Wavelet Extraction from Virtual SSP->SSP Data



Figure 1. The original single shot profile.

Figure 2. Pseudo Shot Profile (an AGC is applied).

Objective: Learn to use 2D Interferometric Interpolation, and extract source wavelet. Study advantages and disadvantages of this method.

Theory: The frequency-domain implementation of the SSP->SSP transform uses the equation

Im(G(A|B)) = k int G(A|x)G(x|B)* dx ,

yet in practice we use bandlimited data D(A|x)=W(f) G(A|x) for the crosscorrelation rather than the Green's function G(A|x), where W(f) is the source wavelet. That is,

Im(GG(A|B)) = |W(f)|2 Im(G(A|B)) = k |W(f)|2 int G(A|x)G(xB)* dx

We can easily extract the actual GG(A|B) from Im(GG(A|B)) (e.g., find time domain version of Im(GG(A|B)), eliminate acausal part for t less than 0, and then Fourier transform result to get GG(A|B).), so that the following is true:

GG(A|B)/D(A|B) = |W(f)|2 G(A|B)/[W(f)G(A|B)] = W(f)*.

In summary, the conjugate of the wavelet spectrum W(f)* can be extracted from SSP data by generating virtual SSP data and then dividing it by the actual data in the frequency domain.

Don't forget to put in a damping parameter e in the denominator to protect yourself against singularities, i.e.,

GG(A|B)/D(A|B) = GG(A|B)D(A|B)*/[|D(A|B)|2 + e]

Skill Learned: Reinforce principles of SSP-> SSP transform, exercise skills in spectral deconvolution, and estimate degradation of wavelet extraction with respect to degradation of aperture and trace spacing.

Procedure:

  1. Make a directory, and load the files: Sigsbee2B_main.m, DirectWaveMute.m, xcorr.m, ProcessMatchedData.m, wigb.m, VirtualShotGenerator.m, and agc.m. A Sigsbee2B synthetic data set is used to test the codes. Use command "unzip data.zip" to extract the data. The data set is very big (300M), UTAM users can read the data from the folder '/uufs/geophys.utah.edu/common/tomofs/www/htdocs/book/interpolate' without downloading it. The csg_fs319.su means the data contains free surface related multiples and 319 is the shot number.
  2. We use the 319th shot gather as the original data and remove the first 20 traces of this shot as the nearoffset missing data.
  3. Type " Sigsbee2B_main " in Matlab to run the program for interpolation using all the shot gathers. And we will get the rough interpolation result, which contain the amplitude and wavelet misfits. We need do some extra work on it to correct the misfits.
  4. Now that you have the virtual and actual SSP data, extract the wavelet by the wavelet extraction procedure. The actual wavelet is a Ricker wavelet. Compare your actual wavelet against estimated wavelet. Which is better, an average wavelet estimated by averaging many extracted wavelets or a estimated single wavelet?
  5. Test the sensitivity of the extracted wavelet to the degradation in trace sampling interval and reduced number of shot gathers. Does the damping term help stabilize the inversion?
  6. Repeat steps 3 and 4 except now use as your input data the Sigsbee data convolved with a ringy source wavelet. You should get as an extracted wavelet the ringy wavelet convolved with the Ricker wavelet.

Questions:

  1. Will the source or receiver radiation pattern affect the shape or amplitude of the source wavelet at far offset traces compared to near offset traces? See Fig. 4.2 in Aki and Richards, 1980.
  2. Will attenuation affect the shape or amplitude of the source wavelet at far offset traces compared to near offset traces? If so, what would your strategy be to find the wavelet for far-offset trace and the wavelet for a near-offset trace?
  3. Is the SSP-SSP transform valid for attenuative media?
  4. What other factors affect the accuracy of the extracted wavelet?