies because on the first\nthree frequencies the transmitting frequencies are\nhigher than the receiving frequencies but on F4 the\ntransmitting frequency is lower than the receiving\nfrequency.\nThe phase resolver rotates through one complete phase\nevery 2.500 meters which is the unit legth for F1 (see\nfigure 1), therefore there is no correction to L1.\nIn order to obtain L2 and L3 (see figure 1) the resolver\nlength is multiplied by 0.9975 for L2 and 0.9524 for L3.\n2.500 X 0.9975 = 2.494 which is the unit length for F2\nand 2.500 X 0.9524 = 2.381 which is the unit length for\nF3 and F4.\nL4 is not reduced to its proper unit length because the\nunit length, in this computation, for F4 is the same\nas F3. If we use the uncorrected value for both frequen-\ncies, the difference \"G\" would be the same as if they\nwere corrected. The term (R-C only) means that if two\nunit lengths were added to L3' because it was smaller\nthan L1, those two unit lengths should not be used when\ncomputing for \"G\" because only F3 and F4 should be\nconsidered when computing for the multiple of 2000 meter\npieces.\n40","GEODIMETER CORRECTIONS\nThe corrections include the geodimeter and mirror eccen-\ntricity (see figure 4), geodimeter zero constant, reflec-\ntor zero constant, and atmospheric correction.\nMirror\nGeodimeter\n+\n+\nEccentric correction\nFigure 4\nGEODIMETER ZERO CONSTANT\nThe electrical center of the geodimeter is not over the\nplumb line when the instrument is plumbed over the mark.\nIt would be very difficult (if not impossible) to measure\nthe eccentric distance with a tape because the light path\npasses over several optics, so the zero constant is estab-\nlished by making a measurement with the instrument over\na precise taped distance and the difference between the\ntaped distance and the geodimeter distance is the geodi-\nmeter zero constant. The tape distance for the Model 6\nseries should be 49.860 meters and it should be 50.210\n41","meters for the Model 8. When those distances are used the\nvalues for \"R\" and \"C\" are practically the same on F1\nand F3 and any small error in the phase resolver will\nbe cancelled.\nGEODIMETER TILT CORRECTION\nThe bases of the Model 6 Geodimeters are leveled with\nthree leveling screws and there is no correction for\ntilt with them. The Model 8 Geodimeter is equipped with\na tilting head that allows the optical center of the\ninstrument to remain fixed when the instrument is rotated\nvertically. The head is equipped with a scale graduated\nin centimeters and a bubble. The scale is to measure\nthe offset of the instrument over the plumb line due to\nan unlevel tripod or stand (see figure 5) . The slope\nof the line being measured has nothing to do with the\ntilt correction. If the tripod or stand is level, there\nwill be no tilt correction. Figure 6 shows a scale rea-\nding of +0.009 m but the optical center of the instru-\nment is over the plumb line and there is no tilt correc-\ntion. The tilt correction is determined as follows: Point\nthe instrument in the general direction of the mirror,\nlevel the bubble with the vertical slow motion screw and\nread the scale. The tilt correction is recorded on the\ngeodimeter eccentricity line.\n42","Figure 5\nGeodimeter with tilt correction\nTO mirror\nTilt corrections+0.009m\nFigure 6\nGeodimeter without tilt correction\nTO\nmirror\n43","2.970in.\n1. 892 in:\n1.390in.\nzero constant\n1.390-2.970 = -1.580 in.\n-0.040m\nFigure 8\nFigure 7\nRetroreflector zero\nLight path through\nconstant computation\nretroreflector\nMIRROR ZERO CONSTANT\nThe mirror zero constant is the distance the retro-\nreflector is eccentric to the plumb line. Figure 7 is\na sketch showing the actual lightpath making three re-\nflective turns through a retroreflector.\nFigure 8 is the mirror zero constant computation. If\nthe plumb line, in figure 8, was at the intersection\nof the two dashed lines there would be no mirror zero\nconstant but the plumb line would be so far behind the\nprism that it would be very clumsy to use in the field.\nThe plumb line is usually set in the center of the mirror\nhousing and the mirror zero constant is computed. The\nlightpath will be twice the thickness of the prism and\nit travels through glass slower than air by a factor\n44","of 0.57. The geodimeter resolves one half the total\ndistance the lightpath travels, so the 1.892 inches\nwill not be doubled. The computation in figure 8 is:\n1.390 - (1.892 X 1.57) = -1.580 inches or-0.040\nmeter.\nWhen using the Model 8 Geodimeter on distances of two\nmiles or less, a diffused mirror is usually used. The\ndiffused mirrors are made of plastic and the average\nrefractive index for plastic is 1.47. The same principle\nis used for computing the zero constant for a diffused\nmirror as it is for a retroreflector.\nATMOSPHERIC CORRECTION\nThe correction for temperature and pressure for the\nobservations on figure 3 are:\nmean pressure = 1172 feet = 754.7 mm (see Table 1)\nmean temperature = 23.9°c\n0.000107925 X 754.7\n= 1.0002742\n(274.2 PPM)\n1 +\n273.2 + 23.9\nThe correction for relative humidity is taken from the\ngraph on Table 2 and it is -0.8 PPM. The total value is:\nTemperature and pressure = 274.2 PPM\n= - 0.8 PPM\nRelative humidity\n273.4 PPM\n45","The standard unit lengths for the geodimeter are compu-\nted using 308.6 PPM for atmosphere. Atmosphere correc-\ntion for the observations on figure 3 are 308.6 - 273.4\n= +35.2 Parts-Per-Million. In the lower right portion\nof figure 3, the computed distance (D) up to this point\nis 9052.378 meters. The atmosphere correction is\n0.009052 X 35.2 = +0.319 meter.\nERRORS\nThere are several ways to introduce errors when measuring\na distance with a geodimeter. Following is a list of a\nfew possible errors that will be discussed:\n1. Frequency count in error\n2. Transient time in instrument\n3. Barometric pressure\n4. Air temperature\n5. Relative humidity\n6. Wrong sign on first phase reading\n7. Wrong phase resolver reading\nFREQUENCY COUNT\nThe Model 6A and 8 Geodimeters have two crystal ovens\nin them. One oven is in the transmitting section and\nthe other one is in the receiving section. An error of\n30 Hz in the transmitting frequency will introduce an\n46","error of one part in one million of the distance measured.\nThe receiving frequency will introduce no error in the\ndistance but will cause spreads between the phase\nvalues. There are several reliable frequency counters\non the market today and much thought and consideration\nshould be given in selecting one of reputable dependa-\nbility.\nIn some older model geodimeters, the oven temperature\nwould cycle through several degrees of temperature causing\nthe modulating frequency to vary. The frequency change\nis proportional to the crystal oven temperature change.\nOn the newer model geodimeters, the crystal oven\ntemperature is very stable allowing the modulating fre-\nquency to be very stable, however, the frequency count\nshould be checked on a reliable frequency counter\noccasionally. If observations are made in bright sun-\nlight 65°F or above it is possible for the crystal\novens to overheat and to be on the safe side the\ninstrument should be shaded.\nTRANSIENT TIME IN INSTRUMENT\nThe geodimeter and mirror should be pointed properly\nwhen measuring a distance, however, if they are not,\nit should introduce no error in the distance if there\nis no transient time in the instrument and it is\n47","operating properly. Figure 9 is an example of how to\ncheck the instrument transient time.\n4\n6\n5\n2\n3\nFigure 9\n226\n228\n228\n222\n227\nPhase 2\n227\n558\n560\n562\n559\n557\nPhase 3\n557\n394\n394\n392\n390\n392\n393\nMean\nThe circles in figure 9 represent the optical receiving\ntube on the geodimeter. The shaded areas represent that\nhalf of the tube covered up. The transient time check\nis as follows: Place a diffused mirror approximately\n50 meters away from the instrument. All observations\nare made on the diffused mirror on F1. The mean values\nfor 2, 3, 4, and 5 should not differ from the mean of\n1 and 6 more than 7 or 8 units.\nThe next step is to measure a distance with the instru-\nment of about 300 meters, using standard observing\nprocedures, on a retroreflector and on a diffused mirror.\nThe two computed distances should agree very closely\nand the difference should not exceed 6 millimeters. If\n48","the instrument passes the two above tests, the receiver\nis functioning properly. If it did not pass the above\ntests, then the pointing of the instrument is critical\nand it should go back to the manufacturer for main-\ntenance.\nBARONETRIC PRESSURE\nIf precise distances are to be established, a good\nquality altimeter or barometer should be used and they\nshould be checked for precision at the local airport,\nhowever, it is possible to check them on the station\nsite if the elevation of the station is known. Following\nis the procedure to check an altimeter:\nFor every 0.10 inch the barometer increases the alti-\nmeter will decrease approximately 90 feet and vice versa.\nBarometric pressure announced on the local weather\nreport is always in inches of mercury reduced to sea\nlevel but you are very seldom at sea level so you will\nhave to compute the difference. If the local weather\nstation announces the barometric pressure is 30.10\ninches and your station elevation is 550.50 feet the\naltimeter should read:\nFrom Table 1, 30.10 in. = 819\nft.\nStation elevation\n= 550.50 ft.\nAltimeter should read\n1,369.50 ft.\nIf the altimeter is checked on the station site without\n49","the tables and we used a conversion factor of 90 feet\nper 0.10 inch of mercury, the computation would be:\n29.90 in. - 30.10 in. = -0.20 in. = -180 ft.\n550 ft. - 180 ft. = 370 ft.\nMost altimeters, by manufacturer's design, read 1000 feet\ntoo high, so we will add the 370 feet to the 1000 feet\nequals 1,370 feet. An error of 90 feet or 0.10 inch\nwill make an error of one part in one million of the\ndistance measured.\nAIR TEMPERATURE\nWhen using electro-optical distance measuring instru-\nments, the air temperature is the most uncertain\ncorrection. According to several tests performed by\nthe National Geodetic Survey, the air temperature is\nfairly constant along the line being measured when\nthe wind is about 3 miles-per-hour or above, however,\nwhen it is calm it would be difficult to obtain the\ncorrect mean temperature of the line by measuring\ntemperatures at the two end terminals. When distances\nare measured from a tripod or a 4-foot stand, the\nmost accurate method of measuring air temperature is\nwith thermistors at least 25 feet above the ground.\nIf thermistors are not used the thermometer should be\nas high above the ground as possible and in the wind.\n50","If distances are measured in the daytime, the thermo-\nmeter or thermistor should be well shaded from the sun\nand in the wind.\nRELATIVE HUMIDITY\nThe correction for relative humidity for light waves\nis very small and very seldom exceeds one part in one\nmillion. It has been found that the average correction\nis 0.4 PPM and this is normally used unless the highest\nprecision is required. If the humidity is taken, it is\nonly necessary to take it at one end of the line.\nWRONG SIGN ON FIRST PHASE READING\nIt is fairly common to read the first sign of the\nfirst phase wrong. If the distance is computed with\na wrong sign, you wouldn't even be in the ball park.\nIt is very easy to detect a wrong sign in the calibrate\ncolumn because they should always be the same. They\ncan be plus or they can be minus and all four frequencies\nshould be the same and they will not change from day\nto day. On most geodimeters, the sign of the first\nphase is plus, but if the wires are reversed on the\nnull meter the signs will reverse and the sign for F1\nin figure 3 would be C = - and R = + but the distance\nwould still compute properly. It is very difficult (if\nnot impossible) to detect a wrong sign in the mirror (R)\n51","column when computing the distance. The observer can\neliminate the problem by going back to the reflector\nposition and double checking the sign of the first phase\nposition after he has completed the sheet.\nWRONG PHASE RESOLVER READING\nIn some cases, the phase resolver is read in error or\nthe value is recorded wrong. One way to detect the error\nis to use a mirror bar (see figure 10) and take one\ncomplete sheet with the mirror plumb over the mark on\nthe zero point, one sheet with the mirror at 0.400 m\nand one sheet with the mirror at -0.400 m. When the\nmirror is at +0.400 m, each phase reading on the mirror\nshould be about 100 units lower than the one on the\nzero point and when the mirror is at -0.400 m each phase\nreading should be about 100 units higher than the ones\non the zero point. By using the mirror bar as shown in\nfigure 10, and measuring distances to all three points,\nany small error in the phase resolver will be cancelled.\n+0.400m\n-0.400m\nTo Geodimeter\nMirror Bar\nFigure 10\n52","LASER SAFETY\nThe author is not an expert on laser safety. It is not\nintended for this article to govern the rules and\nregulations of laser safety, however, several years\nof experience with laser electro-optical distance\nmeasuring instruments indicate a few safety precautions\nshould be observed.\nNOAA, National Geodetic Survey (formerly ESSA) has the\nsafety standards for the Model 4L Geodimeter in the\nSafety Handbook. Most of the standards do not apply\nto the Model 8 because the intensity of light, line\nof polarization of the light, and the eyepieces are\ndifferent. A few laser safety procedures with the\nMode18, which may be helpful, will be discussed.\nThe lasers in the Model 8 Geodimeter are the helium-\nneon continuous gas lasers. The power output of this\ntype laser ranges from 5.0 milliwatts to 6.5 milliwatts.\nThese lasers operate at 6,328 Angstroms (red light)\nand the exit beam diameter is 0.6 millimeter.\nThe laser beam is transmitted through a prism that\nturns the light beam 180° through a quarter-wave plate,\na KDP Cell, a polarizer, and a beam expander that\ncollimates the beam to a diameter of 20 millimeters.\nWhen the beam has been expanded to 20 millimeters, the\n53","power output will range from 1.3 milliwatts to 1.7 milli-\nwatts of light which is almost parallel.\nWhen working or operating in heavily populated areas,\nor where there is danger to the public or someone coming\ninto visual contact with the light at a distance of\nless than 6,000 meters, the geodimeter should be roughly\nput on target before the laser is turned on.\nWhen putting the geodimeter on target, the instrument\nis centered on the mirror by sighting through a 42\npower eyepiece. This is probably the most dangerous\npart of the system and precautions should be taken to\nprotect the eye. On short distances, two miles or less,\ndiffused mirrors should be used. If diffused mirrors\nare not available, the retroreflector should be covered\nwith a thin piece of plastic to diffuse the return\nlight.\nThe laser geodimeter should never be left unattended\nunless the laser is turned off or the instrument is\non interior calibrate.\nPersonnel should never observe the laser light with\nbinoculars, theodolites or any magnifying lens unless\nsome filtering system is used to dim the light.\nLaser danger signs should be displayed in all areas\nwhere the laser geodimeter is being used.\n54","The geodimeter observer and the mirror tender should\nsee that no one comes into visual contact with the\nlaser beam. The hazard zone should be defined as an\narea of 5 meters in radius along the line of sight\nfrom the instrument out to 6,000 meters when the\nlaser is on.\nTechnical specifications furnished by AGA Corporation\nare:\nModel\nLight Source\nRange\nAccuracy\nday\nnight\n5mm + 1mm/km\nTungsten lamp up to 5 km\n15 km\n6A & 6B\nMercury vapor up to 10 km\n5mm + 1mm/km\n25 km\n8\n45 km\n60 km\n5mm + 1mm/km\nLaser\nThe above maximum distances with the Model 6 series are\nprobably based on the most ideal conditions. Past experi-\nence indicates those distances could not be obtained\nunder normal working conditions.\nSeveral lines exceeding 60 kilometers have been measured\nunder ideal conditions at night with the Model 8, how-\never, the daylight range would be about 30% less than\nthat at night.\nAccording to numerous tests, the above stated accuracies\ncan be expected if the geodimeters are operating properly.\n55","TABLE 1\n*\nBarometric Pressure Conversion Tables\nin\nft\nin\nft\nin\nft\nin\nft\nin\nft\nmm\nmm\nmm\nmm\nmm\no\n500\n1000\n29,90\n1500\n2000\n732\n31.\n00\n28.\n80\n773\n759\n745\n787\n73\n30.\n40\n29\n30\n758\n772\n744\n786\n730\n771\n29.\n80\n757\nI\n00\n6\n00\nII\n00\n743\n16\n00\n21\n100\n30\n90\n785\n28.\n70\n729\n770\n756\n30.\n742\n30\n784\n29.\n20\n728\n769\n755\n783\n741\n29.\n70\n30.\n80\n727\n2\n00\n768\n7\n00\n754\n12\n00\n17\n00\n22\n00\n782\n740\n28.\n60\n30.\n20\n726\n767\n753\n29.\n10\n781\n739\n725\n766\n752\n29.\n60\n780\n30.\n70\n738\n3\n00\n8\n00\n13\n00\n18\n00\n23\noo\n724\n765\n28.\n50\n751\n779\n30.\n10\n737\n29.\n00\n764\n723\n750\n778\n736\n29.\n50\n30.\n60\n763\n749\n722\n777\n735\n4\nOO\n28\n40\n9\n00\n14\n00\n19\n00\n24\n00\n30\n00\n762\n748\n776\n721\n28\n90\n734\n761\n775\n747\n720\n29.\n40\n30.\n50\n733\n760\n774\n746\n719\n28.\n30\n5\n00\n29.90\n1000\n15\n00\n20\n00\n25\n00\nNOTE Zero on above tables corresponds to minus 1000 ft. on Smithsonian Meteorological Table 51. (i.e. 1000\n*\nhas been added to tabular values in feet) 56","TABLE 1 ( (continued)\nin\nft\nin\nft\nin\nft\nmm\nin\nft\nin\nft\nmm\nmm\nmm\nmm\n2500\n3000\n3500\n4000\n4500\n680\n718\n705\n692\n667\n679\n717\n704\n69\n27.\n20\n666\n27.70\n26.\n70\n28.\n20\n678\n26.\n20\n2600\n3100\n3600\n41\n00\n46\n00\n716\n703\n690\n665\n677\n702\n715\n689\n664\n676\n10\n27.\n27.\n60\n26\n60\n714\n701\n28.\n10\n688\n26\n10\n663\n27/00\n32\n00\n37/00\n42\n00\n47\n100\n675\n713\n700\n687\n662\n674\n712\n699\n686\n2700\n27.\n.50\n26.\n50\n661\n28.\n00\n673\n711\n698\n26\n00\n2800\n33/00\n3800\n43\n00\n48\n00\n685\n660\n672\n710\n697\n684\n659\n671\n709\n27.\n40\n26.\n90\n696\n27.\n90\n26\n40\n683\n29\n00\n3400\n39/00\n44\n00\n49\n00\n658\n25\n90\n670\n708\n695\n682\n657\n707\n669\n694\n68\n27.\n30\n26\n80\n27.\n80\n656\n706\n26\n30\n668\n693\n30\n00\n35\n00\n40\noo\n45\noo\n50\n00\n57","TABLE 1 (continued)\nin\nft\nin\nft\nin\nft\nin\nft\nin\nft\nmm\nmm\nmm\nmm\nmm\n5500\n5000\n6000\n6500\n7000\n609\n25.|80\n620\n655\n24.\n40\n643\n25.30\n631\n608\n619\n654\n642\n24.\n80\n23.\n90\n630\n607\n61\n00\n7100\n51\n00\n56\n00\n66\n00\n618\n653\n641\n25.\n70\n629\n24.\n30\n606\n25.\n20\n652\n617\n640\n628\n605\n651\n24.70\n616\n639\n52\n00\n57\n00\n62\n00\n67\n00\n72\n00\n23.\n80\n627\n25.60\n604\n650\n615\n638\n24.\n20\n25.\n10\n626\n603\n649\n637\n614\n53\n00\n58\n00\n63\n00\n68\n00\n73\n00\n625\n24.\n60\n23.\n70\n648\n602\n25.\n50\n636\n613\n624\n647\n24.\n10\n25.\n00\n601\n635\n612\n623\n64/00\n54\n00\n59\n00\n69\n00\n74\n00\n646\n600\n634\n24.\n50\n611\n622\n23.\n60\n25.\n40\n645\n633\n599\n610\n24.\n90\n621\n24.\n00\n644\n632\nI\n65/00\n55\n00\n60\n00\n70\n00\n7500598\n58","14\n13\n(Dry Bulb minus Wet Bulb) is represented by the 0° to 30° curves.\n12\n11\nCORRECTION (NEGATIVE) IN 7th DECIMAL PLACE OF REFRACTION\nDry Bulb Temp. and (Dry Bulb minus Wet Bulb)\n10\nCORRECTION FOR HUMIDITY (LIGHT WAVES)\n9\n0°\nArgument for the correction is:\n5°\n8\nTABLE 2\n10\n7\n15\n6\n20\n0°\n25\n5\n5°\n30°\n4\n10°\n15.\n3\n20°\n2\n25°\n30°\n1\n55°\n60°\n65°\n75°\n70°\n80°\n85°\n90°\n95°\n100°\n15°\n30°\n35°\n40°\n45°\n50°\n10°\n20°\n25°","ELECTRONIC DISTANCE MEASURING INSTRUMENTS\nby Raymond IV. Tomlinson\nNOAA, National Ocean Survey\nNational Geodetic Survey\nRockville, Maryland 20852\nTwo basic types of Electronic Distance Measuring Instruments\nare currently being used for obtaining accurate distance\nmeasurements. These types are lightwave and microwave\n(infrared being considered as lightwave as the computations\nare based on the same formulas). Numerous instruments are\navailable and statistical data are shown on Table I with\nservicing and company information addresses shown on Table II.\nThe theory of operation and detailed servicing has been\nneglected since most users will not be performing their own\nservice.\nThe selection of the proper Electronic Distance Measuring\nInstruments is a major problem. Consideration must be given\nto the accuracies desired, operational and maintenance pro-\nblems, personnel, power requirements, size, weight, durability,\nand numerous other things. The first problem to solve is\nwhether to select Microwave or Lightwave equipment. Items to\nbe considered are length of lines, terrain, climate, usability\nand versatility of equipment. In other words, will this\n60","equipment permit an increase in the number of projects that\ncould be undertaken, thereby,increasing the gross business\nand profits? Only the user can make these decisions as\noften the price for a complete system is very nearly the\nsame.\nThe prime concern, regardless of the system purchased, is\noperational personnel. Their advice should be considered\nprior to purchase since often they will be responsible for\nthe system paying for itself or continually be inoperative.\nAs most are well aware, the surveyor purchasing a new elec -\ntronic measuring instrument a few years ago was considered\nfoolish and wasting good money. Today most, if not all the\nlarger projects, which require precision length control, are\nawarded to the surveyor with up-to-date equipment. If a\nbusiness is faltering, one should consider a change and plan a\nprogram to include Electronic Distance Measuring Instruments.\nThe Model T era has passed and those who fail to keep abreast\nof the latest technology, will seldom receive the prime con-\ntracts.\nElectronic Distance Measuring Instruments have really come\ninto its own for control surveys in the past 10 to 20 years.\nNo longer do we have to pull out the standardized tapes\nand perform days of laborious work to measure base lines.\n61","Many people feel that we have only to solve the refractive\nindex problem and then Electronic Distance Measuring Instru-\nments can measure to the accuracy standard of 1 part in 1\nmillion. There is not total agreement in this regard,\nbecause there are numerous additional problems -- instru-\nments delivered without adequate training, trouble shocting\nproblems, proper alignment and/or adjustments, frequency\nchecks, obtaining correct meteorological data, and misuse and\nabuse of equipment. All of these and others lead to blunders\nand/or unnecessary errors.\nBrochures distributed for the various Electronic Distance\nMeasuring Instrument firms have impressive claims regarding\naccuracies and probably these accuracies can be obtained\nunder certain atmospheric conditions and properly tuned\ninstruments. In the general run of the mill control survey\nprojects, the surveyor will seldom have good atmospheric con-\nditions and often uses improperly tuned instruments. There is\nlittle that can be done to change the atmospheric conditions,\nbut better meteorological data can be obtained and the\ninstruments can be kept properly tuned.\nMICROWAVE DISTANCE MEASURING INSTRUMENTS\nMicrowave instruments were first used in the radar method of\naircraft detection, and navigation developed during World\nWar II. The first use of microwaves by surveyors for distance\nmeasurement on the ground became a reality with the develop-\n62","ment of the Tellurometer in 1957. Since that date, other\nmicrowave systems have been developed, but all are essentially\nthe same. The Cubic Corporation introduced their version of\nmicrowaves under the name of Microdist in 1960. Shortly\nthe reafter, the name was changed to Electrotape DM-20, which\nis still in production today. The Wild-Heerbrugg Company\nintroduced the Distomat DI-50 in 1966, which has been replaced\nwith the DI-60. Other microwave systems have since been\ndeveloped for positioning ships, offshore points, etc., but\nnone have the accuracy for precise geodetic surveys.\nMicrowave instruments have an advantage over lightwaves for\ndistance measurement because they can be used day and night,\ncan penetrate fog, clouds and SO forth. They are less\naccurate due to the difficulty in determining the humidity\ncorrections, which have little effect on lightwave instruments.\nSUGGESTED PROCEDURES FOR CHECKOUT AND CALIBRATION\nUpon receiving EDMI certain checks should be made to insure\nthat proper measurements will be obtained. Some of the major\nitems are :\n1. Training of Operating Personnel\na. Trained in proper operating procedures, which\nshould be received from the manufacturer.\nb. Training in precise geodetic measurements\nshould be obtained from experienced geodetic\npersonnel.\n63","2. Selection of Related Equipment\na. The EDMI cost from $4,000 to $10,000 and will\nnot give accurate measurements unless the\nrelated equipment selected is also reliable.\nb. Cheap thermometers, barometers, tripods, etc.,\nwill give nothing but poor results and trouble.\nWhen 5 to 10 thousand dollars is expended for\nthe EDMI, the additional cost of reliable\nmeteorological equipment and good tripods is\nrather insignificant.\nC. Select stable tripods, accurate optical\nplummets , calibrated barometers, physchrometers,\nthermometers, and a proven power supply.\nd. Check barometers, psychrometers, and thermo-\nmeters with a standard at least each month for\nprecision and more often when subject to abuse\nor misuse. These can be checked at the nearest\nairport.\n3. Checking Instrument Constant\na. Microwave and lightwave equipment should be\nchecked over precision base lines at the start\nand end of each project. Measure this line on\nat least two different days and compute the\nresults using meteorological data, difference\nin elevation, mirror constants, etc., but do\n64","not apply any instrument constant and the\ndifference between the EDMI measured distance and\nbase line measurement should be the instrument\nconstant, Microwave measurements should be\nmade over a line length in excess of one mile\nand preferably about 5 miles with each terminal\nwell elevated to prevent ground swing or\ngrazing lines.\nb. If a taped base line is not available, a line of\nabout 5 miles in length should be selected\n(depending on range of your instrument). . This\nline should be measured in two parts and then\noverall. The check is obtained by adding or\nprojecting the 2 short lengths to the overall\nlength. Angles are required for the projection\nif the three points are not in a straight line.\nC\nB\nA\nAC - AB - BC = Instrument Constant\nB\na\nC\nC\nA\na'\nc'\nc'= cos A\na'= cos C\nb-c'-a' = Instrument Constant\n65","SYSTEMATIC AND ACCIDENTAL ERRORS\nSystematic and accidental errors or blunders can often be\neliminated through proper training of personnel. An\nemployee who has been properly trained will know to look\nfor misreadings and avoid the following:\n1. Centering error - not using proper procedures to\nplumb over the stations. Tripods that are loose\nor damaged.\n2.\nImproper pointing - all EDMI should be pointed to\nobtain maximum signal return.\n3. Improper voltage - full charged batteries and/or\nquality generators are a must for all EDMI.\n4. Improper readings - one of the major blunders is\ntaking readings with improper null and nulling with\nimproper signs. Misreadings of 100 are common and\nyet so unnecessary if a little care is exercised.\n5. Improper meteorological data - using cheap thermo-\nmeters, barometers, psychrometers, etc. Readings\nobserved with thermometers in direct sunlight, not\nwetting the wet bulb sock, improper fan ventilation,\nand most of all, not waiting until the thermometers\nhave reached equilibrium before taking the readings.