Far-zone contributions to the gravity field quantities by means of Molodensky’s truncation coefficients

To reduce the numerical complexity of inverse solutions to large systems of discretised integral equations in gravimetric geoid/quasigeoid modelling, the surface domain of Green’s integrals is subdivided into the near-zone and far-zone integration sub-domains. The inversion is performed for the near zone using regional detailed gravity data. The farzone contributions to the gravity field quantities are estimated from an available global geopotential model using techniques for a spherical harmonic analysis of the gravity field. For computing the far-zone contributions by means of Green’s integrals, truncation coefficients are applied. Different forms of truncation coefficients have been derived depending on a type of integrals in solving various geodetic boundary-value problems. In this study, we utilise Molodensky’s truncation coefficients to Green’s integrals for computing the far-zone contributions to the disturbing potential, the gravity disturbance, and the gravity anomaly. We also demonstrate that Molodensky’s truncation coefficients can be uniformly applied to all types of Green’s integrals used in solving the boundaryvalue problems. The numerical example of the far-zone contributions to the gravity field quantities is given over the area of study which comprises the Canadian Rocky Mountains. The coefficients of a global geopotential model and a detailed digital terrain model are used as input data.     

Gravity disturbances in regions of negative heights: A reference quasi-ellipsoid approach

Compilation of the bathymetrically and topographically corrected gravity disturbance, the so called BT disturbance, for the purpose of gravity interpretation/inversion, is investigated from the numerical point of view, with special emphasis on regions of negative heights. In regions of negative ellipsoidal (geodetic) heights, such as the Dead Sea region onshore or offshore areas of negative geoidal heights, two issues complicate the compilation and subsequently the inversion of the BT disturbance. The first is associated with the evaluation of normal gravity below the surface of the reference ellipsoid (RE). The latter is tied to the legitimacy of the harmonic continuation of the BT disturbance in these regions. These two issues are proposed to be resolved by the so called reference quasi-ellipsoid (RQE) approach. New bathymetric and topographic corrections are derived based on the RQE and the inverse problem is formulated based on the RQE. The RQE approach enables the computation of normal gravity by means of the international gravity formula, and makes the harmonic continuation in the regions of negative heights of gravity stations legitimate. The gravimetric inversion is then transformed from the RQE approach back to the RE approach, following the now legitimate harmonic upward continuation of the gravity data to stations on or above the RE. Stripping, the removal of an effect of a known density contrast, is considered in the context of the RQE approach. A numerical case study is presented for the RQE approach in a region of NW Canada.     

Moho depth uncertainties in the Vening-Meinesz Moritz inverse problem of isostasy

We formulate an error propagation model based on solving the Vening Meinesz-Moritz (VMM) inverse problem of isostasy. The system of observation equations in the VMM model defines the relation between the isostatic gravity data and the Moho depth by means of a second-order Fredholm integral equation of the first kind. The corresponding error model (derived in a spectral domain) functionally relates the Moho depth errors with the commission errors of used gravity and topographic/bathymetric models. The error model also incorporates the non-isostatic bias which describes the disagreement, mainly of systematic nature, between the isostatic and seismic models. The error analysis is conducted at the study area of the Tibetan Plateau and Himalayas with the world largest crustal thickness. The Moho depth uncertainties due to errors of the currently available global gravity and topographic models are estimated to be typically up to 1–2 km, provided that the GOCE gravity gradient observables improved the medium-wavelength gravity spectra. The errors due to disregarding sedimentary basins can locally exceed ～2 km. The largest errors (which cause a systematic bias between isostatic and seismic models) are attributed to unmodeled mantle heterogeneities (including the core-mantle boundary) and other geophysical processes. These errors are mostly less than 2 km under significant orogens (Himalayas, Ural), but can reach up to ～10 km under the oceanic crust.     

Comparison of methods for a 3-D density inversion from airborne gravity gradiometry

In this study, we apply Tikhonov’s regularization algorithm for a 3-D density inversion from the gravity-gradiometry data. To reduce the non-uniqueness of the inverse solution (carried out without additional information from geological evidence), we implement the depth-weighting empirical function. However, the application of an empirical function in the inversion equation brings the bias problem of the regularization factor when a traditional Tikhonov’s algorithm is applied. To solve the bias problem of regularization factor selection, we present a standardized solution that comprises two parts for solving a 3-D constrained inversion equation, specifically the linear matrix transformation and Tikhonov’s regularization algorithm. Since traditional regularization techniques become numerically inefficient when dealing with large number of data, we further apply methods which include the Simultaneous Iterative Reconstruction Technique (SIRT) and the wavelet compression combined with Least Squares QR-decomposition (LSQR). In our simulation study, we demonstrate that SIRT as well as the wavelet compression plus LSQR algorithm improve the computation efficiency, while provide results which closely agree with that obtained from applying Tikhonov’s regularization. In particular, the algorithm of wavelet compression plus LSQR shows the best computing efficiency, because it combines the advantages of coefficients compression of big matrix and fast solution of sparse matrix. Similar findings are confirmed from the vertical gravity gradient data inversion for detecting potential deposits at the Kauring (near Perth, Western Australia) testing site.     

