l\n(11:109)\nThe equivalent results for spherical waves are (Tatarski, 1967)\n(11:110)\n=\nBA()01321\n(11:111)\nand with Cn uniform along Z\nV(XL)>>\n(11:112)","References\n11-21\nThe physical interpretation and applications of (11:106) - (11:112) are extensively considered in later\nchapters. (see especially Chapter 25). We attempt in this section to indicate how they are obtained and\nprovide a convenient summary of the results. Extensive details in the derivation of each equation are found\nin the cited references.\n11.3.3 Restrictions on the Solution\nThe solutions for the covariance function, BA(p), and the log-amplitude variance are known to be\nincorrect for optical propagation near the ground. In fact, in the case of 02 > 0.3, the results appear to\nbreak down for both acoustic (Mandics, 1971) and optical beams (Ochs and Lawrence, 1969). There is\nevidence that the diffraction-theory results (Chapter 25) do not apply for such strong-scintillation (large\n02) conditions. This is reflected in the mathematics by the non-convergence of the perturbation series for\nE1 when the strength of turbulence, Cn, or the path length L is sufficiently large. To date much work has\nbeen done to attempt an understanding of this problem with very little accomplishments.\n11.4 References\nBatchelor, G. K., 1953: The Theory of Homogeneous Turbulence. Cambridge University Press, London.\nBlackman, R. B. and J. W. Tukey, 1958: The Measurement of Power Spectra from the Point of View of\nCommunications Engineering. Dover Publ., Inc. New York.\nClifford, S. F. and E. H. Brown (1970), Propagation of sound in a turbulent atmosphere, J. Acoust. Soc.\nAm. 48, 1123-1127.\nClifford, S. F. and J. W. Strohbehn (1970), The theory of microwave line-of-sight propagation through a\nturbulent atmosphere, IEEE Trans. on Ant. and Prop. AP-18, 264-274.\nFried, D. L. (1967), Propagation of a spherical wave in a turbulent medium, J. Opt. Soc. Am. 57, 175-180.\nIshimaru, A. (1969), Fluctuations of a beam wave propagating through a locally homogeneous medium,\nRadio Sci. 4, 295-305.\nKolmogorov, A., 1941: in Turbulence, Classic Papers on Statistical Theory. S. K. Friedlander and L.\nTopper Editors, INTERSCIENCE, New York, 151, 1961.\nLee, R. W. and J. C. Harp (1969), Weak scattering in random media, with applications to remote probing,\nProc. IEEE 57, 375-406.\nLumley, J. L. and H. A. Panofsky, 1964: The Structure of Atmospheric Turbulence. John Wiley and Sons,\nNew York.\nLutomirski, R. F. and H. T. Yura (1971), Propagation of a finite optical beam in an inhomogeneous\nmedium, Appl. Opt. 10, 1652-1658.\nMandics, P. A. (1971), Line-of-sight acoustical probing of atmospheric turbulence, Stanford Electronics\nLabs Tech. Rept. No. 4502-1 (SEL-71-002).\nMonin, A. S. (1962), Characteristics of the scattering of sound in a turbulent atmosphere, Soviet\nPhysics-Acoustics 7, 370-373.\nOchs, G. R. and R. S. Lawrence (1969), Saturation of laser beam scintillation under conditions of strong\natmospheric turbulence, J. Opt. Soc. Am. 59, 226-227.\nPasquill, F., 1961: Atmospheric Diffusions. D. Van Nostrand, New York.","Propagation and Scattering in Random Media\n11-22\nSaleh, A. A. M. (1967), An investigation of laser wave depolarization due to atmospheric transmission,\nIEEE J. Quant. Electron, QE-3, 540-543.\nSchmeltzer, R. A. (1967), Means, variances and covariances for laser beam propagation through a random\nmedium, Quart. Appl. Math. 24, 339-354.\nStrohbehn, J. W. (1966), The feasibility of laser experiments for measuring atmospheric turbulence\nparameters, J. Geophys. Res. 71, 5793-5808.\nStrohbehn, J. W. (1968), Line-of-sight wave propagation through the turbulent atmosphere, Proc. IEEE 56,\n1301-1318.\nStrohbehn, J. W. (1970), The feasibility of laser experiments for measuring the permittivity spectrum of\nthe turbulent atmosphere, J. Geophys. Res. 75, 1067-1076.\nStrohbehn, J. W. and S. F. Clifford (1967), Polarization and angle-of-arrival fluctuations for a plane wave\npropagated through a turbulent medium, IEEE Trans. on Ant. and Prop. AP-15, 416-421.\nTatarski, V. I., 1961: Wave Propagation in a Turbulent Medium. McGraw-Hill Book Co., New York.\n(Translated by R. A. Silverman)\nTatarski, V. I., 1967: Propagation of Waves in a Turbulent Atmosphere. Nauka, Moscow, USSR. (in\nRussian)\nWiener, N., 1949: Extrapolation, Interpolation and Smoothing of Stationary Time Series. John Wiley and\nSons, New York.\nYaglom, A. M., 1962: An Introduction to the Theory of Stationary Random Functions. Prentice-Hall, Inc.\nEnglewood Cliffs, New Jersey.","List of Symbols\n11-23\nList of Symbols\nAo\nAmplitude vector of the unperturbed\nFA\nTwo-dimensional spectrum of the\namplitude fluctuations\nwave\nA1\nAmplitude fluctuation of a plane EM\nFn\nTwo-dimensional spectrum of the\nrefractivity fluctuations\nwave\nBAG\nTwo-dimensional spatial covariance\nFS\nTwo-dimensional spectrum of the\nfunction of the amplitude fluctua-\nphase fluctuations\ntions\nf(r)\nA general function of position\nBjl(2)\nThe correlation tensor of the velocity\nH\nMagnetic field\nfield\nH\nUnperturbed magnetic field\nB(r)\nCovariance function of the refractive\nH1\nPerturbed magnetic field\nindex fluctuations\ni\nRunning index of integers\nBS(p)\nTwo-dimensional spatial covariance\nfunction of the phase fluctuations\nIm\nTake the imaginary part of\nBTT\nCovariance function of the tempera-\nK=(Kx,Ky,\nThree-dimensional spatial wave\nture fluctuations\nnumber\nK2)\nBXX\nCovariance function of the x-directed\nKo\nThree-dimensional wave vector des-\nwind velocity\ncribing the turbulent spatial fre-\nC2\nRefractive index structure constant\nquencies that reinforce the incident\nC7\nTemperature structure constant\nfield in the direction of m\nC2\nWind velocity structure constant\nk\nWavenumber of the radiation\nSpeed of phase propagation of acoustic\nC\nL\nA dimension of the volume containing\nor electromagnetic waves\nrefractivity fluctuations, pathlength\nMean speed of sound in air\nCO\nL0\nOuter scale of turbulence\nDn(r)\nStructure function of the refractivity\nlo\nInner scale of turbulence\nfluctuations\nUnit vector in the scattering direction\nm\n(z,K)\nFourier-Stieltjes measure of the per-\nn\nDummy position variable\nturbed electric field\nn(r)\nIndex of refraction\nda(z,K)\nFourier-Stieltjes measure of the wave\nn1(r)\namplitude fluctuations\nRefractive index fluctuation\nP\ndo(z,K)\nFourier-Stieltjes measure of the wave\nUnperturbed acoustic wave\nphase fluctuations\nP1\nPerturbed acoustic wave field\ndv(z,K)\nFourier-Stieltjes measure of the refrac-\nP(S2)\nProbability distribution function\ntivity fluctuations\np(S2)\nProbability density function\nds\nSolid angle\nr=(x,y,z)\nA position vector\ndo\nScattering cross section\nr'=(x',y',\nA vector describing coordinates inside\nE\nTotal electric field\nz')\nthe volume containing refractivity\nE[]\nExpected value operator\nvariations\nE1\nPerturbed electric field vector\nS\nScattered wave Poynting vector\nE0\nS\nUnperturbed electric field vector\nTotal phase of a plane wave\nE(K)\nThree-dimensional energy spectrum\nSo\nUnperturbed phase of a plane wave","List of Symbols\n11-24\nIncident wave Poynting vector\nSo\nPerturbed phase of a plane wave\nS1\nThe Component of Poynting vector\nSm\nin the direction of m\nA temperature fluctuation from\nT\nbackground\nT\nAverage background tempurature\nTime\nt\nu's\nFluctuations in x-directed wind\nvelocity\nV\nVolume\nV\nGradient operator\n8\nKronecker delta\njl\nDummy position variable\nis\nScattering angle\n0\na\nWavelength\nPosition vector\nP\nThe mean density of the atmosphere\nPo\n02 ox\nThe variance of the log-amplitude\n4\nGeneral function\nThree-dimensional spectrum of the\n$ n (K)\nrefractive index fluctuations\nThe acoustic refractive index spectrum\nnn\nThree-dimensional spectrum of the\nTT\ntemperature fluctuations\nGeneral function\nAngle between m and Ao, logarithm\nX\nof the amplitude fluctuations\nSo\nEnsemble index\nRadian frequency\nw\n0\nAveraging operator","Chapter 12 REMOTE SENSING OF SEA STATE BY RADAR\nDonald E. Barrick\nElectromagnetics Division\nBattelle, Columbus Laboratories\nSeveral radar techniques have evolved over recent years which permit the straightforward\nmeasurement of certain important ocean wave parameters. At MF and HF, the ocean waveheight\nspatial spectrum can be measured directly via the first-order Bragg-scattered signal intensity; a\nvariety of experiments are briefly examined which involve monostatic ground-wave and\nionospheric radars, bistatic HF buoy-shore systems, bistatic LORAN A signal scatter systems, and\nbistatic buoy-satellite systems. The second-order contributions to HF scatter produce a\ncontinuous Doppler return which varies in position and amplitude with sea state. At UHF, it is\npossible to measure indirectly the spatial slope spectrum of the longer ocean waves via\ncross-correlation of simultaneous Bragg-effect returns at two frequencies. Finally, short-pulse\nmicrowave satellite altimeters permit a direct measurement of the significant (or rms) waveheight\nof the sea at the suborbital point via the specular point scatter mechanism. These techniques will\nbe important for (i) detailed oceanographic measurements of the characteristics of sea waves, (ii)\nroutine monitoring of sea state for maritime purposes, and (iii) deduction of wind patterns above\nthe seas for meteorological purposes.\n12.1\nDescription of the Sea Surface\nThe quantitative interpretation of radar scatter from the sea requires the use and appreciation of\ncertain properties of ocean waves. A brief review is undertaken here of the ocean-wave physics and\ncharacteristics which we will need later; also, common oceanographic nomenclature pertaining to ocean waves\nis defined and explained. A readable but detailed treatment of all aspects of ocean wave physics can be found\nin the text by Kinsman (1965); a more elementary introduction to water waves is the concise soft-cover\nbooklet by Bascom (1964).\n12.1.1 Nomenclautre\nSea State. This term as used here refers to the state of the sea, or roughness, as determined by the heights of\nthe largest waves present. Numbers have been assigned to sea states by the International Mariners'\nCodes, and these are related to wave heights in (T12.1).\nSignificant Wave Height. This term is a common maritime descriptor referring to the average of the\nheights-from crest to trough-of the 1/3 highest waves; it is denoted H1/3.\nRMS Wave (or Roughness) Height. This is a term describing root-mean-square height-above the mean surface\nlevel-used in rough surface scatter theories; it is denoted here by h. While there is no exact general\nrelationship between h and H1/3, a common approximation frequently used for wind waves is\nH1/3~2.83 h.\nLength. The length or spatial period of a single ocean wave is the distance from one crest to another; it is\ndenoted L.\nPeriod. Unless denoted otherwise, this refers to the temporal period, and is the length of time it takes two\nsuccessive crests of a single wave to pass one point. It is denoted T.\nSpatial Wavenumber. This is defined in terms of the length of an ocean wave as K = 2n/L.\nTemporal Wavenumber. This radian wavenumber is given in terms of the period by w = 2n/T.\nFetch. The fetch is the horizontal distance over which a nearly constant wind has been blowing.\nDuration. This term refers to the length of time during which a nearly constant wind has been blowing.\nWind Waves. This term refers to a system of ocean waves which is being, or has very recently been, aroused by\nwinds blowing locally above that area of the ocean. Wind waves result in a random appearing ocean\nheight profile.\nFully Developed Seas. This is an equilibrium sea state condition reached after sufficient duration and fetch at a\ngiven wind speed. The estimated duration and fetch versus wind speed required to produce fully\ndeveloped seas is provided in (T12.1)","Remote Sensing of Sea State by Radar\n12-2\nTable 12.1 Deep-Water Wind Waves And Sea State\nWIND VELOCITY (KNOTS)\n30\n40\n50\n60\n70\n4\n5\n6\n7\n8\n9\n10\n20\n7\n1\n2\n3\n4\n5\n6\n8\n9\n10\n11\nBEAUFORT WIND\nMODE\nLIGHT\nLIGHT\nGENTLE\nMODERATE\nFRESH\nSTRONG\nFRESH\nSTRONG\nWHOLE\nRATE\nSTORM\nAND DESCRIPTION\nAIR\nBREEZE\nBREEZE\nBREEZE\nBREEZE\nBREEZE\nGALE\nGALE\nGALE\nGALE\nREQUIRED FETCH\n50\n100\n200\n300\n400\n500\n600\n700\n(MILES)\nREQUIRED WIND\n35\n5\n20\n25\n30\nDURATION (HOURS)\nSIGNIFICANT WAVE HEIGHT*\n4\n1\n2\n6\n8\n10\n15\n20\n25\n30\n40\n50\n60\nWHITE CAPS\n(FEET)\nFORM\n1\n2\n3\n4\n5\n6\n7\n8\nSEA STATE\nMODE\nVERY\nVERY\nAND DESCRIPTION\nSMOOTH\nSLIGHT\nROUGH\nHIGH\nPRECIPITOUS\nRATE\nROUGH\nHIGH\nWAVE PERIOD (SECONDS)\n1\n2\n3\n4\n6\n8\n10\n12\n14\n16\n18\n20\nWAVE LENGTH (FEET)\n20\n40\n60\n80\n100\n150\n200\n300\n400\n500\n600\n800\n1000\n1400\n1800\n60\nWAVE VELOCITY (KNOTS)\n5\n10\n15\n20\n25\n30\n35\n40\n45\n50\n55\nPARTICLE VELOCITY (FEET/SECOND)\n1\n2\n3\n4\n5\n6\n8\n10\n12\n14\n30\n70\nWIND VELOCITY (KNOTS)\n40\n50\n60\n4\n5\n6\n7\n8\n9\n10\n20\n*If the fetch and duration are as great as indicated above, these waveheight and sea state\nconditions exist. If fetch and duration are greater, waveheight can be up to 10% greater.