Fourier coefficients-based stepwise Bayesian inversion for elastic and fracture parameters using azimuthal seismic data

The inversion of elastic and fracture parameters from azimuthal seismic data plays a critical role in charactering naturally fractured reservoirs. We propose a Fourier coefficients-based stepwise Bayesian inversion method to estimate these reservoir parameters with improved accuracy. Utilizing the relationship between the azimuthal PP-wave reflection coefficient and Fourier series expansion, we first derive the Fourier coefficient equation for horizontally transverse isotropic (HTI) media. We then conduct a sensitivity analysis of azimuthal Fourier coefficients with respect to P- and S-wave velocities, density, and fracture weaknesses. The results indicate that the zeroth-order Fourier coefficient exhibits greater sensitivity to P- and S-wave velocities and density compared to normal and tangential fracture weaknesses, whereas the second-order Fourier coefficient is more responsive to fracture weaknesses than the fourth-order Fourier coefficient. Based on these sensitivity observations, we develop a stepwise Bayesian inversion approach that involves (1) computing the cosine and sine components of the Fourier coefficients for the azimuthal seismic data, (2) estimating normal and tangential fracture weaknesses using use the second-order Fourier coefficient within a Bayesian framework, and (3) recovering background P- and S-wave velocities and density using the zeroth-order Fourier coefficient along with the previously estimated fracture weaknesses. Both synthetic and field data applications confirm the robustness and effectiveness of the proposed Fourier coefficients-based stepwise Bayesian inversion method for estimating elastic and fracture parameters in naturally fractured reservoirs.