Anisotropic Anelastic Impedance Inversion for Attenuation Factor and Weaknesses Combining Newton and Bayesian Markov Chain Monte Carlo Algorithms

Abstract

The problem of frequency-dependent seismic inversion for estimating attenuation and anisotropy based on rock physics effective model is reviewed and analyzed. Based on an extended periodically layered medium model, frequency-dependent stiffness parameters of anisotropic anelastic media as a function of attenuation factor and weaknesses are presented. For the case of an interface separating two anisotropic anelastic media, a new parameterized P-to-P reflection coefficient and anisotropic anelastic impedance ${\mathcal {A}}_\mathrm{EI}$ are proposed. Based on the anisotropic ${\mathcal {A}}_\mathrm{EI}$, an inversion workflow of employing frequency components of seismic amplitudes to estimate attenuation factor and weaknesses for anisotropic anelastic media is presented. In the first stage inversion, the anisotropic ${\mathcal {A}}_\mathrm{EI}$ is calculated using frequency components of partially incidence-angle-stacked seismic data; and the second-stage inversion for unknown attenuation factor and weaknesses is implemented combining a Newton method and Bayesian Markov chain Monte Carlo algorithm by calculating first- and second-order derivatives of anisotropic ${\mathcal {A}}_\mathrm{EI}$ with respect to unknown parameters. Tests on synthetic seismic gathers confirm the accuracy of new parameterized reflection coefficient, and the unknown parameters are estimated reliably even in the case of seismic data of signal-to-noise ratio of 2 being employed. Applying the inversion approach and workflow to a field data set acquired over a shale reservoir, reliable attenuation factors and weaknesses that match well log curves are obtained, which are preserved as indicators for identifying potential hydrocarbon-bearing cracked reservoirs.

Article Highlights

• A new parameterized PP-wave reflection coefficient and anisotropic anelastic impedance are derived in terms of attenuation factor and weaknesses

• A two-stage inversion workflow of employing frequency components of seismic data to estimate attenuation factor and weaknesses combining Newton method and Bayesian Markov chain MonteCarlo algorithm

• Reliably estimated attenuation factor and weakness that match well log curves appear to be indicators or identification of potential cracked reservoirs

Appendices

Appendix A: Derivation and simplification of stiffness parameters and perturbations across the interface

In Eq. 2, we present stiffness parameters of the PLM, and under the assumptions that $\frac{h_\mathrm{c}}{H}{\rightarrow }0$ and $\frac{h_\mathrm{b}}{H}{\approx }1$, we simplify the stiffness parameters in terms of weaknesses. Taking $C_{11}$ as an example, we obtain

$$\begin{aligned} \begin{aligned} {{C}_{11}}&=\frac{{{H}^{2}}{{{\mathcal {M}}}_{\text {c}}}{{{\mathcal {M}}}_{\text {b}}}+4{{h}_{\text {c}}}{{h}_{\text {b}}}\left( {{{\mathcal {U}}}_{\text {c}}}-{{{\mathcal {U}}}_{\text {b}}} \right) \left( {{{\mathcal {M}}}_{\text {c}}}-{{{\mathcal {U}}}_{\text {c}}}-{{{\mathcal {M}}}_{\text {b}}}+{{{\mathcal {U}}}_{\text {b}}} \right) }{H\left( {{h}_{\text {c}}}{{{\mathcal {M}}}_{\text {b}}}+{{h}_{\text {b}}}{{{\mathcal {M}}}_{\text {c}}} \right) }\\&{\approx } \frac{1+\frac{4\left( {{{\mathcal {U}}}_{\text {c}}}-{{{\mathcal {U}}}_{\text {b}}} \right) \left( {{{\mathcal {M}}}_{\text {c}}}-{{{\mathcal {U}}}_{\text {c}}}-{{{\mathcal {M}}}_{\text {b}}}+{{{\mathcal {U}}}_{\text {b}}} \right) }{{{{\mathcal {M}}}_{\text {b}}}}{{Z}_{\text{N}}}}{{{Z}_{\text{N}}}+\frac{1}{{{{\mathcal {M}}}_{\text {b}}}}}, \end{aligned} \end{aligned}$$

