# Seismic AVOA Inversion for Weak Anisotropy Parameters and Fracture Density in a Monoclinic Medium
## 2 Methods  
### Elastic Compliance Tensor of Fractured Medium  
In an elastic medium that contains an arbitrary number of sets of fractures, with an arbitrary orientation distribution, by using the divergence theorem and Hooke's law, it can be shown that the elastic compliance tensor of the fractured medium can be written in the following form [29; 30]:  
\[S_{ijkl}\ =\ S_{ijkl}^{0}+\Delta S_{ijkl} \tag{1}\]  
where \(S^{0}\) is the compliance matrix of the medium (including the effects of pores, cracks, and stress, except for those fractures explicitly included in \(\Delta S\)). The subscripts \(i\), \(j\), \(k\), and \(l\) are the free index. The effective excess compliance \(\Delta S_{ijkl}\), due to the presence of the fractures, with rotationally invariant shear compliance, can be written as [27]:  
\[\Delta S_{ijkl}\ =\ \frac{1}{4}\Big{(}\delta_{ik}\alpha_{jl}+\delta_{il}\alpha_{ jk}+\delta_{jk}\alpha_{il}+\delta_{jl}\alpha_{ik}\Big{)}+\beta_{ijkl}. \tag{2}\]  
Here, \(\delta_{ij}\) is the Kronecker delta, \(\alpha_{ij}\) is a second-rank tensor, and \(\beta_{ijkl}\) is a fourth-rank tensor. If the two fracture sets are not orthogonal, but are perfectly aligned (within each set), the compliance tensors become [6]:  
\[\alpha\ =\ N_{V1}\overline{A}_{1}\overline{B}_{\text{T1}}\left[ \begin{array}{ ccc}\sin^{4}\varphi_{1}&\sin^{2}\varphi_{1}\cos^{2}\varphi_{1}&0&0&2\sin^{3} \varphi_{1}\cos\varphi_{1}\\ \sin^{2}\varphi_{1}\cos^{2}\varphi_{1}&\cos^{4}\varphi_{1}&0&0&2\sin\varphi _{1}\cos^{3}\varphi_{1}\\ 0&0&0&0&0\\ \end{array} \right]+ \tag{3}\]  
\[\beta\ =\ N_{V1}\overline{A}_{1}\Big{(}\overline{B}_{\text{N1}}-\overline{B}_{T 1}\Big{)}\left[ \begin{array}{cccccc}\sin^{4}\varphi_{1}&\sin^{2}\varphi_{1} \cos^{2}\varphi_{1}&0&0&0&2\sin^{3}\varphi_{1}\cos\varphi_{1}\\ \sin^{2}\varphi_{1}\cos^{2}\varphi_{1}&\cos^{4}\varphi_{1}&0&0&0&2\sin\varphi _{1}\cos^{3}\varphi_{1}\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 2\sin^{3}\varphi_{1}\cos\varphi_{1}&2\sin\varphi_{1}\cos^{3}\varphi_{1}&0&0&0& 4\sin^{2}\varphi_{1}\cos^{2}\varphi_{1}\\ \end{array} \right]+ \tag{4}\]  
\[N_{V2}\overline{A}_{2}\Big{(}\overline{B}_{N2}-\overline{B}_{T2}\Big{)} \left[ \begin{array}{cccccc}\sin^{4}\varphi_{2}&\sin^{2}\varphi_{2}\cos^{2} \varphi_{2}&0&0&0&2\sin^{3}\varphi_{2}\cos\varphi_{2}\\ 0&\cos^{4}\varphi_{2}&0&0&0&2\sin\varphi_{2}\cos^{3}\varphi_{2}\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 2\sin^{3}\varphi_{2}\cos\varphi_{2}&2\sin\varphi_{2}\cos^{3}\varphi_{2}&0&0&0& 4\sin^{2}\varphi_{2}\cos^{2}\varphi_{2}\\ \end{array} \right]\]  
where \(N_{V}\) is the number of fractures per unit volume (fracture density), \(\overline{A}\) is the average area of the fractures in the set, \(\overline{B}_{T}\) and \(\overline{B}_{N}\) are the (area-weighted) average specific tangential and normal compliances, and \(\varphi\) is the azimuthal angle between the fracture strike and the survey 1-axis. The subscripts 1 and 2 denote the 1 st and 2 nd fracture set, respectively. If the background medium is isotropic, or transversely isotropic, and there are at least two non-orthogonal vertical fracture sets, this leads to monoclinic symmetry of the fractured rock, as shown in Figure 1.  
The additional compliance, due to two or more sets of aligned vertical fractures not aligned with the coordinate system, has the form [6]:  
\[\Delta S\ =\ \left[ \begin{array}{cccccc}\alpha_{11}+\beta_{1111}&\beta_{1122}&0&0& 0&\alpha_{12}+2\beta_{1112}\\ \beta_{1122}&\alpha_{22}+\beta_{2222}&0&0&0&\alpha_{12}+2\beta_{1222}\\ 0&0&0&0&0&0\\ 0&0&0&\alpha_{22}&\alpha_{12}&0\\ 0&0&0&\alpha_{12}&\alpha_{11}&0\\ \alpha_{12}+2\beta_{1112}&\alpha_{12}+\beta_{1222}&0&0&0&\alpha_{11}+\alpha_{ 22}+4\beta_{1122}\end{array} \right] \tag{5}\]  
When the fracture's additional compliance \(\Delta S\) is small, one can directly calculate the stiffness matrix using the background stiffness matrix \(C_{0}\)[2]:  
\[C\ =\ C_{0}-C_{0}\Delta SC_{0}. \tag{6}\]  
We assumed a polar anisotropic background. An isotropic background is a special case of this.