The characteristics of an earthquake can be derived by estimating the source geometries of the earthquake using parameter inversion that minimizes the L2 norm of residuals between the measured and the synthetic displacement calculated from a dislocation model. Estimating source geometries in a dislocation model has been regarded as solving a nonlinear inverse problem. To avoid local minima and describe uncertainties, the Monte-Carlo restarts are often used to solve the problem, assuming the initial parameter search space provided by seismological studies. Since search space size significantly affects the accuracy and execution time of this procedure, faulty initial search space from seismological studies may adversely affect the accuracy of the results and the computation time. Besides, many source parameters describing physical faults lead to bad data visualization. In this paper, we propose a new machine learning-based search space reduction algorithm to overcome these challenges. This paper assumes a rectangular dislocation model, i.e., the Okada model, to calculate the surface deformation mathematically. As for the geodetic measurement of three-dimensional (3D) surface deformation, we used the stacking interferometric synthetic aperture radar (InSAR) and the multiple-aperture SAR interferometry (MAI). We define a wide initial search space and perform the Monte-Carlo restarts to collect the data points with root-mean-square error (RMSE) between measured and modeled displacement. Then, the principal component analysis (PCA) and the

In the past decades, interferometric synthetic aperture radar (InSAR) has been a powerful method to acquire geophysical features such as surface deformation or topography by comparing the phases of at least two complex-valued SAR images obtained from different location or time. Since SAR provides high-resolution images, InSAR can measure the surface deformation with centimetric or even millimetric accuracy. This accurate deformation map enables the observation of ocean and ground surface changes, the measurement of ice drift and glacier elevations, and the analysis of seismic deformation or volcanic activities [

Surface deformation acquired from InSAR measurements provides essential information for studying earthquakes and volcanic activities. To derive the characteristics of an earthquake, we can estimate the source geometries of an earthquake by performing parameter inversion that minimizes the L2 norm of residuals between measured and modeled displacement calculated from a dislocation model [

A dislocation model provides the mathematically calculated coseismic displacement of the earth’s surface. The most popular dislocation model is a rectangular dislocation model (Okada model) [

In general, since the dislocation model cannot express the surface displacement linearly, estimating source geometries is a nonlinear inverse problem that can be solved by a nonlinear least square method. The nonlinear least square method iteratively searches local minima based on the partial derivatives of the objective function that indicates the misfit between observed and synthetic data. Hence, the solution may vary according to the starting point of parameters, and thus, we cannot assure that the solution is a global minimum [

In solving a nonlinear least square problem with the Monte-Carlo restarts, we should designate a search space by setting boundary conditions of each parameter to pick random starting points. Since a nonlinear least square method iteratively explores the parameter search space, the search space size significantly affects the accuracy and execution time of the algorithm. Previous studies [

This paper presents a new search space reduction algorithm based on machine learning techniques to come up with the challenges mentioned above. Our proposed method proceeds as follows:

Initially, assume a wide range of search space and perform the Monte-Carlo restarts of five thousand iterations with randomly initialized starting points

Then, choose the samples that preserve maximum variance and have a low RMSE and project them to the two-dimensional (2D) space using PCA

Decide hyperparameter

Choose the cluster with the lowest mean RMSE and reduce the search space using the cluster

Perform additional Monte-Carlo restarts of five thousand iterations in the reduced search space and determine the final source parameters

Our search space reduction algorithm was compared with the conventional approach that uses geological studies to define the search space in the same study area, the 2017 Pohang earthquake. The evaluation was performed in terms of the size of search space, RMSE calculated from determined source parameters, a 95% confidence interval of each source parameter, and the average iteration number of a nonlinear least square.

This paper is organized as follows. Section

InSAR is the method that extracts geophysical features such as surface topography and deformation by comparing the phase offsets of at least two complex-valued SAR images [

The source geometries of an earthquake can also be estimated by performing parameter inversion that finds the best approximate source parameters having the minimum misfit between the measured displacement and the synthetic 3D surface displacement of a theoretical dislocation model. Typical dislocation models are as follows. Mogi’s model [

To estimate the source parameters using a nonlinear dislocation model, we can use the nonlinear least square method that minimizes the Euclidean norm of residuals between the measured and the modeled displacement. Let us denote the dislocation model as

Source parameters

The nonlinear least square method iteratively explores the search space from a specific starting point to local minima, so the method cannot derive global minima. To avoid local minima and describe uncertainties of parameters, we can use the Monte-Carlo restarts that generate many initial starting points and then solve the nonlinear least square problem for each starting point. Obtaining results from multiple starting points helps to find the global minimum and show the probabilistic descriptions such as standard deviation and histogram. Many studies employed the Monte-Carlo restarts to obtain the best-fit source parameters of earthquakes or volcanic activities [

Our approach used stacking InSAR and multiaperture InSAR (MAI) [

Machine learning has been used widely in remote sensing [

Unsupervised learning techniques have also been used broadly in remote sensing. Principal component analysis (PCA), one of the representatives among them, reduces the dimension of the given dataset. PCA finds a lower dimensional hyperplane that preserves the maximum variance to maintain the maximum amount of original information [

This paper is aimed at reducing the parameter search space for the determination of earthquake source parameters with better computation time efficiency and data visualization. Figure

Schematic workflow of our approach. (a) is a big workflow that estimates earthquake source parameters. (b) shows our machine learning approach, included in (a), to reduce search space.

As the first step of our approach, we set a wide initial search space. Second, we perform the Monte-Carlo restarts of five thousand iterations to associate the RMSE and source parameters in the initial search space. In performing Monte-Carlo restarts, we randomly subsample measured deformation to reduce computation time. Third, we reduce the parameter search space using the PCA and the

Our approach is applied to the 2017 Pohang earthquake [^{2}.

Our study used the Okada model [

Source parameters of the Okada model.

Parameter | Unit | Description |
---|---|---|

km | Distance from the reference point to the east | |

km | Distance from the reference point to the north | |

Depth | km | Depth of source |

Strike | Degree | Angle of fault relative north |

Dip | Degree | Angle between the fault and a horizontal plane |

Length | km | Length of fault |

Width | km | Width of fault |

Rake | Degree | Angle of slip relative to the Width direction |

Slip | mm | Dislocation in rake direction |

Open | km | Dislocation in tensile component |

Fault geometry of the Okada model.

As the first step, our algorithm conducts the Monte-Carlo restarts for a wide initial search space described in Table

Initial search space of source parameters.

Depth (km) | Strike (deg.) | Dip (deg.) | Length (km) | Width (km) | Rake (deg.) | Slip (mm) | |||
---|---|---|---|---|---|---|---|---|---|

LB | -40 | -40 | 1 | 0 | 0 | 0.01 | 0.01 | 0 | -10000 |

UB | 40 | 40 | 60 | 360 | 90 | 120 | 120 | 180 | 10000 |

Then, the Monte-Carlo restarts of five thousand iterations are performed to extract the

Data acquisition in the initial search space using the Monte-Carlo restarts.

Options of MATLAB lsqnonlin() function.

Option | Value |
---|---|

Algorithm | Trust-region-reflective |

MaxIter | 500 |

TolX | 0.01 |

TolFun | 0.01 |

This study uses the PCA and the

PCA procedure for dimensionality reduction of Monte-Carlo results.

PCA enables visualization of high-dimensional data in the low-dimensional space while preserving information of the original data as much as possible. In general, a 2D scatter matrix is used to visualize high-dimensional data, but each element of the scatter matrix represents only partial information. Furthermore, since the number of two-dimensional scatter plots of

For the data projected to the 2D space, we cluster the data using the

If the PCA preserves most of the variance of the Monte-Carlo results and all source parameters are on the same scale, the distances between original data points in the latent 2D space can be directly assessed [

Before the PCA fitting, each parameter is scaled to have a standard normal distribution

To evaluate the effectiveness of our machine learning-based search space reduction algorithm, this paper applied the proposed machine learning-based algorithm to the 2017 Pohang earthquake case [

As mentioned in Section

Explained variance ratio according to the number of samples.

Table

Principal components of each parameter.

Parameters | PC 1 | PC 2 |
---|---|---|

-0.16955 | -0.94018 | |

-0.34902 | 0.14051 | |

Depth | 0.35540 | 0.01463 |

Strike | 0.35263 | 0.05568 |

Dip | 0.34854 | -0.01957 |

Length | -0.35430 | 0.03954 |

Width | -0.33501 | 0.27958 |

Rake | 0.35549 | 0.04171 |

Slip | 0.33640 | -0.10564 |

Samples projected into 2D space.

Then, we performed

Results of

Statistics of each cluster.

Cluster no. | Mean of RMSE (cm) | Standard deviation of RMSE (cm) | Number of samples |
---|---|---|---|

1 | 0.14767 | 151 | |

2 | 0.14934 | 32 | |

3 | 0.14867 | 10 |

We then reduced the search space using the minimum and maximum values of data in each cluster. Tables

Search space determined from seismological approaches.

