The study area (125–122° W, 39–42° N) is located on the west coast of the United States. Since the calculation of RTM-GA at a given point requires elevation data from a surrounding region, the terrain and bathymetric data coverage is extended to 126–121° W, 38–43° N. The topography of the study area and its surroundings is shown in Fig. 1. The study area borders the Pacific Ocean to the west and includes the Central Valley, the Sierra Nevada, and the Cascade Range. The highest elevation point is Mount Shasta, located in the southern Cascade Range, with an elevation of 4316 m. The complex topographic environment of the study area implies that the GA information provided by global gravity field models lacks significant high-frequency GA signals.

XGM2019e is a global gravity field model developed by integrating terrestrial gravity observations with satellite-derived data. In the high-frequency spectral range, noise contributions from both data sources must be carefully addressed. In particular, coastal regions and areas with sparse or missing ground data are more prone to noise-induced distortions in the model signal. To suppress such effects, a weighted smoothing transition strategy is applied. Nevertheless, even after these procedures, the signal-to-noise ratio (SNR) at higher harmonic degrees remains relatively low, leading to attenuation of true signals and thereby constraining the practical use of high-degree gravity field models (Zingerle et al., 2020). Consequently, adopting a limited set of spherical harmonic coefficients instead of all high-degree terms provides a better trade-off between spatial resolution and SNR. Accordingly, the XGM2019e-2159 model, expanded up to degree 2159 and corresponding to a spatial resolution of 5′×5′, is adopted in this study. The model is constructed based on several datasets, including the combined satellite-only gravity field model GOCO06s, ground GA data provided by the National Geospatial-Intelligence Agency, and DTU13 marine GA derived from satellite radar altimetry. The GA model derived from XGM2019e-2159 can be computed via the International Centre for Global Earth Models (ICGEM) website (http://icgem.gfz-potsdam.de/calcgrid, last access: 21 April 2026).

This study uses the high-resolution SRTM V4.1 dataset as the digital elevation model, which was acquired from the Shuttle Radar Topography Mission (SRTM) (https://srtm.csi.cgiar.org, last access: 21 April 2026). This mission was a collaboration between the National Imagery and Mapping Agency and the National Aeronautics and Space Administration. The elevation data of SRTM V4.1 are referenced to the EGM96 geoid, with a spatial resolution of 3^′′ × 3^′′. SRTMV4.1 employs a new interpolation algorithm and supplementary DEM data to fill data voids present in SRTM3, resulting in significantly improved elevation accuracy compared to SRTM3 (Reuter et al., 2007).

The bathymetric data in this study are sourced from GEBCO (General Bathymetric Chart of the Oceans), a project based on the Global Earth System Project. The dataset encompasses global DEM data ranging from grid scale to basin scale, integrating multiple bathymetric data sources, including shipborne echo sounding, satellite altimetry data, and other high-resolution bathymetric measurements. The GEBCO_2024 dataset used in this study was released in July 2024 (https://www.gebco.net, last access: 21 April 2026). It provides globally comprehensive elevation data on a 15^′′ × 15^′′ geographic grid (Tozer et al., 2019).

This study uses the measured gravity data NGS99, published by the National Geodetic Survey (NGS), as the reference dataset (https://www.ncei.noaa.gov/products/gravity-data, last access: 21 April 2026). The NGS99 dataset includes 1 633 499 discrete gravity measurement points. The NGS99 gravity data cover not only the inland regions of the United States but also extend to its coastal areas. Figure 2 illustrates the distribution of NGS99 measured GA values and measurement locations within the study area, comprising 10 797 oceanic points and 3247 land-based points.

According to the definition, RTM represents the difference between the detailed and reference topography. Due to the resolution mismatch between the 3^′′ × 3^′′ topographic data SRTM V4.1 on land and the 15^′′ × 15^′′ bathymetric data GEBCO_2024 in the ocean, and the fact that SRTM V4.1 data is only available on land, the two models must be merged before constructing the residual terrain model. First, bicubic interpolation is applied to interpolate GEBCO_2024 to a 3^′′ × 3^′′ grid, matching the resolution of SRTM V4.1. Then, the terrestrial data in the study area from GEBCO_2024 is removed, and the terrestrial data from SRTM V4.1 is incorporated into GEBCO_2024. This process yields the detailed topography model SRTM-GEBCO required for this study.

The reference topography model is obtained by applying a spatial filtering approach based on the moving average to the SRTM-GEBCO data. The core idea of this method is to replace the value of a target point with the average of all points within a certain neighborhood around it. In essence, it acts as a low-pass filter, removing high-frequency topographic details while retaining low-frequency large-scale topographic variations. To maintain consistency with the spatial resolution of the XGM2019e-2159 gravity field model, we set the resolution of the filtered reference terrain model to 5^′ × 5^′ as well.

