# Rapid Estimation of Earthquake Magnitude and Source Parameters Using Genetic Algorithms
## 2 Materials and Methods
### Inversion of [PERSON]'s Model  
A fault is a large fracture in the Earth's crust that causes two sides of rock blocks to become displaced relative to each other. An earthquake occurs due to a sudden release of energy that causes the fault to slip suddenly. The plane of the fracture where the fault slips is called the fault plane, and is represented by parameters such as strike, dip, depth, and rake [29]. Figure 1a illustrates a fault plane.  
A slip on the fault plane will cause some displacements on the seafloor of the Earth, which is called surface deformation. [PERSON]'s elastic dislocation model, proposed by [PERSON] in 1985, features a set of analytical solutions for surface displacements in a homogeneous half-space. This model translates the effect of a fault-plane slip into the ground displacements represented in \(\left(u_{x},u_{y},u_{z}\right)\)[30], as illustrated in Figure 1b.  
[PERSON]'s model is widely used for co-seismic displacement analysis. The displacement field \(u_{i}(x_{1},x_{2},x_{3})\) due to a dislocation \(\Delta u_{j}(\xi_{1},\xi_{2},\xi_{3})\) across a surface \(\Sigma\) in an isotropic medium is given by the following formula [31]:  
\[u_{i}=\frac{1}{F}\iint_{\Sigma}\Delta u_{j}\bigg{[}\lambda\delta_{jk}\frac{ \delta u_{i}^{n}}{\delta\xi_{n}}+\mu\left(\frac{\partial u_{i}^{j}}{\partial \xi_{k}}+\frac{\partial u_{i}^{k}}{\partial\xi_{j}}\right)\bigg{]}v_{k}d\Sigma. \tag{1}\]  
A fault in an earthquake can be modeled as a collection of rectangular patches, where each patch has the following parameters [32]:  
Figure 1: Basic concepts related to elastic dislocation based on [PERSON]’s model: (a) a fault plane with its parameters; (b) ground displacement as the effect of the slip of a fault plane.  
1. The latitude and longitude of the fault plane's centroid point;
2. The length and width of the fault field (in meters or kilometers);
3. The depth (in meters or kilometers);
4. The strike (0-360\({}^{\circ}\));
5. The dip (0-90\({}^{\circ}\));
6. The rake (\(-\)180-180\({}^{\circ}\));
7. The slip (\(>\)0 m).  
In simple terms, [PERSON]'s model formulates these nine parameters to calculate theoseismic displacement fault values, tilts, and compression of an earthquake. This process can be expressed as a function of the nine parameters, which results in the following set of three-dimensional displacements [32]:  
\[Okada(E,N,depth,strike,dip,length,width,rake,slip)\ =\ [uE,uN,uZ] \tag{2}\]  
where \(E\) and \(N\) are the coordinates of the observation at the geographic reference (east and north) relative to the centroid of the fault plane, and \([uN,\ uE,\ uZ]\) is a set of displacement values at the observation points in the directions of north (\(N\)), east (\(E\)), and vertical (\(Z\)). Elastic dislocation modeling, in particular [PERSON]'s model, requires the assumption of the fault plane's centroid. In actual conditions, the centroid of an earthquake's fault plane can be assumed to be the hypocenter location obtained from the seismic observations.  
For the inversion of [PERSON]'s model, a calculation in the inverse direction of the model is used to find some or all of the earthquake's source parameters in the model function above from the displacement values at the observation points.