(A.1)

where ${{Z}_{\text{N}}}{\approx }\lim \limits _{h_\mathrm{c}\rightarrow 0}\frac{{{h}_{\text {c}}}}{H{{{\mathcal {M}}}_{\text {c}}}}$. In the case of fluid filled compliant layers (fractures or cracks), we assume $\frac{{\mathcal {M}}_\mathrm{c}}{{\mathcal {M}}_\mathrm{b}}{\approx }0$ and $\frac{{\mathcal {U}}_\mathrm{c}}{{\mathcal {M}}_\mathrm{b}}{\approx }0$, and we further simplify $C_{11}$ as

$$\begin{aligned} \begin{aligned} {{C}_{11}}\approx \frac{1+4\left( -{{\kappa }_{\text {b}}} \right) \left( -{{{\mathcal {M}}}_{\text {b}}}+{{{\mathcal {U}}}_{\text {b}}} \right) {{Z}_{\text{N}}}}{{{Z}_{ \text{N}}}+\frac{1}{{{{\mathcal {M}}}_{\text {b}}}}}, \end{aligned} \end{aligned}$$

(A.2)

where ${\kappa }_\mathrm{b}={{\mathcal {U}}_\mathrm{b}}/{{\mathcal {M}}_\mathrm{b}}$.

Using relationship $\delta _{\text{N}}=\frac{{\mathcal {M}}_\mathrm{b}Z_{\text{N}}}{1+{\mathcal {M}}_\mathrm{b}Z_{\text{N}}}$, we rewrite equation A.2 as

$$\begin{aligned} \begin{aligned} {{C}_{11}}\approx {{{\mathcal {M}}}_{\text {b}}}\left[ 1-\left( 1-2{\kappa }_\mathrm{b}\right) ^{2}{\delta _{\text{N}}}\right] . \end{aligned} \end{aligned}$$

(A.3)

Other stiffness parameters are simplified as

$$\begin{aligned}&C_{12}{\approx }\left( {{{\mathcal {M}}}_\mathrm{b}}-2{{{\mathcal {U}}}_\mathrm{b}}\right) \left[ 1-\left( 1-2{{\kappa }_\mathrm{b}}\right) {{\delta }_{\text{N}}}\right] , \nonumber \\&\quad C_{13}{\approx }\left( {{{\mathcal {M}}}_\mathrm{b}}-2{{{\mathcal {U}}}_\mathrm{b}}\right) \left( 1-{{\delta }_{ \text{N}}}\right) , \nonumber \\&\quad C_{33}{\approx }{{{\mathcal {M}}}_\mathrm{b}}\left( 1-{{\delta }_{\text{N}}}\right) . \end{aligned}$$

(A.4)

Similar to the simplification of $C_{11}$, we express $C_{55}$ and $C_{66}$ as

$$\begin{aligned}&{{C}_{55}}=\frac{H{{{\mathcal {U}}}_{\text {c}}}{{{\mathcal {U}}}_{\text {b}}}}{{{h}_{\text {c}}}{{{\mathcal {U}}}_{\text {b}}}+{{h}_{\text {b}}}{{{\mathcal {U}}}_{\text {c}}}}=\frac{1}{\frac{{{h}_{\text {c}}}{{{\mathcal {U}}}_{\text {b}}}}{H{{{\mathcal {U}}}_{\text {c}}}{{{\mathcal {U}}}_{\text {b}}}}+\frac{{{h}_{\text {b}}}{{{\mathcal {U}}}_{\text {c}}}}{H{{{\mathcal {U}}}_{\text {c}}}{{{\mathcal {U}}}_{\text {b}}}}}{\approx } \frac{1}{{{Z}_\text {T}}+\frac{1}{{{{\mathcal {U}}}_{\text {b}}}}}={{{\mathcal {U}}}_{\text {b}}}\left( 1-{{\delta }_{T}} \right) , \nonumber \\&\quad {{C}_{66}}=\frac{{{h}_{\text {c}}}{{{\mathcal {U}}}_{\text {c}}}+{{h}_{\text {b}}}{{{\mathcal {U}}}_{\text {b}}}}{H}{\approx } \frac{{{h}_{\text {c}}}{{{\mathcal {U}}}_{\text {c}}}}{H}+{{{\mathcal {U}}}_{\text {b}}}{\approx } {{{\mathcal {U}}}_{\text {b}}}, \end{aligned}$$

(A.5)

under the assumption that $\frac{h_\mathrm{c}}{H}{\approx }0$.

