In geophysical estimation problems, regularization is
used to solve ill-posed problems by providing additional constraints on the estimated model.
Shaping regularization (Fomel, 2008,2007) implies a mapping of the input model
to the space of acceptable functions. The mapping is controlled by the shaping operator
. In the linear case, the solution of the estimation problem using shaping regularization is defined as:
Shaping regularization in the application to NMO stack utilizes signal from different offsets
to reconstruct a high resolution stack. The model
is in this case a seismic trace at zero-offset and the
data
is a CMP gather. We define the linear operators used in the shaping regularization
scheme as:
We implement PWC stacking in the backward operator of shaping regularization which follows local slopes of a CMP gather. The key idea of this stacking procedure is to start at the farthest offset trace of the gather and make a local slope prediction of the preceding trace using PWC (Figure 1a). The partially corrected trace is then stacked with the uncorrected neighbor, which is the input for the next local prediction (Figure 1b). The process is repeated in the offset direction until the zero offset trace is reconstructed (Figure 1c). This recursive stacking approach results in higher resolution stacks compared to conventional NMO and stack. The procedure is equivalent to computing the zero scale of the seislet transform (Fomel and Liu, 2010). Advantages of PWC stacking include eliminating the effects of “NMO stretch" as well as the problem of non-hyperbolic moveout. The approximate inverse of PWC stacking is defined by predictive painting (Fomel, 2010). This algorithm is comprised of two main steps, namely estimating local slopes of seismic events using plane-wave destruction (PWD) (Fomel, 2002) and spreading information from a seed trace inside a volume. In this application, we use the updated model to spread information across the CMP gather using the estimated dip field.
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Figure 1. Schematic of the PWC stacking algorithm. (a) Stack far offset trace T ![]() ![]() ![]() ![]() |
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In PWC stacking, each seismic trace is predicted from its neighbors that are shifted
along the event slopes. Slopes are estimated by PWD, which minimizes the prediction error to estimate optimal slopes.
PWD can be sensitive to conflicting slopes at far offsets of a CMP gather when the dip is large and
cause PWC stacking to fail in characterizing an optimal stack.
To account for this, we first apply a constant velocity NMO correction to the CMP gather, which results in
smoothly varying slopes without crossing events.
We then estimate the moveout of the corrected seismic events at offset
as follows: