Method for stabilization of low frequencies in tau-p domain filtering and deconvolution

1. A method for analyzing seismic data collected with one or more sensors at a plurality of sensor positions in a range along a dimension of a formation, the method comprising operating a processor to transform seismic data D into tau-p space, the seismic data D representing the formation, said transforming comprising:

for each frequency:

generating frequency dependent transformation matrices R that are configured to transform frequency-offset seismic data D into frequency-domain tau-p seismic data A according to A=(R + R) −1 R + D;

generating an estimate of the number of nonzero eigenvalues of the transformation matrix R + R, the number of nonzero eigenvalues being a function of frequency;

creating a transformation matrix X by combining a noise factor and the transformation matrix R + R, the noise factor being a function of the number of nonzero eigenvalues;

transforming the frequency-offset seismic data D into frequency-domain tau-p seismic data A according to A=(X) −1 R + D; and

outputting a tau-p seismogram indicative of frequency-domain tau-p seismic data A.

2. The method of claim 1 , wherein the frequency-domain tau-p seismic data A is multiplied by the square root of a derivative operator.

3. The method of claim 1 , wherein the seismic data D is collected at sensor positions located in a range along a dimension of the formation.

4. The method of claim 3 , wherein the sensor positions are located in a range along the surface of the formation.

5. The method of claim 3 , wherein the sensor positions are located in a range along the depth of the formation.

6. The method of claim 3 , wherein the number of nonzero eigenvalues is a function of the range of sensor positions.

7. The method of claim 1 , wherein the number of nonzero eigenvalues is a function of a p-domain sampling interval.

8. The method of claim 1 , wherein the noise factor is a function of the sum of the eigenvalues of transformation matrix R + R divided by the number of nonzero eigenvalues.

9. The method of claim 1 , wherein the noise factor is substantially proportional to the average of the nonzero eigenvalues of transformation matrix R + R.

10. An apparatus for analyzing seismic data collected with one or more sensors at a plurality of sensor positions in a range along a dimension of a formation, comprising:

a processor configured to transform seismic data D into tau-p space according to a transformation method, the transformation method comprising:

for each frequency:

generating frequency dependent transformation matrices R that are configured to transform the frequency-offset seismic data D into frequency-domain tau-p seismic data A according to A=(R + R) −1 R + D ;

generating an estimate of the number of nonzero eigenvalues of the transformation matrix R + R, the number of nonzero eigenvalues being a function of frequency;

creating a transformation matrix X by combining a noise factor and the transformation matrix R + R, the noise factor being a function of the number of nonzero eigenvalues;

transforming the frequency-offset seismic data D into frequency-domain tau-p seismic data A according to A=(X) −1 R + D; and

outputting a tau-p seismogram indicative of frequency-domain tau-p seismic data A.

11. The apparatus of claim 10 , wherein the number of nonzero eigenvalues is a function of the range of sensor positions.

12. The apparatus of claim 11 , wherein the number of nonzero eigenvalues is a function of a p-domain sampling interval.

13. The apparatus of claim 10 , wherein the noise factor is a function of the sum of the eigenvalues of transformation matrix R + R divided by the number of nonzero eigenvalues.

14. The apparatus of claim 10 , wherein the sensor positions are in a range along the surface of the formation.

15. The apparatus of claim 10 , wherein the sensor positions are in a range along the depth of the formation.