Seismic Reservoir Modeling. Dario Grana

with associated matrix F; and (iii) the measurement errors ε are Gaussian , with 0 mean and covariance matrix ∑_ε, and they are independent of m; then, the posterior distribution m∣d is also Gaussian with conditional mean μ_m∣d:

      (1.53)

      and conditional covariance matrix ∑_m∣d:

      (1.54)

      For the proof, we refer the reader to Tarantola (2005). This result is extensively used in Chapter 5 for seismic inversion problems.

      Example 1.3

      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.