The elevation of the aforementioned detailed topography model SRTM-GEBCO is denoted as HDET and the elevation of the reference topography model obtained by spatial filtering is denoted as HREF. Then, the elevation of the RTM can be expressed as: 1ΔHRTM=HDET-HREF. The schematic diagram of the RTM for land and marine areas is shown in Fig. 3, where the RTM for the marine area is obtained using the RET method.

The RTM of the study area is obtained using the above method, as shown in Fig. 4. In the marine regions, the RTM is constructed using the RET method. A comparison between Figs. 1 and 4 reveals that the residual terrain elevation exhibits alternating positive and negative values in areas with significant terrain undulation, with a maximum reaching over 800 m and a minimum below -700 m. In contrast, in oceanic regions and relatively flat plains or valleys, the variation in residual terrain elevation is also smaller.

This study refines the Earth's GA model in coastal areas using a spatial domain approach. First, the residual terrain is divided into discrete prism-shaped units, and the gravitational influence of each unit on the observation point is calculated. Then, the cumulative effect of all these units provides the gravitational contribution from the residual terrain at the location of interest. Due to the oscillation of RTM elevations between negative and positive values within a certain area, gravity forward modeling based on the residual terrain model is only required over k prisms in the vicinity of the computation point (Forsberg, 1984; Hirt et al., 2010b; Wang et al., 2024). When computing derivatives of the gravitational potential such as GA and vertical deflections from the RTM, an integration radius of several tens of kilometers is generally sufficient (Hirt et al., 2010a). In this study, an integration radius of 111 km is adopted for forward modeling of the RTM-GA. To fully utilize the detailed topographic data, grid prisms with a side length of 90 m × 90 m are employed.

As shown in Fig. 5, a right-handed Cartesian coordinate system is established with the z axis oriented vertically downward. The gravitational disturbing potential induced at point P by a prism of uniform density can be formulated as: 2T=Gρ∫x1x2∫y1y2∫z1z21ldxdydz. The gravity disturbance can be obtained by computing the partial derivative of the disturbing potential with respect to the vertical direction, and it is given by: 3δg=-∂T∂z=Gρ∫x1x2∫y1y2∫z1z2z-cl3dxdydz. In this formula, G denotes the gravitational constant; ρ is the density of the prism; (xi,yi,zi) (i=1,2) represent the coordinates of the prism's eight corners; and l is the distance from the prism's vertex to the computation point P, where l=(x-a)2+(y-b)2+(z-c)2.

After solving the integral, it can be expressed as: 4δg(a,b,c)=-Gρ{|||(x-a)ln⁡[(y-b)+l]+(y-b)ln⁡[(x-a)+l]-(z-c)arctan⁡(x-z)(y-b)(z-c)l|x1x2|y1y2|z1z2}. The relationship between gravity disturbance and GA can be expressed as: 5Δg=δg-2rT. where r denotes the geocentric radius vector of the calculation point. The gravitational contribution of an individual prism to the computation point P can be computed using the above formula, based on the spatial relationship between P and the prism's vertices. If the integration region contains a total of k prisms, the total gravity anomaly at P due to the residual terrain is obtained by summing the contributions from all individual prisms. This yields the RTM-GA ΔgRTM, which is expressed as: 6ΔgRTM=∑i=1kΔgRTM(i). After the RTM-GA is obtained through gravity forward modeling, the XGM-GA model can be refined, and the truncation errors can be effectively compensated. Let ΔgXGM denote the modeled GA before refinement. After incorporating the ΔgRTM, the refined modeled GA ΔgXGM/RTM can be expressed as: 7ΔgXGM/RTM=ΔgRTM+ΔgXGM. Before performing the calculations, the geodetic coordinate system of the original topographic data needs to be transformed to the local Cartesian coordinate system centered on (123.5° W, 40.5° N) within the study area.

In inland areas, the computation points are located above the detailed topography. In this case, the density of the prism is set to the average crustal density, ρc= 2670 kg m⁻³. However, over the ocean, since both the measured GA and the modeled GA are located on the sea surface, the computation points should be placed at the sea surface rather than on the detailed seafloor topography. Since the average seawater density is ρw= 1030 kg m⁻³, the corresponding prism density should be Δρ=ρc-ρw.

As shown in Fig. 6, when a terrestrial computation point is situated at the land–sea boundary, the integration region includes both land and ocean. To avoid the need to distinguish between different density values in the forward modeling process, this study adopts the RET method proposed by Hirt (2013). In the RET method, seawater is compressed into an equivalent rock mass by multiplying the ocean depth (H<0) with a scaling factor (1-ρw/ρc) ≈ 0.614. With the RET method in RTM forward modeling, both the landmass and the compressed seawater mass can be assigned a uniform density of ρc= 2670 kg m⁻³, eliminating the need to differentiate the density of land and ocean prisms.

