The moments of the impulse response: A new paradigm for the interpretation of transient electromagnetic data

Abstract

We define the nth moment of the transient electromagnetic impulse response as the definite integral with respect to time of the “quadrature” magnetic‐field impulse response weighted by time to the nth power. In this context, the quadrature response is defined as the full impulse response with the in‐phase component (i.e., the delta function component at zero time) removed. The low‐order moments are equivalent to familiar quantities: the zeroth moment (n = 0) is numerically equal to the frequency‐domain inductive limit, and the first moment is the resistive‐limit response. The higher order moments can be of particular benefit: successively they put greater emphasis on the late‐time data, and hence can bring out features in the data that are more conductive or deeper.An advantage of calculating moments (and hence the inductive and resistive limit) is that these data are not strongly dependent on any distortion of the waveform from an ideal impulse. Hence, it is not critical to deconvolve the data prior to estimating the moments.If a conductor has a single exponential decay, the nth moment of the decay is proportional to the nth power of the time constant of the exponential. Thus, it is relatively easy to estimate the time constant from the moments. For a conductive sphere model, the expressions for the moments are more complicated, but are still simpler than the full transient solution or the frequency‐domain solution.In a field example, the high‐order moments emphasize local highly conductive features, but also show the noise present in the late‐time data. A discrete feature on the profile evident in moments 3 through 10 has been modeled as a spherical conductor with its center at 90 m depth, a radius of 45 m, and a conductivity of 9.4 S/m.