A Direct Approach for Local Quasi-Geoid Modeling Based on Spherical Radial Basis Functions Using a Noisy Satellite-Only Global Gravity Field Model

Abstract

The remove–compute–restore (RCR) approach is widely used in local quasi-geoid modeling. However, the classical RCR approach usually does not take into account the noise of the satellite-only global gravity field model (GGM), which may lead to a suboptimal result. This paper presents an approach for local quasi-geoid modeling based on spherical radial basis functions that combines local noisy datasets and a noisy satellite-only GGM. This approach includes an RCR procedure using a satellite-only GGM. This is a direct approach that takes the spherical harmonic coefficients of satellite-only GGM as a noisy dataset and includes the corresponding full-noise covariance matrix in the least-squares estimation, aiming to obtain a statistically optimal local quasi-geoid model. The direct approach goes beyond the indirect approach, which treats the height anomalies generated from the satellite-only GGM as a noisy dataset. However, the generated GGM height anomaly dataset is not an equivalent representation of the satellite-only GGM, which may result in the loss of information from the satellite-only GGM. Through mathematical deduction, we demonstrate the theoretical consistency between the direct approach and the indirect approach. The direct approach also has an advantage over the indirect approach in terms of computational complexity due to the simpler algorithm. We conducted a synthetic closed-loop test with a real data distribution in Colorado, and numerical results demonstrated the advantage of the direct approach in local quasi-geoid modeling. In terms of the root mean square of the differences between the predicted values and the true reference values, the direct approach provided an improvement of approximately 14% compared to the indirect approach.

1. Introduction

Exploring high-precision methods for local quasi-geoid modeling is of great significance for the realization of the International Height Reference System (IHRS) [1]. This is because the disturbing potential values at reference stations need to be determined using the appropriate quasi-geoid modeling method, which is crucial for calculating the geopotential numbers of the IHRS reference stations [2].

Over the past two decades, the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On missions and the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) mission have collected a large amount of satellite-to-satellite tracking (SST) data and gravity gradient data, respectively. Based on these satellite data, many satellite-only global gravity field models (GGMs) have been established, providing the middle- to long-wavelength signal of the Earth’s gravity field. Moreover, with the use of post-fit residual analysis [3], an effective tool for analyzing the noise model of satellite data, the latest satellite-only GGMs are equipped with reliable full-noise covariance matrices (NCMs), such as GOCO05s [4] and GOCO06s [5]. To this day, the remove–compute–restore (RCR) approach [6] is still frequently used in local quasi-geoid modeling. However, the classical RCR approach does not use the satellite-only GGM in the same way as local datasets such as terrestrial and airborne data, thereby neglecting the noise in the satellite-only GGM, which may result in a suboptimal model [7].

Based on the theory of the geodetic boundary-value problems, commonly used methods for local quasi-geoid modeling can be summarized as three types: Stokes/Molodensky integral and its various modified formulas, least-squares collocation (LSC), and the spherical radial basis function (SRBF) method [8,9,10]. In theory, for all three types of methods, some publications have already considered the combination of local datasets with a noisy GGM in the RCR procedure. The spectral combination approach [11,12,13] includes the modified Stokes kernel and employs the error degree variances of the GGM, but generally does not employ the full NCM of the GGM. The recently proposed residual LSC (RLSC) approach [14,15] is an improvement upon the standard LSC approach, and the modified formula of LSC includes the full NCM of the GGM. Compared to covariance matrices derived from error degree variances or empirical covariance fitting, using an anisotropic and location-dependent full NCM in local quasi-geoid modeling facilitates realistic error model estimation. In this paper, we focus on the combination of local datasets with a noisy GGM in the SRBF method.

SRBFs have good localization properties, and it is easy to combine heterogeneous datasets using SRBFs. Therefore, SRBFs are a powerful tool for local quasi-geoid modeling. In particular, when using least-squares techniques to estimate the SRBF model, it is possible to include the NCMs of the datasets and provide the NCM of the estimated parameters, which is of significant importance for uncertainty analysis of the geopotential numbers at the IHRS reference stations. For the SRBF method, there are currently two statistically optimal ways to combine local datasets and a noisy satellite-only GGM (or satellite datasets): (1) directly using the original satellite datasets along with their NCMs [16]; (2) using the height anomalies computed on the Earth’s surface via spherical harmonic (SH) synthesis from the satellite-only GGM, with the corresponding NCM propagated from the NCM of the satellite-only GGM’s SH coefficients [7,17,18]. In the first approach, the complex functional model and the huge amount of data tend to constrain the use of low-low and high-low SST data. The second approach (referred to as the “indirect approach” for brevity) has been recently proposed. The indirect approach first computes the height anomalies using a satellite-only GGM at pre-defined grid points on the Earth’s surface, treating them as a low-resolution GGM dataset. Then, by combining the GGM dataset and the high-resolution local datasets, a two-scale SRBF model is established. There are two main problems with the indirect approach: (1) the generated GGM height anomaly dataset is not an equivalent representation of the satellite-only GGM, making this conversion non-lossless; (2) it is necessary to sufficiently expand the local data area to compute the coarse-scale model, whereas, in practice, it cannot be guaranteed that the neighboring area of the target area has available high-resolution datasets. Moreover, the NCM of the GGM height anomaly dataset is extremely ill-conditioned, increasing the difficulty of parameter estimation [17].

