###### Abstract
The increase in traveltime with offset is exactly hyperbolic and is described by the well-known NMO formula [3], that reads  
\[t^{2}(h)=t_{0}^{2}+\frac{h^{2}}{v_{NMO}^{2}}, \tag{1}\]  
with \(h\) being the half-offset, i.e., half of the offset, \(t_{0}\) being the ZO traveltime and \(v_{\text{NMO}}\) being the NMO velocity. If we correct the traveltimes respectively, we obtain a flat NMO corrected event that we can stack in offset direction. Using an extra- and interpolated continuous velocity model, this process is applied for all traveltimes of the stacked ZO trace to be constructed. Figure 1 shows the different stages of NMO velocity analysis and CMP stacking, using an example from ground penetrating radar. In Figure 1a, a CMP gather with six distinct reflection events is displayed, overlain by those hyperbolas that provide the closest fit to these reflections. The latter were determined by coherence analysis, whereas semblance [4] was used to measure the coherence of the data along various candidate hyperbolas. During this process, the semblance spectrum depicted in Figure 1b, was obtained, where subsequently the best fitting velocities were picked for the six reflections and extrapolated and interpolated to a continuous NMO velocity model. Figure 1c, shows the CMP gather after NMO correction. Stacking this gather in offset direction and placing the stack results at the traveltimes which the velocity model predicts for zero-offset produces the stacked ZO trace depicted in Figure 1d. To better cope with dipping reflectors, Dip-Moveout Correction (DMO) (see [5] and references given therein) was later added to CMP stacking. NMO correction with subsequent CMP stacking was often reviewed, updated, and extended (see, e.g., [7; 8; 9; 10]). The Common Reflection Surface (CRS) stack method (see [11; 12; 13] and references given therein) constitutes one of the latest and probably the most successful generalizations of this technique. The CRS method extends CMP stacking towards the off-CMP dimension by adding a relative midpoint coordinate to the traveltime  
Figure 1: **(a)** A CMP gather with six different reflections and their hyperbolic approximations (red lines); **(b)** the semblance spectrum, where the coherence of the data along hyperbolas associated with a variety of NMO velocities is depicted over time and best fitting velocities were picked for the six reflections and extrapolated and interpolated to a continuous velocity model (black line); **(c)** the CMP gather after NMO correction using this model; **(d)** the stacked trace. Figure taken from [6]. approximation. It provides a second-order approximation of the two-way traveltime of rays, reflected on a common reflection surface (3D seismic) or common reflection segment (2D seismic), respectively. The latter is depicted in Figure 2a as the red area around the normal incidence point R of the central ZO ray (straight blue line), which is defined by its emergence location \(x_{0}\) and traveltime \(t_{0}\). The same situation, but with inverted time axis, can be found in Figure 2b, where a collection of neighboring CMP gathers from near-surface seismic data is depicted side-by-side, in order to illustrate the continuation of reflection events in both midpoint and offset direction. For 2D acquisition on a plane surface without topography overlying an inhomogeneous but isotropic medium, according to [15], the hyperbolic CRS stacking operator in midpoint displacement, \(\Delta x_{m}\), and half-offset, \(h\), coordinates can be written as  
\[t^{2}(\Delta x_{m},h)=\left[t_{0}+\frac{2\sin\alpha}{v_{0}}\Delta x_{m}\right] ^{2}+\frac{2t_{0}\cos^{2}\alpha}{v_{0}}\bigg{[}\frac{\Delta x_{m}^{2}}{R_{N}}+ \frac{h^{2}}{R_{NIP}}\bigg{]}, \tag{2}\]  
with \(\Delta x_{m}=x_{m}-x_{0}\). The near-surface velocity in the vicinity of \(x_{0}\), \(v_{0}\), is assumed to be constant and a priori known.