The geoid-quasigeoid separation (GQS) traditionally uses the Bouguer anomalies to approximate the difference between the mean gravity and normal gravity along the plumb line. This approximation is adequate in flat and low elevation areas, but not in high and rugged mountains. To increase the accuracy, higher order terms of the corrections (potential and gravity gradient) to the approximation were computed in Colorado where the 1 cm geoid computation experiment was conducted. Over an area of 730 km by 560 km where the elevation ranges between 932 and 4,385 m, the potential correction (Pot. Corr.) reaches −0.190 m and its root mean square (RMS) is 0.019 m. The gravity gradient correction is small but has high variation: the RMS of the correction is merely 0.003 m but varies from −0.025 to 0.020 m. In addition, the difference between the Bouguer gravity anomaly and gravity disturbance causes about a 0.01 m bias and a maximum correction of 0.02 m. The total corrections range from −0.135 to 0.180 m, with an RMS value of 0.019 m for the region. The magnitude of the corrections is large enough and is not negligible considering today’s cm-geoid requirement. After the test in Colorado, the complete GQS term is computed in 1′ × 1′ grids for the experimental geoid 2020 (xGEOID20), which covers a region bordered by latitude 0–85° north, longitude 180–350° east. Over the land areas, the RMS of the GQS is 0.119 m and the maximum reaches 1.3 m. The RMS of the GQS increases with respect to the height until 4,000 m, then decreases unexpectedly. At the highest peaks (5,500–6,000 m) of Denali and Mount Logan, the RMS of the GQS ranges between 0.08 and 0.189 m. The small GQS at these high peaks are caused by steep slopes around the peaks that produce large Pot. Corr. caused by the topography. In addition, the higher order correction terms reach half of a meter in those peaks.

The geoid-quasigeoid separation (GQS) has traditionally used the Bouguer anomalies to approximate the difference between the mean gravity along the plumb line and the mean normal gravity (Heiskanen and Moritz

The rigorous formulation of the GQS is expanded into a Taylor series of the Bouguer disturbance in

Computation of the topographic effects on gravity and potential are split into near- and far-zone computations. It is shown in

Section

The sGQS has been computed by using the Bouguer anomaly Δ_{
B
} (Heiskanen and Moritz

For a better approximation, Flury and Rummel (

The GQS in terms of the Bouguer disturbance is derived and given in

In this article, we use Equation (

The first, second, and third terms on the right side of Equation (

The following datasets were used:

The xG20DEM in 3″ grid spacing along latitude and longitude directions (Krcmaric 2022). This DEM combined TanDEM-X (Wessel et al.

Terrestrial gravity data consisting of 1,633,376 point-gravity data provided by the National Geospatial-Intelligence Agency and 135,290 NGS point gravity data over the ocean areas, and the DTU15 (Andersen et al.

The xGEOID19B (Li et al.

The simple and complete GQS were computed using the abovementioned data-sets. Note that the gravity dataset used in the Colorado experiment was a subset of the terrestrial gravity (Wang et al.

Computations of the topographic effects on gravity and potentials were split into near-zone and far-zone contributions. It is shown in

The terrain correction was computed at every location of the terrestrial gravity observations using the xG20DEM elevation model in 3″

The difference between the Bouguer gravity disturbance and Bouguer gravity anomaly can be computed as

Because the correction terms have meaningful contribution only in high and rugged mountains, the Colorado region (35° ≤

Topographic height of xG20DEM in the Colorado area, spatial resolution 3″. The elevation has a mean value of 2,014 m and a STD DEV value of 614 m. The minimum height is 929 m and the highest peak is 4,381 m. The red line represents the GSVS17 (van Westrum et al.

The xGEOID19B ranges from −26.98 to −12.53 m of geoid height in this region. Thus, the gravity differences computed from equation (

Gravity difference correction to the GQS. Units in meters. Mean value = −0.011, Min = −0.020, Max = −0.007, Std Dev = 0.002, and RMS = 0.012.

Potential correction. Unit in meters. Mean value = −0.004, Min = −0.190, Max = 0.117, STD DEV = 0.018, and RMS = 0.019.

Contribution of the gravity gradients correction to the separation. Unit in meters. Mean value = 0.000, Min = −0.025, Max = 0.020, Std Dev = 0.003, and RMS = 0.003.

Difference between the complete and sGQS. Units in meters. Mean value = −0.08, Min = −0.135, Max = 0.180, Std Dev = 0.017, and RMS = 0.019.

The topographic potential is computed at the Earth’s surface and on the geoid using a 1′

The G.G. Corr. is a second-order term and it is plotted in

Similar to the G.D. Corr., the G.G. Corr. is small in terms of Std Dev and RMS values, and it ranges from −0.025 m to 0.020 m. In other words, the correction of gravity gradients is statistically small, but it can have a few cm contributions pointwise.

