Seismic Reservoir Modeling. Dario GranaЧитать онлайн книгу.

with associated matrix F; and (iii) the measurement errors ε are Gaussian bold-italic epsilon tilde script upper N left-parenthesis bold-italic epsilon semicolon bold 0 comma bold sigma-summation Underscript epsilon Endscripts right-parenthesis

bold-italic epsilon tilde script upper N left-parenthesis bold-italic epsilon semicolon bold 0 comma bold sigma-summation Underscript epsilon Endscripts right-parenthesis

, with 0 mean and covariance matrix ∑_ε, and they are independent of m; then, the posterior distribution m∣d is also Gaussian bold-italic m bar bold-italic d tilde script upper N left-parenthesis bold-italic m semicolon bold-italic mu Subscript m bar d Baseline comma bold sigma-summation Underscript m bar d Endscripts right-parenthesis

bold-italic m bar bold-italic d tilde script upper N left-parenthesis bold-italic m semicolon bold-italic mu Subscript m bar d Baseline comma bold sigma-summation Underscript m bar d Endscripts right-parenthesis

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 script upper N left-parenthesis upper V Subscript upper S Baseline semicolon mu Subscript upper S Baseline comma sigma Subscript upper S Superscript 2 Baseline right-parenthesis 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:

upper V Subscript upper P Baseline equals 2 upper V Subscript s Baseline plus epsilon period

We then assume that the measurement error is Gaussian distributed script upper N left-parenthesis epsilon semicolon mu Subscript epsilon Baseline comma sigma Subscript epsilon Superscript 2 Baseline right-parenthesis 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 script upper N left-parenthesis upper V Subscript upper S Baseline semicolon mu Subscript upper S bar upper P Baseline comma sigma Subscript upper S bar upper P Superscript 2 Baseline right-parenthesis with mean μ_S∣P:

mu Subscript upper S bar upper P Baseline equals 2 plus 0.495 times left-parenthesis 3.5 minus 2 times 2 right-parenthesis equals 1.75 k m slash normal s

and standard deviation σ_S∣P:

sigma Subscript upper S bar upper P Baseline equals StartRoot 0.0625 minus 0.495 times 2 times 0.0625 EndRoot equals 0.025 k m slash normal s period

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:

mu Subscript upper S bar upper P Baseline equals 2 plus 0.495 times left-parenthesis 4.5 minus 2 times 2 right-parenthesis equals 2.25 k m slash normal s

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.

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