What is the depth of investigation of a resistivity measurement? Enrique Gómez-Treviño 1 and Francisco J. Esparza 1 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 differ- ent picture for a layered earth. The concept of elementary con- tributions 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 sen- sitivity; 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 con- tribution or as the local sensitivity. To further support the effec- tiveness of the concept, we applied it beyond its traditional application to homogeneous media. We developed an approxi- mate 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. INTRODUCTION The theory of resistivity soundings is very well developed. The mathematical formulation of Stefanescu et al. (1930) and the fast computation of sounding curves using linear filter theory by Ghosh (1971) very much characterize the solution of the key issues of the forward problem. It is now possible to reproduce, in practically no time, sets of master curves that were a privilege to have a few dec- ades ago. However, then and now, the analysis of master curves reveals basic issues that are universally acknowledged, but that are not readily addressed from the way the curves are computed. The first thing that stands out is that the depth of investigation in- creases with the separation between potential and current electro- des. This is inferred from the fact that a deep interface gradually makes itself present in the sounding curve as the separation in- creases. However, this inference is not readily available from the way the curves are calculated. For a pole-pole array with a separa- tion r between the current and potential electrodes, the apparent resistivity ρ a ðrÞ due to a resistivity variation ρðzÞ, where z repre- sents depth, is computed as ρ a ðrÞ¼ Z ∞ 0 Kðλ; ρðzÞÞJ 0 ðλrÞdλ: (1) The function Kðλ; ρðzÞÞ is called the resistivity transform and represents the solution of a differential equation, λ is the variable of integration, and J 0 is the Bessel function of the first kind and order zero. The function Kðλ; ρðzÞÞ is integrated over the variable λ, leaving ρðzÞ and z implicit for any separation r. Because there is no integral over depth, the depth of investigation for a given sep- aration is not readily available from equation 1. This peculiarity was recognized early on after the work of Stefanescu et al. (1930) by Evjen (1938), who proposes what he calls depth factors on the basis of the image solution to the potential of a point source. His results did not received much attention until they were revived as the depth of investigation characteristics by Roy and Aparao (1971) using an Manuscript received by the Editor 12 July 2013; revised manuscript received 3 October 2013; published online 17 February 2014. 1 CICESE, División de Ciencias de la Tierra, Ensenada, Baja California, Mexico. E-mail: egomez@cicese.mx; fesparz@cicese.mx. © 2014 Society of Exploration Geophysicists. All rights reserved. W1 GEOPHYSICS, VOL. 79, NO. 2 (MARCH-APRIL 2014); P. W1–W10, 12 FIGS. 10.1190/GEO2013-0261.1