The combination of all three corrections, or the difference between the complete and sGQS, is plotted in

From all the above results, we have the following conclusions:

The largest correction comes from the potential difference term in Equation (

Contributions of the other two corrections are small. The G.G. Corr. is about 0.003 m in terms of RMS value, but it can reach 0.03 m pointwise in the extreme. The gravity correction (free-air vs gravity disturbance) has a −0.011 m bias and its extreme values reach −0.02 m because of the high elevation of the region.

In summary, the contribution of the correction terms to the GQS reaches 0.180 m in the Colorado area of steep topography where the average height of topography is 2,017 m and the highest peak reaches 4,385 m, and the RMS of the corrections is 0.019 m.

In the end, it is necessary to point out that the Bouguer anomaly is the main term in the GQS: it is an order larger than the correction terms. The simple separation (using the Bouguer anomalies only) is plotted in

The sGQS. Units in meters. Mean value = −0.458, Min = −1.376, Max = −0.126, Std Dev = 0.230, and RMS = 0.513.

As a comparison, the cGQS in Colorado is plotted in

The complete GQS. Units in meters. Mean value = −0.454, Min = −1.257, Max = −0.127, Std Dev = 0.223, and RMS = 0.506.

Comparing

After the regional runs for Colorado, the GQS was computed for the xGEOID20 region. Over ocean areas, the elevation is zero, therefore making the separation zero. Thus, the statistical analysis was on land only. To have a full picture, the statistics of the sGQS, cGQS and the correction terms are listed in (

Statistics of the simple separation (sGQS) and complete separation (cGQS), potential correction (Pot. Corr.), gravity gradient correction (G.G. Corr.), and gravity difference correction (G.D. Corr.), land only. Number of samples: 15,252,799. Units are in meters

sGQS | cGQS | Pot. Corr. | G.G. Corr. | G.D. Corr. | |
---|---|---|---|---|---|

Mean value | −0.057 | −0.055 | 0.001 | −0.000 | 0.002 |

Std Dev | 0.119 | 0.119 | 0.006 | 0.002 | 0.011 |

RMS | 0.132 | 0.131 | 0.006 | 0.002 | 0.011 |

Min | −1.348 | −1.275 | −0.144 | −0.090 | −0.031 |

Max | 1.082 | 1.296 | 0.590 | 0.102 | 0.065 |

Statistically, the sGQS and cGQS are very close. The corrections are around a cm or less in terms of the RMS, but the corrections can reach 0.590 and 0.102 m pointwise for the potential and the G.G. Corrs., respectively. Large corrections happen in high mountains (

RMS values of the GQS as a function of height. Units in meters

Elevation (m) | No. of pts | sGQS | cGQS | Pot. Corr. | G.G. Corr. | G.D. Corr. |
---|---|---|---|---|---|---|

0–500 | 8,778,198 | 0.013 | 0.015 | 0.001 | 0.000 | 0.002 |

500–1,000 | 2,544,283 | 0.055 | 0.058 | 0.003 | 0.001 | 0.005 |

1,000–1,500 | 1,376,066 | 0.143 | 0.147 | 0.007 | 0.002 | 0.009 |

1,500–2,000 | 1,051,370 | 0.236 | 0.239 | 0.010 | 0.002 | 0.015 |

2,000–2,500 | 807,425 | 0.318 | 0.315 | 0.012 | 0.003 | 0.023 |

2,500–3,000 | 553,678 | 0.293 | 0.285 | 0.014 | 0.005 | 0.034 |

3,000–3,500 | 132,939 | 0.513 | 0.490 | 0.025 | 0.010 | 0.040 |

3,500–4,000 | 7,792 | 0.754 | 0.717 | 0.088 | 0.008 | 0.023 |

4,000–4,500 | 806 | 0.451 | 0.387 | 0.149 | 0.010 | 0.023 |

4,500–5,000 | 187 | 0.481 | 0.431 | 0.254 | 0.013 | 0.027 |

5,000–5,500 | 48 | 0.423 | 0.293 | 0.374 | 0.016 | 0.032 |

5,500–6,000 | 7 | 0.433 | 0.136 | 0.522 | 0.016 | 0.033 |

0–6000 | 15,252,799 | 0.132 | 13.1 | 0.006 | 0.002 | 0.011 |

It has been assumed that the magnitude of GQS would increase with height, and the largest correction would be found around the highest mountains. Statistical analysis of the computed GQS supports this assumption partially, but with some modification.

