# Feasibility of Identifying Shale Sweet Spots by Downhole Microseismic Imaging
## 2 Data and Methods  
The Poisson's ratio or Vp/Vs ratio is an important indicator for the reservoir lithology and fluid properties [4]. And P-wave velocity helps characterize the reservoir structure [23]. Also, the conventional technique of taking the ratio of P- and S-wave velocities cannot ensure that the area is well resolved, in which the P- and S-wave raypaths are not identical [39]. In this section, we therefore attempt to describe an alternating method to estimate the 3D P-wave velocity as well as Poisson's ratio and corresponding event locations using the P- and S-wave arrival times recorded by the receivers in borehole. Analogous to the alternating strategy, we separate the whole inversion system into two parts: one is to solve for the event locations and origin time using a 3D grid search approach, and the other one is to invert the 3D P-wave velocity and Poisson's ratio models simultaneously in a modified traveltime tomography. Hence, the modified traveltime tomography and the 3D grid search are iteratively employed in an alternating way until the arrival time misfit or residual becomes negligible. In the following, we shall explicitly introduce the approaches employed in the alternating method. We apply the 3D grid search method to optimize event location parameters before updating the P-wave velocity and Poisson's ratio since the grid search approach does not require an initial guess of the locations and is able to avoid trapping into a local minimum. In microseismic data processing, the observed arrival time \(t_{ij}\) from a microseismic event \(i\) to a receiver \(j\) within the borehole is formulated as:  
\[t_{ij}=t_{i}^{0}+T^{ij}. \tag{1}\]  
where \(t_{ij}\) is the arrival time of either the observed P- or S-wave. \(t_{i}^{0}\) is the origin time of the event \(i\). \(T^{ij}\) is the P- or S-wave traveltime from a microseismic event \(i\) to a receiver \(j\). We define the normalized misfit function of the 3D grid search approach [38,40] as Equation (2), and origin time \(t_{i}^{0}\) is found by averaging the arrival times in Equation (3), then the P- or S-wave traveltime can be computed using Equation (4):  
\[t_{ij}^{\text{res}}=\sqrt{\frac{\sum_{j=1}^{n}(t_{ij,p}-t_{i}^{0}-T_{ij,p}^{ \text{cal}})+\sum_{j=1}^{n}(t_{ij,s}-t_{i}^{0}-T_{ij,s}^{\text{cal}})}{2n-3}}, \tag{2}\]  
\[t_{i}^{0}=\frac{1}{2n}\left(\sum_{j=1}^{n}(t_{ij,p}-T_{ij,p}^{\text{cal}})+ \sum_{j=1}^{n}(t_{ij,s}-T_{ij,s}^{\text{cal}})\right), \tag{3}\]  
\[T_{ij}=t_{ij}-t_{i}^{0}. \tag{4}\]  
where \(t_{ij}^{\text{res}}\) represents the overall traveltime misfit or residual of the grid search, \(t_{ij,p}\) and \(t_{ij,s}\) mean the observed P- and S-wave arrival times from event \(i\) to receiver \(j\), respectively, \(T_{ij,p}^{cal}\) and \(T_{ij,s}^{cal}\) denote the calculated traveltime data of the P and S waves, \(T_{ij}\) stands for the observed traveltime, and \(n\) is the number of receivers. We apply the 3D wavefront raytracing method [41] to calculate the synthetic P- and S-wave traveltimes. Note that in the S-wave raytracing, the utilized S-wave velocity model is converted from the initial or updated P-wave velocity and Poisson's ratio models, which are obtained in the aforementioned traveltime tomography. After determining the event parameters using 3D grid search, we utilize the searched event locations to invert for the P-wave velocity and Poisson's ratio models with the computed traveltimes of the P and S waves.