2.5-D/3-D resistivity modelling in anisotropic media using Gaussian quadrature grids

Abstract

S U M M A R YWe present a new numerical scheme for 2.5-D/3-D direct current resistivity modelling in heterogeneous, anisotropic media. This method, named the 'Gaussian quadrature grid' (GQG) method, cooperatively combines the solution of the Variational Principle of the partial differential equation, Gaussian quadrature abscissae and local cardinal functions so that it has the main advantages of the spectral element method. The formulation shows that the GQG method is a modification of the spectral element method but does not employ the constant elements or require the mesh generator to match the Earth's surface. This makes it much easier to deal with geological models having a 2-D/3-D complex topography than using traditional numerical methods. The GQG technique can achieve a similar convergence rate to the spectral element method. We show it transforms the 2.5-D/3-D resistivity modelling problem into a sparse and symmetric linear equation system that can be solved by an iterative or matrix inversion method.Comparison with analytic solutions for homogeneous isotropic and anisotropic models shows that the error depends on the Gaussian quadrature order (abscissa number) and the subdomain size. The higher the order or the smaller the subdomain size that is employed, the more accurate are the results obtained. Several other synthetic examples, both homogeneous and inhomogeneous, incorporating sloping, undulating and severe topography, are presented and found to yield results comparable to finite element solutions involving a dense mesh.

2.5D Resistivity Inversion in Anisotropic Media: Numerical Experiments

“…Zhou et al (2009) presented the Gaussian quadrature grid numerical scheme for 2.5D resistivity modeling in which the Variational Principle was applied to (1) to reformulate the problem in functional (Ψ) form: The quantity G  is the spatially Fourier transformed Green's function, which is equal to the impedance (or resistance) U/I.…”

Electric Potential and Fréchet Derivatives for a Uniform Anisotropic Medium with a Tilted Axis of Symmetry

“…They have to be calculated numerically such as by a finite element or finite difference scheme. We recently presented a new Gaussian quadrature grid scheme for evaluation of the Green's functions, which enables easy incorporation of surface topography and medium anisotropy (ZHOU et al, 2009). The quantity w p appearing above is the product of the Gaussian weights (for each coordinate direction) at the Gaussian point p. The volume is divided into a 3-D Gaussian quadrature grid and a Gaussian quadrature formula used to calculate the volume integral arising in the weak form of solution of the partial differential equation.…”

“…The sensitivity functions (Fréchet derivatives) are particularly important in optimised experimental design (STUMMER et al, 2004) and in actual inversion of resistivity data (ZHOU and GREENHALGH, 1999). Few papers in the resistivity modelling and inversion literature incorporate anisotropy (DAS, 1995;YIN and WIEDELT, 1999;PAIN et al, 2003, HERWANGER et al, 2004LI and SPITZER, 2005;KIM et al, 2006;ZHOU et al, 2009). They offer purely numerical solutions-mainly finite element and finite difference-for the potential in general inhomogeneous media.…”

Comparison of DC sensitivity patterns for anisotropic and isotropic media

“…It was given in terms of the Green's functions, which must be calculated anyway as part of the forward modelling. Accurate and efficient schemes for calculating the Green's functions in an anisotropic and inhomogeneous earth are only recently available (Li and Spitzer, 2005;Zhou et al, 2009). …”