# Direct Inversion Method of Brittleness Parameters Based on Reweighted Lp-Norm
## 2 Materials and Methods Used for Experimentation  
### Reweighted Lp Method  
First, the Zoeppritz approximate equation [7] used in this manuscript is introduced, and its expression is as follows:  
\[R(\theta)=a(\theta)R_{\_}E\rho+b(\theta)R_{\_}\sigma+c(\theta)R_{\_}\rho \tag{1}\]  
\[a(\theta)=\frac{1}{4}\sec^{2}\theta-2k^{2}\sin^{2}\theta \tag{2}\]\[b(\theta)=\frac{1}{4}\sec^{2}\theta\frac{(2k^{2}-3)(2k^{2}-1)^{2}}{k^{2}(4k^{2}-3 )}+2k^{2}\sin^{2}\theta\frac{1-2k^{2}}{3-4k^{2}} \tag{3}\]  
\[c(\theta)=\frac{1}{2}+2k^{2}\sin^{2}\theta-\frac{1}{2}\sec^{2}\theta \tag{4}\]  
where \(\theta\) is the angle of incidence; \(E\rho\) is [PERSON]'s modulus multiplied by density; \(\sigma\) is Poisson's ratio; \(\rho\) is density; k is the ratio of shear to compressional wave velocity (\(k=\frac{V_{\rho}}{V_{\rho}}\)); and \(R\_{\_}{E\rho}\), \(R\_{\_}{\sigma}\), and \(R\_{\_}{\rho}\) are the reflectivity of the \(E\rho\), Poisson's ratio, and density, respectively. To obtain the required parameters from the seismic data, it is necessary to establish the relationship between the seismic data and the parameters. Taking \(\rho\) as an example, the reflection coefficient of \(\rho\) can be approximated as follows:  
\[R\_{\_}{\rho}=\frac{\rho_{\_}{i+1}-\rho_{\_}{i}}{\rho_{\_}{i+1}+\rho_{\_}{i}} \approx\frac{1}{2}\Delta\ln(\rho)=\frac{1}{2}\left[\ln(\rho_{\_}{i+1})-\ln( \rho_{\_}{i})\right]=\frac{1}{2}(L_{\_}{i+1}-L_{\_}{i}) \tag{5}\]  
where \(L_{\_}{i}=\ln(\rho_{\_}{i})\).  
\(R\_{\_}{\rho}\) is represented by the following matrix:  
\[R\_{\_}{\rho}=\frac{1}{2}DL_{\_}{\rho} \tag{6}\]  
where \(D\) is the difference matrix, and \(L_{\_}{\rho}\) is the natural logarithm of density.  
According to Equation (1) with the use of a wavelet kernel matrix \(W(\theta)\), the seismic record \(S(\theta)\) can be shown as follows:  
\[\begin{bmatrix}S(\theta)\\ \vdots\\ S(\theta_{\_}{M})\end{bmatrix}=\frac{1}{2}\begin{bmatrix}W(\theta_{\_}1)a( \theta_{\_}1)D&W(\theta_{\_}1)b(\theta_{\_}1)D&W(\theta_{\_}1)c(\theta_{\_}1 )D\\ \vdots&\vdots&\vdots\\ W(\theta_{\_}{M})a(\theta_{\_}M)D&W(\theta_{\_}M)b(\theta_{\_}M)D&W(\theta_{\_ }M)c(\theta_{\_}M)D\end{bmatrix}\begin{bmatrix}L_{\_}{\_}{\rho}\\ L_{\_}{\sigma}\\ L_{\_}{\rho}\end{bmatrix} \tag{7}\]  
where \(L_{\_}{\rho}\) is the natural logarithm of the \(E\rho\), \(L_{\_}{\sigma}\) is the natural logarithm of Poisson's ratio, and \(L_{\_}{\rho}\) is the natural logarithm of density.  
The forward equation based on an approximate linear formula can be abbreviated as follows:  
\[S=WBDL=AL \tag{8}\]  
where \(B\) is the matrix consisting of Equations (2)-(4), and \(L\) is the matrix of the natural logarithms of the \(E\rho\), [PERSON]'s ratio, and density.  
As shown in the introduction, we first introduced a weighted Lp-norm into the inversion of brittleness parameters. This method can solve the shortcomings of L1 regularization in inverse objective function. The regularization term is weighted and constrained by the Lp-norm, which yields results with better sparsity and improves the inversion accuracy. The reweighted Lp method for inversion parameters can be defined as follows:  
\[Lp(L)=\lambda(w_{\_}{i}\|L\|_{\_}1) \tag{9}\]  
where \(\left|\right|\)\({}_{1}\) is the \(L1-norm\), \(\lambda\) is the regularization factor, and \(w_{\_}{i}\) is the weight value.  
\[(w_{\_}{i})^{l+1}=\frac{1}{\left\|\left(L_{\_}i\right)^{l+1}\right\|_{\_}{p}^{ p}+\varepsilon} \tag{10}\]  
where \(\left|\right|\)\({}_{p}\) is the \(Lp-norm(0\leq p<1)\), and \(\varepsilon\) is a constant greater than 0. \(\varepsilon\) can ensure the stability of the algorithm, that is, if \(L\) has a value of zero in a round of iterations, there is a non-zero value in the corresponding position in the next iteration.