However, when using the RET method to obtain the RTM-GA at computation points in coastal land areas, the process of “compressing” seawater into an equivalent rock mass by multiplying the ocean depth (H<0) by a coefficient factor of 0.614 causes a vertical shift in the mass center of the marine prisms. The gravitational effect of a prism at a computation point is closely related to their relative distance and position. Therefore, the displacement of prism positions caused by the RET method introduces corresponding errors. Hirt (2013) also acknowledged the existence of this error and considered its effect acceptable in shallow coastal areas, hence no correction was applied. In this study, a mass center offset correction (MCOC) was applied to the marine prisms compressed by the RET method, as shown in Fig. 6.

Let HA and HB denote the elevations of the detailed topography and reference topography before compression, respectively. After applying the RET method, the elevations of the compressed detailed topography and reference topography are denoted as HC and HD, respectively. After the mass center offset correction, they are represented as HQ and HW, with their relationships expressed as follows: 8HC=0.614HAHD=0.614HB,9HQ=HA+HB2+HC-HD2=1.614HA+0.386HB2HW=HA+HB2-HC-HD2=0.386HA+1.614HB2.

During the computation of the RTM-GA, a “non-harmonic” issue may arise when a computation point is located below the reference topography, as exemplified by point P2 in Fig. 3a. For these points, the directly forward-modeled gravity potential is non-harmonic, necessitating a harmonic correction to satisfy the harmonic condition. This study employs the harmonic correction (HC) of the condensation method, which condenses the residual terrain mass between the computation point and the reference surface into an infinitely thin mass layer directly beneath the computation point. The purpose of this method is to transform this internal gravity field functional into a downward continued harmonic gravity field functional. This ensures that no residual mass remains above the condensed P2 point (Forsberg and Tscherning, 1981).

The harmonic correction formula of the condensation method can be expressed as: 10HC=4πGρcΔh, where ρc is the average crustal density, and Δh=HDET-HREF with Δh<0. The modeled GA corrected by RTM can be expressed as: 11ΔgXGM/RTM=ΔgXGM+ΔgRTMΔgXGM/RTM=ΔgXGM+ΔgRTM+HCΔh≧0Δh<0. Since the computation points over the ocean are located on the sea surface, harmonic correction is not required in oceanic regions.

Figure 7 presents the workflow for enhancing the XGM-GA model through the method described above.

As shown in Fig. 8, the 1^′ × 1^′ RTM-GA model is computed based on the residual terrain data using Eqs. (2)–(6). Figure 9 shows the XGM-GA models before and after correction based on RTM gravity forward modeling. For simplicity, the RTM-corrected XGM2019e-2159 gravity anomaly model is referred to as the XGM/RTM-GA model.

A statistical analysis of the GA from the XGM model, RTM model, and XGM/RTM model within the study area is summarized in Table 1.

Based on Figs. 8 and 9 as well as Table 1, it can be concluded that both the RMS and the standard deviation (SD) of the XGM/RTM-GA model are higher than those of the XGM-GA model. The RTM-GA model primarily reflects small-scale, high-frequency GA, which are closely associated with terrain undulations. The XGM-GA model shows a smooth spatial distribution, representing large-scale gravity variations. The XGM/RTM-GA model contains both the large-scale low-frequency information of the XGM model and the small-scale high-frequency information of the RTM model, retaining the overall trend while enhancing local details.

To verify the reliability of the results, the accuracy was evaluated using the NGS99 measured GA data. Figure 10 presents the RTM-GA computed at the locations of the measured points. By comparing Figs. 10 and 1, it can be seen that points with large absolute values of RTM-GA are mainly located in mountainous areas with significant terrain undulations, indicating that more rugged terrains contain richer high-frequency gravity signals.

The differences between the measured GA and the XGM-GA at the measured points within the study area are calculated, as shown in Fig. 11a. It is evident that the largest differences are concentrated in areas with rugged terrain within the study area, with the maximum absolute difference exceeding 100 mGal. By adding the RTM-GA to the XGM-GA, the XGM/RTM-GA is obtained, and the difference between it and the measured GA is shown in Fig. 11b.

A comparison between Fig. 11a and b reveals that, following RTM correction, the difference between the modeled GA and the measured GA is significantly reduced in the central and northern regions with higher elevations, with a more uniform spatial distribution and a marked reduction in the number of high-amplitude areas. Incorporating the computed RTM-GA into the XGM-GA model can compensate for its truncation error, as the RTM-GA reflects high-frequency information that the XGM model cannot represent. The comparative statistical results between the XGM-GA and the measured GA, as well as between the XGM/RTM-GA and the measured GA at land, ocean, and all measured points, are presented in Table 2.