\nSwell. When wind waves move out of the area in which they were originally excited by the winds, or after\nwinds have ceased to blow, these waves change their shape and settle down to what is known as\n\"swell\". Swell appears less random and more nearly sinusoidal, of great length, and with great width\nalong the crestlines. The usual period of swell is from six to sixteen seconds. Swell, while an occasional\nphenomenon, can arise from storm areas thousands of miles distant.\nDeep-Water Waves. When the water is sufficiently deep that the effect of the bottom on the propagation\ncharacteristics of the waves can be neglected, they are called \"deep-water\" waves. Generally, if the\ndepth is greater than 1/2 the length of a given wave, the deep-water approximation is valid. Except near\nbeaches, ocean waves are deep-water waves, and we utilize this assumption throughout this chapter.\nGravity Waves. This term refers to waves in which the chief restoring force upon the perturbed water mass is\ngravity. Waves whose lengths, L, are greater than 1.73 cm (Phillips, 1966) are gravity waves. Since\ngravity waves are the essence of sea state, they are the only types of waves considered in this chapter.\nCapillary Waves. This term refers to waves in which the chief restoring force acting on the perturbed water\nmass is surface tension. Less than 1.73 cm in length, they are not important for most of the topics of\nthis chapter.\n12.1.2 Wind Wave Surface Height and Slope Distributions\nPatterns of wind waves having various lengths, heights, and directions of motion interact to form a\nrandom-appearing surface. Hence the quantitative characteristics of such a surface are best described\nstatistically. One of the statistical functions frequently occurring in the analysis of radio wave interactions\nwith the sea is the probability density function of the surface height and its spatial derivatives (or slopes).\nPhysically, the probability density function p(x) is defined such that p(x)dx is the probability that the random\nvariable lies in the interval dx between X - dx/2 and X + dx/2.","Description of the Sea Surface\n12-3\np(s)\np(5)\nGram-Charlier\nGram-Charlier\n0.40\nTotal number of\nGaussian\n0.40\nTotal number of\nGaussian\ndata points 11,786\nMeasured frequency\ndata points\n12,634\nMeasured frequency\n0.35\nMean wind\ndistribution\n0.35\nMean wind\ndistribution\nspeed (m/sec) 4.63\nSkewness 0.168\nspeed (m/sec) 7.05g\nSkewness 0.045\n0.30\nKurtosis\n0.010\n0.30\nKurtosis 0.029\nJULY RECORDS\n0.25\nNOVEMBER\n0.25\nRECORDS\n0,20\n0.20\n0,15\n0.15\n0.10\n0,10\n0.05\n0.05\no\no\n-3.5-3.0-2.5-2.0 -1.5 -1.0-0.5 o 0.5 1.0 1.5 2.0 2,5 3.0 3.5\n+3.5-3.0-2.5-2.0 -1.5 -1.0-0.5 o 0.5 1.0 1.5 2.0 2.5 30 3.5\nNormalized Height, 5/h\nNormalized Height 5/h\nFigure 12.1 Measured versus model probability density functions for sea surface height (after MacKay,\n1959).\nFor the sea surface height, 5, above the mean sea level, MacKay (1959) found from detailed analyses of\nmeasured wave records that the height probability density function is nearly Gaussian (or normal). This yields\n(12:1)\nwhere h is the rms height of the surface, i.e., where < > denotes average. The actual\ndensity for the sea height cannot be truly Gaussian for two reasons: (i) For Gaussian distributed waves, there is\nalways some finite-albeit small-probability that very large waveheights can occur, whereas for the sea, wave\nbreaking occurs when the heights and slopes exceed certain critical values. (ii) The Gaussian function is\nsymmetric, whereas the sea height is not truly symmetric about the mean. This can be seen from looking at the\nsea surface profile, which tends to have sharp pointed peaks (for 5>0), but rounded shallow troughs (for\n5 <0). Thus the sea surface profile would not look the same upside down, whereas a true Gaussian variable\nwould.\nFor the latter reason, the true height probability density function is slightly better matched by a\nGram-Charlier model than by the Gaussian, as shown in (F12.1), after MacKay (1959). The difference is very\nslight, however, and for most analytical purposes the Gaussian model is entirely adequate+ The Gaussian\nheight distribution will be assumed and used throughout this chapter.\nIf the height distribution for the sea were truly Gaussian, then the distribution of its spatial derivatives\n(i.e., the slopes) would also be Gaussian, because a linear operation on a Gaussian random variable (e.g.,\ndifferentiation) produces another Gaussian random variable. The actual slope distributions for the sea are again\nalmost-but not quite-Gaussian. We take the x-axis as horizontal and pointing in the dominant wind direction\n(i.e., along the downwind direction), and the y-axis as horizontal and pointing in the crosswind direction. Then\nCox and Munk (1954), using glitter point photography to measure the directional slopes, find that\nis (=as/dy) in the crosswind direction is symmetric, but 5x (= as/dx) is skewed toward the upwind direction,\nprobably due to wind stress. This is shown in (F12.2). Thus the departure from Gaussian is again slight, and\nwhile it could be important in applications involving radar scatterometers looking near the vertical with high\nangular resolution (Nathanson, 1971), the difference is ignored in this chapter. Also noteworthy from the\nfigures is the fact that the observed rms slope in the upwind-downwind direction is not significantly different\nfrom that in the crosswind direction. It can be shown analytically that 5x and are uncorrelated at any given\npoint on the ocean. Therefore, we take the following for the joint probability density function for the surface\nslopes:\n(12:2)\nt One case where one might desire a higher order correction to the Gaussian model accounting for the skewness is in a detailed\nanalysis of the short pulse return from a radar altimeter.","12-4\nRemote Sensing of Sea State by Radar\nCrosswind\n5°\n5°\nObserved\ndistribution\nGaussian\ndistribution\n10°\n10°\n5°\n0°\n15°\n15°\n10°\n5°\n20°\n20°\nUp/Downwind\n15%\nObserved\ndistribution\n15°\nGaussian\n20°\n20°\ndistribution\n25°\n25°\n-3\n-2\no\n-\n2\n3\nNormalized Slope , 25x/s\nor 2 5y/s\nFigure 12.2 Measured versus model probability density functions for sea surface slopes (after Cox and Munk,\n1954).\nwhere, as explained above, we take s2 S being the total\nrms slope of the ocean surface at a given point.\n12.1.3 First-Order Gravity Wave Dispersion Relationship\nThe equation for the surface height, 5, of a deep-water gravity wave is obtained from hydrodynamic\ntheory (Kinsman, 1965). Generally, the wave surface height, 5, is a function of the two orthogonal horizontal\ncoordinates x,y and of time, t, i.e., 5(x,y,t). This function-as well as the velocity potential and stream\nfunction-must satisfy Laplace's second-order differential equation; in addition, they satisfy two boundary\nconditions at the free surface: (i) the kinematic condition and (ii) the dynamic condition. While Laplace's\ndifferential equation is linear, the boundary conditions are not. Thus an exact solution is difficult to obtain.\nThe common method of solving these equations is to expand all of the functions into a perturbational\nseries. Then the nonlinear boundary conditions are ordered into several equations, each containing terms of a\nhigher order of magnitude. The ordering (or perturbational) parameter is the height of a wave divided by its\nlength; this quantity is always very small for gravity waves. The lowest-order equations to emerge are linear\nand can be solved for the first-order height, is of the surface. Second and higher-order solutions for 5 can also\nbe obtained, and will be discussed in a later section. As the height of the water wave decreases, the first-order\nsolution becomes increasingly valid because higher-order terms decrease in magnitude. Hence, the first-order\nsolution is also referred to as the small-amplitude approximation for water waves.\nThe first-order solution, is has several distinctive characteristics. It can consist of the superposition of\nan arbitrary number of sinusoids of different amplitudes, spatial lengths, and directions. But-unique to water\nwaves-each sinusoid of a given wavelength (or wavenumber) moves at a distinct phase velocity. The\nrelationship for the first-order phase velocity, V, is obtained also from the lowest-order surface boundary","Description of the Sea Surface\n12-5\nconditions. It is (for gravity waves)\nTE\n(12:3)\nwhere g is the acceleration of gravity (~9.81 m/s), and K 2n/L is the total spatial wavenumber for the wave\nof length L.\nAnother way of stating (12:3) is to relate the temporal wavenumber, w, of the wave to its spatial\nwavenumbers; this is commonly called a dispersion relationship in physics. It is\n(12:4)\nwhere we assume for generality that the wave is moving in a direction whose angle 0 with respect to the x-axis\nis given by tan-superscript(1) (Ky/Kx). The total wavenumber magnitude, K, is thus the square root of the sum of the\nsquares of the X- and y-directed spatial wavenumbers K X and Ky.\nFrom (12:4) we can obtain still another commonly seen first-order expression relating the wavelength\nto the period:\n==(2a)L\n(12:5)\nTable 12.2 Relationship Between Period, Length, And Phase Velocity\nOf Small Amplitude Gravity Waves\nPeriod, T,\nWave Length, L\nVelocity, V\nseconds\nfeet\nmeters\nknots\nmeters/second\n6\n184\n56\n18.1\n9.3\n8\n326\n100\n24.1\n12.4\n10\n512\n156\n30.2\n15.5\n12\n738\n225\n36.2\n18.6\n14\n1000\n305\n42.4\n21.8\n16\n1310\n400\n48.6\n25.0\nThus we see for first-order gravity waves a unique square root relationship between the water\nwavelength and its temporal characteristics such as its velocity and period. Table (12.2) provides a ready\nconnection between these quantities for the longer, higher waves which generally comprise sea state. It will be\nseen later that this square-root dispersion relationship forms the basis for several unique radar experiments\ninvolving scatter from sea waves.\n12.1.4 Waveheight Spectrum of Wind Waves\nThe statistical quantity developed by oceanographers to relate the height of ocean waves to their length\nis the waveheight spectrum. It will be seen later that this spectrum also appears directly in radar scatter\ntheories. The most general form for this spectrum contains two spatial wavenumbers (Kx,) and one temporal\nwavenumber (w) to describe the waveheight $(x,y,t) as a function of its three independent variables; we denote\nit as S(Kx,Ky,w).\nFor a random-like system of wind waves, we assume that the dominant wind and wave direction is in\nthe +x direction. Then we can express the surface height in a Fourier series as a sum of traveling waves:\nayio\n(12:6)\n=\nm,n=-00","Remote Sensing of Sea State by Radar\n12-6\nHere a = 2n/Lf and W = 2n/Tf, where Lf and Tf are the wavelength and period of the fundamental components\nin the expansion. The wavenumbers of each sinusoid are then Kx = am, Ky = an, w = wk. The first summation\nin (12:6) is more general, assuming no particular dispersion relationship. Because first-order water waves are\nconstrained to follow the dispersion relationship expressed by (12:4), however, it is possible to simplify this to\na double summation over two independent indices (or wavenumbers); the third is given in terms of the first\ntwo as\n(12:7)\n=\nwhere sgn(u) = +1 depending upon whether its argument u is +.\nThe first-order spatial/temporal average waveheight spectrum can now be written in terms of the\nFourier coefficients of the expansion (Barrick, 1972):\n(m'=\nn' = - n\n(12:8)\n=\n0 for other m',n',k'\n(2)\nand\n

