Introduction

In seismic data processing, common midpoint (CMP) stacking is one of the most fundamental processes that combines NMO-corrected traces across a CMP gather to produce a single trace with a higher signal-to-noise ratio (Rashed, 2014). Many problems arise with the assumptions and principles that set the foundation for conventional CMP stacking. Traditional stacking assumes that the NMO-corrected gather has perfectly aligned seismic reflections (Yilmaz, 2001). However, NMO correction is an approximation that assumes the travel-time as a function of offset follows a hyperbolic trajectory in a CMP gather, which may fail in common geologic settings that involve velocity variations or anisotropy. NMO correction also causes undesirable distortions of signals on a seismic trace known as “NMO stretch”, which lowers the frequency content of the corrected reflection event at far offsets (Claerbout, 1985). This violates the assumption of a uniform distribution of phase and frequency of seismic reflections across the corrected gather. Common procedures to eliminate this stretching effect involve muting samples with severe distortions. This causes a decrease in fold and may destroy useful far-offset information essential for amplitude variation with offset (AVO) analysis (Swan, 1988). Inaccuracy in stretch muting with residual “stretching” effects may also produce a lower-amplitude and lower-resolution stack (Miller, 1992).

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$^2$), 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$^2$ 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.