In this study, we propose a direct approach for combining local datasets and a noisy satellite-only GGM. In the direct approach, the SH coefficients of the satellite-only GGM and their full NCM are directly treated as a noisy dataset. Compared to the first two approaches, the direct approach is simpler and more efficient. This is because SRBFs are constructed based on SHs, and there exists a close theoretical connection between them [19]. Moreover, the SRBF coefficients and SH coefficients have an explicit analytical relationship [20]. To the best of our knowledge, no publication has studied the direct approach. One important reason may be that the analytical relationship between the SRBF coefficients and the SH coefficients relies on the orthogonality of the surface SHs over the unit sphere. If the target area is on a global scale, the direct approach can certainly be adopted. However, can the direct approach still be adopted if the target area is on a local scale? Furthermore, what is the connection between the direct approach and the indirect approach? This paper aims to address the two research questions.

This paper is organized as follows. In Section 2, we present the direct approach and revisit the indirect approach, demonstrating their theoretical consistency. In Section 3, we conduct a synthetic closed-loop test. In Section 4, we analyze and discuss the experimental results. Finally, we provide the conclusion and give the outlook in Section 5.

2. Methodology

2.1. SHs and SRBFs

Let x be a three-dimensional coordinate vector of any point outside the sphere ${\mathrm{\Omega }}_R$ with reference radius R, and $\widehat{x}$ denotes the normalized vector of x. The disturbing potential of point x has the following approximate SH expansion:

$$T(\mathit{x})=\frac{GM}{R}{\displaystyle \sum _{l=0}^L{\displaystyle \sum _{m=-l}^l{\left(\frac{R}{\left|x\right|}\right)}^{l+1}{\overline{c}}_{l,m}}}{\overline{Y}}_{l,m}(\widehat{\mathit{x}})$$

(1)

where GM denotes the product of the gravitational constant and mass of the Earth, and L denotes the maximum SH degree. ${\overline{c}}_{l,m}$ and ${\overline{Y}}_{l,m}$ denote the 4π fully normalized SH coefficients and the surface SH of degree l and order m, respectively. We can rewrite Equation (1) as

$$T(\mathit{x})=\frac{GM}{R}\mathit{h}(\mathit{x})\mathit{C}$$

(2)

wherein

$$\mathit{h}(\mathit{x})=\left({\left(\frac{R}{\left|\mathit{x}\right|}\right)}^1{\overline{Y}}_{0,0}(\widehat{\mathit{x}}),{\left(\frac{R}{\left|\mathit{x}\right|}\right)}^2{\overline{Y}}_{1,-1}(\widehat{\mathit{x}}),\dots ,{\left(\frac{R}{\left|\mathit{x}\right|}\right)}^{L+1}{\overline{Y}}_{L,L}(\widehat{\mathit{x}})\right)$$

(3)

$$\mathit{C}={\left({\overline{c}}_{0,0},{\overline{c}}_{1,-1},\dots ,{\overline{c}}_{L,L}\right)}^{\mathrm{T}}.$$

(4)

Let z be a three-dimensional coordinate vector of a point on the sphere ${\mathrm{\Omega }}_r$ with radius r, ${\mathit{z}}_i,i=1,2,\dots ,I$ uniformly covering the whole sphere ${\mathrm{\Omega }}_r$, and the sphere ${\mathrm{\Omega }}_r$ is completely enclosed by the topographic masses. The disturbing potential at any point x outside the sphere ${\mathrm{\Omega }}_r$ can be approximated by a series expansion of the SRBFs:

$$T(\mathit{x})={\displaystyle \sum _{i=1}^I{\beta }_i}B(\mathit{x},{\mathit{z}}_i)$$