\n(12:9)\n0 for other m', n'\nAgain, using the first-order dispersion relationships it is possible to express the more general\nS(Kx,Ky,w) in terms of the directional spatial spectrum, S(Kx,Ky):\n(12:10)\nwhere W+ is given in (12:7) and s(u) is the Dirac impulse function of argument u. The normalization here is\nsuch that the mean-square surface height is\n(12:11)\nOne can define a non-directional temporal spectrum S(w) as follows:\n(12:12)\n=\nIt turns out that oceanographers can conveniently measure s(w) directly in a number of ways. For a review of\nthese techniques, see Kinsman (1965). Many have reported detailed observations of s(w) for wind-driven\nocean waves. Others have attempted to fit empirical laws to these observations to relate the spectrum to the\nwind speed. One such set of observations is shown in (F12.3a) taken from Moskowitz (1964); these carefully\nselected spectra for deep-water waves are fully developed only at wind speeds below 30 knots, however.\nMoskowitz notes that on the open seas, the fetch and duration are rarely sufficient for winds above 30 knots\nthat the sea will reach a fully developed condition. Thus observed spectra at these higher winds will usually be\nlower than models developed for fully developed seas.\nSeveral semi-empirical models for wind-wave spectra enjoy popularity; among them are the\nNeumann-Pierson, the Pierson-Moskowitz, and the Phillips spectra (Kinsman, 1965). These differ chiefly in the\nform postulated for the lower-end cutoff. Because of its mathematical simplicity and for general lack of\ndetailed information about the cutoff (which is observed to be quite steep in the absence of swell), we shall\nhere employ the Phillips model when this function is needed for quantitative estimates. Furthermore, since\nobservations indicate that a specific directionality is difficult to justify (Phillips, 1966; Munk and Nierenberg,","12-7\nDescription of the Sea Surface\n1969), we shall assume that the model is semi-isotropic. This means that all directions in the +x half-space are\nequally favored in amplitude by the waves. This model then has the form\n+ for K III X 2 + K X 2 > g/u2\nfrom\nS(Kx,Ky) =\n(12:13)\nfor k = 1/kx2 < g/u2\n6.65x10\n1200\n1200\n40 knots\nfor\nS(f)\no for f g/u\ns(w) =\n(12:14)\n0 for w < g/u\nwhere again the energy is distributed symmetrically for tw. Figure (12.3b) shows plots of (12:14) for\ncomparison with spectra observed by Moskowitz.\nPhysically, our Phillips model implies that the wind does not affect the shape of the spectrum in the\nequilibrium region. As the wind increases it merely drives the cutoff lower, piling up more energy beneath the\nspectrum (and increasing the rms waveheight). This assumes of course that one waits until the seas are fully\ndeveloped at a given wind speed. The reason for this effect on the lower-end cutoff can be explained simply.\nThe longest (and hence fastest) waves which can be excited by the wind are those whose phase velocity, V,\nmatches the wind speed u. The length of these waves is given by the dispersion relationship (12:3):\n[(gLco)/(2)] = (g/k co) =v2 = u2. = Solving this for KCO, we obtain g/u2, the sinusoid with the smallest (or\ncutoff) wavenumber which is excited by the wind with speed u.","Remote Sensing of Sea State by Radar\n12-8\n12.1.5 RMS Height and Slope of Wind Waves\nThe rms (or significant) waveheight, as mentioned previously, is the essence of sea state. The rms slope\nof water waves, while not as directly indicative of sea state, appears frequently in scatter theories, especially\nspecular point theories for microwave frequencies. Hence it is desirable to have quantitative estimates of the\ndependence of each on wind speed for fully developed seas.\nThe mean-square waveheight, h2 can be obtained directly from (12:11) by using the Phillips spectrum\n(12:13). This gives\n(12:15)\nwhere B = 0.005 (a dimensionless constant), windspeed (m/s), and g is the acceleration gravity (9.81 m/s2).\nThe mean-square slope can be obtained in a similar manner. Since we have already assumed a\nsemi-isotropic directional pattern for the Phillips spectrum, we have <\nwhere s2 is the total slope at a point on the surface. We obtain a result for -after integration of the Phillips\nspectrum-wh depends upon the upper (as well as the lower) bound on the spectrum. If one is interested\nonly in the slope of the gravity waves, then it makes sense to take as the upper limit 0.038 m-1, the\nboundary between the gravity and capillary wave regions. We then have\n(12:16)\nOften in the specular point theories applicable at microwave and higher frequencies, the slopes of the\ncapillary waves do in fact affect the magnitude of the scatter. In this case, the mean-square slope should\ninclude these capillaries. Phillips (1966) and Miles (1962) show that for high winds, about half of the\nmean-square slope comes from the capillaries, and one needs to add a term to (12:16) for u >: 5 m/s to\naccount for viscous dissipation. This correction term is\n&\n(12:17)\nwhere B' =\nA simpler empirical relationship derived from (F4.17) of Phillips (1966) can be obtained which\nincludes the effect of capillary slopes as well as the slopes of the gravity waves. It is valid roughly for\n1 m/s < u < 15 m/s.\n(12:18)\n=\n12.2 MF/HF Radar Scatter from the Sea\nOne of the more thoroughly established radar techniques for remote sensing of sea wave characteristics\nuses frequencies in the MF and HF regions. Recent quantitative theories, confirmed by a variety of\nexperimental configurations, lend considerable credence to the concept. We review first the physical\nmechanism and theoretically predicted echo strength, and then apply these results to several monostatic and\nbistatic concepts at MF and HF. Supporting experimental data for these techniques is presented where\navailable.\n12.2.1 Predicted Magnitude and Physical Nature of Sea Echo at MF/HF\nSea echo at frequencies below VHF has been observed by radars since World War II. Crombie (1955)\nappears to have been the first to correctly deduce the physical mechanism responsible for this sea scatter.\nBased upon HF experimental observations of the backscatter Doppler signal spectrum, he noted that-in\ncontrast with a typical noiselike clutter-th sea echo always appeared at a discrete frequency shift above and\nbelow the HF carrier. These discrete Doppler shifts could not be produced by all of the ocean waves\nilluminated by the radar, since according to (12:3) waves of different lengths move at different velocities and\nhence would produce echoes at many Doppler shifts. Thus, working backwards and calculating the ocean wave","MF/HF Radar Scatter from the Sea\n12-9\nvelocity from the observed discrete Doppler shift, and then the length of the ocean wave traveling at this\nvelocity, he arrived at the following rather startling result: The only ocean wave from the entire spectrum\npresent which produces backscatter at HF has a wavelength precisely one-half the radar wavelength and is\nmoving directly toward and/or away from the radar. The observed Doppler shift of the sea return was seen to\nincrease with the square root of the carrier-frequency-rather than in direct proportion, as with a discrete\nmoving target-further confirming this explanation (following the square-root relationship between velocity\nand length of gravity waves, as given in (12.3)). Hence the experimentally deduced mechanism was seen to be\n\"Bragg scatter\", the same phenomenon responsible for scatter of X-rays in crystals and light rays from\ndiffraction gratings and holograms.\nQuantitative theoretical analyses of the scatter problem lagged these experimental deductions by\nseveral years. Peake (1959) appears to have been the first to reduce the classic statistical boundary\nperturbation theory of Rice (1951) to 0°, the normalized average scattering cross section per unit area for\na\nslightly rough surface. Barrick and Peake (1968) noted that this result, when interpreted, shows that scatter is\nproduced by the Bragg mechanism, in agreement with Crombie's deductions. Based upon a deterministic\nanalysis of backscatter from sinusoidal waves, Wait (1966) independently obtained a result which was\nexplainable via Bragg scatter.\nNo attempt was made until very recently to apply these scatter theories to the sea, which, as we have\nseen in the preceding section, has a unique waveheight spectrum and simple first-order dispersion relationship\nbetween spatial and temporal ocean wavenumbers. Barrick (1970, 1972) and Crombie (1971) both have\nobtained quantitative predictions for the scattered signal spectrum for sea echo, including the temporal\nvariation and the dispersion relationship for the ocean waveheight. The results and notation of Barrick are\nsomewhat more general and will be employed in this chapter; Crombie's solution for backscatter agrees both\nquantitatively and qualitatively with Barrick's, serving as an independent check.\nThe technique used by Barrick was initially applied by Rayleigh to scatter of acoustic waves from a\nsinusoidal surface. It was generalized by Rice to permit the analysis of the average electromagnetic signal\nintensity scattered from a randomly rough surface. Basically, one employs a Fourier series expansion for the\nsurface, as given in (12:6), and then expands the three components of the electromagnetic field above the\nsurface into the same type of series with the same wavenumbers (am, an wk), but with unknown coefficients.\nThese coefficients are then determined by enforcing the boundary conditions at the surface. The fields at the\nboundary are expanded in a perturbational manner, permitting an ordering of the terms and a straight-forward\nsolution for the unknown field coefficients. Mathematical details are found in Rice (1951), Peake (1959), and\nBarrick (1970, 1972).\nThis boundary perturbation approach requires the assumption of the following limitations in order to\nbe mathematically valid: (i) the height of the surface must be, small in terms of the radio wavelength, (ii)\nsurface slopes must be small compared to unity, and (iii) the impedance of the surface medium must be small\nin terms of the free space wave impedance. These conditions are all satisfied by the sea below mid-VHF.\nThe solutions obtained from the Rice perturbation technique possess some similarity to those obtained\nearlier by Davies (1954) for a slightly rough surface using a physical optics technique. The perturbation results\nare superior, however, for two reasons: (i) they contain polarization dependence and correctly predict\nnear-grazing scatter for vertical polarization, whereas physical optics does not, and (ii) they are mathematically\nvalid in the low-frequency limit (as wavelength approaches infinity), whereas the physical optics\napproximation will eventually fail its inherent requirement that surface radii of curvature be much larger than\nwavelength.\nBefore giving the solutions for the scattering coefficients of the sea, we first review the radar range\nequations for average received power and its spectral density scattered from a patch of sea of area dS:\ndPR(w)\no(w) W/rad/s\nPTGTGR12\nF4FR dS X\n(12:19)\n(4n)3\nRRR2\ndPR\nW,","Remote Sensing of Sea State by Radar\n12-10\nwhere PT is the transmitted power, RR and RT are the ranges from the scattering patch ds to the receiver and\ntransmitter respectively, and A is the radar wavelength. The quantities FT and FR are the Norton attenuation\nfactors between the patch and the transmitter and receiver for TM propagation near the surface; they account\nfor any propagation losses greater than the normal free-space (1/R2) spreading losses, and hence approach\nunity for a perfectly conducting flat earth. One must be cautious in defining the antenna gains GT and GR in\nthe direction of the scattering patch. For ground-wave or line-of-sight propagation to and from the patch, GT\nand GR must be the equivalent free-space gains of the antenna; that gain is less than its measured gain in the\npresence of the conducting ground by 6 dB. For example, a vertical quarter-wave monopole fed against the\nground would have an equivalent free-space gain for use in (12:19) of -0.85 dB rather than +5.15 dB. For\nover-the-horizon ionospheric propagation to the patch, however, one employs the normal gains of the antenna\nmeasured in the presence of the ground (e.g., +5.15 dB for the quarter-wave monopole)t.\nThe actual average scattering cross section for the patch of sea within the radar resolution cell of area\ndS(m²) is then o°dS(m²). Hence, oo is the average scattering cross section of the sea per unit area. Its\ncounterpart in the equation for received power spectral density is o(w), the average scattering cross section per\nunit area per rad/s bandwidth. The normalization used here is such that\nReferring to (F12.4) which defines the incidence and scattering angles at the sea surface patch ds, we\ncan write the following expressions for o(w) and o0 for vertically incident and vertically scattered polarization\n(Barrick, 1972):\nZ\nor\n^\nIncidence\nOs\nA\ndirection\nS or V\n40\nS\nor h\n^\nScattering\n.\n0\ndirection\n0\ni\nor\nV\nX\n(x,y)\nIlluminated\nrough surface area\nL\n-y\nFigure 12.4 Local geometry near scattering patch.\nt Any ionospheric attenuation losses could in this case be absorbed in a factor similar to FFR.","MF/HF Radar Scatter from the Sea\n12-11\nwv(w)\n= W[ko(sin is sin Oi), ii), ko ko sin sin Os sin 145, w - wol 4s],\n(12:20)\nwhere ko = 2n/A is the radio wavenumber, wo is the radiant carrier frequency, and W(Kx,Ky,w), W(Kx,Ky) are\nthe first-order spatial/temporal and spatial waveheight spectra of the sea, respectively; they are related to the\nspectra defined in (12.1.4) byt\n(12:21)\nEquation (12:20) was derived assuming that the scattering patch is perfectly conducting. For the sea at\nMF/HF, this approximation is quite valid. The only seemingly confusing issue is the fact that when either the\nincident or scatter polarization state is vertical, and when the propagation angle for that polarization state\napproaches grazing, the scattered power remains finite. For a finitely conducting surface medium, the result\nanalogous to (12:20) would always approach zero at grazing. This apparent difference is reconciled by Barrick\n(1972); the effect of finite conductivity for vertical polarization is separated from the scattering cross section\nand expressed as the Norton attenuation factors, FT and FR, in (12:19). Since vertical polarization is the only\nmode which can propagate efficiently as a ground-wave near the sea at MF/HF, most experiments would\nlogically employ vertical (or TM) if one or the other paths to the scatter patch grazes the sea surface. Hence,\none can handle the analysis of such an experiment by treating the sea surface patch as perfectly conducting,\nand then accounting for the finite conductivity by employing the Norton factors FT and/or FR, depending\nupon whether the incident and/or scatter states are vertically polarized.