# Microseismic Location Error Due to Eikonal Traveltime Calculation
## 4 Discussion  
We use a rather simple horizontally layered model in order to emphasize how numerical errors from traveltime computation propagate into the location error. If a systematic bias is present in computed traveltimes, it changes the curvature of the hodograph and effectively moves the event location to a different depth. The mislocation distance can be counter-intuitive. If we consider the value of traveltime error, which did not exceed 10 ms for the regular FSM, then a traveltime of 10 ms corresponds to the distance of 40 m, which is how far a P-wave will travel in a medium with average P-wave velocity of 4000 m/s. However, the location error was three times larger and reached 120 m. In the general case, when the velocity model includes lateral heterogeneities, the event's location will be also shifted in the horizontal direction. Therefore, when choosing an appropriate method for traveltime computation it is important to consider not only the maximal time error but also its variations with offset and propagation direction.  
We also note that velocity model is calibrated before locating microseismic events: velocity is adjusted in a way that one or several events with a-priori known locations (check shots, string shots, etc.) are imaged to their correct positions. This calibration will reduce location error associated with inaccurate traveltime computation. However, because there is a strong dependence of the depth error on the horizontal event location, model calibration will improve location accuracy only locally close to the reference event used for calibration.  
We have demonstrated that traveltimes, computed with FSM for the regular eikonal equation, introduce significant bias when used for location of microseismic events; they can exceed 100 m for realistic scenarios. The events are shifted deeper and the shift depends on the distance from the center of the acquisition setup. The reason of the bias is a strong dependence of the traveltime error on the propagation direction, which changes the move-out of the arrival times and leads to event mislocations. The FSM for factored eikonal equation effectively addresses this issue by providing significantly more accurate traveltimes. Alternatively, factored eikonal equation can be solved using physics-informed neural networks. Event locations, obtained with this method, have similar accuracy to the factored FSM. However, the location error is more random and less systematically biased. Moreover, PINNs also allow the flexibility of benefiting from transfer learning for updated velocity models, in addition to source positions, as demonstrated by [PERSON] et al. [9]. Furthermore, since the PINN eikonal solver uses Tensorflow at the backend, it allows easy deployment of computations across a variety of platforms (CPUs, GPUs) and architectures (desktops, clusters). On the contrary, significant effort needs to be spent on adapting the FSM algorithms to benefit from different computational platforms or architectures.  
Conceptualization, methodology, software, and visualization D.A. and U.b.W.; writing--original draft preparation, D.A., U.b.W.; revisions to the manuscript and supervision of the research work, L.E. All authors have read and agreed to the published version of the manuscript.  
This research was partly supported by the Polish National Science Centre grant no UMO-2018/30/Q/ST10/00680.  
Not applicable.  
Data Availability StatementData associated with this research are available and can be obtained by contacting the corresponding author.  
We are grateful to [PERSON] from King Abdullah University of Science and Technology (KAUST) for consulting on PINN eikonal solver. We also appreciate kind support of the King Fahd University of Petroleum and Minerals (KFUPM). Authors have greatly benefited from discussions with [PERSON] of GFZ Potsdam.  
The authors declare no conflict of interest.