What is the depth of investigation of a resistivity measurement?

Abstract

Ever since the first computation of resistivity sounding curves, there has been the impression that somehow they are averages of the vertical resistivity profile. This prompted the idea to represent apparent resistivity as an integral over depth and to define depth of investigation using the integrands of the integrals as elementary contributions. However, elementary contributions for a boundary value problem cannot be uniquely defined and are not physically meaningful. Many practical applications that have been derived from this approach might be at stake regarding their theoretical basis. On the other hand, a sensitivity function has a definite physical meaning and it is uniquely defined, but it offers a different picture for a layered earth. The concept of elementary contributions must then be abandoned as not real, as some respected scholars have suggested, or it must be put on solid ground if we are going to continue using it. Our claim is that any definition of elementary contributions must comply with the concept of sensitivity; otherwise, it must be discarded not because it might be proved wrong, but because we cannot have multiple functions pretending to represent the depth of investigation of a resistivity measurement. We determined that both concepts can be unified and reconciled into a single formulation. That is, one and the same function of depth can be interpreted as an elementary contribution or as the local sensitivity. To further support the effectiveness of the concept, we applied it beyond its traditional application to homogeneous media. We developed an approximate formula for computing apparent resistivity as a weighted average of the vertical resistivity profile. The formula works in the way of a toy model; it is an approximation, but it provides immediate insights into how a vertical resistivity profile relates to its sounding curve.