In Eq. 5, we show the frequency-dependent stiffness parameters of the extended PLM model, and using the frequency-dependent stiffness parameters, we next show perturbations in stiffness parameters in the case of an interface separating two anisotropic anelastic media. Again taking $C_{11}$ as an example, we obtain the perturbation ${\varDelta }C_{11}$ as

$$\begin{aligned} \begin{aligned} \varDelta {{C}_{11}}(f) {\approx }&\left( {{M}_{0}}+{\varDelta } M \right) \left[ 1+2\eta\; \left( f \right) \left( {{{{\mathcal {Q}}}_{\text{P}0}}}+\varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) -{{\left( 1-2g \right) }^{2}}\left( {{\delta }_{\text{N}}}_{0}+\varDelta {{\delta }_{\text{N}}} \right) \right] \\&-{{M}_{0}}\left[ 1+2\eta\; \left( f \right) {{{{\mathcal {Q}}}_{\text{P}0}}}-{{\left( 1-2g \right) }^{2}}{{\delta }_{\text{N0}}} \right] \\ =&{{M}_{0}}\left[ 2\eta\; \left( f \right) \left( \varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) -{{\left( 1-2g \right) }^{2}}\left( \varDelta {{\delta }_{\text{N}}} \right) \right] \\&+\varDelta M\left[ 1+2\eta\; \left( f \right) \left( {{{{\mathcal {Q}}}_\text{P0}}}+\varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) -{{\left( 1-2g \right) }^{2}}\left( {{\delta }_{\text {N0}}}+\varDelta {{\delta }_{\text{N}}} \right) \right] , \end{aligned} \end{aligned}$$

(A.6)

where $M_{0}$, ${{{\mathcal {Q}}}_{\text{P}0}}$ and $\delta _{\text{N}0}$ represent P-wave modulus, attenuation factor and weakness of background, and ${\varDelta }M$, ${\varDelta }{{{\mathcal {Q}}}_{\text{P}}}$ and ${\varDelta }\delta _{\text{N}}$ represent changes in P-wave modulus, attenuation factor and weakness. Under the assumption of isotropic elastic background (${{\mathcal {Q}}}_\text{P0}=0$ and $\delta _\text{N0}=0$), we further derive $\varDelta {C}_{11}(f)$ as

$$\begin{aligned} \begin{aligned} \varDelta {{C}_{11}}(f) {\approx }&{{M}_{0}}\left[ 2\eta\; \left( f \right) \left( \varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) -{{\left( 1-2g \right) }^{2}}\left( \varDelta {{\delta }_{\text{N}}} \right) \right] \\&+\varDelta M\left[ 1+2\eta\; \left( f \right) \left( \varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) -{{\left( 1-2g \right) }^{2}}\left( \varDelta {{\delta }_{\text{N}}} \right) \right] , \end{aligned} \end{aligned}$$

(A.7)

and under the assumption that ${\varDelta }M$, ${\varDelta }{{{\mathcal {Q}}}_{\text{P}}}$ and ${\varDelta }{\delta _{\text{N}}}$ are not too large, we neglect the term proportional to ${\varDelta }M{\varDelta }{{{\mathcal {Q}}}_{\text{P}}}$ and ${\varDelta }M{\varDelta }{\delta _{\text{N}}}$ to further simplify $\varDelta {C}_{11}$ as

$$\begin{aligned} \begin{aligned} \varDelta {{C}_{11}}(f)\approx \varDelta M+{{M}}\left[ 2\eta\;\left( f \right) \left( \varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) -{{\left( 1-2g \right) }^{2}}\left( \varDelta {{\delta }_{\text{N}}} \right) \right] , \end{aligned} \end{aligned}$$

