A numerical study of residual terrain modelling (RTM) techniques and the harmonic correction using ultra-high-degree spectral gravity modelling

Abstract

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 ($4\pi G\rho H_{\text{P}}^{\text{RTM}} )$ may reach amplitudes of ~ 10 mGal for points located deep inside the reference topography. We further show that the popular RTM approximation ($2\pi G\rho H_{\text{P}}^{\text{RTM}} - {\text{TC}}$) 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.

Appendix 1

1.1 Results for the European Alps area

Table 4 reports the descriptive statistics of all gravity components and their differences over the European Alps. The RTM-A is in ~ 0.6 mGal RMS agreement with the baseline solution. For RTM-B, the agreement deteriorates to the level of ~ 1.8 mGal RMS (reflecting the mass simplification error) and RTM-C to the level of ~ 3.2 mGal RMS (computation point inconsistency error). For RTM-D, the RMS-differences w.r.t. the baseline solution are ~ 12.6 mGal and maximum errors exceed 50 mGal (cf. Table 4). Focussing on RTM-A and points with positive (negative) RTM elevations, the agreement with the baseline solution is 0.21 mGal (0.77 mGal). In the latter case, maximum errors may reach amplitudes of up to ~ 7.5 mGal, reflecting the approximative character of the harmonic correction. Overall, the error level associated with the various RTM approximations is somewhat lower over the European Alps (Table 4) than the Himalayas (Table 3) which is explained by the different ruggedness of the test areas.

Table 4 Descriptive statistics of the constituents NI, SGM, HF of the RTM (baseline solution and variants A, B, C, D) and their differences over the 2° × 2° test area “European Alps” (45°–47° latitude and 7°–9° longitude)