Explicit expressions for the Fréchet derivatives in 3D anisotropic resistivity inversion

Abstract

We have developed explicit expressions for the Fréchet derivatives or sensitivity functions in resistivity imaging of a heterogeneous and fully anisotropic earth. The formulation involves the Green’s functions and their gradients, and it is developed from a formal perturbation analysis and by means of a numerical (finite-element) method. A critical factor in the equations is the derivative of the electrical conductivity tensor with respect to the principal conductivity values and the angles defining the axes of symmetry. The Fréchet derivative expressions were derived for the 2.5D and 3D problems using constant-point and constant-block model parameterizations. Special cases such as an isotropic earth and tilted transversely isotropic (TTI) media emerge from the general solutions. Numerical examples were investigated for various sensitivities as functions of dip angle and strike of the plane of stratification in uniform TTI media.

Comparison of DC sensitivity patterns for anisotropic and isotropic media

“…In a recent paper (Greenhalgh et al, 2009a) we derived the Fréchet derivatives for general 3D/2.5D anisotropic heterogeneous media. Equations were given using both the constant point and the constant block medium parameterisation, and were presented for the pole-pole array in terms of the Green's functions for the electric potential from both the true source G A and the adjoint source G M (potential electrode position).…”

“…Whilst a number of papers have appeared on how to calculate the Fréchet derivatives in resistivity inversion (McGillivray and Oldenburg, 1990;Park and Van, 1991;Sasaki, 1994;Loke and Barker, 1995;Spitzer, 1998;Zhou and Greenhalgh, 1999) they are all based on an isotropic medium assumption. Only recently, in a companion paper (Greenhalgh et al, 2009a), did we present a mathematical formulation for calculating the Fréchet derivatives for a general anisotropic heterogeneous 3D or 2.5D medium. It was given in terms of the Green's functions, which must be calculated anyway as part of the forward modelling.…”

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

“…Normally they are computed for all subsurface positions and for all electrode combinations to show the sensitivity behaviour with each configuration and facilitate updates in the conductivity estimates during a nonlinear inversion of electrical resistivity data. Using the constant block approximation for conductivity specification, GREENHALGH et al (2008a), using a formal perturbation analysis and the self-adjoint nature of the differential operator, derived the following formula for the derivatives of the potential with respect to the model parameters m b in the most general 3-D anisotropic, heterogeneous medium:…”

“…In a recent paper we presented a general formulation for calculating the electric potential and Fréchet derivatives in an arbitrary 3-D anisotropic, heterogeneous medium (GREENHALGH et al, 2008a). It was based on a new Gaussian quadrature grid formulation for calculating the 3-D Green's functions.…”

Resistivity inversion of transversely isotropic media

“…There are many algorithms and methods available in the literature for this purpose (e.g., Pain et al, 2003;Herwanger et al, 2004;LaBrecque et al, 2004;Kim et al, 2006;Greenhalgh et al, 2009b;Wiese et al, 2015). The forward 2D DC FD method can be formulated as the following matrix notation: d=G(m) (11) where G is a nonlinear forward operator, m is a parameter vector, and d is an observation vector (Meju, 1994).…”