\nEFFECTS OF ATMOSPHERE AND INTERFERENCE\nOne of the largest sources of error in electronic distance\nmeasuring is due to the uncertainty in the refractive index,\n66","In order to obtain the best possible value of the refractive\nindex, atmospheric parameters should be taken at terminals\nof the line and at points along the line. For lines under\n5 miles in length, an average of the atmospheric parameters\ntaken at terminals of the lines (if taken properly) should\ngive the refractive index correction accurate to within 1 cm.\nfor lightwave instruments and 1-3 cm. for microwave instruments.\nThe relative humidity has a very small effect en the results\nfrom instruments using lightwaves, seldom more than a fraction\nof one part per million of the distance measured. The\nhumidity effect in the refraction computation involving\nmicrowave instruments is much greater than that for lightwaves.\nFor microwave instruments, a measurement under normal con-\nditions, (t = 20°C, p = 760 mm Hg and e = 10 mm Hg), an error\nof 1°C between the dry and wet bulb temperatures will pro-\nduce an error of 7 parts per million in the distance,\nThe effect of this 1°C error increases considerably as the\ntemperature increases, rising to about 17 p.p.m. at a temper-\nature of 45°C.\nNumerous things cause interference in the measurement of\ndistances with lightwave or microwave instruments. The\nfollowing should be eliminated if possible:\nReflection and ground swing from improper line clearance\nReflection over water\n67","Line of sight broken during measurement\nMeasuring in rain, blowing sand, foggy, etc.\nRadio communication on same frequency\nMeasuring through uncleared line of sight\nMeasuring while near ricrowave towers\nMeasuring through or underneath power lines\nExcessive heat waves along line\nLoose power connections\nSUGGESTED FIELD PROCEDURES TO DETECT ERRORS IN THE FIELD\nNumerous points could be listed here, but it is assumed\npersonnel have been trained in the procedures for plumbing\nand instrument operational procedures. In other words, the\nfollowing field procedures are suggested for fully trained\nobservers who have the basic knowledge for obtaining the accur-\nacy desired.\nLightwave Instruments - Sources of error relate to the fol-\nlowing: Alignment of optics, frequency checks, calibration,\nzero constant, proper voltage, and meteorological data.\nAlignment of optics is very critical to achieve accurate\nmeasurements. This includes alignment of the Kerr Cell,\nKDP Cell, Photomultiplier for transit time, etc.\nFrequency checks for the three frequencies shall be taken\nevery two or three months for lower accuracy work and each\n68","week for high precision work. This check is necessary due\nto frequency drift, which must be monitored to obtain pre-\ncision length measurements.\nA trained observer can minimize calibration curve errors by\nwatching the spread of the phase readings and the spread of\nthe three frequencies on D1, D2, and D3. The spread\nbetween any of the four phase readings should not exceed 20\nunits on the earlier Model nor 50 units for the later Model\ninstruments. The spread between the means of phase 1-4 and\n2-3 should not exceed 4 divisions. For precision work, the phase\nlimits should not exceed 10 and 25 units, respectfully, nor 2\ndivisions between the mean of phase 1-4 and 2-3. When the\nfrequencies contain a spread of more than 6 cm between D1, D2,\nand D3, the instrument should be recalibrated as it becomes\nvery difficult to resolve the proper 5 and 100 meter intervals\nof the AX400 and BX21.\nCalibration curves and zero constants should be checked if\nthe optics are realigned or the photomultiplier moved or\nreplaced.\nProper voltage is a must for accurate measurements and\nquality generators and fully charged batteries will enhance\nthe quality and production.\nIlumidity is not a major factor with lightwaves, and for most\nprojects a correction of +0.4 ppm of the distance measured\n69","can be used, but good dry bulb readings and altireter read-\nings should be obtained. Thermometers should be placed in\na shaded area 6 to 8 feet above ground level for lower order\nwork and 20 feet or more for high precision projects.\nA complete set of measurements for first- and second-\norder work should consist of three complete measurements.\nOne each with the prism over true center, set-up and set-\nback 0.4 meter. The set-ups and set-backs are easily\naccomplished by fabricating a mirror bar (Fig. 2) with holes\ndrilled at +0.4 and -0.4 meter from true center.\n0.4m.\nTrue Center\n0.4m.\n+\nMirror Bar\nFigure 2\nFor high precision work, spur lines, open end traverses,\netc., , 4 complete measurements should be made at +0.4,\n-0.4 and two from true center, one of which should be on\nhigh calibration and one on low calibration. These set-ups\nand set-backs permit covering more of the delay line, and\nwhere possible, high and low calibration readings should be\ntaken for high precision work. The spread between these\nmeasurements on one night should not exceed 2 cm. If two\n70","nights' measurements are used for high precision work,\nthe spread between the mean of the two nights should not\nexceed 2.5 cm for lines less than 8 kilometers in length and\n1,7 cm + 1 part per million for lines longer than 8 kilo-\nmeters.\nMicrowave Instruments - Microwave instruments are electroni-\ncally sound, have well balanced circuits with precision\ncrystal ovens. The major drawbacks are obtaining represent-\native humidity corrections for the longer lines, reflections\nand ground swing on the shorter lines. Reflection and ground\nswing can give ambiguous readings, but this has been greatly\nreduced in some of the more recent models due to the higher\nfrequency employed and one and one-half degree beam width.\nReflections when measuring over water can be a problem unless\nthe terminal points are elevated for proper line clearance.\nWhen measuring from a tripod on a shore point, it is some-\ntimes helpful to place the master unit behind or partially\nblocked by a sand dune. This will chop off the bottom\nportion of the radio wave and allow the top portion the\nnecessary clearance across the water. Some skeptics will\nargue that this will give erroneous readings, but several\nexperiences have not shown an appreciable loss of accuracy.\nThese instruments, like any other Electronic Distance\nMeasuring Instruments, Theodolite, etc., should receive\n71","preventative maintenance. Microwave instruments are too\nfrequently sent to the manufacturer for repairs. If the\ncompany was contacted by telephone and the malfunction\nexplained, the majority of the repairs could be performed in\nthe field.\nA complete measurement with Microwave equipment should con-\nsist of a measurement from both ends of the line with a mini- -\nmum of ten fine readings and course readings at the beginning\nand end of the measurements. Observers should take these\nmeasurements independent of each other as it has been proven\nthat one observer making the setting and then switching to\nthe other observer for a reading results in incorrect measure-\nments.\nThe two measurements should agree in the field by less than\n0.1 meter for lines under 10,000 meters. Lines in excess\nof 10,000 meters should agree within 1:100,000. The spread\nof the fine readings should not exceed 4 cm. Observers\nshould avoid stations near high voltage lines, microwave\ntowers, and grazing lines. When any of these factors are\nencountered, notes should be entered in the field records.\nBatteries should be well charged and lines should not\nbe measured when the battery reading is less than specified\nin manufacturer's manual.\n72","When measuring to spur lines or open end traverse points, the\nfollowing method should be used to prevent angular and distance\nblunders.\nEXAMPLE FOR SPUR LINE MEASUREMENTS\n*\nJOHN OFFSET\nWOODS\nJOHN\n*\n*Main line of traverse or progress of project\nObserve horizontal angles at all three points.\nMeasure JOHN-WOODS and JOHN OFFSET-WOODS with EDM Instruments.\nMeasure JOHN to JOHN OFFSET with standardized tape.\nThe closure of the triangle (in seconds) multiplied by the\noffset distance (in meters) should not exceed 400.\nMETEOROLOGICAL DATA\nOne of the largest sources of error in electronic distance\nmeasuring is due to the uncertainty in the refractive index.\nIn order to obtain the best possible value of the refractive\nindex, meteorological data must be taken very accurately at\n73","the terminals of the line and at points along the line for\nmore precision. For lines under 5 miles in length, an aver-\nage of the atmospheric parameters taken at the terminals of\nthe\nlines should give the refractive index correction accurate\nwithin 1 cm for lightwave instruments. This accuracy can\nonly be obtained if you meet the following requirements:\n1. Temperature readings to be taken 6 to 10 feet\nabove the ground to reduce ground radiation.\nPsychrometers protected from direct sunlight\nand shaded from heat reflections and nearby illum-\ninated objects.\n2. Dry and wet bulb thermometers should be accurate\nand checked for dependability with a standard each\nweek.\n3. Psychrometers should be slightly tilted and pointed\ninto the wind to increase the air velocity around\nthe thermometer bulbs and to insure adequate\nventilation. It is very important to have a motor\nsufficiently powered to keep air moving past the\nbulbs at a rapid pace. Poor circulation past the\nwet bulb can be a source of serious error.\n4. When wetting the wet bulb sock, care should be\ntaken to prevent water drops from forming a\nbridge between the wet bulb and the shield as this\nreduces the proper circulation.\n74","5. Barometers should be systematically calibrated with\na mercury standard or to station barometers that\nhave been calibrated.\n6. Barometers should be placed in a shaded area and\nprotected from gusty wind conditions. These in-\nstruments should be handled with care and suitable cush-\nioned carrying cases used for transporting them.\nCare must be exercised in shipment as some are\ndamaged during air shipments if they are not locked\nor sealed in a pressurized container.\nRegardless of the precision thermometers, barometers, etc.,\non hand, the data obtained is worthless unless proper pro-\ncedures are used. This comes right back to training, as\ntrained observers know that correct meteorological data\nare just as important as having the instrument centered\nover the station.\nReading the dry and wet bulbs should be done when the\nthermometers have reached equilibrium, not before or after,\nas such readings will be erroneous. For microwave instruments\nan error of 1°C between dry and wet bulb readings under\nnormal conditions (t=20°C, P=760 mn Hg and e=10 mm Hg) will\nproduce an error of 7 parts per million in distance. This\n1°C error at a temperature of 45°C will produce an error of\n17 parts per million. These errors are actually blunders\n75","and are often due to lack of personnel training. Everyone\nwishes to obtain the maximum accuracies from his equipment\nand to do this, we must spend the time and money for train-\ning our personnel.\nRegardless of the type or accuracy of instrument used in a\nsurvey, it is of little value unless it is operated by\nqualified observing personnel. Recently the National Geo-\n-\ndetic Survey received traverse data obtained by a state\nhighway department using theodolites and a Model 4D Geodi-\nmeter. It was requested that the data be evaluated to\nlocate blunders as the loops failed to close by three to five\nhundred feet in X and Y. Upon reviewing the Geodimeter field\nrecords, it was noted there were excessive spreads between\nthe D1, D2, and D3, and it was very difficult to resolve\nthe nearest multiple of 5 meters and 100 meters, because the\nAX400 and BX21 values often fell halfway and thus were\nambiguous. Two employees from the state highway department\nalong with their 4D instrument came to Rockville for three\ndays of training. The Geodimeter was checked and found out\nof alignment and over 200 cycles away from the operating\nfrequency. The operators stated the instrument had been in\ncontinuous use for six years, had received only one day of\ninstruction, and were tcld \"they could measure accurately\nany time they could null the instrurent.\" They did not\n76","know how to calibrate the instrument or make the necessary\nalignment adjustments. A similar situation was encountered\nwith another state highway department, which employed\nTellurometers and inexperienced operators were obtaining\nreadings that were the reverse of the proper readings.\nNormally, a company will give one or two days of training\nand instruction when their representative delivers the\ninstruments, but to obtain the maximum from your instruments\nand protect your investment, some companies have a five-\nday training course for $100 to $250, plus employees\nexpenses. This type of training does pay dividends in the\nlong run!\n77","REFRACTIVE INDEX COMPUTATIONS\nThe value for the velocity of light used in the reduction\nof electronic distance measurements, 299, 792.5 km/sec in\nvacuo, was adopted by the International Association of\nGeodesy in 1957. In the development of any type of EDM\nequipment, some measuring frequency and a unit length has\nbeen adopted. If the unit lengths are used as constants\nin the reductions, a constant value for the refractive index\nformula is obtained from,\nV\nC =4UF\n, where, V = velocity of light\nU = unit length\nF = measuring frequency\nUsing frequency 1 for the Geodimeter, where F = 29,970,000/c\nand U = 2.5 meters, C is then 1.000 30864, or 308.64 X 10\nThis value of C is referred to as the preset, or assumed,\nindex value. Index values for DI-10 = 1.0002819, H P-3800=\n1.0002783, MA-100 = 1.000274, and 76 = 1.0003086. In any case,\nthe assumed value should be consistent with the instrument fre-\nquency, unit length, and the adopted value for the velocity of light.\nFor Instruments Using Light Waves and Infrared\nThe Barrell and Sears formula for group velocity of the\nrefractive index of light in the atmosphere at O degrees\ncentigrade, 760 mm of Mercury pressure, and 0.03% Carbon\nDioxide is:\n4.8864\n+ 0.068 A 4 ) 10-6\nng = 1 + 2 287.604 + 2\nwhere A is the wavelength of light expressed\nin microns.\nDue to variations in temperature, pressure, and humidity,\nthe refractive index changes to a new value na'\n- - 5.5 e 10~0 -8\n\"\np\n760\n1\n+\n273.2\n273.2\n77a","p is the pressure in mm of mercury, t is degrees\nCentigrade, and e is vapor pressure in mm or mercury.\nFor computational purposes, the formula above reduces to,\n0.359474 (n g - 1) p - 1.5026 e 10-5\nna =\n273.2 + t\n273.2 + t\nIf N X 10-6 = 0.359474 (n, - 1), then values of N\ng\nfor various values of a are,\nN\nA\n109.460\nmercury vapor lamp\n0.5500\n109.129\nstandard lamp\n0.5650\n0.6328 red laser light\n107.925\n105.72\n0.8750 infrared\n105.59\n0.9000 infrared\n105.57\n0.9050\ninfrared\n105.54\ninfrared\n0.9100\n105.496\ninfrared\n0.9200\n105.450\ninfrared\n0.9300\nThe basic formula for computing the refractive index correction\nin parts per million of the distance, is used in the form,\nN p\n15 e\ncorr'n (ppm) = I -\n+\n273.2 + t\n273.2 + t\nwhere,\nI = assumed index\nN = constant for given value of A\np = pressure in millimeters\nt = Centigrade temperature\ne = vapor pressure in mm.\nNOTE: An average value\nExamples:\nof 0.4 ppm is used for\nthe humidity correction\nMeasured = 950.000 meters\nDI-10\nand the term involving\nI = 281.9\ne is not used.\nN = 105.72\np = 752.9\nt = 26.0°C\n105.72 X 752.9\nN p\n= 266.0\n=\n273.2 + 26.0\n273.2 + t\n77b","= 950.000 X 10-6 (281.9 - 266.0)\nCorr'n in meters\n0.000950 x (+15.9) = 0.015\n=\nSlope distance\n= 950.000 + 0.015 = 950.015 meters\nH-P 3800\nMeasured 1199.9890 meters\nI = 278.7\nN = 105.46\nP = 754.9\nt = 26.6°C\n105.54 X 754.9\nN p\n= 265.8\n=\n273.2 +t\n273.2 + 26.6\nCorr'n in meters = 1199.9890 X 10-6 (278.7 - 265.8)\n0.001200 X (12.9) = +0.0155\n=\n= 1199.9890 + 0.0155 = 1200.0045 meters\nSlope distance\nMeasured 1650.0203 meters\nMA-100\nI = 274.4\nN = 105.450\np = 758.2\nt = 28.0°C\nInst. Constant = - 0.0144\nPrism Constant\n= - 0.0270\n105.450 X 758.2 = 265.4\nN p\n=\n273.2 +t\n273.2 + 280\nCorr'n in meters = 1650.0203 X 10-6 (274.4 - 265.4)\n= 0.001650 x(+9.0)=0.0148 =\n= 1650.0203 + 0.0148 - 0.0144 - 0.0270 =\n1649.9937 meters\nMeasured 1649.823 meters\nRANGER\nI = 310.4\nN = 107.925\np = 758.2\nt = 19.5°C\nInst. Constant = 0.165\nPrism Constant = - 0.028\n107.925 X 758.2\nN p\n= 279.6\n=\n273.2 +t\n273.2 + 19.5\nI 1649.823 X 10-6 (310.4 - 279.6)\nCorr'n\n0.001650 X (+30.6) = + 0.0505\n=\nSlope distances = 1649.823 + 0.051 + 0.165 - 0.027 =\n1650.012 meters\n77c","FOR INSTRUMENTS USING MICRO-WAVES\nThe Essen and Froome formula for the refractive index of\nthe atmosphere using electromagnetic waves was provi-\nsionally adopted by the International Association of Geodesy\nin 1960. This formula is,\n(nr - 1)106 103.49\n(p - e) + + 86 . 26 (1 + 5748) T\ne\n.\nin which,\nT = temperature in degrees Kelvin\np = pressure in millimeters\ne = vapor pressure in mm\nA modified form of this formula is,\n(nr - 1)106 - 103.46 + p t + (273,2 + t)2 490,814.24 e\nwhere, t = Centigrade temperature\nIf the temperatures are in Farenheit, atmospheric and vapor\npressures in inches of mercury, the above formula is,\n(nr - 1)106= 459.7 + p t (459.7 + t)\n4730 + 40,394,200,\nor\nI - (kP m + 1.805k2 (e'\n- rpm(t-t'))) m\nwhere:\nI = preset, or assumed Index\nP. = mean pressure in inches of Hg.\nt m = mean dry-bulb temp. in OFFICE\nt 1 = mean wet-bulb temp. in °F.\nk = 4730/(459.688 +t)\ne = saturation vapor pressure in inches.\nr = 0.000367 (1 + (t' - 32)/1571)\nThe terms k, e; and r are tabulated in Tellurometer\nManual, Publication 62-1, Revised edition. t is used\nas argument for k. t' is used as argument for e' and\nr. (Table on page 77g)\n77d","Example:\nMRA - 3\nmeasured = 11620.019 meters\nI\n= 325 .\nn\nt\n= 50.9.F\n= 48.1 F\nt\nSheet\n9.264\ne' = 0.337\n= 0.000371\nr\n1 k\n=\nk2\nk P\ne'\nP\nt-t'\nr\n325.0- 9.264x29.09 + 1.805x85.8217 0.337-(0.000371x29.09x2.8)\n325.0- 269.49 + (154.908 X 0.307) =\n325.0- 269.49 + 47.56) = 8.0 ppm.\n77e","U.S. DEPARTMENT OF COMMERCE\nNOAA FORM 76-61\nNATIONA L OCEANIC AND ATMOSPHERIC ADMINIST RATION\n(6-71)\nTELLUROMETER OBSERVATIONS\nMODEL MRA 3 MK-2\nDATE\nLOCALITY\nSTATE\n1/15/72\nCity\nNEW YORK\nHEIGHT OF INST.\nINST. NUMBER\nSTATION\n1.62 m.\n1012\nSCM\nHEIGHT OF INST.\nINST. NUMBER\nTO STATION\n1014\n1.70 m.\nEMPIRE\nOBSERVER\nCHIEF OF PARTY\nPROJECT NUMBER\nC.B.D.\nA. B. C.\nW12\nA\nMEAN A\nB\nC\nD\nE\nA\nMETEOROLOGICAL DATA\nF\nR\nP\n0.04\nT\nT\n14\n61\n20\n0.15\n10\nTIME\nDB\nWB\n6\n48.0\n29.08\n2\n0.02\n0.10\n1\n1\nMASTER\n52.0\n0.02\n0.15\n48.0\n29.10\n50.0\nREMOTE\n9.99\n0.00\n48.0\n29.08\n51.5\nMASTER\n9.96\n0.00\n48.5\n29.10\n50.0\nREMOTE\n9.89\n116.36\n9.97\nSUM\n203.5\n192.5\n9.95\n0.00\n48.1\n50.9\n29.09\nMEAN\n0.06\n9.93\n0.04\n9.99\n61\n20\n0.02\n0.10\n10\n15\n6\nSUM\n2\n1\n1\nMEAN\n0.019\nASSUMED INDEX 325\nFIELD INDEX\n11620.019\nDISTANCE\nASSUMED - FIELD\n(\n)\n0.093\nDISTANCE x 10-6 XI 8\nMETEOROLOGICAL CORRECTION\n)-\nFIXED CORRECTION\n11620.113\nSLOPE DISTANCE\nNOAA FORM 76-61 (6-71) SUPERSEDES C&GS FORM 5135 WHICH MAY BE USED.\n77f\n* U.S. GOVERNMENT PRINTING OFFICE: 1971-769372/404 REG.#6","l\ne'\nk\nr\n/\ne'\nk\nr\nI\ne'\nr\nk\n.000\n.000\n.000\n-40\n0.004\n350\n11.270\n15\n0.081\n363\n9.964\n70\n0.739\n376\n8.930\n-39\n.004\n11.243\n10\n.085\n9.943\n71\n.765\n8.913\n-38\n.004\n351\n11.217\n17\n.089\n9.922\n72\n.791\n8.896\n-37\n.005\n11.190\n18\n.093\n364\n9.902\n73\n.818\n377\n8.879\n-36\n.005\n11.164\n19\n.098\n9.881\n74\n.846\n8.863\n-35\n0.005\n11.137\n20\n0.103\n9.861\n75\n0.875\n8.846\n-34\n.005\n352\n11.111\n21\n.108\n9.840\n76\n.905\n8.830\n-33\n.006\n11.085\n22\n.113\n365\n9.819\n77\n.935\n378\n8.813\n-32\n.006\n11.060\n23\n.119\n9.799\n78\n.967\n8.797\n-31\n.007\n11.033\n24\n.124\n9.779\n79\n.999\n8.780\n-30\n0.007\n353\n11.008\n25\n0.131\n9.759\n80\n1.032\n8.764\n-29\n.007\n10.982\n26\n.137\n366\n9.739\n81\n1.067\n8.748\n-28\n.008\n10.957\n27\n.143\n9.719\n82\n1.102\n379\n8.732\n-27\n.008\n10.931\n28\n.150\n9.699\n83\n1.138\n8.716\n-26\n.009\n10.906\n29\n.157\n9.679\n84\n1.175\n8.700\n-25\n0.009\n354\n10.881\n30\n0.165\n367\n9.659\n85\n1.214\n8.684\n-24\n.010\n10.856\n31\n.172\n9.639\n86\n1.253\n380\n8.668\n-23\n.011\n10.831\n32\n.180\n9.620\n87\n1.294\n8.652\n-22\n.011\n10.806\n33\n.188\n9.600\n88\n1.335\n8.636\n-21\n.012\n355\n10.782\n34\n.195\n9.581\n89\n1.378\n8.620\n-20\n0.013\n10.757\n35\n0.203\n368\n9.561\n90\n1.422\n381\n8.605\n-19\n.013\n10.733\n36\n.212\n9.542\n91\n1.467\n8.589\n-18\n.014\n10.709\n37\n.220\n9.523\n92\n1.514\n8:574\n-17\n.015\n356\n10.684\n38\n.229\n9.504\n93\n1.561\n8.557\n-16\n.016\n10.660\n39\n.238\n369\n9.485\n94\n1.610\n8.543\n-15\n0.017\n10.636\n40\n0.248\n9.466\n95\n1.661\n382\n8.527\n-14\n.018\n10.013\n41\n.257\n9.447\n96\n1.712\n8.512\n-13\n.019\n10.589\n42\n.268\n9.428\n97\n1.766\n8.496\n-12\n.020\n357\n10.565\n43\n.278\n370\n9.409\n98\n1.820\n8.481\n-11\n.021\n10.542\n44\n.289\n9.391\n99\n1.876\n383\n8.466\n-10 0.022\n10.518\n45\n0.300\n9.372\n100\n1.933\n8.451\n- 9\n.023\n10.495\n46\n.312\n9.353\n101\n1.992\n8.436\n- 8\n.025\n358\n10.472\n47\n.324\n371\n9.335\n102\n2.053\n8.421\n- 7\n.026\n10.448\n48\n.336\n9.317\n103\n2.115\n384\n8.406\n- 6\n.027\n10.425\n49\n.349\n9.298\n104\n2.179\n8.391\n- 5 0.029\n10.402\n50\n0.362\n9.280\n105\n2.244\n8.376\n- 4\n.030\n359\n10.380\n51\n.376\n9.262\n106\n2.311\n8.361\n- 3\n.032\n10.357\n52\n.390\n372\n9.244\n107\n2.380\n385\n8.347\n- 2\n.034\n10.334\n53\n.405\n9.226\n108\n2.450\n8.332\n- 1\n.036\n10.312\n54\n.420\n9.208\n109\n2.523\n8.317\n0\n0.038\n360\n10.289\n55\n0.436\n9.190\n110\n2.597\n8.303\n+ 1\n.040\n10.267\n56\n.452\n373\n9.172\n111\n2.673\n8.288\n2\n.042\n10.245\n57\n.468\n9.154\n112\n2.751\n386\n8.274\n3\n.044\n10.223\n58\n.486\n9.137\n113\n2.831\n8.259\n4\n.046\n10.200\n59\n.503\n9.119\n114\n2.913\n8.245\n5\n0.049\n361\n10.179\n60\n0.522\n374\n9.101\n115\n2.996\n8.230\n6\n.051\n10.157\n61\n.540\n9.084\n116\n3.082\n387\n8.216\n7\n.054\n10.135\n62\n.560\n9.067\n117\n3.170\n8.202\n8\n.057\n10.113\n63\n.580\n9.049\n118\n3.261\n8.188\n9\n.060\n362\n10.092\n64\n.601\n9.032\n119\n3.353\n8.174\n10\n0.063\n10.070\n65\n0.622\n375\n9.015\n120\n3.448\n388\n8.160\n11\n.066\n10.049\n06\n.644\n8.998\n12\n.069\n10.028\n67\n.667\n8.980\n13\n.073\n363\n10.006\n68\n.690\n8.963\n14\n.077\n9.985\n69\n.714\n376\n8.946\n15\n.081\n363\n9.964\n70\n.739\n376\n8.930\n77g","APPROX.\n3,900\n4,000\n8,500\n4,200\n7,000\n17,500\n12-15,000\n4,100\n49,500\n8,000\n17,000\n20,000\n4,000\n8,000\n10,000\n7,000\n9,000\n$18,000\n11,500\n9,000\n9,500\nPRICE\n+ 1 cm + 1 P/100,000\n+ 5 mm + 1 P/100,000\n+ 1 cm + 1 P/300,000\n5 mm + 1 P/100,000\n+ 1.5 cm + 3 PPM\n+ 1.5 cm + 3 PPM\n+ 1 mm or o PPM\nACCURACY\n+ 5 mm + 1 PPM\n+ 1 cm + 1 PPM\n+ 5 mm + 1 PPM\n+ 5 mm + 1 PPM\n+ 1 cm + 3 PPM\n+ 3 mm + 3 PPM\n+ 5 m + 2 PPM\n6 mm + 1 PPM\n5 mm + 1 PPM\n5 mm + 1 PPM\n5 mm + 1 PPM\n5 mm + 1 PPM\n+ 2 to 5 mm.\n+ 0.05 ft.\n+ 1 cm\nSTATISTICAL DATA FOR ELECTRONIC DISTANCE MEASURING INSTRUMENTS\n150 to 100,000 ft.\nUp to about 3 km\n15 m. to 25 km.\n15 m. to 15 km.\n15 m. to 25 km.\n15 m. to 60 km.\n20 m. to 50 km.\n50 m. to 50 km.\n100 m. to 50 km.\n75 m. to 65 km.\n25 m. to 50 km.\n15 m. to 15 km.\nO to more than\nO to 1,350 m.\n1 to 2,000 m.\n1 to 2,000 m.\nO to 3,000 m.\nO to 2,000 m.\n1 to 3,000 m.\nO to 3,000 m.\nUp to 80 km\nO to 60 km\nRANGE\n6 km\nMercury Vapor\nMercury Vapor\nTABLE 1\nMicrowave\nMicrowave\nMicrowave\nMicrowave\nMicrowave\nMicrowave\nInfrared\nInfrared\nInfrared\nInfrared\nInfrared\nInfrared\nTungsten\nTungsten\nEMMISION\nSOURCE\nLaser\nLaser\nLaser\nLaser\nLaser\nLaser\nLaser Sys. & Elec.\nLaser Sys. & Elec.\nLaser Sys. & Elec.\nCubic Corporation\nCubic Corporation\nScintrex Limited\nHewlett-Packard\nAGA Corporation\nAGA Corporation\nAGA Corporation\nAGA Corporation\nAGA Corporation\nSpectra-Physics\nTellurometer\nMANUFACTURER\nTellurometer\nTellurometer\nTellurometer\nTellurometer\nWild\nWild\nMicro-Ranger\nRangemaster\nElectrotype\nINSTRUMENTS\nAkkuranger\nGeodolite\nLightwave\nMicrowave\nH-P 3800\nInfrared\nCubitape\nCA-1000\nMRA-101\nMA-100\nRanger\nDI-10\nDI-60\nMRA-3\nMRA-4\n700\n6B\n76\n6A\n8","TABLE II\nINSTRUMENT SERVICE OR INQUIRIES CONSULT THE FOLLOWING:\nFirm Names\nInstruments Manufactured &\nServiced\nGeodimeters, Models 4B, 4D, 6,\nAGA Corporation\n6A, 6B, 8, 76, & 700\n550 County Avenue\nSecaucus, New Jersey 07094\nCubic Corporation\nAutotape, Electrotape DM-20,\nP. O. Box 769\n& Cubitape DM-60\nSan Diego, California 92112\nHP3800A and HP3800B\nHewlett-Packard\nLoveland Divisions\nP. O. Box 301\nLoveland, Colorado 80537\nLaser Systems & Electronics\nRanger, Ranger II, Rangemaster,\nBox 248\n& Micro-Ranger\nTullahoma, Tennessee 37388\nGeodolite 3G\nSpectra-Physics\n1250 West Middlefield Road\nMountain View, California 94040\nMRA-1, MRA-2, MRA-3, MRA-101,\nTellurometer, Inc.\nMRA-4, MRB-2, MA-100, CA-1000\n170 Finn Court\nFarmingdale, Long Island,\nNew York 11735\nDistomat DI-10, Distomat DI-50,\nWild Heerbrugg Instruments, Inc.\n465 Smith Street\n& Distomat DI-60\nFarmingdale, Long Island,\nNew York 11735\nScintrex Limited\nAkkuranger MkII\n222 Snidercroft Road\nConcord, Ontario, Canada\n79","RECTANGULAR COORDINATE SYSTEMS\nJoseph F. Dracup\nSupervisory Geodesist\nGeodesy Division\nNational Ocean Survey\nNational Oceanic and Atmospheric Administration\nU. S. Department of Commerce\nRockville, Maryland 20852\nIntroduction\nRectangular coordinate systems are probably as old\nas the surveying profession. The Romans employed such\nsystems in the layouts of their cities and military encamp-\nments, and they have been in more or less general use since\nthat time. In the United States, the public land surveys\nwere defined in rectangular coordinates, and as early as\n1846, Simeon Borden, who initiated triangulation surveys\nin Massachusetts, proposed that the state be divided into\nfive tangent plane systems. Coordinates based on a tangent\nplane system can have a direct relationship to national\nnetwork geographic coordinates, but the projection is not\nconformal except at the origin and is limited to a rather\nsmall area. The State coordinates systems, on the other\nhand, are conformal and cover a much larger area. In\naddition to these grid systems, Universal Transverse\nMercator coordinates are sometimes used in this country.\nAlthough other coordinate systems are employed in various\n80","countries, the following discussions will deal only with\nthose noted above.\nCoordinate Systems\nTangent Plane Coordinates - Prior to the development\nof the State coordinate system, several cities, including\nNew York, employed this map projection which can be described\nas an approximate azimuthal equidistant projection. Although\nsome cities continue to use this system, its use today\nshould be restricted only to those areas to which the\nnational network has not been extended or for special\npurpose engineering surveys such as those required for\nbridges, tunnels, etc., where a temporary system of plane\ncoordinates of high accuracy and limited extent is required.\nIt is very important to note here, that in no case should\nthis plane coordinate system be extended more than twenty\nor twenty-five miles from the origin.\nAs noted in the Introduction, this projection is not\nconformal, and perhaps a definition of this term is in\norder. In a Conformal projection the scale at any particular\npoint is constant in all directions, and small figures on\nthe surface of a sphere retain their original forms on the\nmap. Although this condition is met for the point selected\nas the origin, provided the system is confined to the recom-\nmended area size, the scale at the other points is not the\n81","same in all directions, being zero in the direction of\nthe origin and varying between points.