The concept of moments of the impulse response, introduced by Smith and Lee (2000), provides another means for rapid interpretation and modelling of transient electromagnetic data. This concept builds on developments made by Macnae et al. (1998Macnae et al. ( , 1999 in modelling the inductive and resistive limits. In fact, the inductive limit is the zeroth-order moment and the resistive limit is the first-order moment.
ABSTRACTThe impulse response moments of a conductive sphere in free space excited by a uniform magnetic field can be used to approximate the moments of a sphere in a dipolar field. The numerical computations are straightforward and the approximation is especially good for higher-order moments. The greatest discrepancy is seen on the zeroth-order moment when the radius of the sphere is large. It is possible to improve the accuracy for the zeroth-order moment by modelling the large-radius sphere (in a dipole field) as the combined response of multiple small-radius spheres (each in a locally uniform field). The small spheres are closely packed inside the larger sphere. The discrepancy can be reduced to less than 15 % in this manner.The sphere in a uniform field can also be used to approximate the response of a body that has its currents constrained to flow in a plane with a specific orientation. This means that plate-like bodies or anisotropic spheres can also be modelled.The third-order moment has been calculated from data acquired during a MEGATEM airborne electromagnetic survey of the Reid-Mahaffy test site. There is an anomalous response in the thirdorder moment that can be modelled by a sphere at 170 m depth with a conductivity of 15 S/m and a radius of 40 m. The currents flowing in the sphere are constrained to flow in a vertical plane. This model is consistent with the geology of the area and a hole drilled to test the anomalous zone.
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.
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