Elastic Impedance Variation with Angle and Azimuth Inversion for Brittleness and Fracture Parameters in Anisotropic Elastic Media

Abstract

Young’s modulus, Poisson’s ratio, and fracture excess compliances, which are related to rock brittleness and natural fractures, can be used to evaluate the hydraulic fracturing and infer the optimized sweet spots in unconventional reservoirs. We aim to characterize the elastic properties of rock brittleness and compliance from the observable wide-azimuth seismic data via the inversion of Young’s modulus, Poisson’s ratio, and excess compliances. Using the linear slip model, we first derive the perturbations in stiffness components in terms of Young’s modulus, Poisson’s ratio, and excess compliances for the case of weak anisotropy and small contrasts in elastic properties across the interface. Based on the relationship between scattering function and reflection coefficient in weakly anisotropic media, we then derive a linearized PP-wave reflection coefficient and an azimuthal elastic impedance (EI) equation as a function of Young’s modulus, Poisson’s ratio, density, and excess compliances. Finally, we develop an EI variation with incident angle and azimuth inversion method to estimate the Young’s modulus, Poisson’s ratio, and excess compliances in a Bayesian framework. The approach is implemented in a two-step inversion: azimuthal EI inversion and estimation of model parameters. A synthetic test demonstrates that the model parameter can be reasonably estimated even containing moderate noise. A field data set test reveals that the inversion results agree well with the well log interpretation.

Appendices

Appendix 1: Linearized PP-Wave Reflection Coefficient and Azimuthal Elastic Impedance in an Orthorhombic Anisotropic Medium

A horizontally layered rock permeated by a single set of aligned, vertical fractures is equivalent to an orthorhombic anisotropic medium (Schoenberg and Helbig 1997; Bakulin et al. 2000b). Following Pan et al. (2018a), we derive the stiffness components of an orthorhombic anisotropic medium based on weak-anisotropy assumption in terms of Young’s modulus, Poisson’s ratio, and excess compliances

$$\begin{aligned} C_{11} & = \frac{{E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}} - \frac{{E^{2} \left( {1 - \sigma } \right)^{2} }}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}Z_{\text{N}} + \frac{{2E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\varepsilon_{b} , \\ C_{12} & = \frac{E\sigma }{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}} - \frac{{E^{2} \sigma \left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}Z_{\text{N}} + \frac{{2E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\varepsilon_{b} - \frac{2E}{{\left( {1 + \sigma } \right)}}\gamma_{b} , \\ C_{13} & = \frac{E\sigma }{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}} - \frac{{E^{2} \sigma \left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}Z_{\text{N}} + \frac{{E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\delta_{b} , \\ C_{22} & = \frac{{E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}} - \frac{{E^{2} \sigma^{2} }}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}Z_{\text{N}} + \frac{{2E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\varepsilon_{b} , \\ C_{23} & = \frac{E\sigma }{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}} - \frac{{E^{2} \sigma^{2} }}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}Z_{\text{N}} + \frac{{E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\delta_{b} , \\ C_{33} & = \frac{{E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}} - \frac{{E^{2} \sigma^{2} }}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}Z_{\text{N}} \\ C_{44} & = \frac{E}{{2\left( {1 + \sigma } \right)}},\text{ }\quad {\text{and}} \\ C_{55} & = \frac{E}{{2\left( {1 + \sigma } \right)}} - \frac{{E^{2} }}{{4\left( {1 + \sigma } \right)^{2} }}Z_{\text{V}} \text{,}\quad {\text{and}} \\ C_{66} & = \frac{E}{{2\left( {1 + \sigma } \right)}} - \frac{{E^{2} }}{{4\left( {1 + \sigma } \right)^{2} }}Z_{\text{H}} - \frac{E}{{\left( {1 + \sigma } \right)}}\gamma_{b} , \\ \end{aligned}$$

(16)

where $Z_{\text{N}}$, $Z_{\text{V}}$, and $Z_{\text{H}}$ denote the normal, vertical, and horizontal tangential excess compliances, respectively, and $\varepsilon_{b}$, $\delta_{b}$, and $\gamma_{b}$ denote the Thomsen’s (1986) weak-anisotropy (WA) parameters, which are used to characterize the vertical horizontally isotropic (VTI) medium.