(A.8)

where we approximately replace $M_{0}$ with M. Perturbations in other stiffness parameters, $\varDelta {C}_{12}$, $\varDelta {C}_{13}$ and $\varDelta {C}_{33}$, are simplified as

$$\begin{aligned}&\varDelta {{C}_{12}} (f)\approx \varDelta M-2\varDelta \mu -\left( M-2\mu \right)\; \left( 1-2g \right)\; \varDelta {{\delta }_{\text{N}}} \nonumber \\&\quad +2M\eta\; \left( f \right) \left( \varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) -4\mu \eta \left( f \right) \varepsilon \left( \varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) , \nonumber \\&\quad \varDelta {{C}_{13}}(f)\approx \left( \varDelta M-2\varDelta \mu \right) -\left( M-2\mu \right)\; \varDelta {{\delta }_{\text{N}}} \nonumber \\&\quad +2M\eta \left(\; f \right) \left( \varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) -4\mu \eta\; \left( f \right) \;\varepsilon \;\left( \varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) , \nonumber \\&\quad \varDelta {{C}_{33}}(f)\approx \varDelta M+2M\eta\; \left( f \right) \varDelta {{{\mathcal {Q}}}_{\text{P}}}-M\varDelta {{\delta }_{\text{N}}}. \end{aligned}$$

(A.9)

The perturbation in stiffness parameter $C_{55}$ expressed as

$$\begin{aligned}&\varDelta {{C}_{55}}(f)\approx \left( {{\mu}_{0}}+\varDelta \mu \right) \left[ 1+2\eta \left( f \right)\varepsilon \left( {{{{\mathcal {Q}}}_{0}}}+\varDelta {{{\mathcal{Q}}}_{\text{P}}} \right) -\left( {{\delta }_{\text{T}}0}+\varDelta {{\delta }_{\text{T}}} \right) \right] \nonumber\\&\quad -{{\mu }_{0}}\left[ 1+\frac{2\eta \left( f \right)\varepsilon }{{{Q}_{0}}}-{{\delta }_{\text{T}0}} \right] \nonumber\\&\quad \approx {{\mu }_{0}}\left[ 2\eta \left( f \right)\varepsilon \left( \varDelta {{{\mathcal {Q}}}_{\text{P}}} \right)-\left( \varDelta {{\delta }_{\text{T}}} \right) \right] \nonumber\\&\quad +\varDelta \mu \left[ 1+2\eta \left( f \right)\varepsilon \left( {{{{\mathcal {Q}}}_{0}}}+\varDelta {{{\mathcal{Q}}}_{\text{P}}} \right) -\left( {{\delta}_{\text{T}0}}+\varDelta {{\delta }_{\text{T}}} \right) \right] ,\end{aligned}$$

(A.10)

and again under the assumption of isotropic elastic background and small perturbations in S-wave modulus, attenuation and anisotropy, we further simplify ${\varDelta }C_{55}$ as

$$\begin{aligned}&\varDelta {{C}_{55}}(f)\approx \left( {{\mu}_{0}}+\varDelta \mu \right) \left[ 1+2\eta \left( f \right) \varepsilon \left( {{{{\mathcal {Q}}}_{0}}}+\varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) -\left( {{\delta}_{\text {T}0}}+\varDelta {{\delta}_{\text{T}}} \right) \right] \nonumber \\&\quad -{{\mu}_{0}}\left[ 1+\frac{2\eta \left( f \right) \varepsilon}{{{Q}_{0}}}-{{\delta}_{\text{T}0}} \right] \nonumber \\&\quad \approx {{\mu}}\left[ 2\eta \left( f \right) \varepsilon \left( \varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) -\left( \varDelta {{\delta}_{\text{T}}} \right) \right] +\varDelta \mu , \end{aligned}$$

(A.11)

where we approximately replace $\mu _{0}$ with $\mu$.

