Transforming a time-domain electromagnetic signal to a frequency-domain electromagnetic response using regularization inversion

Abstract

Wide applications of time-domain electromagnetic (TEM) data require 3D inversion. A possible strategy is to use the developed 3D inversion algorithms in frequency-domain (FD) electromagnetic (EM) methods. Thus, the key of the strategy is how to transform the time-domain ([Formula: see text]-[Formula: see text]) EM signal into the FD. An inversion algorithm has been developed to transform the [Formula: see text]-[Formula: see text] signal into a corresponding FD response. In this method, a step-off current is presumed. Under this assumption, the Fourier transform relating the EM FD response to the [Formula: see text]-[Formula: see text] signal becomes a sine or cosine transformation. Using the polynomial approximation method, the transformation turns into a linear equation. From a set of [Formula: see text]-[Formula: see text] signals, FD responses could be obtained by solving these linear equations in the least-squares sense. To reduce the nonuniqueness of the solution, and enhance the solution stability, an additional smoothness constraint on the FD response is imposed, thus converting the minimization problem into a regularization inversion problem. The algorithm is applied to synthetic and field vertical magnetic data in the in-loop TEM surveying mode. The numerical results show that in the entire audio-frequency range, the relative errors between the inversed and theoretical FD responses of the real and imaginary parts are almost all less than 1%, with the largest discrepancy of 5% occurring at high frequencies. There are two significances behind our work: First, the possibility of accurately transforming [Formula: see text]-[Formula: see text] response into FD response in audio-frequency range is coming into true, thereby (from the mathematical perspective) implementing the equivalence between the responses of the EM method in the time domain and the FD. Second, the algorithm provides a new approach to interpret TEM data in 3D mode by using developed 3D FEM inversion techniques.