Local gravity field modeling using basis functions of harmonic nature and vector airborne Gravimetry, Case Study: Gravity field modeling over north-east of Tanzania region


1 Ph.D. Student, Department of Geodesy, Faculty of Surveying and Geomatic Engineering, K. N. Toosi University of Technology, Tehran, Iran

2 Assistant Professor, Department of Geodesy, Faculty of Surveying and Geomatic Engineering, K. N. Toosi University of Technology, Tehran, Iran


Many different methods for gravity field modelling have been investigated, among which, the harmonic expansion has been widely used due to harmonic nature of gravity potential field that satisfies Laplace equation in an empty space. This method, however, cannot reach to a high resolution in a gravity field, and suffers from omitting the high frequency gravity signals and therefore it is not appropriate for local gravity field modelling. To overcome this drawback and recover high frequency features of gravity field, appropriate basis functions with local support should be used. One of the methods for local gravity field modeling based on local harmonic function is spherical cap harmonic analysis. In this method, the Dirikhlet boundary value problem for Laplace equation is solved for boundary conditions on the surface of a spherical cap which results in Eigen expansion of the solution in terms of the associated Legendre function of non-integer degree and integer order. Another method that can be used for local gravity field modeling is rectangular harmonic analysis. In this method, Laplace equation is solved in a local Cartesian coordinate system and boundary conditions are applied on a plane area which. In this approach, trigonometric functions are used as basis functions.
In this study, the problem of local gravity field modeling based on both spherical cap, and rectangular harmonic expansion is investigated. Also, a numerical study is conducted to show the performance of each method for local gravity field modeling. To do so the observations of vector airborne gravimetry in the northwest of Tanzania in Highland region are used to derive the coefficients of each model. The low-frequency part of observed gravity field is removed from the data using EGM2008 geo-potential model, and the resulting residual gravity field is considered for local modelling. Since the governing equations for determination of the coefficients suffer from an ill-conditioning problem, it is necessary to apply some regularization schemes to find the optimum solution. Here, the Tikhonov regularization method is utilized to obtain the regular solution. In this study, the edge effect for each model is also analyzed. To show this effect, the results of models are compared with the observations of gravity at some control points distributed both within the study area and its margin. It should be noted that the maximum degree of expansion in harmonic series, plays an important role in appropriate fitting of local gravity field models to the gravity data and it has significant effects on the computational task of determining the coefficients of each model. For this purpose, local gravity field modelling is calculated with different value of maximum degree of expansion and then regarding to the result (accuracy of local gravity model by comparing with control points), appropriate value of maximum degree of expansion for each model is determined.
Finally the results of two models are compared to each other to show the performance of each models in local gravity field modeling. The results of this study reveal that ASHA has the ability to model local gravity with accuracy of about 1 mGal, and RHA method in the best situation can just achieve to a 3 mGal accuracy, although the convergence rate in RHA model is faster than ASHA model. Also by comparing the edge effect on each models, it is seen that the edge effect in two models and in all directions occurred but in a Z direction of RHA model that are more significant than the other directions in two models and one may conclude that the edge effect of RHA are much larger than that of ASHA. Finally, the result obtained shows that ASHA model can have better results for local gravity modelling.


Main Subjects

Alldredge, L., 1981, Rectangular harmonic analysis applied to the geomagnetic field, Journal of Geophysical Research: Solid Earth, 86, 3021-3026.
De Santis, A., 1992, Conventional spherical harmonic analysis for regional modelling of the geomagnetic field, Geophysical research letters, 19, 1065-1067.
De Santis, A. and Torta, J., 1997, Spherical cap harmonic analysis: a comment on its proper use for local gravity field representation, Journal of Geodesy, 71, 526-532.
De Santis, A., Torta, J. and Lowes, F., 1999, Spherical cap harmonics revisited and their relationship to ordinary spherical harmonics, Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy, 24, 935-941.
Hansen, P. C., 1999, The L-curve and its use in the numerical treatment of inverse problems.
Haines, G., 1985, Spherical cap harmonic analysis, Journal of Geophysical Research: Solid Earth, 90, 2583-2591.
Hwang, J. S., Han, H.C., Han, S. C., Kim, K. O., Kim, J. H., Kang, M. H. and Kim, C. H., 2012, Gravity and geoid model in South Korea and its vicinity by spherical cap harmonic analysis, Journal of Geodynamics, 53, 27-33.
Jiang, T., Li, J., Dang, Y., Zhang, C., Wang, Z. and Ke, B., 2014, Regional gravity field modeling based on rectangular harmonic analysis, Science China. Earth Sciences, 57, 1637.
Liu, J., Chen, R., Wang, Z. and Zhang, H., 2011, Spherical cap harmonic model for mapping and predicting regional TEC, GPS solutions, 15, 109-119.
Malin, S., Düzgit, Z. and Baydemir, N., 1996, Rectangular harmonic analysis revisited, Journal of Geophysical Research: Solid Earth, 101, 28205-28209.
Razin, M. R. G. and Voosoghi, B., 2017, Regional ionosphere modeling using spherical cap harmonics and empirical orthogonal functions over Iran, Acta Geodaetica et Geophysica, 52, 19-33.
Schwarz, K., Kern, M. and Nassar, S., 2002., Estimating the gravity disturbance vector from airborne gravimetry Vistas for Geodesy in the New Millennium: Springer, 199-204.
Thébault, E., Schott, J., Mandea, M. and Hoffbeck, J., 2004, A new proposal for spherical cap harmonic modelling, Geophysical Journal International, 159, 83-103.
Thébault, E., Schott, J. and Mandea, M., 2006, Revised spherical cap harmonic analysis (R‐SCHA): Validation and properties, Journal of Geophysical Research: Solid Earth, 111.
Younis, G., 2013, Regional gravity field modeling with adjusted spherical cap harmonics in an integrated approach: TU Darmstadt.
Younis, G. K., Jäger, R. and Becker, M., 2013, Transformation of global spherical harmonic models of the gravity field to a local adjusted spherical cap harmonic model. Arabian Journal of Geosciences, 6, 375-381.
Younis, G., 2015, Local earth gravity/potential modeling using ASCH, Arabian Journal of Geosciences, 8, 8681-8685.