\nTo develop this grid system, a basic framework of\ntriangulation, traverse, and/or trilateration is established\nand geographic positions computed for the points involved.\nInasmuch as this is a purely local system, the geographic\nposition of only one point in the net is required to\ninitiate the computation of the other points. This\nposition can be obtained by any available source, even\nby scaling from a map. Orientation is usually more\ncritical, the accuracy of which would depend upon various\ncircumstances and generally should be obtained by astro-\nnomical observations or connections to the national net.\nOnce the basic framework is adjusted, one point is\nselected as the origin and plane coordinates are deter-\nmined using the formulae given below. Generally, it is\nbest to select some point near the center of the system\nas the origin and to assign coordinate values of sufficient\nnumerical size to insure that negative coordinates do not\noccur. In some cases, however, the following sample com-\nputation, for example, the origin may be defined as having\n0,0 coordinates and the coordinates for the other points\nreferred to the quadrant in which they appear.\n82","The computation of local coordinates from geographic\npositions, and vice versa, can readily be made with the\nfunctions listed in Special Publication No. 241, \"Natural\nTables for the Computation of Geodetic Positions.\" (p. 84A)\nAs an example, assume that triangulation station\nBOGART is the origin, and its plane coordinates are O, O.\nThe plane coordinates of station ROSSVILLE in relation\nto BOGART are required. The following data are given:\nLatitude\nLongitude\n40° 36' 07.281\nBOGART\n°0\n74° 06' 58.125\n40° 32' 39.250\nROSSVILLE 1\n13'07.843\n^1\n0° 06' 09.'718\nAN\n369.718 (west)\nor\nThe formulas are:\nX = AX\"/H, y = 1 - yo + V (x/10,000)2\nFrom the tables in Sp. Pub. No. 241, take out yo and\n1 (meridional arcs) for °0 and 1, H for 1 and V for yo.*\nH (8 dec. pl.)\n= 0.0424 9577\nyo = 4,496,166.595\nV (app.) (4 dec. pl.) = 6.7099\ny1 4,489,749.774\ncorr. (-.642) (.0213) = -.0137\nV(x/10,000)2 = +5.068\nV (corrected)\n6.6962\nX = \"/H = 8,700.113 meters\ny = -6,411.753 meters\nor 28,543.62 ft. (west)\nor 21,035.89 ft. (south)\n* Note: V should be taken out for 1 but, in making\nmany computations based on one origin, it is easier to use\n83","one basic value of V and correct it by multiplying AV by\n(y1 - yo), pointed off four decimal places, and applying\nthe result algebraically.\nAs an example of the inverse computation: assume that\nthe local plane coordinates, X and y, of ROSSVILLE and the\ngeographic position origin, BOGART, are known. Required is\nthe geographic position of ROSSVILLE. The distance from\nx2 + y2 = 35,457.68 ft.\nBOGART to ROSSVILLE may be taken as\nor 10,807.522 meters. The tangent of the azimuth from\nBOGART to ROSSVILLE is x/y or 1.356 9010. The azimuth is\nthen 53° 36' 38.8. Using this length and azimuth and the\ngeographic position of BOGART, compute the geographic position\nof ROSSVILLE, as explained on page 84, Sp. Pub. No. 241, for\ncomputation over short lines. (pp. 84A and 84B)\nIt would always be recommended that the computations\nand adjustment of the primary network be carried out at\nthe sea level surface, and then if actual ground distances\nare required in the operations, to raise the coordinates\nto the average elevation of the project by multiplying\nthe coordinates by a factor. For most work this factor\n20,906,000 + h\nwould be, if the elevations are in feet,\n,\n20,906,000\n6,372,000 + h for elevations in meters, with \"n\" being\nor\n6,372,000\nthe elevation in feet or meters, respectively. A further\n84","COMPUTATION OF GEODETIC POSITIONS\nDiff.\nMeridional\nDifference\nDiff.\nV1 for\nH\nSin\nV\nAV\nK\nLat.\nper\nper\nares\nper\nV and K\nsecond\nsecond\n(meters)\nsecond\no\n0.0959\n4,429,318.908\n30.842450\n0.0421\n56492\n170.88\n0.642 78761\n371.35\n4,420,000\n6.54918\n2091\n40 00\n0.0421\n66745\n171.03\n0.643\n01042\n371.25\n4,430,000\n6.57009\n2096\n0.0960\n01\n4,431,169.455\n30.842533\n0.0421\n77007\n171.17\n0.613\n23317\n371.15\n4,440,000\n6.59105\n2102\n0.0961\n02\n4,433,020.007\n30.842633\n4,131,870.565 30.842717; 0.0421 87277\n171.32\n0.643\n45586\n371.08\n4,450,000\n6.61207\n2108 0.0962\n03\n4,436,721.128 30.842817 0.0421 97556\n171.45\n0.643\n67851\n370.98\n4,460,000\n6.63315\n2113 0.0963\n04\n4,438,571.697 30.842900 0.0122 07843\n171.60\n0.643 90110\n370.88\n4,470,000\n6.65428\n2119 0.0964\n40 05\n4,440,422.271 30.813000, 0.0122 18139\n171.75\n0.644 12363\n370.80\n4,480,000\n6.67547\n2125 0.0966\n06\n2130; 0.0967\n4,442,272.851 30.843083. 0.0422 28444\n171.88\n0.644 34611\n370.70\n4,490,000\n6.69672\n07\n2135 0.0968\n4,414,123.436 30.843167 0.0422 38757\n172.03\n0.644 56853\n370.62\n4,500,000,\n6.71802\n08\n4,445,974.026 30.843267 0.0422 49079\n172.17\n0.644\n79090\n370.53\n4,510,000\n6.73937\n2142 0.0969\n09\n4,447,824.622 30.843350 0.0422 59409\n172.32\n0.645\n01322\n370.43\n4,520,000\n6.76079\n2147 0.0970\n40 10\n4,449,675.223 30.843433 0.0422 69748\n6.78226\n172.47\n0.645\n23548\n370.35\n4,530,000\n2154 0.0972\n11\n4,451,525.829 30.813533 0.0422 80096\n172.60\n0.645\n45769\n370.25\n4,540,000\n6.S0380\n2158 0.0973\n12\n4,453,376.441 30.843617 0.0422 90452\n172.75\n0.645\n67984\n370.17\n4,550,000\n6.82538\n2165, 0.0974\n13\n4,455,227.05S 30.843717 0.0423 00817\n172.90\n0.645\n90194\n370.07\n14\n4,457,077.681 30.843800 0.0423 11191\n173.05\n0.616\n12398\n369.98\n40 15\nCorrection for H\n173.18\n0.646\n34597\n369.88\n16\n4,458,928.309\n30.843900\n0.0423\n21574\n4,460,778.943 30.843983. 0.0423 31965\n173.33\n0.646\n56790\n369.80\nSeconds\nCor.\n17\n4,462,629.582 30.844067 0.0423 42365\n173.48\n0.646\n78978\n369.70\n0 & 60\n0\n18\n30.844167\n0.0423 52774\n173.62\n0.647\n01160\n369.62\n10 & 50\n-1\n19\n4,464,480.226\n20 & 40\n-1\n30.844250\n0.0423 63191\n173.77\n0.647\n23337\n369.53\n40 20\n4,406,330.876\n30 & 30\n-1\n0.647\n45509\n369.43\n21\n4,468,181.531\n30.844350\n0.0423\n73617\n173.92\n4,470,032.192 30.844433 0.0423 84052\n174.05\n0.647\n67675\n369.33\n22\n4,471,882.858 30.844517 0.0423 94495\n174.20\n0.647 89835\n369.25\n23\n0.0424 04947\n174.35\n0.648\n11990\n369.17\n24\n4,473,733.529\n30.844617\n30.844700\n0.0424 15408\n174.50\n0.648\n34140\n369.07\n40 25\n4,475,584.206\nCorrection for V\n368.97\n30.844800\n0.0424 25878\n174.65\n0.048 56284\n26\n4,477,434.888\n368.88\nDecimal\nCor.\n30.844883\n0.042-1 36357\n174.78\n0.648 78422\n27\n4,479,285.576\n368.80\n0 & 10\n0\n30.844967\n0.0424 46844\n174.93\n0.649 00555\n28\n4,481,136.269\n0.649 22683\n368.70\n1 & 9\n0\n29\n4,482,986.967\n30.845067\n0.0424 57340\n175.08\n2 & 8\n0\n175.23\n0.649 44805\n368.60\n4,484,837.671\n30.845150\n0.0424 67845\n40 30\n3 & 7\n-1\n0.649 66921\n368.52\n0.0424 78359\n175.38\n31\n4,486,688.380\n30.845250\n4 & 6\n-1\n0.649 S9032\n368.43\n0.0424 88882\n175.53\n32\n4,488,539.095\n30.845333\n5 & 5\n-1\n4,490,389.815\n30.845417\n0.0424 99414\n175.67\n0.650 11138\n368.33\n33\n368.25\n30.845517\n0.0425 09954\n175.82\n0.650 33238\n34\n4,492,240.5-10\n30.845600\n0.0125 20503\n175.97\n0.650 55333\n368.15\n4,494,091.271\n40 35\n0.650 77422\n368.05\n30.845700\n0.0425 31061\n176.12\n36\n4,495,942.007\n176.27\n0.650 99505\n367.97\n30.845783\n0.0425 41628\n37\n4,497,792.749\n176.40\n0.651 21583\n367.88\n38\n4,499,643.496\n30.815883\n0.0425 52204\n0.0425 627SS\n176.55\n0.651 43656\n367.78\n39\n4,501,494.249\n30.845967\n176.70\n0.651 65723\n367.68\n4,503,345.007\n30.816067\n0.0425 73381\n40 40\n30.846150\n0.0425 S3983\n176.85\n0.651 S7784\n307.60\n4,505,195.771\n41\n0.652 09840\n367.52\n30.846233\n0.0425 94591\n177.00\n42\n4,507,046.540\n0.0426 05214\n177.17\n0.652 31891\n367.42\n43\n4,508,897.314\n30.816333\n!\n0.652 53936\n367.32\n0.0426 15844\n177.30\n4,510,748.094\n30.846417\n44\nCORRECTION FACTORS\n0.652 75975\n367.23\n4,512,598.879\n30.846517\n0.0126 26482\n177.45\n40 45\n(Corrections to unity\n0.652 98009\n367.15\n0.0426 37129\n177.60\n4,514,449.670\n30.846600\n46\nin 7th decimal place.)\n0.653 20038\n367.03\n30.846683\n0.0426 17785\n177.75\n47\n4,516,300.466\n0.653 42060\n366.97\nCorrection for x\n-1/fb\n30.846783\n0.0126 58-150\n177.90\n48\n4,518,151.267\n0.653 64078\n366.87\nCorrection for y\n+fa\n30.846867\n0.0426 69124\n178.05\n49\n4,520,002.074\nAre-sinc correction\n+V(Va)/15\n178.20\n0.653 86090\n366.77\n4,521,852.886\n30.816967\n0.0426 79807\n40 50\n178.35\n0.651 0S096\n366.68\n30.847050\n0.0426 90499\n51\n4,523,703.704\n0.654 30097\n366.58\n30.8471330\n0.0127 01200\n178.48\n52\n4,525,554.527\n178.65\n0.651\n52092\n366.48\n30.847233\n0.0427 11909\n53\n4,527,405.355\n366.40\n0.0427 22628\n178.80\n0.654\n74081\n54\n4,529,256.189\n30.847317\nLat.\nf\nF\n0.0427 33356\n178.95\n0.654\n96065\n366.32\n40 55\n4,531,107.028\n30.S47417\no\n0.0427\n41093\n179.10\n0.655\n18014\n366.22\n4,532,957.873\n30.847500\n56\n10-12\n0.655\n40017\n366.13\n40 00\n8.2035\n0.739X\n30.847600\n0.0427\n54839\n179.23\n4,534,808.723\n57\n0.655\n61985\n366.03\n40 30\n8.2025\n0.736\n0.0427 65593\n179.40\n58\n4,536,659.579\n30.847683\n179.55 0.655 83947\n365.93\n41 00\n8.2016\n0.732\n0.0427 76357\n59\n4,538,510.440\n30.847767\n0.656 05903\n0.0427 87130\n41 00\n4,540,361.306\n84A","38.8\n00.5\n00.00\n0.65042\n36\n180\n53\n233\n4,489,749.775\n6.696\n0.7569\n281","discussion on this subject will be made later in the\nsection concerned with project datum coordinates.\nState Plane Coordinates - In 1933, at the request of\na North Carolina Highway Department engineer, Dr. Oscar S.\nAdams of the United States Coast and Geodetic Survey under-\ntook a feasibility study for a plane coordinate system\nutilizing the geodetic data derived from the national net-\nwork. As a result of this study, coordinate systems were\ndeveloped for each state.\nSeveral factors entered into the selection of a\nparticular system or systems for each State and the\npartitioning or zoning required. It was decided for\nreasons that will be given later, that the Lambert con-\nformal map projection with two standard parallels would\nbe used for those states which extended primarily east\nand west, and the transverse Mercator projection for\nthose states whose larger dimensions were north and south.\nIn addition, it was considered desirable that the zone\nboundaries followed county lines, and with one exception\nthis condition was satisfied. A standard of 1:10,000 was\nset as the maximum scale reduction, thus limiting the zone\nwidths to 158 miles. Also considered were the scale reductions\nin the vicinity of metropolitan areas, and with few exceptions\nthe zone designs are such that scale reductions in these\nregions are smaller than the standard.\n85","Care was taken to insure sufficient overlap and to\navoid negative coordinate values; but in two instances\nsmall islands lying some distance off the coast were\noverlooked with the result that points on these islands\nhave negative coordinates.\nIn preparing the tables for the transverse Mercator\nsystems, a small corrective term which would come in only\nat the extremities of a particularily wide zone was neglected.\nExcept in northern Nevada, where the Central zone does not\nextend and the East and West zones meet, the neglecting\nof this term is of no consequence. At worst it could\nintroduce errors of 0.1 to 0.2 feet in relating geodetic\ndistances to grid values and might affect the grid azimuths\nby a few hundredths of a second. Since this would occur\nonly when the points involved are widely separated, the\nerror introduced is of little or no significance. When\nthe transverse Mercator systems for Alaska Zones 2-9 were\ndeveloped, this corrective term was included.\nPrior to 1939 Los Angeles County was included as part of\nCalifornia Zone 5, but many surveys had been computed on\na local plane system which consisted of three zones desig-\nnated as A, B, and C. The origins of these zones were all\nat latitude 34° 08' with the central meridians for A, B,\nand C being at longitude 118° 58', 118° 20', and 117° 42',\nrespectively. Although these coordinate systems were con-\nsidered to be computed on the polyconic projection, the\ncomputations were made using the formulae for Tangent Plane\nCoordinates as described on pp. 81-85. It was contended\nthat for the small areas involved, the differences are\ninsignificant and for all intents and purposes this is\ntrue. While this grid system had been rather extensively\n86","employed, the need for three zones (required to maintain the\nirreconcilable scale error at better than 1:100,000) made\nfor an awkward situation. Consequently, zone 7 was devel-\noped covering all of Los Angeles County with a maximum\ncomputable scale reduction of 1:87,300. The origin of zone\nB which was located at latitude 34° 08' and longitude 118°\n20' with previously assigned values for X = 1,186,692.58\nand Y = 1,160,926.74 was selected as the origin for zone\n7, and to distinguish between the systems, 3,000,000.00\nfeet was added to these origin coordinates.\nAs finally developed, nine states contain but one\nzone; in two states, Florida and New York, both plane\ncoordinate systems were employed; and by accepting a slightly\nlarger scale reduction standard (as was done in several\ninstances), a state as large as Texas was covered by five\nzones. In later years, plane coordinate systems were\ndeveloped for Alaska and Hawaii and most possessions.\nAlaska employs both projections, and in addition, an\noblique Mercator projection for southeast Alaska, and is\ndivided into ten zones. Guam utilizes a tangent plane\nsystem.\nRecently, Michigan proposed that the state be placed\non a Lambert system rather than the original transverse\nMercator projection. In addition, it was requested that\nthe system be referenced at an elevation of 800 feet (the\naverage elevation of the state) rather than sea level as\n87","are all other state systems. Although it was understandable,\nwith the large population centers spread east to west\nacross southern Michigan, that the Lambert projection\nwould be more satisfactory, there was some reluctance\nto reference the system at other than sea level. However,\nthe proposal was accepted and plane coordinates for all\nnational network stations in Michigan are now available\non both systems.\nWith the discovery of oil off the Gulf coasts of\nLouisiana and Texas, the south zone for Louisiana was\nextended to cover the original discovery area. However,\nnegative coordinates resulted in some instances. As the\noil leases moved further south into the Gulf of Mexico,\nan additional extension to this zone could not be satis-\nfactorily made and it was necessary to develop a new grid\nzone. The solution was rather simple. The south zone\nof Texas covered much of the latitude range required and\nby simply shifting the central meridian from 98° o 30' to\n91° 20' and extending the tables to cover the entire area,\nthe Louisiana offshore zone was created. In this regard,\nit may be of interest to know that the State systems can\nbe shifted anyplace in the world, north or south of the\nequator without additional computations, by shifting the\ncentral meridian to that locality providing the area is\nwithin the latitude range of the tables and the basic\n88","surveys are computed on the Clarke Spheroid of 1866.\nActually, an area could be covered by using a system\nfor one State zone for a part, a second State zone for\nanother part, etc. The latitude bands for the Lambert\nprojection extend around the world, and the latitude range\nof the transverse Mercator systems can be satisfactorily\nextended to within a few degrees of the poles. As a\ncase in point, the UTM coordinates which will be described\nlater are nothing more than the transverse Mercator system\nwith the equator as the origin.\nTables are available for the direct and inverse com-\nputation of coordinates for all systems using desk calcu-\nlators. With the exception of Alaska, the computational\nformat for each projection is identical. For Alaska the\n2-1/2 minute intersection tables are employed using an\ninterpolation technique which is fully explained in the\npublications.\nCoordinates for the tangent plane system used on\nGuam are computed using Special Publication No. 241, as\ndescribed previously.\nAlthough each State coordinate projection table\ncontain examples of the direct and inverse computations,\nsample computations on the Lambert and transverse Mercator\nsystem are given here for handy reference (Figures 1, 2, 3,\n4). The formulae are straightforward and no detailed\n89","State - Zone Kansas-North\nY=R6-R cos O\nX= R sin O + C\nC = 2,000,000.00\nRb=25,979,068.57\nGrid Az. = Geod. Az. - e\nsin e\nX\nLatitude\nR\nStation\nLongitude\nO\nCOS O\nY\n38 58 52.096\n25,743,138.15\n+0.01326 78253\nI\n2,341,555.46\nROBBINS\n96 47 54.567\n+0 45 36.7657\n19786\n0.99991\n238,196.37\nGrid azimuth to azimuth mark\n36\n42\n39 58 41.957\n25,379,925.63\n-0.00651 51579\n1,834,645.78\n2\nCOOPER\n98 35 23.954\n2 23.8573\n0.99997 87762\n599,681.60\n256° 58 07.\nGrid azimuth to azimuth mark\nFigure 1 - Plane Coordinates on Lambert Projection\nl= 0.63271 48646\nSTATE-ZONE Kansas - North\nStation ROBBINS\n25,979,068.57\n-2,000,000.00\nR\nC\n- 238,196.37\n2,341,555.46\nx\nY\n25,740,872.20\n+ 341,555.46\nx'=x-C\nRo-y\n+2736.7657\n+0.01326 89933\ntan\n0\n+0 45 36\".7657\n+4325.433\nAl=01\n0\n+1° 12' 05\".433\n0.99991 19786\nAl\ncos 0\n98° 00' 00\"000\nR=(Rb-y) - cos A\n25,743,138.15\nCentral Meridian\n96° 47' 54\"567\n38° o 58 52.096\nr C. M.-Ar\nStation COOPER\n25,979,068.57\nRb\nC\n-2,000,000.00\n- 599,681.60\n1,834,645.78\nx\nY\n25,379,386.97\n- 165,354.22\nRo-y\nx'=x-C\n-1343.8573\n-0.00651 52961\ntan 9=x'(Ro-y)\n0\n-2123.954\n-0° 22 , 23.8573\nAl=0-1\n0\n-0° 35' 23\".954\n0.99997 87762\nAl\ncos 0\n98° 00' 00\"000\n25,379,925.63\n:=(Ro-y) cos 0\nCentral Meridian\n98° 35 23.954\n39° 58 , 41.957\n1=C.M.-AM\nFigure 2 - Geodetic Positions from Lambert Coordinates\n90","State\nZone\nIdaho\nEast\nCentral meridian\n112° 10' 00\".000\nStation\nPINHEAD\nWALKER\n43°\n48\n07\".616\n43°\n26\".260\n35'\n42\n29.824\n111\n35.516\n112\n1\n22\nAA = Central mer.-)\n+ 0° 27' 30\".176\n- 0° 12' 35\".516\nAX\"\n+ 1,650.176\n- 755\"516\n( As\") 100\n272.308\n57.080\n73.336360\n73.594594\nH\n1.230701\n1.230273\nV\nb\n-0.418\n-0.357\n+1.175\na\n+0.595\n+ 121,017.48\n55,601.64\n-\n335.07\n70.21\n778,234.67\nTabular y\n701,147.74\n621,017.48\n444,398.36\nX\n778,569.74\n701,217.95\ny\nAa\"\n+ 1,142\"21\n- 520.93\nAa\n0°\n02.2\n0°\n08'\n40.9\n19'\n+\n-\nGeod. Az. to Az. Mk.\n26\n16.7\n42.8\n53\n200\n33\n14\n42\n24\nGrid. Az. to Az. Mk.\n53\n07\n200\nx =x'+500,000\nH and V = Tab. H and Tab. V.\ndecrease increase H. Annumerically\ny=Tab. y\nWhen ab is\ng increases AX\" sin & numerically\nGrid Az. = Geod. Az. -Aa\nFigure 3 - Plane Coordinates on Transverse Mercator\nProjection\n91","STATE - ZONE Idaho - East\nStation WALKER\nX\n621,017.48\nY\n778,569.74\nC\n500,000.00\nP(10,800)2+\n-\n335.08\n-\nx'\n+ 121,017.48\nYo\n778,234.66\n2.28775\nP\n+ 1,650 \"\nApprox.A.=\nd\n+ 0.03\nA1=(x'+ab) - H\n+ 1,650.176\nH\n73.336360\nA1\n+ 0\n30.176\n27\nb\n- 0.357\na\n+ 1.175\nCentral Meridian\n112\n10\n00.000\n43\n48\n07.616\n1= C.M.-A1\n111°\n42\n29.824\nStation PINHEAD\nX\n444,398.36\nY\n701,217.95\nC\n500,000.00\nP(10,000)++\n-\n70.21\n-\nx'\n55,601.64\n701,147.74\nYo\n2.27087\nP\n- 756\"\nApprox.A)=x'-H\nd\n+ 0.01\nA1=(X'+ab) - H\n- 755.516\n73.594594\n- 0°\nH\n35.516\nAl\n12\n-\n- 0.418\nb\n+ 0.595\na\nCentral Meridian\n112\n10\n00.000\n43\n35'\n26.260\n1= C.M.-A1\n112\n22'\n35.516\n+, decrease\nx'\nWhen ab is\nnumerically\n, increase\nFigure 4 - Geodetic Positions from Transverse Mercator\nCoordinates\n92","detailed instructions are presented. * Tables for the com-\nputations on the transverse Mercator projection contain\nall data necessary, but an additional table containing\nsines, cosines, and tangents to ten decimal places is\nrequired for the Lambert projection. A table containing this\ninformation from 0° to 6° (Special Publication No. 246) is\navailable from the Government Printing Office at a moderate\ncost. This range (0° to 6°) is more than sufficient for\nthe computations within the area limitations as employed\nfor the system.\n* See Addendum, p. 104.\nFor those who wish to program the computations on\nelectronic computers, there is available from the GPO\nPublication 62-4, \"Plane Coordinates by Automatic Data\nProcessing.\" This publications contains the constants\nrequired for the computations involved in all the systems\ndevised to date by the Coast and Geodetic Survey\n(now known as the National Ocean Survey) and complete\ndescriptive examples. Computations can be carried out\nbeyond the range of the published tables even to computing\nnegative values; but it must be remembered that should these\ndata be used to control surveys, the scale factors increase\nrapidly beyond the published range and equally important the\nsecond term corrections also significantly increase in size.\nAlthough the programs are not included because of the variety\n93","and capacity of available computers, no information has\nbeen received of any great programming difficulties. As\na matter of fact, these formulae have been programmed very\nsuccessfully on a number of the small programmable desk\ntype computers.\nNone of the conformal map projections used in the\nnational grid systems can be truly illustrated graphically\nand the proofs can only be demonstrated mathematically. For\nthose who may be interested, the mathematical details are\ngiven in several publications dealing with map projections,\none of which (Special Publication No. 251) is given in the\nlist of references. In general, the primary concern of\nmost practicing surveyors is the scale reduction and this\ncan be shown graphically, in a broad sense, as illustrated\nin Figures 5, 6, 7. Of lesser concern at this time, but of\nincreasing importance as more accurate surveys are made, is\nthe second term correction. This correction will be con-\nsidered in some detail in the chapter on computations.\nFigure 5 illustrates the Lambert projection with two\nstandard parallels. This projection can be viewed as a\ncone, cutting or being secant to the sphere (circle) repre-\nseting the earth at these parallels. The axis of the cone\ncoincides with the polar axis of the reference spheroid.\nTo simplify the explanation of the scale reduction, consider\nchord distances passing through the polar axis at right angles\n94","SCALE EXACT\nSTANDARD PARALLEL\nSTANDARD PARALLEL\nSCALE EXACT\nFigure 5 - Lambert Projection - Cone Secant to Sphere\nas being geodetic and the corresponding distances on the\ncone as grid values. As can be seen, geodetic distances\nare longer than the corresponding grid distances when lo-\ncated between the standard paralleAs, with the opposite\nbeing true for distances north and south of these parallels.\nAlong the standard parallels, geodetic and grid distances\nare identical. In other words, since any measured distance\nis a geodetic quantity, these distances should be made\nsmaller by applying a scale factor before using in a\n95","computation involving Lambert plane coordinates for points\nwithin the standard parallels. Similarily, the distances\nshould be increased for points outside these parallels.\nThe maximum negative scale reduction is along the central\nparallel or yo of the projection which is approximately\nmidway between the standard parallels.\nThe Lambert projection was selected for those states\nextending mostly east and west because the scale varies\nwith the latitude and is not effected by longitude differences.\nThe transverse Mercator projection is a so called\n\"developed cylinder\" type as shown in a very simple form\nby Figures 6 and 7. The spheres (circles) represent the\nearth and the rectangles the cylinder. In Figure 6, the\ncylinder is tangent to the sphere at the poles and the\nscale is exact along any meridian selected as the central\nmeridian. Grid meridians are parallel to this meridian\nand do not intersect at the poles as do geodetic meridians.\nAs a result, the scale varies in this projection with the\ndistance between these meridians and the central meridian.\nThis condition may be seen by visualizing the chord distances\npassing through the line representing the equator at right\nangles as geodetic values and the corresponding distances\non the cylinder as grid distances. Obviously, the grid\ndistances are longer with the exception of those along the\ncentral meridian. This projection, of course, is perfectly\n96","EQUATOR\nFigure 6 - Transverse Mercator Projection, Cylinder Tangent\nto Sphere at Poles\nEQUATOR\nFigure 7 - Transverse Mercator Projection, Cylinder Secant to\nSphere\n97","valid; however, a greater area within the same scale\nstandard can be utilized by applying a scale reduction\non the central meridian thus creating a secant type\ncondition similar to that used for the Lambert projection.\nIncidentally, one standard parallel, or the cone tangent\nto the sphere, could have been employed for the Lambert\nprojection but for the same reason noted above this was\nnot done.\nFigure 7 illustrates the transverse Mercator pro-\njection as used in the State coordinate systems. By\nplacing a scale reduction on the central meridian, the\nscale is exact along the meridians where the cylinder\nintersects the sphere. Between these meridians, grid\ndistances are smaller than geodetic distances, and beyond\nthese meridians the opposite condition exists.\nAs noted previously, the scale varies east and west\nof the central meridian, and for this reason the transverse\nMercator projection was selected for those states extending\nprimarily north and south.\nUniversal Transverse Mercator Coordinates - This grid\nsystem has been adopted by the Defense Department for world-\nwide military mapping purposes, but on occasion it has been\nemployed by civilian agencies. The system has the equator\nas the origin and extends in 6° bands eastward from the 180°\nmeridian, thus establishing 60 zones within the area between\n98","latitudes 80° S and 84° N. The central meridian is always\nthe middle meridian of a zone. For example, zone 17 is\nbetween 78° and 84° west longitude, and the central meridian\nis 81° longitude. The scale reduction along the central\nmeridian is 1:2,500 and exact 180,000 meters (about 112\nmiles) east and west of the central meridians.\nThe actual map projection is for all intents and\npurposes the same as utilized in the State coordinate\nsystems, but with a considerably larger scale reduction\non the central meridian, and as a consequence of the wider\nzones significantly larger second term corrections at the\nextremities. Also, the coordinates are in meters while\nin the State systems these values are in feet.\nSample computations are shown in Figures 8 and 9. The\nformulas are easy to follow in a relatively simple straight-\nforward computation, and no detailed instructions will be\ngiven here. * The Roman numerials and alphabetical notations\nrefer to the tabulated functions.\n* See Addendum, p. 104.\n99","FE\nUNIT\nCENTRAL MERIDIAN\nZONE\nSPHEROID\nMeter\n500,000 m.\n99° 00'\n14\nClarke of 1866\nLOLITA (Texas)\nStation\nBASSETT (Texas)\nStation\nA\n96°32'35\"983\n28°50'04.