The relationship between the spectral function and the underlying conductivity structure in 1‐D magnetotellurics

SUMMARY The frequency response c (ω) in 1‐D magnetotellurics admits a well‐known integral representation with kernel 1/(λ+ i ω) and non‐negative spectral function w (λ), λ≥ 0 . The purpose of this paper is to elucidate the hidden, but fundamental relationship between w (λ) and the underlying conductivity structure σ( z ) . The most important criterion for classifying the conductivity structure is the existence of moments of w (λ) : if all moments exist, σ( z ) consists of a finite or infinite number of thin sheets; the sheet parameters are obtained from orthogonal polynomials associated with the weight function w (λ) . If no moment (or only a finite number of moments) exists, σ( z ) contains sections with a piecewise continuous conductivity structure, possibly covered by thin sheets. In both cases, the spectrum may be continuous, completely discrete or a mixture of both. The great variety of possible spectral functions is illustrated by a plethora of examples. The present investigation has no immediate impact on practical inversion because the unstable determination of w (λ) is mostly circumvented in the inversion of experimental data. Therefore, the rich morphology of the spectral function generally has escaped our attention.