Hypothesis and Inversion Method

The VELEST program is a FORTRAN77 routine that has been designed to derive 1-D velocity model. It was modified, implemented some methods and tested several times since 1976 (Kissling, 1995). Now, it can be used to simultaneously invert a large number of earthquake travel time data for hypocentral and velocity model parameters — P- and S-wave velocities. Because this inversion is a non-linear problem, the solution is obtained iteratively. One-iteration consists of solving both the forward and inverse problem once. The briefly procedures as following flow chart:

Figure 20. The procedure flow chart of minimum 1-D model. (Kissling, 1995)

4.2.1.1 Travel Time of a Single Ray

The travel time between a seismic source and a receiver for a given velocity model with ray-tracing is a forward problem. And, the arrival times of seismic waves generated by an earthquake and observed at the seismic stations are a function of the hypocentral parameters and the velocity model:

0 0 0 0

( , , , , ( , , )) t=F t x y z V x y z

Consequently, the residual travel time is a function of the differences between the estimated and the true hypocentral and velocity parameters:

0 0 0 0

( , , , , ( , , ))

res obs calc

t =t −t =F dt dx dy dz dV x y z

Applying a first-order Taylor series expansion to the equation, a linear relationship between the residual travel time and adjustments to the hypocentral and velocity parameter is obtained:

This formula contains forward problem and inverse problem.

4.2.1.2 Coupled Hypocenter Velocity Model Problem

The determination of the unknown hypocentral parameters and the velocity parameters from a set of arrival time is a coupled hypocenter-velocity-parameter problem:

x y z coordinates of hypocenter ( , , )

δ partial derivative of travel time with respect to jth hypocentral parameter

k

F m δ

δ partial derivative of travel time with respect to kth velocity model parameter

Continuously, the formula is written in matrix notation and separated into two matrices representing the hypocentral and the velocity model parameters. And it was added by an error vector that contains remaining part of the travel time residual. The remaining part is what cannot be explained by adjustments to the unknown parameters. The relation can be written as follows:

t=Hh+Mm e+ =Ad+e

And the method used for the formula is solved by full inversion of the damped least squares matrix.

Although the VELEST algorithm can simultaneously invert P- and S-wave arrival times for 1-D P- and S-wave velocity models. In this study, 1-D P-wave velocity model was the only one to be determined by the VELEST algorithm. Then, the Wadati-diagram was applied to get the Vp/Vs ratio in the target area, and the 1-D S-wave velocity model could be obtained. The procedures to obtain a “minimum 1-D P-wave model” followed

t i the ith travel time residual

n the total number of observations for this event

t vector of travel time residuals

H matrix of partial derivatives of travel time with respect to hypocentral parameters h vector of hypocentral parameter adjustments

M matrix of partial derivatives of travel time with respect to model parameters m vector of model parameter adjustments

e vector of travel time errors A matrix of all partial derivatives

d vector of hypocentral and model parameter adjustments

the guidelines proposed by Kissling et al. (1994). The brief guidelines for a minimum 1-D model as following:

Step 1. Establishing the “a Priori 1-D Model” based on the previous study

Step 2. Establishing the Geometry and the Velocity Intervals of Potential 1-D Model Step 3. Relocation and Final Selection of Events

Step 4. Calculation of Minimum 1-D Model

4.3 3-D Image Tomography

Traditionally, seismic arrival times are used to produce a velocity model, especially the arrival time data of body waves. Vp and Vs are modeled from P and S arrivals, respectively. Thurber (1993) proposed an approach for Vp/Vs ratio from ts-tp. In this study, computer algorithms (SIMUL2000) developed by Thurber (1983, 1993) and Eberhart-Phillips (1990) (documentation provided by Evans et al. (1994)).

4.3.1 Hypothesis and Inversion Method

The SIMUL2000 program is a damped-least-squares and full matrix inversion intended for using with natural local earthquakes, with or without shots and blasts. The iterative damped-least-squares method is applied to obtain a solution for the 3-D structures of Vp and Vp/Vs ratio, and simultaneously achieves new hypocenter solutions.

P arrivals and S-P times were used in this study. The conceptual approach parallels that of Aki and Lee (1976) and the velocity of the medium is parameterized by assigning at a large number of discrete points in three dimensions. The velocity at a given point is determined by interpolation among the surrounding grid points.

Usually, the problem is formulated as over-determined. So, the damped-least-squares solution to the linearized problem is obtained from

(

)

m = M M + L

M t

Among these parameters, the matrix M is constructed according to the parameter separation techniques of Pavlis and Booker (1980). After the equation is solved, the velocity parameter changes are applied to the model and the earthquakes are individually relocated in the new model. The simultaneous inversion is repeated iteratively and the F test (DeGroot, 1975) is used to select a stopping point for the procedure.

This method uses the pseudo-bending ray-tracing algorithm (Um and Thurber, 1987) to find the rays and calculates the travel times between the events and seismic stations.

The algorithm utilizes direct minimization of the travel time: an initial path estimate is perturbed by using a geometric interpretation of the ray equations, and the travel time along the path is minimized in a piecewise fashion. The perturbation is iteratively performed until the travel time converges.