Prestack Bayesian Linearized Inversion with Decorrelated Prior Information

Abstract

The statistical correlation between the three elastic parameters of the P- and S-wave velocities and density is significant for stabilizing the prestack inversion. In the prestack Bayesian linearized inversion (BLI), the prior correlation of the three elastic parameters is included in a multivariate Gaussian distribution, and is represented by the cross-variograms between the three model parameters. However, the cross-variograms are roughly calibrated from certain sparse existing data (such as well-log data), which may produce statistical error and reduce inversion accuracy. To address this issue, this work proposes a decorrelated Bayesian linearized inversion (DBLI) by integrating the BLI with a decorrelation strategy. The decorrelation utilizes principal component analysis to obtain independent model parameters with zero covariances. Since the cross-variograms of the model parameters are no more than their covariances according to the derivation, the cross-variograms between the three independent model parameters are also zero. Thus, the estimation of the cross-variogram is unnecessary in DBLI, thereby avoiding the statistical error produced in the prior correlation characterization. The contribution of DBLI can be summarized by two main aspects. First, DBLI enables one to avoid the problem of reduced inversion stability and accuracy caused by statistical error, which is verified by tests on both the theoretical model and field data. Second, the derived relationship between the covariance and the cross-variogram is a potential contribution to both the geophysical inversion and geostatistical modeling.

Appendix 1

This appendix shows the derivation of the CCR. The CCR for the P- and S-wave velocities (${\mathbf{v}}_{p}$ and ${\mathbf{v}}_{s}$) is taken as an example.

1.1 Expressions of Covariance and Cross-Variogram Sill

The covariance $c_{ps}$ between ${\mathbf{v}}_{p}$ and ${\mathbf{v}}_{s}$ can be expressed as

$$ c_{ps} = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {{\mathbf{v}}_{p} \left( i \right) - E\left( {{\mathbf{v}}_{p} } \right)} \right]\left[ {{\mathbf{v}}_{s} \left( i \right) - E\left( {{\mathbf{v}}_{s} } \right)} \right]} {\kern 1pt} \; = \;E\left( {{\mathbf{v}}_{p}^{T} {\mathbf{v}}_{s} } \right) - E\left( {{\mathbf{v}}_{p} } \right)E\left( {{\mathbf{v}}_{s} } \right), $$

(A-1)

where E represents the mean of a vector, and $E\left( {{\mathbf{v}}_{p} } \right)$ and $E\left( {{\mathbf{v}}_{s} } \right)$ represent the means of ${\mathbf{v}}_{p}$ and ${\mathbf{v}}_{s}$, respectively.

The sill of the cross-variogram $\gamma_{ps}$ between ${\mathbf{v}}_{p}$ and ${\mathbf{v}}_{s}$, $f_{ps}$, can be expressed as

$$ f_{ps} = \frac{1}{{2\left( {N - h} \right)}}\sum\limits_{j = 1}^{N - h} {\left[ {{\mathbf{v}}_{{\mathbf{p}}} \left( j \right) - {\mathbf{v}}_{{\mathbf{p}}} \left( {j + h} \right)} \right]} \left[ {{\mathbf{v}}_{{\mathbf{s}}} \left( j \right) - {\mathbf{v}}_{{\mathbf{s}}} \left( {j + h} \right)} \right], $$

(A-2)

where h is the range of the $\gamma_{ps}$ (h is commonly fixed for one variogram or cross-variogram), and $n - h$ is the number of dot pairs with the spacing distance h in the N sampling points.

As for ${\mathbf{v}}_{p}$ and ${\mathbf{v}}_{s}$, the two assumptions can be denoted by

• (I) $N \gg h$, i.e., the range h can be ignored compared with the number of sampling points N.

• (II) $c_{ps} \ge 0$, i.e., the correlation coefficients between different model parameters are non-negative.