The perturbation in stiffness parameter $C_{66}$ simplifies as

$$\begin{aligned} \begin{aligned} \varDelta {{C}_{66}}(f)\approx&2{\mu }\eta\; \left( f \right)\; \varepsilon\; \left( \varDelta {{{\mathcal {Q}}}_{\text{P}}} \right) +\varDelta \mu . \end{aligned} \end{aligned}$$

(A.12)

Appendix B: PP-wave reflection coefficient in anisotropic anelastic media

Shaw and Sen (2004, 2006) propose an approach of employing the scattering function that is related to perturbations in stiffness parameters to calculate reflection coefficients in anisotropic media. Following Shaw and Sen (2004, 2006), we use simplified perturbations in frequency-dependent stiffness parameters shown in Appendix A to derive P-to-P reflection coefficient in anisotropic anelastic media. The scattering function in anisotropic anelastic media is calculated as

$$\begin{aligned} \begin{aligned} S(f)=\varDelta \rho \cos 2\theta +\sum \limits _{i=1,j=1}^{i=6,j=6}{{\chi }_{ij}}~{\varDelta {{C}_{ij}}(f)}, \end{aligned} \end{aligned}$$

(A.13)

where $\varDelta \rho$ represents perturbation in density across the reflection interface, $\theta$ is the angle of P wave, and $\chi _{ij}$ is a function of $\theta$ and azimuth $\phi$, which is given by

$$\begin{aligned} \begin{aligned}&{{\chi }_{11}}=\frac{{{\sin }^{4}}\theta {{\cos }^{4}}\phi }{{{\alpha }^{2}}},{{\chi }_{12}}=\frac{{{\sin }^{4}}\theta {{\sin }^{2}}\phi {{\cos }^{2}}\phi }{{{\alpha }^{2}}},{{\chi }_{13}}=\frac{{{\sin }^{2}}\theta {{\cos }^{2}}\theta {{\cos }^{2}}\phi }{{{\alpha }^{2}}},\\&{{\chi }_{21}}={{\chi }_{12}},{{\chi }_{22}}=\frac{{{\sin }^{4}}\theta {{\sin }^{4}}\phi }{{{\alpha }^{2}}},{{\chi }_{23}}=\frac{{{\sin }^{2}}\theta {{\cos }^{2}}\theta {{\sin }^{2}}\phi }{{{\alpha }^{2}}},\\&{{\chi }_{31}}={{\chi }_{13}},{{\chi }_{32}}={{\chi }_{23}},{{\chi }_{33}}=\frac{{{\cos }^{4}}\theta }{{{\alpha }^{2}}},\\&{{\chi }_{44}}=\frac{-4{{\sin }^{2}}\theta {{\cos }^{2}}\theta {{\sin }^{2}}\phi }{{{\alpha }^{2}}},{{\chi }_{55}}=\frac{-4{{\sin }^{2}}\theta {{\cos }^{2}}\theta {{\cos }^{2}}\phi }{{{\alpha }^{2}}},{{\chi }_{66}}=\frac{4{{\sin }^{4}}\theta {{\sin }^{2}}\phi {{\cos }^{2}}\phi }{{{\alpha }^{2}}}, \end{aligned} \end{aligned}$$

(A.14)

where $\alpha$ is P-wave velocity of background. The P-to-P reflection coefficient is derived as

$$\begin{aligned} \begin{aligned} {{R}_\mathrm{PP}}(\theta ,f)=\frac{1}{4\rho {{\cos }^{2}}\theta }&\left. [ \varDelta \rho \cos 2\theta +\left( {{\chi }_{11}}+{{\chi }_{22}} \right) \varDelta {{C}_{11}}(f)+2{{\chi }_{12}}\varDelta {{C}_{12}}(f)\right. \\&\left. +2{{\chi }_{13}}\varDelta {{C}_{13}}(f)+2{{\chi }_{23}}\varDelta {{C}_{13}}(f)+{{\chi }_{33}}\varDelta {{C}_{33}}(f)\right. \\&\left. +\left( {{\chi }_{55}}+{{\chi }_{44}}(f) \right) \varDelta {{C}_{55}}(f)+{{\chi }_{66}}\varDelta {{C}_{66}}(f)\right. ]. \end{aligned} \end{aligned}$$

