Two-parameters residual-moveout analysis for wave-equation migration velocity analysis |

As a quality measurements of the gradient information, I compute the correlation across the angle axis between the RMO function that would be computed by picking the maxima of the coherency spectra and the RMO function computed using the gradient.

For the ``Taylor'' RMO the reference RMO function is computed as follows:

where are the coordinates of the power-spectrum maximum. The RMO function computed from the gradient of the power spectrum is,

and the correlation is computed as

I compare this correlation function over a range of with the correlation function computed by

where the one-parameter RMO is computed as follows

Figure 5 compares the correlation functions (panel a) and (panel b) for the first CIG analyzed (Figure 1a). The asterisk superimposed onto the plots of the correlation functions is located at the maximum of the power spectrum displayed in Figure 2a. The coordinates of this maximum are used to evaluate the moveout according to equation 4. Accurate gradient directions correspond to positive correlation (plotted in white in the figure), whereas potentially misleading gradient directions correspond to negative correlation (plotted in black in the figure).

The correlation functions are mostly positive over a wide range of parameters , indicating that a velocity estimation method based on these RMO functions is likely to have good global convergence properties. In particular, the positive correlation functions at indicates that the gradient computed starting from the migrated CIG shown in Figure 2a would be accurate, even if this CIG is far from being flat.

The correlation functions shown in Figure 5 are very similar. Therefore, the global convergence of the velocity estimation would be robust independently of whether the one-parameter or the two-parameter RMO function is used.

CorrShift-TP
Correlation functions
corresponding to the CIG shown in
Figure 1a
for:
a) the ``Taylor'' two-parameter RMO function (equation 6),
and
b) the one-parameter RMO function (equation 7).
Figure 5. |
---|

CorrShift-OP
Correlation functions
corresponding to the CIG shown in
Figure 1a
for:
a) the ``Orthogonal'' two-parameter RMO function (equation 11),
and
b) the one-parameter RMO function (equation 12).
Figure 6. |
---|

Similar correlation analysis of the RMO function can be performed when applying the ``Orthogonal'' RMO instead of the ``Taylor'' RMO. In this case the reference RMO function is computed as follows:

where are the coordinates of the corresponding power-spectrum maximum. The RMO function computed from the gradient of the power spectrum is,

and the correlation is computed as

This correlation is compared with the correlation

where the one-parameter RMO is computed as follows

Figure 6 compares the correlation functions (panel a) and (panel b) for the first CIG analyzed (Figure 1a). The asterisk superimposed onto the plots of the correlation functions is located at the maximum of the power spectrum displayed in Figure 2b. The coordinates of this maximum are used to evaluate the moveout according to equation 9. As for the previous figure, accurate gradient directions correspond to positive correlation (plotted in white in the figure), whereas potentially misleading gradient directions correspond to negative correlation (plotted in black in the figure).

In this case the correlation functions shown in Figure 6 are not as similar as in the previous case. In particular, the black area around the value in Figure 6a indicate that the two-parameter RMO analysis would provide unreliable gradients. This problem is related to the diagonal artifacts visible in the power spectrum shown in in Figure 2a. These artifacts are caused by the fact that the second term in the ``Orthogonal'' RMO function has an extremum in the middle of the angular range, in contrast with the other RMO functions that have an extremum at normal-incidence. This mid-range extremum causes spurious local maxima of the spectrum at depths different than the normal incidence depth of the imaged reflector. These artifacts are much weaker when I averaged the power spectrum over a thinner depth interval (30 m) than the one used for computing the function displayed in Figure 6a. The new averaging window is of thickness comparable to the image of the reflector. Figure 7a shows the power spectrum obtained with this thinner averaging window, and Figure 7b corresponds to the two-parameters correlation function, which is a substantial improvement with respect to the one shown in Figure 6a.

CorrShift-OP-narrow
Panel a):
Two-parameter stack-power spectra resulting from RMO analysis
of the CIG shown in
Figure 1a
obtained using a thinner averaging window (30 m) than the one
used to compute the spectrum shown in
Figure 2b.
Panel b):
Correlation function
for the ``Orthogonal'' two-parameter RMO function
obtained using the thinner averaging window.
Figure 7. |
---|

CorrShift-TP-aniso
Correlation functions
corresponding to the CIG shown in
Figure 3a
for:
a) the ``Taylor'' two-parameter RMO function (equation 6),
and
b) the one-parameter RMO function (equation 7).
Figure 8. |
---|

CorrShift-OP-aniso
Correlation functions
corresponding to the CIG shown in
Figure 3a
for:
a) the ``Orthogonal'' two-parameter RMO function (equation 11),
and
b) the one-parameter RMO function (equation 12).
Figure 9. |
---|

Two-parameters residual-moveout analysis for wave-equation migration velocity analysis |

2011-09-13