\nIf one or the other or both polarization states are not vertical, one can modify (12:20) in the following\nmanner to give the other three cross sections and spectra for the seatt: and\nohh(wh) are obtained by replacing the factor (sin O sin Os - cosys) in (12:20) by (cos e; sin gs 2,\nand (cos Oi cos Os sin 4s)2, respectively. Thus the dependence of scatter upon the nature of\nthe roughness is the same for any polarization state; it is contained in the surface height spatial/temporal\nspectrum.\nPhysically (12:20) is interpreted as follows. The ocean spatial wavenumbers (Kx,Ky) which are\nproducing scatter are given in terms of the radio wavenumber, ko, and observation angles Oj, Os, is by\ncos sin 0j) and Ky=ko sin Os sin P.S. These, however, are precisely the wavenumbers\nrequired of a diffraction grating which is to scatter a wave incident from Oi into a direction Os, 4s. Hence, the\ntheory shows that the ocean surface produces scatter by the simple Bragg mechanism, which confirms the\nexperimental deductions of Crombie (1955). Furthermore (12:20) implies that, in order to measure the\ndirectional spectrum of the sea, one can measure the sea echo (i.e., o(w) or 00) and vary ko, Oi, 0 or is in\nwhatever manner is most convenient experimentally. Different schemes which vary one or more of these\nquantities are to be examined in the following subsections.\nHaving the radar range equation (12:19) and the expression for the sea scatter cross sections (12:20),\none can now analyze any monostatic or bistatic configuration by integrating (12:19) over the area illuminated\nwithin the radar range cell and/or beam. Examples will be considered later. To obtain estimates for the\nreceived sea-scatter power magnitude for quantitative system design, one can employ the Phillips spectrum\n(12:13) in (12:20). Use of this spectrum for sufficiently high wind speeds (such that u2 gk\nsin Oi - 2 sin 0j sin Os cos is + sin2 Os) provides an upper limit on received power; for winds and seas which\nare lower, the received power will be always less than this amount.\nt We apologize for the inconsistency in notation for the waveheight spectrum. Unfortunately, oceanographers independently\nestablished the convention using S, while scattering analysts adopted the Rice convention based upon W. Both are\ncurrently\nfound in the literature, depending upon the discipline preferred by the user. Consequently, we employ both here and give the\nconnection between them to facilitate reference to other works.\ntt The first subscript always refers to the polarization state of the incident wave, while the second denotes the state for the\nscattered wave of interest.","Remote Sensing of Sea State by Radar\n12-12\n12.2.2 Backscatter MF/HF Experiments\nWe consider in this section possible backscatter experiments employing MF/HF radars. These are in\nalmost all cases either surface-based ground-wave or ionospheric sky-wave configurations. In both situations,\n45 and Oj, Os are sufficiently close to grazing (i.e., within 20°) that the sin 0 factors appearing in (12:20)\ncan be replaced by unity. We then have\nW(-2kg,0,w - wo)\n= 16nk X X\n(12:22)\nIt is understood of course that the waveheight spectrum wavenumbers Kx and Ky (in the x,y directions) are\ndefined at the scattering patch dS. Hence, as one integrates (12:19) over the surface S, (12:22) implies that the\nx-y axis at the patch remain constant with respect to the radar line of sight, and therefore must rotate as the\nline of sight changes in azimuthal position on the ocean. Thus, as one swings the radar beam by 90°, he is not\nonly looking at a different patch of ocean, but with the Kx, Ky wavenumber positions in (12:22) interchanged.\nLet us first calculate ov(w) and OVE based upon the fully developed Phillips spectrum model, and\ncompare these predictions with experimental evidence. Using (12:13) in (12:22), we have\nov==0.02=-17dB\n(12:23)\nIt was initially assumed that waves were traveling only in the +x direction (away from the radar). If waves are\nalso moving into the - half-space (toward the radar), then we have an impulse function at = +12gko\nas well as the one at w=wo-1/28kg shown above. Thus one sees that in general, according to the first-order\ntheory, all of the energy backscattered is contained at two discrete Doppler shifts (+v2gk0) from the carrier.\nSecondly, the magnitude of ovy the average backscattered cross section per unit area, has as its upper limit\n-17 dB, as defined according to (12:19).\nLet us now compare both of these predictions with experimental evidence based upon ground-wave\nradar configurations. One set of recent ground-wave measurements of sea backscatter was made by Headrick of\nthe Naval Research Laboratory (Barrick, 1972) at 10.087 MHz, in which he obtained measurements of ovv. In\nthe experiment, two vertical monopoles were located near Annapolis, Maryland, on the upper Chesapeake Bay.\nSpectral processing permitted separation of water-wave scatter from stationary ground clutter echoes. The\nsignal format used provided a 20 nmi range resolution cell. The Norton attenuation factor FR (=FT) was\ncalculated for four range cells at different distances on the bay using the pertinent water conductivity (i.e.,\n~ 2 mho/m).\nData were recorded and processed on February 4, 1969, a day on which a moderate wind was blowing\nfrom the north. Waves receding from the radar were observed to be stronger due to the wind, and water waves\nof the Bragg scatter length N/2 (15 m in this case) were estimated to be fully developed. The average received\npower from the water was processed at four ranges down the bay: 45, 55, 67, and 75 nmi. Propagation to all\nof these points was via groundwave since they were all below the radio horizon; thus one must compare\nmeasurements with at grazing incidence, as given in (12:22) or (12:23). With the water area within each\nresolution cell (i.e., ds) estimated from maps of the bay, this factor-as well as the Norton attenuation\nfactors-were removed from the radar equation. This yielded experimental values for OVE of -17 dB at all four\nrangest\nThe fact that the 15 m long water waves were fully developed (only a 9.4 knot wind is required to\narouse waves of this length) means that the backscatter might have been expected to approach the Phillips\nsaturation estimate in (12:23) as an upper limit. The agreement between measured and predicted values of\nnot only lends credence to the theory, but confirms the oceanographic estimate of the \"Phillips saturation\nconstant\", B = 0.5 10- used in (12:13).\n+ Headrick employs the actual antenna gains rather than their effective free space gains. Hence his reported values of -29 dB\nwith (12:19) correspond to of -17 dB by our definition -6dB caused by each antenna.","12-13\nMF/HF Radar Scatter from the Sea\nAs further evidence of the validity of the first-order theory for ocean-wave scatter, we cite recent HF\nmeasurements by Crombie et al. (1970) from Barbados Island in the West Indies. Again the antennas were\nlocated near the water SO that propagation to ranges beyond the horizon was via ground wave. In this case we\nexamine Crombie's received signal spectrum; a very high spectral resolution of 0.002 Hz was obtained with\ndigital signal processing. Backscatter was received with broad-band vertical monopole antennas from the\nhalf-space toward the east.\nShown in (F12.5) are the relative received power spectra measured simultaneously on August 15, 1969,\nat 2.9 and 8.37 MHz from the range cell at 45 km. Coherent processing at a 0.5 Hz offset (removed in the\nfigures here) permits both negative and positive shifts above the carrier to be observed. The first-order peaks\n(corresponding to our impulse functions in (12:23)) occur as predicted at +0.174 Hz from the 2.9 MHz carrier\nand +0.296 Hz at 8.37 MHz. The relative strength of the positive spike over the negative spike at both\nfrequencies agrees with the dominant wind direction in this area; trade winds from the east should excite\nwest-moving water waves, producing a positive Doppler shift. Lesser spikes in the records at 0.0 Hz, +0.25 Hz\nfor 2.9 MHz and at +0.42 Hz for 8.37 MHz are attributed by Crombie as due to higher-order hydrodynamic\nand electromagnetic contributions. Theory of such processes is examined in a later subsection.\nWe turn attention now to two ground-wave backscatter experiments which can be used to measure the\nwaveheight spectrum of the ocean. In both cases, one must vary the frequency (and hence the Bragg\nwavenumber, 2kg) which samples the ocean waveheight spectrum in (12:22) over its significant lower end. If\nwe assume that the spectrum cuts off at a wavenumber somewhere near g/u2 (g = 9.81 m/s2 u = wind speed,\nm/s,) then for higher winds and seas one must use lower frequencies. A plot of the backscatter radar frequency\nrequired versus wind speed is given in (F12.6); one of course should actually employ a frequency lower than\nthis (by possibly 20 percent) to ascertain the spectral behavior below cutoff.\nThe first experiment, discussed and tested by Crombie (1971), employs azimuthally omnidirectional\nantennas on the coast. Using pulsed signals and a range gate set at 22.5 km from the radar, Crombie obtained\nthe average received power at as many as eight frequencies, ranging between 1.7 MHz and 12.3 MHz. With such\nan experiment, one is simultaneously observing sea scatter from a semi-circular annulus, and it is assumed that\nthe sea is relatively homogeneous over such a circle (i.e., that the directional ocean waveheight spectrum is\nessentially constant over 45 km). This assumption is reasonable for on-shore winds and waves; for off-shore\nwinds, however, the limited fetch does not permit the waves nearer the shore to build up as high as those more\ndistant. Crombie (1971) notes that even at 100 km, higher off-shore waves may still not be fully developed.\nHence if one spectrally processes the signals and employs the energy only in the Doppler line above the carrier,\nthe homogeneity assumption should be valid.\nSince the antennas in this experiment are azimuthally omni-directional and since the received energy\nfrom everywhere in the semi-circular annulus occurs at the same + Doppler shift, it is not possible to obtain\nthe directional waveheight spectrum. One can obtain the non-directional waveheight temporal spectrum, which\nwas defined in (12:12) and exemplified in (F12.3). To do this and relate s(w) to the received power, PR, at\ncarrier frequency fo, one must integrate the second version of (12:19) over the semi-circular annulus, using\n(12:22) and (12:12). The one-sided spectral result is\n(12:24)\nwhere C is the free-space radio wave velocity and AR is the width of the range-resolution cell. It must be noted\nthat several of the factors on the right side of (12:23) may vary with frequency, including antenna gains,\ntransmitted power, and the Norton attenuation factors.\nShown in (F12.7) are several such temporal spectra reduced by Crombie from observations off\nBarbados Island in the West Indies (plotted versus Hertz rather than rad/s). In the upper two, he was able to\ncompare the predicted significant waveheight (obtained from integrating the area under the spectrum) with\nlaser profilometer measurements of significant waveheights; the agreement is good. In the bottom plot,\nCrombie shows also a Pierson-Moskowitz model spectrum predicted for a 20 knot wind and fully developed\nseas for comparison. Crombie cautions that the spectra to the left of the line marked \"minimum observed\nfrequency\" are estimated; in the upper pair of records, the estimated portion is a substantial portion of the\narea under the curves. This points up the importance of employing a sufficiently low (MF) frequency if one\nwishes to obtain spectral detail in higher sea states.\nA variation of the above backscatter experiment can provide the directional rather than merely the\nnon-directional waveheight spectrum. The azimuthally omni-directional antenna is placed on a moving ship at","12-14\nRemote Sensing of Sea State by Radar\n-0.4\n-0.2\n0.0\n0.2\n0.4\n-0.4\n0.6\n-0.2\n0.0\n0.2\n0.4\n0.6\nFrequency Shift From Carrier, f-fo , Hz\nFrequency Shift From Carrier, f-fo , Hz\nFigure 12.5 Measured backscatter signal Doppler spectra of HF sea echo (after Crombie et al., 1970) (a)\n2.9 MHz carrier frequency (b) 8.37 MHz carrier frequency.