Impacts of Using the Rigorous Topographic Gravity Modeling Method and Lateral Density Variation Model on Topographic Reductions and Geoid Modeling: A Case Study in Colorado, USA

Abstract

Until now, the prismatic mass approximation of the topography and the constant density assumption have been mostly utilized in topographic reductions, which are rough approximations of reality and can be avoided. In this study, the more rigorous tesseroidal mass representation of topographic masses and the global lateral topographic density variation model UNB_TopoDens are considered in topographic reductions. Three tesseroidal modeling methods based on different combinations of numerical tesseroidal approaches are developed for precise topographic gravity modeling. The computational performances of the classic prismatic modeling method and new tesseroidal modeling methods in computing the residual terrain modeling (RTM), terrain correction (TC), full topographic, and Airy-Heiskanen (AH) model-based isostatic effects are tested in the Colorado area with rugged topography. In addition, the improvement of computational efficiency achieved by applying the OpenMP parallelizing technique and the contribution of considering the UNB_TopoDens model are investigated. Then, the RTM effects are applied to local geoid modeling to see the geoid model changes caused by using rigorous tesseroidal modeling methods and by considering lateral density variations. The main numerical findings are: (1) The application of the OpenMP parallelization can significantly reduce the computational time, while the efficiency improvement rate depends on the number of used threads; (2) The modeling method effect on the computation of the RTM, TC, full topographic, and AH isostatic effects is smaller than the lateral density variation effect; (3) In the case of using the RTM reduction, the use of the tesseroidal modeling method instead of the prismatic modeling method can cause geoid model differences at the millimeter level, while almost the same standard deviations are obtained by comparing the geoid models to the GSVS17 and historical GNSS-leveling data; (4) the differences in the geoid height due to lateral density variations can reach a magnitude of about 8 cm when using the RTM reduction scheme, while the validation of geoid models at the GSVS17 GNSS-leveling benchmarks revealed that the geoid considering the UNB_TopoDens model has a slightly larger standard deviation than the one using a constant density of $2.67\;{\text{g}}/{\text{cm}}^{3}$.

Appendix

The statistics of the RTM, TC, full topographic, and AH isostatic effects computed by the TC77/TC90, TCTESSV1a, TCTESSV1b, and TCTESSV2 methods considering the UNB_TopoDens model, as well as their differences with respect to the corresponding results computed based on the constant density model with the value of $2.67\;{\text{g}}/{\text{cm}}^{3}$ are given in Tables 11, 12, 13, 14. Through observing the values in these four tables and Table 9, we can see that the computed topographic effects and the influences of lateral density variations are quite similar among all topographic gravity modeling methods.

Table 11 The same as for Table 9 but using the TC77/TC90 method

Table 12 The same as for Table 9 but using the TCTESSV1a method

Table 13 The same as for Table 9 but using the TCTESSV1b method

Table 14 The same as for Table 9 but using the TCTESSV2 method

The differences of the geoid models corresponding to TCTESSV2 and TCTESSV3, TCTESSV1a and TCTESSV3 are illustrated in Figs.

Fig. 15

Differences of the geoid models. The used RTM effects are computed by TCTESSV2 and TCTESSV3 using the constant density model with the value of $2.67\;{\text{g}}/{\text{cm}}^{3}$

15 and

Fig. 16

Differences of the geoid models. The used RTM effects are computed by TCTESSV1a and TCTESSV3 using the constant density model with the value of $2.67\;{\text{g}}/{\text{cm}}^{3}$

16, respectively. Because the RTM effects computed by TCTESSV2 and TCTESSV3 are almost the same, the resulting geoid differences are so small that can be neglected, with the mean value of zero, the SD of about 0.018 mm, the minimum and maximum values of about -1 mm and 1 mm (see Fig. 15). Because of the larger differences between the RTM effects computed by TCTESSV1a and TCTESSV3 than those between TCTESSV2 and TCTESSV3, slightly larger geoid differences having the mean value of about zero, the SD of about 0.39 mm, the minimum and maximum values of about -2.2 mm and 3.4 mm are observed (see Fig. 16). Through analyzing Figs. 9, 10, 15, and 16, we find that the geoid changes caused by using different topographic gravity modeling methods are at the millimeter level in the Colorado area.