Читать книгу Seismic Reservoir Modeling - Dario Grana - Страница 30

We illustrate the Bayesian approach for linear inverse problems in a geophysical application. We assume that the model variable of interest is S‐wave velocity V_S and that a measurement of P‐wave velocity V_P is available. The goal of this exercise is to predict the conditional probability of S‐wave velocity given P‐wave velocity.

We assume that S‐wave velocity is distributed according to a Gaussian distribution with prior mean μ_S = 2 km/s and prior standard deviation σ_S = 0.25 km/s (). We assume that the forward operator linking P‐wave and S‐wave velocity is a linear model of the form:

We then assume that the measurement error is Gaussian distributed with mean μ_ε = 0 and standard deviation σ_ε = 0.05 km/s ().

If the available measurement of P‐wave velocity is V_P = 3.5 km/s, then the posterior distribution of S‐wave velocity given the P‐wave velocity measurement is Gaussian distributed with mean μ_S∣P:

and standard deviation σ_S∣P:

If the available measurement of P‐wave velocity is V_P = 4.5 km/s, then the mean μ_S∣P of the posterior distribution is:

and the standard deviation is σ_S∣P = 0.025 km/s.

The posterior standard deviation does not depend on the measurement but only on the prior standard deviation of the model variable and the standard deviation of the error.