(5)

where ${\beta }_i$ denotes the SRBF coefficient. SRBF is defined as

$$B(\mathit{x},{\mathit{z}}_i)={\displaystyle \sum _{l=0}^L\frac{2l+1}{4\pi }}{\left(\frac{r}{\left|\mathit{x}\right|}\right)}^{l+1}{B}_l{P}_l({\widehat{\mathit{x}}}^{\mathrm{T}}{\widehat{\mathit{z}}}_i)$$

(6)

where $B_l$ and $P_l$ denote the Legendre coefficient and the Legendre polynomial of degree l, respectively. Based on the orthogonality relations of the surface SHs on the unit sphere, the following SHs addition theorem holds [21]:

$$P_l({\widehat{\mathit{x}}}^{\mathrm{T}}{\widehat{\mathit{z}}}_i)=\frac{1}{2l+1}{\displaystyle \sum _{m=-l}^l{\overline{Y}}_{l,m}(\widehat{\mathit{x}}){\overline{Y}}_{l,m}({\widehat{\mathit{z}}}_i)}.$$

(7)

Substituting Equations (6) and (7) into Equation (5) yields the following equation:

$$T(\mathit{x})={\displaystyle \sum _{i=0}^I{\beta }_i}{\displaystyle \sum _{l=0}^L{\displaystyle \sum _{m=-l}^l\frac{1}{4\pi }{\left(\frac{r}{\left|\mathit{x}\right|}\right)}^{l+1}{B}_l}}{\overline{Y}}_{l,m}(\widehat{\mathit{x}}){\overline{Y}}_{l,m}({\widehat{\mathit{z}}}_i)$$

(8)

which can be rewritten as

$$T(\mathit{x})=\frac{1}{4\pi }\mathit{f}(\mathit{x})\mathit{G}\mathit{\beta }$$

(9)

wherein

$$\mathit{f}(\mathit{x})=\left(B_0{\left(\frac{r}{\left|\mathit{x}\right|}\right)}^1{\overline{Y}}_{0,0}(\widehat{\mathit{x}}),B_1{\left(\frac{r}{\left|\mathit{x}\right|}\right)}^2{\overline{Y}}_{1,-1}(\widehat{\mathit{x}}),\dots ,B_L{\left(\frac{r}{\left|\mathit{x}\right|}\right)}^{L+1}{\overline{Y}}_{L,L}(\widehat{\mathit{x}})\right)$$

(10)

$$\mathit{g}(\mathit{z})={\left({\overline{Y}}_{0,0}(\widehat{\mathit{z}}),{\overline{Y}}_{1,-1}(\widehat{\mathit{z}}),\dots ,{\overline{Y}}_{L,L}(\widehat{\mathit{z}})\right)}^{\mathrm{T}}$$

(11)

$$\mathit{G}(\mathit{z})=\left(\mathit{g}({\mathit{z}}_1),\mathit{g}({\mathit{z}}_2),\dots ,\mathit{g}({\mathit{z}}_I)\right)$$

(12)

$$\mathit{\beta }={({\beta }_1,{\beta }_2,\dots ,{\beta }_I)}^{\mathrm{T}}.$$

(13)

Define $\mathit{Y}(\mathit{x})$ as a ${(L+1)}^2\times 1$ vector, and

$$\mathit{Y}(\mathit{x})={\left({\overline{Y}}_{0,0}(\widehat{\mathit{x}}),{\overline{Y}}_{1,-1}(\widehat{\mathit{x}}),\dots ,{\overline{Y}}_{L,L}(\widehat{\mathit{x}})\right)}^{\mathrm{T}}.$$

(14)

Due to the orthogonality of surface SHs on the unit sphere, the integral over the sphere ${\mathrm{\Omega }}_{\left|x\right|}$ of the product of $\mathit{Y}(\mathit{x})$ and $\mathit{Y}{(\mathit{x})}^{\mathrm{T}}$ is a unit matrix, namely

$${\displaystyle \underset{{\mathrm{\Omega }}_{\left|x\right|}}{\int }\mathit{Y}(\mathit{x})}\mathit{Y}{(\mathit{x})}^{\mathrm{T}}\mathrm{d}{\mathrm{\Omega }}_{\left|x\right|}(\mathit{x})=\mathit{I}$$

(15)

where $\mathrm{d}{\mathrm{\Omega }}_{\left|x\right|}(\mathit{x})$ denotes the surface element of the sphere ${\mathrm{\Omega }}_{\left|x\right|}$. Now, calculate the integral of the product of $\mathit{Y}(\mathit{x})$ and $T(\mathit{x})$ over the sphere ${\mathrm{\Omega }}_{\left|x\right|}$:

$${\displaystyle \underset{{\mathrm{\Omega }}_{\left|x\right|}}{\int }\mathit{Y}(\mathit{x})}T(\mathit{x})\mathrm{d}{\mathrm{\Omega }}_{\left|x\right|}(\mathit{x}).$$

