###### Abstract
It plays the role of a constant of proportionality, allowing to parametrize a hyperbolic traveltime surface by three independent kinematic properties of the wavefield measured in \(x_{0}\): \(\alpha\), the emergence angle of the central ZO ray and \(R_{NIP}\) and \(R_{N}\), the wavefront radii of two notional eigenwaves known as normal incident point (NIP) wave and normal (N) wave, as illustrated in Figure 3.  
Figure 2: (**a**) CRS stacking surface in the midpoint-offset domain, displayed above the corresponding 2D velocity medium composed of two constant velocity layers, separated by a dome shaped reflector. The gray curves are the forward modeled common-offset traveltimes for this interface. The stacking surface is depicted in red and spans over an entire collection of CMP gathers, a so-called CRS super-gather. All amplitudes summed along the red surface are assigned to the point \(P_{0}=(x_{0},t_{0})\), where \(x_{0}\) is the coincident source and receiver coordinate and \(t_{0}\) is the traveltime of the central ZO ray, depicted as a straight blue line (Figure modified from [14]). (**b**) A collection of neighboring CMP gathers from near-surface seismic data, showing the continuation of reflection events in both midpoint and offset direction. These so-called CRS parameters represent the generalized counterpart to the NMO velocity which can be expressed as  
\[v_{NMO}=\sqrt{\frac{2v_{0}R_{NIP}}{t_{0}\cos^{2}\alpha}}. \tag{3}\]  
Looking closer at how CRS stacking is actually implemented, we observe a basic difference to the conventional NMO/DMO correction plus CMP stack. Following the paradigm of data-driven imaging, an automated search for the optimum stacking parameter triple \((\alpha,\ R_{NIP},\ R_{N})\) is performed for every sample of the output section instead of stacking with NMO velocities extrapolated and interpolated from a limited number of visually detectable reflection events, as portrayed in Figure 1. In other words, for every ZO sample \(P_{0}=(x_{0},t_{0})\), those values for \(\alpha\), \(R_{NIP}\), and \(R_{N}\) are searched for which parameterize the CRS traveltime surface, given by Equation (2), in such a way that it fits best to a measured reflection in the data. If \(P_{0}\) does not lie on a reflection event, the attribute triple with the maximum coherence will nevertheless achieve only a relatively low coherence value and stacking will not result in a constructive summation of amplitudes. A positive side-effect of treating every sample of the reflection event independently is that the so-called NMO-stretch (see Figure 1c, event four), which deteriorates the resulting stack section, is avoided [17]. This is because, unlike the conventional NMO velocity model shown in Figure 1b, the NMO velocity calculated from the CRS stacking parameters is generally neither constant for the entire duration of a reflection event nor does it increase due to velocity extra- or interpolation. Instead, it generally decreases with time from the top to the bottom of the event [16]. Why do we need NMO velocities that decrease over the wavelet to avoid the NMO stretch? Our hypothesis is that this results from the slightly unphysical way traveltimes are measured, according to which the last sample of the wavelet was not only recorded later but also has a larger traveltime than the first sample and thus has apparently traveled with a lower speed. Consequently, the NMO velocity must decrease towards the tail end of reflection events, as the apparent traveltime increases while the depth of the reflecting interface remains constant. This can also be seen from Equation (3), assuming a homogeneous layer with velocity \(v_{0}\) over a horizontal interface. In this case, the ZO ray emerges vertically, i.e., \(\alpha=0\). The NMO velocity is equal to \(v_{0}\) if, and only if, \(t_{0}\) is the exact time between the firing of the shot and the arrival of the head of the wavelet at the surface; only then \(\frac{2R_{NIP}}{t_{0}}=v_{0}\) holds.