Handling Negative Values for the Logarithmic Objective Function in Acoustic Laplace-Domain Full-Waveform Inversion Using Real Variables

The Laplace-domain waveform inversion is a full-waveform inversion method that recovers large-scale subsurface models. The inversion updates subsurface model parameters to minimize the differences between the modeled and the observed wavefields in the Laplace domain. The inversion results can be used as an accurate initial model for subsequent high-resolution waveform inversions. Pure Laplace-domain wavefields can be obtained by transforming the time-domain signals using the Laplace transform of real variables. The real Laplace transform is mathematically identical to the Fourier transform using the imaginary angular frequency; however, the Laplace transform using only real variables is computationally more efficient than that using complex variables. The Laplace-transformed wavefields are real-valued signals, and thus, it is natural to use real values in the Laplace-domain waveform inversions. However, the real logarithm function in the logarithmic objective function cannot handle negative values. Inversions using complex logarithms can solve this problem, but they demand more memory and computations than those required for inversions using real variables only. We suggest a simple method to overcome the negative-value problem for the real logarithm in the objective function. By taking the absolute values of the negative signals in the logarithmic objective function, we can obtain inversion results from inversions using real variables only that are equivalent to those from inversions using complex variables. We demonstrate the proposed method using the Society of Exploration Geophysicists (SEG)/European Associations of Geoscientists & Engineers (EAGE) salt model and a field data set. The inversions using real variables only took less than 22% of the time of the inversion using complex variables in the numerical examples.