The Moho surface can be determined according to isostatic theories and the recent Vening Meinesz-Moritz (VMM) theory of isostasy has been successful for this purpose. In this paper, we will study this method from a theoretical prospective and try to find its connection to the Airy-Heiskanen (AH) and Vening Meinesz original theories. We develop Jeffrey’s inverse solution to isostasy according to the recent developments of the VMM method and compare both methods in similar situations. We will show that they are generalisations of the AH model in a global and continuous domain. In the VMM spherical harmonic solution for Moho depth, the mean Moho depth contributes only to the zero-degree term of the series, whilst in Jeffrey’s solution it contributes to all frequencies. We improve the VMM spherical harmonic series further so that the mean Moho can contribute to all frequencies of the solution. This modification makes the VMM global solution superior to the Jeffrey one, but in a global scale, the difference between both solutions is less than 3 km. Both solutions are asymptotically-convergent and we present two methods to obtain smooth solutions for Moho from them.

Elastic thickness (Te) is one of mechanical properties of the Earth's lithosphere. The lithosphere is assumed to be a thin elastic shell, which is bended under the topographic, bathymetric and sediment loads on. The flexure of this elastic shell depends on its thickness or Te. Those shells having larger Te flex less. In this paper, a forward computational method is presented based on the Vening Meinesz–Moritz (VMM) and flexural theories of isostasy. Two Moho flexure models are determined using these theories, considering effects of surface and subsurface loads. Different values are selected for Te in the flexural method to see by which one, the closest Moho flexure to that of the VMM is achieved. The effects of topographic/bathymetric, sediments and crustal crystalline masses, and laterally variable upper mantle density, Young's modulus and Poisson's ratio are considered in whole computational process. Our mathematical derivations are based on spherical harmonics, which can be used to estimate Te at any single point, meaning that there is no edge effect in the method. However, the Te map needs to be filtered to remove noise at some points. A median filter with a window size of 5° × 5° and overlap of 4° works well for this purpose. The method is applied to estimate Te over South America using the data of CRUST1.0 and a global gravity model.

The sub-lithospheric stress due to mantle convection can be computed from gravity data and propagated through the lithosphere by solving the boundary-value problem of elasticity for the Earth's lithosphere. In this case, a full tensor of stress can be computed at any point inside this elastic layer. Here, we present mathematical foundations for recovering such a tensor from gravitational tensor measured at satellite altitudes. The mathematical relations will be much simpler in this way than the case of using gravity data as no derivative of spherical harmonics or Legendre polynomials is involved in the expressions. Here, new relations between the spherical harmonic coefficients of the stress and gravitational tensor elements are presented. Thereafter integral equations are established from them to recover the elements of stress tensor from those of the gravitational tensor. The integrals have no closed-form kernels, but they are easy to invert and their spatial truncation errors are reducible. The integral equations are used to invert the real data of the gravity field and steady-state ocean circulation explorer (GOCE) mission, in November 2009, over the South American plate and its surroundings to recover the stress tensor at a depth of 35 km. The recovered stress fields are in good agreement with the tectonic and geological features of the area.

Different approximations are used in Moho modelling based on isostatic theories. The well-known approximation is considering a plate shell model for isostatic equilibrium, which is an oversimplified assumption for the Earth’s crust. Considering a spherical shellmodel, as used in the Vening Meinesz-Moritz (VMM) theory, is a more realistic assumption, but it suffers from different types of mathematical approximations. In this paper, the idea is to investigate such approximations and present their magnitudes and locations all over the globe. Furthermore, we show that the mathematical model of Moho depth according to the VMM principle can be simplified to that of the plate shell model after four approximations. Linearisation of the binomial term involving the topographic/bathymetric heights is sufficient as long as their spherical harmonic expansion is limited to degree and order 180. The impact of the higher order terms is less than 2 km. The Taylor expansion of the binomial term involving the Moho depth (T) up to second order with the assumption of T-2 = TT0, T-0 is the mean compensation depth, improves this approximation further by up to 4 km over continents. This approximation has a significant role in Moho modelling over continents; otherwise, loss of frequency occurs in the Moho solution. On the other hand, the linear approximation performs better over oceans and considering higher order terms creates unrealistic frequencies reaching to a magnitude of 5 km in the Moho solution. Involving gravity data according to the VMM principle influences the Moho depth significantly up to 15 km in some areas.

5. Comparison of gravimetric and mantle flow solutions for sub-lithopsheric stress modeling and their combination

Eshagh, Mehdi

et al.

University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.

Steinberger, Bernhard

Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Telegrafenberg, Potsdam, Germany & Centre for Earth Evolution and Dynamics (CEED), University of Oslo, Postboks 1028 Blindern, Oslo, Norway.

Tenzer, Robert

Hong Kong Polytechnic University, Department of Land Surveying and Geo-Informatics, 11 Yuk Chai Rd, Hung Hom, Hong Kong.

Tassara, Andrés

Universidad de Concepción, Departamento de Ciencias de la Tierra, Facultad de Ciencias Químicas, Victor Lamas 1290, Concepción, Chile.

6. Spectral combination of spherical gravitational curvature boundary-value problems

Pitonak, Martin

et al.

University of West Bohemia, NTIS—New Technologies for the Information Society, Faculty of Applied Sciences, Technicka´ 8, 306 14 Plzen, Czech Republic.

Eshagh, Mehdi

University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.

Sprlak, Michal

University of Newcastle, School of Engineering, Faculty of Engineering and Built Environment, Callaghan, NSW 2308, Australia.

Tenzer, Robert

The Hong Kong Polytechnic University, Department of Land Surveying and Geo-informatics, 181 Chatham Road South, Hung Hom, 999077 Kowloon, Hong Kong.

Novak, Pavel

University of West Bohemia, NTIS—New Technologies for the Information Society, Faculty of Applied Sciences, Technicka´ 8, 306 14 Plzen, Czech Republic.

Four solutions of the spherical gravitational curvature boundary-value problems can be exploited for the determination of the Earth’s gravitational potential. In this paper we discuss the combination of simulated satellite gravitational curvatures, that is, components of the third-order gravitational tensor, by merging these solutions using the spectral combination method. For this purpose, integral estimators of biased-and unbiased-types are derived. In numerical studies, we investigate the performance of the developed mathematical models for the gravitational field modelling in the area of Central Europe based on simulated satellite measurements. First, we verify the correctness of the integral estimators for the spectral downward continuation by a closed-loop test. Estimated errors of the combined solution are about eight orders smaller than those from the individual solutions. Second, we perform a numerical experiment by considering the Gaussian noise with the standard deviation of 6.5 x 10(-17) m(-1) s(-2) in the input data at the satellite altitude of 250 km above the mean Earth sphere. This value of standard deviation is equivalent to a signal-to-noise ratio of 10. Superior results with respect to the global geopotential model TIM-r5 (Brockmann et al. 2014) are obtained by the spectral downward continuation of the vertical-vertical-vertical component with the standard deviation of 2.104 m(2) s(-2), but the root mean square error is the largest and reaches 9.734 m(2) s(-2). Using the spectral combination of all gravitational curvatures the root mean square error is more than 400 times smaller but the standard deviation reaches 17.234 m(2) s(-2). The combination of more components decreases the root mean square error of the corresponding solutions while the standard deviations of the combined solutions do not improve as compared to the solution from the vertical-vertical-vertical component. The presented method represents a weight mean in the spectral domain that minimizes the root mean square error of the combined solutions and improves standard deviation of the solution based only on the least accurate components.