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# Methodology

The first stage in generating the migration algorithm is to develop an analytic expression for the propagation of source wavefields through the earth. To do this, two approximations are employed. First, I assume that earthquake wavefronts at teleseismic distances (between 30and 103 in epicentral distance from source) can be approximated by plane waves. This approximation holds because at these distances and array lengths there is little curvature in the earthquake wavefront. The second approximation is that the source propagates through a vertically-stratified, v(z) medium. This approximation is consistent with the notion that earth structure `visible' at teleseismic frequencies (i.e. between 0.005 and 4 Hz) is predominately v(z). Accordingly, the source wavefront is parameterized with constant horizontal ray parameters in a Cartesian system. This enables an analytic calculation of the planar source wavefield at all times. When discussing the source wavefield henceforth, I am referring to its analytic expression.

To enable the use of the survey-sinking algorithm, teleseismic data need to be reconfigured to an approximate equivalent of zero-offset geometry. One way that this can be achieved is to exploit the fact that planar teleseismic wavefields sweep across recording arrays in a linear fashion. This leads to a LMO of first arrivals in the recorded teleseismic section, with the degree of moveout being dependent on the incident wave's orientation to the strike of the array, as shown in Figure .

 Geometry Figure 1 Diagram of a simple teleseismic scattering model. a) Earth model consisting of a planar discontinuity and point scatterer. b) P-S forward-scattering expected for a source parallel to receiver array-axis. Conversions are parallel to the source wavefront, and the diffraction is symmetric about the scattering point. c) As in b) but with source located perpendicular to receiver array-axis.

Thus, as an initial approximation, I apply an adjoint LMO operator to remove the LMO recorded in the data. (The processing flow is depicted in Figure ).

In this new reference frame, zero time is defined by the arrival of the teleseismic source at each station. Since all arrivals now occur at zero-time, this may be considered as a zero-offset experiment. Note that reflections and mode conversions (i.e. P- to S-wave) from horizontal discontinuities are flattened by this transformation. One drawback, though, is that diffracted hyperbolas symmetric in the initial reference frame are now skewed to one side of the hyperbola apex. The degree to which they are skewed is dependent on the angle of the applied adjoint LMO (see Figure ). While this may not be an optimal physical representation, two observations support the use of this approximation. First, coherent events observed in the receiver wavefield predominately consist of planar reflected and converted events. These types of scattered energy are properly migrated by a zero-offset wave-equation method. Second, little-to-no LMO shift is applied to plane waves arriving from directions nearly or exactly coincident with the strike-axis of the recording array. In these cases planar events (plus diffractions if present) are properly migrated by a zero-offset algorithm.

 Flow Figure 2 Processing Flow.

Next: Double square root equations Up: Shragge: Teleseismic PSM Previous: Shragge: Teleseismic PSM
Stanford Exploration Project
7/8/2003