(16)

Substituting Equation (2) into Equation (16) yields the following equation:

$${\displaystyle \underset{{\mathrm{\Omega }}_{\left|x\right|}}{\int }\mathit{Y}(\mathit{x})}T(\mathit{x})\mathrm{d}{\mathrm{\Omega }}_{\left|x\right|}(\mathit{x})=\frac{GM}{R}{\mathit{U}}_1\mathit{C}$$

(17)

where ${\mathit{U}}_1$ is a diagonal matrix, and

$${\mathit{U}}_1=\mathrm{diag}\left({\left(\frac{R}{\left|\mathit{x}\right|}\right)}^1,{\left(\frac{R}{\left|\mathit{x}\right|}\right)}^2,{\left(\frac{R}{\left|\mathit{x}\right|}\right)}^2,{\left(\frac{R}{\left|\mathit{x}\right|}\right)}^2,{\left(\frac{R}{\left|\mathit{x}\right|}\right)}^3,\dots ,{\left(\frac{R}{\left|\mathit{x}\right|}\right)}^{L+1}\right).$$

(18)

Substituting Equation (9) into Equation (16) yields the following equation:

$${\displaystyle \underset{{\mathrm{\Omega }}_{\left|x\right|}}{\int }\mathit{Y}(\mathit{x})}T(\mathit{x})\mathrm{d}{\mathrm{\Omega }}_{\left|x\right|}(\mathit{x})=\frac{1}{4\pi }\mathit{B}{\mathit{U}}_2\mathit{G}\mathit{\beta }$$

(19)

where diagonal matrix $\mathit{B}=\mathrm{diag}(B_0,B_1,B_1,B_1,B_2,\dots ,B_L)$, and

$${\mathit{U}}_2=diag\left({\left(\frac{r}{\left|\mathit{x}\right|}\right)}^1,{\left(\frac{r}{\left|\mathit{x}\right|}\right)}^2,{\left(\frac{r}{\left|\mathit{x}\right|}\right)}^2,{\left(\frac{r}{\left|\mathit{x}\right|}\right)}^2,{\left(\frac{r}{\left|\mathit{x}\right|}\right)}^3,\dots ,{\left(\frac{r}{\left|\mathit{x}\right|}\right)}^{L+1}\right).$$

(20)

By comparing Equation (17) and Equation (19), the analytical relationship between the SH coefficients and the SRBF coefficients can be obtained:

$$\mathit{C}=\frac{1}{4\pi }\frac{R}{GM}{\mathit{U}}_1^{-1}\mathit{B}{\mathit{U}}_2\mathit{G}\mathit{\beta }=\mathit{N}\mathit{\beta }$$

(21)

wherein

$$\mathit{N}=\frac{1}{4\pi }\frac{R}{GM}\left[\begin{array}{ccc}{B}_0{\left(\frac{r}{R}\right)}^1{\overline{Y}}_{0,0}({\widehat{\mathit{z}}}_1)& \cdots & B_0{\left(\frac{r}{R}\right)}^1{\overline{Y}}_{0,0}({\widehat{\mathit{z}}}_I)\\ ⋮& \ddots & ⋮\\ B_L{\left(\frac{r}{R}\right)}^{L+1}{\overline{Y}}_{L,L}({\widehat{\mathit{z}}}_1)& \cdots & B_L{\left(\frac{r}{R}\right)}^{L+1}{\overline{Y}}_{L,L}({\widehat{\mathit{z}}}_I)\end{array}\right]$$

(22)

is a ${(L+1)}^2\times I$ matrix. If the SRBF coefficients are known, Equation (21) can be used to determine the SH coefficients. It should be noted that in this case, I should be larger than ${(L+1)}^2$, i.e., the number of SRBFs is “admissible” [19].

2.2. Direct Approach

Since Equation (21) relies on the orthogonality of surface SHs on the unit sphere, the SRBF model should be considered at a global scale. However, in local modeling, SRBFs are constrained within the local parameterization area ${\mathrm{\Omega }}_{\mathrm{P}}$ of the sphere ${\mathrm{\Omega }}_r$, and the SRBFs outside the area ${\mathrm{\Omega }}_{\mathrm{P}}$ are ignored, which is equivalent to setting the SRBF coefficients outside the area ${\mathrm{\Omega }}_{\mathrm{P}}$ to zero [22]. By merging the zero coefficients and the coefficients inside the area ${\mathrm{\Omega }}_{\mathrm{P}}$, the overall SRBF model is defined on the whole sphere ${\mathrm{\Omega }}_r$. In the synthesis step, zero coefficients automatically vanish, therefore no additional setup is required in practice. The SRBF coefficient vectors inside and outside the area ${\mathrm{\Omega }}_{\mathrm{P}}$ are denoted as ${\mathit{\beta }}_{\mathrm{A}}$ and ${\mathit{\beta }}_{\mathrm{B}}$, respectively, where ${\mathit{\beta }}_{\mathrm{B}}$ is a zero vector. Thus, we have

