# Seismic AVOA Inversion for Weak Anisotropy Parameters and Fracture Density in a Monoclinic Medium
## 2 Methods
### PP-Wave Reflection Coefficient and Generalized Anisotropy Parameters  
In this study, the elastic contrast between the overburden and the reservoir was assumed to be small. In this situation, the plane-wave/P-wave reflection coefficient for a plane separating media, with arbitrary elastic symmetry and with WA, can be written in the form [32]  
\[ \begin{array}{l}R_{pp}(\theta,\phi)\ =\ R_{pp}^{iso}(\theta)+\frac{1}{2} \Delta\varepsilon_{z}+\frac{1}{2}\left[\begin{array}{c}\left(\Delta\delta_{x }-8\frac{\overline{V}_{z}^{2}}{\overline{V}_{p}^{2}}\Delta\gamma_{x}\right) \cos^{2}\phi+\\ \left(\Delta\delta_{y}-8\frac{\overline{V}_{z}^{2}}{\overline{V}_{p}^{2}} \Delta\gamma_{y}\right)\sin^{2}\phi+\\ 2\left(\Delta\Delta\varepsilon_{z}-4\frac{\overline{V}_{z}^{2}}{\overline{V}_ {p}}\Delta\varepsilon_{45}\right)\cos\phi\sin\phi-\Delta\varepsilon_{z} \end{array}\right]\sin^{2}\theta+\\ \frac{1}{2}\Big{[}\Delta\varepsilon_{x}\cos^{4}\phi+\Delta\varepsilon_{y}\sin ^{4}\phi+\Delta\delta_{z}\cos^{2}\phi\sin^{2}\phi+2\Big{(}\Delta\varepsilon_{ 16}\cos^{2}\phi+\Delta\varepsilon_{26}\sin^{2}\phi\Big{)}\cos\phi\sin\phi \Big{]}\sin^{2}\theta\tan^{2}\theta\end{array}  \tag{8}\]  
Figure 1: Two non-orthogonal vertical fracture sets with a vertical transverse isotropic (VTI) background. where \(R_{PP}^{iso}(\theta)\) denotes the weak-contrast reflection coefficient at an interface separating two slightly different isotropic media, and the generalized Thomsen anisotropy parameters are given by  
\[ \begin{array}{l}\delta_{x}\ =\ \frac{A_{13}+2A_{55}-V_{p}^{2}}{V_{p}^{2}}, \delta_{y}\ =\ \frac{A_{23}+2A_{44}-V_{p}^{2}}{V_{p}^{2}},\delta_{z}\ =\ \frac{A_{12}+2A_{66}-V_{p}^{2}}{V_{p}^{2}},\\ X_{z}\ =\ \frac{A_{36}+2A_{45}}{V_{p}^{2}},\varepsilon_{16}\ =\ \frac{A_{16}}{V_{p}^{2}}, \varepsilon_{26}\ =\ \frac{A_{36}}{V_{p}^{2}},\varepsilon_{45}\ =\ \frac{A_{45}}{V_{s}^{2}}, \varepsilon_{x}\ =\ \frac{A_{11}-V_{p}^{2}}{2V_{p}^{2}},\\ \varepsilon_{y}\ =\ \frac{A_{22}-V_{p}^{2}}{2V_{p}^{2}},\varepsilon_{z}\ =\ \frac{A_{33}-V_{p}^{2}}{2V_{p}^{2}}, \gamma_{x}\ =\ \frac{A_{55}-V_{s}^{2}}{2V_{s}^{2}},\gamma_{y}\ =\ \frac{A_{44}-V_{s}^{2}}{2V_{s}^{2}}\end{array}  \tag{9}\]  
where \(V_{p}\) and \(V_{s}\) are the P- and S-wave velocities of the background isotropic medium, respectively, and \(A_{ij}\ =\ \frac{C_{ij}}{\rho}\) is the density-normalized elastic stiffness. Incorporating Equations (5) and (7) and substituting the density-normalized elastic stiffness of Equation (9), we obtained a new expression of the generalized Thomsen anisotropy parameters, for the monoclinic medium, in terms of WA parameters and the compliance tensors. These parameters are presented in Appendix A. Under the assumption of small compliance tensors and WA parameters, we neglected the product terms of the compliance tensors and the weak anisotropic parameters. By substituting the generalized WA parameters in Equation (8) with Equation (A1), a P-wave reflection coefficient, in terms of WA parameters and the compliance tensors, was obtained:  
\[ \begin{array}{l}R_{pp}(\theta,\phi)\ =\ R_{PP}^{iso}(\theta)+R_{PP}^{adi}( \theta)+F_{1}(\theta)\delta^{psd}+F_{2}(\theta)\varepsilon+F_{3}(\theta,\phi )\alpha_{11}+F_{4}(\theta,\phi)\alpha_{12}+\\ F_{5}(\theta,\phi)\alpha_{22}+F_{6}(\theta,\phi)\beta_{1111}+F_{7}(\theta, \phi)\beta_{1112}+F_{8}(\theta,\phi)\beta_{1122}+F_{9}(\theta,\phi)\beta_{122 2}+F_{10}(\theta,\phi)\beta_{2222}\end{array}  \tag{10}\]  
where the pseudo weak anisotropic parameter \(\delta^{psd}\ =\ \sqrt{M-\mu+2M\delta}\). The second term on the righthand side is only a function of offset because the background is VTI, which does not depend on the azimuth. Equations for sensitivities, \(\mathbf{F}\), are given in Appendix B. In order to realize the direct inversion of fracture density, the linear P-wave reflection coefficient, in terms of weak anisotropic parameters and fracture density, could be obtained by combining Equations (3) and (4) and Equation (10).