\"095\n101°43'18\"013\n0°04'48\".992\nA\nAX\n(I)\n3189471.376\n2.27.24.017\n2 43 18.013\nat\n+\n(I)\n3327500.243\n3168.101\np2\n0.7821664\n0.9600106\n(II)\n3252.483\nD2\n(II)\n0.61178\np4\n0.92162\n(III)\n2.259\n2.247\n(III)\np+\nA6\n+ 0.001\nA6\n+ 0.001\n0.8844017\n270990.581\n0.9798013\n(IV)\np\n267699.763\n(IV)\np\np3\n0.691749\n(V)\n0.940620\n57.201\n52.581\np3\n(V)\nB5\n0.002\n- 0.006\nB5\n739704.10\n3191950.74\n237657.97\nE\n3330624.73\nN\nE\nN\nFigure 8 - U.T.M. Grid Coordinates From Geographic Coordinates\nThe computations are made using the following formulae:\nNorth of Equator N = (I) + (II)p2 + (III)p4 + A6\nSouth of Equator N 10,000,000 (I) + (II)p2 + (III)p 4 + A6\n= -\nE' = (IV)p + (v)p3 + B5\nE' is added to produce E if the point is east of the\ncentral meridian, subtracted if it is west.\nThe Roman numerials, A6, and B5 refer to functions\nappearing in the tables: or is the central meridian minus\nthe longitude of the point; and \"p\" is (0.0001) (AX\").\n100","SPHEROID\nCENTRAL MERIDIAN\nZONE\nUNIT\nFE\nClarke of 1866\n99° 00'\n14\nMeter\n500,000 m.\nBASSETT (Texas)\nLOLITA (Texas)\nStation\nStation\n3330624.73\n237657.97\n3191950.74\nE\n739704.10\nN\nN\nE\n30°06'30\"5044\n$\n28°51'24\"6631\n'''\n1476.219\n0.06882334\n1403.181\nq2\n0.05745806\n(VII)\n(VII)\n92\n18.12\n0.0047367\n16.95\n(VIII)\nq4\n0.0033014\n(VIII)\n94\n- 0.001\nD6\nO\nD\n37365.884\n0.26234203\n36909.550\n0.23970410\n(IX)\n9\n(IX)\n9\n256.583\n0.0180553\n243.648\nq3\n(X)\n0.0137729\n(X)\nq 3\n+ 0.0042\nE5\n+ 0.0021\nE5\n01'41\"5124\n+ 2 43 18.0134\n01'20\"5681\nAl - 2 27 24.0168\nAO\nAo\n-\nAX\n-\n30°04'48\".9920\n101°43'18\"0134\n28°50'04\".0950\n96°32'35\".9832\nA\nA\nFigure 9 - Geographic Coordinates From U.T.M. Grid Coordinates\nThe computations are made using the following formulae:\nLatitude (0) = 01 - (VII)a2 + (VIII) q4 - D6\nLongitude Difference (A\"X) = (IX)q - (x)q3 + 5\nLongitude (x) = Central Meridian + AN\nWhen E is less than 500,000, AX is added to the Central\nMeridian to produce 1, and subtracted when it is more.\nThe Roman numerials, ' ' , D6 and E5 refer to functions\nappearing in the tables; and \"g\" is (0.000001)(E').\n101","In addition to the publications concerned with com-\nputations on the Clarke Spheroid of 1866 as given in the\nReferences, there are tables available in the TM 5-241 -\nseries for the Australian National, Bessel, Clarke of 1880,\nEverest, and International spheroid and the extension of\nthe tables to 84° north latitude for several of these\nspheroids.\nSurveys computed on this grid system are carried out\nfollowing the same procedures as when employing State plane\ncoordinates. All distances must be in meters, of course.\nAs noted previously, the scale reductions and second-term\ncorrections may be much larger than those for the State\nsystems, and an inspection, at the very least, should be\nmade to ascertain whether it is necessary to apply these\ncorrections.\nSome consideration is being given to adopting a similar\ngrid system as a replacement for the State systems when a\nnew adjustment of the national geodetic network is made. In\norder to maintain a maximum scale reduction equivalent to,\nor as is more likely smaller than the present standard used\nin the State systems, the zone widths will be considerably\nless than that used in the UTM system. As a point of interest,\nthe new adjustment will probably be initiated in this decade\nand hopefully completed in the early 1980's.\n102","Project Datum Coordinates - Occasionally for some\nengineering and special projects, it is pertinent that\nthe grid system be related to actual horizontal ground\ndistances. As noted earlier, tangent plane coordinates\ncan be satisfactorily used in such situations, however,\nthis would involve the more difficult geodetic computations\nin establishing the basic framework. For some areas, even as\nlarge as a moderately sized county, State plane coordinates\nmay be raised to the average elevation of the project, and\nby applying the average scale factor the resulting distances\nare essentially at ground level. There are three con-\nsiderations that must be fully taken into account and understood\nbefore anyone employs such grid systems, however. One, the\nscale factor: since the average scale factor is used, the\nmaximum deviation from the true scale will be at the fringes\nof the project, however, should the area be large, significant\nerrors may be introduced elsewhere as well. An investigation\nshould always be made to assure that these errors are\nacceptable within the survey requirements. Two, elevations:\nwhere there are large differences of elevations, this type\nof grid system can prove troublesome. It must be remembered\nthat for each 21 feet variance from the average elevation,\nan error of about one part in a million is introduced. An\nerror of this magnitude is insignificant, of course. However,\nshould the difference be about 400 feet, the error would\n103","approach 1:50,000, and in many instances this would be\nunacceptable. It must always be kept in mind that the\nscale and elevation factors are used in combination and\nthe combined error must be considered in determining\nwhether an acceptable accuracy results. Three, documen-\ntation: to aid in reducing misinterpretations and costly\nmistakes, all computations, maps, drawings, etc., must\ncarry explanatory notations in sufficient detail.\nIn general, to obtain the best results, the basic net\ncomputations should be carried out at the sea level\nreference and the resulting coordinates divided by the\ncombined average elevation-scale factor. It has been\nsuggested by some that rather than divide the \"X\" coordinate\nby the combined factor, that the X' value be used and then\nthe constant for the zone added. Either method is entirely\nsatisfactory.\nProject datum coordinates would seldom be recommended.\nOften the sole purpose is to eliminate the sea level and scale\nfactor computations. In all candidness, the amount of time\nsaved in such instances is rather small, and the possibility\nof problems developing at a later date is quite large.\nAddendum: Step-by-step computations for the conversion of\nlatitudes and longitudes to State plane coordinates and UTM\ncoordinates and the inverse solutions are included in the\npaper entitled, \"The National Geodetic Survey, Its Products\nand Their Application to Local Surveying Needs\" which is\nincluded with the Workshop published material.\n104","CORRECTIONS TO MEASURED LENGTHS\nAND COMPUTATION OF GRID AZIMUTHS\nJoseph F. Dracup\nSupervisory Geodesist\nGeodesy Division\nNational Ocean Survey\nNational Oceanic and Atmospheric Administration\nU. S. Department of Commerce\nRockville, Maryland 20852\nFor distance measurements, the corrections involving\nthe instrument and mirror constants and eccentricities are\ngenerally straightforward and no further comment will be\noffered except to express a word of caution - be sure\nof the signs and that all corrections are included.\nCorrections for weather conditions or the meteorological\neffect, on the other hand, are often not full understood;\nthe data are incorrectly obtained and equally often in-\ncorrectly applied. The graphs furnished by the manufacturer\nto determine this correction will easily provide the necessary\ncorrections and are recommended for general use, but one\nmust remember that paper shrinks and periodically precise\ncomputations should be made and compared with the graphical\nresults. Corrective terms, if significant, can then be\napplied to the corrections obtained from the graphs.\nPlastic computers supplied by some manufacturers are fine\nwhen first put into use, but as time goes on, the components\nloosen and erroneous data may result. Too often the field\nequipment may be in excellent operating condition, but\n105","the end results may be something less than desired,\nsimply because the office computing devices are not kept\nin the same condition.\nComputations made on electronic computers,\nproperly programmed, will give results that are correct\nfor the data entered--but often these results do not reflect\nthe accuracies of the observations primarily because less\nthan adequate care was taken in obtaining the meteorological\ndata. Tabulated below are the errors to lengths resulting\nfrom meteorological observations:\nElectro-Optical\nMicrowave\nTemperature\n1 PPM\nDry Bulb 1°C\n1 PPM\nDifference in Dry\n1 PPM to 17 PPM\n*\nand Wet Bulbs 1°C\nBarometer 0.1 inch or\n1 PPM\n1 PPM\nAltimeter 100 feet\n* The correction for humidity may be ignored or\ntaken at 0.4 PPM.\nAs may be seen, the humidity correction for micro-\nwave equipment can be quite significant and it is imperative\nthat accurate dry and wet bulb temperatures be obtained at\nboth ends of the lines. For measurements made over steep\nlines, the barometric or altimeter observations are critical.\nAs an example, a difference of elevation of 1000 ft. intro- -\nduces an error of about 5 PPM if these observations are made\nat one end only. To many, these possible errors may seem\n106","small and irrelevant, yet carelessness leads to more\ncarelessness, and inaccuracies of 1 to 2 PPM soon evolve\ninto much larger errors.\nAs final comments regarding this subject, it is rather\nludicrous to pay thousands of dollars for distancing\nequipment and then buy the cheapest necessary accessories\nsuch as thermometers and barometers. Equally ludicrous\nis the fact that many surveyors actually believe they are\nsaving money by ignoring or estimating the fundamental\ncorrections to electronic distance measurements. Nothing\ncan be further from the truth, the time spent in obtaining\nthe meteorological observations is measured in but fractions\nof a minute.\nIn addition to the meteorological corrections, the\nmeasured distances should be corrected to the horizontal\n(slope or inclination correction), and generally to sea\nlevel if the measurements are to be used in State plane\ncoordinate computations; and occasionally when the lines\nare long (10 miles or more) for the arc correction.\nForms such as that shown on the next page are quite handy.\nNote that all forms given in the Workshop courses are not\ncluttered and seldom are in combination with other pro- -\ncesses involved in the total computation. Long experience\nhas shown that forms designed for specific tasks with\nsufficient space for the necessary computations are much\n107","Form 603-5A\nCORRECTION TO BE APPLIED TO GEODIMETER SLOPE DISTANCE\nGEODIMETER STATION\nMIRROR STATION\nElevation of Geodimeter Station\nM\nhl\nHeight of Geodimeter\nM\nh2\nhl\nM hl-h2\nM (h1-h2) 2\nElevation of Mirror Station\nHeight of Mirror\nM hl + h2\nh2\nM\nD*\nD2\nP**\n2P\nMean latitude of base line\nAzimuth of base line\n( hl + h2 ) D\nSea Level\nM\n2P\nD - VD - (h1 - h2)2\nSlope\nM\nArc\n1.027 D3 X 10-15\nM\n+\nSum of Corrections\nM\n*Geodimeter slope distance in meters\nRadii of curvature of the earth's surface\nFrom: Pages 58 through 65, Geodimeter Manual, Publication 62-2\nArc Correction Table\nCor.\nD\nCor.\nD\nCor.\nD\nCor.\nD\nCor.\nD\nCor.\nD\nkm\nkm\nkm\nkm\nm\nkm\nkm\nm\nm\nm\nm\nm\n51\n.136\n21\n.010\n31\n.031\n41\n.071\n1\n.000\n11\n.001\n52\n.144\n22\n.011\n32\n.034\n42\n.076\n2\n12\n.000\n.002\n33\n.037\n43\n.082\n53\n.153\n3\n13\n.000\n.002\n23\n.012\n34\n.040\n44\n.088\n54\n.162\n.003\n24\n.014\n4\n.000\n14\n45\n.094\n55\n.171\n.003\n25\n.016\n35\n.044\n5\n.000\n15\n46\n.100\n56\n.180\n26\n36\n.048\n6\n.000\n16\n.004\n.018\n27\n.020\n37\n.052\n47\n.107\n57\n.190\n7\n.000\n17\n.005\n28\n.022\n38\n.056\n48\n.114\n58\n.200\n8\n18\n.006\n.000\n.061\n49\n29\n.025\n39\n.121\n59\n.211\n9\n.001\n19\n.007\n30\n40\n.066\n50\n.128\n60\n.222\n.028\n10\n.001\n20\n.008\n108","better than those which use all available space to squeeze\nrelevant and irrelevant data on a single sheet.\nThe primary discussion here will be directed towards\nthe slope correction. Corrections to sea level are only\nbriefly mentioned since a fuller discussion will be found\nin the section on Computations and Adjustment.\nThe slope corrections may be determined from vertical\nangle observations or from differences of elevations. For\nthe more precise surveys, differences of elevations are\npreferred providing these values reflect the difference of\nelevation between the actual heights of the instruments\nand/or reflectors. Note on form 603-5A the heights of the\ninstruments and mirror are added to the elevations of the\nrespective ground points prior to making the reduction\ncomputations. To many surveyors whose operations are\ncarried out using tripods, all more or less the same\nheight, this additional refinement may seem unnecessary,\nand generally this is true; however in some instances\nrather small differences in the heights of the tripods\ncan significantly effect the reduced distances. For\nexamples: a difference of one foot in the tripod heights\non a 100-foot line differing in elevation by 10 feet, will\nproduce an error in the reduced length by 0.1 ft. or 1:1,000.\nSimilarly, an error of 2 feet in the heights where the\nelevation difference is 200 feet on a line, 5000 feet in\nlength will introduce an error of 0.08 ft. in the hori-\nzontal distance. The important issue here--why not employ\n109","correct practices regardless of the desired accuracy of\na survey? The time required to do so is exceedingly small.\nWhenever the cosines of the vertical angles between\npoints are used to obtain horizontal distances, two important\nconsiderations must always be kept in mind.\nOne, are there differences in the tripod heights as dis-\ncussed above? Two, have the observations been corrected\nfor curvature and refraction? This correction is\nhas no effect\nnegligible for short lines or/where small differences\nin elevations occur, but can be very significant in\nsome instances as will be shown in the following examples.\nThese examples cover most of the situations encountered\nin reducing slope distances to the horizontal.\n(1) Elevations: A = 280.6 ft. B = 518.4 ft.\nSlope distance A to B = 7456.35 ft. Difference of\nelevation A-B = 227.8 ft.\nis S If (h)2, where \"S\" is\nThe exact formula\nthe horizontal distance; \"D\" the slope distance; and \"n\"\n(7456.35)2 2 - (227.8) 2\nthe difference in elevation. Therefore S\n1\n= 7453.168 ft.\n(2) Vertical angles observed. The theoretical obser-\nvation at A to B = + 1° 19' 54.4 and the cosine is\n0.99957774 with the resulting reduced distance being\n7453.201 ft. But this differs from the exact reduction\nby 0.033 ft. and is due to ignoring the correction to\n110","the observed vertical angles for curvature and re-\nfraction. This correction K (in seconds) - 0.004231\nX D when \"D\" is in feet or 0.01388 X D when D is in meters.\nSince the constant includes, in addition to the correction\nfor curvature, the coefficient of refraction which\nvaries according to meteorological conditions, the\nformula used is approximate (an average value of 0.071\nfor the coefficient of refraction was used);\nbut it is sufficiently accurate for most purposes. The\ncorrection is added to angles of increasing elevation\nand subtracted from depression angles. In this case\nK\" = 7456 X 0.004231 = 31\".5, and the corrected vertical\nangle A to B = + 1° 39' 54.4 + 31.5 or + 1° 40' 25\".9.\nThe cosine of this angle = 0.99957329 and multiplying\nthis value by the slope distance the horizontal distance\nis 7453.168, which checks the exact reduction.\nThe computation of \"K\" is not necessary if reciprocal\nvertical angles are observed since the mean of the obser-\nvations A-B and B-A eliminates the correction for curvature\nand refraction. But as a check on the observations, the\nvalue should be computed. The difference between the\nvertical angles is 2K + N where N should seldom exceed\n20\" if the observations are made on the same day and at\napproximately the same time. The theoretical observation\nB to A =-1° 40' 57.4 and the mean observed value is\n1° 40' 25\".9 checking the corrected observations A-B and B-A.\n111","(3) Occasionally the vertical angle observations are\nmade at different heights of instruments (t) and signals\n(o) than the length measurements. These varying heights\npresent no problem if elevations for the monuments are\nfirst determined (see publication G-56 in list of references)\nbut the observations must be corrected if the cosine\nformula is to be used.\nLet us assume that at A the theodolite (t) was 5- -\nfeet higher than the distancing measuring equipment and at\nB the signal (o) was 15 feet higher. The formula to\ncorrect the observations is C (in seconds) = - (t-o) X 206265\nD\nor in this case C = - (-10) X 206265 = +276.6.\n7456\nThe theoretical observed vertical angle A-B in this\ncase is + 1° 441 30.9 and the corrected angle is:\n+ 1° 441 30\".9\n36.6\n04\n*C = -\n31.5\nK = +\n+ 1° 40' 25\".8 which checks the corrected angle\nin (2) above by 0.1.\n* Applied with opposite sign for angles of increasing\nelevation.\nIn the most accurate surveys the sea level correction\nform 603-5A ) should be made using a value of \"P\" deter-\n(\nmined for the mean latitude of the points involved and\nthe mean azimuth (geodetic) between these points. Tables\n112","are available for \"P\", however, for the average length\nof line involved in most local surveys, the sea level\nreduction described in the section on Computations\nand Adjustment is sufficiently accurate. But in those\ncases where the lines are long in areas of higher\nelevation, the more precise computation should be con-\nsidered.\nGrid Azimuths\nGrid azimuths may be determined directly from the\nplane coordinates of the points involved or through the\nreduction of geodetic azimuths. The formulae follow:\n(1) Using plane coordinates:\nTan a 2A-B or Cot a A-B = Y A . B\nwhere A and B are the points involved and \"a\" is the\ngrid azimuth of the line. The signs of XA - XB and\nYA - YB correspond to the quadrant in which the azimuth\nappears with an azimuth of 0° 00' 00\" due south,\n90° 00' 00\" due west, 180° 00' 00\" due north, and\n270° 00' 00 due east. Geodetic azimuths in the United\nStates are referenced clockwise from the south with the\nquadrants as given above, however, many surveyors perfer\nto orient surveys from the north and this is generally\nacceptable, but in the following discussions the azimuths\nfollow the geodetic practice.\n113","It is generally considered best when using the\nformulae given above, because of the number of significant\nfigures involved, to utilize the value for Tan a or Cot a\nwhich is less than one. When the azimuth is in the first\nand third quadrant, the angle corresponding to the tangent\nor cotangent should be determined and added to 0° or 180°,\nrespectively. Azimuths in the second and fourth quad-\nrants are obtained by determining the angle for the co-\nfunction and adding this value to 90° or 270°, respectively.\nThe cofunction of the tangent is the cotangent and vice versa.\n(2) When geodetic azimuths are available, the cor-\nresponding grid azimuth = Geod. Az A-B - Aa (or 0) + second\nterm correction. Aa is the mapping angle for the transverse\nMercator system and 0 for the Lambert system. The second\nterm corrections are described in the section on Computations\nand Adjustment.\na\" = Sin (CM - X)\" + g where and X are the latitude\nand longitude respectively of the point, CM is the longitude\nof the central meridian for a zone and values for \"g\" are given\nin a tabular form in the projection tables. The \"g\" term may be\nignored when (CM - X)\" is less than 2500\". Aa\" may also be\ncomputed using the formula Aa\" = MX' - e, where M is a\nfunction of the Y coordinate of a point. Tables for M and\ne are given in the projection tables. As an example, the\ntable for Arizona is given on p. 118.\n114","The 0 angle may be computed using the longitude for\na point and interpolating from the table given with the\nprojection tables. As an example, a partial table for\nUtah south zone corresponding to the longitude range for\nthe numerical example that follows is given on p. 119.\nThe O angle may also be computed by the formula\nTan O R where Rb is a constant for the zone\ninvolved.\nNumerical examples for the transverse Mercator and\nLambert systems follow:\nArizona Central Zone\n(1) A 0 = 33° 19' 11.1287\nX = 482,449.72\nY = 843,845.64\nA = 111 58 26.8321\nX = 506,105.19\n56.1137\nB\n33\n15\nQ\n=\nY = 824,132.48\nA = 111 53 48.0940\na = = 0.83334468. Azimuth is in fourth\n-23,655.47\nquadrant, interpolate for tan 0.83334468 = 39° 48' 21.5\nand add to 270° or Grid Azimuth A-B = 309° 48' 21.5.\nWhen geodetic azimuths are available, the grid azimuths may\nbe derived as follows: Using the formula Aa = Sin 0 (CM-X)\nAu at A = 0.549311 (111° 55' 00.0000 - 111° 58' 26\"8321) = -113.6\nAa: at B = 0.548521 (111° 55' 00.0000 - 111° 53' 48.0940) = + 39.4\n129°\n49'\n00.9\nGeod Az A-B 309° 46' 27.9\nGeod Az B-A\n39.4\n- Aa at A + O 01 53.6\n- Aa at B\n-\nGrid Az B-A 129 48 21.5\nGrid Az A-B 309 48 21.5\nThe second term correction is less than 0.1 and was\nneglected in this example.\n115","Utah South Zone\nX = 2,235,545.34\n(2) A 0 = 38° 21' 47..76185\n618,804.51\nA = 110 40 42.75141\nY =\nX = 2,259,464.19\n38\n16\n43.30415\nB\n0\n=\n588,225.29\nA = 110 35 46.18097\nY =\nXA - = -23,918.85\nYA - YB = +30,579.22\n-23,918.85 = 0.78219294. Azimuth is in\nTan\n=\nA-B\n+30,579.22\nfourth quadrant, interpolate for cot 0.78219294 = 51° 58' 03.8\nand add to 270° or Grid Azimuth A-B = 321° 58' 03.88\nWhen geodetic azimuths are available, the grid azimuths\nmay be derived as follows:\n0 at A (from tables) + 0° 30' 11.9\nO at B (from tables) + O 33 13.6\nFor those who wish to compute the O angles using the\ncoordinates, Rb = 27,432,812.88 for Utah south zone.\nGeod Az B-A 142° 31' 18.5\nGeod Az A-B 322° 28' 14.6\n- O at B - O 33 13.6\n- 0 at A - O 30 11.9\n01.1\n*\n01.1\n*\n+\n-\nGrid Az B-A 141 58 03.8\nGrid Az A-B 321 58 03.8\nSee explanation with\n* Second term corrections.\nComputation and Adjustment Section. Utah south\nzone Yo = 406,857.53\n116","TRANSVERSE MERCATOR PROJECTION\nTABLE FOR g\nsa 11 = sin (AA\") + g\nor\"\nLati-\ntude\n4000\"\n6000\"\n0\"\n1000\"\n2000\"\n3000\"\n5000 H\n24°\n0.00\n0!00\n0!02\n0.07\n0.17\n0!33\n0!58\no\no\n0.02\n0.07\n0.17\n0.34\n25\n0.59\n26°\n0.08\n0.18\n0.60\n0.00\n0.00\n0.02\n0.35\n0.61\n0.08\n0.18\n0.35\n27\no\no\n0.02\n0.62\n28\n0\no\n0.02\n0.08\n0.18\n0.36\n0.63\n0.02\n0.08\n0.19\n0.37\n29\no\no\n0.64\n30\no\no\n0.02\n0.08\n0.19\n0.37\n0.64\n0.08\n31°\n0.37\n0.00\n0.00\n0.02\n0.19\n0.65\n0.19\n0.38\n32\no\no\n0.02\n0.08\n0.19\n0.38\n0.65\n33\no\no\n0.02\n0.08\n0.65\n34\no\n0,02\n0.08\n0.19\n0.38\no\n0.08\n35\no\n0.02\n0.19\n0.38\n0.65\nO\n0.65\n36°\n0.08\n0.19\n0.38\n0.00\n0.00\n0.02\n0.65\no\n0.02\n0.08\n0.19\n0.38\n37\n0\n0.65\n38\no\no\n0.02\n0.08\n0.19\n0.38\n0.64\n0.02\n0.08\n0.19\n0.37\n39\no\no\n0.64\n40\n0.08\n0.19\n0.37\nO\n0\n0.02\n0.63\n41°\n0.00\n0.02\n0.08\n0.19\n0.37\n0.00\n0.63\n42\n0.02\n0.08\n0.18\n0.35\no\no\n43\n0.36\n0.62\n0.08\n0.18\no\no\n0.02\n44\n0.61\n0.08\no\n0.02\n0.18\n0.35\no\n0.60\n45\n0.08\n0.18\n0.35\no\no\n0.02\n0.34\n46\n0.59\n0.02\n0.07\n0.17\n0.00\n0.00\n47\n0.33\n0.58\n0.02\n0.07\n0.17\no\no\n48\n0.07\n0.17\n0.33\n0.56\nO\no\n0.02\n0.16\n0.32\n0.55\n49\n0.02\n0.07\no\no\n0.16\n0.54\n0.31\n0.02\n0.07\n50\n0.00\n0.00\n]\n[\n3\n(A)\")\n3\nC (sin 1\") CO8\n+\nF\ng =\n2\n2A\nA, C and F are position factors.\n117","TRANSVERSE MERCATOR PROJECTION\nArizona\nsa = Mx' - e\nEast and central zones\nWest zone\nM\nAM\ny\nM\nAM\n644\n0\n0.005\n9177\n644\n0.005\n9175\n648\n100,000\n0.005\n9821\n648\n0.005\n9819\n0469\n200,000\n0.006\n652\n0467\n652\n0.006\n0.006\n300,000\n656\n1121\n0.006\n655\n1119\n400,000\n0.006\n659\n1777\n0.006\n1774\n660\n0.006\n2436\n664\n500,000\n0.006\n2434\n663\n600,000\n0.006\n667\n3100\n0.006\n668\n3097\n0.006\n700,000\n3767\n672\n0.006\n672\n3705\n800,000\n0.006\n4439\n676\n0.006\n4437\n675\n0.006\n680\n900,000\n5115\n0.006\n680\n5112\n0.006\n684\n1,000,000\n0.006\n685\n5795\n5792\n1,100,000\n6479\n0.006\n689\n0.006\n639\n6477\n1,200,000\n0.006\n693\n7168\n0.006\n693\n7166\n0.006\n1,300,000\n7861\n698\n0.006\n7859\n698\n1,400,000\n0.006\n8559\n703\n0.006\n8557\n702\n0.006\n9262\n1,500,000\n0.006\n707\n9259\n707\n1,600,000\n0.006\n9969\n0.006\n712\n9066\n712\n1,700,000\n0.007\n0681\n717\n0678\n0.007\n717\n1,800,000\n0.007\n1398\n722\n0.007\n1395\n722\n1,900,000\n0.007\n2120\n727\n0.007\n2117\n727\n2847\n2,000,000\n0.007\n2844\n732\n0.007\n732\n2,100,000\n0.007\n3579\n737\n0.007\n3570\n737\n2,200,000\n4316\n0.007\n742\n4313\n0.007\n742\n2,300,000\n0.007\n5058\n0.007\n5055\ne\nx'\n400,000\n200,000\n300,000\ny\n500,000\nO\n0.6\n0.0\n0.1\n0.3\n1,000,000\n0.0\n0.1\n0.4\n0.8\n2,000,000\n0.0\n0.2\n0.5\n1,0\n118","LAMBERT PROJECTION FOR UTAH (SOUTH)\nTable II (Cont'd).\n1\" of long. as 0.61268734 of O\nO\nLong.\n0\nLong.\nO\nLong.\n40.5674\n36\n4610760\n111°\n-0°\n038\n17'\n12.7194\n111°\n01'\n+0°\n26'\n+0°\n39'\n110°\n04\n17.3287\n09.3147\n37\n-0\n02\n17\nto\n35.9582\n27\nto\n38\n04\n54.0899\n-0\n16\n38\n32.5535\n03\n+0\n59.1969\n28\n+0\n37\n30.8512\n05\n39\n-0\n04\n15\n55.7923\nto\n22.4357\n29\n+0\n37\n07.6124\n40\n-0.\n06\n15\n19.0310\n45.6744\n05\n+0\n36\n30\n+0\n44.3737\n41°\n06\n14\n42.2698\n-0\n068\n111°\n+0\n36\n08.9132\n111°\n110°\n31'\n+0\n42\n21.1349\n14\nof\n07\n05.5085\n07\n+0\n32\n+0\n35\n32.1520\n43\n07\n57.8961\n28.7473\n8\n13\n08\nto\n34\n55.3907\n33\n40\n44\n34.6574\n-0\n08\n12\n51.9861\n34\n09\nto\n18.6295\n34\n+0\n11.4186\n45\n-0\n09\n15.2248\n12\n41.8682\n10\n+0\n35\nto\n33\n46\n48.1799\n38.4636\n09\nill\n111°\n8\n11°\n+0\n11\n36\n33\n05.1070\n110°\nto\n47\n24.9411\n-\n10\n28.3458\n11\n01.7023\n12\n+0\n32\n37\n+0\n48\n24.9411\n-0\n11\n01.7023\nto\n10\n51.5845\n13\n38\n+0\n31\n49\n38.4636\n48.1799\n-\n11\n14\nto\n09\n14.8233\n39\n+0\n31\n15.2248\n8\n12\n11.4186\n50\n15\n+0\n09\n38.0620\n40\nto\n30\n51.9861\n34.6574\n12\n111°\n518\n16\n-\n+0\n08\n41'\n01.3008\n111°\n30\n110°\n+0\n28.7473\n-0\n13\n57.8961\n52\n+0\n07\n24.5395\n17\n42\nto\n29\n14\n05.5085\n21.1349\n53\n-0\n43\n47.7783\n18\n+0\n07\nto\n28\n14\n42.2698\n44.3737\n54\n-0\n06\n44\n19\nto\n11.0171\n40\n28\n06\n55\n-0\n15\n19.0310\n07.6124\nto\n45\n20\n34.2558\n+0\n27\n568\n55.7923\n15\n30.8512\n111°\n8\n461\n26\n57.4946\n111°\n21'\nto\n05\n110°\nto\n32.5535\n-0\n16\n04\n54.0899\n57\n22\n+0\n47\n26\n20.7334\n+0\n09.3147\n04\n58\n-0\n17\n17.3287\n48\n43.9721\n23\n+0\n+0\n25\n46.0760\n17\n40.5674\n59\n-0\n24\n+0\n03\n49\n25\n07.2109\n+0\n18\n22.8372\n03.8062\n112°\n00\n-\n30.4496\n25\n+0\n03\n24\n50\n40\n59.5985\n27.0450\n01°\n18\n112°\n-9\n53.6884\n26\n+0\n02\n111°\n23\n110°\n52.\nto\n19\n36.3597\n02\n-0\n50.2837\n27\n+0\n01\n23\n16.9271\n52\n+0\n13.1210\n03\n-0\n20\n13.5225\n40.1659\n28\nto\n01\n53\nto\n22\n49.8822\n20\n36.7612\n04\n-0\n00\n29\n+0\n54\n03.4047\n+0\n22\n26.6434\n05\n-\n21\n00.0000\n26.6434\n30\no\n00\n55\nto\n21\n03.4047\n06\n22\n36.7612\n112°\n-\n49.8822\n31\n00\n56\n111\n-\n110°\n+0\n20\n40.1659\n22\n07\n-0\n13.5225\n01\n32\n-0\n13.1210\n57\nto\n20\n16.9271\n23\n08\n-0\n50.2837\n01\n33\n-0\n36.3597\n19\n58\nto\n53.6884\n23\n09\n-0\n27.0450\n34\n02\n-0\n18\n59.5985\n59\n40\n30.4496\n24\n10\n-0\n03.8062\n-\n03\n35\n18\n22.8372\n+0\n111°\n00\n119","COMPUTATIONS AND ADJUSTMENTS\nJoseph F. Dracup\nSupervisory Geodesist\nGeodesy Division\nNational Ocean Survey\nNational Oceanic and Atmospheric Administration\nU. S. Department of Commerce\nRockville, Maryland 20852\n120","ERRATA\nLecture Notes for Use At Workshops on Surveying\nInstrumentation and Coordinate Computation\npage 110 - (2) Fourth line from bottom, + 1° 19' 54.4\nshould be+1° 39 54.4.\npage 163 - Second sentence should read, \"When employing\ncondition equations involving a mixture of angle\nand distance measurements, the formation of the\nequations is probably simpler when the weights\nare expressed in the same measure.\npage 192 - Length A = 0.9154 ft., not 0.29 ft.; p = 1.1933,\nnot 1.1891.