BAND-LIMITED GREEN'S FUNCTIONS

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BAND-LIMITED GREEN'S FUNCTIONS

A simple way of estimating a traveltime that is valid in the seismic frequency band is to compute the full wavefield and then pick the maximum energy arrival at each location. However this method is ridiculously expensive. If the modeling is performed in the frequency domain, it requires a finite-difference solution to the wave equation for every frequency in the data. In contrast a fast, explicit, eikonal solver [(van Trier and Symes,1991)] requires only one finite-difference calculation.

My method is based on a scheme for reducing the number of frequencies that are extrapolated, whilst still retaining the ability to pick the true maximum energy event. I still model the full frequency band but I reduce the number of frequencies by increasing the the sampling in frequency. When transformed back to the time domain the wavelet shapes and traveltimes of the true wavefield are not lost. The wavefield is merely replicated a regular intervals, it is aliased in time. If the aliases do not overlap, and the approximate position of the true alias is known then it can be uniquely retrieved.

In order to track the correct alias I perform the calculation in a polar coordinate frame. If I know the correct traveltime at one radius I can predict the position of the correct alias of the wavefield at the next radius. Given that knowledge, I can pick the correct maximum-energy traveltime. By extrapolating both the traveltime field, and the wavefield, outwards from the origin I am able to overcome the problems caused by aliasing. Figure 1 shows the wavefield at one radius for a medium with a circular velocity anomaly. The top left frame used all 64 frequencies, the top right frame used 32, the bottom left used 8 and the bottom right used 4. It is clear that if we know which is the true alias, it can be separated from the others in all the plots except the one created using 4 frequencies.

In my algorithm a small number of frequencies (8-16) in the seismic frequency band are extrapolated outwards from the source location using a paraxial one-wave equation in polar coordinates. The traveltime and wavefield are both known at the origin and they are extrapolated outwards to fill the whole space. At each radius the wavefield is parameterized by a traveltime/amplitude/phase triplet. The traveltime is chosen to correspond to the traveltime of the maximum energy event at each location.

Figure 1: [*] [*][IMAGE tex2html_wrap]

The algorithm at each radius is as follows.

1) Calculate the wavefield at the new radius for the sparsely sampled set of frequencies.

2) Choose a time window centered around the traveltime from previous radius.

3) Calculate a sampled time domain representation in the window by slow Fourier transform.

4) Pick the maximum energy sample.

5) Use a quadratic fit to find the traveltime of the local peak of the energy function.

6) Calculate the amplitude, and phase at this traveltime.

The cost of this algorithm is surprisingly modest. In a constant velocity model it costs about 8 times as much as an explicit finite-difference solution to the eikonal equation. In a complex model the cost of the two algorithms is about the same. In the complex velocity model the explicit eikonal solver must use very small radial steps to remain stable. The band-limited Green's function calculation is based on a stable wavefield extrapolator so the grid, and hence the cost, is independent of model complexity.

The band-limited Green's functions have several desirable properties:

1) They can be calculated in any slowness model, there is no smoothness constraint.

2) The solution is found at every point in the subsurface (no shadow zones).

3) The maximum energy arrival is found rather than the first arrival.

4) The solution is an estimate of the Green's function in the seismic frequency band not the solution at very high frequency.

5) Traveltime, amplitude and phase are calculated.

It also has some limitations:

1) The traveltime field is discontinuous. This makes it harder to interpolate.

2) No explicit rays or takeoff angles are calculated. They must be inferred from the traveltime gradients.


[<-] [ ^ ] [->] [Previous Page] [Next Page]
Previous: Introduction
Up: Maximum energy traveltimes calculated in the seismic frequecy band
Next: TRAVELTIMES IN THE MARMOUSI MODEL
Previous Page: Introduction
Next Page: TRAVELTIMES IN THE MARMOUSI MODEL

© 1994 Copyright by Dave Nichols

dave@sep.Stanford.EDU
Fri Feb 3 01:19:43 PST 1995