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Seismic Reservoir Modeling. Dario Grana

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Seismic Reservoir Modeling - Dario Grana

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      In geophysical inverse problems, we often assume that the physical relation f in Eq. () is linear and that the prior distribution P(m) is Gaussian (Tarantola 2005). These two assumptions are not necessarily required to solve the Bayesian inverse problem, but under these assumptions, the inverse solution can be analytically derived. Indeed, in the Gaussian case, the solution to the Bayesian linear inverse problem is well‐known (Tarantola 2005). If we assume that: (i) the prior distribution of the model is Gaussian, i.e. bold-italic m tilde script upper N left-parenthesis bold-italic m semicolon bold-italic mu Subscript m Baseline comma bold sigma-summation Underscript m Endscripts right-parenthesis, where μm is the prior mean and ∑m is the prior covariance matrix; (ii) the forward operator f is linear 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, 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 with conditional mean μm∣d:

      (1.53)bold-italic mu Subscript m bar d Baseline equals bold-italic mu Subscript m Baseline plus bold sigma-summation Underscript m Endscripts bold upper F Superscript upper T Baseline left-parenthesis bold upper F bold sigma-summation Underscript m Endscripts bold upper F Superscript upper T Baseline plus bold sigma-summation Underscript epsilon Endscripts right-parenthesis Superscript negative 1 Baseline left-parenthesis bold-italic d minus bold upper F bold-italic mu Subscript m Baseline right-parenthesis

      and conditional covariance matrix ∑m∣d:

      (1.54)bold sigma-summation Underscript m bar d Endscripts equals bold sigma-summation Underscript m Endscripts minus bold sigma-summation Underscript m Endscripts bold upper F Superscript upper T Baseline left-parenthesis bold upper F bold sigma-summation Underscript m Endscripts bold upper F Superscript upper T Baseline plus bold sigma-summation Underscript epsilon Endscripts right-parenthesis Superscript negative 1 Baseline bold upper F bold sigma-summation Underscript m Endscripts period

      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 VS and that a measurement of P‐wave velocity VP 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 μS = 2 km/s and prior standard deviation σS = 0.25 km/s (sigma Subscript upper S Superscript 2 Baseline equals 0.0625). 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

      If the available measurement of P‐wave velocity is VP = 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 μS∣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 σS∣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 VP = 4.5 km/s, then the mean μS∣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 σS∣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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