Regional gravity field refinement for (quasi-) geoid determination based on spherical radial basis functions in Colorado

Abstract

This study presents a solution of the ‘1 cm Geoid Experiment’ (Colorado Experiment) using spherical radial basis functions (SRBFs). As the only group using SRBFs among the fourteen participated institutions from all over the world, we highlight the methodology of SRBFs in this paper. Detailed explanations are given regarding the settings of the four most important factors that influence the performance of SRBFs in gravity field modeling, namely (1) the choosing bandwidth, (2) the locations of the SRBFs, (3) the type of the SRBFs as well as (4) the extensions of the data zone for reducing the edge effect. Two types of basis functions covering the same spectral range are used for the terrestrial and the airborne measurements, respectively. The non-smoothing Shannon function is applied to the terrestrial data to avoid the loss of spectral information. The cubic polynomial (CuP) function which has smoothing features is applied to the airborne data as a low-pass filter for filtering the high-frequency noise. Although the idea of combining different SRBFs for different observations was proven in theory to be possible, it is applied to real data for the first time, in this study. The RMS error of our height anomaly result along the GSVS17 benchmarks w.r.t the validation data (which is the mean results of the other contributions in the ‘Colorado Experiment’) drops by 5% when combining the Shannon function for the terrestrial data and the CuP function for the airborne data, compared to those obtained by using the Shannon function for both the two data sets. This improvement indicates the validity and benefits of using different SRBFs for different observation types. Global gravity model (GGM), topographic model, the terrestrial gravity data, as well as the airborne gravity data are combined, and the contribution of each data set to the final solution is discussed. By adding the terrestrial data to the GGM and the topographic model, the RMS error of the height anomaly result w.r.t the validation data drops from 4 to 1.8 cm, and it is further reduced to 1 cm by including the airborne data. Comparisons with the mean results of all the contributions show that our height anomaly and geoid height solutions at the GSVS17 benchmarks have an RMS error of 1.0 cm and 1.3 cm, respectively; and our height anomaly results give an RMS value of 1.6 cm in the whole study area, which are all the smallest among the participants.

Combination of different observation types through a multi-resolution representation of the regional gravity field using the pyramid algorithm and parameter estimation

“…SRBFs (1) can act as high-pass, low-pass or band-pass filters, depending on the chosen Legendre coefficients B n , and a harmonic function F(x) can be filtered by it through a spherical convolution (Schmidt et al 2007;Liu et al 2020b). In case of using a band-limited SRBF, e.g., a spherical scaling function B(x, x k ) =: i (x, x k,i ), which means the Legendre coefficient B n =: φ n,i > 0 for degree n = 0, 1, .…”

“…where K i and d k,i are the number of grid points and the corresponding series coefficients, respectively, at the level i. As explained in Liu et al (2020b), the coefficients d k,i do not depend on the choice of the SRBFs, as soon as the SRBFs are band-limited up to the same degree, i.e., their Legendre coefficients are equal to 0 for all degree values n > 2 i − 1. Thus, the same set of unknown coefficients d k,i at level i can be used with both the spherical scaling function and the spherical wavelet function of the same level.…”

Quasi Geoid and Geoid Modeling with the Use of Terrestrial and Airborne Gravity Data by the GGI Method—A Case Study in the Mountainous Area of Colorado

“…The geoid model was determined on the basis of the quasigeoid model, based on relation (28). The 𝑔̅ values necessary to calculate the geoid to quasigeoid separation (27) were calculated according to formula (31). The gravity values 𝑔 at each 1′ × 1′ grid point were determined by the interpolation of the complete Bouguer anomalies (CBA) using the kriging method.…”

“…An alternative to the RCR method is the method developed at the Royal Institute of Technology in Stockholm (KTH method), which does not require gravity reduction, but consists of a least squares modification of Stokes' formula [18][19][20][21]. An alternative solution to that presented above is the method based on spherical radial basis functions (SRBFs) [22,23], which have been applied in gravity field modeling, e.g., [24][25][26][27]. The established approaches to regional geoid and quasigeoid model determination were utilized in the Colorado geoid experiment, and a detailed description can be found in the publications on this experiment, while a general description and comparison are included in [28].…”

Unified Land–Ocean Quasi-Geoid Computation from Heterogeneous Data Sets Based on Radial Basis Functions

“…Wu et al adopt zero-order and first-order Tikhonov regularization to solve ill-posed equations and prove that firstorder regularization has better performance [29]. Based on the Tikhonov regularization, Liu et al analyze the defects of the L-curve method and variance component estimation (VCE) in determining regularization parameters and then propose two combined methods, VCE-Lc and Lc-VCE, which are proven to be superior to traditional methods [30,31]. The spatial positions of RBFs have a great influence on the modeling result.…”