Quasi-analytical method for frequency-to-time conversion in CSEM applications

Abstract

Frequency-to-time transformations are of interest to controlled-source electromagnetic methods when time-domain data are inverted for a subsurface resistivity model by numerical frequency-domain modeling at a selected, small number of frequencies whereas the data misfit is determined in the time domain. We propose an efficient, Prony-type method using frequency-domain diffusive-field basis functions for which the time-domain equivalents are known. Diffusive fields are characterized by an exponential part whose argument is proportional to the square root of frequency and a part that is polynomial in integer powers of the square root of frequency. Data at a limited number of frequencies suffice for the transformation back to the time. In the exponential part, several diffusion-time values must be chosen. Once a suitable range of diffusion-time values are found, the method is quite robust in the number of values used. The highest power in the polynomial part can be determined from the source and receiver type. When the frequency-domain data are accurately approximated by the basis functions, the timedomain result is also accurate. This method is accurate over a wider time range than other methods and has the correct late-time asymptotic behavior. The method works well for data computed for layered and 3D subsurface models.

Frequency-to-time Transformation by a Diffusion Expansion Method

“…따라서 (3)식의 선형 방정식으로부터 α mn 을 계산 할 수 있다. 일반적으로 M = 3, N = 2 정도이면 정확한 값을 나타내는 것으로 보고된 바 있다(Slob and van den Burg, 1999;Tehrani et al, 2012). 일단 α mn 이 결정되면, (8)식을 사 용하여 충격 반응을, (14)식을 사용하여 계단 반응을 계산할 수 있다.…”