# Microseismic Location Error Due to Eikonal Traveltime Calculation
## 2 Methods
### FSM for the Factored Eikonal Equation  
In order to address this issue and obviate the singularity at the source location, [PERSON] et al. [8] suggested applying the FSM to the factored eikonal equation. The idea is to use an analytical solution \(T_{0}\) of the eikonal equation for a simplified model and then search for correction \(\tau\) for this solution. The simplified solution can be the traveltime in a homogeneous model with the same velocity as at the source position or just a distance  
Figure 1: 3D rectangular mesh around the current point \({\bf C}\).  
function between the source and a grid point. A multiplicative factored decomposition suggests that for a model characterized by the slowness  
\[S(\mathbf{x})=S_{0}(\mathbf{x})\alpha(\mathbf{x}), \tag{4}\]  
where \(S_{0}(\mathbf{x})\) is a reference slowness in the simplified model and \(\alpha(\mathbf{x})\) is a multiplicative factor. The solution of eikonal equation can be represented as  
\[T(\mathbf{x})=T_{0}(\mathbf{x})\tau(\mathbf{x}) \tag{5}\]  
where \(T_{0}(\mathbf{x})\) is the solution of eikonal equation in the simplified model \(S_{0}(\mathbf{x})\) and:  
\[|\
abla T_{0}|^{2}=S_{0}^{2}(\mathbf{x}), \tag{6}\]  
the \(\alpha(\mathbf{x})\) is a known scaling factor between the simplified and original model, and \(\tau(\mathbf{x})\) is the unknown traveltime correction. Substitution of (4) and (5) into (2) yields the factored eikonal equation:  
\[T_{0}^{2}|\
abla\tau|^{2}+2T_{0}\tau(\
abla T_{0},\
abla\tau)+\tau^{2}|\
abla T _{0}|^{2}-a^{2}S_{0}^{2}(\mathbf{x})=0. \tag{7}\]  
This equation can be solved at every grid point \(\mathbf{C}\) of the model in a similar way as a regular eikonal equation by considering the same eight tetrahedrons, substituting the first order finite difference approximation for \(\
abla\tau\):  
\[\
abla\tau\approx(\tau_{\mathrm{C}}-\tau_{W},\tau_{\mathrm{C}}-\tau_{S},\tau_ {c}-\tau_{D})\cdot h^{-1}, \tag{8}\]  
and applying causality condition:  
\[\tau_{\mathrm{C}}T_{0}(\mathbf{C})>\tau_{\mathrm{S}}T_{0}(\mathbf{S}),\ \tau_{ \mathrm{C}}T_{0}(\mathbf{C})>\tau_{W}T_{0}(\mathbf{W}),\ \tau_{\mathrm{C}}T_{0}(\mathbf{C})>\tau_{D}T_{0}(\mathbf{D}). \tag{9}\]  
When there are no real roots of the quadratic equation for \(\tau_{\mathrm{C}}\) or they do not satisfy the causality condition, it is reduced to the 2D case and solved at each face of the tetrahedron separately. If the 2D case does not yield a suitable solution either, the information of \(\tau\) is propagated along the edges of the selected triangle using the method of characteristics [8]. For the triangle \(\Delta CWD\) the method will yield two solution candidates:  
\[\tau_{WC}=\frac{S(\mathbf{C})h+\tau_{W}T_{0}(\mathbf{C})}{T_{0}(\mathbf{C})+T _{0x}\delta x},\ \tau_{DC}=\frac{S(\mathbf{C})h+\tau_{D}T_{0}(\mathbf{C})}{T_{0}(\mathbf{C})+T _{0z}\delta z}. \tag{10}\]  
Here \(\delta x\) and \(\delta z\) are coordinate shifts from the point C to the points W and D, respectively, i.e., either \(+h\) or \(-h\) depending on the face of tetrahedron that is currently considered.  
In the last step, the solution candidate with a minimal value is selected and used to update \(\tau\) at the current grid point. The procedure is repeated for all grid points during the eight sweeps as in the FSM for the regular eikonal equation. The last difference between these two methods is in the initialization procedure: in the case of FSM for a factored eikonal equation a value of \"1\" instead of zero is assigned to the array of \(\tau\) values at the point-source location.