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Residual terrain modelling (RTM) plays a key role for short-scale gravity modelling in physical geodesy, e.g. for interpolation of observed gravity and augmentation of global geopotential models (GGMs). However, approximation errors encountered in RTM computation schemes are little investigated. The goal of the present paper is to examine widely used classical RTM techniques in order to provide insights into RTM-specific approximation errors and the resulting RTM accuracy. This is achieved by introducing a new, independent RTM technique as baseline that relies on the combination of (1) a full-scale global numerical integration in the spatial domain and (2) ultra-high-degree spectral forward modelling. The global integration provides the full gravity signal of the complete (detailed) topography, and the spectral modelling that of the RTM reference topography. As a main benefit, the RTM baseline technique inherently solves the “non-harmonicity problem” encountered in classical RTM techniques for points inside the reference topography. The new technique is utilized in a closed-loop type testing regime for in-depth examination of four variants of classical RTM techniques used in the literature which are all affected by one or two types of RTM-specific approximation errors. These are errors due to the (1) harmonic correction (HC) needed for points located inside the reference topography, (2) mass simplification, (3) vertical computation point inconsistency, and (4) neglect of terrain correction (TC) of the reference topography. For the Himalaya Mountains and the European Alps, and a degree-2160 reference topography, RTM approximation errors are quantified. As key finding, approximation errors associated with the standard HC (4πGρHPRTM) may reach amplitudes of ~ 10 mGal for points located deep inside the reference topography. We further show that the popular RTM approximation (2πGρHPRTM-TC) suffers from severe errors that may reach ~ 90 mGal amplitudes in rugged terrain. As a general conclusion, the RTM baseline technique allows inspecting present and future RTM techniques down to the sub-mGal level, thus improving our understanding of technique characteristics and errors. We expect the insights to be useful for future RTM applications, e.g. in geoid modelling using remove–compute–restore techniques, and in the development of new GGMs or high-resolution augmentations thereof.

Gravity forward modeling provides important high-resolution information for the development of global gravity models, and can also be applied in many studies, e.g., topographic/isostatic effects computation and Bouguer anomaly maps compilation. In this paper, we present efficient spectral forward modeling approaches in the spheroidal harmonic domain, based on a single layer with constant density or volumetric layers with laterally varying density. With the binomial series expansion applied in spheroidal harmonic gravity forward modeling, the computational cost of these approaches is much lower than similar approaches. In both layering cases, we derive topographic potential models up to degree and order (d/o) 2190 by applying the approaches proposed here. Our methodology is evaluated by comparing these outcome models with other similar topographic potential models derived from spherical harmonic solutions. We find that topographic potentials from spheroidal and spherical harmonic approaches are in great agreement. Finally, the model named EHFM_Earth_7200 with a maximum degree of 7200 was derived by a layer-based approach. The evaluations by ground-truth data show that EHFM_Earth_7200 improves GO_CONS_GCF_2_DIR_R6 by 4% over Antarctica, and improves EGM2008 by ~ 34% over northern Canada. A global map of Bouguer gravity anomaly was also compiled with EHFM_Earth_7200 and EGM2008. As the main conclusion of this work, the new model EHFM_Earth_7200 is beneficial for investigating and modeling the Earth’s external gravity field, the new approaches have comparable accuracy to spherical harmonic approaches and are more suitable for practical use with guaranteed convergence regions because they are performed in the spheroidal harmonic domain.

In recent years, the fundamental quantity of the gravitational field has been extended from gravitational potential, gravitational vector, and gravitational gradient tensor to gravitational curvature with its first measurement along the vertical direction in laboratory conditions. Previous studies numerically identified the near-zone and polar-region problems for gravitational curvature of a tesseroid, but these issues remain unresolved. In this contribution, we derive the new third-order central and single-sided difference formulas with one, two, and three arguments using the finite difference method. To solve these near-zone and polar-region problems, we apply a numerical approach combining the conditional split, finite difference, and double exponential rule based on these newly derived third-order difference formulas when the computation point is located below, inside, and outside the tesseroid. Numerical experiments with a spherical shell discretized into tesseroids reveal that the accuracy of gravitational curvature is about 4–8 digits in double precision. Numerical results confirm that when the computation point moves to the surface of the tesseroid, the relative and absolute errors of gravitational curvature do not change much, i.e., the near-zone problem can be adequately solved using the numerical approach in this study. When the latitude of the computation point increases, the relative and absolute errors of gravitational curvature do not increase, which solves the polar-region problem with this stable numerical approach. The provided Fortran codes at https://github.com/xiaoledeng/xtessgc-xqtessgc will help with potential applications for the gravitational field of different celestial bodies in geodesy, geophysics, and planetary sciences.

