Construction of pseudo-Hessian for vertical seismic profile full waveform inversion

Abstract

The pseudo-Hessian matrix has been widely developed and applied to full waveform inversion to reduce the computational cost of obtaining the Hessian and its inverse matrix. Because the conventional pseudo-Hessian matrix assumes a surface seismic profiling geometry, this conventional method is compatible only with horizontally deployed sources and receiver geometries and cannot reflect the geometrical spreading effect of vertically deployed receivers. To enable the pseudo-Hessian to properly consider vertically deployed receiver locations and the geometrical spreading inherent in vertical seismic profiling data, we re-formulate part of the pseudo-Hessian matrix formulation. We verify our new scheme by directly calculating the Hessian matrices for a simple 1D model and conducting inversion tests with synthetic data sets. A comparison to the approximated Hessian matrix confirms that the vertical seismic profiling pseudo-Hessian provides better agreement with the approximated Hessian than the conventional pseudo-Hessian. In addition, we describe two numerical tests using multi-offset synthetic vertical seismic profiling data sets. When we apply the conventional pseudo-Hessian matrix to invert the synthetic data, we find that the inversion process cannot properly invert far offset data. However, we find that the pseudo-Hessian regarding vertical seismic profiling geometry allows the inversion process to invert far offset data and produces fewer artifacts than the conventional method. This study demonstrates that the vertical seismic profiling pseudo-Hessian matrix represents an alternative method that is necessary for the calculation of the pseudo-Hessian matrix for full waveform inversion with vertical seismic profiling using the Gauss–Newton method.