The concept of moments of the electromagnetic response is useful in electromagnetic interpretation. Analytic formulae exist for low-order moments of a few conductivity models, enabling source parameters such as time constant, depth, conductance and conductivity to be estimated from the measured moments of the electromagnetic response. However, most models for which analytic formulae exist have conductivity varying abruptly as a function of depth or position. In this paper, we have derived a procedure that allows moments of any order to be calculated for a conductivity which has finite conductance but can otherwise vary arbitrarily with depth. The horizontal loop transient electromagnetic step response is computed as a sum of residues. Integration of the step response over time yields a mathematical expression for a moment of any order. We illustrate the procedure for a Gaussian conductivity function which varies smoothly with depth. The Gaussian model produces results that agree in specific limits with the thin sheet, thick sheet and half-space cases.

中文翻译：

---

具有有限电导的一维地球瞬态电磁响应的多阶矩——理论

电磁响应矩的概念在电磁解释中很有用。一些电导率模型的低阶矩存在解析公式，可以根据电磁响应的测量矩估计时间常数、深度、电导和电导率等源参数。然而，存在解析公式的大多数模型的电导率作为深度或位置的函数突然变化。在本文中，我们导出了一个程序，该程序允许为具有有限电导但可以随深度任意变化的电导计算任何阶的矩。水平环瞬态电磁阶跃响应计算为残差的总和。阶跃响应随时间的积分产生任意阶时刻的数学表达式。我们说明了随深度平滑变化的高斯电导率函数的过程。高斯模型产生的结果与薄板、厚板和半空间情况在特定限制上一致。