# Microseismic Location Error Due to Eikonal Traveltime Calculation
## 3 Numerical Results
### Homogeneous Model  
In the first example, we consider a homogeneous model with P-wave velocity \(v_{p}=4000\) m/s and 20 m grid spacing. Here, we test only FSM for the original eikonal equation to demonstrate the issue in the simplest case. We select a symmetrical star-shaped acquisition geometry with each receiver line having 2000 m length and 20 m spacing (Figure 3). The target zone where we search for events with diffraction stacking is located at 2000 m to 2500 m depth range and spans from \(-1450\) m to 1450 m in both horizontal directions. We generate a set of synthetic seismograms corresponding to events (explosive-single polarity) regularly distributed on a horizontal plane at 2200 m depth inside the target zone. We then locate the events with the diffraction stacking method using travelltimes computed with FSM for the original eikonal equation and compare true and diffraction stacking locations using the methodology of [PERSON] et al. [1].  
The traveltime computation error of FSM times for this model does not exceed 7 ms (relative to the analytical solution). As was mentioned previously, if this error was constant across the monitoring array, it would not affect location accuracy at all and only result in a time shift of the origin time. However, Figure 4 shows that the depth location error reaches 150 m in the center of the acquisition setup and decreases to 20-40 m upon reaching the edge of the tested area for source locations. This error distribution results from the fact that the traveltime error is offset dependent and sources in the center are most affected. Positive values of the depth error mean that events are imaged deeper than their true location. The horizontal error is positive if the estimated event location is closer to the center of the model than the true one. Generally sources are located deeper and closer to the center as a result of the traveltime error.  
Figure 3: Map view (**a**) and side view (**b**) of the acquisition setup and target zone used in diffraction stacking.  
Comparison of hodographs of the FSM solution with exact traveltimes in the homogeneous model provides an insight into the error behavior (Figure 5). First, we observe that the hodograph curvature of FSM is different from the exact solution, meaning a dependence of the traveltime error on the offset. This curvature is compensated by moving events deeper in the center of the array, resulting in a vertical error. At the same time, the apex of the hyperbola does not change and as a result the horizontal location error remains low except for the boundaries of the target zone where depth error trades off with offset. This observation is consistent with previous publications--the depth of detected microseismic events is the most sensitive parameter for surface monitoring.  
Second, the number of receivers with significant moveout change is different for the event in the center of the model (Figure 5a) and for the event close to the edge of the target zone (Figure 5b). This results in larger errors in the center of the model. Note that the hodographs presented in figures are adjusted by a constant time shift in a way that minimizes the rms time error. The absolute traveltime errors are minimal at zero offset (i.e., at the apex of hyperbola).  
We do not present results of FSM and PINN for a factored eikonal equation in the homogeneous model, as it converges to the exact solution within the machine precision.  
Figure 4: Location error in meters for the vertical (**a**) and horizontal (**b**) directions after using fast sweeping method (FSM) for the regular eikonal equation in a homogeneous medium.  
Figure 5: Comparison of exact traveltimes and results of FSM in a homogeneous model for event in the center of the model (**a**) and close to the edge of the target zone (**b**).