*2.1. Velocity Spectrum*

The velocity spectrum [15] is a 2D image generated by scanning and stacking (or correlating) energy along different velocities on each CDP gather using the similarity criterion. The x and y axes are the velocity and two-way travel time, respectively. The velocity spectrum is composed of a series of energy clusters, and the velocity information is determined according to the abscissa position of the energy clusters. Most of the current research on velocity spectrum focuses on improving the resolution of velocity spectrum and then improving the accuracy of velocity pickup [23–29]. The application of deep learning in velocity analysis almost focuses on the automatic pickup of energy clusters in velocity spectrum [30–32].

In addition to the position and velocity information that can be used for velocity analysis in conventional seismic processing, velocity spectrum is also a 2D image that can reflect formation characteristics. Velocity spectra also have advantages in identifying lateral characteristics of formation. The reasons are as follows: firstly, the interference of complex underground structures and noise always leads to incomplete, staggered, and amplitude varying hyperbolas, while the velocity spectrum scans and superimposes the events along the hyperbola, which can attenuate the non-hyperbolic noise to a certain extent. Secondly, in most cases, the energy clusters in velocity spectrum corresponds to the interfaces. Because the formation interfaces have relatively coherent structure in the underground space, the velocity spectra of adjacent CDP points have similarity, and the characteristics of the formation interface can be obtained from the velocity spectra by this similarity. The velocity spectrum Formula (1) is as follows:

$$S = \frac{\sum\_{j=0}^{M} \left(\sum\_{i=1}^{N} A\_{i,j}\right)^2}{I \sum\_{j=0}^{M} \sum\_{i=1}^{N} \left(A\_{i,j}\right)^2} \tag{1}$$

where *M* is the length of the time window, *N* is the offset length, *I* is the number of seismic traces in the CDP gather, and *Ai*,*<sup>j</sup>* is the amplitude at offset *i* and time *j*. According to Formula (1), if all seismic traces are the same, *S* equals to 1. If each seismic trace is a random value, *S* approaches 0. Only when the scanning velocity is equal to the normal move-out (NMO) velocity, the waveforms of each trace are the most similar, in-phase stacking is realized within the time window, and *S* is close to 1. Here, we use real seismic data after conventional data processing and prestack migration to show the consistency between energy clusters in velocity spectrum and formation interfaces. Figure 1a is the common reflection point (CRP) gather at the location of CDP number 30 (CDP 30), which has undergone NMO removal processing with *v* = 1500 m/s. Figure 1b is the velocity spectrum generated from the gather shown in Figure 1a. The x and y axes are the velocity and two-way travel time, respectively. The center of the energy clusters in velocity spectrum corresponds to the maximum value of stacking energy. Figure 1c is the stack section. It can be seen that the centers of the energy clusters correspond to the events on the seismic section. Each energy cluster corresponds to an obvious formation interface. Figure 2a–c are velocity spectra of CDP 32, CDP 34, and CDP 36, respectively. The three velocity spectra have high similarity. With the similarity of the energy clusters in lateral, the positions of the formation interfaces can be obtained by tracking the energy clusters.

**Figure 1.** Consistency between energy clusters in velocity spectrum and formation interfaces. (**a**) CDP gather at CDP 30; (**b**) velocity spectrum generated from the gather shown in (**a**); (**c**) stack section.

**Figure 2.** Similarity of adjacent velocity spectra. (**a**) Velocity spectrum of CD P32; (**b**) velocity spectrum of CDP 34; (**c**) velocity spectrum of CDP 36.
