# Feasibility of Identifying Shale Sweet Spots by Downhole Microseismic Imaging
## 2 Data and Methods
Based on a priori information, the linear approximation of the traveltime residuals can be expressed for the perturbation of the P-wave velocity or Poisson's ratio model parameters as:  
\[r_{ij}=T_{ij}-T_{ij}^{\text{cal}}=\sum_{l=1}^{L}\frac{\partial T_{ij}}{ \partial m_{l}}\Delta m_{l}. \tag{5}\]  
where \(r_{ij}\) is the P  
To directly invert for the Poisson's ratio model, we replace \(m_{l}^{S}\) with \((m_{l}^{P}\cdot m_{l}^{S})\) and take advantage of the relation between \(m_{l}^{r}\) and \(m_{l}^{r}\), \(m_{l}^{v}=(m_{l}^{r^{2}}-2)/2/(m_{l}^{r^{2}}-1)\) so that Equation (7) can be transformed step by step as follows:  
\[r_{ij}^{S} =\sum_{l=1}^{L}\frac{\partial T_{ij}^{S}}{\partial m_{l}^{S}} \Delta m_{l}^{S} \tag{8}\] \[=\sum_{l=1}^{L}\frac{\partial T_{ij}^{S}}{\partial m_{l}^{S}}( \Delta m_{l}^{P}\cdot m_{l}^{r})\] (9) \[=\sum_{l=1}^{L}\frac{\partial T_{ij}^{S}}{\partial m_{l}^{S}}( \Delta m_{l}^{P}\cdot m_{l}^{r}+m_{l}^{P}\cdot\Delta m_{l}^{r})\] (10) \[=\sum_{l=1}^{L}\frac{\partial T_{ij}^{S}}{\partial m_{l}^{S}} \left(\Delta m_{l}^{P}\cdot m_{l}^{r}+\frac{m_{l}^{P}(m_{l}^{r2}-1)^{2}}{m_{l }^{r}}\cdot\Delta m_{l}^{v}\right). \tag{11}\]  
where \(m_{l}^{r}\) denotes the ratio of \(m_{l}^{P}\) and \(m_{l}^{S}\). \(m_{l}^{v}\) denotes the [PERSON]'s ratio. By combining Equations (6) and (11), we invert P- and S-wave traveltimes for P-wave velocity and [PERSON]'s ratio models at the same time. From the perspective of the inverse problem, we can define the objective function of the traveltime tomography as  
\[\Phi(m)=\|d-G(m)\|_{2}^{2}+\tau\|R(m-m_{0})\|_{2}^{2}. \tag{12}\]  
where \(d\) is the traveltime corresponding to \(T_{ij}\), which is viewed as the observed traveltime on traveltime tomography, and \(G(m)\) denotes the calculated or predicted traveltime in the current model \(m\) by 3D wavefront raytracing. For the stability of the solutions of the inverse problem, the Tikhonov regularization term is added in the above function. \(\tau\) is the regularization parameter, and \(R\) is a 3D discrete derivative operator. Here, \(R\) plays the role of Tikhonov regularization in the form of the second-order model derivative operators. Tikhonov regularization minimizes spatial derivatives of the model to help choose a minimum-structure solution because such a smooth solution is the highest probability solution mathematically. \(m_{0}\) is the initial model. Thus, \(m-m_{0}\) means that the inverted model can keep the structure feature (e.g., 3D interfaces) of the initial model. To solve the above equation and obtain the update of the model parameters, we use the Gauss-Newton method to linearize Equation (12) and apply the conjugate gradient method [42] to iteratively invert for the update of the Vp and Poisson ratio. During the performance of the alternating method, both approaches are iteratively employed to update the corresponding event locations as well as the Vp and Poisson's ratio models. To clearly understand the alternating method, we summarize the entire procedure of the alternating workflow as the pseudocode given below (Algorithm 1). ```
Step 1: Initialize \(P\)-wave velocity \(m^{P}\) and Poisson's ratio \(m^{v}\). Step 2: Setup survey geometry and record arrival time data. for each iteration from 1 to MaxLoops do  Step 3: Conduct a 3D grid search for event locations \(X\). Step 4: Compute origin times using Equation (3). Step 5: Calculate traveltimes using Equation (4). Step 6: Update \(m^{P}\) and \(m^{v}\) based on traveltime tomography. if termination criteria are met then  Step 7: Return results \(X\), \(m^{P}\), and \(m^{v}\) and stop. endif endfor
```  
**Algorithm 1** Workflow of the alternating inversion.