Waveform inversion using a back-propagation algorithm and a Huber function norm

Abstract

Waveform inversion faces difficulties when applied to real seismic data, including the existence of many kinds of noise. The [Formula: see text]-norm is more robust to noise with outliers than the least-squares method. Nevertheless, the least-squares method is preferred as an objective function in many algorithms because the gradient of the [Formula: see text]-norm has a singularity when the residual becomes zero. We propose a complex-valued Huber function for frequency-domain waveform inversion that combines the [Formula: see text]-norm (for small residuals) with the [Formula: see text]-norm (for large residuals). We also derive a discretized formula for the gradient of the Huber function. Through numerical tests on simple synthetic models and Marmousi data, we find the Huber function is more robust to outliers and coherent noise. We apply our waveform-inversion algorithm to field data taken from the continental shelf under the East Sea in Korea. In this setting, we obtain a velocity model whose synthetic shot profiles are similar to the real seismic data.