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5.1 Summary

In this thesis, a parameterization method using radial basis functions was developed for regional gravity field modeling. It comprises (1) the design of the RBFs, including the choice of the spatial bandwidths and horizontal positions of the RBFs, and (2) the parameter determination. Moreover, the spatial bandwidths of the RBFs are determined by the type, spectral bandwidths, and radial distances of the RBFs.

Due to the availability of analytical expressions, the point mass RBFs were selected for representing the regional gravity field by fitting to different kinds of gravity data, leading to the so-called point mass method, which was applied together with the RCR technique. As a result, the input data were residuals with the low and very-high frequencies removed. To avoid errors caused by the mismatch of the spectral bandwidths between the RBFs and the input data, the point mass RBFs neglecting the low and very-high degree terms in the series expansions were preferred. Since the removal of the very-high degree terms had nearly no effect on the spatial bandwidths of the RBFs, only the terms from degree 0 to degreeNmin−1 were set to be zero (see Eq. (3.6)), yielding the reduced point mass RBFs. Accordingly, the point mass RBFs containing all degree terms were called the full RBFs. Both RBFs were tested numerically.

For the determination of the positions of the point mass RBFs (i.e., radial distances and horizontal positions), two cases were examined. In the first case, the point mass RBFs were located at the grid points. This point mass method was named PM-FIX. The employed grid type was the geographic grid. Afterwards, several model factors for designing the grid had to be considered. They were the grid extent, grid spacing, and grid depth. Furthermore, the grid formation was also of importance for achieving a good approximation of the gravity field. All these factors were investigated numerically.

In the second case, the related point mass method was called PM-FRE, in which the number and the positions of the RBFs were completely or partly unknown. A search process was developed for finding the RBFs, in which the magnitudes and positions of the RBFs were estimated simultaneously by solving a series of small-scale nonlinear problems. This search process aimed at minimizing the residuals between the predicted and observed gravity values (i.e., data misfit). Before starting it, several model factors (e.g., initial depth and depth limits, optimization direction, etc.) had to be defined properly so that a good approximation can be guaranteed. They were all numerically investi-gated and discussed. Due to the depth limits on the selected point mass RBFs, the nonlinear problem to be solved in the search process was bound-constrained. Consequently, the choice of a suitable optimization algorithm was necessary. Among the four tested iteration algorithms (i.e., LM, NLCG, L-BFGS, and L-BFGS-B), the L-BFGS-B algorithm proved to be the most proper one. The search process was usually terminated by satisfying a defined data misfit, or by satisfying a given number of point mass RBFs, which is defined based on the number of observations or by testing different choices.

Sometimes, the criterion for stopping the search process was realized by considering the data misfit as a function of the number of RBFs. In this case, if the data misfit decreased very slowly, the search can be stopped accordingly. After the search process, a set of point mass RBFs with known positions and magnitudes are obtained. Because the point mass RBFs were selected and estimated individually, a readjustment of the magnitudes of all found RBFs based on the whole input data was carried out while keeping the positions of the RBFs fixed. This led to the two-step approach of PM-FRE, which is one of the major innovations of this thesis.

No matter whether PM-FIX or PM-FRE was used, the final magnitudes of the point mass RBFs

were obtained by solving a linear equation system in the least-squares sense. In the case of ill-posedness, Tikhonov regularization was applied. Three approaches were compared for choosing the proper regularization parameter. Although the empirical approach provided the best solutions with the smallest errors when comparing to the independent data, the VCE technique proved to be more elegant. Besides its capability for the determination of the regularization parameter, it was also well suited for the data combination.

The applicability of this regional approach was demonstrated by analyzing the results of five nu-merical tests with synthetic and real data. The first nunu-merical test aimed at investigating the effect of different choices of model factors on the solutions. As a result, a computation procedure of PM-FIX was proposed for practical applications. Regarding PM-FRE, an empirical rule was developed for choosing the initial depth and depth limits and the radial-direction optimization was preferred in the search process. In the second step, the reduced RBFs are highly recommended for the readjust-ment of the magnitudes, leading to an unconstrained solution. Alternatively, the use of full RBFs for constructing the linear equation system and the adding of the constraints in the adjustment are also able to provide good solutions, resulting in a constrained solution. The unconstrained solution performed slightly better than the constrained one, and also was easier to calculate. Therefore, the former was preferred. By means of numerical comparisons between PM-FIX and PM-FRE, the lat-ter outperformed the former. Therefore, only PM-FRE was used in the remaining numerical tests.