Using the derived stiffnesses in Eq. (16), we ignore the items that proportional to $\Delta E\left( {Z_{\text{N}} + \Delta Z_{\text{N}} } \right)$, $\Delta \sigma \left( {Z_{\text{N}} + \Delta Z_{\text{N}} } \right)$, $\Delta E\left( {Z_{\text{V}} + \Delta Z_{\text{V}} } \right)$, $\Delta \sigma \left( {Z_{\text{V}} + \Delta Z_{\text{V}} } \right)$, $\Delta E\left( {Z_{\text{H}} + \Delta Z_{\text{H}} } \right)$, $\Delta \sigma \left( {Z_{\text{H}} + \Delta Z_{\text{H}} } \right)$, $\Delta E\left( {\varepsilon_{b} + \Delta \varepsilon_{b} } \right)$, $\Delta \sigma \left( {\varepsilon_{b} + \Delta \varepsilon_{b} } \right)$, $\Delta E\left( {\delta_{b} + \Delta \delta_{b} } \right)$, $\Delta \sigma \left( {\delta_{b} + \Delta \delta_{b} } \right)$, $\Delta E\left( {\gamma_{b} + \Delta \gamma_{b} } \right)$, $\Delta \sigma \left( {\gamma_{b} + \Delta \gamma_{b} } \right)$, $\left( {\Delta E} \right)^{2}$ or $\left( {\Delta \sigma } \right)^{2}$, we further derive the perturbations in stiffnesses in terms of Young’s modulus, Poisson’s ratio, and excess compliances for the case of weak anisotropy, weak excess compliances, and small contrasts in elastic properties across the interface

$$\begin{aligned} \Delta C_{11} & = \frac{{\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\Delta E + \frac{{2E\sigma \left( {2 - \sigma } \right)}}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}\Delta \sigma - \frac{{E^{2} \left( {1 - \sigma } \right)^{2} }}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}\Delta Z_{\text{N}} + \frac{{2E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\Delta \varepsilon_{b} , \\ \Delta C_{12} & = \frac{\sigma }{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\Delta E + \frac{{E\left( {2\sigma^{2} + 1} \right)}}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}\Delta \sigma - \frac{{E^{2} \sigma \left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}\Delta Z_{\text{N}} + \frac{{2E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\Delta \varepsilon_{b} - \frac{2E}{{\left( {1 + \sigma } \right)}}\Delta \gamma_{b} , \\ \Delta C_{13} & = \frac{\sigma }{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\Delta E + \frac{{E\left( {2\sigma^{2} + 1} \right)}}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}\Delta \sigma - \frac{{E^{2} \sigma \left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}\Delta Z_{\text{N}} + \frac{{E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\Delta \delta_{b} , \\ \Delta C_{22} & = \frac{{\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\Delta E + \frac{{2E\sigma \left( {2 - \sigma } \right)}}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}\Delta \sigma - \frac{{E^{2} \sigma^{2} }}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}\Delta Z_{\text{N}} + \frac{{2E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\Delta \varepsilon_{b} , \\ \Delta C_{23} & = \frac{\sigma }{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\Delta E + \frac{{E\left( {2\sigma^{2} + 1} \right)}}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}\Delta \sigma - \frac{{E^{2} \sigma^{2} }}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}\Delta Z_{\text{N}} + \frac{{E\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\Delta \delta_{b} , \\ \Delta C_{33} & = \frac{{\left( {1 - \sigma } \right)}}{{\left( {1 + \sigma } \right)\left( {1 - 2\sigma } \right)}}\Delta E + \frac{{2E\sigma \left( {2 - \sigma } \right)}}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}\Delta \sigma - \frac{{E^{2} \sigma^{2} }}{{\left( {1 + \sigma } \right)^{2} \left( {1 - 2\sigma } \right)^{2} }}\Delta Z_{\text{N}} \\ \Delta C_{44} & = \frac{1}{{2\left( {1 + \sigma } \right)}}\Delta E - \frac{E}{{2\left( {1 + \sigma } \right)^{2} }}\Delta \sigma ,\text{ } \\ \Delta C_{55} & = \frac{1}{{2\left( {1 + \sigma } \right)}}\Delta E - \frac{E}{{2\left( {1 + \sigma } \right)^{2} }}\Delta \sigma - \frac{{E^{2} }}{{4\left( {1 + \sigma } \right)^{2} }}\Delta Z_{\text{V}} \quad {\text{and}} \\ \Delta C_{66} & = \frac{1}{{2\left( {1 + \sigma } \right)}}\Delta E - \frac{E}{{2\left( {1 + \sigma } \right)^{2} }}\Delta \sigma - \frac{{E^{2} }}{{4\left( {1 + \sigma } \right)^{2} }}\Delta Z_{\text{H}} - \frac{E}{{\left( {1 + \sigma } \right)}}\Delta \gamma_{b} , \\ \end{aligned}$$

