TDEM by FDEM

Abstract

Time-domain electromagnetic measurements of induction currents are useful for geophysical prospecting in shallow sea water and on land. We review the complexity of several numerical modelling schemes. A multigrid solver makes frequency-domain modelling followed by a Fourier transform an appealing choice. Examples are included. DOI: 10.2529/PIERS060907142708 Controlled-source electromagnetic measurements of induction currents in the earth can provide resistivity maps for geophysical prospecting. In marine environments, the current source often employs one or a few frequencies. In shallow sea water or on land, the response of air is dominant and time-domain measurements are more appropriate. Because electromagnetic signals in the earth are strongly difiusive, direct interpretation of measured data can be di-cult. Inversion of the data for a resistivity model may provide better results. Therefore, an e-cient modelling and inversion algorithm is required. In the frequency domain, the multigrid method allows for a reasonably fast solution of the discretized equations (1,2). On equidistant or mildly stretched grids, the number of iterations required to solve the equations at a given frequency is about 10, independent of the number of unknowns. Only with stronger stretching does the number of iterations increase. A more powerful method based on semi-coarsening and line-relaxation (3) is less sensitive to grid stretching but the required computer time per iteration is much larger. For time-domain modelling, there are a number of methods. The simplest is explicit time stepping, but this is rather costly. The Du Fort-Frankel method (4) is more e-cient, but involves an artiflcial light speed term. Implicit methods are only e-cient if a fast solver is available. Drushkin and Knizhnerman (5,6) proposed a technique based on Lanczos reduction and matrix exponentials. Time-domain solutions can also be obtained from a frequency-domain code after a Fourier transform. An example for horizontally layered media can be found in, for instance, (7). Here, the computational cost of these methods is compared by complexity analysis. This provides an estimate of the cost as a function of the number of unknowns, but without the actual constants. The next section compares the various methods. The frequency-domain method appears to be attractive. Examples that highlight some of the issues when using a frequency-domain method are included.