Through the second and the third tests, it was shown that PM-FRE can recover the gravity anoma-lies from the observed geoid heights, providing better solutions than LSC. We also found that the use of full RBFs works fine for computing the unconstrained solution, indicating that the choice of spectral bandwidths of the RBFs depends on the type of modeled gravity field quantities. If they were in the short-wavelength domain, the use of reduced RBFs is not necessary. In addition, an iterative procedure based on VCE was developed for dealing with the case of several data groups with the same data type but different accuracies. The numerical results confirmed the advantage of the iterative procedure. On the basis of the three numerical tests, the “optimal” strategy for PM-FRE was obtained. In the fourth numerical test, a set of synthetic terrestrial and airborne gravity disturbances were used for the regional modeling. When using terrestrial data only, the solution of PM-FRE was compared to the LSC solution, showing a smaller RMS error of the modeled disturbing potential at ground. When considering both terrestrial and airborne data, the solutions obtained by using different schemes for finding the point mass RBFs were compared and validated, exhibiting a good agreement between each other. In the last numerical test in the Auvergne area using a large number of real gravity observations, both gravity and height anomaly solutions were calculated with small RMS (or STD) errors by PM-FRE. Two combination approaches were successfully applied to eliminate the discrepancies between the gravimetric and GPS/leveling-derived height anomalies.

In general, it can be concluded that, a good representation of the gravity field over a particular research area can be achieved by using PM-FRE as was demonstrated by five numerical tests. In comparison to the traditional integral method, PM-FRE is expected to predict all gravity field quan-tities with the use of resulting point mass RBFs. In all numerical tests carried out in this thesis, PM-FRE provides good gravity and height anomaly solutions. Comparing to PM-FIX or a related parameterization method, in which the RBFs are at the nodes of defined grids, PM-FRE is able to adapt the spatial distribution of the RBFs to the given gravity field characteristics. As can be seen from the Auvergne test case, more point mass RBFs are selected in the regions with rough gravity field features and fewer RBFs in areas with smooth structures. The advantage of PM-FRE with respect to LSC is that a smaller number of unknowns has to be estimated either in the nonlinear problems in the first step or in the linear problem in the second step. As a consequence, PM-FRE can be applied to some large-scale test cases (e.g., Auvergne test case) with the use of personal computer, whereas LSC can not. However, some effort is required for finding the point mass RBFs in the first step of PM-FRE, as a series of small-scale nonlinear problems has to be solved. Regarding the linear problem

in the second step, the computational complexity of the entire process of PM-FRE is quite high.

5.2 Outlook

Despite the satisfactory results already obtained by PM-FRE in this thesis, more numerical tests using real data sets are recommended for further testing of this method. There is still potential for further improvements of the method. In the current version, the Tikhonov regularization matrix used in the second step is simply taken as the identity matrix. The solution accuracy of PM-FRE might be improved if the identity regularization matrix is replaced by a diagonal matrix, of which the diagonal elements are appropriately tailored according to the signal content in the areas around the point mass RBFs. The respective information can be obtained from the available digital terrain models or a priori gravity field solutions. Similar studies have been carried out by Eicker (2008) and Eicker et al. (2014).

In this thesis, the point mass RBFs to be selected in a layer have the same initial depths and depth limits when using either the point mass search model SMA or SMB. A more reasonable case would be that each point mass RBF has its own initial depth and depth limits, which could be determined based on the density of the data points or the signal content in the area around the RBF. Further investigations are necessary.

So far, we have tested the performance of PM-FRE for regional modeling in the case of using a single data type with the same accuracy or different accuracies. Here arises the question how to deal with the case of several data groups with different data types. If the number of data points for one data group is much larger than the others, the point mass RBFs can be searched for based on only this data group; such a case has been verified in the Auvergne test case for the combination of gravity and GPS/leveling data. Otherwise, the search process becomes more complicated. A possible scheme is that the search process is carried out individually based on each data group. Suppose that there are a set of marine gravity data and a set of altimetry data, the first set of point mass RBFs can be obtained based on only the marine gravity data, then similar procedures are repeated for generating the second set of point mass RBFs based on the altimetry data. Afterwards, all searched point mass RBFs are readjusted with the input of the two data sets in the weighted least-squares. The weight for each data group can be determined by VCE or manually fixed by the user based on available a priori information. This is subject to further work.

Although Tenzer and Klees (2008) demonstrated that similar gravity solutions can be obtained when using different types of RBFs (i.e., different shape coefficients) if the spatial bandwidths of the RBFs are chosen properly, it is also of interest to apply the concept of PM-FRE to other types of RBFs. The approach offers various possibilities for further applications due to its flexible and modular structure.