(A.15)

Substituting perturbations in stiffness parameters to equation A.15, we derive $R_\mathrm{PP}(\theta ,f)$ as

$$\begin{aligned} \begin{aligned} {{R}_\mathrm{PP}}(\theta ,f)=&\frac{\Delta \rho \cos 2\theta }{4\rho {{\cos }^{2}}\theta }\\&+\frac{1}{4M{{\cos }^{2}}\theta }\left. [ \Delta M-8{{\sin }^{2}}\theta {{\cos }^{2}}\theta \Delta \mu + 2M\eta \left( f \right) \Delta {{\mathcal {Q}}_{\text{P}}}\right. \\&\left. -8{{\sin }^{2}}\theta {{\cos }^{2}}\theta \mu \eta \left( f \right) \varepsilon \Delta {{\mathcal {Q}}_{\text{P}}} -M{{\left( 1-2g{{\sin }^{2}}\theta \right) }^{2}}\Delta {{\delta }_{\text{N}}} +4{{\sin }^{2}}\theta {{\cos }^{2}}\theta \mu \Delta {{\delta }_\mathrm{T}} \right. ]\\ =&\frac{\Delta \rho \cos 2\theta }{4\rho ~{{\cos }^{2}}\theta }+\frac{\Delta M}{4M{{\cos }^{2}}\theta }-2g~{{\sin }^{2}}\theta \frac{\Delta \mu }{\mu }\\&+\left( \frac{1}{2{{\cos }^{2}}\theta }-2g~{{\sin }^{2}}\theta ~\varepsilon \right) \eta \left( f \right) \Delta {{\mathcal {Q}}_{\text{P}}}\\&-\frac{1}{4}{{\sec }^{2}}\theta {{\left( 1-2g~{{\sin }^{2}}\theta \right) }^{2}}\Delta {{\delta }_{\text{N}}}+g{{\sin }^{2}}\theta \Delta {{\delta }_{\text{T}}}. \end{aligned} \end{aligned}$$

(A.16)

Appendix C: Algorithm of Bayesian Markov chain Monte Carlo (MCMC)

Following Chen et al. (2017), we express the posterior probability distribution function (PDF), $P\left( \mathbf {m}|\mathbf {d}\right)$, as

$$\begin{aligned} \begin{aligned} P\left( \mathbf {m}|\mathbf {d}\right) =P\left( \mathbf {d}|\mathbf {m}\right) P\left( \mathbf {m}\right) , \end{aligned} \end{aligned}$$

(A.17)

where $P\left( \mathbf {d}|\mathbf {m}\right)$ is the likelihood function, and $P\left( \mathbf {m}\right)$ is the prior probability function. In the present study, we assume both the likelihood function and the prior probability are consistent with Gaussian distribution; hence the posterior PDF is expressed as