$$\mathit{\beta }=\left[\begin{array}{c}{\mathit{\beta }}_{\mathrm{A}}\\ {\mathit{\beta }}_{\mathrm{B}}\end{array}\right],\mathrm{with}\left\{\begin{array}{c}{\mathit{\beta }}_{\mathrm{A}}=({\beta }_1,{\beta }_2,\dots ,{\beta }_k{)}^{\mathrm{T}}\\ {\mathit{\beta }}_{\mathrm{B}}=({\beta }_{k+1},{\beta }_{k+2},\dots ,{\beta }_I{)}^{\mathrm{T}}=\mathbf{0}\end{array}\right.$$

(23)

where k is the number of SRBFs inside the area ${\mathrm{\Omega }}_{\mathrm{P}}$. Substituting Equation (23) into Equation (5), Equation (21) is replaced by

$$\mathit{C}=\mathit{N}\mathit{\beta }={\mathit{N}}_{\mathrm{A}}{\mathit{\beta }}_{\mathrm{A}}+{\mathit{N}}_{\mathrm{A}}{\mathit{\beta }}_{\mathrm{A}}={\mathit{N}}_{\mathrm{A}}{\mathit{\beta }}_{\mathrm{A}}$$

(24)

wherein

$${\mathit{N}}_{\mathrm{A}}=\frac{1}{4\pi }\frac{R}{GM}\left[\begin{array}{ccc}{B}_0{\left(\frac{r}{R}\right)}^1{\overline{Y}}_{0,0}({\widehat{\mathit{z}}}_1)& \cdots & B_0{\left(\frac{r}{R}\right)}^1{\overline{Y}}_{0,0}({\widehat{\mathit{z}}}_k)\\ ⋮& \ddots & ⋮\\ B_L{\left(\frac{r}{R}\right)}^{L+1}{\overline{Y}}_{L,L}({\widehat{\mathit{z}}}_1)& \cdots & B_L{\left(\frac{r}{R}\right)}^{L+1}{\overline{Y}}_{L,L}({\widehat{\mathit{z}}}_k)\end{array}\right].$$

(25)

In fact, Equation (24) has been used in previous studies. For example, Wittwer [23] employed Equation (24) to analyze the noise spectrum of the terrestrial dataset and compared it with the noise spectrum from the satellite-only GGM dataset, demonstrating the improvement of GRACE data for the local gravity field model; Bucha et al. [22] used Equation (24) to convert band-limited SRBF coefficients into SH coefficients (also known as the correction signal) and added the correction signal to the original GGM, resulting in an SH model specifically tailored for the Slovak Republic. However, Equation (24) is used in reverse in the direct approach, i.e., the SH coefficients are used to estimate the SRBF coefficients. When using the RCR approach, the modeling object is the residual disturbing potential, as the middle- to long-wavelength signal provided by the satellite-only GGM is removed from the disturbing potential. It is also for this reason that, in the direct approach, the mathematical expectations of the residual SH coefficients of the residual disturbing potential are zero within the corresponding spectral band. Thus, the following functional model can be obtained:

$$\mathbf{0}=E\left\{{\mathit{C}}_{\mathrm{res}}\right\}={\mathit{N}}_{\mathrm{A}}{\mathit{\beta }}_{\mathrm{A}},D\left\{{\mathit{C}}_{\mathrm{res}}\right\}={\mathit{D}}_{\mathrm{C}}$$

(26)

where ${\mathit{C}}_{\mathrm{res}}$ denotes the residual SH coefficient vector, $E\left\{·\right\}$ denotes the expectation operator, and $D\left\{·\right\}$ denotes the dispersion operator. The NCM of the residual SH coefficient vector ${\mathit{D}}_{\mathrm{C}}$ is none other than the NCM of the SH coefficients of the satellite-only GGM. By combining Equation (26) and the functional models of the local datasets, the overall linear Gauss–Markov model is obtained for estimating the SRBF coefficients.