\nThe changes in \"A\" and subsequently to \"p\" were\ndue to a mistake in the original computation,\nand rather than change the value for \"kp\" which\nis used in further computations, it was necessary\nto revise the numerical value for \"A\" in the\nfollowing paragraph as well as to modify the first\nstatement. In addition, \"A\" would not generally\nbe given to more than three decimal places as\nwas necessary here.\n'For observation equations, the weights for\nvarious observational data, that is directions,\nlengths, and azimuths, need not be based on\nprecisions expressed in the same measure such\nas seconds, as was used in the previous example\nwhere condition equations were employed. Nor\nwas it necessary in that example, but the CO-\nefficients involved with the length measurements\nwould be numerically different. For the measured\nlength (p. 198) used in this survey, the \"A\"\nvalue was determined using 0.5 ft. + 1:90,000\nof the length as the criterion.","COMPUTATIONS AND ADJUSTMENTS\nTraverse computations, whether made on a local system\nor using State plane coordinates, are for all intents and\npurposes identical. It is true that to obtain the best\nresults on the State plane coordinate systems, lengths\nshould be corrected by a scale factor and reduced to sea\nlevel, and in some instances second term corrections should\nbe applied to the observed angles prior to computation.\nThese are rather simple corrections to derive and apply.\nHowever, the scale and sea level factors, in particular,\nhave long been given as the principal reasons for not\nemploying the State coordinate systems. It is under-\nstandable that the distances shown on a plat should be\nthose at the elevation plane of the measurements. These\nvalues, however, can easily be obtained after a survey\nhas been computed and adjusted on the State system. As\na matter of fact, the scale and sea level factors can\noften be ignored and ground distances used in computing\nthird order (1:5,000) or lesser accurate surveys in those\nareas where elevations are 2000 feet or less. In some\ninstances, the effects of the scale and sea level factors\nmay counterbalance one another to a great extent, and\ndistances at elevations of as much as 4000 feet may be\nused as measured.\nIn the following description of the computation and\nadjustment of a simulated traverse survey in southern\n121","Wisconsin, the explanation of each operation is concise.\nMore complete details will be found in publications\ngiven in the list of references. A sketch of the traverse\nis shown by Figure (1). . The configuration violates many\nof the rules for a good network design but this was done\npurposely in order to emphasize the differences in\nadjustment techniques. The dotted curved lines will be\nused to show the relationship between observed and grid\nangles.\nyo\nAzimuth Mark\n5\n1\n2\n6\n4\nAzimuth Mark\n3\nFigure 1\n122","Second Term Corrections: Any observed angle is a\ngeodetic quantity. In those cases where a survey has been\nmade to a high quality and the computations are carried\nout on a plane, these observations should be corrected\nby the second term, or T-t correction, in order to reduce\nthem to grid angles. The basic formulae are given in\neach of the State projection tables. However, the following\napproximate formula is sufficiently accurate for most\npurposes: the correction at each end of a line If\n2.36 (AX) (AY) X 10-10.\nFor the transverse Mercator system,\nAX is the difference between the X value on the central\nmeridian (usually 500,000 feet) and the mean X value of\nthe two points involved. AY is the difference in the Y\ncoordinates for the two points. For the Lambert projection,\nAX is the difference in the X coordinates of the two points\nand AY is the difference between the yo value (given with\nthe tables) and the mean Y value for the points involved.\nApproximate coordinates are required to make the com-\nputations, but these values given to the nearest one\nthousand feet are adequate. For the Wisconsin south zone,\nthe yo = 510,702.41. The corrections need to be computed\nfor one end of the line only as the value at the opposite\nend, using the approximate formula, is identical. No\ncorrections are necessary to the observations to the azimuth\nmarks in this example because the distance from the stations\n123","to these marks are usually quite short and the correction\nwould be zero. If more distant points were used, cor-\nrections should, of course, be applied. The computations\nof the second term corrections for the traverse example\nare tabulated in Table 1.\n*\n*\n2.36(AX)(AY)\nAY\nAX\nFrom\nTo\n1\".\n0.158\n3.095\n1\n2\n0.008\n3.162\n0.1\n2\n3\n1.2\n4\n0.160\n3.211\n3\n0.008\n3.148\n0.1\n4\n5\n0.8\n0.114\n6\n3.127\n5\nTable 1\n* Coordinate values were available to hundreds of\nfeet and were used, although as noted earlier coordinates\nto the nearest thousand feet are sufficient.\nAlthough the signs of the corrections to the observed\nangles can be obtained from the computation, or through\nthe azimuth of the lines, it is much easier to determine\nthese signs in the following manner. Visualize the\nobservations (geodetic) as being a curved line concave\ntowards the central meridian for surveys computed on the\ntransverse Mercator projection and towards the yo (central\nparallel) for the Lambert system. The dotted curved lines\nin Figure (1) represent the observed values, and the\nstraight lines the grid quantities. Since the angles were\n124","measured clockwise in the direction of the traverse\nwest to east, it can be seen the geodetic angles are\nlarger than their corresponding grid angles in every\ncase. The correction to each angle, in this example,\nis simply the sum of the corrections for the two lines\ninvolved at each point as follows:\nGrid Angle\nCorr'n\nObserved Angle\nPoint\n17\".2\n-1.1\n441\n18\"3\n90°\n1\n54.0\n-1.2\n265\n15\n55.2\n2\n25.6\n82\n48\n26.9\n-1.3\n3\n07.3\n08.6\n-1.3\n4\n105\n03\n45.3\n46.2\n-0.9\n304\n33\n5\n38.7\n-0.8\n245 17 39.5\n6\nTable 2\nAlthough taken individually, the above corrections\nare quite small; they are accumulative and would effect\nthe azimuth closure by 6\"6. Whenever a long traverse is\nlocated in an area which is near the limits of a zone,\nthe second term corrections can, on occasion, be rather\nlarge, and it is always prudent to compute the correction\nbetween the terminal points. The accumulated corrections\nin a traverse between these points cannot be less than\ntwice the correction for one end. For example, the second-\nterm correction between points 1 and 6 is 3\"3. The value\n125","at the opposite end of the line is the same and the total\ncorrection is therefore 6\"6, which is equal, in this case,\nto the sum of the second-term corrections at all the points\nin the traverse.\nScale and Sea Level Corrections: Any length measure-\nment made on the surface of the earth is a geodetic distance\nat the mean elevation of the points involved. The plane\ncoordinates which have been determined from the geographic\nposition of points in the national network, with one\nexception (the Lambert system for Michigan), are based\non a sea-level reference. Accordingly, to obtain the\nbest results for surveys made at some other elevation\nsurface, the lengths should be reduced to the datum\nreference plane. In addition, geodetic lengths should\nbe corrected for the scale differential between the actual\nlength and the same length as represented on a map pro-\njection. As mentioned previously, these corrections may\nbe neglected in some instances for lower order surveys,\nbut generally should be applied where accuracies of 1:10,000\nor better are anticipated.\nFor the transverse Mercator projection, the scale is\nconstant along the central meridian and varies as the X\nvalues extend east and west from this meridian. The scale\nfor the Lambert system is constant along any parallel and\nvaries in a north-south direction. These values are tabu-\nlated in the State projection tables. The scale factors\nare a function of the X' values for the transverse Mercator\n126","system and the latitude of the points for the Lambert\nprojection.\nWhile in theory each length should be corrected for\nthe mean scale factor between the points, in actual\npractice, for limited surveys, a mean scale factor com-\nputed from the positional data for the terminal control\npoints will usually suffice. For this sample computation\nthe mean latitude is about 42° 3215, and by interpolating\nbetween the scale factors, as given in the Wisconsin south\nzone tables, for 32' and 33' a value of 1.0000442 was\ndetermined. Although each of the measured distances\ncould be multiplied by this factor, with the results\nbeing grid distances at the elevation plane of the\nmeasurements, it is easier to compute a combined scale\nand sea level factor and make the reductions in one step.\nThis operation will be explained shortly.\nA distance measured between two points at an elevation\nabove sea-level will be longer than the corresponding dis-\ntance between the same two points should these points be\nplumbed to the sea level surface. This may be easily seen\nby referring to Figure (2) below. The following formula,\nwhile an approximation, may be used to determine the dif-\nference between ground level and sea level lengths for most\nSxh\nwhere \"S\" is the measured distance,\nsurveys:\n20,906,000'\n\"n\" is the mean elevation of the line, and 20,906,000 is\n127","LAMBERT PROJECTION FOR WISCONSIN (SOUTH)\nTable I.\nScale in\nScale\ny's\nTabular\nR\nLat.\nunits of\nexpressed\ndifference\n(feet)\ny value on\n7th place\nfor 1 sec.\nas a\ncentral\nof logs\nratio\nof lat.\nmeridian\n(feet)\n(feet)\n1.0002282\n101.24717\n+990.9\n22,672,134.66\n0\n42° 00'\n1.0002212\n6,074.83\n101.24700\n+960.7\n22,666,059.83\n01\n1.0002143\n101.24650\n+930.8\n12,149.65\n22,659,985.01\n02\n1.0002075\n101.24617\n18,224.44\n+901.2\n22,653,910.22\n03\n101.24567\n1.0002008\n+872.1\n22,647,835.45\n24,299.21\n04\n1.0001942\n101.24533\n+843.3\n22,641,760.71\n30,373.95\n05\n1.0001876\n+814.8\n36,448.67\n101.24500\n22,635,685.99\n42° 06\n101.24467\n+786.7\n1.0001811\n42,523.37\n22,629,611.29\n07\n1.0001748\n101.24433\n48,598.05\n+759.0\n22,623,536.61\n08\n101.24400\n+731.6\n1.0001685\n22,617,461.95\n54,672.71\n09\n1.0001622\n101.24367\n+704.6\n22,611,387.31\n60,747.35\n10\n1.0001561\n101.24333\n+677.9\n42°\n22,605,312.69\n66,821.97\n11'\n+651.6\n1.0001500\n72,896.57\n101.24300\n22,599,238.09\n12\n1.0001441\n101.24267\n+625.7\n78,971.15\n13\n22,593,163.51\n+600.1\n1.0001382\n85,045.71\n101.24250\n14\n22,587,088.95\n1.0001324\n+574.9\n101.24217\n22,581,014.40\n$1,120.26\n15\n1.0001266\n97,194.79\n101.24183\n+550.0\n420 16:\n22,574,939.87\n101.24167\n22,568,865.36\n103,269.30\n+525.5\n1.0001210\n17\n101.24133\n+501.4\n1.0001155\n22,562,790.86\n109,343.80\n18\n101.24100\n+477.6\n1.0001100\n22,556,716.38\n115,418.28\n19\n1.0001046\n101.24083\n+454.2\n121,492.74\n20\n22,550,641.92\n101.24067\n+431.1\n42°\n22,544,567.47\n127,567.19\n1.0000993\n21'\n101.24033\n+408.4\n1.0000940\n133,641.63\n22\n22,538,493.03\n101.24017\n+386.0\n1.0000889\n22,532,418.61\n139,716.05\n23\n101.24000\n+364.0\n1.0000838\n22,526,344.20\n145,790.46\n24\n+342.4\n151,864.86\n1,0000788\n101.23983\n22,520,269.80\n25\n42° 268\n22,514,195.41\n+321.1\n1.0000739\n157,939.25\n101.23950\n1.0000691\n22,508,121.04\n164,013.62\n101.23933\n+300.2\n27\n1.0000644\n22,502,046.68\n+279.7\n170,087.98\n101.23917\n28\n1.0000598\n176,162.33\n101.23917\n+259.5\n22,495,972.33\n29\n1.0000552\n182,236.68\n101.23883\n+239.7\n22,489,897.98\n30\n101.23867\n+220.2\n1.0000507\n42°\n22,483,823.65\n188,311.01\n31'\n1.0000463\n101.23550\n+201.1\n22,477,749.33\n194,385.33\n32\n1.0000420\n+182.3\n200,459.64\n101.23850\n22,471,675.02\n33\n1.0000378\n+164.0\n34\n22,465,600.71\n101.23833\n206,533.95\n1.0000336\n+145.9\n22,459,526.41\n101.23817\n212,608.25\n35\n128","the mean radius (R) of the Earth for the United States.\nAll values are in feet. In the more exact formula, \"S\"\nwould be the sea level distance and \"R\" would be computed\nfor the mean latitude and azimuth of each line.\nB\nA\nGround Level\nh\nA\nSea Level\nB'\n-\nR\nCenter of Earth\nFigure 2\nThe sea level reduction formula may be put in the\nh\nGenerally\nmore convenient factor form, 1. -\n20,906,000\na mean elevation for an entire traverse can usually be\nused without any great loss in accuracy. For example,\nif the maximum deviation from a mean elevation is 418 feet,\nall distances would be correct within 1:50,000 in as far\nas the sea level correction is concerned.\nIn the traverse serving as the example, the average\nelevation was assumed to be 750 feet above sea level.\nUsing the reduction formula given last, the sea level\n750\n= 0.9999641.\nfactor was computed as follows, 1.\n20,906,000\n129","This factor and the scale factor, computed previously,\nwere combined by multiplication (1.0000442 X 0.9999641)\nand the resulting combined factor = 1.0000083 was obtained.\nEach of the measured lengths was multiplied by this factor,\nwith the results as shown below (table 3). The measured\nlengths are horizontal distances in feet.\nGrid Lengths\nMeasured Lengths\nTo\nFrom\n15,766.07\n15,765.94\n1\n2\n13,004.33\n13,004.22\n3\n2\n16,293.03\n16,292.89\n4\n3\n11,487.03\n11,486.93\n4\n5\n14,655.39\n14,655.27\n6\n5\nTable 3\nComputation of Grid Azimuths: As shown in table 4,\nthe preliminary grid azimuths are computed using the fixed\ngrid azimuth between Station 1 and its Azimuth Mark and\nthe grid angles. The closure is obtained by subtracting\nthe preliminary grid azimuth Station 6 to Azimuth Mark\nfrom the fixed value. The closure is divided by the\nnumber of angles (-10\"8/6 = -1\"8), and each grid angle\nis corrected by this amount. A recomputation is then made\nto obtain azimuths corrected for closure.\n130","CORRECTION FOR\nCORRECTED AZIMUTH\nSTATION\nPRELIMINARY AZIMUTH\nCLOSURE\n\"\n\"\no\nTO\nFROM\n*\n180\n20\n31.2\n180\n20\n31.2*\nAzimuth Mk.\n1\n44\n15.4\n-1.8\n44\n90\n17.2\n90\nL\n46.6\n04\n271\n04\n48.4\n271\n2\n1\n04\n46.6\n04\n48.4\n91\n91\n1\n2\n-1.8\n265\n15\n52.2\n54.0\n265\n15\nL\n38.8\n356\n42.4\n20\n356\n20\n3\n2\n38.8\n176\n42.4\n20\n176\n20\n2\n3\n-1.8\n82\n48\n23.8\n82\n48\n25.6\nL\n02.6\n08.0\n259\n09\n4\n259\n09\n3\n02.6\n08.0\n79\n09\n79\n09\n4\n3\n05.5\n-1.8\n105\n03\n07.3\n105\n03\nL\n08.1\n184\n12\n184\n12\n15.3\n4\n5\n08.1\n4\n12\n4\n15.3\n4\n12\n5\n43.5\n-1.8\n304\n33\n45.3\n304\n33\nL\n308\n45\n51.6\n308\n46\n00.6\n6\n5\n51.6\n128\n45\n128\n46\n00.6\n5\n6\n36.9\n-1.8\n245\n17\n38.7\n245\n17\nL\n28.5\n14\n03\n*\n14\n03\n39.3\nAzimuth Mk.\n6\n-10.8\nClosure\n=\nGrid Azimuths\n*\nFixed\nL\nTable 4\nOn occasion where lines significantly shorter than the\naverage are involved, an unequal distribution of the azimuth\nclosure might be considered. The angles which include the\nshort lines would receive larger corrections than those\nwhere the distances are more nearly equal. Depending upon\nvarious circumstances, the corrections to the angles involving\nthe short lines could be larger by perhaps as much as a\nfactor of ten. Some caution must be exercised in making\n131","this decision, however, since a continuous tilt will be\nintroduced in the traverse unless counterbalanced elsewhere.\nAzimuths rather than bearings are preferred because\nangles can be added or subtracted without considering\nthe quadrant involved. It should also be noted that\nazimuths determined in the national network are referenced\nin a clockwise manner with 0° 00' 00\" being due south.\nThe azimuths corrected for closure will be used to\ncompute the plane coordinates of the traverse points\nwhich will be adjusted by the Compass and Transit Rules.\nIn the Least Squares adjustments, the coordinates are\ncomputed using the uncorrected preliminary azimuths and\nan azimuth equation is included in the simultaneous\nsolution with the position equations.\nAdjustment Methods: Traverses may be adjusted by\nthe Compass and Transit Rules and by the Method of Least\nSquares. All other methods are either a combination or\nmodified versions of these procedures. More often than\nnot, these other methods are employed to obtain some\ndesired result. The sample problem will be adjusted by\nthe three methods noted above and a comparison of the\nresults made. The angle observations and distance\nmeasurements were given equal weight in the adjustments.\nA weighted least squares computation is not difficult\nto process, but the application of weights to Compass\n132","and Transit Rule adjustments, while feasible, might produce\nrather arbitrary results because of the independent manner\nin which the closures are adjusted.\nThe Compass Rule adjustment is shown in Table 5. This\nmethod is usually employed when the angles and distances\nhave been considered to be of equal accuracy. Unfortunately\nthe adjusted results do not always justify this consideration.\nSuch is the case in this particular traverse, as may be seen\nin the comparison of the results (Table 13).\nIt is assumed that all surveyors are familiar with the\ncomputations shown in Table 5 and no detailed explanation\nwill be given. The closure in X is +3.43 ft., and in Y\n-3.31 ft., or a total closure of 4.77 ft. or 1:14,900. The\nlengths are the basis for the adjustment and the two factors\ndetermined are as follows: \"X\" correction = +3.43/71205.85 =\n+0.4817 per 10,000 ft., and \"Y\" correction = -3.31/71025.85 =\n-0.4648 per 10,000 ft. The corrections to the coordinates\nare accumulated values. For example, the correction to the\npreliminary \"X\" coordinate for point 4 = +0.4817 (1.5766 +\n1.3004 + 1.6293) If +2.17 ft.\nThe Transit Rule adjustment method is often suggested\nfor use in those cases where the angles are thought to be\nmore accurate than the distances. It is contended that this\nmethod will produce smaller corrections to the angles. This\ncontention is, of course, not completely true in all cases\n133","193,406.90\n- 3.31\n193,403.59\n- 0.73\n201,037.13\n188,058.65\n191,126.76\n- 2.09\n202,582.91\n- 2.63\n202,580.28\n201,334.92\n188,059.99\n- 1.34\n191,124.67\n201,037.86\nAccumulative Corrections for Closure\nFeet\nV\nGRID COORDINATES\n2,242,758.60\n+ 3.43\n2,242,762.03\n2,231,331.38\n2,231,334.10\n+ 2.72\n2,214,487.84\n2,214,489.23\n2,230,489.64\n2,230,491.81\n2,213,658.63\n+ 1.39\n+ 2.17\n2,197,895.36\n+ 0.76\n2,213,659.39\nPreliminary Coordinates\nFeet\nI\nAdjusted Coordinates\nTRAVERSE COMPUTATION BY LATITUDES AND DEPARTURES\n**\n-12977.87 -\n***\n*\n- 9176.01\n+11456.15\n- 297.06\n+ 3066.77\nLATITUDE\nFixed\nFixed\nFeet\n+ 15763.27\n+ 841.74\n+11427.22\n+16001.80\n+ 829.21\n*\n**\n***\nDEPARTURE\nFeet\nU.S. DEPARTMENT OF COMMERCE\nCOAST AND GEODETIC SURVEY\nTable 5\nCOMPASS RULE\nCOS AZIMUTH\n0.62611855\nSIN AZIMUTH\n0.07327736\n0.99731160\n0.77972788\n0.99982248\n0.99796501\n0.98212573\n0.18822607\n0.01884165\n0.06376401\nGRID DISTANCE\n11487.03\n14655.39\n16293.03\n13004.33\n15766.07\nFeet\n51.6\n356 20 38.8\n02.6\n184 12 08.1\n46.6\nPLANE AZIMUTH\n45\n09\n04\n308\n259\n271\nSTATION\n6\n5\n3\n4\n2\n1\nFORM 738\n(3-28-28)","since the angles will usually be distorted should a large\nclosure exist perpendicular to the general direction of\nthe survey. In this particular example, however, the\npremise held, even though there is a rather large closure\nin \"Y\" in a line running essentially west to east. But\nthe smaller corrections to the angles are more a result\nof the geometry of the traverse, with two long lines\nperpendicular to the direction of the survey, then anything\nelse.\nThe preliminary computations required to make the\nTransit Rule adjustment are identical to those for the\nCompass Rule adjustment, and the complete computation is\nshown in Table 6. For the Transit Rule adjustment, the\nDepartures and Latitudes are used for distributing the\nclosures. The Departures are summed without regard to\nsign, and the \"X\" correction factor is determined by\ndividing the \"X\" closure by this sum. Similarly the \"Y\"\ncorrection factor is found from the sum of the Latitudes\nand \"Y\" closure. The following factors were obtained: \"X\"\ncorrection = +3.43/44863.24 = +0.7645 per 10,000 ft., and\n\"Y\" correction = -3.31/36973.86 If -0.8952 per 10,000 ft.\nAs was the case with the Compass Rule adjustment, the\ncorrections shown are accumulated values. For example, the\ncorrection to the preliminary \"Y\" coordinate for point 4 =\n-0.8952 (0.0297 + 1.2978 + 0.3067) = -1.46 ft.\n135","- 0.03.\n202,580.42\n- 1.19\n188,058.80\n193,406.90\n201,037.86\n201,037.83\n188,059.99\n191,126.76\n- 1.46\n191,125.30\n202,582.91\n- 2.49\n193,403.59\n201,334.92\n-3.31\nUSCOMM-DC 5270\nAccumulated Corrections for Closure\nFeet\ny\nGRID COORDINATES\n2,213,658.63\n2,213,659.84\n2,214,487.84\n2,230,489.64\n2,242,758.60\n+ 2.56\n2.231.333.94\n+ 3.43\n2,242,762.03\n2,197,895.36\n+1.21\n+ 1.27\n2,214,489.11\n+ 2.49\n2,230,492.13\n2,231,331.38\nPreliminary Coordinates\nFeet\n=\n*** Adjusted Coordinates\nTRAVERSE COMPUTATION BY LATITUDES AND DEPARTURES\n**\n*\n***\n- 297.06\n-12977.87\nLATITUDE\n+ 3066.77\n+11456.15\n- 9176.01\nFixed\nFixed\nFeet\n+ 16001.80\n+ 841.74\n+ 11427.22\n+ 15763.27\n829.21\n*\n**\nDEPARTURE\nFeet\nU.S. DEPARTMENT OF COMMERCE\nCOAST AND GEODETIC SURVEY\nTable 6\nTRANSIT RULE\n+\nCOS AZIMUTH\nSIN AZIMUTH\n0.99731160\n0.77972788\n0.99982248\n0.01884165\n0.06376401\n0.99796501\n0.98212573\n0.07327736\n0.62611855\n0.18822607\nGRID DISTANCE\n13004.33\n16293.03\n11487.03\n14655.39\n15766.07\nFeet\n51.6\n271 04 46.6\n356 20 38.8\n259 09 02.6\n184 12 08.1\nPLANE AZIMUTH\n45\n308\nSTATION\n4\n5\n3\n6\n1\n2\nFORM 738\n(3-28-58)","The Compass and Transit Rules are the most commonly\nused techniques for balancing traverses. In the past,\nwhen very few local surveys were accomplished to even\nsecond-order accuracy, their use was acceptable and\njustified. But to apply these methods to the adjustment\nof higher grade surveys, as is done daily, makes little\nsense in this age of electronic computers. The best\nmethod, Least Squares, for adjusting any survey data has\nlong been available but seldom used except by geodetic\norganizations. There is nothing mystic about Least\nSquares, nor is great mathematical knowledge required to\ncarry out the computations. Once the technique is learned,\nthe additional time required to adjust a simple traverse\nrun between two control points would seldom be more than\none-half day if the computations are made on a desk cal-\nculator and only a very few minutes if a computer is available.\nThe adjustment method which will be described in the following\npages was developed by Buford K. Meade, Chief, Triangulation\nBranch, National Ocean Survey (Coast and Geodetic Survey),\nand was presented in a paper entitled, \"The Practical Use\nof the Oregon State Plane Coordinate System\". This paper\nwas given at the Surveying and Mapping Conference, Oregon\nState University, Corvallis, Oregon, March 1964. Copies\nare available from the author.\n137","There are two Least Squares methods which may be used\nto adjust traverse and the results will be identical regard-\nless of whether the computations are carried out on the\nreference spheroid; or, when all corrections are applied,\non the plane. One method employs condition equations and\nthe other observation equations.\nThree condition equations are required for a simple\ntraverse run between two control points. Any number of\nnew points may be included without increasing the number\nof equations to be solved. For each additional control\npoint used, the number of equations is increased by three\nif each control point includes an azimuth tie and two if\nno azimuth tie is made.\nThe observation equation method requires a minimum of\ntwo normal equations to be solved for each new point.\nObviously this method would not be recommended because in\na simple traverse, as noted above, containing 20 new points,\n40 equations would need to be solved while only 3 condition\nequations would be required to obtain the same results. In\naddition, the computation of the observation equations take\na considerably greater effort than the formation of the\ncondition equations. In the example here, only the con-\ndition method on a plane will be described.\nThe preliminary computations are identical to those\nmade for the Compass and Transit Rule adjustment except\n138","193,405.806\n- 2.216\nUSCOMM-DC 5270\n201,334.92\n193,403.59\nFeel\ny\nGRID COORDINATES\n2,242,758.450\n+ 3.580\n2,242,762.03\n2,197,895.36\nFoot\nI\nTRAVERSE COMPUTATION BY LATITUDES AND DEPARTURES\n+ 3066.354\n+11456.119\n- 9176.510\n- 297.196\n-12977.881\nLATITUDE\nFeet\n+16001.884\n+ 842.139\n+11426.816\n+15763.269\n+ 828.982\nDEPARTURE\nFeet\nPRELIMINARY COMPUTATION\nU.S. DEPARTMENT OF COMMERCE\nCOAST AND GEODETIC SURVEY\nLEAST SQUARES\nTable 7\nCOS AZIMUTH\nSIN AZIMUTH\n0.99730904\n0.77970056\n0.62615257\n0.01885038\n0.06374659\n0.99796611\n0.98213066\n0.18820036\n0.07331217\n0.99982231\nGRID DISTANCE\n16293.03\n11487.03\n14655.39\n13004.33\n15766.07\nFeet\n46 00.6\n48.4\n42.4\n08.0\n184 12 15.3\nPLANE AZIMUTH\n20\n04\n09\n308\n356\n259\n271\nSTATION\n3\n4\n5\n6\n1\n2\nFORM 738\n(3-28-58)","that the azimuth closure is not distributed and the pre-\nliminary azimuths uncorrected for the azimuth closure\n(Table 4) are used with the grid lengths to obtain the\npreliminary Departures and Latitudes. In this computation\nit is not necessary to tabulated the preliminary coordinate for\nthe new stations. These computations are shown in Table 7.\nThe closures in X and Y are slightly different than those\npreviously obtained because the uncorrected grid azimuths\nare used.\nThe three basic conditions are for the closures in\nazimuth, X and Y are as follows:\n(1) O =\n(2) O = - (Yn (XK - X 1 ) Vs\n(3) O = +206265 (Yn + (YK - Y1)Vs\nwhere, Va and VS represent the corrections to be applied to\nthe angles and distances,\nAn and Af are the computed and fixed azimuths at the\nterminal point,\ni and k represent the initial and computed coordinates\nfor consecutive points,\nXn and Yn are the computed coordinates of the terminal\npoint,\nx\nXf and Y f are the fixed coordinates of the terminal point,\nXC and YC are the computed coordinates.\n140","The azimuth equation for the example is as follows:\n(1) O = +10.8 +1.00 (1) +1.00 (2) +1.00 (3) +1.00 (4)\n+1.00 (5) +1.00 (6)\nThe first item in the equation is the (An - Af) value\nand is usually referred to as the constant term. This\nvalue was obtained from the data appearing on Table 4 and\nwas computed thusly:\n(An = 14° 03' 39.3) - (Af If 14° 03' 28\"5) If +10\".8.\nThe angle corrections or unknowns which will be determined\nfrom the solution of the equations are identified by the\nnumbers in parenthesis. These numbers correspond to those\nassigned the traverse points (Figure 1). The coefficients\nfor the angle corrections in the azimuth equations are\nalways assigned the value 1.00.\nThe formation of the equations for the closures in X\nand Y follows:\n(2) O = +206265 (Xn - Xf) + (Yn - Y1)(1) + (Yn - Y2)(2) +\n- Y3)(3) + (Yn - Y4)(4) + (Yn - (55)(5)\n(X2 - X1)(1-2) + (x3 - X2)(2-3) +\n+\n- X4)(4-5) + (Xn - x5)(5-6) -\nIn computing the coefficients involving the Y coordinates,\nthe computed Latitude values may be substituted for the\ncoordinates. (Yn - Y1) is the algebraic sum of the Latitudes;\n( Yn - Y2) = the algebraic sum of the Latitudes (Yn - Y1) minus\nLatitude The other coefficients for the angle corrections\nthe\nmaybe obtained in a similar manner. The coefficients for the\n141","corrections to the lengths are the computed Departures\nbetween the points indicated. The constant or closure\nterm 206265 (Xn - Xf) = 206265X (-3.580). It is not\nnecessary to carry the elements involved in the computation\nof the coefficients to more than values in feet. The com-\nplete X correction equation for the example follows:\n(2) o -738428.700 -7929 (1) -7632 (2) +5346 (3)\n+2280 (4) -9177 (5) +15763 (1-2)\n+829 (2-3) +16002 (3-4) +842 (4-5)\n+11427 (5-6)\nIn order to reduce the size of the numbers used in\na computation on a desk calculator, the equation should\nbe divided by 2000.