Construction of conductance bounds from magnetotelluric impedances

Abstract

Whereas any finite set of impedance data does not constrain the electrical conductivity σ (z) at a fixed level z in a 1D-model, the conductance function S (z₂) as the depth-integrated conductivity from the surface to the depth z₂ will be constrained. Assuming only the non-negativity of σ (z), it is shown that for a given depth z₂ the models generating the lower and upper bound of S (z₂) consist of a sequence of thin sheets. The determination of the positions of the thin sheets and their conductances leads to a system of nonlinear equations. As a limitation the present approach requires the existence of a model, which exactly fits the data. The structure of the extremal models as a function of z₂ is discussed in examples with a small number of frequencies. Moreover, it is shown that any set of complex 1D impedances for M frequencies can be represented by a partial fraction expansion involving not more than 2M (positive) constants. For exactly 2M constants there are two complementary representations related to the lower and upper bound of S (z₂). For the simple one-frequency case, a more general extremal problem is briefly considered, where the admitted conductivities are constrained by a priori bounds σ _– (z) and σ ₊ (z) such that σ _– (z) ≤ σ (z) ≤ σ ₊ (z). In this case, the extremal models for S (z₂) consist of a sequence of sections with alternating conductivities σ _– (z) and σ ₊ (z). The sharpening of conductance bounds by incorporating a priori information is illustrated by an example.

          ARK: https://n2t.net/ark:/88439/y078099

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