2.3. Indirect Approach

In the indirect approach, the satellite-only GGM is first used to compute the height anomalies at pre-defined grid points on the Earth’s surface. According to Bruns’ formula [21], the height anomaly is defined as

$$\zeta (\mathit{x})=\frac{T(\mathit{x})}{\gamma ({\mathit{x}}^{\prime })}$$

(27)

where $\gamma ({\mathit{x}}^{\prime })$ denotes the normal gravity of point x on the telluroid. According to Equation (2), the height anomaly computed using the satellite-only GGM can be expressed as

$$\zeta (\mathit{x})=\frac{GM}{R\gamma ({\mathit{x}}^{\prime })}\mathit{h}(\mathit{x})\mathit{C}.$$

(28)

Based on Equation (9), the following observation equation can be established:

$$\zeta (\mathit{x})=\frac{1}{4\pi \gamma ({\mathit{x}}^{\prime })}\mathit{f}(\mathit{x})\mathit{G}\mathit{\beta },\mathrm{with}\mathit{\beta }=\left[\begin{array}{c}{\mathit{\beta }}_{\mathrm{A}}\\ {\mathit{\beta }}_{\mathrm{B}}\end{array}\right].$$

(29)

Obviously, Equation (29) implicitly includes a constraint condition that sets the vector of the SRBF coefficients outside the area ${\mathrm{\Omega }}_{\mathrm{P}}$ to a zero vector, i.e., ${\mathit{\beta }}_{\mathrm{B}}=({\beta }_{k+1},{\beta }_{k+2},\dots ,{\beta }_I{)}^{\mathrm{T}}=\mathbf{0}$.

Calculating the integral of the product of $\mathit{Y}(\mathit{x})$ and $\mathit{\zeta }(\mathit{x})$ over the sphere ${\mathrm{\Omega }}_{\left|x\right|}$ based on Equation (17) and Equation (19), respectively, we have

$${\displaystyle \underset{{\mathrm{\Omega }}_{\left|x\right|}}{\int }\mathit{Y}(\mathit{x})}\zeta (\mathit{x})\mathrm{d}{\mathrm{\Omega }}_{\left|x\right|}(\mathit{x})=\frac{GM}{R\gamma ({\mathit{x}}^{\prime })}{\mathit{U}}_1\mathit{C}=\frac{1}{4\pi \gamma ({\mathit{x}}^{\prime })}\mathit{B}{\mathit{U}}_2\mathit{G}\mathit{\beta }.$$

(30)

For any $\zeta ({\mathit{x}}_j),j=1,2,\dots ,J$, Equation (30) holds. Since Equation (24) can also be derived from Equation (30), the direct approach and the indirect approach are theoretically equivalent. However, the computational procedure of the indirect approach is complex.

3. Experiment Settings

To validate the performance of the direct approach, we conducted a simulation experiment with Colorado (USA) as the core area and compared the results with the classical RCR approach and the indirect approach. The reason for conducting the simulation experiment is that noise-free control data are advantageous for interpreting the results.

3.1. Datasets

The study area (also referred to as the data area) belongs to the Rocky Mountains, and it is bound by 100.8°W to 111.2°W and 33.8°N to 41.2°N. The rugged terrain increases the difficulty of the quasi-geoid modeling. We use the high-resolution EIGEN-6C4 [24] model complete to degree 2190 to generate gravity disturbance data and add white Gaussian noise with a standard deviation (STD) of 1 mGal to simulate the local dataset. Due to the knowledge of the actual terrestrial dataset within the area between 102°W and 110°W and 35°N and 40°N, the 20,000 data points within this area simulate the distribution of the actual dataset [2]. Furthermore, there are an additional 9000 data points randomly distributed in other areas of the study area. Figure 1 illustrates the topography of the study area as well as the distribution of data points.

3.2. Classical RCR Approach

To better understand this section, please first read Appendix A to learn about the two-scale SRBF model used in the indirect approach. This is because the single-scale model shown in Equation (A5) is the model established using the classical RCR approach. Therefore, the main task of the classical RCR approach is the estimation of the coefficient set $\left\{{\beta }_{i,1}\right\}$.

RCR procedure. To reduce the errors of linearization and spherical approximation, we removed the GOCO05s model complete to degree 280 and the XGM2016 [25] model from degree 281 to 719 from the local gravity disturbance data. In mountainous areas, topographic effects also play an important role [22,26], so we further removed the topographic model dV_ELL_Earth2014 [27] from degree 720 to 2190 from the local gravity disturbance data. After the GGM reduction and topographic reduction, the STD of the local gravity disturbance dataset is reduced from 37.80 mGal to 8.11 mGal. Figure 2 shows the spatial pattern of the local gravity disturbance data.

