Oldenburg made an excellent example of the application of linearized inverse theory to invert dc resistivity sounding data to fit a continuous vertical variation of resistivity.

In the Introduction he mentioned that the Frechet kernels for resistivity are the same as the depth investigation characteristic function (DIC) used by Roy and Apparao (1971). In the second part of the paper, he showed that it is so for a uniformly conducting half space. He mentioned that the electrostatic analog which was used (by Roy and Apparao) becomes quite complex when a layered medium is introduced, and that the extension to a continuous ρ(z) would be a difficult task (p. 623).

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