From Table 2, it can be seen that the accuracy of the GA model is improved on both land and sea after applying the RTM correction. At the land measurement points, the RMS of the difference between the measured GA and the XGM-GA decreased by 14.62 mGal, with an improvement rate (IR) of 52.1 %. At the ocean measurement points, the RMS of the difference decreased by 0.45 mGal, with an IR of 7.1 %. Overall, at all measurement points, the RMS of the difference decreased by 6.33 mGal, with an IR of 43.4 %. According to the statistics, the accuracy of the XGM-GA model is lower in land areas and higher in ocean regions, while the RTM correction is significantly more effective in land areas than in ocean regions.

To investigate the correction effect of the RTM method at different elevations, the measured GA points were classified into five categories based on elevation. The elevation ranges for these five categories are [-3400, -1600 m), [-1600, 0 m), [0, 500 m), [500, 1500 m), and [1500, 4000 m), with 5035, 5762, 1498, 1360, and 389 points, respectively. The RTM-GA was calculated separately for these five categories of points, and the XGM-GA and XGM/RTM-GA were compared with the measured GA for each category. The statistical results are shown in Table 3.

From the table above, it can be seen that the higher the elevation on land, the greater the difference between the modeled and measured values; the deeper the ocean, the smaller the difference at the sea surface between the modeled and measured values. This suggests that areas with greater terrain undulation on land have lower modeled GA accuracy, as the terrain contributes to the omission of high-frequency information in the modeled GA. At sea surface measurement points, the shallower the water, the closer the seafloor terrain is to the measurement point. Consequently, the gravitational effect produced by the residual seafloor topography at these points is greater, leading to better correction results with the RTM method. The RMS values at these five types of points were reduced by 0.20, 0.58, 14.39, 15.25, and 15.61 mGal, respectively. It is evident that as the elevation increases, the RTM correction effect also increases.

Figure 12 shows the distribution histograms of the differences between the measured GA and both the XGM-GA and XGM/RTM-GA at all measured points in the study area. Statistical analysis shows that 93.75 % of the differences between the measured GA and XGM-GA fall within ±30 mGal, while this proportion increases to 99.15 % for the differences between the measured GA and XGM/RTM-GA. These findings demonstrate a notable improvement in data quality resulting from the application of the RTM method.

When a terrestrial computation point is located in a coastal area, the integration region includes both land and ocean. The standard calculation requires distinguishing rectangular prisms with different densities for land and ocean, which reduces computational efficiency and increases the computational burden. To avoid the need to distinguish between different density values during the forward modeling process, this study employs the RET method, compressing seawater into an equivalent rock mass while incorporating mass center offset correction.

A 6 km landward buffer zone was established along the coastline to identify measured points where the integration region includes both land and ocean. In the study area, 264 land-coastal points were selected for forward modeling of RTM-GA using the RET method. The calculations considered three cases: residual terrain in the land region, residual terrain in the ocean region, and residual terrain in the entire integration region. The statistical results are presented in Table 4.

The statistical results indicate that using residual terrain on both land and ocean can refine the XGM-GA to varying degrees. Compared to the measured GA, the RMS difference is reduced by 6.27 mGal when considering only residual terrain on land (setting RTM elevation over the ocean to zero). When considering only residual terrain in the ocean (setting RTM elevation over land to zero), the RMS reduction is 0.71 mGal. When considering all residual terrain, the RMS difference is reduced by 7.01 mGal.

It is noteworthy that when only the ocean residual terrain is considered, the RMS is reduced by only 0.34 mGal without applying the MCOC, whereas it decreases by 0.71 mGal when the correction is applied. When both marine and terrestrial residual terrain are considered, the application of the MCOC further reduces the RMS by 0.25 mGal compared to the case without applying it. This indicates that introducing the mass center offset correction in the RET method can effectively enhance the accuracy of the model.

To more comprehensively reflect the effects of RTM correction on the XGM-GA, a power spectral density (PSD) analysis was conducted in the frequency domain for both the XGM-GA model and the XGM/RTM-GA model. The results are shown in Fig. 13. PSD analysis is a commonly used method for evaluating signal characteristics in the frequency domain, which effectively reveals the energy distribution of data across different frequencies. At the same wavelength, a higher PSD value indicates that the model contains greater energy and more complete information.

According to statistical analysis, the PSD of the XGM-GA and XGM/RTM-GA models are nearly identical in the medium-to-long-wavelength range, where wavelengths exceed 39.5 km. In the wavelength range of 18.7 to 39.5 km, the PSD of the XGM/RTM model exhibits a slight increase compared to that of the XGM model. For wavelengths shorter than 18.7 km, the PSD of the XGM/RTM model shows a substantial enhancement relative to the XGM model. These results indicate that the XGM-GA model contains more high-frequency information, confirming that the RTM method effectively compensates for the deficiency of high-frequency components in the model and reduces its truncation error.