\n1000\n500\n400\n300\n200\n100\n50\n40\n30\n20\n10\n5\n4\n3\n2\nI\n0.5\no\n2\n4\n6\n8\n10\n12\n14\n16\n18\n20\n22\n24\n26\n28\n30\n32\n34\nWind Speed, knots\nFigure 12.6 Solid curve gives frequency necessary to observe lower end (cutoff) of gravity wave spectrum for\nnear-grazing backscatter; dashed curve gives frequency limit where slightly rough surface model fails\nmathematically for given wind speed.","Description of the Sea Surface\n12-15\nJuly II, 1969 1415 hr\n1.0\nH1/3\n0.8m\n(Convair 990 H1/3=1.0m)\no\nJuly 14, 1969 1718 hr\nH1/3 =0.8m\n0.5\n(Convair 990 H1/3= 0.6m)\no\nJuly 13, 1969 1115 hr\n0.5\nH1/3=0.5m\no\nJuly 12 1969 1056 hr\nMoskowitz\n1.0\n(20 knots\nH1/3 0.67m\nwind)\no o\n0.1\n0.2\n0.3\n0.4\nFrequency , Hz\nFigure 12.7 Temporal waveheight spectra deduced from MF|HF ground-wave backscatter observations (after\nCrombie, 1971).\nsea. In this case the sea return does not appear at the previous single set of unique Bragg Doppler shifts around\nthe carrier, for the ship's motion introduces a different Doppler bias on the return from each point around the\nrange ring. This Doppler shift is Wsh = 2wo vs h cos A/c, where Vsh is the ship speed and 0 is the angle from the\nship's bow to a point on the range ring. We assume again that the ocean waves are moving predominantly into\none half-space and that the \"sea state\" is homogeneous over the area of the ring. Then the one-sided power\nspectral density from a given element of area on the ring (i.e., ds = RARAA) is found from (12:19) and\n(12:22) to be\n24PTGTGRF*ARA6\nS(2k - 0) ,\n(12:25)\nwhere Wm = 2worsh/c is the maximum ship-induced Doppler shift and w+=1/2gko is the Doppler shift from\nthe gravity waves. To obtain the average power density spectrum, PR(w), from (12:25) we must integrate over\n0. The resulting one-sided spectrum PR(w) is seen to consist of two \"pedestals\", each centered at 2gk.\nThus one can relate the spatial directional waveheight spectrum, S(Kx,Ky) to, say, PR(w) within the positive","Remote Sensing of Sea State by Radar\n12-16\npedestalatwo+ 2gkasfollows:\n(12:26)\nn=w-wo-1/28kg, and where it is understood that PR(w) is non-zero only within the pedestal region,\nwhere\ni.e., Inl >1/8kg); in this case, n-w-wo. Thus by measuring the received sea echo signal\nspectrum at the satellite, and knowing its velocity, position, and orbital plane, one can obtain the directional\nwaveheight spectrum for spatial wavenumbers kx=ko cos 4, , Ky = k o s in y, for all I by noting that cos\nI\n= n/w dm and sin =1/1-n2/wam One mustevary the carrier frequency, however, to sample the\nspectrum at a different total wavenumber, As before, there is an ambiguity, in that\nwaves crossing the circle at +4 cannot be distinguished from waves crossing the circle at Ruck et al. (1971)\ndiscuss several techniques for removing this ambiguity.","Remote Sensing of Sea State by Radar\n12-26\nAs an illustration of what the received signal spectrum will look like at 5 and 10 MHz for fully\ndeveloped semi-isotropic waves describable by the Phillips spectrum (12:13), we consider the following\nexample. The antenna on the satellite is assumed to be a half-wave dipole with gain 1.64, and the \"free-space\"\ngain of the quarter-wave monopole transmitting antenna is taken to be 0.82. The satellite is at an altitude of\n300 km and moving with velocity 8000 m/s. We select a pulse length T = 10 us, yielding a clutter ring width\nART = km. Let us select a time delay t - td = 50 us, corresponding to RT = 15 km. Then we obtain the\nspectra shown in (F12.17). At the top of these plots, the angle I of ocean wave directions producing the echo\nat that Doppler shift is given. The maximum Doppler shift from the satellite is fdm = 6.67 Hz at 5 MHz, which\nis considerably larger than the ocean-wave Doppler gk of 0.161 Hz; hence the neglect of ocean-wave-\ninduced Doppler shifts in this experiment seems reasonable. Also, there is no need for spectral processing\nresolution less than about 0.5 Hz, which alleviates the data handling requirements aboard the satellite.\nThe satellite should receive and process the direct signal as well as the sea-reflected signal. The direct\nsignal will serve (a) as a time reference, (b) as a Doppler (frequency) reference, and (c) to calibrate and remove\nany unknown path loss through the ionosphere from the sea return. For the example considered in the\npreceding paragraph, the direct signal power received at the satellite is of the order of 10-9 PT to 101 PT,\ndepending upon the gain pattern of the transmitting antenna in the direction of the satellite. This compares\nwith a total maximum received sea echo power of approximately 2.5 PT at 5 MHz.\nThe most serious limitations on a system such as this are imposed by the ionosphere. Orbital altitudes\ngreater than 200 km will often not allow penetration of the lower HF frequencies through the ionosphere to\nthe satellite. The F2 layer of the ionosphere is the densest and if the satellite is orbiting above it (i.e., above\n300 km), then the following conclusions concerning ionospheric limitations were determined by Ruck et al.\n(1971).\n(a) The operating frequency of the sensor must be confined to the range 3.5 to 30 MHz. (This permits\nsensing of ocean waves with lengths between 10 and 100 m.) Propagation conditions favorable to the system\nexist at night between 0 and 6 hr local time. At such times the minimum ionospheric penetration frequency\nranges from 3.5 to 5 MHz depending upon the season and sunspot cycle. Operation throughout the rest of the\nday can take place at frequencies as low as 9-10 MHz.\nRestriction of operation at the lowest frequencies to a six-hour period every day, however, may not\nlimit the utility of the sensor for the following reason. At the lowest frequencies, the longest ocean waves are\nbeing observed (i.e., greater than 40 m). However, these longer ocean waves require greater times (i.e., of the\norder of 24 h) to build up and die down (T12.1). Thus the heights of these longer waves will not change\nappreciably over times less than a day, and their observation once a day should be sufficient.\n(b) During normal ionospheric conditions, the (excess) absorption loss due to passage through the\nionosphere will be less than approximately 15 dB providing the operating frequency exceeds the minimum\npenetration frequency by 0.5 MHz.\n(c) The noise environment encountered by the satellite will be that due to cosmic noise, with a\nmaximum effective noise temperature of about 4 X 106 K at 3 MHz.\nBased on these loss and noise considerations, the study concludes that adequate signal-to-noise ratios\ncan be obtained with average transmitter power output levels of the order of 10 W for a satellite in a 400 km\norbit.\n12.3 Second-Order HF Sea Echo\nIt was noted several places previously in this chapter (when comparing the first-order Bragg scatter\ntheory with measured sea echo spectra) that the observed records often contain smaller-but non-negligible-\npeaks at Dopplers other than the first-order lines. Also, Barnum (1971) observes an overall higher \"floor\"\nunder the sea spectra than would be expected from normal processor and noise clutter. He has confirmed that\nthis \"floor\" is produced by sea echo by (a) looking at land scatter for comparison, and (b) shutting off the\ntransmitter and observing the \"floor\" due to system and external noise. This \"floor\" and the higher-order\npeaks are not predicted from the first-order theory developed previously; one must go to a higher-order","Second-Order HF Sea Echo\n12-27\n180° 154° 143° 134° 127° 120° 114° 107° 102° 96° 90° 84° 78° 73° 66° 60° 53° 46° 37° 26°\n0°\n4.0\n3.0\nw PR(w)\n\"dm Palw)\n2.0\n1.5\n1.0\n0.9\n0.8\n0.7\n0.6\n0.5\n0.4\n0.3\n0.2\n0.15\n0.1\n0.09\n0.08\n0.07\n0,06\n6\n7\n-7\n-6\n-5\n-4\n-3\n-2\n-I\no\nI\n2\n3\n4\n5\nf-fo\n180°154°\n143°\n134°\n127°\n120°\n114°\n107°\n102°\n96°\n90°\n84°\n78°\n73°\n66°\n60°\n53°\n46°\n37°\n26°\n0°\n10.0\n9.0\n8.0\n7.0\n6.0\n\"dm Pr(w)\n5.0\nPT\n4.0\n3.0\n2.0\n1.5\nx\n1.0\n0,9\n0.8\n0.7\n0.6\n0.5\n0.4\n0.3\n0.2\n0.15\n-14\n-12\n-10\n-8\n-6\n-4\n-2\no\n2\n4\n6\n8\n10\n12\n14\nf-fo\nFigure 12.17 Predicted received signal spectra for bistatic sea scatter of F12.16; computed for fully aroused\nPhillips isotropic waveheight spectrum (a) 5 MHz carrier frequency (b) 10 MHz carrier frequency.","Remote Sensing of Sea State by Radar\n12-28\nanalysis of scattering and ocean wave interaction processes. Such spectral contributions could ultimately pose\na limitation on the performance of any radar system designed for first-order Bragg scatter, and hence they\nshould be qualitatively and quantitatively understood.\nA more positive reason for examining these higher order sea echo spectral contributions is that they\nmay in themselves provide additional information about the state of the sea. This suggestion was made by\nHasselmann (1971), Crombie (1971), and Barrick (1971b).\nTwo obvious sources for this higher-order return suggest themselves: (a) the second-order terms for\nscatter from the Rice boundary perturbation theory, and (b) the second-order terms from the hydrodynamic\nequations describing the water surface height. To simplify matters here, we consider backscattering at grazing\nincidence with vertical polarization over a perfectly conducting sea. Then it can be shown that the average\nsecond-order backscatter cross section per unit surface area per rad/s bandwith, ov(w), to be used with\n(12:19) is:\n(12:31)\n=\n= sgn (k1x)VgK1;W2=sgn(2x)\nVgk2;n=w-wo is the Doppler shift from the carrier; 8(x) is the Dirac impulse function of argument x; and\nW(K) = W(Kx,Ky) is the directional spatial waveheight spectrum of the ocean, as defined in (12:21).\nFor the second-order electromagnetic contributions alone, is found to be\nTEM1\n(12:32)\nwhere A is the normalized impedance of the sea surface, as discussed by Barrick (1971a).\nThe second-order hydrodynamic effects produce\n()\n(12:33)\nwhere i = V-1 and wB=1/2gko is the first-order Bragg Doppler shift. The total T which must be used in the\nintegral to account for both types of second-order effects is\n(12:34)","Second-Order HF Sea Echo\n12-29\n10 MHz\nFirst Order Bragg Doppler,\no° 17 dB\n40 Knot Wind,\no = 20.6 dB\nElectromagnetic and\nHydrodynamic\nSingularities at 2\nElectromagnetic\nSingularity at\n23/4\n30 Knot Wind,\n2\nx10\noo = -24.8 dB\n20 Knot Wind,\noo =29.0 dB\n15 Knot Wind,\noo =-32.6 dB\n2\nO.IXIO\n-0.4\n-0.2\no\n0.2\n0.4\n0.6\n0.8\n1.0\n1.2\nI.4\n1.6\n1.8\n2.0\n2.2\n2.4\nf f\no\nNormalized Doppler Shift\ng/\nFigure 12.18 Predicted Doppler spectrum of first and second-order near-grazing sea backscatter at 10 MHz\nfor propagation in upwind direction (Phillips semi-isotropic waveheight spectrum assumed).\nThe above integral clearly shows that a double-scatter Bragg process is responsible for the second-order\nsea return. The scattered radio wavenumber, -kox, is equal to K 1 + K2 + kox, where the last term is the\nincident radio wavenumber. The frequency, w, of the scattered field is identically W1 + W2 + wo. In the case\nof the electromagnetic second-order effects, an ocean wavetrain with wavenumber K1 scatters the radio\nenergy along the surface to a second wavetrain with wavenumber K2, which redirects it back toward the\nsource; the intermediate radio wave can be either propagating or evanescent. In the case of the hydrodynamic\neffects, two ocean wavetrains produce second-order ocean waves with wavenumbers K1 +K2; these latter\nocean waves are not freely propagating because they do not satisfy the first-order gravity wave dispersion\nrelationship, but they do produce radar scatter.