Dislocation model

INTRODUCTION

Dislocation model had first presented by Okada (see Okada, Y. (1992). Internal deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America, 82(2), 1018-1040.) and provide a powerful way for finite fault inversion. With assuming the surface (static)observations can be linearly related to the slip on fault, the unknown slip could be solved by the linear equation.

METHOD

Given the dimension to a fault (e.g. fault length, width, position, number of subfault), static observations can be considered as the total contribution of slip on each subfault with the component of strike-slip dip-slip and tensile (i.e. the CLVD). For a normal earthquake, only strike-slip and dip-slip are considered as reasonable earthquake mechanism. Thus, the Green's function is shown with the form of (strike,0,0) and (0,dip,0), in the disloc3d module (you can get the source code from: http://www.avo.alaska.edu/Software/Kilauea/disloc3d.html) then you get the "G" in the following equation.

Because this is not a totally linear equation, the next step will be dealing with the nonlinear part by grid-searching. The nonlinear parameters including (1)fault size & number of subfault, (2)Laplace operator (smoothing the adjacent slip)(3)position of the fault (X,Y,Z), the (4)strike & dip.

First, the fault size can be determined by empirical law (e.g. 20 km*20 km for a Mw6.0 earthquake), because fitting quality is proportional to the fault size, a given size should be reasonable. Number of subfaults are also depend on the empirical law and the total number of observations. For example, given the number of 6*6 subfault, the total uncertainty would be 36+36 for strike-slip and dip-slip. Therefore, if total observations do not outweigh this value and we do not given another constrain, this will be an underdetermine problem. Luckily, the Laplace operator provides additional constrain to the slip behavior. Laplace here is aimed for correcting the "unreasonable" discontinuous slip behavior, such as from 1-cm slip to 100-cm slip then to 10-cm slip, in the nearby 3-connected subfault. However, there is a trade off between smoothing and fitting.

The next step is to determine the location (XYZ), strike and dip. With applying the grid searching method, the position and fault orientation can be found. All the slip can be inverted as the linear equation: Gm=d,

m=(G^TG)^-1G^Td

Parameters:

Strike:8 Dip:30

Fault size:30 km

Max depth:30 km

Mw6.0

Max slip:74 mm