\n(3) o If +206265 (Yn - Ye) - (Xn X1)(1) + (Xn - X2)(2) +\n- - X4)(4) + -\n+\n(Y2 Y1)(1-2) + (Y3 -Y2)(2-3)+(Y4- Y3) -\n(3-4) + (Y5 - Y4)(4-5) + (Yn Y5)(5-6)\nThe coefficients computed using the X coordinates\nmay be determined using the same procedure as described\npreviously for the Y coordinates, Similarly, the length\ncoefficients are the computed Latitudes between the points\nindicated. The constant or closure term 206265 (Yn - Yf) =\n206265X (+2.216). The complete Y correction equations for\nthe example follows:\n142","(3)\n(2)\n-28271\n(1)\n-29100\n-44863\n+457083.240\n(3)\nO\n=\n(1-2)\n(5)\n(4)\n-11427\n-297\n-12269\n-12978 (2-3) +3066 (3-4) +11456 (4-5)\n-9177 (5-6)\nThe equation should be divided by 5000 for the same\nreason as noted previously.\nThe equations are now tabulated vertically as shown\nin Table 8. . This tabulation is known as the Correlate\nequations and will be used to form the Normal equations,\nand in the computation of the corrections to the angles\nand lengths.\nCORRELATE EQUATIONS\nCorrections\nEquations\nCorr'n\nAdopt\nV's\n3\n2\n1\nNo.\n+ 4\".34\n+ 4\"340\n-3.964\n-8.973\n+1.00\n1\n- 2.46\n- 2.466\n-5.820\n-3.816\n+1.00\n2\n+ 8.85\n+ 8.851\n-5.654\n+2.673\n+1.00\n3\n- 1.089\n- 1.09\n-2.454\n+1.140\n+1.00\n4\n-11.788\n-11.79\n-2.285\n-4.588\n+1.00\n5\n- 8.65\n- 8.648\n6\n+1.00\nds in ft.\n+ 1.095\n+14.332\n+7.882\n-0.059\n1-2\n+ 0.414\n+ 6.570\n-2.596\n+0.414\n2-3\n+ 1.030\n+13.039\n+0.613\n+8.001\n3-4\n- 0.244\n4.381\n+2.291\n+0.421\n4-5\n+ 1.024\n+14.411\n-1.835\n+5.714\n5-6\nTable 8\nThe Normal Equations are formed using the Correlate\nEquations in the following manner and tabulated as shown\n143","in Table 9. Step 1 - square each term in equation 1 and\ntabulate the result.\nStep 2 - multiply each term in equation 1 by its\ncorresponding term in equation 2 and tabulate the result.\nStep 3 - multiply each term in equation 1 by its\ncorresponding term in equation 3 and tabulate the result.\nThe values in the \"n\" column are the constant terms of\nthe equations.\nStep 4 - square each term in equation 2 and tabulate\nthe result.\nStep 5 - multiply each term in equation 2 by its\ncorresponding term in equation 3 and tabulate the result.\nStep 6 - square the terms in equation 3 and tabulate\nthe result. All multiplications are made algebraically\nand summed in each step.\nNORMAL EQUATIONS\nn\n1\n2\n3\n+6\n-8.555\n-25.186\n+10.8\n+218.909783\n+44.195122\n-369.214350\n+173.332558\n+91.416648\nTable 9\nThe Forward Solution is shown in Table 10, and an\nexplanation follows:\nStep 1 - arrange the Normal Equations from Table 9\nas shown.\nn,\nStep 2 - divide the terms for equations 2 and 3 and\n144","on the first line, by the term on the far left and change\nthe sign: -8.555/+6 =+1.425833; -25.186/+6 =+4.197667;\nand +10.8/+6 = -1.800000.\nIn the following steps the actual multiplications\nwill be shown. The same pattern would be followed in\nall other adjustments.\nStep 3 - (-8.555) X (+1.425833) + (+218.909783) =\n+206.711782.\nStep 4 - (-8.555) X (+4.197667) + (+44.195122) = +8.284081.\nStep 5 - (-8.555) X (-1.800000) + (-369.214350) =\n-353.815350. The resulting quantities are known as the\nreduced normal terms.\nStep 6 - divide the reduced normal terms for equation 3\nandn by the reduced normal term for 2 and change the signs:\n+8.284081/+206.711782 = -0.040076; and -353.815350/+206.711782 =\n+1.711636.\nStep 7 - (-25.186) X (+4.197667) + (+8.284081) X\n(-0.040076) + (+173.332558) = +67.278124\nStep 8 - (-25.186) X (-1.800000) + (+8.284081) X\n(+1.711636) + (+91.416648) = +150.930779.\nStep 9 - divide the reduced normal term for n by the\nreduced normal term for equation 3 and change the sign:\n+150.930779/+67.278124 = -2.243386.\nThe cross multiplications made in Steps 4, 5, and 8\ncan be made in either direction. For example Step 4 could\n145","be made in the following manner: (+1.425833) X (-25.186) +\n(+44.195122) = +8.284092. The results are slightly dif-\nferent because of the significant figures involved. These\nsmall differences will not effect the final result. As a\nmatter of fact, 4 decimal places are sufficient for the\nnormal and reduced normal terms, and 5 decimal places for\nthe divided quantities. Also, two decimal places will\nusually be sufficient for the Correlate equations.\nFORWARD SOLUTION\nn\n3\n2\n1\n+10.8\n-25.186\n-8.555\n+6\n-1.800000\n+4.197667\n+1.425833\n-369.214350\n+44.195122\n+218.909783\n-353.815350\n+8.284081\n+206.711782\n+1.711636\n.040076\n+91.416648\n+173.332558\n+150.930779\n+67.278124\n-2.243386\nTable 10\nThe results of the Back Solution are shown in Table 11.\nThese quantities are identified as \"C\" Is and were computed\nas follows:\nStep 1 - c3 = Step 9 of the Forward Solution = -2.243386.\nStep 2 - C2 = (+1.711636) + (-0.040076) X C3 = +1.801542.\nStep 3-C1- = (-1.800000) + (+4.197667) X c3 +\n(+1.425833) X C2 = -8.648289.\nGenerally 4 or 5 decimal places for the C's will be all that\nis required.\n146","C1 -8.648289\nC2 = +1.801542\nC3 = -2.243386\nTable 11\nThe C's are substituted in the Correlate equations\n(Table 8) as described below, and the corrections to the\nangles and lengths determined. Each correction is the sum\nof the multiplications of the C's by their corresponding\ncoefficients. For examples: the angle correction for\nstation 2 = +1.00 X (C1) + (-3.816) X C2 + (-5.820) X C3 =\n2.466; the length correction for the distance 3-4 = +8.001 X\nC2 + (+0.613) X C3 = 13.039.\nThe computed length corrections are in angular measure\nand must be converted to feet before being applied to the\nmeasured grid distances by the following formula:\nV\"S\" X (distance in feet)\nFor example, the\nds in feet =\n206265\n15766\ncorrection in feet to the length 1-2 = +14.332 206265 X =\n+1.095 ft.\nThe corrections to the angles are adopted to hundredths\nof seconds to fit the azimuth closure and applied to the\nangles in the preliminary azimuth computation (Table 4).\nThe final adjusted azimuths are determined using the corrected\nangles. These values are not shown in Table 4, but will be\nfound in the computation of the adjusted coordinates (Table 12).\nThe corrections to the lengths are applied and are shown also\nin Table 12.\n147","191,124.78\n202,580.62\n193,403.59\nUSCOMM-DC 5270\n201,334.92\n201,037.37\n188,059.07\nFeet\nGRID COORDINATES\n2,230,491.66\n2,231,334.32\n2,242,762.03\n2,197,895.36\n2,213,659.72\n2,214,488.61\nFeet\nI\nTRAVERSE COMPUTATION BY LATITUDES AND DEPARTURES\n-12978.301\n+ 3065.715\n+11455.836\n9177.032\n297.549\nLATITUDE\nFixed\nFixed\nFeet\n+\n+11427.710\n+ 828.890\n+15764.357\n+16003.055\n+ 842.657\nDEPARTURE\nFeet\nU.S. DEPARTMENT OF COMMERCE\nCOAST AND GEODETIC SURVEY\nFINAL COMPUTATION\nTable 12\nLEAST SQUARES\nCOS AZIMUTH\n0.99982192\n0.01887142\n0.06373749\n0.99796670\n0.98214044\n0.07335878\n0.62614444\n0.18814927\n0.99730561\n0.77970708\nSIN AZIMUTH\n13004.744\n16294.060\n11486.786\n14656.414\n15767.165\nGRID DISTANCE\nFeet\n184 12 24.94\n58.45\n271 04 52.74\n18.73\n356 20 44.28\nPLANE AZIMUTH\n09\n45\n259\n308\nSTATION\n3\n4\n5\n6\nFORM 738\n1\n2\n(3-28-55)","The final coordinates are computed using standard\npractices. Note the corrected lengths and computed\nDepartures and Latitudes are shown to 3 decimal places.\nIn general usage, these computations may be made to 2\ndecimal places, but the computed coordinates for the\nterminal point could differ from the fixed values by\nseveral hundredths of feet because of round-offs. For the\nexample the check was 0.001 ft. in X and Y.\nA comparison of the results is tabulated in Table 13.\nAs might be expected, a review of the total survey corrections\n( (angles and lengths) as expressed by the averages values shows\nquite conclusively that the results obtained from the Least\nSquares solution are superior to those resulting from the\nother methods. In this particular traverse, the smallest\nangular corrections were obtained from the Transit Rule\ncomputation, and the smallest length corrections from the\nCompass Rule adjustment. The opposite is true when the\nmaximum corrections in angle and length are considered.\nThere are significant differences in the adjusted\nplane coordinates with the maximum of 0.70 ft. being between\nthe \"Y\" coordinates for point 2, as computed by the Compass\nand Transit Rules. For many mapping projects these dif-\nferences are not too important but could present problems\nshould these points be used to control other surveys of\nequal accuracy.\n149","COMPARISONS\nAngles\nPlane Coordinates\nA-O\nC-O\nT-O\nPoint\nA-C\nA-T\n+ 4\"3\n+ 7\".6\n- 1.7\n1\nFIXED\nFIXED\n-20.6\n- 1.7\n+0.33\n-0.12\n- 2.5\n2 AX\n-0.46\n+0.24\nAY\n+ 8.9\n+18.8\n+ 4.3\n-0.62\n-0.50\n3 AX\n+0.42\n+0.27\nAY\n- 2.4\n-0.47\n- 5.5\n- 1.1\n4 AX\n-0.15\n-0.52\nAY\n+0.11\n+0.38\n-11.8\n-11.2\n- 3.1\n5 AX\n+0.22\n+0.34\n+0.20\nAY\n8.6\n6\n3.0\n- 3.1\nFIXED\nFIXED\n6.2\n10.6\n3.2\nAverage\nDISTANCES\nFrom To\nA-O\nC-O\nT-O\nft.\nft.\nft.\n+1.21\n1 - 2\n+1.10\n+0.77\n+0.41\n+0.65\n+1.17\n2 - 3\n+0.62\n+1.14\n3 - 4\n+1.03\n4 - 5\n-0.24\n-0.50\n-1.02\n5 - 6\n+0.98\n+1.19\n+1.02\nAverage 0.76\n0.70\n1.15\n*\n19,500\n20,200\n12,300\nA = Least Squares\nC = Compass Rule\nT = Transit Rule\nO = Observed or measured\n* = Average 1: part in correction\nTable 13\n150","In the final analysis, some may not be overly impressed\nwith the concluded superiority of the method of Least Squares.\nAn example could have been set up to make this more positive,\nbut this was not the intent. The real value of the method\nof Least Squares is better shown in a complicated network\nand/or where the observed quantities are considered to be\nof different weight.\n151","ADJUSTMENTS USING WEIGHTS\nWeights: With the requirements for higher accuracy\nsurveys ever increasing, more surveyors are employing one-\nsecond theodolites and distance measuring equipment.\nAlthough these devices, properly used, can produce almost\nany required accuracy, they will seldom, if ever, produce\nresults of exact equal reliability, and in many instances\nthe variances are of sufficient size to justify the use\nof equating factors or weights in the adjustment of the\ndata.\nAs noted previously, no great mathematical ability\nis required to employ a least squares adjustment, and this\ncondition is unchanged when weights are introduced into\nthe adjustment. Weights may be defined in a broad sense\nas simply the relative worth of various measurements.\nUsually these values are obtained from a statistical evalu-\nation of the data, but any good approximation of the relative\naccuracies of the various survey components is acceptable.\nIf an analysis is made of the observed data, the\nWeight where SE is the Standard Error of the\nmean and is defined by SE VELVE\nIn this\nthe formula =\nformula E (v2) is the sum of the squares of the residuals\n(v's) and \"n\" is the number of observations. The residuals\nare the differences between the observed quantities and\nthe mean values. Several examples follow.\n152","(1) Two observations are made with a 5\" rejection\nlimit. The maximum SE in this case would be\n+ 5\" as may be seen by the following computation.\nOnly the seconds of the observations have been\ntabulated.\nv2\nV\n11\".3\n+5.0\n25\nObs. no. 1\n-5.0\n25\n21.3\nObs. no. 2\n16.3\nMean obs.\n2(2-1)==5\"\n50\nSE = +\n(2) Assume four observations are made and each obser-\nvation differs from the mean by exactly 4\" ; the\nSE =\nor + 2\"3\n(3) A distance is measured in feet on three occasions\nwith the following results:\nv2\nV\n0.0169\n3259.87\n-0.13\nMeas. no. 1\n0.0081\n3259.65\n+0.09\nMeas. no. 2\n0.0016\n+0.04\n3259.70\nMeas. no. 3\n3259.74\nMean\n0.0266\n+ 0.07 ft.\nSE\nIf\n=\n3(3-1)\n153","Some caution must be exercised in accepting such\nvalues in all cases as indicating the true relative worth\nof the observations. For examples: (1) assume a number\nof traverse angles were observed twice and in no case were\nthere differences in excess of 1\" - the maximum SE would be\n+ 0.5; (2) the corresponding distances were also measured\ntwice and the differences did not exceed 0.1 ft. - the\nmaximum SE being + 0.05 ft. In the first example the SE\nis most likely unreliable, however, in the second the SE\ncould be considered reasonable, depending, of course, on\nthe length of the line and the equipment used.\nIn higher quality surveys, it is presupposed that the\nspecifications are such that the resulting SE's are mean-\ningful, however, even in these cases a considered judgement\nmust occasionally be made. As an example, if the lines are\nshort, say about 1000 feet, centering errors of 0.003 ft.\nin the instrument and signals can produce a maximum error\nin an angle of about 3\", yet the SE's may be the same or\neven smaller than that obtained over much longer lines.\nIt must also be kept in mind, whenever steep slopes are\ninvolved, that relatively small errors in the differences\nof elevation can produce errors in the reduced distances\nmuch larger than that which might be indicated by the SE's.\nObviously in these examples, weights determined from the\nSE's may not be good measures of relative worth.\n154","Tabulated below are average Standard Error values\nfor the number of horizontal observations as indicated\nand a 5\" rejection limit from the mean. Also included\nare similar values for electronic distancing equipment,\nbut it must be stressed that such results can only be\nobtained when precise procedures are followed and the\natmospheric data reflects the average conditions along\nthe lines. In those cases where these conditions are\nnot rigorously met, SE's 2 to 5 times larger are\nprobably closer to the true value. No estimates are\ngiven for the new short range equipment, however, pre-\nliminary evaluations give good indication that the\nadvertised accuracy can generally be achieved.\nDistancing Equipment\nMicro-wave\nElectro-Optical\nHorizontal Observations\n1.5 cm. (0.05 ft.)\n1 cm. (0.03 ft. )\n1.5\n4 Positions\n+ three parts per\n+ one part per\nmillion of the\nmillion of the\n0.8\n8 Positions\nlength\nlength\n16 Positions 0.5\nIn theory, each observation and measurement should be\nweighted according to its SE; however, in practice an average\nweight for each observational component of a survey may be\nused providing the following conditions are met:\n(1) All observations and measurements are secured\nusing prescribed specifications. If varying\nprocedures and acceptance limits are employed,\nindividual weights would be in order.\n155","(2) No significant differences in the length of\nthe lines. In general, the accuracy of hori-\nzontal angles observed over lines, one mile\nor longer in length, would be more or less\nthe same. The accuracy would only be impaired\nslightly at one half mile, but would decline\nrapidly as the distance shortens beyond this\nrange. A similar situation exists with\nrespect to distances measured electronically,\nbut as the distances shorten, the differences\nin relative accuracy between angles and lengths\nincreases rapidly. This is mostly due to the\nfact that distance accuracies are defined as a\nconstant plus a variable such as 0.05 ft. +\n1:300,000. The following table gives some\nindication of these differences.\nB\n1: Part In\n1: Part In\nA\nDistance\nin seconds\n(feet)\n60\"\n103\"\n2,000\n0.05\n100\n15\n21\n10,000\n0.05\n500\n10\n10\n20,000\n0.05\n1,000\n5\n3\n70,000\n0.07\n5,000\n5\n2\n0.08\n125,000\n10,000\n5\n1\n167,000\n0.12\n20,000\nA - Distance accuracy based on 0.05 ft. + 1:300,000\nof distance.\n156","B - Estimated accuracy of horizontal angles observed\nusing four positions and a five-second rejection\nlimit.\nNOTE: The values given in columns A and B refer to\nestimated accuracy and not precision as ex-\npressed by the Standard Error. The distance\naccuracies (A) do not refer to any particular\nequipment and in some instances may be smaller\nthan can be obtained unless precise procedures\nare employed. The estimated angle accuracies\n(B), on the other hand, are probably on the\nhigh side.\nNumber of Condition Equations: In a traverse, not\ninvolving loops, the number of condition equations is\nequal to 3(N-1) + M - U, where N is the number of fixed\ncontrol points, M is the number of observed azimuths,\nand U is the number of fixed control points without azimuth\ncontrol. For complicated traverse networks, the number of\nequations may be more easily determined by the following\nformula: CE = 3(L - S)+ M - U, where L is the number of\nlengths, S is the number of new points, and M and U are\nthe same as noted above. This formula, of course, is\napplicable for all traverses. One word of caution, however,\nto obtain the proper number of equations at least one point\n157","must be fixed or considered fixed. As an example of\na point being considered fixed, take a traverse network\nconsisting of one loop involving only four new points,\nfour measured distances, and one observed azimuth CE =\n3(4-3) + l - l = 3. Since the point considered fixed\ndoes not have a fixed azimuth, the \"U\" term of the equation\nmust be included also.\nWhen azimuth control is available at the control\npoints, the equations are in groups of three - one for\nthe azimuth closure, one for the X closure, and one for\nthe Y closure. To avoid any confusion, the azimuth equations\nshould follow the same routes as the X and Y equations. In\nthose cases where there is no azimuth control at a fixed\npoint, only the X and Y equations are necessary, of course.\nWhen observed azimuths are included, one additional equation\nis introduced for each of these azimuths and any route may\nbe taken except that a fixed azimuth must be included, if\navailable, in at least one of the equations where more than\none observed azimuth is employed.\nIn a complicated network, the route selections for\nthe equations may seem difficult to ascertain, but if three\nsimple rules are followed no problems should be encountered.\n(1) All fixed points, including the azimuths at\nthese points, must be used at least once in\nthe formation of the equations. But fixed\npoints without azimuth control should not be\nused as a terminal point more than once.\n158","(2) All observed azimuths and measured lengths\nmust be used at least once in the formation\nof the equations.\n(3) Each set of equations must contain some terms\nnot appearing in any other set. This restriction\ndoes not apply to routes required specifically\nfor observed azimuths.\nFigure (3) is an example of a rather complicated network.\n4\n2\n3\n1\n6\n7\n5\n9\n10\n8\nFixed Control Point\nNew Station\nAzimuth to Azimuth Mark\nObserved Azimuth\nFigure 3\n159","In this network (Figure 3), there are four fixed\ncontrol points (but only two have azimuth control), six\nnew stations, thirteen measured distances, and one observed\nazimuth. The number of condition equations is 3(13-6) + 1-2 - 20.\nIt should be noted that only the fixed and junction points\nare shown in this example, any number of new points between\nthese junction points may be introduced without changing\nthe number of equations provided there are no multiple\nconnections between these points.\nThe \"buildup\" method of selecting routes should be\nmade. In this method the equations involving the fixed\ncontrol would be selected first and then the other routes\nfollowing the rules given previously. The selection would\nbe as follows:\nTotal Equations\nRoute\nEquations\n2\n1-2-3-4\nX, Y\n1-5-8\n2\nX, Y\n3\n1-5-8-9-10\nAz, X, Y\n3\n1-2-3-7-10\nAz, X, Y\n3\n1-2-6-9-10\nAz, X, Y\n3\n1-2-6-7-10\nAz, X, Y\n3\n1-5-6-9-10\nAz, X, Y\n1\nAz\n1-2-3\n20\nNumerous other routes could have been selected that would\nfulfill the rules.\n160","Preliminary computations are then made over the selected\nroutes to obtain the closures and preliminary coordinates\nnecessary to form the equations.\nIf the directions rather than angles are employed in\na traverse adjustment, the number of equations is not\nincreased but the number of terms in each equation is\nincreased by about one-half since each angle is the dif-\nference between two directions. The use of directions\nhas some advantages when several angles are composed of\nobservations over long and short lines and there are large\nuncertainties in the accuracy of these measurements. No\nexample is given for a traverse adjustment using directions\nin this paper, but a document to be released shortly will\ncontain programs for adjusting traverse which uses directions.\nThe sample adjustment of a triangulation figure will be\nmade using directions, however.\nThe number of observation equations has been previously\ndiscussed and additional comments will be found in the\nsection concerned with adjustment of triangulation.\nTwo examples of a weighted adjustment will be presented.\nThe first, by condition equations, is recommended for\ntraverse and the second, using the method of variation of\ncoordinates which employs observation equations, for tri-\nangulation Both computations will be made on the plane, using\nState plane coordinates, but the procedures are applicable\nto other grid systems.\n161","Condition Equations : This example is a simulated\nin western Georgia composed of two traverse lines\ntraverse forming a \"T\" configuration with fixed control on the north,\nand south. In addition, an azimuth was considered\neast, being observed at the junction of the two lines (3-4).\nas 4 is a sketch of the survey and the curved lines\nFigure represent the geodetic angle observations and the straight\nthe grid values. Since Georgia uses the transverse\nlines Mercator plane system, the curved lines are concave towards\nthe central meridian.\n4\n6\n3\n7\n2\n8\n16\n9\nA\nFixed Control Points\nA\nNew Stations\nObserved Azimuth\nFigure 4\n162","The first step is to determine the Weights for the\nvarious quantities. When employing condition equations\ninvolving a mixture of angle and distance measurements,\nthe weights must be in the same measure. This does not\npresent any problem in determining the weights for the\nangles and azimuths since the SE's or estimated accuracies\nwould be in seconds, but this would not be the case for\nthe lengths. Inasmuch as there is no great variances in\nthe distances (Table 17), a mean value of 14,900 ft. was\nused with the SE assumed to be 0.1 ft. + 1:100,000 of the\nSE X 206265\ndistance or 0.25 ft. Using the formula SE\" =\nDistance\nor SE\" \" = 0.25 206265, the value was found to be 3\"5.\n14,900\nThe SE's for the angles and azimuths are given below.\nThe computation of the weights is as follows:\nkp*\nSE\n(SE)2\np\n=\n1.00\n0.2500\n2\".O\nAngles\n0.33\n0.0816\n3.5\nLength\n0.44\n0.1111\n3.0\nAzimuth\nIn this tables the values in the second column\nare the weights \"p\", but to simplify future\ncomputations they should be multiplied by a\nfactor \"k\". This factor should be such that the\nweight to be applied to the majority of the\nelements involved is 1.00. For this example\na factor of 4 served the purpose.\n163","Using the formulae presented earlier, the Number of\nCondition Equations in this survey are CE = 3(2) + 1 = 7\n3(8-6) + 1 = 7. The routes selected for the equa-\nor\ntions involving the control points were 1-3-5 and 1-3-9.\nThe route for the azimuth equation containing the observed\nazimuth is 1-3. Other routes could have been selected,\nof course.\nThe Second Term Corrections are now computed as described on\npp. 123-124 remembering that the survey is to be computed\nusing transverse Mercator coordinates and AX is the mean\nX' of the two points involved and AY is the difference in\nthe Y's. Since the observations at the terminal stations\nare over longer lines than usually associated with azimuth\nmarks, corrections must be determined for these lines. The\ncomputations are shown in Table 14.\n2.36(AX)(AY)\nAY\nAX\nFrom\nTo\n1.2\n0.135\n1\n2\n3.791\n-\n4.4\n3.691\n0.510\n1 - 5\n0.2\n3.541\n0.025\n1 - 9\n1.4\n3.763\n0.155\n2 - 3\n3.728\n0.126\n1.1\n3 - 4\n0.2\n3 - 6\n3.696\n0.025\n0.8\n3.634\n0.093\n4 - 5\n0.484\n3.9\n3.419\n5 - 9\n3.578\n0.126\n1.1\n6 - 7\n3.460\n0.7\n0.090\n7 - 8\n0.6\n0.074\n8 - 9\n3.333\nTable 14\n164","The second term corrections are applied to the Observed\nAngles to obtain the Grid Angles (Table 15) as explained on pp.\n124-125. In this case the total corrections are not all of\nthe same sign nor are the individual corrections added\nalgebraically and a difference approach must be taken\nthan previously. As examples: the correction geodetic\nto grid for the line 1-9 is +0.2, and 1-2 it is +1.2. If\ndirections were observed, the clockwise angle at 1 from\n9 to 2 would be direction 1-2 minus direction 1-9, there-\nfore, the correction would be +1.2 - (+0\".2) or 1.0.\nSimilarly, the clockwise angle at 3 between 2 and 6 would\nbe corrected by the correction 3-6 minus the correction\n3-2 or +0.2 - (-1.4) = +1.6.\nWt. = p\nGrid Angle\nCorr'n\nObserved Angles\nPoint\n9\n60\".4\n+1\".0\n1.0\n291°12'59\"4\n1\n13.4\n1.0\n+2.6\n165 33 10.8\n2\n1.0\n55.9\n200 25 53.4\n+2.5\n3\n42.5\n1.0\n210 04 40.6\n+1.9\n4\n57.8\n1.0\n-3.1\n273 42 00.9\n5\n9\n2\n1.0\n+1.6\n52.3\n254 19 50.7\n3\n1.0\n37.0\n-0.9\n239 53 37.9\n6\n10.4\n1.0\n-1.8\n167 24 12.2\n7\n1.0\n20.1\n174 19 21.4\n-1.3\n8\n20.4\n1.0\n+3.3\n28 15 17.1\n9\n5\nTable 15\n165","The computation of the Scale and Sea Level Factors and\nthe reduction of the Horizontal Measured Distances is shown\nTables 16 and 17. The discussion appearing on pp. 126-130\nin\nis applicable here except that the survey is being com-\nputed on the transverse Mercator grid system and individual\ncorrections to each length have been determined. In inter-\npolating for the scale factor from the tables, X' values\nto the nearest one hundred feet will normally be sufficiently\naccurate. As a case in point, a difference of one in the\nsixth decimal place of the scale factor will produce only\na one part per million change in the length.