\nOne of the two integration processes can be done in closed form, this resulting from the impulse-function occurring in the\n+ integrand. The remaining integration is done numerically because of the comple form of the integrand.","Remote Sensing of Sea State by Radar\n12-30\nTo illustrate the nature of both of these second-order effects at HF, we perform the integration\nindicated in (12:31) numerically+ for the following example. The frequency chosen is 10 MHz, and we employ\nthe Phillips semi-isotropic wind-wave model of (12:13) for W(R), the first-order waveheight spectrum. Using\ndifferent windspeeds as the parameter and considering propagation in both the upwind and crosswind\ndirections, we obtain plots shown in (F12.18) and (F12.19) for the signal spectrum w The first-order\ndiscrete Doppler lines are also shown for reference; they will be present at 10 MHz for the Phillips model at\nwinds greater than 9-1/2 knots. Also shown is the normalized radar cross section for each spectrum,\n'vv(w)dw), where the spectra plotted here are two-sided, i.e., ov(w) = ow(w)\nThe figures illustrate that second-order continuous Doppler sidebands do occur and their magnitude\nwill depend upon sea state. These sidebands contain continuous, integrable singularities (of the square-root-\ntypes) at positions of V2 and 2 Superscript(3) times the first-order Bragg line. The V2 singularity is due to both\nelectromagnetic and hydrodynamic second-order effects. Electromagnetically, it is due to higher-order Bragg\nscatter, i.e., from ocean waves of length L = A (rather than N/2). These ocean waves travel at a speed V2\ngreater than those at N/2, and hence the V2 spike; this is a \"grating lobe\" effect occurring for larger diffraction\ngrating spacings. Hydrodynamically, the V2 singularity is due to the second spatial harmonic of the trochoidal\nwave profile of fundamental length L = A; this second harmonic is of length N/2, producing first-order Bragg\nscatter, but it travels at the same phase speed as the fundamental to which it is attached. The phase speed of\nthe fundamental is V2 greater than the normal ocean wave with length N/2, and hence the 2 hydrodynamic\ncontribution. Finally, the 23 singularity is due to a \"corner reflector\" electromagnetic effect. This occurs\nwhen the two sets of (non-evanescent) scattering ocean waves pass through 45° with respect to the\npropagation direction. The total Doppler shift from these two sets of ocean waves, W1 + W2 =\nvsin a), is maximum at a = 45°, i.e., WB. Thus a condition of mathematical stationarity occurs for the\nDoppler shift when a passes through 45°.\nAs deduced previously from (F12.18) and (F12.19), the second-order received sea echo spectrum\nincreases both in its amplitude and in its proximity to the first-order Bragg lines with increasing wind speed.\nConversely, for a given wind speed, the same second-order spectrum increases with increasing carrier\nfrequency. The common parameter for each curve in the figures is g/(2kou2), where u is the wind speed and\nko (=2nfo/c) is the radar wavenumber. Thus for a given value of this parameter, the same spectrum curve can\nbe obtained by doubling wind speed and reducing frequency by a factor of four, or if the frequency is\nincreased by a factor of four, by halving the wind speed.\nAt present, we have no conclusive experimental validation of the theory because of a general lack of\naccurate HF sea echo spectrum measurements. Several available records, however, exhibit many of the\nprincipal features of our predicted second-order spectra. Crombie's ground-wave measurements at 8.37 MHz\n(F12.5) shows a definite second peak above the positive first-order line; its position is greater than the\nfirst-order line by a factor of 1.36. From (F12.18), this would correspond to seas aroused by a wind speed of\nabout 21-22 knots. Also evident in this record by Crombie is a third peak which occurs at about 1.69 times the\nfirst-order line; the 23/4 (= 1.682) singularity appears to explain this peak. Barnum's sky-wave backscatter\nmeasurements (F12.8) at 25.75 MHz also appear to contain higher-order peaks beyond the first-order line. He\nestimates that these occur at 1.35, 1.70, and 2.00 times the first-order line. Again, the first higher order peak\nat 1.35 is explainable by seas driven by about 13-knot winds, while the second peak is quite close to the\npredicted singularity. As with Crombie's record, the height of the third peak near 23 /4 is less than that of\nthe second, while the height of the second is considerably less than that of the first-order Bragg line; these\nfeatures all agree with the predicted second-order spectral behavior.\nHasselmann (1971) has suggested (based upon several approximations which were examined in detail\nby Stewart (1971)) that the second-order sea echo spectra above and below the first-order Bragg lines should\nbe symmetric reproductions of the first-order temporal nondirectional waveheight spectrum of the sea,\ncentered about the first-order lines. Our more detailed derivations show that these second-order spectra are not\nsymmetric about the Bragg lines; this is especially true for the crosswind case, where no energy at all appeared\nabove the Bragg line. However, these second-order contributions do possess some of the features of the\nfirst-order waveheight temporal spectrum centered around the Bragg lines. For example, they become higher in\nt The angle a here is (180° - )/2, where is the bistatic angle between the incident and the first-scattered radio wave.\nBackscatter produced by such a double interaction process requires that the bistatic angle between the first-scattered and\nsecond-scattered (i.e., backscattered) radio wave be .","UHF Indirect Bragg Scatter\n12-31\namplitude and move in closer to the Bragg lines with increasing wind speed and/or frequency. Hence we agree\nwith Hasselmann's basic conclusions that sea state can be deduced at higher HF frequencies by examining the\nfeatures (i.e., strength and position) of the second-order peaks in the sea echo backscatter spectrum. Upon\nfurther confirmation by measurements, this technique may prove to be quite valuable in remote sensing of sea\nstate, especially with ionospheric radars which are restricted in their operation to the upper HF region.\n12.4 UHF Indirect Bragg Scatter Using Two Frequencies\nA technique currently under development for measuring the slope spectrum of the longer gravity waves\nwill be briefly examined here. This concept employs the correlation between the sea return at two closely\nspaced UHF frequencies as a measure of the larger and longer ocean waves present. The interpretation of the\nfinal result of the derivation shows that the mechanism yielding the slope spectrum of the surface resembles\nBragg scatter; the surface slope spectrum is evaluated, however, not at the wavenumber corresponding to the\ncarrier frequency, but at the \"beat\" wavenumber corresponding to the difference between the two UHF\nfrequencies.\nFirst-Order Bragg\n10 MHz\nDopplers, oo = - 17 dB\n40 Knot Wind,\noo = 20.4 dB\n30 Knot Wind,\noo = -23.7 dB\n1x10-2 2\n20 Knot Wind,\noo = 27.0 dB\n15 Knot Wind,\no° 29.8 dB\n-1.4\n-1.2\n-1.0\n-0.8\n-0.6\n-0.4\n0.2\no\n0.2\n0.4\n06\n0.8\n1.0\n1.2\n1.4\n-\nf-fo\nNormalized Doppler Shift\ng/\nFigure 12.19 Predicted Doppler spectrum of first and second-order near-grazing sea backscatter at 10 MHz\nfor propagation in crosswind direction (Phillips semi-isotropic waveheight spectrum assumed).","Remote Sensing of Sea State by Radar\n12-32\nBefore summarizing the analysis, let us briefly discuss the physics behind the anticipated behavior. HF\nscatter from the sea has been conclusively shown, both theoretically and experimentally, to be due to the\nBragg (or diffraction grating) effect. The theory shows that a mathematical upper limit of frequency (for a\ngiven waveheight) can be expected, beyond which the perturbation approach used should not be valid; this\nupper frequency in terms of wind speed is given in (F12.6). Several recent experimental efforts, however, have\nestablished that the Bragg mechanism produces non-specular sea scatter at UHF, microwave, and even\nmillimeter-wave frequencies. Wright (1968) deduced this from his signal spectra-as well as from quantitative\ncomparisons of oo for various polarizations with the previously developed theory; he observed scatter from\nwaves generated in a controlled wind tank. Guinard and Daley (1970) established that the Bragg mechanism\nalso explained-even quantitatively-the microwave scatter they observed on the sea. In the latter case, much\nlarger and longer ocean waves are present; yet measurements have confirmed that the much smaller wavelets\nactually producing microwave backscatter are those whose lengths are N/2 sin 0), where 0 is the angle of\nincidence from the vertical. Again, their results measured for 00 and hh (the average backscatter cross\nsections per unit area) agree quantitatively with theoretical predictions based on a slightly rough surface, i.e.,\n(12:20), (12:22), and (12:23), in their dependence on polarization, incidence angle, and saturation effect in\nthe wind-wave equilibrium region. Such agreement is apparent over most aspect angles, as long as one stays\naway from the specular direction (i.e., the vertical for backscatter) and grazing incidence (where shadowing\nbecomes significant).\nBarrick and Peake (1968) and Wright (1968) explained this behavior by considering the surface at these\nhigher frequencies to be a \"composite\", made up of two or more scales of roughness. Thus, one has the\nBragg-scattering wavelets riding on top of the longer and higher gravity waves. With this model, one obtains\ntwo regions of scatter: the quasi-specular region and the diffuse region. Near the specular direction,\nbackscatter is produced via reflections from many specular points, or facets, oriented normal to the line of\nsight. For the sea, this type of backscatter dominates out to 10-15° from the vertical; its magnitude and\nbehavior is predictable from both physical and geometrical optics approaches. Farther away from the specular\ndirection, scatter is predictable via the Bragg mechanism, as though the smaller wavelets riding on the larger\nwaves were really the only ones present. The magnitude and polarization dependence of this \"diffuse\" scatter\nfollows (12:20) for the slightly rough surface.\nValenzuela (1968) first noted that the magnitude of the return from a slightly rough surface (i.e., the\nBragg scatter) does have some dependence upon the local incidence and scattering angles, as seen in\n(12:20)-even though the dependence may be weak for some polarization states over a large range of angles.\nHence the effect of the longer gravity waves under the Bragg-scattering wavelets should be seen as a \"tilting\nplane\", modulating the amplitude of the Bragg scatter because of the slope of the larger-scale wave\nunderneath. Let us take as an example a uniform Bragg-scattering wavetrain on top of a single larger and longer\nsinusoidal wave. Now imagine a short radar pulse, less in its spatial length than one-quarter the wavelength of\nthe longer sea wave, propagating along the surface and backscattering via the Bragg mechanism from the\nwavelets. Due to the slope of the longer sinusoidal wave and hence the periodic variation of the local angle of\nincidence to the pulse as it propagates along, the radar receiver should see a return which is amplitude\nmodulated in a periodic manner by the longer gravity wave. If one analyzed the spectrum of this amplitude\nmodulated signal, he would be able to relate the result to the slope of the larger wave at its own spatial\nfrequency or wavenumber. Thus one could, with such a short-pulse experiment, measure the slope spectrum of\nthe longer gravity waves by Fourier transforming the received signal strength and looking at its spectrum. This\ntechnique was examined recently by Soviet investigators (Zamarayev and Kalmykov, 1969).\nThe concept to be analyzed here is quite similar to that described above. By using two frequencies,\nhowever, and cross-correlating their received powers, one eliminates the Fourier transform process required for\nthe short-pulse technique. Nearly CW signals can be used. The bandwidth required (i.e., frequency separation\nhere) is much the same as for the short-pulse, however, because both techniques are essentially employing\nspatial range resolution to distinguish the slopes of the underlying longer gravity waves. By eliminating the\nspectral analysis process and the short pulse requirement, we feel that the two-frequency correlation concept\noffers a possibly more tractable sensing tool.\nThe two-frequency correlation concept was analyzed in Ruck et al., (1971) in detail. The derivation\nthere was meant to establish quantitatively some of the features of the correlated power. To facilitate the\nanalysis, the following assumptions were imposed: (a) Only backscatter was considered. (b) The surface was\ntaken as perfectly conducting. (c) Horizontal polarization was examined for incidence and backscatter. (d) A","UHF Indirect Bragg Scatter\n12-33\n0\npulse\nwidth\nAdvancing radar\nresolution cell at\nMean surface\nposition c/ N\nlevel\nFigure 12.20 Physical picture of specular point scatter. Specular points within radar resolution cell are shown\nhighlighted.\none-dimensional random surface was analyzed, corrugated along the plane of incidence. (e) The incidence\nangle region was selected to be not too close to the specular direction (i.e., the vertical), but yet not SO close to\ngrazing that shadowing is a problem. (f) The slopes of both the large-scale and small-scale sea-wave components\npresent were assumed small.