1.2 Simplification of Cross-Variogram Sill

$f_{ps}$ in Eq. (A-2) can be written as

$$ \begin{aligned} f_{ps} & { = }\frac{1}{{2\left( {N - h} \right)}}\sum\limits_{j = 1}^{N - h} {{\mathbf{v}}_{p} \left( j \right){\mathbf{v}}_{s} \left( j \right)} + \frac{1}{{2\left( {N - h} \right)}}\sum\limits_{j = 1}^{N - h} {{\mathbf{v}}_{p} \left( {j + h} \right){\mathbf{v}}_{s} \left( {j + h} \right)} \\ & \quad - \frac{1}{{2\left( {N - h} \right)}}\sum\limits_{j = 1}^{N - h} {{\mathbf{v}}_{p} \left( j \right){\mathbf{v}}_{s} \left( {j + h} \right)} - \frac{1}{{2\left( {N - h} \right)}}\sum\limits_{j = 1}^{N - h} {{\mathbf{v}}_{p} \left( {j + h} \right){\mathbf{v}}_{s} \left( j \right)} . \\ \end{aligned} $$

(A-3)

Then, according to assumption (I) that $n \gg h$, the first and second terms in Eq. (A-3) are transformed to

$$ {\kern 1pt} \frac{1}{{2\left( {N - h} \right)}}\sum\limits_{j = 1}^{N - h} {{\mathbf{v}}_{p} \left( j \right){\mathbf{v}}_{s} \left( j \right)} \approx \frac{1}{{2\left( {N - h} \right)}}\sum\limits_{j = 1}^{N - h} {{\mathbf{v}}_{p} \left( {j + h} \right){\mathbf{v}}_{s} \left( {j + h} \right)} \approx \frac{1}{2}E\left( {{\mathbf{v}}_{p}^{T} {\mathbf{v}}_{s} } \right). $$

(A-4)

In order to simplify the third and fourth terms of Eq. (A-3), four subsequences of ${\mathbf{v}}_{p}$ and ${\mathbf{v}}_{s}$ are defined as

$$ \left\{ {\begin{array}{*{20}l} {v_{{p1}} = v_{p} \left( 1 \right),{\kern 1pt} \;v_{p} \left( 2 \right), \ldots ,\;v_{p} \left( {N - h} \right),} \hfill \\ {v_{{p2}} = v_{p} \left( {h + 1} \right),{\kern 1pt} \;v_{p} \left( {h + 2} \right), \ldots ,\;{\kern 1pt} v_{p} \left( N \right),} \hfill \\ {v_{{s1}} = v_{s} \left( {h + 1} \right),\;{\kern 1pt} v_{s} \left( {h + 2} \right), \ldots ,\;v_{s} \left( N \right),} \hfill \\ {v_{{s2}} = v_{s} \left( 1 \right),{\kern 1pt} \;v_{s} \left( 2 \right), \ldots ,\;v_{s} \left( {N - h} \right).} \hfill \\ \end{array} } \right. $$

(A-5)

Then, the third and fourth terms of Eq. (A-3) are converted to

$$ \left\{ \begin{aligned} \frac{1}{{2\left( {N - h} \right)}}\sum\limits_{j = 1}^{N - h} {{\mathbf{v}}_{p} \left( j \right){\mathbf{v}}_{s} \left( {j + h} \right)} { = }\frac{1}{{2\left( {N - h} \right)}}\sum\limits_{i = 1}^{N - h} {{\mathbf{v}}_{p1} \left( i \right){\mathbf{v}}_{s1} \left( i \right)} \approx \frac{1}{2}E\left( {{\mathbf{v}}_{p1}^{T} {\mathbf{v}}_{s1} } \right), \hfill \\ \frac{1}{{2\left( {N - h} \right)}}\sum\limits_{j = 1}^{N - h} {{\mathbf{v}}_{p} \left( {j + h} \right){\mathbf{v}}_{s} \left( j \right){ = }\frac{1}{{2\left( {N - h} \right)}}\sum\limits_{i = 1}^{N - h} {{\mathbf{v}}_{p2} \left( i \right){\mathbf{v}}_{s2} \left( i \right) \approx } \frac{1}{2}E\left( {{\mathbf{v}}_{p2}^{T} {\mathbf{v}}_{s2} } \right)} , \hfill \\ \end{aligned} \right. $$

