Subcrustal stress induced by mantle convection can be determined by the Earth's gravitational potential. In this study, the spherical harmonic expansion of the simplified Navier–Stokes equation is developed further so satellite gradiometry data (SGD) can be used to determine the subcrustal stress. To do so, we present two methods for producing the stress components or an equivalent function thereof, the so-called S function, from which the stress components can be computed numerically. First, some integral estimators are presented to integrate the SGD and deliver the stress components and/or the S function. Second, integral equations are constructed for inversion of the SGD to the aforementioned quantities. The kernel functions of the integrals of both approaches are plotted and interpreted. The behaviour of the integral kernels is dependent on the signal and noise spectra in the first approach whilst it does not depend on extra information in the second method. It is shown that recovering the stress from the vertical–vertical gradients, using the integral estimators presented, is suitable, but when using the integral equations the vertical–vertical gradients are recommended for recovering the S function and the vertical–horizontal gradients for the stress components. This study is theoretical and numerical results using synthetic or real data are not given.

The spherical harmonic expressions of the horizontal sub-crustal stress components induced by the mantle convection are convergent only to low degrees. In this paper, we use the method of stress (S) function with numerical differentiation and present a formula for determining the degree of convergence from the mean Moho depth. We found that for the global mean Moho depth, 23 km, this convergence degree is 622 and for Iran, 35 km, it is 372. Also, three methods are developed and applied for computing the sub-crustal stress, (1) direct integration with a spectral kernel limited up to the degree of convergence, (2) integral inversion with a kernel having closed-form formula without any frequency limit, and (3) solving an integral equation with limited spectral kernel to the convergence degree. The second method has no divergence problem and its kernel function is well behaving so that the system of equations from which the S function is determined is stable, and no regularisation is needed to solve it. It should be noted that for using this method the resolution of the recovery should be higher than 0.5° × 0.5°, otherwise the recovered S function and correspondingly the stress components will have smaller magnitude than those derived from the other two methods. Our numerical studies for stress recovery in Iran and its surrounding areas show that the methods, which use the limited spectral kernels to the convergence degree, deliver consistent results to that of the spherical harmonic expansion.

University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering. Department of Geodesy, K.N.Toosi University of Technology, Tehran.

The Earth’s gravity potential can be determined from its second-order partial derivatives using the spherical gradiometric boundary-value problems which have three integral solutions. The problem of merging these solutions by spectral combination is the main subject of this paper. Integral estimators of biased- and unbiased-types are presented for recovering the disturbing gravity potential from gravity gradients. It is shown that only kernels of the biased-type integral estimators are suitable for simultaneous downward continuation and combination of gravity gradients. Numerical results show insignificant practical difference between the biased and unbiased estimators at sea level and the contribution of far-zone gravity gradients remains significant for integration. These contributions depend on the noise level of the gravity gradients at higher levels than sea. In the cases of combining the gravity gradients, contaminated with Gaussian noise, at sea and 250 km levels the errors of the estimated geoid heights are about 10 and 3 times smaller than those obtained by each integral

Elastic thickness (Te) is a parameter representing the lithospheric strength with respect to the loading. Those places, having large values of elastic thickness, flexes less. In this paper, the on-orbit measured gravitational gradients of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) mission are used for determining the elastic thickness over Africa. A forward computational method is developed based on the Vening Meinesz-Moritz (VMM) and flexural theories of isostasy to find a mathematical relation between the second-order derivative of the Earth’s gravity field measured by the GOCE satellite and mechanical properties of the lithosphere. The loading of topography and bathymetry, sediments and crystalline masses are computed from CRUST1.0, in addition to estimates of laterally-variable density of the upper mantle, Young’s modulus and Poisson’s ratio. The second-order radial derivatives of the gravitational potential are synthesised from the crustal model and different a priori values of elastic thickness to find which one matches the GOCE on-orbit gradient. This method is developed in terms of spherical harmonics and performed at any point along the GOCE orbit without using any planar approximation. Our map of Te over Africa shows that the intra-continental hotspots and volcanoes, such as Ahaggar, Tibesti, Darfur, Cameroon volcanic line and Libya are connected by corridors of low Te. The high values of Te are mainly associated with the cratonic areas of Congo, Chad and the Western African basin.

n this study we investigate the lithospheric stresses computed from the gravity and lithospheric structure models. The functional relation between the lithospheric stress tensor and the gravity field parameters is formulated based on solving the boundary-value problem of elasticity in order to determine the propagation of stresses inside the lithosphere, while assuming the horizontal shear stress components (computed at the base of the lithosphere) as lower boundary values for solving this problem. We further suppress the signature of global mantle flow in the stress spectrum by subtracting the long-wavelength harmonics (below the degree of 13). This numerical scheme is applied to compute the normal and shear stress tensor components globally at the Moho interface. The results reveal that most of the lithospheric stresses are accumulated along active convergent tectonic margins of oceanic subductions and along continent-to-continent tectonic plate collisions. These results indicate that, aside from a frictional drag caused by mantle convection, the largest stresses within the lithosphere are induced by subduction slab pull forces on the side of subducted lithosphere, which are coupled by slightly less pronounced stresses (on the side of overriding lithospheric plate) possibly attributed to trench suction. Our results also show the presence of (intra-plate) lithospheric loading stresses along Hawaii islands. The signature of ridge push (along divergent tectonic margins) and basal shear traction resistive forces is not clearly manifested at the investigated stress spectrum (between the degrees from 13 to 180).