## 2 Data-Driven Strategies for Stacking Parameter Estimation
### Hybrid Global Diffraction/Local Reflection Search  
The N-wave is related to the so-called exploding reflector experiment. It emanates at the reflector with the curvature of a circular reflector element surrounding the NIP and propagates to the surface, where it reaches the ZO location \(x_{0}\) with the radius of curvature \(R_{N}\). The NIP-wave instead, starts from a point source at the NIP and reaches \(x_{0}\) with the radius of curvature \(R_{NIP}\). From these definitions, it follows that for a diffraction point \(R_{N}\) is equal to \(R_{NIP}\). Therefore, substituting \(R_{N}\) with \(R_{NIP}\) transforms Equation (2) into the CRS diffraction operator ([19]), which reads  
\[t^{2}(\Delta x_{m},h)=\bigg{[}t_{0}+\frac{2\sin\alpha}{v_{0}}\Delta x_{m} \bigg{]}^{2}+\frac{2t_{0}\cos^{2}\alpha}{v_{0}}\bigg{[}\frac{\Delta x_{m}^{2} +h^{2}}{R_{NIP}}\bigg{]}, \tag{4}\]  
with \(\Delta x_{m}=x_{m}-x_{0}\).  
Regarding global parameter search algorithms that employ a spatial operator, the computationally most efficient is the data-driven Common Diffraction Surface (CDS) stack developed by [42]. The CDS stack method utilizes the two-parameter diffraction operator given by Equation (4), but does not involve a search for \(\alpha\). Instead, the optimum \(R_{NIP}(\alpha)\) is determined for each \(\alpha\) value on a predefined grid, before stacking in a DMO-like manner over all operators defined by the resulting \(\alpha\), \(R_{NIP}(\alpha)\) combinations, thereby reducing computational effort and resolving the conflicting dip problem. In [42], \(R_{NIP}\) is renamed \(R_{CDS}\) because, when fitting the diffraction operator to a reflection event, the resulting value for \(R_{CDS}\) has not the exact physical meaning of \(R_{NIP}\), but represents a more generic stacking parameter.  
Moving a step closer to the global \(1\times 3\) parameter optimization, ref. [43] introduced a global \(2+1\) parameter search method. It begins with a two-parameter search using the CRS diffraction operator, followed by a one-parameter search utilizing the full CRS reflection operator while maintaining the two previously determined parameters constant.  
As a middle ground between the global simultaneous three-parameter search and the pragmatic search strategy, we propose a global simultaneous diffraction search as presented by [43], followed by a local three-parameter optimization that starts with as initial value for \(R_{N}\). In other words, the hybrid optimization algorithm that we propose in this paper consists of a global search assuming a diffraction, followed by a local search assuming a reflection. This makes sense considering that an arbitrarily located diffraction point and an arbitrarily located and oriented circular reflector segment are just two different levels of approximation of an unknown arbitrarily located, shaped and oriented subsurface reflector. Using \(R_{N}=R_{NIP}\) as initial guess, is well justified by the NIP wave theorem [44], which states that the reflection traveltimes in the CMP gather are, up to the second order in half-offset \(h\), equal to the diffraction traveltimes, which correspond to a diffractor at the normal incidence point (NIP) of the associated normal incidence ray. Furthermore, a sensibility analysis conducted by [45], depicted in Figure 4, shows that the CRS traveltime depends much more on the values of \(\alpha\) and \(v_{NMO}(\alpha,\,R_{NIP})\) than on the value of the reflector curvature \(R_{N}\).  
Calculating the computational effort required for the global optimization, with the same assumptions that we made in the previous section, we have 100 coherence calculations for the simultaneous two-parameter search and end up with a computational effort that is about 16.6 times bigger than the effort of the pragmatic search strategy (see Table 1). In return, we obtain for the velocity analysis all benefits provided by a spatial operator that covers several CMP gathers.