Parameterization area. To control edge effects, we refer to [9,28] to set the width between the parameterization area (the area covered by SRBFs) and the data area to ${0.2}^{\circ }$. Thus, the parameterization area is bound by ${100.6}^{\circ }\mathrm{W}$ to ${111.4}^{\circ }\mathrm{W}$ and ${33.6}^{\circ }\mathrm{N}$ to ${41.4}^{\circ }\mathrm{N}$.

Basis functions. In this study, the Shannon function is employed, with the Legendre coefficients defined as $B_l=1,\mathrm{for}l=1,2,\dots ,L$, ensuring that the signal is not smoothed within the corresponding spectral band. The maximum SH degree L of the basis functions is set to 2190, consistent with the SH degree of the local gravity disturbance data. The SRBFs are placed at the commonly used Reuter grid [20,29] points and the control parameter of the Reuter grid is set to 2190.

Model estimation. The functional model of the local gravity disturbance data can be represented as a linear Gauss–Markov model:

$$E\left\{{\mathit{y}}_1\right\}={\mathit{A}}_1{\mathit{\beta }}_1,D\left\{{\mathit{y}}_1\right\}={\mathit{D}}_1$$

(31)

where ${\mathit{y}}_1$ denotes the vector of the local gravity disturbance dataset. ${\mathit{A}}_1$ and ${\mathit{D}}_1$ denote the design matrix and the NCM of the vector of the local gravity disturbance dataset, respectively. The vector ${\mathit{\beta }}_1$ consists of the model coefficient set $\left\{{\beta }_{i,1}\right\}$, which is estimated using zero-order Tikhonov regularization [30]. Once the model coefficient set has been estimated, the single-scale model of the residual disturbing potential is given by Equation (A5). The residual height anomaly at any point within the target area can be computed based on the Bruns’ formula. To obtain the full signal of the height anomaly, it is necessary to restore the height anomaly signal from (1) the GOCO05s model complete to degree 280, (2) the XGM2016 model from degree 281 to 719, and (3) the dV_ELL_Earth2014 model from degree 720 to 2190.

3.3. Direct Approach

The experimental settings of the direct approach are the same as the classical RCR approach, except for the addition of a noisy satellite-only GGM dataset. Although the GOCO05s model complete to degree 280 is employed in the RCR procedure, the noisy GGM dataset consists of the SH coefficients of the GOCO05s model up to degree 230 and the corresponding NCMs. The purpose of this setup is to make the noisy GGM dataset used in the direct approach and the indirect approach consistent (see Section 3.4).

According to Equations (24) and (26), the functional model of the residual SH coefficients can be represented as a linear Gauss–Markov model:

$$\mathbf{0}=E\left\{{\mathit{C}}_{\mathrm{res}}\right\}={\mathit{A}}_C{\mathit{\beta }}_1,D\left\{{\mathit{C}}_{\mathrm{res}}\right\}={\mathit{D}}_C$$

(32)

where ${\mathit{A}}_C$ denotes the design matrix of the residual SH coefficients. Combining Equations (31) and (32) yields an overall linear Gauss–Markov model:

$$E\left\{\mathit{y}\right\}=\mathit{A}{\mathit{\beta }}_1,D\left\{\mathit{y}\right\}=\mathit{D}$$

(33)

wherein

$$\mathit{y}=\left(\begin{array}{c}{\mathit{y}}_1\\ {\mathit{C}}_{\mathrm{res}}\end{array}\right),\mathit{A}=\left(\begin{array}{c}{\mathit{A}}_1\\ {\mathit{A}}_C\end{array}\right),\mathrm{and}\mathit{D}=\left(\begin{array}{cc}{\mathit{D}}_1& \mathbf{0}\\ \mathbf{0}& {\mathbf{D}}_C\end{array}\right).$$

(34)

The model coefficients are still estimated using zero-order Tikhonov regularization.

3.4. Indirect Approach

The experimental settings of the indirect approach are inspired by [17,18]. The first step of the indirect approach is to estimate a single-scale model using the local gravity disturbance dataset and the classical RCR approach. The second step is to estimate a coarse-scale model using two datasets: one dataset consists of the height anomalies generated from the low-pass filtered GOCO05s model, and the other dataset consists of the height anomalies generated from the low-pass filtered single-scale model. The NCMs of the two datasets are both calculated according to the law of covariance propagation. It is worth noting that the generated GGM height anomalies do not contain any signal higher than degree 230. The reason is that the commission error of the GOCO05s model increases exponentially with the degree, and truncating the GOCO05s model at degree 230 is a compromise selection (see [7,18] for more details). Finally, the high-pass filtered single-scale model [18] is merged with the coarse-scale model to obtain an overall two-scale SRBF model. More details on the experimental settings of the indirect approach can be found in Appendix B.