\nBased upon these assumptions, the variance of the backscattered power densities at two frequencies, fb\nand fa, was obtained, i.e., Var [P(Ak)] III P, where Ak=kb-ka2(f-fa)/c Thus, Ak\ncan be considered a \"beat\" or difference wavenumber. The length and width of the surface patch (assumed\nsquare) subtended in the radar cell is L, and the angle of incidence from the vertical is 0. Then the result for\nthe variance was found to be\nVar[P(Ak)]\n[\n+\n(12:35)\nwhere R is the distance from the scattering patch to the far-field point, E is the electric field strength of the\nplane wave incident on the surface, and Z is the free-space wave impedance. The quantity W(KX) is the\none-dimensional spatial waveheight spectrum of the sea surface in the wave-length range around one meter;\nWSL(Kx) is the one-dimensional spatial waveslope spectrum of the larger gravity waves. The remaining\nquantities appearing in (12:35) are qz=2 = cos 0, S = dL/dx = slope of the larger-scale component of the\nsurface, s = 2s cos 0 - 2 sin 0, and - sin 0)2 + cos 0 S sin 0] X [cos 0 + S sin 0].\nIn order to obtain numerical estimates of the relative magnitudes of the first and second terms in\n(12:35), we must evaluate the quantity in square brackets and its derivative at zero slope. To do this, a form\nfor the waveheight spectrum of the smaller-scale (Bragg-scattering) ocean waves must be assumed. By selecting\nour operating frequency in the UHF band (viz., near 1 GHz), we ensure that these Bragg-scattering ocean\nwaves are of the order of 30 cm in length. Such waves are still gravity waves (in contrast to capillary waves)\nand hence should follow the Phillips model in the saturation region. Furthermore, these shorter gravity waves\nrequire winds greater than only 1-1/2 knots to excite them; hence, they are nearly always present. On the\nother hand, their build-up time is of the order of 10 minutes, in contrast to capillary waves which build up and\ndie down in a matter of seconds; therefore they should exist rather uniformly and stably over times and areas\nwhich are significant in making the measurements. This is the reason that 1 GHz is proposed as the operating\nfrequency.","Remote Sensing of Sea State by Radar\n12-34\nOne can readily convert the two-dimensional Phillips model (12:13) to a one-dimensional version. The\nresult is\nw(*x)\n(12:36)\nwhere Kx >0 (i.e., the above spectrum is one-sided), and as before, B = 0.005 (dimensionless).\nUpon evaluation of the indicated factors in (12:35), we obtain\nVar\n(12:37)\nLet us now interpret the two terms in (12:35) and (12:37). The first term is merely the Fourier\ntransform of the range resolution cell illumination pattern on the surface (i.e., here we assumed uniform\nillumination over the cell of length (L/2) sin 0) evaluated at spatial frequency 2Ak. If the range cell is\nbeam-limited rather than pulse-limited, one might approximate the cell illumination by a uniform pattern\nbetween the half-power points of the antenna beam pattern on the surface. If one uses a more realistic\nillumination pattern along this cell, the (sin x)/x function will be replaced by the Fourier transform of the\nactual pattern. With a properly selected and tapered illumination function, the first term can be kept very\nsmall, SO long as AkL sin 0 is large compared to unity. In a pulse-limited situation, this means making the pulse\nlength, T, sufficiently long that 2 sin 0 >>1 over the range of Af used in the experiment.\nThe second term in (12:35) and (12:37) contains the desired information about sea state, as\nrepresented in the waveslope spectrum of the longer gravity waves. Their waveheight spectrum is readily\nrelated to WSL(Kx) as Hence, measurement of the waveslope spectrum by sweeping\nfrequency (and thus varying Ak) can be directly transformed into waveheight spectral information. It is\ndesirable to select 0, the incidence angle, SO that the magnitude of the second term is enhanced with respect to\nthe first. For horizontal polarization and backscatter, a poor choice would be 0 at or near 37.8° from the\nvertical, for this makes the factor in parentheses multiplying the second term identically zero. On the other\nhand, a value of 0 near 60° will usually result in the second term being larger than the first for WSL non-zero\nand near its equilibrium (saturation) value.\nThe argument of the term containing the waveslope spectrum would lead one to think that a\nBragg-effect scatter were occurring at the beat wavenumber 2Ak sin 0, rather than at the carrier wavenumber\nk or kb. Hence we refer to this as an indirect Bragg-scatter measurement. By sweeping Af from 2-20 MHz, one\nshould be able to obtain sea state information by measuring the magnitude, shape, and cutoff of the larger\ngravity wave slope spectrum. The two frequencies can be generated quite simply by using a balanced\nmodulator near the output of the transmitter. Since the scattered power is correlated in the receiver, it is not\nnecessary to maintain phase coherence of the two signals through the receiver channels. Hence the equipment\nrequirements should present no significant obstacles.\nThis technique, examined here for backscatter and horizontal polarization, can be used for other\npolarization states and in bistatic arrangements (so long as one avoids the specular reflection direction). The\nanalysis is currently being extended to include three-dimensional scatter from two-dimensionally rough,\nnon-perfectly conducting surfaces. The basic nature of the results are not expected to differ from those\nexamined here, however. Up to the present, this technique has not been tested experimentally; hence we can\noffer no measured data for validation of the concept. Plans are underway to test the technique in the near\nfuture.\n12.5 Sea State Effects on a Microwave Radar Altimeter Pulse\nAs a final tool for remote sensing of both geodetic and sea state information, we discuss the microwave\nradar altimeter. Decisions by NASA to fly short-pulse altimeters in both the GEOS-C and Skylab series of","Sea State Effects\n12-35\nsatellites have recently accelerated theoretical and experimental efforts on radar altimetry. A sufficiently clear\npicture is presently available-both from analysis and experimental data-of the basic interaction process\nbetween the pulse and the sea surface. At and near the (vertical) sub-altimeter point, microwave scatter is\nproduced by specular points which are distributed in height, thus having some obvious relationship to sea\nstate. In most cases of interest, the effective radar spatial pulse width is smaller than the ocean waveheights it\nencounters on the surface; hence a stretching of the pulse will occur due to sea state. When the purpose of the\nexperiment is to find the instantaneous mean sea level (for geodetic reasons) to an accuracy of less than one\nmeter, one is faced with the problem of finding this position in a received echo distorted by sea state effects;\nhence one must remove such effects from the signal. On the other hand, one may wish to use the radar\naltimeter as a sea state sensor; in this case, he would like to know how to relate sea state to the received pulse\ndistortion. Both problems are examined here.\nThe first subsection discusses the specular point-theory of sea scatter and obtains the distribution of\nthese points as a function of waveheight. The second subsection then applies the specular-point scattering\nmodel to the radar altimeter configuration and determines a simple closed-form solution for the altimeter\nreturn for the case of Gaussian pulse and beam widths. The final subsection simplifies this result for certain\nlimiting altimeter configurations commonly used in practice, and compares the model with measured data.\n12.5.1 Specular Point Distribution and the Scattering Model\nFor the microwave frequencies at which an altimeter will operate, scatter from the sea within the\nnear-vertical region directly beneath the altimeter is quasi-specular in nature. This means that backscatter is\nproduced by specular or glitter points on the surface whose normals point toward the radar. Such scatter\npersists only to about 10-15° away from the vertical, since gravity waves can seldom maintain slopes greater\nthan this amount before they break and dissipate energy. A physical picture of the specular points illuminated\nwithin a short pulse radar cell advancing at an angle 0 with respect to the mean surface normal is shown in\n(F12.20).\nSpecular point scatter is readily predictable from geometrical and/or physical optics principles, and has\nbeen analyzed by Kodis (1966) and Barrick (1968). Here the theory is extended to include the height of the\nsurface, since the short radar pulse may not illuminate the entire peak-trough region at a given time. As the\nstarting point, we note from elementary geometrical optics principles that the field scattered by N specular\npoints (expressed in terms of the square root of the backscatter cross section) is\n?\ncos\n(12:38)\nwhere gi is the Gaussian curvature at the i-th specular point, i.e., gi = PiiP2il, with P1i and P2i as the principal\nradii of curvature at this point. Also, ij is the height of the i-th specular point above the mean surface, 0 is the\nangle of incidence from the vertical, and ko = 2n/A is the free-space radar wavenumber, A being the\nwavelength.\nWe now square and average the above equation with respect to the phase, fij, noting that\ncos 0 will be uniformly distributed between zero and 2n as long as the sea waveheight is\ngreater than the radar wavelength. We then rewrite the remaining single summation in integral form as\na\ndistribution of specular points versus height, i, above the surface and versus Gaussian curvature, g. The average\nradar cross section per unit surface area per unit height, n°(5), can then be written in terms of the average\nspecular point density, n(s,g) as follows (details are found in Ruck et al, 1971):\n(12:39)\nwhere n(s,g) is the average number of specular points within the height interval S - ds/2 to S + d's/2 and with\nGaussian curvatures between g - dg/2 and g dg/2. The quantity n°(s) is related to 0°, the average\nbackscatter cross section per unit area as 0°=/\" 70(5) d's. Thus a short pulse having a radar resolution cell\nof width AS will produce, on the average, a radar cross section per unit area of n°(S)","12-36\nRemote Sensing of Sea State by Radar\nThe specular point density, n, can readily be determined (almost by inspection) from the work of\nBarrick (1968) preceding Equation (7) of that paper; one merely includes height, 5, in the probability\ndensities. The following result-applicable for backscatter-is obtained for the integrand of (12:39):\nn(s,g)gdg = sec4 0 d$xxd$xyd$yy\n(12:40)\nwhere\nare the partial derivatives of the surface height up to second order, are the\nsurface slopes required at a specular point (these latter slopes are known geometrical quantities). The quantity\np(x1,..., xn) is the joint probability density function for the random variables X1\nXn.\nThe integration over 5xx xy, and yy can now be performed. Furthermore, since the height 5 and the\nslopes ysp at any point are uncorrelated (as discussed in 12.1.2), and since we intend to employ\nGaussian distributions for the surface height and slopes (also discussed in 12.1.2), we can finally express the\nscatter per unit height as the product of the height and slope density functions:\n(12:41)\nwhere p(s) is as given in (12:1) and p(5xsy is given in (12:2). Also, the required total slope at the specular\npoint to be used in (12:2) is VSxsp+sysp = tan = 0.\n12.5.2 Application to Satellite Altimeter\nWe now apply (12:41) to the problem depicted in (F12.21). An altimeter at height H emits a spherical\npulse which in turn sweeps past a spherical earth. The spatial pulse width for a backscatter radar is AS = CT/2,\nwhere c is the free space radio wave velocity and T is the time width of the pulse (compressed, if applicable).\nAs our time reference, we choose t =0 to be the time that the center of the signal, reflected from the\nuppermost cap of a smooth spherical earth, returns to the receiver. In terms of the angles shown in (F12.21),\nthe height, is to a point at the center of the cell above the mean sea surface can then be written as\nor\n,\nor\n(12:42)\nwhere it is assumed that H<> tp, we have the situation depicted in (F12.22). The return at a given time is obtained from the area in\nthe circular annulus subtended by the pulse. This is referred to as pulse-limited operation. The return in the\nlimit ts >> may then be obtained from (12:46) as\n(12:48)\nThe above equation shows that the return consists of a rapid rise near t = 0, as expressed by the\nquantity in square brackets, followed by a very gradual exponential decay to zero. In this case, the leading\nedge of the return contains the desired information about the mean surface position and/or sea state; the","Sea State Effects\n12-39\nTransmitted pulse\nTransmitted\nReturn for beam-\nG2o(t)\nG20 (t)\npulse\nReturn for pulse-\nlimited operation\nlimited operation\nYB /2\n4B\n/2\nH\nCT/2\nCT/2\nEarth\nEarth\nAdvancing radar\nAdvancing radar\nresolution cell\na\nresolution cell\nPulse-Limited Altimeter\nBeam-Limited Altimeter\nFigure 12.22 Two modes of altimeter operation and the resulting signals.