Far-zone gravity field contributions corrected for the effect of topography by means of molodensky’s truncation coefficients

A spectral representation of the topographic corrections to gravity field quantities is formulated in terms of spherical height functions. When computing the far-zone contributions to the topographic corrections, various types of the truncation coefficients are applied to a spectral representation of Newton’s integral. In this study we utilise Molodensky’s truncation coefficients in deriving the expressions for the far-zone contributions to topographic corrections. The expressions for computing the far-zone gravity field contributions corrected for the effect of topography are then obtained by combining the expressions for the far-zone contributions to the gravity field quantities and to the respective topographic corrections, both expressed in terms of Molodensky’s truncation coefficients. The numerical examples of the far-zone contributions to the topographic corrections and to the topography-corrected gravity field quantities are given over the study area situated in the Canadian Rocky Mountains with adjacent planes. Coefficients of the global elevation and geopotential models are used as the input data.     

A regional gravimetric Moho recovery under Tibet using gravitational potential data from a satellite global model

The Moho information under Tibet Plateau is important for a better understanding of the geodynamic processes associated with the continental collision of the Indian and Eurasian tectonic plates and subsequent formation of Himalayan and Tibetan orogens. However, under the central and western parts of Tibet, the existing Moho models are still relatively inaccurate due to a sparse and irregular distribution of seismic surveys. To overcome this problem, the gravimetric data could be used to interpolate the Moho information, where seismic data are missing. In this study, we apply the gravimetric method for a regional Moho recovery under Tibet. Compared to existing methods that use either the gravity or gravity-gradient data, the method presented here utilizes a more generic definition based on a functional relation between the Moho depth and the gravitational potential. Since the gravity and gravity-gradient data have more regional support than the potential field, a numerical test is conducted to find an optimal data area extension that is needed to solve a regional inversion problem in order to reduce errors caused by disregarding the far-zone contribution. Our analysis shows that for the potential field such extension should be at least 25°, while 5° for the gravity and only about 1° for the gravity gradient. The comparison of our gravimetric result with the CRUST1.0 seismic model shows differences at the level of expected accuracy of the gravimetric method of about 5 km and without the presence of significant bias.     

Combined Gravimetric–Seismic Crustal Model for Antarctica

The latest seismic data and improved information about the subglacial bedrock relief are used in this study to estimate the sediment and crustal thickness under the Antarctic continent. Since large parts of Antarctica are not yet covered by seismic surveys, the gravity and crustal structure models are used to interpolate the Moho information where seismic data are missing. The gravity information is also extended offshore to detect the Moho under continental margins and neighboring oceanic crust. The processing strategy involves the solution to the Vening Meinesz-Moritz’s inverse problem of isostasy constrained on seismic data. A comparison of our new results with existing studies indicates a substantial improvement in the sediment and crustal models. The seismic data analysis shows significant sediment accumulations in Antarctica, with broad sedimentary basins. According to our result, the maximum sediment thickness in Antarctica is about 15 km under Filchner-Ronne Ice Shelf. The Moho relief closely resembles major geological and tectonic features. A rather thick continental crust of East Antarctic Craton is separated from a complex geological/tectonic structure of West Antarctica by the Transantarctic Mountains. The average Moho depth of 34.1 km under the Antarctic continent slightly differs from previous estimates. A maximum Moho deepening of 58.2 km under the Gamburtsev Subglacial Mountains in East Antarctica confirmed the presence of deep and compact orogenic roots. Another large Moho depth in East Antarctica is detected under Dronning Maud Land with two orogenic roots under Wohlthat Massif (48–50 km) and the Kottas Mountains (48–50 km) that are separated by a relatively thin crust along Jutulstraumen Rift. The Moho depth under central parts of the Transantarctic Mountains reaches 46 km. The maximum Moho deepening (34–38 km) in West Antarctica is under the Antarctic Peninsula. The Moho depth minima in East Antarctica are found under the Lambert Trench (24–28 km), while in West Antarctica the Moho depth minima are along the West Antarctic Rift System under the Bentley depression (20–22 km) and Ross Sea Ice Shelf (16–24 km). The gravimetric result confirmed a maximum extension of the Antarctic continental margins under the Ross Sea Embayment and the Weddell Sea Embayment with an extremely thin continental crust (10–20 km).