(17)

where $\Delta E$, $\Delta \sigma$, $\Delta \varepsilon_{b}$, $\Delta \delta_{b}$, $\Delta \gamma_{b}$, $\Delta Z_{\text{N}}$, $\Delta Z_{\text{V}}$, and $\Delta Z_{\text{H}}$ denote the contrasts in Young’s modulus, Poisson’s ratio, Thomsen’s WA parameters, and excess compliances across the interface.

Similarly, we then derive the linearized PP-wave reflection coefficient of an orthorhombic anisotropic medium

$$\begin{aligned} R_{\text{PP}}^{\text{ORT}} (\theta ,\phi ) & = \left( {\frac{{\sec^{2} \theta }}{4} - 2g\sin^{2} \theta } \right)\frac{\Delta E}{{\bar{E}}} + \left( {\frac{{\sec^{2} \theta }}{4}\frac{{\left( {2g - 3} \right)\left( {2g - 1} \right)^{2} }}{{g\left( {4g - 3} \right)}} + 2g\sin^{2} \theta \frac{1 - 2g}{3 - 4g}} \right)\frac{\Delta \sigma }{{\bar{\sigma }}} \\ & \quad + \,\left( {\frac{1}{2} - \frac{{\sec^{2} \theta }}{4}} \right)\frac{\Delta \rho }{{\bar{\rho }}} + \frac{{\sin^{2} \theta \tan^{2} \theta }}{2}\Delta \varepsilon_{b} + \frac{{\sin^{2} \theta }}{2}\Delta \delta_{b} \\ & \quad - \,\left( {\frac{{\bar{E}\left( {1 - \bar{\sigma }} \right)}}{{\left( {1 + \bar{\sigma }} \right)\left( {1 - 2\bar{\sigma }} \right)}}\frac{{\sec^{2} \theta }}{4}\left[ {2g\left( {\sin^{2} \theta \sin^{2} \phi + \cos^{2} \theta } \right) - 1} \right]^{2} } \right)\Delta Z_{\text{N}} \\ & \quad + \,\left( {\frac{{\bar{E}g}}{{2\left( {1 + \bar{\sigma }} \right)}}\sin^{2} \theta \cos^{2} \phi } \right)\Delta Z_{\text{V}} - \left( {\frac{{\bar{E}g}}{{2\left( {1 + \bar{\sigma }} \right)}}\sin^{2} \theta \tan^{2} \theta \sin^{2} \phi \cos^{2} \phi } \right)\Delta Z_{\text{H}} , \\ \end{aligned}$$

(18)

where $\bar{E}$, $\bar{\sigma }$, and $\bar{\rho }$ represent the averages over the interface, and $g = {{\left( {1 - 2\bar{\sigma }} \right)} \mathord{\left/ {\vphantom {{\left( {1 - 2\bar{\sigma }} \right)} {\left( {2 - 2\bar{\sigma }} \right)}}} \right. \kern-0pt} {\left( {2 - 2\bar{\sigma }} \right)}}$.