$$\begin{aligned} \begin{aligned} P\left( \mathbf {m}|\mathbf {d}\right)\; {\propto }\;\exp&\left. \{ -\Sigma \frac{\left[ \mathbf{{d}}-\mathbf{{G}}\left( \mathbf{{m}}\right) \right] ^{T}\left[ \mathbf{{d}}-\mathbf{{G}}\left( \mathbf{{m}}\right) \right] }{2{\sigma ^{2}_\mathbf{\mathrm{e}}}}\right. \\&\left. -\Sigma \frac{\left( \mathbf{{M}}-{\psi _\mathbf{{M}}}\right) ^{T}\left( \mathbf{{M}}-{{\psi _\mathbf{{M}}}}\right) }{2{\sigma ^{2}_{\mathbf{{M}}}}} -\Sigma \frac{\left( {\varvec{\mu }}-{{\psi _{\varvec{\mu }}}}\right) ^{T}\left( {\varvec{\mu }}-{{\psi _{\varvec{\mu }}}}\right) }{2{\sigma ^{2}_{{\varvec{\mu }}}}}\right. \\&\left. -\Sigma \frac{\left( {\varvec{\rho }}-{\psi _{\varvec{\rho }}}\right) ^{T}\left( {\varvec{\rho }}-{{\psi _{\varvec{ \rho }}}}\right) }{2{\sigma ^{2}_{{{\varvec{\rho }}}}}} -\Sigma \frac{\left( \mathbf{{Q}_{\text{n}}}-{{\psi _{\mathbf{{Q}_\text{n}}}}}\right) ^{T}\left( \mathbf{{Q}_{\text{n}}}-{{\psi _{\mathbf{{Q}_\text{n}}}}}\right) }{2{\sigma ^{2}_{{{\mathbf{{Q}_{\text{n}}}}}}}}\right. \\&\left. -\Sigma \frac{\left( {\varvec{\delta }_\mathrm{N}}-{\psi _{{\varvec{\delta }_\mathrm{N}}}}\right) ^{T}\left( {\varvec{\delta }_\mathrm{N}}-{\psi _{{\varvec{\delta }_\mathrm{N}}}}\right) }{2{\sigma ^{2}_{{\varvec{\delta }_{\text{N}}}}}} -\Sigma \frac{\left( {\varvec{\delta }_\mathrm{T}}-{\psi _{{\varvec{\delta }_\mathrm{T}}}}\right) ^{T}\left( {\varvec{\delta }_\mathrm{T}}-{\psi _{{\varvec{\delta }_\mathrm{T}}}}\right) }{2{\sigma ^{2}_{{\varvec{\delta }_{\text{T}}}}}} \right. \}, \end{aligned} \end{aligned}$$

(A.18)

where $\psi _\mathbf{{M}}$, $\psi _{\varvec{\mu }}$, $\psi _{{\varvec{\rho }}}$, $\psi _\mathbf{{Q}_{\text{n}}}$, $\psi _{{{\varvec{\delta }}_{\text{N}}}}$, and $\psi _{{{\varvec{\delta }}_{\text{T}}}}$ represent average values of corresponding vectors respectively, $\sigma ^2_\mathbf{{M}}$, $\sigma ^2_{\varvec{\mu }}$, $\sigma ^2_{{\varvec{\rho }}}$, $\sigma ^2_\mathbf{{Q}_{\text{n}}}$, $\sigma ^2_{{{\varvec{\delta }}_{\text{N}}}}$, and $\sigma ^2_{{{\varvec{\delta }}_{\text{T}}}}$ represent variance values of corresponding vectors respectively, and $\sigma ^2_\mathbf{{e}}$ represents the variance of errors/noises between input and modeled data.

In the Bayesian MCMC algorithm, we determine whether the generated candidate $\mathbf{{m}}^{'}$ should be accepted or not based on the calculation of a acceptance probability

$$\begin{aligned} \begin{aligned} P_\mathrm{ac}\left( {\textbf {m}}^{'}\right) =\min \left[ 1,\frac{P\left( \mathbf{{m}}^{'}|{\textbf {d}}\right) }{P\left( {\textbf {m}}_{0}|{\textbf {d}}\right) }\right] , \end{aligned} \end{aligned}$$

(A.19)

where $P\left( \mathbf{{m}}^{'}|{\textbf {d}}\right)$ and $P\left( \mathbf{{m}}_{0}|\mathbf {d}\right)$ are posterior probabilities that are computed using equation A.18 for the generated candidate and the initial guess, respectively.

Given a generated candidate ${\textbf {m}}^{'}$, we compute the acceptance probability $P_\mathrm{ac}\left( {\textbf {m}}^{'}\right)$ using equation A.19 and compare $P_\mathrm{ac}(\mathbf{{m}}^{'})$ with a random value $\beta$ generated in the range of $\left[ 0,1\right]$. We only accept the candidate that makes $P_\mathrm{ac}\left( {\textbf {m}}^{'}\right)$ be larger than $\beta$.