4. Results and Discussion

4.1. Results

The height anomalies generated from the EIGEN-6C4 model up to degree 2190 are defined as truth values, with a resolution of $5^{\prime }\times 5^{\prime }$ distributed in the target area (between ${103}^{\circ }\mathrm{W}$ and ${109}^{\circ }\mathrm{W}$ and ${36}^{\circ }\mathrm{N}$ and ${39}^{\circ }\mathrm{N}$). We compare the quasi-geoid models calculated by different approaches with the truth values.

Table 1. Statistics of the differences between the quasi-geoid models and truth values (unit: cm).

Approach	RMS	STD	Max	Min	Mean
approach-1	1.82	1.70	6.60	−10.21	−0.66
approach-2	1.49	1.49	7.39	−8.31	−0.04
approach-3	1.73	1.73	7.71	−8.95	−0.08

Note. approach-1 = classical RCR approach; approach-2 = direct approach; approach-3 = indirect approach.

lists the statistics of the differences between the local quasi-geoid models and the truth values, while Figure 3 displays the spatial patterns of the differences. The numerical results show that the direct approach has superior performance. Compared to the classical RCR approach, the direct approach reduces the root-mean-square (RMS) error of the quasi-geoid model from 1.82 cm to 1.49 cm, which means that the model accuracy is improved by 18.1%. Figure 3a shows that in the northwest, northeast, and southeast of the target area, the quasi-geoid model calculated by the classical RCR approach exhibits large errors, which are caused by the sparse data distribution in these sub-areas (see Figure 1b). Instead, Figure 3b shows that the quasi-geoid model calculated by the direct approach effectively controls the prediction errors in the northwest and northeast of the target area and shows minor improvements in other areas as well, except for the southeast of the target area. The improvements of the direct approach on the local quasi-geoid model reflect the benefit of using satellite-only GGM as a noisy dataset.

In terms of the RMS error of the quasi-geoid model, the indirect approach shows a slight improvement compared to the classical RCR approach (from 1.82 cm to 1.73 cm). Figure 3a,c show that in the northwest and northeast of the target area, the quasi-geoid model calculated by the indirect approach is slightly better than the quasi-geoid model calculated by the classical RCR approach. However, the situation is reversed in the central part of the target area. The indirect approach does not show the significant advantage of using satellite-only GGM as a noisy dataset. Figure 3b,c indicate that the quality of the quasi-geoid model calculated by the indirect approach is inferior to that calculated by the direct approach, especially in the central and eastern parts of the target area. In terms of the RMS error of the quasi-geoid model, the indirect approach is also inferior to the direct approach (1.73 cm vs. 1.49 cm).

4.2. Discussion

In Section 2, we demonstrated the theoretical equivalence of the direct approach and the indirect approach. However, in this study, the numerical results show that the direct approach outperforms the indirect approach. Slobbe et al. [18] attributed the poor performance of the indirect approach to the unreasonable relative weights between heterogeneous datasets. However, we tried different relative weights, but the results of the indirect approach were still not as good as the direct approach. The reason why the indirect approach does not perform as well as the direct approach may be related to the generation of the height anomaly dataset using satellite-only GGM. We generated very dense GGM height anomaly data and sufficiently expanded the parameterization area of the SRBFs, as shown in Appendix B. Nevertheless, the RMS of the differences between the height anomalies derived from the residual disturbing potential model estimated using the generated GGM height anomaly data and the control data is still greater than 0.3 cm. This may be related to the rough topography within the study area. Therefore, the generated height anomaly data are not an equivalent representation of the satellite-only GGM, which may lead to the loss of information from the satellite-only GGM. Instead, the direct approach directly uses the SH coefficients of the satellite-only GGM as a noisy dataset, effectively utilizing the information from the satellite-only GGM, which significantly improves the quality of the local quasi-geoid model.

5. Conclusions

In this study, we proposed an approach that directly uses the SH coefficients and the corresponding NCM of the satellite-only GGM as a noisy dataset and combines the GGM dataset and the local datasets to estimate the local quasi-geoid model. We have theoretically demonstrated the feasibility of the direct approach through mathematical deduction. Compared to the classical RCR approach, the direct approach achieves the optimal combination of local datasets with a noisy satellite-only GGM. We revisited the indirect approach and revealed the theoretical consistency between the direct approach and the indirect approach. However, compared to the indirect approach, the direct approach is more attractive due to its reliable results and low computational complexity. In the future, we will apply the direct approach to actual quasi-geoid modeling.