In a similar way, we also derive the azimuthal EI equation in an orthorhombic anisotropic medium

$$\begin{aligned} {\text{EI}}_{\text{PP}}^{\text{ORT}} \left( {\theta ,\phi } \right) & = \left[ E \right]^{{\frac{{\sec^{2} \theta }}{2} - 4g\sin^{2} \theta }} \left[ \sigma \right]^{{\frac{{\sec^{2} \theta }}{2}\frac{{\left( {2g - 3} \right)\left( {2g - 1} \right)^{2} }}{{g\left( {4g - 3} \right)}} + 4g\sin^{2} \theta \frac{1 - 2g}{3 - 4g}}} \left[ \rho \right]^{{1 - \frac{{\sec^{2} \theta }}{2}}} \\ & \quad \cdot \exp \left[ \begin{array}{l} \left[ {\sin^{2} \theta \tan^{2} \theta } \right]\varepsilon_{b} + \left[ {\sin^{2} \theta } \right]\delta_{b} \hfill \\ - \left\{ {\frac{{\bar{E}\left( {1 - \bar{\sigma }} \right)}}{{\left( {1 + \bar{\sigma }} \right)\left( {1 - 2\bar{\sigma }} \right)}}\frac{{\sec^{2} \theta }}{2}\left[ {2g\left( {\sin^{2} \theta \sin^{2} \phi + \cos^{2} \theta } \right) - 1} \right]^{2} } \right\}Z_{\text{N}} \hfill \\ + \left[ {\frac{{\bar{E}g}}{{\left( {1 + \bar{\sigma }} \right)}}\sin^{2} \theta \cos^{2} \phi } \right]Z_{\text{V}} - \left[ {\frac{{\bar{E}g}}{{\left( {1 + \bar{\sigma }} \right)}}\sin^{2} \theta \tan^{2} \theta \sin^{2} \phi \cos^{2} \phi } \right]Z_{\text{H}} \hfill \\ \end{array} \right]. \\ \end{aligned}$$

(19)

Appendix 2: Decorrelation of Model Parameters

To decorrelate the model parameters, we first calculate the covariance matrix ${\varvec{C}}_{{\varvec{m}}}$ of model parameters

$${\varvec{C}}_{{\varvec{m}}} = \left[ {\begin{array}{*{20}c} {\sigma_{{L_{E} }}^{2} } & {\sigma_{{L_{E} L_{\sigma } }} } & {\sigma_{{L_{E} L_{\rho } }} } & {\sigma_{{L_{E} Z_{N} }} } & {\sigma_{{L_{E} Z_{T} }} } \\ {\sigma_{{L_{E} L_{\sigma } }} } & {\sigma_{{L_{\sigma } }}^{2} } & {\sigma_{{L_{\sigma } L_{\rho } }} } & {\sigma_{{L_{\sigma } Z_{N} }} } & {\sigma_{{L_{\sigma } Z_{T} }} } \\ {\sigma_{{L_{E} L_{\rho } }} } & {\sigma_{{L_{\sigma } L_{\rho } }} } & {\sigma_{{L_{\rho } }}^{2} } & {\sigma_{{L_{\rho } Z_{N} }} } & {\sigma_{{L_{\rho } Z_{T} }} } \\ {\sigma_{{L_{E} Z_{N} }} } & {\sigma_{{L_{\sigma } Z_{N} }} } & {\sigma_{{L_{\rho } Z_{N} }} } & {\sigma_{{Z_{N} }}^{2} } & {\sigma_{{Z_{N} Z_{T} }} } \\ {\sigma_{{L_{E} Z_{T} }} } & {\sigma_{{L_{\sigma } Z_{T} }} } & {\sigma_{{L_{\rho } Z_{T} }} } & {\sigma_{{Z_{N} Z_{T} }} } & {\sigma_{{Z_{T} }}^{2} } \\ \end{array} } \right],$$