\nH= 435 km\nH= 435 km\n4B 3°\n4/8 3°\n0.9\n0.9\nT<15 nsec\nT<15 nsec\n0.8\n0.8\n0.7\n0.7\n0.6\n0.6\n0.5\n0.5\n0.4\n0.4\n0.3\n0.3\ne-'\n0.2\n0.1\n0.1\n-100-90-80-70-60-50-40-30 -20 -10 O 10 20 30 40 50 60 70 80 90 100\n-100 -90-80 -70-60-50-40-30 -20 -10 0 10 20 30 40 50 60 70 80 90 100\nTime, nanoseconds\nTime, nanoseconds\nFigure 12.23 Leading edge of averaged altimeter\nFigure 12.24 Derivative of leading edge of averaged\noutput versus time for pulse-limited operation.\naltimeter output versus time for pulse-limited\noperation.","Remote Sensing of Sea State by Radar\n12-40\nFlight #14\nRun 12\nH = 10 kft\nT = 20 nsec\ntr = 21 nsec\ntr =21 nsec\nMeasured wind = 12 knots\nCalculated wind = 14.1 knots\nFlight 16\nRun 9\nH = 10 kft\nT = 20 nsec\ntr = 30 nsec\ntr = 30 nsec\nMeasured wind = 22 knots\nCalculated wind = 21.2 knots\nFigure 12.25 Measured aircraft altimeter responses. Wind speeds inferred from rise times are compared to\nobserved wind speeds.\nH1/3= 3.1 ft\nH1/3 = 5.2 ft\n24\n28\n32\n36\n40\n20\n24\n28\n32\n36\n40\nTime, nanoseconds\nTime, nanoseconds\nMeasured radar response\nMeasured wavestaff response\nCalculated response\nFigure 12.26 Measured (after Yaplee et al., 1971) altimeter (impulse) responses versus calculated using\nbeam-limited model.","References\n12-41\nleading edge (normalized) is proportional to [1+d(t/tp)] Shown in (F12.23) are curves of this leading edge\nas a function of significant waveheight (related to rms waveheight in 12.1.1). Also shown on these curves are\nthe wind speeds required to fully arouse wind waves to these heights. The parameters chosen are typical of the\nNASA Skylab satellite altimeter. The figure shows that the mean surface position can be found from the mean\nreturn (in the absence of noise) by locating the half-power point on the leading edge.\nFor sea-state determination, one could determine the rise time of the leading edge. A possibly simpler\ntechnique would be to differentiate the mean return near the leading edge, producing a pulse-type signal as\nshown in (F12.24). This can be obtained from (12:48) as\n(12:49)\nThus the width of this pulse is directly proportional to the effective pulse width, tp. If the altimeter pulse\nwidth, T, is kept small with respect to the expected stretching due to roughness (i.e., less than about 15 ns),\nthen the width of this differentiated return is directly proportional to significant (or rms) waveheight. Thus an\norbiting altimeter such as this could very simply monitor the significant waveheight of the oceans along its\norbital path.\nAs limited validation of the model, we show results obtained by Raytheon (Ruck et al, 1971) from an\naircraft altimeter at 10,000 ft. The pulse width of 20 ns and beamwidth of 5° resulted in a nearly pulse-limited\noperation. Surface wind speeds were reported as 12 and 22 knots during two of the flights; these are compared\nin (F12.25) with wind speeds deduced from the rise times, tr, of the leading edge (assuming wind-driven waves\nin which waveheight is related to wind speed through (12:15)). The agreement seems quite reasonable.\n(b) Beam-Limited Operation. If the antenna beamwidth and/or altimeter height are sufficiently small,\nthen the illumination geometry shown in (F12.22) will result. Here, ts <\nensemble average of random variable X\na\nradar wavelength\neffective spatial altimeter pulse with\nvariable in elliptical coordinate\nXW\nu\nafter stretching by ocean waves\nsystems\nZ\nwave impedance of free space\nprincipal radii of curvature at point on\nP1,P2\n(~ 120 ohms)\nsurface\noo\ngrazing angle\naverage scattering cross section of sea\na\nper unit area\n'EM\ncoupling factor for electromagnetic\nsecond-order sea scatter terms\no(w)\naverage scattering cross section of sea\nper unit area per rad/s bandwidth\nTH\ncoupling factor for hydrodynamic\nsecond-order sea scatter terms\nradar temporal pulse width\nT","Remote Sensing of Sea State by Radar\n12-46\nerror function of argument X\n(x)\nazimuthal angle in bistatic radar near\nI\nsurface between incidence and\nscatter plane (4 = 180° for\nbackscatter); also, in radar\naltimeter, angle at earth center\nbetween altimeter and surface\nscattering patch\nazimuthal angle of scattered radar\nis\nwave from incidence plane\nhalf-power beamwidth of radar\nY B\naltimeter antenna\nangle of radar altimeter between\n4\nvertical and scattering point on\nearth\ntemporal wavenumber of ocean wave\nw\n(= 2n/T)\nradian wavenumber of radio carrier\nwo\nfrequency\nradian Doppler shift for near-grazing\nWB\nbackscatter (=1/2kog)\nmaximum satellite-induced (radian)\nw dm\nDoppler shift\nmaximum ship-induced (radian)\nwm\nDoppler shift\nradian Doppler shift from sides of\nWs\nellipse in bistatic radar","Chapter 13 ATMOSPHERIC MOTION BY DOPPLER RADAR\nD. A. Wilson and L. J. Miller\nWave Propagation Laboratory\nEnvironmental Research Laboratories\nNational Oceanic and Atmospheric Administration\nThis chapter reviews the principles and applications of Doppler radar to the study of\natmospheric motion. The first part of the chapter describes the amplitude and phase\ninformation contained in the backscattered signal and how this information can be extracted\nand processed to obtain backscattering cross sections and radial velocities of targets in the\natmosphere. The second part of the chapter then describes how this reflectivity and velocity\ninformation can be used in the investigation of the structure and motion fields associated with\natmospheric phenomena. It is seen that Doppler radar can be usefully employed as a research\ntool in several areas of meteorology including small scale turbulence, cloud physics, and the\ndynamics of convective storms and large scale storms.\n13.0 Introduction\nRadars transmit radio frequency (r-f) energy which is intercepted and reradiated by land, sea, and\natmospheric targets. Changes in the received signal from that transmitted give clues to important target\nparameters such as scattering cross section as well as indirect measurements of atmospheric absorption and\nattenuation. Radar, as used in the field of meteorology, allows one to detect the presence of objects in the\natmosphere, to recognize their character, and to determine their position in three-dimensional space.\nDoppler radar differs from conventional radar in that it is a very accurate phase measuring device, allowing\none to determine the position of the targets as a function of time; thus, their velocities can be determined.\n13.1\nBasic Radar Principles\nMost signals can be represented by the real or imaginary part of\ns(t) = A(t) exp\n(13:1)\nwhere the amplitude A, frequency w, and phase can be functions of time and j is the V-1. Some coding\nschemes for information transmission and reception are amplitude, frequency, or phase modulation. These\nare generated by varying A (AM), w (FM), or (PM) with time. Radar return signals are quite complex\nsince generally they consist of combined modulation. To extract information, therefore, it is necessary to\ndemodulate the return signal by appropriate means.\nRadiated signals can be of a continuous wave (CW) nature or pulsed. An additional transmission\nform is the so-called pulsed CW whereby CW source characteristics are maintained pulse-to-pulse. This\nchapter is restricted to a pulsed CW coherent Doppler radar. This system maintains constant amplitude,\nfrequency, and phase of the transmitted signal. Interpretation of amplitude and phase fluctuations of the\nr-f return leads to the Doppler or power density spectrum as a measure of target radial speed.\n13.1.1 The Backscattered Signal from a Point Target\nA point target (small cross-sectional area) within the scattering volume defined by the antenna\nbeamwidth and pulse duration T, see (F13.1), returns a signal whose instantaneous voltage is\nE(t) = A(t) cos [wct+(t)\n(13:2)\n,","Atmospheric Motion by Doppler Radar\n13-2\nwhere wc=2nfc is the constant carrier frequency and is the phase relative to the carrier phase. If the\ntarget is fixed, the phase is constant and a function of the distance r from the radar. A moving target\nhaving radial velocity VR returns a signal whose phase varies with time and is given by\n(13:3)\n+ VRt)\n,\nwhere l is the incident radiation wavelength and RO is the initial distance. For narrow beam radiation\npatterns, the radial velocity is approximately the projection of the vector velocity on the radar beam axis.\nr\nCT/2\n0\no\nFigure 13.1 Scattering volume at distance r from a pulsed radar having a pulse duration T and beamwidths\n0 and in the horizontal and vertical, respectively.\n13.1.2 The Backscattered Signal from Distributed Targets\nWhen the scattering volume contains N point targets, the return signal is the superposition of\nindividual returns. The instantaneous return voltage is then\nN\n(13:4)\nE(t) = An(t) cos\n,\nn=1\nwhere A is the amplitude and is the phase of the return signal from the th scatterer. The above\nn\nn\nexpression assumes secondary scattering effects are negligible compared to the first order scattering. In\naddition, target motion must be statistically independent and targets must move freely for several radar\nwavelengths (Lhermitte, 1968c) for a clear definition of individual phases. A more complicated expression\nis needed when these conditions are not satisfied. With the possible exception of heavy rain, snow, or hail,\nthe above expression is valid for atmospheric scattering.","Basic Radar Principles\n13-3\n13.1.3 The Radar Equation\nA target having a cross-sectional area AC C located at a distance r from the radar will intercept an\namount of power, PtGA/², where Pt is the transmitted power and G is the transmitting antenna gain\nfactor. If the target reradiates isotropically, the return power at the receiver is (Battan, 1959)\n(13:5)\nfor a receiving antenna having effective area A e The relationship between effective area and gain is (Kraus,\n1950)\n(13:6)\nA semiempirical expression for antenna gain of a paraboloidal reflector with aperture Ap is (Battan, 1959)\n(13:7)\nThis expression is sufficiently accurate for most weather radars.\nSince most targets do not scatter isotropically, it is convenient to introduce the backscattering cross\nsection o, defined as \"the area intercepting that amount of power, which, if scattered isotropically, would\nreturn an amount of power equal to that actually received\" (Battan, 1959), that is,\nPower reflected toward the receiving\n)\naperture per unit solid angle\n(Incident power density per 4n steradians)\nSubstituting backscatter cross section for geometric cross section and replacing the effective area with\n(13:6), the return power, (13:5), becomes\nPr=\n(13:8)\nwhere the constant K,=P+G2X/64m3 depends only on the particular radar system used and not the\nscatterer. For N targets where °n is the cross section of the nth scatterer, on the average, the return power\nis\n(13:9)\nwhere r is the range to the center of the scattering volume. The above expression assumes random phase of\nthe individual return voltages. A slightly more useful meteorological form is obtained by using the average\nradar cross section per unit volume and multiplying by the volume, V, effectively illuminated. This leads to\n(13:10)","Atmospheric Motion by Doppler Radar\n13-4\nThe quantity is called the radar reflectivity. The effective volume for distances much greater\nthan a pulse length is approximately\nv = nroger\n(13:11)\nwhere 0, are, respectively, the off-axis horizontal and vertical beam angles (assumed to be at most a few\ndegrees) and C (~3 X 108 m sec -1) is the propagation speed. The above expressions assume constant gain\nacross the antenna beam. Approximating the antenna pattern by a Gaussian beam (Lhermitte, 1963;\nNathanson and Reilly, 1968), the gain is\n(13:12)\nG(0,0)\nwhere 00,00 are the standard deviations of the two-way pattern (assumed to be at most a few degrees) and\nGo is the on-axis gain factor. Accounting for gain variations across the beam, the exact form of the radar\nequation, (13:9), becomes\n(13:13)\nIntroducing the radar reflectivity n = n(r, 0, 0), the summation can be expressed as a volume integral over\nthe pulse or contributing region SO that\n(13:14)\nUsing the Gaussian beam approximation over a volume having uniform reflectivity, integration leads to\n(Probert-Jones, 1962)\n(13:15)\nwhere ln2 is the natural logarithm of 2. Equation (13:15) has been grouped according to the constant\n(c/10242n2), the measurable radar parameters and target parameters (n/I2).\n13.1.3.1 Radar Backscattering Cross Section and the Weather Radar Equation\nA general treatment of plane wave scattering by a sphere was made by Mie (1908) and later restated\nby Stratton (1941), Goldstein (1946), and Kerr (1951). Ryde (1946) presented a theoretical paper\non Mie scattering applied to radar echoes from water and ice particles. From these theoretical\nconsiderations, it has been shown (Gunn and East, 1954) that the backscattering cross section for particle\ndiameters D <