A quasi-linear inverse problem with a priori information about model parameters is formulated in a stochastic framework by using a singular value decomposition technique for arbitrary rectangular matrices.
In many geophysical inverse problems, we have a priori information from which a most plausible solution and the statistics of its probable error can be guessed. Starting from the most plausible solution, the optimization of model parameters is made by the successive iteration of solving a set of standardized linear equations for corrections at each step. In under-determined cases, the solution depends inherently on the initial guess of model parameters, and then uncertainties in the solution are evaluated by the covariances of estimation error which results not only from the random noise in data but also from the probable error in the initial guess.
A best linear inverse operator which minimizes the variances of estimation errors within the framework of a generalized least-squares approach is directly obtained from the "natural inverse" of Lanczos for a coefficient matrix by regarding an eigenvalue in the inverse as zero if it is smaller than unity. This provides a theoretical basis on the "sharp cutoff approach" of Wiggins and also Jackson in their general inverse formalisms.

View full abstract

In this paper, we treat quasi-static displacement fields due to an axially symmetric load applied on the surface of a composite medium which consists of elastic layers overlying a stratified viscoelastic half-space. Integral representations of the surface displacements are obtained from those in the associated elastic problem by applying the correspondence principle of linear viscoelasticity. General features of the viscoelastic displacements are examined from various aspects through the computation of numerical examples for a three-layered model composed of an elastic surface layer, an intervening low viscosity layer and a viscoelastic substratum with relatively high viscosity.
Uplift phenomena in two tectonically different regions, Fennoscandia and Lake Bonneville (the Basin and Range province), are analyzed to investigate regional difference in rheological structure of the earth's crust and upper mantle. The uplift data in Fennoscandia are well interpreted by a three-layered model which consists of a 100km thick elastic surface layer (lithosphere) and a 100-200km thick low viscosity layer (asthenosphere) overlying a relatively high viscous (1.4×10²² poise) substratum. The viscosity of the asthenosphere is estimated as 1.4×10²⁰-1.4×10²¹ poise. In the case of Lake Bonneville, the uplift data are adequately explained by taking the thickness of the lithosphere as 30-40km and the viscosity of the asthenosphere as less than 1.0-2.0×10²¹ poise. The lower bound of the asthenosphere is obscure from the present analysis. These results show that there exists notable differences in the thickness of the lithosphere between two tectonically different regions, but not in the viscosity of the asthenosphere.

View full abstract