Quantum mechanics

Tectonics

Gaussian

Inverse transform sampling

Statistics

Amplitude versus offset

Wave equation

Ultrasonic sensor

Acoustics

Surface wave

Seismology

Geology

Acoustic impedance

Physics

Amplitude

Inversion (geology)

Mathematics

Optics

Reflection coefficient

Mathematical analysis

Geometry

Cauchy distribution

Probability density function

Azimuth

Seismic inversion

Geophysics

The estimation of P- to S-wave velocity ratio (PSR) in the subsurface has many applications in gas-bearing reservoir prospecting, lithology discrimination, and anomalous pore-pressure prediction. Conventionally, it is estimated with the P- and S-wave velocities/moduli/impedances that are directly obtained from prestack seismic data using the existing reflection coefficient equation (e.g., the Aki-Richards approximation). However, this indirect inversion method creates cumulative errors for the estimated PSR results. To eliminate cumulative errors, we first develop a novel generalized elastic impedance that has more explicit physical meaning compared to conventional elastic impedance. Then, we derive a linear P-wave reflection coefficient equation in terms of the PSR, P-wave velocity, and density under the assumption of weak contrast. Furthermore, a robust amplitude-variation-with-offset inversion method is constructed with the proposed reflection coefficient equation in the Bayesian framework. Cauchy and Gaussian probability distributions are used as the prior probability distribution of the model parameter and likelihood function, respectively, to predict the maximum posterior probability solution for PSR, P velocity, and density. Synthetic and field examples illustrate that the proposed direct inversion method performs with higher accuracy compared with the indirect method that inverts for the P- and S-wave velocities and also illustrate the feasibility of our method for inverting the three parameters even with strong noise. Our inverted results conform to the drilling results, which validate the robustness of the proposed direct inversion method.

Amplitude-variation-with-offset inversion using P- to S-wave velocity ratio and P-wave velocity