Research Of Pseudo-spectral Method For2-D Electromagnetic Diffusion In Time Domain

Electromagnetic modeling and propagation at low frequencies (EM diffusion) isused in a number of applications, such as geothermal exploration, EM induction inboreholes and logging while drilling, evaluation of hydrocarbon by mappingsub-seafloor resistivity, geo-electrical surveys for groundwater and mineralexploration and magnetotelluric problems.The common schemes of electromagnetic modeling are finite-difference,finite-element and spectral methods. The finite-difference method demands time-stepsatisfies the stringent stability condition. The explicit finite-element method has a lowaccuracy and the implicit finite-element method has significant computation effort. Sothey are restricted to plane layers or have finite accuracy.The spectral method is used in electromagnetic modeling in the latter. It showsthat the spectral method is an efficient tool of solving equations of partial derivativewith respect to the time variable according to the recent study. A standard spectralscheme uses a finite difference approach to approximate time derivative. In view ofthe drawback of the finite difference approach, we convert time derivative toevolution operator and compute spatial derivatives with the staggered Fourierpseudo-spectral method or Chebyshev pseudo-spectral method.The pseudo-spectral method has infinite accuracy in space in theory. The FFTused in the calculation is very time-saving. The approximation of evolution operatorwith Chebyshev expansion overcomes two drawbacks: low accuracy and stringentstability condition, since the error in time decays exponentially. It does not need innerproducts. This is highly beneficial, especially in parallel computing.This paper presents the basic principal of EM modeling in time domain andintroduces the staggered Fourier pseudo-spectral and Chebyshev pseudo-spectral methods which used to approximate spatial derivatives in detail. The errors result ofthe approximation of evolution operator with Chebyshev expansion is given in thepaper. Mirroring the study of seismology, the paper proposes that the spatialderivative is approximated by Chebyshev pseudo-spectral method in the verticaldirection which is not periodic and Fourier pseudo-spectral method in the horizontaldirection. Consequently, the demand of periodic boundary condition can be improved.At the end, spectral method is used in solve initial-value problems and EM diffusionproblems of transient source. Some numerical solutions of simple models arecompared with analytical solutions which show good approximations. And somenumerical solutions of complex models are also given， including response ofconductor and response of overburden which influence response of conductorqualitatively. The result corresponds to the theoretical analysis.