(20)

where the diagonal elements denote the variances $\sigma_{{L_{E} }}^{2}$, $\sigma_{{L_{\sigma } }}^{2}$, $\sigma_{{L_{\rho } }}^{2}$, $\sigma_{{Z_{N} }}^{2}$, and $\sigma_{{Z_{T} }}^{2}$ of model parameters, and the off-diagonal elements or covariances characterize the correlation of model parameters. Using the singular value decomposition (SVD) method, the parameter covariance matrix ${\varvec{C}}_{{\varvec{m}}}$ is decomposed as (Downton 2005)

$${\varvec{C}}_{{\varvec{m}}} = \varvec{u\sum u}^{\text{T}} = \varvec{u}\left[ {\begin{array}{*{20}c} {\sigma_{1}^{2} } & 0 & 0 & 0 & 0 \\ 0 & {\sigma_{2}^{2} } & 0 & 0 & 0 \\ 0 & 0 & {\sigma_{3}^{2} } & 0 & 0 \\ 0 & 0 & 0 & {\sigma_{4}^{2} } & 0 \\ 0 & 0 & 0 & 0 & {\sigma_{5}^{2} } \\ \end{array} } \right]\varvec{u}^{\text{T}} ,$$

(21)

where $\varvec{u}$ represents the eigenvector, and $\varvec{\sum }$ represents the diagonal matrix of eigenvalues, in which all the values are real and positive (and can be presented as real numbers squared $\sigma_{i}^{2} ,\text{ }i = 1,2, \ldots ,5$).

We define the inverse of single-interface eigenvector $\varvec{u}$ as

$$\varvec{u}^{ - 1} = \left[ {\begin{array}{*{20}c} {u_{11} } & {u_{12} } & {u_{13} } & {u_{14} } & {u_{15} } \\ {u_{21} } & {u_{22} } & {u_{23} } & {u_{24} } & {u_{25} } \\ {u_{31} } & {u_{32} } & {u_{33} } & {u_{34} } & {u_{35} } \\ {u_{41} } & {u_{42} } & {u_{43} } & {u_{44} } & {u_{45} } \\ {u_{51} } & {u_{52} } & {u_{53} } & {u_{54} } & {u_{55} } \\ \end{array} } \right].$$

(22)

For the case of $J$ interfaces, the single-interface eigenvector $\varvec{u}$ can be extended as

