## 4 Discussion  
In addition to seismic data compression and noise reduction, SMD also provides a new method of seismic data analysis. Rather than a list of displacements distributed in space and time to describe incoming waves, we have pairs of singular vectors paired with shift vectors. In an ideal scenario, for each arriving wave the column vector represents the waveform, the row vector represents the amplitudes at different receiver locations and the shift vector represents relative arrival times. Even though the ideal case is rarely achieved, we believe that the results of SMD provide an excellent advance in the realm of seismic analysis especially with its application of machine learning. With SMD, identifying features that can be used for building models is facilitated when noise-reduced data is represented in the SMD-compressed format. The application of machine learning to data compression was explored in [24], which applied SVD to synthetic data and developed a model for estimating source location and orientation. However, SMD may provide greater opportunities for machine learning application. It is instructive to provide an example of how the SMD algorithm can help estimate physical properties such as elastic wave velocities. To this end, herein we predict the average acoustic velocity (\(\alpha\)) in our model (seawater) by analyzing the results of the SMD algorithm. Specifically, we use curvature of the shift vectors, and the zero offset time to estimate the average velocity of the reflected waves. Since we have a controlled source, a wave's zero offset time is simply it's row position in the first column of the data matrix multiplied by the time difference between consecutive recordings (\(t_{d}\)). Since the row position of a wave can be determined from the shift vector and the first element of the column vector, our wave velocity estimation is obtained solely from the data stored in SMD results. To derive the formula for wave velocity we must make several assumptions about our surroundings. First, we assume that the reflecting surface (the ocean floor) is horizontal, and that the depth of the ocean floor (\(z_{f}\)) is significantly larger than the horizontal distance from source to the receiver (\(x\ll z_{f}\)). This gives us the expression for the total distance travel by the reflected waves (\(d\)):  
\[d\approx 2z_{f}+\frac{x^{2}}{4z_{f}}. \tag{11}\]  
Since we are considering the average acoustic velocity (\(\alpha\)) in our model, we can rewrite the expression (11) in terms of the reflected wave travel-time (\(T\)):  
\[T\approx\frac{2z_{f}}{\alpha}+\frac{x^{2}}{4z_{f}\alpha}. \tag{12}\]  
The zero offset time \(T_{0}\) is defined as travel-time \(T\) at zero horizontal distance (\(x=0\)):  
\[T_{0}\approx\frac{2z_{f}}{\alpha}. \tag{13}\]  
Taking the second derivative of the expression (12) with respect to horizontal position \(x\) gives:  
\[\frac{\partial^{2}T}{\partial x^{2}}\approx\frac{1}{2z_{f}\alpha}. \tag{14}\]  
Using expression (13) to substitute \((2z_{f})\) with \((T_{0}\alpha)\) in expression (14) gives:  
\[\alpha\approx\left(\frac{\partial^{2}T}{\partial x^{2}}T_{0}\right)^{\frac{- 1}{2}} \tag{15}\]  
Expression (15) can be used to calculate the acoustic velocity if we can express both \(\frac{\partial^{2}T}{\partial x^{2}}\) and \(T_{0}\) in terms of SMD results.