(A-6)

Thus, $f_{ps}$ is converted to

$$ f_{ps} = E\left( {{\mathbf{v}}_{p}^{T} {\mathbf{v}}_{s} } \right) - \frac{1}{2}E\left( {{\mathbf{v}}_{p1}^{T} {\mathbf{v}}_{s1} } \right) - \frac{1}{2}E\left( {{\mathbf{v}}_{p2}^{T} {\mathbf{v}}_{s2} } \right). $$

(A-7)

1.3 Difference between Covariance and Cross-Variogram Sill

Combining Eqs. (A-1) and (A-7), the difference between $c_{ps}$ and $f_{ps}$ is expressed as

$$ c_{ps} - f_{ps} = \frac{1}{2}E\left( {{\mathbf{v}}_{p1}^{T} {\mathbf{v}}_{s1} } \right) + \frac{1}{2}E\left( {{\mathbf{v}}_{p2}^{T} {\mathbf{v}}_{s2} } \right) - E\left( {{\mathbf{v}}_{p} } \right)E\left( {{\mathbf{v}}_{s} } \right). $$

(A-8)

Based on assumption (I), $E\left( {{\mathbf{v}}_{s1} } \right) \approx E\left( {{\mathbf{v}}_{s2} } \right) \approx E\left( {{\mathbf{v}}_{s} } \right)$. Then, it is approximately expressed as

$$ \begin{aligned} & c_{{ps}} - f_{{ps}} \approx \frac{1}{2}E\left( {{\mathbf{v}}_{{p1}}^{T} {\mathbf{v}}_{{s1}} } \right) + \frac{1}{2}E\left( {{\mathbf{v}}_{{p2}}^{T} {\mathbf{v}}_{{s2}} } \right) - \frac{1}{2}E\left( {{\mathbf{v}}_{{p1}} } \right)E\left( {{\mathbf{v}}_{{s1}} } \right) - \frac{1}{2}E\left( {{\mathbf{v}}_{{p2}} } \right)E\left( {{\mathbf{v}}_{{s2}} } \right), \\ & \quad = \frac{1}{2}\left( {c_{{ps1}} + c_{{ps2}} } \right), \\ \end{aligned} $$

(A-9)

where $c_{ps1}$ is the covariance between ${\mathbf{v}}_{p1}$ and ${\mathbf{v}}_{s1}$, and $c_{ps2}$ is the covariance between ${\mathbf{v}}_{p2}$ and ${\mathbf{v}}_{s2}$. The nodes in ${\mathbf{v}}_{p1}$ do not absolutely correspond to the nodes in ${\mathbf{v}}_{s1}$ due to the time shifting caused by the range h. Thus, $c_{ps1}$ and $c_{ps2}$ are approximately less than $c_{ps}$. However, according to the assumptions (I) and (II), the correlation between ${\mathbf{v}}_{p1}$ and ${\mathbf{v}}_{s1}$ should still be positive, and the correlation between ${\mathbf{v}}_{p2}$ and ${\mathbf{v}}_{s2}$ should also be positive, i.e.,

$$ \begin{aligned} c_{ps} & \ge c_{ps1} \ge 0, \\ c_{ps} & \ge c_{ps2} \ge 0. \\ \end{aligned} $$

(A-10)

It can then be observed from Eq. (A-9) that the relationship between $f_{ps}$ and $c_{ps}$ is

$$ c_{ps} \ge f_{ps} \ge 0. $$

(A-11)

Therefore, the CCR in Eq. (5) is derived.