SUMMARY Parker–Oldenburg's method is perhaps the most commonly used technique to estimate the depth of density interface from gravity data. To account for large density variations reported, for instance, at the Moho interface, between the ocean seawater density and marine sediments, or between sediments and the underlying bedrock, some authors extended this method for variable density models. Parker–Oldenburg's method is suitable for local studies, given that a functional relationship between gravity data and interface geometry is derived for Earth's planar approximation. The application of this method in (large-scale) regional, continental or global studies is, however, practically restricted by errors due to disregarding Earth's sphericity. Parker–Oldenburg's method was, therefore, reformulated also for Earth's spherical approximation, but assuming only a uniform density. The importance of taking into consideration density heterogeneities at the interface becomes even more relevant in the context of (large-scale) regional or global studies. To address this issue, we generalize Parker–Oldenburg's method (defined for a spherical coordinate system) for the depth of heterogeneous density interface. Furthermore, we extend our definitions for gravity gradient data of which use in geoscience applications increased considerably, especially after launching the Gravity field and steady-state Ocean Circulation Explorer (GOCE) gravity-gradiometry satellite mission. For completeness, we also provide expressions for potential. The study provides the most complete review of Parker–Oldenburg's method in planar and spherical cases defined for potential, gravity and gravity gradient, while incorporating either uniform or heterogeneous density model at the interface. To improve a numerical efficiency of gravimetric forward modelling and inversion, described in terms of spherical harmonics of Earth's gravity field and interface geometry, we use the fast Fourier transform technique for spherical harmonic analysis and synthesis. The (newly derived) functional models are tested numerically. Our results over a (large-scale) regional study area confirm that the consideration of a global integration and Earth's sphericty improves results of a gravimetric forward modelling and inversion.
SUMMARY Temporal variations in the Earth's gravity field can be used for monitoring of lithospheric deformations. The network of continuously operating gravity stations is required for this purpose but a global coverage by such network is currently extremely sparse. Temporal variations in long-wavelength part of the Earth's gravity field have been, however, observed by two satellite missions, namely the Gravity Recovery And Climate Experiment (GRACE) and the GRACE Follow-On (GRACE-FO). These satellite gravity observations can be used to study long-wavelength deformations of the lithosphere. Consequently, considering the lithosphere as a spherical elastic shell and solving the partial differential equation of elasticity for it, the stress, strain and displacement inside the lithosphere can be estimated. The lower boundary of this shell is assumed to be stressed by mantle convection, which has a direct relation to the Earth's gravity field according to Runcorn's theory. Changes in gravity field lead to changes in the sublithospheric stress and the stress propagated throughout the lithosphere. In this study, we develop mathematical models in spherical coordinates for describing the stress propagation from the sublithosphere through the lithosphere. We then organize a system of observation equations for finding a special solution to the boundary-value problem of elasticity in the way that provides a stable solution. In contrast, models presented in previously published studies are ill-posed. Furthermore, we use constants of the solution determined from the boundary stresses to determine the strain and displacements leading to these stresses, while in previous studies only the stress has been considered according to rheological properties of the lithosphere. We demonstrate a practical applicability of this theoretical model to estimate the stress–strain redistribution caused by the Sar-e-Pol Zahab 2018 earthquake in Iran by using the GRACE-FO monthly solutions.
Global navigation satellite systems (GNSS) techniques, such as GPS, can be used to accurately record vertical crustal movements induced by seasonal terrestrial water storage (TWS) variations. Conversely, the TWS data could be inverted from GPS-observed vertical displacement based on the well-known elastic loading theory through the Tikhonov regularization (TR) or the Helmert variance component estimation (HVCE). To complement a potential non-uniform spatial distribution of GPS sites and to improve the quality of inversion procedure, herein we proposed in this study a novel approach for the TWS inversion by jointly supplementing GPS vertical crustal displacements with minimum usage of external TWS-derived displacements serving as pseudo GPS sites, such as from satellite gravimetry (e.g., Gravity Recovery and Climate Experiment, GRACE) or from hydrological models (e.g., Global Land Data Assimilation System, GLDAS), to constrain the inversion. In addition, Akaike’s Bayesian Information Criterion (ABIC) was employed during the inversion, while comparing with TR and HVCE to demonstrate the feasibility of our approach. Despite the deterioration of the model fitness, our results revealed that the introduction of GRACE or GLDAS data as constraints during the joint inversion effectively reduced the uncertainty and bias by 42% and 41% on average, respectively, with significant improvements in the spatial boundary of our study area. In general, the ABIC with GRACE or GLDAS data constraints displayed an optimal performance in terms of model fitness and inversion performance, compared to those of other GPS-inferred TWS methodologies reported in published studies.
We apply a newly developed method to estimate the Moho depths and density contrast beneath the Himalayas, Tibet and Central Siberia. This method utilizes the combined least-squares approach based on solving the inverse problem of isostasy and using the constraining information from the seismic global crustal model (CRUST2.0). The gravimetric forward modeling is applied to compute the isostatic gravity anomalies using the global geopotential model (GOCO02S)
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