# Microseismic Location Error Due to Eikonal Traveltime Calculation

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Diffraction Stacking

#### 2.2. FSM for the Eikonal Equation

#### 2.3. FSM for the Factored Eikonal Equation

#### 2.4. Eikonal Solution Using Physics-Informed Neural Networks (PINNs)

- a deep neural network (DNN) approximation of the unknown traveltime field $\tau (x,y,z)$,
- a differentiation algorithm, i.e., automatic differentiation in this case, to evaluate the partial derivatives of $\tau (x,y,z)$ with respect to the spatial coordinates $(x,y,z)$,
- a loss function incorporating the underlying eikonal equation sampled on a collocation grid, an optimizer to minimize the loss function by updating the neural network parameters.

## 3. Numerical Results

#### 3.1. Homogeneous Model

#### 3.2. Heterogeneous Model

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**A randomly initialized deep neural network (DNN) is trained on a set randomly selected training points ${\mathbf{x}}^{*}$ with a given slowness model $S\left({\mathbf{x}}^{*}\right)$. The network is minimized with additional inputs at these training points including the known traveltime field ${T}_{0}\left({\mathbf{x}}^{*}\right)$ and its spatial derivative $\nabla {T}_{0}\left({\mathbf{x}}^{*}\right)$. Once the DNN is trained, it is evaluated on a regular grid $\left(\mathbf{x}\right)$ to yield an estimate of the unknown traveltime $\widehat{\tau}\left(\mathbf{x}\right)$ which is then multiplied by ${T}_{0}\left(\mathbf{x}\right)$ providing the estimated traveltime solution $\widehat{T}\left(\mathbf{x}\right)$.

**Figure 3.**Map view (

**a**) and side view (

**b**) of the acquisition setup and target zone used in diffraction stacking.

**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**).

**Figure 7.**Location error of diffraction stacking in meters using regular FSM (

**a**,

**b**), factored FSM (

**c**,

**d**), and physics-informed neural network (PINN) (

**e**,

**f**) in laterally homogeneous model.

**Figure 8.**Comparison of traveltime errors obtained with FSM for the factored eikonal equation and using PINN for an event at the center of the model.

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**MDPI and ACS Style**

Alexandrov, D.; Waheed, U.b.; Eisner, L. Microseismic Location Error Due to Eikonal Traveltime Calculation. *Appl. Sci.* **2021**, *11*, 982.
https://doi.org/10.3390/app11030982

**AMA Style**

Alexandrov D, Waheed Ub, Eisner L. Microseismic Location Error Due to Eikonal Traveltime Calculation. *Applied Sciences*. 2021; 11(3):982.
https://doi.org/10.3390/app11030982

**Chicago/Turabian Style**

Alexandrov, Dmitry, Umair bin Waheed, and Leo Eisner. 2021. "Microseismic Location Error Due to Eikonal Traveltime Calculation" *Applied Sciences* 11, no. 3: 982.
https://doi.org/10.3390/app11030982