Several algorithms were developed to improve CMP stacking and enhance resolution
of stacked sections by reducing stretching effects.
Claerbout (1992) described inverse NMO stack, which recasts
NMO correction and stacking as an inversion process in the constant velocity case. This approach
combines conventional NMO and stack into one step by solving a set of simultaneous equations using
iterative least-squares optimization. Sun (1997) extended Claerbout's idea to the case of depth-variable
velocity. The inverse NMO stack operator applied depends on hyperbolic moveout relation and can
be employed to remove non-hyperbolic events and random noise. Trickett (2003) uses a variation of
Claerbout's inverse NMO stack in his stretch-free stacking method to avoid “NMO stretch".
Trickett's results tend to be higher frequency but noisier
than a conventional stack. Multiple other algorithms have been proposed that aim to reduce
NMO stretching effects (Byun and Nelan, 1997; Rupert and Chun, 1975; Zhang et al., 2013; Masoomzadeh et al., 2010; Perroud and Tygel, 2004; Hilterman and Schuyver, 2003; Hicks, 2001; Kazemi and Siahkoohi, 2011).
Wisecup (1998) introduced random sample interval imaging (RSI), which
maps the CMP gather into the “after NMO space” using the exact moveout times and no
interpolation. The NMO-corrected values are collected in the stack, rather than
summed, where the input sample values are mapped to their correct time values in the stack.
Shatilo and Aminzadeh (2000) proposed a constant NMO correction strategy, which applies a constant NMO shift
within a finite time interval that is equal to the wavelet length of a trace. This approach eliminates wavelet
stretch and preserves higher frequencies than the conventional method, resulting in a higher resolution stack. However,
samples that exist in overlapping time windows are used twice during the correction, resulting in
an amplitude distortion. Stark (2013) discussed the idea of signal recovery beyond the conventional Nyquist
frequency using an approach similar to the RSI
algorithm. The method proposed is an output-driven process,
where the stack is defined as a merge trace and has a potentially higher sampling rate than the input
traces. Using this approach, the final stacked sections are not necessarily limited to the data-collected Nyquist frequencies.
More recently, Ma et al. (2015) proposed a stacking technique based
on a sparse inversion algorithm that computes the stack directly from a CMP gather by solving an optimization
problem using principles of compressive sensing. This method eliminates the stretch effect of conventional
CMP stacking and improves resolution in the stacked section. Silva et al. (2015) introduced a recursive stacking approach using local slopes to compute a stack without stretching effects. In our previous work (Regimbal and Fomel, 2015), we proposed a method that
computed NMO and stack in an iterative fashion using shaping regularization to achieve a higher resolution
stack that avoids the effects of “NMO stretch".
In this paper, we extend the method of shaping NMO stack (Regimbal and Fomel, 2015) further by introducing recursive stacking using plane-wave construction (PWC) (Fomel and Guitton, 2006) in the backward operator of the shaping regularization scheme (Fomel, 2007). PWC stacking is equivalent to computing the zero scale of the seislet transform (Fomel and Liu, 2010). Shaping regularization implies a mapping of the input model to a space of acceptable models. The shaping operator is integrated in an iterative inversion algorithm and provides explicit control on the estimation result. We start by reviewing shaping regularization in the context of NMO and stack and define the operators used in recursive PWC stacking. We test this approach on synthetic examples to demonstrate the algorithm's ability to minimize stretching effects and improve resolution. We then apply this method to a 2-D field dataset from the North Sea and achieve noticeable resolution improvements in the stacked section in comparison with conventional NMO stack.