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We have presented a new technique to invert for reflectivity
models while properly handling the irregular sampling of seismic
data. The technique is based on the method of least
squares and consists of a two-step solution for imaging.
The data is equalized in a first stage with an inverse
filter and an imaging operator is then
applied to the preconditioned data to invert for a model.
The equalization filter corrects the imaging operator for the interdependencies
between data parameters. Each element of the filter is
a mapping between two data elements. It reconstructs a data trace
with given input geometry
at the geometry of the other data element.
The filter is therefore a symmetric AMO matrix with diagonal
elements being the identity
and the off-diagonal elements being the trace to trace AMO transforms.
We tested the effectiveness of the method in the 2D case for the application of
partial stacking by offset continuation.
The equalization step followed by imaging
has proved to correct and equalize the processing for
the effects of fold variations by properly handling the amplitude and phase
of the data.
**input24
**

Figure 1 Irregularly sampled constant offset sections with 600 and 800m offsets.

**input68
**

Figure 2 Irregularly sampled constant offset sections with 1000 and 1200m offsets

**amo-600
**

Figure 3 Offset continuation of irregularly sampled constant-offset section from 600m to 850m.

**amo-1200
**

Figure 4 Offset continuation of irregularly sampled constant-offset section from 1200m to 850m.

**fold
**

Figure 5 Fold distribution in the input data. The maximum fold is 7 and minimum is 0.

**nmo-stack
**

Figure 6 Partial stack after NMO and normalization by the CMP fold in each bin.

**amo-stack
**

Figure 7 Partial stack after AMO to a common offset of 850m.

**equal-stack
**

Figure 8 Output of the two-step inversion at an effective offset of 850 meters.

**amo-stack-win
**

Figure 9 Window on the dipping events from the amo-stack result.

**equal-stack-win
**

Figure 10 Window on the dipping events from the two-step inversion.

**amp-amo
**

Figure 11 Peak amplitudes on the flat reflector from the amo-stack result.

**amp-norm
**

Figure 12 Peak amplitudes on the flat reflector from the two-step solution.

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** Up:** Chemingui and Biondi: Handling
** Previous:** Synthetic results
Stanford Exploration Project

11/11/1997