Time evolution of the electric field by the rapid expansion method in controlled-source electromagnetic (CSEM) applications
I address the problem of modelling the low-frequency, time-domain controlled-source electromagnetic (CSEM) data by the rapid expansion method (REM). The CSEM method is an active EM exploration method that is recognized as complementary to the seismic method, with the focus on determining subsurface electric resistivity. Interpretation of CSEM data relies on an iterative forward modelling process to search for the model that best fits the data. Therefore, forward modelling is an essential part of the interpretation process. REM is an explicit time-domain forward modelling method that solves the diffusive EM field based on a Chebyshev expansion of the time operator. The temporal estimator is accurate to the Nyquist frequency and temporal numerical dispersion can be mitigated. I present several extensions of the REM algorithm to generalize its use in various environments. I show the response from the Earth-air interface can be modelled by solving the air field explicitly in the Chebyshev domain. I show that transverse isotropic anisotropy can be included in the modelling with the manipulation of the conductivity tensor. I show that by introducing another fictitious series of Chebyshev polynomials, the updating of Chebyshev terms is equivalent to coupled EM wave equations in a vacuum. EM wavefield modelling techniques can therefore be transferred to the Chebyshev domain, and I show the use of perfectly matched layers, a well-established absorbing boundary condition designed for EM waves, to solve the numerical boundary problems in the Chebyshev method. I have made two improvements to the numerical efficiency of REM modelling of CSEM data. First, I develop a workflow to solve the 3D electric field by REM but with a 2D model. If the earth model can be simplified to 2D structures, the computational cost to achieve a 3D solution can be reduced by an order of magnitude. Secondly, the code has been parallelized by graphic processing units (GPU), and the performance can be improved by a factor of over 100, compared with the serial REM code implemented in C. The developed new functionalities make the REM algorithm an accurate forward modeller that solves the time-domain electric field efficiently in various environments. Subsequent CSEM inversion studies can therefore benefit from the method to extract resistivity model from full-bandwidth CSEM field data, which should bring us closer to the true subsurface.