Geophysical forward modeling serves as a fundamental theoretical approach for characterizing subsurface structures and material properties, essentially involving the computation of gravity responses at surface or spatial observation points based on a predefined density distribution. With the rapid development of data-driven techniques such as deep learning in geophysical inversion, forward algorithms are facing increasing demands in terms of computational scale, observable types, and efficiency. To address these challenges, this study develops an efficient forward modeling method based on voxel discretization, the enabling rapid calculation of gravity anomalies and radial gravity gradients on multiple observational surfaces. Leveraging the parallel computing capabilities of graphics processing units (GPU), together with tensor acceleration, Compute Unified Device Architecture (CUDA) execution, and Just-in-time (JIT) compilation strategies, the method achieves high efficiency and automation in the forward computation process. Numerical experiments conducted on several typical theoretical models demonstrate the convergence and stability of the calculated results, indicating that the proposed method significantly reduces computation time while maintaining accuracy, thus being well-suited for large-scale 3D modeling and fast batch simulation tasks. This research can efficiently generate forward datasets with multi-view and multi-metric characteristics, providing solid data support and a scalable computational platform for deep-learning-based geophysical inversion studies.

The accurate computation of gravitational effects from topographic and atmospheric masses is one of the core issues in gravity field modeling. Using gravity forward modeling based on Newton’s integral, mass distributions are generally decomposed into regular mass bodies, which can be represented by rectangular prisms or polyhedral bodies in a rectangular coordinate system, or tesseroids in a spherical coordinate system. In this study, we prefer the latter representation because it can directly take the Earth’s curvature into account, which is particularly beneficial for regional and global applications. Since the volume integral cannot be solved analytically in the case of tesseroids, approximation solutions are applied. However, one well-recognized issue of these solutions is that the accuracy decreases as the computation point approaches the tesseroid. To overcome this problem, we develop a method that can precisely compute the gravitational potential V and vector Vx,Vy,Vz on the tesseroid surface. In addition to considering a constant density for the tesseroid, we further derive formulas for a linearly varying density. In the near zone (up to a spherical distance of 15 times the horizontal tesseroid dimension from the computation point), the gravitational effects of the tesseroids are computed by Gauss–Legendre quadrature using a two-dimensional adaptive subdivision technique to ensure high accuracy. The tesseroids outside this region are evaluated by means of expanding the integral kernel in a Taylor series up to the second order. The method is validated by synthetic tests of spherical shells with constant and linearly varying density, and the resulting approximation error is less than 10-4m2s-2 for V, 10-5mGal for Vx, 10-7mGal for Vy, and 10-4mGal for Vz. Its practical applicability is then demonstrated through the computation of topographic reductions in the White Sands test area and of global atmospheric effects on the Earth’s surface using the US Standard Atmosphere 1976.

The truncation coefficient is widely utilized in non-global coverage computations of geophysics and geodesy and is always altitude dependent. As the two commonly used calculation methods for truncation coefficients, i.e., the spectral form and the recursive formula, both suffer from decreasing precision caused by high-altitude, leading to slow convergence for the former and numerical instability recursion for the latter. The asymptotic expansion mathematically converges with increasing degree and can precisely compensate for the shortcomings of the two methods. This study introduces asymptotic expansion to accurately compute the truncation coefficient for the spectral gravity forward modeling to a high degree. The evaluation at the whole altitudes and whole integral radii indicates that the proposed method has the following advantages: (i) The calculation precision increases with increasing degree and is altitude independent; (ii) the accurate calculation can be supported by a double-precision format; and (iii) the calculation can be conducted nearly without extra time cost with increasing degree. Generally, asymptotic expansion is used to calculate the high degree truncation coefficients, while the truncation coefficients at low degrees can be calculated using spectral form or recursive formulas in multiprecision format as a supplement; and the available range of asymptotic expansion is provided in the appendix.