\nCombined\nScale\nElevation\nElevation\nFactor\n(feet)\nFactor\nFactor\nX'\nStation\n1.0000665\n0.9999721\n381356\n0.9999056\n1974\n1\n1.0000626\n0.9999893\n376825\n0.9999267\n1532\n2\n1.0000616\n1.0000097\n0.9999481\n375714\n1085\n3\n1.0000568\n1.0000203\n369943\n764\n0.9999635\n4\n1.0000458\n1.0000297\n356807\n336\n0.9999839\n5\n0.9999803\n363404\n1.0000512\n6\n1482\n0.9999291\n0.9999829\n1.0000421\n0.9999408\n352234\n7\n1237\n0.9999417\n1.0000322\n8\n1892\n0.9999095\n339705\n1.0000224\n0.9999179\n326924\n2185\n0.9998955\n9\nTable 16\n166","TRANSVERSE MERCATOR PROJECTION FOR GEORGIA (EAST & WEST)\nxi\nScale in\nScale\nX'\nScale in\nScale\n(feet)\n(feet)\nunits of\nunits of\nexpressed\nexpressed\n7th place\n7th place\nas a\nas a\nof logs\nof logs\nratio\nratio\n-434.3\n-282.0\n0.9999000\n175,000\n0.9999351\no\n-434.2\n-273.2\n5,000\n0.9999000\n180,000\n0.9999371\n-433.8\n-264.1\n10,000\n0.9999001\n185,000\n0.9999392\n15,000\n-433.2\n-254.8\n0.9999003\n190,000\n0.9999413\n-432.3\n-245.2\n20,000\n0.9999005\n195,000\n0.9999435\n-431.2\n0.9999458\n25,000\n0.9999007\n200,000\n-235.4\n-429.8\n30,000\n0.9999010\n205,000\n-225.3\n0.9999481\n-428.2\n0.9999014\n35,000\n210,000\n-215.0\n0.9999505\n-426.3\n-204.4\n40,000\n0.9999018\n215,000\n0.9999529\n45,000\n-424.2\n-193.6\n0.9999023\n220,000\n0.9999554\n-421.9\n50,000\n0.9999029\n225,000\n-182.5\n0.9999580\n-419.3\n55,000\n0.9999035\n230,000\n-171.2\n0.9999606\n-416.4\n60,000\n0.9999041\n235,000\n-159.7\n0.9999632\n65,000\n-413.3\n0.9999048\n240,000\n-147.9\n0.9999659\n70,000\n-409.9\n0.9999056\n245,000\n-135.8\n0.9999687\n-406.3\n0.9999064\n75,000\n250,000\n-123.5\n0.9999716\n-402.5\n80,000\n0.9999073\n255,000\n-110.9\n0.9999745\n85,000\n-398.4\n0.9999083\n260,000\n- 98.1\n0.9999774\n90,000\n-394.0\n265,000\n0.9999093\n- 85.1\n0.9999804\n-389.4\n95,000\n0.9999103\n270,000\nwe 71.8\n0.9999835\n-384.6\n0.9999114\n100,000\n0.9999866\n275,000\nare 58.2\n105,000\n0.9999126\n- 44.4\n-379.5\n280,000\n0.9999898\n110,000\n-374.1\n- 30.4\n0.9999139\n285,000\n0.9999930\n115,000\n-368.5\n0.9999151\n290,000\n- 16.1\n0.9999963\n120,000\n-362.7\n0.9999165\n295,000\n- 1.5\n0.9999997\n-356.6\n125,000\n0.9999179\n300,000\n+ 13.3\n1.0000031\n130,000\n-350.3\n0.9999193\n305,000\n+ 28.3\n1.0000065\n135,000\n-343.7\n+ 43.6\n0.9999209\n310,000\n1.0000100\n140,000\n-336.8\n0.9999224\n315,000\n+ 59.1\n1.0000136\n145,000\n0.9999241\n+ 74.9\n-329.7\n320,000\n1.0000172\n-322.4\n0.9999258\n325,000\n+ 91.0\n150,000\n1.0000210\n1.0000247\n-314.8\n155,000\n0.9999275\n330,000\n+107.3\n160,000\n+123.8\n1.0000285\n-307.0\n0.9999293\n335,000\n+140.6\n1.0000324\n165,000\n340,000\n-298.9\n0.9999312\n345,000\n+157.6\n1.0000363\n170,000\n-290.6\n0.9999331\n167","TRANSVERSE MERCATOR PROJECTION FOR GEORGIA (EAST & WEST)\nX1\nScale in\nScale\n(feet)\nunits of\nexpressed\n7th place\nas a\nof logs\nratio\n1.0000403\n+174.9\n350,000\n1.0000443\n355,000\n+192.4\n1.0000484\n360,000\n+210.2\n365,000\n+228.2\n1.0000525\n+246.5\n1.0000568\n370,000\n1.0000610\n+265.0\n375,000\n1.0000653\n380,000\n+283.8\n1.0000697\n385,000\n+302.8\n1.0000742\n390,000\n+322.1\n+341.6\n395,000\n1.0000787\n400,000\n+361.4\n1.0000832\n405,000\n+381.4\n1.0000878\n410,000\n+401.6\n1.0000925\n415,000\n+422.1\n1.0000972\n420,000\n+442.9\n1.0001020\n425,000\n+463.9\n1.0001068\n430,000\n4485.2\n1.0001117\n435,000\n1.0001167\n+506.7\n440,000\n+528.5\n1.0001217\n445,000\n1.0001268\n+550.5\n450,000\n+572.7\n1.0001319\n455,000\n+595.2\n1.0001370\n460,000\n1.0001423\n+618.0\n465,000\n+641.0\n1.0001476\n470,000\n+664.2\n1.0001529\n475,000\n+687.7\n1.0001.583\n480,000\n1.0001638\n+711.4\n485,000\n+735.4\n1.0001693\n490,000\n1.0001749\n+759.7\n495,000\n1.0001806\n+784.2\n1.0001863\n+808.9\n500,000\n1.0001920\n+833.9\n505,000\n1.0001978\n510,000\n+859.1\n+884.6\n1.0002037\n515,000\n1.0002096\n+910.4\n520,000\n.\n168","Measured\nMean\nCombined Factor\nGrid Distance\nWt. = p\nFrom To\nDistance\n14,238.75\n0.9999807\n0.33\n14239.02\n1\n2\n15,520.25\n0.33\n15520.26\n0.9999995\n2\n3\n13,895.46\n0.33\n4\n13895.25\n1.0000150\n3\n12,564.06\n0.33\n6\n12564.12\n0.9999950\n3\n16,110.92\n0.33\n16110.52\n4 5\n1.0000250\n16,803.11\n0.9999816\n0.33\n6 7\n16803.42\n15,434.82\n0.9999623\n0.33\n8\n15435.40\n7\n14,766.27\n0.9999298\n0.33\n8\n14767.31\n9\nTable 17\nThe computation of the Azimuth Closures is shown in\nTables 18, 19, 20. The computed azimuths will be used in\nthe computation of the X and Y Closures as shown in Tables\n21 and 22. The fixed positions, fixed azimuths, and ob-\nserved azimuth 3-4 are given in the tables. More particulars\nare given on pp. 130-132 and 138-140.\nNote that the angle designations are tabulated on\nthe azimuth computations. This is necessary because at\nthe junction station, point 3, a distinction between the\nangles used in the computations must be made. Should\ndirections, rather than angles be used, the numbering\nsystem employed would take this into account.\nFor obvious reasons, when determining the X and Y\nclosures by hand computations, a minimum effort should be\nexpended. The computations shown in Tables 21 and 22\n169","13.6\n00.4\n14.0\n14.0\n13.4\n27.4\n27.4\n52.3\nPRELIMINARY AZIMUTH\n37.0\n19.7\n19.7\n10.4\n56.7\n56.7\n07.1\n07.1\n20.1\n27.2\n27.2\n20.4\n47.6\n148 18 53.0\n- 5.4\n\"\n20\n13\n33\n33\n33\n06\n06\n19\n26\n26\n53\n19\n19\n24\n44\n44\n19\n03\n03\n15\n18\n267\n291\n198\n18\n165\n184\n4\n254\n258\no\n78\n239\n318\n138\n167\n305\n125\n174\n300\n120\n28\n148\n*\n*\nTable 19\nTO\n9\n2\n1\n3\n2\n6\n3\n7\n6\n8\n7\n9\n8\n5\n** Angle Designation\nFixed Azimuth\nSTATION\n3b\n**\n1\n2\n6\n7\n8\n9\nFROM\n1\n1\n2\n2\n3\n3\n6\n6\n7\n7\n8\n8\n9\n9\n*\nL\nL\nL\nL\nL\nL\nL\nL\n14.0\n13.4\n13.6\n00.4\n14.0\n27.4\n27.4\n23.3\n23.3\n55.9\n42.5\n05.8\n05.8\n47.8\n03.6\n53.0\nPRELIMINARY AZIMUTH\n+ 10.6\n20\n13\n33\n33\n33\n06\n06\n25\n32\n32\n04\n37\n37\n41\n19\n18\n267\n291\n198\n18\n165\n184\n4\n200\n204\n24\n210\n234\n54\n273\n328\n328\n*\n*\nTable 18\nTO\n9\n2\n1\n3\n2\n4\n3\n4\n5\n9\n** Angle Designation\nFixed Azimuth\nSTATION\n3a\n**\n1\n2\n4\n5\nFROM\n1\n1\n2\n2\n3\n3\n4\n4\n5\n5\n*\nL\nL\nL\nL\nL\nL","STATION\nPRELIMINARY AZIMUTH\nTO\n\"\nFROM\n267\n13.6\n20\n1\n**\n9\n*\n00.4\nL\n291\n13\n1\n198\n14.0\n33\n2\n1\n18\n14.0\n33\n1\n2\n165\n13.4\n33\nL\n2\n184\n06\n27.4\n3\n2\n4\n06\n27.4\n2\n3\n200\n25\n55.9\nL\n3a\n4\n204\n32\n23.3\n3\n16.4\n204\n32\nObserved Az.\nFixed Az\n+ 06.9\nAngle Designation\nL\nTable 20\nreflect this reasoning. Also, the sines and cosines of\nthe azimuths were omitted because the computations were\nmade on a desk size programmable computer with built-in\ntrigonometric functions.\nThe Formation of the Condition Equations is shown in\nTable 23. These equations, with one exception, are computed\nusing the basic equations as given on p. 140 and explained on\nThe one exception being that the azimuth\n140-143.\npp.\nequation involving the observed azimuth is in the form\no - An - (AO + K) + E (Va), where AO is the observed\nazimuth (i-k), and K is the correction to be determined in\nthe adjustment for this azimuth. The other terms of the\nequation are as identified on p. 140. If for some reason\nthe observed azimuth is to be held fixed, the \"K\" would be\n171","1,519,363.268\n- 2.492\n1,519,365.76\n1,468,415.31\nUSCOMM-DC 5270\nEget\ny\nGRID COORDINATES\n143,192.926\n+ 2.486\n118,643.92\n143,190.44\nFeet\nx\n+ 12640.324\n+ 15480.383\n+ 13498.688\n+ 9328.563\nLATITUDE\nFixed\nFixed\nFeet\nPRELIMINARY COORDINATE COMPUTATION\n+13135.435\n+ 5771.136\n+ 4530.720\n+ 1111.715\nDEPARTURE\nFeet\nTable 21\nLOG COS AZIMUTH\nLOG SIN AZIMUTH\nLOG DEPARTURE\nLOG LATITUDE\nLOG DISTANCE\nGRID DISTANCE\n13895.46\n16110.92\n14238.75\n15520.25\nFeet\n204 32 23.3\n198 33 14.0\n184 06 27.4\n234 37 05.8\nPLANE AZIMUTH\nSTATION\n3\n4\n2\n1\n5\n17\n2","1,470,949.682\n+ 2.702\n1,497,394.381\n173,077.31 1,470,946.98\nUSCOMM- DC 5270\nFeet\ny\nGRID COORDINATES\n- 1.613\n173,075.697\n124,286.355\nFeet\nx\n-12552.166\n- 9014.575\n- 7395.977\n+ 2518.019\nLATITUDE\nFixed\nFeet\n(From Table 21)\n+ 12780.542\n+ 11170.839\n+ 12528.811\nPRELIMINARY COORDINATE COMPUTATION\n+ 12309.150\nDEPARTURE\nFeet\nLOG COS AZIMUTH\nLOG SIN AZIMUTH\nLOG DEPARTURE\nTable 22\nLOG LATITUDE\nLOG DISTANCE\nGRID DISTANCE\n15434.82\n14766.27\n16803.11\n12564.06\nFeet\n300 03 27.2\n318 19 56.7\n305 44 07.1\n258 26 19.7\nPLANE AZIMUTH\nSTATION\n9\n8\n6\n7\n3\n173","FORMATION OF CONDITION EQUATIONS\n1) O = + 10.6 + 1.00(1) + 1.00(2) + 1.00(3a) + 1.00(4) + 1.00(5)\n2) O = - 5.4 + 1.00(1) + 1.00(2) + 1.00(3b) + 1.00(6) + 1.00(7)\n+ 1.00(8) + 1.00(9)\n3) O = + 6.9 - 1.00(3-4) + 1.00(1) + 1.00(2) + 1.00(3a)\n4) 0 = + 2.486 X 206265 + (Y5 - Y1)(1) + (Y5 - Y2)(2)\n+ ( - Y3)(3a) + ( - Y4)(4)\n; 10,000\n+ (X2 - X X ) (1-2) + (x3 - x2)(2-3)\n+ (x4 x3)(3-4) + (x5 - X4)(4-5)\n0 = + 51.2775 + 5.095(1) + 3.745(2) + 2.197 (3a) +0.933(4)\n0.453(1-2) + 0.111(2-3) + 0.577(3-4)\n+\n+ 1.314(4-5)\n5) O = - 2.492 X 206265 - (x5 -\n- (X5 - X3)(3a) - (X5 - X4)(4)\n10,000\n+ (Y2 - Y1)(1-2) + (Y3 - Y2)(2-3)\n+ (Y4 - Y3)(3-4) + (Y5 - Y4)(4-5)\no = - 51,4012 - 2.455(1) - 2.002(2) - 1.891(3a) - 1.314(4)\n+ 1.350(1-2) + 1.548(2-3) + 1.264(3-4)\n+ 0.933(4-5)\n6) O = - 1.613 X 206265 + ( 99 - Y1)(1) + ( 99 - Y2)(2) + ( 9 - Y3)(3b)\n+ ( 9 - Y6)(6) + ( , - Y7)(7) + ( - Yg)(8)\n10,000\n+ (X2 X1)(1-2) + (x3 x2)(2-3)\nr\n+ (x6 - x3) (3-6) + (x, - x6)(6-7)\n+ (Xg xy)(7-8) + (xg - xg)(8-9)\nTable 23\n174","o = - 33.2705 + 0.254(1) - 1.096(2) - 2.644(3b) - 2.896(6)\n- 1.641(7) - 0.740(8) + 0.453(1-2)\n+ .111(2-3) + 1.231(3-6) + 1.117(6-7)\n+ 1.253(7-8) + 1.278(8-9)\n7) o = + 2.702 X -\n10,000\n-\n-\n+ -\n+\n+ (Y8- 7(7-8)(y -8)(8-9)\no +55.7328 - 5.443(1) - 4.990(2) - 4.879(3b) - 3.648(6)\n- 2.531(7) - 1.278(8) + 1.350(1-2) + 1,548(2-3)\n+ 1.252(3-6) - 1.255(6-7) - 0.901(7-8)\n- 0.740(8-9)\n(Table 23 - Continued)\nbe eliminated and the basic equation would be used. As may\nbe seen from the above equation, the sign of the \"K\" term\nis minus with a coefficient of 1.00 generally assigned in\nthe form -1.00(1-k) or in this example, -1.00(3-4) as\nshown in equation 3, Table 23.\nThese equations conform to the routes as selected on\np. 164. To save a step, the constant terms and coefficients\nfor the X and Y equations were divided for the reasons given\non pp. 142-143 prior to tabulating the values in Table 23\n175","The equations are tabulated vertically (Correlate\nEquations) as shown in Table 24. This tabulation contains\nadditional columns, a/p and W not shown in Table 8, p. 143.\nThe numbers appearing in the column labeled a/p are\nobtained by dividing some convenient number \"a\" by the\nweight \"p\" assigned the observations identified by the\nCorrection Number column. Weights for the observations\ninvolved in this adjustment were determined earlier (p. 163).\nAlthough an examination of these weights showed that for\npractical purposes two choices for \"a\" would be acceptable\n(1.00 or 0.33), it was decided to use 1.00. In this manner\nthe values a/p for the angles were 1.00; 3.03 for the lengths;\nand 2.27 for the observed azimuth. If an \"a\" of 0.33 had\nbeen used, a/p for the lengths would have been 1.00; the\nangles 0.33; and the azimuth 0.75. The a/p values are\nalways considered positive in the various operations in\nwhich they are involved.\nThe \"E\" column is computed by summing the coefficients\nalgebraically, the values for a/p not included. For\nexamples: (1) the \"E\" for the line involving correction\nno.1 = +1.00 + 1.00 + 1.00 + 5.095 - 2.455 + 0.254 - 5.443 =\n+0.451; (2) W for line 2-3 = +0.111 + 1.548 + 0.111 + 1.548 =\n3.318. By introducing this column, the computations of the\nnormal equations and the forward solution are self checking\noperations. This self checking feature is not restricted\n176","ds in ft.\n+1.841 +9.0259 +0.608\n-0.310\n+1.231 +0.252 +1.483 +5.3173 +0.324\n+1.117 -1.255 -0.138 +18.4818 +1.506\n+1.253 -0.901 +0.352 +16.0670 +1.202\n+1.278 -0.740 +0.538 +14.7401 +1.055\n+0.453 +1.350 +0.453 +1.350 +3.606 +1.9792 +0.137\n+0.111 +1.548 +0.111 +1.548 +3.318 +4.6277 +0.348\n-0.740 -1.278 -1.018 +0.6894 +0.69\n+1.000 -1.6779 -1.68\n-7.42\n+2.306 -6.8493 -6.85\n-2.644 -4.879 -6.523 +7.7305 +7.73\n-2.72\n+6.44\n-2.896 -3.648 -5.544 +3.4749 +3.48\n-1.641 -2.531 -3.172 +2.6511 +2.65\nAdopt\n-5.28\n1.00 +1.00 +1.00 +1.00 +3.745 -2.002 -1.096 -4.990 -1.343 -2.1933 -2.19\nCorrections\n-3.9750\n-7.4176\n-2.7194\n+6.4366\n1.00 +1.00 +1.00 +1.00 +5.095 -2.455 +0.254 -5.443 +0.451 -5\"2746\nV's\n-1.000\n+2.247\n+1.000\n+0.619\nW\n7\n6\nCORRELATE EQUATIONS\n+1.264\n+0.933\n-1.314\n-1.891\nEquations\n5\n+1.314\n+0.577\n+0.933\n+2.197\n4\nTable 24\n-1.00\n+1.00\n3\n+1.00\n+1.00\n+1.00\n+1.00\n+1.00\n2\n+1.00\n1.00 +1.00\n+1.00\n1\n3.03\n3.03\n3.03\n3.03\n3.03\n3.03\n3.03\nof\n3.03\n1.00\n1.00\n1.00\n2.27\n1.00\n1.00\n1.00\n1.00\na/p\nCorr'n\n3a\n3b\n7-8\n8-9\n2-3\n3-4\n4-5\n3-6\n6-7\n1-2\n8\n9\n3-4\n4\n5\n6\n7\n1\n2\nNo.","to weighted adjustments, it could have been used in the\nprevious example given on pp. 143-146 but was omitted on\norder to reduce the length of the explanation given on\nthose pages.\nThe \"Corrections\" columns are identical to those\npreviously described (p.147) but the actual computation\nis somewhat different and will be described later.\nThe Normal Equations (Table 25) are formed in the\nsame manner as described on pp. 143-144 except that each\nmultiplication is also multiplied by its corresponding\na/p. In addition, the check column \"E\" is introduced.\nA few examples involving a/p and W follow:\n(1) An inspection reveals that the a/p values for\nthe angles is 1.00 and therefore the formation of the\nnormal equations for equations 1 and 2 are identical\nto the procedures explained previously. W is are obtained\nby treating the W terms in the correlate equations as\nif they were part of other equations and then the con-\nstant terms \"n\" are added to the result. For equation\n1, W = (+1.00)(+0.451) + (+1.00)(-1.343) + (+1.00)(+2.306)\n+ (+1.00)(+0.619) + (+1.00)(+1.000) + (+10.6) If +13.633.\nFor equation 2, W If (+1.00)(+0.451) + (+1.00)( -1.343) +\n(+1.00)(-6.523) + (+1.00)(-5.544) + (+1.00)(-3.172) +\n(+1.00)(-1.018) + (+1.00)(+1.000) + (-5.4) = -21.549.\n178","+ 72.4200\n- 11.2773\n+151.1622\n+ 22.1265\n+ 10.584\n+ 13.633\n- 21.549\nW\n+55.7328\n-51.4012\n-33.2705\n+51.2775\n+ 6.9\n+10.6\n- 5.4\nn\n- 44.0460\n+ 36.1355\n+ 25.4301\n+121.5448\n- 10.433\n- 10.433\n- 22.769\n7\n+ 3.9443\n- 2.1513\n+38.6209\n- 8.763\n- 0.842\n- 0.842\n6\nNORMAL EQUATIONS\nTable 25\n-17.0880\n+35.5991\n6.348\n7.662\n4.457\n5\n+52.5808\n+11.970\n+ 8.840\n+11.037\n4\n+3.00\n+2.00\n+5.27\n3\n+2.00\n+7.00\n2\n+5.00\n1","(2) The a/p column enters the computation with equation\n3 but only effects the normal terms for 3 and W since the\nformation of the other normal terms are not involved with\nthe coefficient 3-4. The normal terms for columns 3 and W\nwere computed as follows:\n(3) If (+1.00)2 + (+1.00)2 + (+1.00)2 + (-1.00)2(2.27) If +5.27\nE If (+1.00)(+0.451) + (+1.00)(-1.343) + (+1.00)(+2.306)\n+ (-1.00)(-1.000)(2.27) + (+6.9) = +10.584\n(3) The formation of the other normal equations will\nalso involve different values for a/p. In this sample\nadjustment, the formation is simplified because only two\nvalues for a/p are involved and the normal terms may be\nformed by summing two partial sets of multiplications as\nfollows:\n(4) = (+5.095)2 + (+3.745)2 + (+2.197)2 + (+0.933)2/x(1.00)\n+ (7+0.453)2 + (+0.111)2 + (+0.577)2 + (+1.314)2\nX (3.03) = +52.5808\nLet us assume the a/p values differ for each observed\nquantity. The formation of the normal term 4 on 5 would\nthen involve the following multiplications. The values\nfor a/p are as given in Table 24.\n(4) X (5) If + (+5.095)(-2.455)(1.00) + (+3.745)(-2.002)(1.00)\n+ (+2.197)(-1.891)(1.00) + (+0.933)(-1.314)(1.00)\n+ (+0.453)(+1.350)(3.03) + (+0.111)(+1.548)(3.03)\n+ (+0.577)(+1.264)(3.03) + (+1.314)(+0.933)(3.03)\n= -17.0880.\n180","The computation of the remaining normal equation\nterms are made in a similar fashion.\nThe normal equations are checked by summing all the\nterms vertically and horizontally for each equation including\nthe \"n\" term but excluding the \"W\" term. This sum should\nequal the W term exactly if all decimal places are used.\nIn this case only four decimals are retained and the sum\nof the normal terms may not equal the computed \"W\" term\nby as much as +0.0002. Where this occurred in this example,\nonly the corrected values for W are shown. Several examples\nare given below:\n(1) (1) = + 5.00(1) + 2.00(2) + 3.00(3) + 11.970( (4)\n7.662(5) - 0.842(6) - 10.433(7) + 10.6(n) =\n+13.633 which checks the W value (+13.633)\n(2) (4) = +11.970(1) + 8.840(2) + 11.037(3) + 52.5808(4)\n-17.0880(5) - 2.1513(6) - 44.0460(7)\n+51.2775(M) = +72.4200 which checks the\nW value (+72.4200)\nNOTE: The numerical notations have no meaning except\nto indicate to the reader the location of the\nterms in the normal equations used in the example.\nThe computation of the Forward Solution is given in\nTable 26 and was performed using the same technique described on\npp. 144-146. The \"W\" column is handled in an identical\nmanner as the other columns. To employ the self checking\n181","- 2.87020\n+ 1.04577\n4.03791\n+ 4.35519\n- 1.74896\n+ .19860\n- 26.8423\n+151.1622\n+ 72.4200\n- 11.2773\n- 4.3778\n+ 22.1265\n+119.7659\n- 27.0022\n+ 5.8884\n+ 51.6019\n- 2.7266\n+ 10.584\n+ 13.633\n- 21.549\nW\n-\n+ 1.55484\n1.69289\n1.94112\n+ 2.04619\n3.03791\n- .52985\n-33.2705\n-52.5204\n+55.7328\n+90.1055\n+ 1.7839\n+51.2775\n+30.4357\n-51.4012\n-42.7870\n- 2.120\n- 9.64\n+10.6\n- 5.4\n+ 6.9\nn\n+\n-\n.00042\n+ 2.99932\n- .22288 + .52682\n.28705\n.73551\n- 44.0460\n+ 16.2123\n+121.5448\n+ 29.6604\n- 5.1607\n36.1355\n+ 25.4301\n+ 0.0107\n+ 2.0866\n- 18.5958\n- 1.7737\n- 10.433\n- 10.433\n- 22.769\n7\n+\n+\n-\n-\n.25746\n- .00701\n+ 1.35906\n+25.6674\n+ 3.9443\n+ .1684\n- 8.4262\n+ 0.7504\n- 2.1513\n+ 4.6287\n+ 0.1545\n+38.6209\n- 0.842\n- 8.763\n- 0.842\n6\n-\nFORWARD SOLUTION\n- .20690\n+ .22455\n+ .46667\n+22.0424\n+ 1.5324\n- 1.3922\n- 1.5712\n-17.0880\n+ 3.7197\n+35.5991\n- 7.662\n- 6.348\n- 4.457\n5\nTable 26\n.65355\n- .98972\n+ 3.3322\n+52.5808\n+17.9785\n+11.970\n- 2.394\n+ 8.840\n+ 4.052\n+11.037\n4\n-\n- .12903\n+3.3668\n- .600\n+3.00\n+2.00\n+0.80\n+5.27\n3\nBACK SOLUTION\n-1.67793\n+4.16118\n+6.43657\n+3.26765\n-3.95301\n+2.04747\n-3.03791\n- .400\n+2.00\n+7.00\n+6.20\nTable 27\nC's\n2\n+5.00\n1\n4\n1\n2\n3\n5\n6\n7","feature sum the normal terms for the first equation and\nthe reduced normals for all other equations prior to dividing\nthe terms. These sums should equal the computed Evalues\nwithin +0.0002 generally, but in a large solution checks of\n+0.0010 may be satisfactory. The W should be changed\nto agree with the sum. The sum of the divided terms should\ncheck the W term by +0.00002 if five decimal places are\nused and within +0.000002 should the computations be carried\nto six decimal places. In summing the divided terms, -1.00000\nmust be included to account for the division of the diagonal\nterm by the negative value of itself. Several examples\nare given below:\n(1) Equation 1 - the terms are identical to those\nin the Normal equation which was previously checked. Divided\nterms sum IN -1.0000(1) - 0.4000(2) - 0.6000(3) - 2.3940(4)\n+ 1.5324(5) + 0.1684(6) + 2.0866(7) - 2.1200( (m) =\n-2.7266 which checks the W value (-2.7266). .\n(2) Equation 3 - Reduced normal sum If + 3.3668(3)\n+ 3.3322(4) - 1.5712(5) + 0.7504(6) - 1.7737(7)\n+ 1.7839(N) = +5.8884 which checks theZ value\n(+5.8884). Divided terms sum = - 1.00000(3)\n- 0.98972(4) + 0.46667(5) - 0.22288(6) + 0.52682(7)\n- 0.52985(N) = -1.74896 which checks the W\nvalue (-1.74896).\n183","(3) Equation 6 - Reduced normals sum If + 25.6674(6)\n+ 0.0107(7) 52.5204(N) = -26.8423 which\nchecked the W value (-26.8424) by -0.0001.\nThe corrected W term is shown in the table.\nDivided terms sum = - 1.00000(6) - 0.00042(7)\n+ 2.04619(N) = +1.04577 which checks the W\nvalue (+1.04577).\nThe numerical notations, as noted previously, are\ngiven only to indicate the location of the terms in the\nsolution.\nThe results (C's) of the Back Solution are shown in\nTable 27. These values were computed using the identical\nmethod given on p. 146. This computation is also made self\nchecking by using the reduced normal terms as shown below.\nIf \"C's\" to five decimal places are used, the sum should\nseldom differ from zero by more than +0.00002 of the\ndiagonal term for the equation, but occasionally may\napproach +0.00003 of the diagonal term. Example (3)\nbelow is a case in point.\n(1) Equation 7 o = + 90.1055 + (+29.6604) X C7 II -0.00013\nallowable = 29.6604 X 0.00002 If +0.00059.\no = - 42.7870 + (+16.2123) X C7 +\n(2) Equation 5\n(+0.1545) X C6 + (+22.0424) X C5 = +0.00022\nallowable = 22.0424 X 0.00002 = +0.00044\n184","(3) Equation 2 o = - 9.64 + (-18.5958) X C7 + (-8.4262)\nC6 + (-1.3922) X C5 + (+4.052) X C4\n+ (+0.80) x C3 + (+6.20) X C2 = +0.00014\nallowable = 6.20 X 0.00002 = +0.00012.\nA further check on the \"C's\" may be obtained by\nmultiplying each \"C\" by its corresponding \"n\" term (the\noriginal and not the reduced value) and summing the results.\nThis sum should check the summation of the \"n\" column\nmultiplications carried out in the same manner as used\nfor the equations. In this sample computation, the sum\nof the C's multiplied by their n terms is -554.1864, and\nthe sum of the n column multiplications is -554.1840. This\ncheck (0.0024) is considered satisfactory for the number\nof significant figures carried in the computations.\nThe \"C's\" are substituted in the Correlate Equations\n(Table 24) as described on p. 147 and the sum of these\nmultiplications is then multiplied by the corresponding\na/p to produce the Corrections to the Angles, Azimuths,\nand Lengths. Since a/p is 1.00 for the angles, the\ncomputations are identical to those previously described\nand no example will be given. The computations required\nfor the azimuth correction and the correction to the\nlength 1-2 follows:\n185","Azimuth 3-4 = (-1.00)(+3.26765)(2.27) = -7\"4176\nLength 1-2 = (+0.453)(-3.95301) + (+1.350)(+4.16118)\n+ (+0.453) (+2.04747) + (+1.350)(-3.03791\nX (3.03) = +1.9792\nThe corrections to the angles and azimuth are adopted\nto fit the azimuth closures and the corrections to the\nlength are converted to feet as described on p. 147 and\nthe final coordinate computations are made using the\nadjusted values. These computations are shown in Tables\n28 and 29. The check is exact to three decimal places in\nX and Y for the route 1-3-5 and 0.001 in X and Y for the\nroute 1-3-9. The sines and cosines for the adjusted\nazimuths are not shown as the computations were made on\na desk size programmable computer. Since this computer's\nbuilt-in feature contains these trigonometrical functions\nto ten decimal places, anyone checking the Latitudes\nand Departures using eight place functions may find\nslight differences (+0.001 ft.) in the tabulated values.\n186","1,497,395.014\n1,510,036.292\n1,481,914.244\n1,519,365.76\n1,468,415.31\nUSCOMM-DC 5270\nFeet\nY\nGRID COORDINATES\n130,056.028\n123,174.338\n124,285.517\n143,190.44\n118,643.92\nFeet\nx\n+ 9329.468\n+15480.770\n+12641.278\n+13498.934\nLATITUDE\nFixed\nFixed\nFeet\n+ 5770.511\n+13134.412\n+ 4530.418\n+ 1111.179\nDEPARTURE\nFINAL COORDINATE COMPUTATIONS\nFeet\nLOG COS AZIMUTH\nTable 28\nLOG SIN AZIMUTH\nLOG DEPARTURE\nLOG LATITUDE\nLOG DISTANCE\n16110.610\n13896.068\n15520.598\n14238.887\nGRID DISTANCE\nFeet\n234 36 48.76\n198 33 08.72\n184 06 19.93\n204 32 08.98\nPLANE AZIMUTH\nSTATION\n4\n5\n3\n1\n2\n187","1,497,395.014\n1,478,343.923\n1,499,913.082\n1,487,359.589\n1,470,946.98\nUSCOMM-DC 5270\nFeet\ny\nGRID COORDINATES\n136,594.988\n124,285.517\n147,766.601\n160,296.108\n173,077.31\nFeet\nx\n+ 2518.068\n12553.493\n+ 9015.666\n7396.944\nLATITUDE\nFeet\nFixed\nFrom Table 28\n+\n+12309.471\n+11171.612\n+12529.507\n+12781.201\nFINAL COORDINATE COMPUTATIONS\nDEPARTURE\nFeet\nLOG COS AZIMUTH\nLOG SIN AZIMUTH\nLOG DEPARTURE\nTable 29\nLOG LATITUDE\nLOG DISTANCE\n12564.384\n16804.616\n15436.022\n14767.325\nGRID DISTANCE\nFeet\n258 26 19.96\n318 20 00.44\n305 44 13.49\n300 03 34.28\nPLANE AZIMUTH\nSTATION\n3\n6\n7\n8\n9\n188","VARIATION OF COORDINATES\nTo demonstrate the Variation of Coordinates method\nof adjustment, a simulated triangulation net along the\nIndiana-Ohio boundary consisting of a quadrilateral with\none measured length and one observed azimuth will serve\nas the example. Thus all types of observation equations\nused in this adjustment method will be employed. Also,\ndirections rather than angles will be used. The com-\nputations will be made on the Indiana east zone transverse\nMercator grid system.\nMany of the computational procedures are identical\nto those described in the two adjustment examples pre-\nviously presented and no further comments will be made\nexcept in those instances where modifications or variations\nare such that additional explanations seem necessary.\nA sketch of the survey is shown in Figure 5. Stations\n2\n10\n4\n3\nFigure 5\n189","1 and 2 are considered fixed, 3 and 4 are new stations,\nand it is assumed the distance and azimuth between 3 and\n4 are observed quantities.\nThe basic equations are given below:\nDirection dai-k = N + + 21 + 206265 N + (AYdX - AYdX + AXAXK)\nor\nN + Z + 2 006265\n(Cos adX - Sin adY - Cos adX + Sin aDY\nLength dsi-k = (AXdX + AYdY AXdXX - AYdY )\nor\nN + Sin adX + Cos adY; - Sin adX - dY\n16x\nAzimuth\nAYdX\n+\n-\nor\nN + 206265 206265 (Cos adX - Sin adY - Cos adX k + Sin ady\n\"N\" equals the computed quantity minus the observed value;\n\"Z1\" is the orientation or correlation unknown for the\ndirections at the \"i\" point and is assigned a coefficient\nof 1.00; \"s\" is the length of the line; \"AX\" = X-XX; -\n\"AY\" = Y1 - Yk; \"Sin a\" and \"Cos a\" are the sine and cosine,\nrespectively, of the azimuth of the line \"1\" to \"k\" and\n\"ax\", \"dy\", \"dxk\", \"ayk\" are the corrections to the assumed\ncoordinates. Z, dX, dYi, dxk, and dY will be determined in the\nsolution of the equations.\n190","Observation equations are prepared for each observed\ndirection, observed azimuth, and measured length. These\nequations resolve into three normal equations for each new\noccupied point and two normal equations for new intersected\npoints. The solution of the normal equations produces cor-\nrections to the assumed coordinates (dxdY) for each new\npoint and the orientation or correlation correction (Z) for\nthe new occupied points. The solution (C's) of the Z equations\nare the corrections to the initial computed azimuths used to\ndetermine the \"N\" terms and also are responsible for the di-\nrection V's at each new point summing to zero (Table 34). In any\nLeast Squares adjustment using directions, whether condition\nequations or observation equations are employed, the adjustment\ncorrections (v's) at each new point must sum to zero. The\nresults, of course, will be the same regardless of the adjust-\nment method. In a weighted adjustment, the sum of the pv's\nequal zero, but in the example given here the weights are the\nsame for all directions and this condition cannot be shown.\nThe \"Z\" equations may be eliminated prior to preparing\nthe \"Table for Formation of Normal Equations\" (Table 34), but\nthe procedure is rather complicated if corrections to directions\nare desired. Equations can be prepared in which corrections to\nangles, rather than directions, are determined. This method\nreduces the number of equations to be solved by one at each\npoint, however, additional arithmetic exercises are required\nand little time is saved and it is not generally recommended.\nExamples of this adjustment method are given in Special Pub-\nlication No. 193 (see references).\n191","The computation of the Weights (p) follows:\np = 12\nA\nkp**\nA\n1\"5\n0.4444\nDirections\n1.00\n3\".0\nAzimuth\n0.1111\n0.25\n1.1891\n2.68\nLength\n0.29 ft.*\n\"A\" is the Standard Error or estimated accuracies of the\nobservations.\nFor observation equations it is not necessary that the\nweights be based on precisions in the same measure,\nsuch as seconds, which is required when condition\nequations are employed. For the measured length\n(p.198) used in this survey, the \"A\" value was\ndetermined using 0.1 ft. + 1:100,000 as the criterion.\nk = 2.25\nComputation of the Second Term Corrections is shown\nin Table 30. The values at each end of the lines are given\nwith the correct sign to reduce the opportunities for\nmistakes occurring when applying these quantities to the\nobserved directions to obtain the Grid values. In addition,\nthe correct signs will simplify the explanation of equating\nthe grid corrections to the angles of a triangle with the\nspherical excess of the figure. It is for this purpose\nthat the values are given to hundredths of seconds. The\napproximate nature of the formulae would preclude using\nanything more than tenths of seconds in practice, however.\n192","The signs of the corrections are easily obtained\nfrom the sketch. At each point whenever the straight\nlines representing the grid directions are to the right\n(clockwise) of the curved lines representing the\ngeodetic values, the signs are plus; to the left (counter-\nclockwise) the signs are minus. .