$$\varvec{U}^{ - 1} = \left[ {\begin{array}{*{20}c} {u_{11} } & 0 & \cdots & {u_{12} } & 0 & \cdots & {u_{13} } & 0 & \cdots & {u_{14} } & 0 & \cdots & {u_{15} } & 0 & \cdots \\ 0 & {u_{11} } & 0 & 0 & {u_{12} } & 0 & 0 & {u_{13} } & 0 & 0 & {u_{14} } & 0 & 0 & {u_{15} } & 0 \\ {} & {} & {} & {} & {} & {} & {} & \cdots & {} & {} & {} & {} & {} & {} & {} \\ {u_{21} } & 0 & \cdots & {u_{22} } & 0 & \cdots & {u_{23} } & 0 & \cdots & {u_{24} } & 0 & \cdots & {u_{25} } & 0 & \cdots \\ 0 & {u_{21} } & 0 & 0 & {u_{22} } & 0 & 0 & {u_{23} } & 0 & 0 & {u_{24} } & 0 & 0 & {u_{25} } & 0 \\ {} & {} & {} & {} & {} & {} & {} & \cdots & {} & {} & {} & {} & {} & {} & {} \\ {u_{31} } & 0 & \cdots & {u_{32} } & 0 & \cdots & {u_{33} } & 0 & \cdots & {u_{34} } & 0 & \cdots & {u_{35} } & 0 & \cdots \\ 0 & {u_{31} } & 0 & 0 & {u_{32} } & 0 & 0 & {u_{33} } & 0 & 0 & {u_{34} } & 0 & 0 & {u_{35} } & 0 \\ {} & {} & {} & {} & {} & {} & {} & \cdots & {} & {} & {} & {} & {} & {} & {} \\ {u_{41} } & 0 & \cdots & {u_{42} } & 0 & \cdots & {u_{43} } & 0 & \cdots & {u_{44} } & 0 & \cdots & {u_{45} } & 0 & \cdots \\ 0 & {u_{41} } & 0 & 0 & {u_{42} } & 0 & 0 & {u_{43} } & 0 & 0 & {u_{44} } & 0 & 0 & {u_{45} } & 0 \\ {} & {} & {} & {} & {} & {} & {} & \cdots & {} & {} & {} & {} & {} & {} & {} \\ {u_{51} } & 0 & \cdots & {u_{52} } & 0 & \cdots & {u_{53} } & 0 & \cdots & {u_{54} } & 0 & \cdots & {u_{55} } & 0 & \cdots \\ 0 & {u_{51} } & 0 & 0 & {u_{52} } & 0 & 0 & {u_{53} } & 0 & 0 & {u_{54} } & 0 & 0 & {u_{55} } & 0 \\ {} & {} & {} & {} & {} & {} & {} & \cdots & {} & {} & {} & {} & {} & {} & {} \\ \end{array} } \right]_{5J \times 5J} .$$

(23)

Here $\varvec{U}^{ - 1}$ is the inverse of decorrelation matrix $\varvec{U}$ of multiple interfaces. Using the transformation of the coefficient matrix ${\varvec{G}}$ and model parameter vector ${\varvec{m}}$

$$\left\{ {\begin{array}{*{20}l} {{\varvec{G}}^{{\prime }} = {\varvec{G}}\varvec{U}} \hfill \\ {{\varvec{m}}^{{\prime }} = \varvec{U}^{ - 1} {\varvec{m}}} \hfill \\ \end{array} } \right.,$$

(24)

and Eq. (12) yields

$$\varvec{d} = {\varvec{G}}^{{\prime }} {\varvec{m}}^{{\prime }} .$$

(25)

The covariance matrix ${\varvec{C}}_{{{\varvec{m^{\prime}}}}}$ after the transformation then becomes

$${\varvec{C}}_{{{\varvec{m^{\prime}}}}} = \left[ {\begin{array}{*{20}c} {\sigma_{1}^{2} } & 0 & 0 & 0 & 0 \\ 0 & {\sigma_{2}^{2} } & 0 & 0 & 0 \\ 0 & 0 & {\sigma_{3}^{2} } & 0 & 0 \\ 0 & 0 & 0 & {\sigma_{4}^{2} } & 0 \\ 0 & 0 & 0 & 0 & {\sigma_{5}^{2} } \\ \end{array} } \right],$$

(26)

where all the off-diagonal elements become to be zero, which indicates that the model parameters after decorrelation are mutually independent.

Appendix 3: Calculation of Excess Compliances Using Fracture Density

According to the relationship between the excess compliances and fracture density (Bakulin et al. 2000a), we derive the gas-filled, or dry, excess compliances expressed by Young’s modulus $E$, Poisson’s ratio $\sigma$, and fracture density $e$

$$Z_{\text{N}} = \frac{{16e\left( {1 - \sigma^{2} } \right)}}{3E},$$

(27)

and

$$Z_{\text{T}} = \frac{{32e\left( {1 - \sigma^{2} } \right)}}{{3E\left( {2 - \sigma } \right)}}.$$

(28)