## 2 Data-Driven Strategies for Stacking Parameter Estimation
### Pragmatic Search Strategy Plus Event-Consistent Smoothing  
Around the year 2000, when the CRS stacking method was in its infancy, it was unfeasible to carry out for every sample of the ZO section a global simultaneous optimization of the full three-parameter reflection operator given by Equation (2). To overcome this impediment, ref. [12] developed and later also implemented a pragmatic search strategy. This cascaded strategy utilizes a sequential approach involving three global one-parameter line searches: the initial one conducted on individual CMP gathers, followed by two additional searches performed on a preliminary stack section generated as an outcome of the first search. Hoping that, by doing so, the proximity of the global coherence maximum is reached, the parameter search is concluded by a local three-parameter optimization utilizing the flexible polyhedron (Simplex) search by [27]. As a result, only local parameter optimization and stacking was carried out using the spatial operator.  
It is evident that the success of this cascaded approach depends very much on the first step, called automatic CMP stack, where a preliminary ZO section is generated. However, the automatic CMP stack risks failing in cases of low signal-to-noise ratio and insufficient CMP fold; a risk especially high for small travel times where the range of usable offsets is very limited. Stacking along the NMO operator given by Equation (1), with only a few traces and unreliable NMO velocities, can create artificial gaps in the ZO image of reflection events or even result in a failure to image them at all.  
The problems with outliers and fluctuations in the parameter values came fully into focus when a velocity model inversion method, called NIP wave tomography, was developed by [19], utilizing picked values of \(\alpha\) and \(R_{NIP}\). To compensate for suboptimal parameters in limited areas of certain CMPs, ref. [28] introduced so-called event-consistent smoothing of stacking parameters, a post processing algorithm that takes advantage of the fact that, due to the predominantly layered structure of the subsurface, a smooth behavior of interfaces, and consequently also of stacking parameters, can be expected. [PERSON], European Space Agency astronaut and commander of the International Space Stationin 2018, implemented a first version of this approach as a student assistant, while studying geophysics at the University of Karlsruhe, Germany.  
The smoothing algorithm is based on the combined application of mean and median filters within a rectangular box aligned with the underlying reflection event. The information on the slopes of events in the time domain is derived from the CRS parameters and allows avoidance of the mixing of intersecting events. As a result, the algorithm works in an event-consistent manner. It most often provides, even in case of conflicting dip situations, significantly improved parameter values. Any harm caused by the smoothing can be remediated by the subsequent application of local simultaneous three-parameter optimization. Typically, several iterations of smoothing and local optimization are carried out to obtain the best possible results.  
The method was successfully employed in [29; 30; 31], and in various other examples. It proved to be especially beneficial in the case of complex near-surface conditions, which often lead to a strongly variable data quality [32]. In general, it can be concluded that, for hydrocarbon exploration (deep targets and large CMP fold), the shortcomings of the pragmatic search plus event-consistent smoothing are not very significant compared to the efficiency benefit obtained with respect to a global simultaneous search. However, for near-surface data, the same logic works but in the opposite direction: if the number of traces per CMP is low, the simultaneous search becomes easily affordable and highly beneficial for a stable and reliable estimation of the stacking parameter, particularly for complex structures and data with low signal-to-noise ratio.