\n2.36AXAY\nFrom To\nAX\nAY\n2.414\n0.128\n-0.73\n1\n2\n2.267\n0.268\n+1.43\n1\n3\n4\n2.444\n0.148\n1\n+0.85\n2\n1\n+0.73\n2.439\n0.396\n+2.28\n2\n3\n4\n2.616\n0.276\n2\n+1.70\n3\n-1.43\n1\n3\n-2.28\n2\n4\n2.469\n3\n0.120\n-0.70\n4\n1\n-0.85\n4\n2\n-1.70\n4\n3\n+0.70\nTable 30\nTo show the relationship of the second term corrections\nand the spherical excess of a figure, the triangle involving\nstations 1-2-3 will serve as an example. The numbering\nsequence is also identical to the numerical Identifiers\nprinted on the Computation of Plane Triangles form (Figure 38).\n193","In addition, these Identifiers are used to determine the\norder in which the directions are utilized to obtain the\nangles and this order relates to the signs of both grid\nand adjustment corrections as will be shown.\nGrid\nCorr'n\nGrid Angle\nA\nStation\nObserved Angle\n99° 48' 12.2\n+2.1\n14\"3\n1\n1\n40.9\n-1.6\n2\n2\n33\n01\n39.3\n47 10 12.9\n-0.9\n12.0\n3\n3\n180° 00' 06\".0\n-0.4\n05.6\nSum\nA = Identifiers\nThe form shown in Table 38 is designed for triangles\nprepared in a clockwise direction, the standard procedure\nin all geodetic organizations and recommended for general\nuse.\nUsing the Identifiers, the angles are always computed\nin the same manner as follows:\nAngle at station 1 = direction 1-3 minus direction 1-2\nAngle at station 2 = direction 2-1 minus direction 2-3\nAngle at station 3 = direction 3-2 minus direction 3-1\nIn this particular example, the Identifiers and the\nstation numbers are the same and as a result it is not\nnecessary to substitute the actual angle identifications\nfor the Identifiers in the grid correction computations\nthat follows:\n194","Adopted\nGrid Correction\nDesignation\nAngle\n+ 1.43 + 0.73 = + 2.16\n+2.1\n(1-3) minus (1-2)\n1\n-1.6\n(2-1) minus (2-3)\n+ 0.73 - 2.28 = 1.55\n2\n(3-2) minus (3-1) - 2.28 + 1.43 = - 0.85\n-0.9\n3\nSum 0.24\n-0.4\nSpherical Excess of Triangle = 0.23\nThe difference in the sum of the adopted corrections\nand the computed spherical excess of 0.2 is of no consequence.\nHowever, the spherical excesses of all triangles could be\ncomputed, adopted to fit the area of the figure, and then\nthe grid corrections could be adopted to fit these values;\nbut this effort would seldom if ever be justified for\nmost local surveys.\nA second example, for the triangle 4-1-2, is given below:\nAdopted\nGrid Correction\nDesignation\nIdentifiers\n-0\".8\n- 1.70 + 0.85 = -0.85\n(4-2) minus (4-1)\n1\n+1.6\n+ 0.85 + 0.73 = +1.58\n(1-4) minus (1-2)\n2\n(2-1) minus (2-4) + 0.73 - 1.70 = -0.97\n-1.0\n3\nSum = -0.24\n-0.2\nSpherical Excess of Triangle = 0.24\nThe lists of directions are given in Table 31. Lists\nof directions are the summation of angles referenced to\none station usually with an initial value of 0° 00' 00..0.\nDirection, in this instance, does not refer to bearings\nor azimuth. Observations made using the method of positions\n195","and the technique employed by many geodetic groups will\nproduce such lists with little or no office work required.\nThe angles are obtained by subtracting one direction from\nanother as shown below.\nThe angles are obtained using the procedures given\nin the description concerning the grid corrections. Two\nexamples follow:\nThe observed geodetic angle at station 1 between\n4\nstations 2 and/is 40° 32' 13.8; the corresponding grid\nangle is 40° 32' 14.7 minus 359° 59' 59.3 or 40° 32' 15\".4.\nSimilarly at station 1 between stations 4 and 3 the geodetic\nangle is 99° 481 12.2 minus 40° 32' 13.8 or 59° 15' 58.4,\nand the gria angle is 99° 48' 13.6 minus 40° 32' 14.7 or\n59° 15' 58.9.\nThe angle designations are (1-4) minus (1-2) and (1-3)\nminus (1-4). Since (1-2) is a fixed line, the designation\nis omitted when applying adjustment corrections, but would\nbe used in obtaining the grid corrections.\nLIST OF DIRECTIONS\nStation = 1\nGrid\nGrid\nTo\nObs.\nGeod.\nDir.\nCorr'n\nDir.\n59\".3\n0°\n00.0\n-0.7\n2\n00'\n13.8\n14.7\n40\n+0.9\n4\n32\n13.6\n48\n+1.4\n12.2\n3 99\n196","LIST OF DIRECTIONS (Continued)\nStation = 2\nGrid\nGrid\nDir.\nGeod.\nDir.\nCorr'n\nTo\nObs.\n00\".O\n01.7\n4\n0°\n+1.7\n00'\n57.6\n48\n49\n55.3\n+2.3\n3\n36.9\n81\n36.2\n+0.7\n51\n1\nStation = 3\nGrid\nGrid\nDir.\nGeod.\nDir.\nCorr'n\nTo\nObs.\n58.6\n0°\n00.0\n-1.4\n00'\n1\n10.6\n47\n10\n12.9\n-2.3\n2\n02.8\n-0.7\n02.1\n4\n81\n52\nStation = 4\nGrid\nGrid\nDir.\nGeod.\nDir.\nCorr'n\nTo\nObs.\n00\".7\n00.0\n0°\n+0.7\n00'\n3\n02.8\n38\n-0.9\n01.9\n52\n1\n13.6\n96\n28\n15.3\n-1.7\n2\nTable 31\nTabulated below are the fixed plane coordinates for\nstations 1 and 2 and the preliminary values for new stations\n4. (Indiana East Zone)\n3 and\n(2) X = 758,634.25\n(1) X = 724,211.02\nY = 845,965.49\nY = 833,160.26\n(4) X = 764,640.90\n(3) X = 729,230.10\nY = 818,338.03\nY = 806,329.05\n197","The reductions of the observed azimuth and measured\nlength to grid values follow:\nGrid Azimuth\nA\nObserved Azimuth\n-Aa\nFrom\nTo\n251° 15' 55.14\n14.3\n251° 471 10\"14\n-31'\n-0.7\n4\n3\nA = second term correction 3-4. Aa is computed from X' and Y\nfor station 3 using data given in projection tables. For the\nLambert projection, the 0 angle is required and this may be\ncomputed by using the longitude scaled from a map, providing\nthis can be done within 2\"-3\" of longitude or 200 to 300 feet\nand the data given in the projection tables. The precise\nX'\nGrid azi-\ncomputation may be made as follows: Tan O =\nRb - Y\nmuth is given to hundredths of seconds but in practice should\nnot be carried to more than tenths unless precise computation\nof Da and \"A\" is made.\nGrid Length\nB\nMeasured Length\nFrom\nTo\n37,387.405\n0.9999741\n37388.373\n4\n3\nB = Combined factor based on a mean elevation for stations\nft.\n3 and 4 of 1100/(0.9999474) and the mean X' for the stations\nof 246936 (1.0000267).\nCOMPUTATION OF ELEMENTS FOR FORMATION OF EQUATIONS\ns superscript(2) X 10- 8\nAzimuth\nTan or Cot*\nAY\nAX\nFrom To\n249°35'42\"391\nFixed\n0.37199385*\n-12805.23\n-34423.23\n1\n2\n7.4510\n349 24 16.671\n0.18706126\n- 5019.08\n+26831.21\n1\n3\n18.5427\n-40429.88 +14822.23 0.36661573* 290 08\n01.431\n4\nl\n11.059 24.3565\n+29404.15 +39636.44 0.74184639\n2\n3\n347 44 01.965 7.9936\n6006.65 +27627.46 0.21741593\n2 4\n13.9814\n-35410.80 -12008.98 0.33913326* 251 15\n59.375\n3 4\nTable 32\n198","TRANSVERSE MERCATOR PROJECTION\nIndiana\nBoth zones\nsa = Mx' - e\nM\nAM\ny\n5541\n751\n0.007\n0\n6292\n757\n0.007\n100,000\n763\n7049\n0.007\n200,000\n768\n7812\n0.007\n300,000\n8580\n774\n0.007\n400,000\n780\n9354\n0.007\n500,000\n786\n0.008\n0134\n600,000\n793\n0.008\n0920\n700,000\n798\n0.008\n1713\n800,000\n805\n0.008\n2511\n900,000\n812\n3316\n0.008\n1,000,000\n818\n4128\n0.008\n1,100,000\n825\n0.008\n4946\n1,200,000\n831\n0.008\n5771\n1,300,000\n839\n0.008 6602\n1,400,000\n846\n0.008 7441\n1,500,000\n0.008 8287\n1,600,000\ne\nX'\n400,000\n300,000\n200,000\n100,000\ny\n0.5\n0.2\n0.0\n0.0\nO\n0.6\n0.2\n0.0\n500,000\n0.0\n0.6\n0.2\n0.1\n0.0\n1,000,000\n0.7\n0.3\n0.1\n0.0\n1,500,000\n199","TRANSVERSE MERCATOR PROJECTION\nINDIANA\nBoth zones\nx\nScale in\nScale\nScale\nX'\nScale in\n(feet)\nunits of\nexpressed\n(feet)\nunits of\nexpressed\n7th place\n7th place\nas a\n8.8 a\nof logs\nratio\nof logs\nratio\n-144.8\n0.9999667\n175,000\n+ 7.3\n1.0000017\n0\n+ 16.1\n-144.7\n1.0000037\n0.9999667\n180,000\n5,000\n1.0000058\n-144.3\n0.9999668\n185,000\n+ 25.1\n10,000\n+ 34.4\n1.0000079\n-143.7\n0.9999669\n190,000\n15,000\n+ 44.0\n-142.5\n0.9999671\n195,000\n1.0000101\n20,000\n1.0000124\n-141.7\n0.9999674\n25,000\n200,000\n+ 53.8\n1.0000147\n-140.3\n0.9999677\n205,000\n+ 63.9\n30,000\n+ 74.2\n1.0000171\n35,000\n-138.7\n0.9999681\n210,000\n+ 84.7\n0.9999685\n215,000\n1.0000195\n40,000\n-136.9\n1.0000220\n45,000\n-134.8\n0.9999690\n220,000\n+ 95.5\n+106.6\n-132.4\n0.9999695\n225,000\n1.0000245\n50,000\n+117.9\n1.0000271\n55,000\n-129.8\n0.9999701\n230,000\n+129.4\n1.0000298\n60,000\n-126.9\n0.9999708\n235,000\n+141.2\n65,000\n1.0000325\n-123,8\n0.9999715\n240,000\n245,000\n+153.2\n1.0000353\n70,000\n-120.5\n0.9999723\n+165.5\n-116.9\n1.0000331\n75,000\n0.9999731\n250,000\n+178.0\n1.0000410\n80,000\n-113.0\n0.9999740\n255,000\n1.0000439\n0.9999749\n85,000\n-108.9\n260,000\n+190.8\n265,000\n1.0000469\n-104.6\n+203.9\n90,000\n0.9999759\n+217.2\n1.0000500\n95,000\n-100.0\n0.9999770\n270,000\n+230.7\n1.0000531\n0.9999781\n275,000\n100,000\n- 95.2\n+244.5\n1.0000563\n105,000\n280,000\n- 90.1\n0.9999793\ncan 84.7\n285,000\n+258.5\n1.0000595\n110,000\n0.9999805\n+272.8\n1.0000628\n115,000\n- 79.1\n0.9999818\n290,000\n1.0000662\n+287.3\n120,000\none 73.3\n0.9999831\n295,000\n1.0000696\n- 67.2\n0.9999845\n300,000\n+302.1\n125,000\n130,000\nand 60.9\n0.9999860\n305,000\n+317.1\n1.0000730\n1.0000765\n- 54.3\n135,000\n0.9999875\n310,000\n+332.3\n140,000\n- 47.5\n+347.8\n1.0000801\n0.9999891\n315,000\n+363.6\n145,000\nSIND 40.4\n1.0000837\n0.9999907\n320,000\n+379.6\n1.0000874\n0.9999924\n325,000\n150,000\n- 33.1\n1.0000912\n0.9999941\n330,000\n+395.9\n155,000\nwe 25.5\n+412.4\n1.0000950\n335,000\n160,000\n- 17.7\n0.9999959\n+429.2\n1.0000988\n340,000\n165,000\n- 9.6\n0.9999978\n446.2\n1.0001027\n345,000\n170,000\n- 1.3\n0.9999997\n+463.4\n1.0001067\n350,000\n200","The computation of the various quantities required\nfor forming the equations is given in Table 32. AX and AY\nare the differences From station minus To station; Tan A1-K\n=\nAX\nor Cot A1-k - X x = the use of the value which is less\nAY\nthan one is generally considered best especially when small\nangles are involved; the Azimuth is derived from either the\ntangent or cotangent with the quadrant determined as shown\nbelow; s2 X 10-8 is AX2 + AY2 and pointing off twelve decimal\nplaces to the left.\n180°\nAX+\nAX-\nAY-\nAY-\n90°\n270°\nAX+\nAX-\nAY+\nAY+\n0°\n360\nTo obtain the \"N\" terms (Table 33), the observed\nazimuths are computed by the following formula:\nAZi-k If AZO - Do + DK' where AZO is the computed azimuth\n(from Table 32) to the initial station, or in the case\nwhere the observations are at a fixed control point, the\nazimuth to another fixed point. Do is the observed grid\ndirection corresponding to this azimuth. Dk is the observed\ngrid direction to the point for which the observed azimuth\nis required. The \"N\" term for the observation equation to\n201","the initial station is always zero since Do and Dk are\nidentical and there are no observation equations between\nfixed points. If the initial direction is 0° 00' 00.00\nthe subtraction of the Do direction is of course omitted\nwhen computing the \"N\" terms. In this regard, the lists\nof grid directions can always be referred to an initial\ndirection of 0° 00' 00.0 by subtracting the initial\ndirections, in the case of new stations, from the other\ndirections and for the fixed points, subtract the directions\nto a fixed point from the other quantities. The Computed\nAzimuth i-k is from Table 32, and N1-k is this value minus\nthe Observed Azimuth\ni-k\nAlso included in this table is the computation of the\n\"N\" terms for the observed azimuth and measured length. The\ncomputed azimuth is taken from Table 32 and the computed\nlength is determined from the formula AX2 + AY2, , using the\nAX and AY for the line from Table 32 also.\nCOMPUTATION OF N TERMS\nComputation of\nN1-K\nObserved Az i-k\nComputed Az i-k\nFrom\nTo\nFixed Azimuth\n1 2\nAz (1-2) 249° o 35'\n1\n3\n-D(1-2)-359\n59\n59.3\n+D(1-3)+99 48 13.6\n+19.980\n56.691\n349° 241 16\".671\nAz(1-3) 349 23\nAz(1-2)\n42.391\n249\n35\n1\nT\n-D(1-2)-359 59 59.3\n+D(1-4)+ 40 32 14.7\n+ 3.640\nz(1-4) 290 07 57.791\n290 08 01.431\n202","Computation of\nObserved AZi-k\nComputed Az\nNi-k\nFrom\nTo\ni-k\nFixed Azimuth\n2\n1\nAz (2-1) 69° 35'\n42\".391\n2\n3\nD(2-1) - 81 51 36.9\n+D(2-3) + 48. 49 57.6\n+ 7.968\n36° 34' 11.059\n(2-3)\n36\n34\n03.091\nAz\nAz (2-1)\n69\n42.391\n35\n4\n2\n-D(2-1)-\n81\n51\n36.9\n-\n+D(2-4)+ o 00 01.7\nAz(2-4) 347 44 07.191\n- 5.226\n347\n44\n01.965\nAz(3-1) 169 24 16.671\n16.671\n0.000\n169\n24\n3\n1\nAz(3-1) 169 24 16.671\n3\n2\n-D(3-1)-359 59 58.6\n+D(3-2) + 47 10 10.6\nAz(3-2) 216 34 28.671\n-17.612\n216\n34\n11.059\nAz(3-1) 169 24 16.671\n4\n3\n-D(3-1)-359 59 58.6\n+D(3-4)+ 81 52 02.1\n-20.796\n:(3-4) 251 16 20.171\n59.375\n251\n15\n71\n15\n59.375\n0.000\n4\n3\n71\n15\n59.375\nz(4-3)\n71\n15\n59.375\n4\n1\n-D(4-3)- o 00 00.7\n+D(4-1)+ 38 52 01.9\n+ 0.856\nAz(4-1) 110 08 00.575\n110 08 01.431\nAz(4-3) 71 15 59.375\n4\n2\n-D(4-3)- o 00 00.7\n+D(4-2)+96 28 13.6\nAz(4-2) 167 44 12.275\n-10.310\n167 44 01.965\n55.14\n+ 4.235\nObs\nAz\n251\n15\n4\n251\n15\n59.375\n3\nComputed\nN1-k(ft.)\n(ft.)\nMeasured Length\nLength,\n'i-k\n+ 4.310\n37,387.405\n4\n37,391.715\n3\nTable 33\n203","The Table for the Formation of Normal Equations (Table 34)\nis computed using the basic equations (p. .190) and the data\nin Tables 32 and 33. Table 34 is used in the same manner\nas the Correlate Equations for condition equations to form\nthe Normal Equations and the corrections to the observed\nquantities.\nTo form this table, the first step is to divide 206265\nfor the line\nby s superscript(2) /and then multiply the corresponding AX and AY to\ndetermine the coefficients for each direction observation\nequation. Only two multiplications are required for the\ndirection observation equations over both ends of a line\nsince the coefficients for dX and dX k are identical\nnumerically but opposite in sign, and this is also true\nfor dY i and dYK. The signs of the direction coefficients\nmay be obtained from the table below:\ndY\ndX\ndX\ndY\nAY1-K\nk\nk\ni-k\n+\n+\n+\n+\n-\n-\n+\n+\n+\n-\n-\n-\n+\n+\n-\n-\n-\n-\n+\n+\n+\n-\n-\nTo simplify the determination of the decimal point, s ² is\ngiven in the form of s2 X 10-8; and to compensate for\nthis decimal change, move the decimal point for 206265,\nAX and AY four places to the left. The values for\nare as follows, s superscript(2) X 10-8 from Table 32:\n20.6265\ns2 X 10-8\n204","From To\nFrom To\n2.7683\n2 - 4\n2.5804\n1 - 3\n1 - 4\n1.1124\n3 - 4\n1.4753\n0.8469\n2 - 3\nA few examples of this computation follow:\n(1) 1-3 = Since station 1 is fixed, there are no Z, dX,\nand Y1 quantities and only the \"k\" portion of\nthe equation is used\n= 0.00Z(1) + 0.00dX1 + 0.00dY1 - (2.7683)(2.6831)3\n(2.7683)(0.5019)3 or - 7.428(dx3) - 1.389(d3) -\nFor direction 3-1 the Z equation comes in and the\nequation is +1.00(Z3) - 7.428(dx3) - 1.389(dY3).\nThe \"N\" terms from Table 33 are tabulated with their\nrespective equation.\n(2) 3-4 = + 1.00(Z3) - (1.4753) (1.2009)3 + (1.4753)(3.5411)3\n+ (1.4753)(1.2009) - (1.4753) (3.5411)4\nor + 1.00(Z3) - 1.772(ax3) + 5.224(dY3) + 1.772(dX4)\n-5.224(ay4)\n4-3 = = - 1.772(dx3) - + 5.224(dY3) + 1.00 (24) + 1.772(dX4)\n5.224(dY)\nCoefficients for the azimuth observation equation are\nidentical to the direction equation coefficients for the\nline 3-4. The coefficients for the length observation\nequation are computed using the basic formula (p.190) and\nthe data in Table 32.\n205","0.000 - 7.817 +1.041 +1.04\nAdopt\n+19.980 +11.163 -2.233 -2.23\n-1.649 -4.497 + 3.640 - 2.506 +0.793 +0.80\n+ 7.968 + 7.101 +2.686 +2.69\n-7.129 -1.550 - 5.226 -13.905 +0.428 +0.43\n-17.612 -17.479 +0.360 +0.35\n+1.772 -5.224 -20.796 -19.796 -1.402 -1.40\n+1.00 -1.649 -4.497 + 0.856 - 4.290 +1.511 +1.51\n+1.00 -7.129 -1.550 -10.310 -17.989 -1.153 -1.16\n-1.772 +5.224 +1.00 +1.772 -5.224 0.000 +1.000 -0.358 -0.36\n+1.772 -5.224 + 4.235 + 4.235 +0.375 +0.38\nCorrections\n+0.947 +0.321 + 4.310 + 4.310 +0.623 ft.\nV's\nW\nN\ndY4\nTABLE FOR FORMATION OF NORMAL EQUATIONS\ndX4\nTable 34\n24\n-3.357 +2.490\n2 1.00 +1.00 -3.357 +2.490\n+5.224\n-7.428 -1.389\n3 l 1.00 +1.00 -7.428 -1.389\n4 1.00 +1.00 -1.772 +5.224\n-0.321\ndY3\n-1.772\n-0.947\nax3\nZz\nFrom To Wt.=p\n1.00\n1.00\n1.00\n1.00\n1.00\n1.00\n1.00\n0.25\n4 2.68\n3\n4\n3\n4\n1\n2\n3\n4\n3\n3\n4\n4\n4\nA 3\nL3\n1\n1\n2\n2","The \"N\" terms are from Table 33 and the weights as\ndetermined earlier. Note that the weight (p) is used\nrather than a/p. \"E\" is the summation of each equation\nincluding \"N\" but excluding the weights.\nThe Normal Equations are shown in Table 35 and were\ncomputed as described previously on pp.178-181. The \"n\"\nterm in computed using the \"N\" terms as though these values\nwere another equation. For example, the \"n\" for dx3 is:\n-7.428)(+19.980) + (-3.357)(+7.968) + (-7.428)(0.000)\n+ (-3.357)(-17.612) + (-1.772)(-20.796) + (-1.772)(0.000)\n+ (-1.772)(+4.235))0.25) + (-0.947)(+4.310)(2.68) = -92.0007.\nThe Forward and Back Solutions are given in Tables 36\nand 37.\nTo obtain the Corrections to the Observed Quantities\n(V's), the \"C\" values are substituted in Table 34 and each sum\nis added to the \"N\" term for the equation involved as shown\nin the following examples:\n1-3 = (-7.428)(+2.70519) + (-1.389)(+1.52565) + 19.980 = 2.233\n3-4 = (+1.00)(+23.25472) + (-1.772)(+2.70519) + (+5.224)(+1.52565)\n+ (+1.772)(-1.01139) + (-5.224)(+1.00394 - 20.796 =\n-1\".402\nL3-4 = (-0.947)(+2.70519) + (-0.321)(+1.52565) + (+0.947)(-1.01139)\n+ (+0.321)(+1.00394) + 4.310 If +0.623 ft. (Note cor-\nrection is in feet.)\n207","2.01094\n.00394\n+ 1.76243\n+ 3.38393 + 2.79490 + 4.11899 + 9.29781\n.21287\n+ 15.03067\n0.1460\n-126.8560\n- 13.4963\n- 23.2464\n+176.3645\n+218.0871\n+165.9572\n+ 30.4767\n-158.2635\n- 21.279\n- 45.092\nW\n-126.8560\n+\n+ 30.4767\n+218.0871\n+176.3645\n+\n-\n-\n- 21.279\n- 45.092\n+ 1.00394\n.89636\n2.81480\n.76324\n12.80267\nW\n- 37.1826\n10.2983\n+ 79.3064\n+ 73.9743\n+110.6775\n-158.5810\n48.3918\n92.0007\n-252.7638\n9.454\n- 38.408\nn\nTable 36\n-158.5810\n+ 79.3064\n+110.6775\n92.0007\n- 38.408\n- 9.454\n+\n+\n+\n-\n-\n-\n-\nn\n.11458\n.79572\n1.74133\n.02063\n+106.9300\n+ 37.0366\n+ 16.9176\n9.4556\n+ 20.0134\n61.6790\n50.4510\n6.9878\n1.8525\n- 11.271\n5.224\ndY4\n+ 20.0134\n- 61.6790\n+106.9300\n+ 16.9176\n5.224\n- 11.271\n+\n-\n+\n+\n-\n+\n-\n-\n-\n-\ndY4\n.02284\n.59067\n.26047\n+ 82.5273\n9.4684\n- 2.0514\n+ 20.0134\n+ 16.5145\n8.4605\n+116.5521\n7.006\nFORWARD SOLUTION\n+ 1.772\nNORMAL EQUATIONS\n-\ndx4\n9.4684\n+ 20.0134\n+116.5521\n+ 1.772\n- 7.006\nTable 35\ndX4\n+\n-\n-\n-\n-\n-\n+ .01973\n..08562\n+5.4288\n+2.5002\n-\n+5.224\n+3.000\n-1.772\n-1.772\nZ4\n-1.772\n+5.224\n+3.000\nZ4\n2.10833\n.11557\n+77.9378\n+63.4032\n-16.0962\n+10.3781\n+ 6.325\n-16.0962\n+77.9378\n+ 6.325\ndY3\ndY,\n-\n4.18567\n+142.3576\n+ 89.7981\n+142.3576\n+ 3.50243\n- 1.01139\n+ 1.00394\nBACK SOLUTION\n+23.25472\n2.70519\n+ 1.52565\n- 12.557\n12.557\ndX,\nC's\nTable 37\n+\n-\n+3.00\n+3.00\nZz\nZz\nax3\ndY3\nZ4\ndX4\ndY","The Final Adjusted Coordinates are obtained by\nadding the dX and dY values (C's) to their respective\npreliminary coordinates.\nYp = 806,329.05\n1.526\n806,330.576\nXF 729,232.805\nYF\n(4) x x 764,640.90\nx = 818,338.03\n818,338.03\n- 1.004\n- 1.011\n764,639.889\n818,339.034\nYF\nXF\nTo check the adjustment, the observed angles in the\ntriangles should be taken out using the lists of directions\nand tabulated as shown on Table 38. The corrections (V's)\nare then adopted to fit the triangle closures and the tri-\nangle sides computed using the Law of Sines. The angle\ncorrections are determined in the following manner as\npreviously described for the Grid corrections (pp. 193-195). @\nTriangle 4-3-1\nIdentifier\nCorr'n angle 4 = V4-1 minus V4-3 = (+1.51)-(-0.36) = +1.87\n1\nCorr'n angle 3 = V3-4 minus V3-1 = (-1.40)-(+1.04) = -2.44\n2\nCorr'n angle I = V1-3 minus V1-4 = (-2.23)-(+0.80) = -3.03\n3\nSum -3.60\nThe triangle computation is self checking as each\nside appears in two triangles. Note also that the computed\nside 3-4 = 37,388.028 is equal to the measured grid length\nplus the adjustment correction (37,387.405 + 0.623 = 37,388.028).\n209","As a final check on the adjusted coordinates, the values\nare computed using the adjusted angles and lengths from\nTable 38. This computation is also self checking and\nis shown in Table 39. The form is self explanatory. Note\nthat the computed coordinates check those obtained by applying\nthe \"C\" IS to the preliminary values (p.209). Also that the\nadjusted azimuth 3-4 251° 15' 55.52 checks exactly the\nobserved azimuth plus the adjustment correction\n(251° 15' 55.14 + 0.38 = 251° 15' 55\".52).\n210","COMPUTATION OF PLANE TRIANGLES\nForm 25g\nS2-3\nsin(2)\nS 1-3 = sin(1) X\nDEPARTMENT OF COMMERCE\nU. $ S. COAST AND GEODETIC SURVEY\n(Ed. June 1950)\nsin(3)\nIndiana\nX\n=\nState:\nPLANE\nOBSERVED ANGLE\nCORR'N\nSTATION\nANGLE\nIdentifiers\n36727.819\n2-3\n-0.68\n47\n11.32\n0.73337177\n3\n10\n12.0\n1\n48\n0.98539793\n14.3\n-2.23\n12.07\n1\n99\n2\n-2.69\n36.61\n0.54503179\n39.3\n2\n33\n01\n3\n49349.482\n-5.6\n1-3\n05.6\n27295.609\n1-2\n27295.609\n2-3\n+1.87\n0.62752177\n03.07\n4\n38\n52\n01.2\n1\n-2.44\n01.06\n0.98994224\n81\n52\n03.5\n3\n2\n0.85954487\n58.9\n55.87\n-3.03\n59\n15\n1\n3\n43059.982 +1\n-3.6\n1-3\n37388.027 +1\n03.6\n1-2\n49349.482\n2-3\n0.99363094\n28\n-0.80\n12.10\n4\n96\n12.9\n1\n0.56923863\n-1.76\n49.74\n34\n41\n51.5\n3\n2\n58.16\n+2.26\n0.75279212\n48\n49\n55.9\n2\n3\n28271.696\n-0.3\n1-3\n37388.028\n00.3\n1-2\n36727.819\n2-3\n0.84435138\n-2.67\n09.03\n36\n11.7\n4\n57\n1\n0.64995002\n+0.80\n16.20\n15.4\n40\n32\n1\n2\n0.98992420\n-0.43\n34.77\n81\n51\n35.2\n2\n3\n28271.697 -1\n-2.3\n1-3\n43059.984 -1\n02.3\n1-2\nM 2755\nTable 38\n211","- 27626.456\n818,339.034\n- 39634.914\n806,330.576\n845,965.49\n845,965.49\nM-2756-2\n05.78\n58.16\n07.62\n07.62\n05.78\n42.39\n05.78\n36.61\n00.00\n00.00\n+0.97717719\n+0.80314752\n44\n44\n34\n49\n34\n34\n35\n01\n00\n00\n8 cos a,\n-s cos aq\nYs\n180\nY1\n180\nYs\nY1\n36\n48\n347\n167\n69\n-33\n36\n216\nS = 28271.696\n8 = 49349.482\ncos a2\ncos a2\n2\n4\n3\n2\n3\n1\n3\n1\n6005.638\n764,639.888\n+1\n- 29401.445\n729,232.805\nto 2\nto 3\nto 1\nto 2\nto 1\nto 3\n&\n-0.21242583\n758,634.25\n&\n+0.59578022\n758,634.25\nPLANE COORDINATE COMPUTATION, TRIANGULATION\n(For calculating machine computation)\nsin a2\n-8 sin a2\nsin a\n3\n3\n1\n3\n3\n1\nsin a2\nX3\nX1\nXs\nX1\n3d/\n3d/\na,\na's\na2\na\na's\na\n+ 12008.458\n818,339.034\nTable 39\n806,330.576\n- 26829.684\n806,330.576\n833,160.26\n12.10\n05.78\n49.74\n55.52\n55.52\n42.39\n12.07\n54.46\n54.46\n11.32\n00.00\n00.00\n-0.32118457\n+0.98293041\n28\n34\n41\n15\n15\n00\n23\n23\n10\n35\n48\n00\ncos a\n-s cos a1\n180\n180\nYe\nY1\nY2\nY1\n+ 34\n96\n216\n251\n71\n249\n+ 99\n349\n169\n47\n37388.028\n27295.609\ncos a\ncos a1\nFirst Angle of Triangle\nFirst Angle of Triangle\n3\n4\n3\n1\n8 =\n=\n2\n1\n2\nS\n1\n+ 35407.084\n764,639.889\nto 2\nto 3\n729,232.805\nto 1\nto 2\n+ 5021.785\n729,232.805\nto 3\nto 1\n&\n&\n724,211.02\nsin a1 -0.94701662\nU. S. COAST AND GEODETIC SURVEY\n-0.18397775\nDEPARTMENT OF COMMERCE\n(Ed. June 1950)\nForm 26g\n-8 sin a,\n8 sin a\n2\n2\n1\n2\n2\n1\nsin a1\nX2\nX1\nX2\nX1\n2d L\n2d L\na1\nai\na1\na'i\na\na","THEODOLITES\nBibliography\nBaker, L. S. , Specifications for Horizontal Control Marks,\nESSA Technical Memorandum C&GSTM-4*\nDracup, Joseph F., , Suggested Specifications for Local\nHorizontal Control Surveys, ACSM Control Surveys\nDivision Technical Monograph No. CS - 1\nGossett, F. R., , Manual of Geodetic Triangulation, USC&GS\nSpecial Publication No. 247**\nHoskinson, Albert J. and Duerksen, J. A., Manual of\nGeodetic Astronomy, USC&GS Special Publication No. 237**\nMitchell, H. C., , Definition of Terms Used in Geodetic and\nOther Surveys, USC&GS Special Publication No. 242**\nPoling, Arthur A. Jr., , Astronomical Azimuths for Local\nControl, Surveying and Mapping Vol. XXVII, No. 4,\nalso available from NGS\nAvailable through the Superintendent of Documents,\nGovernment Printing Office, Washington, D. C. 20402\n213","ELECTRONIC DISTANCE MEASURING INSTRUMENTS\nBibliography\nKaro, H. Arnold, Adm. (Ret.), Use of the Geodimeter and\nTellurometer in Geodetic Measurements, presented at\nthe Second United Stations Regional Cartographic\nConference for Asia and the Far East, Tokyo, Japan\n(October 1958) *\nMeade, Buford K. , Comparison of Electronic Distance\nMeasuring Devices, presented at the Conference on\nSurveying and Mapping, Auburn University, Auburn,\nAlabama (November 1967\nSaastamoinen, J. J. (Editor), Surveyors Guide to Electro-\nMagnetic Distance Measurement, Canadian Institute of\nSurveying (University of Toronto Press)\nShelton, Clarence A. , Season's Report - Geodimeter Measurements,\nJuly to December 1953*\nTomlinson, Raymond W., Operational and Field Problems of\nElectronic Distance Measuring Equipment, presented\nACSM, Washington, D. C. (March 1969) and Texas\nSurveyors Association Short Course, San Antonio,\nTexas (October 1970)*\nTomlinson, Raymond W. Operations and Performance of New\nDistance Measuring Instruments, presented at ASCE,\nWashington, D. C. (July 1969) and published in ASCE\nJournal of the Surveying and Mapping Division (Sept. 1970)*\nTomlinson, Raymond W. and Burger, Thomas C., Electronic\nDistance Measuring Instruments, ACSM Control Surveys\nDivision Technical Monograph, No. CS-2 (March 1971)*\nTomlinson, Raymond W., Short Range Electronic Distance\nMeasuring Instruments, presented at ACSM, San Francisco,\nCalifornia (September 1971)*\nTomlinson, Raymond W., Electronic Distance Measuring Instru-\nments for Surveying, presented at New Hampshire Society\nof Professional Engineers, Bedford, New Hampshire\n(October 1971) *\nVarious publications from the AGA Corporation; Carl Zeiss\n(U. S. Distribution, Keuffel and Esser Company); Cubic\nCorporation; Hewlett-Packard; Jena Optics; Kern Instru-\nments, Inc. Laser Systems and Electronics, Inc.\nTellurometer, Inc. ; and Wild-Heerbrugg Company.\n* Also available from NGS.\n214","COMPUTATIONS AND ADJUSTMENTS\nBibliography\nSines, Cosines, and Tangents, 0° -6 o , for use\nin computing Lambert plane coordinates, USC&GS Special\nPublication No. 246**\nState Plane Coordinate Projection Tables\n(available for all states except Alaska. Computations\nfor Alaska are made using the 2 1/2 -minute intersection\ntables, USC&GS Publication 65-1, Part 49 for zone 1,\nPart 50 for zones 2-9 and Part 51 for zone 10**)\nUSC&GS Special Publications**. Tables are also\navailable for most U. S. Possessions.\nUniversal Transverse Mercator Grid Tables for\nLatitudes 0° -80° : Clarke 1866 Spheroid (Meters):\nVolume 1: Transformation of Coordinates from Geo-\ngraphic to Grid. Department of the Army Technical\nManual TM5-241-4/1**\nUniversal Transverse Mercator Grid Tables for\nLatitudes 0° -80° , Clarke 1866 Spheroid (Meters):\nVolume II: Transformation of Coordinates from Grid\nto Geographic. Department of the Army Technical\nManual TM5-241-4/2**\nAdams, O. S. and Claire, C. N., Manual of Plane Coordinate\nComputation, USC&GS Special Publication No. 193**\nClaire, C. N., State Plane Coordinates by Automatic Data\nProcessing, ESSA (C&GS) Publication 62-4**\nKaufman, H. P. , The ABC of Triangulation Adjustment, USC&GS\nPublication G-45. Available from NGS.\nMeade, B. K., The Practical Use of the Oregon State Plane\nCoordinate System, Proceedings of the 1964 Surveying\nand Mapping Conference, Oregon State University,\nCorvallis, Oregon, also available from NGS.\nMitchell, H. C. and Simmons, L. G., The State Coordinate\nSystems (A Manual for Surveyors), USC&GS Special\nPublication No. 235**\nPoling, A. C., , Elevations from Zenith Distances (Vertical\nAngles), USC&GS Publication G-56, available from NGS.\n215","Computations and Adjustments (continued)\nSimmons, L. G., , Geodetic and Grid Angles-State Coordinate\nSystems, ESSA Technical Report C&GS 36**\nThomas, P. D., Conformal Projections in Geodesy and\nCartography, USC&GS Special Publication No. 251**\nAvailable through the Superintendent of Documents,\nGovernment Printing Office, Washington, D. C. . 20402\nNOTE: On October 3, 1970, a governmental reorganization\ncreated the National Oceanic and Atmospheric\nAdministration, NOAA, replacing the Environmental\nScience Services Administration, ESSA. The Coast\nand Geodetic Survey, which was a component of ESSA,\nhas been renamed the National Ocean Survey (NOS) in\nNOAA. A major component of the NOS is the Office\nof the National Geodetic Survey (NGS).\n216\nGPO 930-611"]}