Parameters | Depth (km) | Strike (deg.) | Dip (deg.) | Length (km) | Width (km) | Rake (deg.) | Slip (mm) | ||
---|---|---|---|---|---|---|---|---|---|

LB | 4.8 | -12.0 | 4.1 | 220.0 | 33.0 | 4.0 | 4.0 | 120.0 | 10 |

UB | 10.0 | -6.0 | 5.7 | 235.0 | 55.0 | 6.5 | 6.7 | 150.0 | 300 |

Reduced search space of three clusters.

Parameters | Cluster 1 | Cluster 2 | Cluster 3 | |||
---|---|---|---|---|---|---|

LB | UB | LB | UB | LB | UB | |

6.26911 | 6.41034 | 6.27024 | 6.37718 | 6.22817 | 6.3297 | |

-7.9826 | -7.87204 | -8.2007 | -8.06074 | -8.10795 | -7.98743 | |

Depth (km) | 4.83734 | 5.19459 | 5.28377 | 5.54763 | 5.18088 | 5.49678 |

Strike (deg.) | 224.9835 | 231.7157 | 232.4768 | 236.8167 | 231.5858 | 236.5544 |

Dip (deg.) | 42.61413 | 44.68897 | 44.99918 | 46.47305 | 44.35434 | 46.34576 |

Length (km) | 4.19294 | 4.92414 | 1.13009 | 3.04056 | 1.7581 | 4.19284 |

Width (km) | 5.7689 | 6.45088 | 4.67679 | 5.51209 | 5.34835 | 6.08954 |

Rake (deg.) | 127.4311 | 131.1753 | 132.5249 | 135.0445 | 131.1863 | 135.0024 |

Slip (mm) | 170 | 230 | 330 | 950 | 220 | 520 |

Comparison of search spaces between conventional approach and our approach.

Table

Search space size derived from conventional seismological studies and our proposed algorithm.

Parameters | Conventional approach | Our approach | Reduction ratio (%) |
---|---|---|---|

5.2 | 0.1412 | 97.3 | |

6.0 | 0.1106 | 98.2 | |

Depth (km) | 1.6 | 0.3573 | 77.7 |

Strike (deg.) | 15.0 | 6.7322 | 55.1 |

Dip (deg.) | 22.0 | 2.0748 | 90.6 |

Length (km) | 2.5 | 0.7312 | 70.8 |

Width (km) | 2.7 | 0.6820 | 74.7 |

Rake (deg.) | 30.0 | 3.7442 | 87.5 |

Slip (mm) | 290 | 60 | 79.3 |

As the final step, we conducted additional five thousand iterations of Monte-Carlo restarts in the reduced search space based on Cluster 1 and determined the source parameters as the best-fit parameter of the results of the Monte-Carlo restarts with a 95% confidence interval. Table

Determined parameters and RMSE of conventional approach and our approach with a 95% confidence interval.

Parameters | Conventional approach | Our ML approach |
---|---|---|

Depth (km) | ||

Strike (deg.) | ||

Dip (deg.) | ||

Length (km) | ||

Width (km) | ||

Rake (deg.) | ||

Slip (mm) | ||

RMSE (cm) | 0.147637 | 0.147634 |

Figure

Histogram of determined parameters.

Figure

Final results of source modeling.

For the computational efficiency validation, the average iterations of a nonlinear least square from the Monte-Carlo restarts were also compared. As shown in Table

Average number of iterations for nonlinear least square from the Monte-Carlo restarts.

Conventional approach | Our approach |
---|---|

4.93 | 2.25 |

This paper presented a new search space reduction algorithm using machine learning techniques for the earthquake source parameter determination. Our algorithm proceeded in the order of the Monte-Carlo restarts, PCA,

We compared the proposed approach with the conventional approach that defines the parameter search space by referring to seismological studies in terms of size of the search space, RMSE, a 95% confidence interval of Monte-Carlo results, and the average iteration number of nonlinear least square from the Monte-Carlo restarts. The experimental results for the 2017 Pohang earthquake test case showed that our approach achieves significant reductions in search space size and a 95% confidence interval size for all source parameters, i.e., about 55.1~98.1% and 60~80.5%, respectively. Also, the average iteration number of a nonlinear least square from our reduced search space was much smaller than that of a conventionally defined search space. Consequently, the results demonstrated that the proposed approach effectively reduces the parameter search space size and the computational burden.

Our future challenge is to develop the automatic search space reduction algorithm that uses the hierarchical clustering algorithm for determining the number of clusters.

The surface deformation maps were provided by Prof. Jung and Mr. Baek. They have generated the maps from the ALOS PALSAR-2 interferometric data that were provided through the JAXA’s ALOS-2 research program (RA4, PI No. 1412).

The authors declare that they have no conflicts of interest.

This work was supported by the 2019 sabbatical year research grant of the University of Seoul.