Bathymetry data over lake areas are not included in the current and previous NGS (National Geodetic Survey) geoid models. Lake surfaces are simply treated as land surfaces during the modeling regardless of the apparent density difference between water and rock, resulting in artificial masses that distort the model from the actual gravity field and the corresponding geoid surface. In this study, compiled high-resolution bathymetry data provided by National Centers for Environmental Information are used to identify the real volume of water bodies. Under the mass conservation principle, two strategies are deployed to properly account the water body bounded by the mean lake surface and the bathymetry indicated lake floor into the current NGS geoid modeling scheme, where the residual terrain modeling method is used to account for topographic effects. The first strategy condenses water bodies into equivalent rock masses, with the cost of changing the geometrical shape of the water body. The second one keeps the shape of the water body unchanged but replaces the water and rock densities inside each topographical column bounded by the geoid surface and the mean lake surface by an averaged density. Both strategies show up to 1-cm geoid changes when compared with the previous geoid model that does not consider bathymetric information. All three geoid models are evaluated by local GNSS/Leveling benchmarks and multi-year-multi-mission altimetry indicated mean lake surface heights. The results show that both strategies can improve the geoid model precision. And the second strategy yields more realistic results. Graphical Abstract

We present an accurate method for the calculation of gravitational potential (GP), vector (GV), and gradient tensor (GGT) of a tesseroid, considering a density model in the form of a polynomial up to cubic order along the vertical direction. The method solves volume integral equations for the gravitational effects due to a tesseroid by the Gauss–Legendre quadrature rule. A two-dimensional adaptive subdivision technique, which automatically divides the tesseroids near the computation point into smaller elements, is applied to improve the computational accuracy. For those tesseroids having small vertical dimensions, an extension technique is additionally utilized to ensure acceptable accuracy, in particular for the evaluation of GV and GGT. Numerical experiments based on spherical shell models, for which analytical solutions exist, are implemented to test the accuracy of the method. The results demonstrate that the new method is capable of computing the gravitational effects of the tesseroids with various horizontal and vertical dimensions as well as density models, while the evaluation point can be on the surface of, near the surface of, outside the tesseroid, or even inside it (only suited for GP and GV). Thus, the method is attractive for many geodetic and geophysical applications on regional and global scales, including the computation of atmospheric effects for terrestrial and satellite usage. Finally, we apply this method for computing the topographic effects in the Himalaya region based on a given digital terrain model and the global atmospheric effects on the Earth’s surface by using three polynomial density models which are derived from the US Standard Atmosphere 1976.

Gravity forward modeling as a basic tool has been widely used for topography correction and 3D density inversion. The source region is usually discretized into tesseroids (i.e., spherical prisms) to consider the influence of the curvature of planets in global or large-scale problems. Traditional gravity forward modeling methods in spherical coordinates, including the Taylor expansion and Gaussian–Legendre quadrature, are all based on spatial domains, which mostly have low computational efficiency. This study proposes a high-efficiency forward modeling method of gravitational fields in the spherical harmonic domain, in which the gravity anomalies and gradient tensors can be expressed as spherical harmonic synthesis forms of spherical harmonic coefficients of 3D density distribution. A homogeneous spherical shell model is used to test its effectiveness compared with traditional spatial domain methods. It demonstrates that the computational efficiency of the proposed spherical harmonic domain method is improved by four orders of magnitude with a similar level of computational accuracy compared with the optimized 3D GLQ method. The test also shows that the computational time of the proposed method is not affected by the observation height. Finally, the proposed forward method is applied to the topography correction of the Moon. The results show that the gravity response of the topography obtained with our method is close to that of the optimized 3D GLQ method and is also consistent with previous results.

We present a brief review of gravity forward algorithms in Cartesian coordinate system, including both space-domain and Fourier-domain approaches, after which we introduce a truly general and efficient algorithm, namely the convolution-type Gauss fast Fourier transform (Conv-Gauss-FFT) algorithm, for 2D and 3D modeling of gravity potential and its derivatives due to sources with arbitrary geometry and arbitrary density distribution which are defined either by discrete or by continuous functions. The Conv-Gauss-FFT algorithm is based on the combined use of a hybrid rectangle-Gaussian grid and the fast Fourier transform (FFT) algorithm. Since the gravity forward problem in Cartesian coordinate system can be expressed as continuous convolution-type integrals, we first approximate the continuous convolution by a weighted sum of a series of shifted discrete convolutions, and then each shifted discrete convolution, which is essentially a Toeplitz system, is calculated efficiently and accurately by combining circulant embedding with the FFT algorithm. Synthetic and real model tests show that the Conv-Gauss-FFT algorithm can obtain high-precision forward results very efficiently for almost any practical model, and it works especially well for complex 3D models when gravity fields on large 3D regular grids are needed.