To inspect the unexpected small GQS in detail, the Bouguer anomaly (BA), gravity gradient of the Bouguer disturbance G1, the terrain correction (TC) and the GQS terms for seven cells are listed in

Gravities and GQS in cells higher than 5,500 m

Mount Logan | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Latitude (degree) | Longitude (degree) | Orth H (m) | BA (mGal) | G1 (mGal) | TC (mGal) | sGQS (m) | cGQS (m) | Pot. Corr. (m) | G.G. Corr. (m) | G.D. Corr. (m) |

60.56667 | −140.4000 | 5,589 | −79.9 | 8.1 | 146.1 | −0.455 | 0.150 | 0.548 | 0.023 | 0.034 |

60.58333 | −140.4500 | 5,531 | −76.1 | 9.5 | 106.6 | −0.428 | 0.105 | 0.472 | 0.027 | 0.034 |

60.58333 | −140.4333 | 5,624 | −78.6 | 8.7 | 112.0 | −0.450 | 0.118 | 0.508 | 0.025 | 0.035 |

Mountain Denali | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Latitude (degree) | Longitude (degree) | Orth H (m) | BA (mGal) | G1 (mGal) | TC (mGal) | sGQS (m) | cGQS (m) | Pot. Corr. (m) | G.G. Corr. (m) | G.D. Corr. (m) |

63.06667 | −151.0167 | 5,715 | −73.9 | 0.7 | 157.4 | −0.430 | 0.173 | 0.570 | 0.002 | 0.031 |

63.06667 | −151.0000 | 5,768 | −74.0 | 0.7 | 158.8 | −0.435 | 0.189 | 0.590 | 0.002 | 0.032 |

63.08333 | −151.0333 | 5,549 | −73.6 | 0.5 | 129.6 | −0.416 | 0.093 | 0.477 | 0.001 | 0.030 |

63.08333 | −151.0167 | 5,589 | −73.7 | 0.5 | 123.6 | −0.420 | 0.088 | 0.475 | 0.001 | 0.031 |

The steep slopes around the peaks cause large terrain corrections and large Pot. Corrs. that compensate for the contribution of the Bouguer disturbances, resulting in small GQS. The Pot. Corr. is around a half meter for those cells, which has the same magnitude as that of the sGQS. The G.G. Corr. is in mm level for the peaks of Denali, but over 2 cm at the peaks of Mount Logan. The G.D. Corrs. at the peaks are around 3 cm.

The GQS is a combination effect of gravity anomalies, the terrain roughness, and the height of the topography. Thus, the extreme values of GQS happen not necessarily at the highest mountains. The information for the extreme values is listed in

Information for the maximum and minimum values of the GQS separation

Latitude (degree) | Longitude (degree) | Orth H (m) | BA (mGal) | G1 (mGal) | TC (mGal) | sGQS (m) | cGQS (m) | Pot. Corr. (m) | G.G. Corr. (m) | G.D. Corr. (m) |
---|---|---|---|---|---|---|---|---|---|---|

19.81667 | −155.4667 | 3,992 | 265.2 | 9.7 | 45.6 | 1.082 | 1.296 | 0.161 | 0.020 | 0.033 |

37.85000 | −107.4500 | 3,756 | −331.3 | −10.0 | 2.7 | −1.270 | −1.275 | 0.032 | −0.019 | −0.018 |

The maximum separation happens at Mauna Kea, Hawaii. The cell has an elevation of 3,992 m (xG20DEM), and the Bouguer anomaly is positive 265 mGals. The Pot. Corr. to the GQS reaches 16 cm. The minimum separation is at Uncompahgre Peak, Colorado with the height of 3,756 m (xG20DEM) where the terrain seems moderate, indicated by a small terrain correction of 2.7 mGal (

The cGQS in the experimental geoid region. Units in meters. Decimeter to meter separations happen in the rocky mountains. The large corrections in Greenland are due to sparse surface gravity observations, resulting in large errors in the Bouguer anomalies. Improvement will be made in the future by using the ARCGP gravity and the ice thickness data in the area.

Three correction terms were added to the simple GQS based on the Bouguer gravity anomalies. The correction terms are first tested in Colorado. The results show that the corrections are small in terms of RMS value (0.019 m), but reach a few cm to decimeters (0.19 m) pointwise. The corrections are significant for the cm geoid computations and must be included in computations.

The cGQS was then computed for xGEOID20. Statistically, more than half of the 1′

The magnitude of GQS is a combination effect of gravity anomalies, terrain roughness, and height of the topography (

The far-zone topographic effects are in the second and higher order terms, which are neglected in this computation. The higher order of vertical derivatives of Bouguer disturbance is also omitted. Even if they are considered as negligible for most cases, the effects will be addressed in the next experimental geoid computations to make sure that they are below 1-cm pointwise.

First, we define the Bouguer disturbing potential

Denoting the vertical derivatives of the potentials as

The Bouguer disturbance along the plumb line can be expanded into a Taylor series as

Equation (

Integrating the Bouguer gravity disturbance in Equation (

From Equation (

The mean gravity disturbance

The mean gravity along the plumb line is

The rigorous GQS is then

Equation (

Computation of topographic effects to gravity and potential are global integrals. For practical computations, it can be split into near-zone (NZ) and far-zone (FZ) computations:

The potential of FZ contribution can be expanded into a Taylor series as

It is necessary to point out that the topographic potential of NZ contribution cannot be expanded into a Taylor series such as Equation (

Substituting Equations (

For convenience, we define the order of approximation by the power of the elevation

Equation (