## 4 Discussion
In Section 2.2.2 it is explained that the row position (\(p_{j}\)) of a wave in any column \(j\) is equal to the sum of the \(j^{th}\) element of the shift vector (\(\mathbf{s}\)) and the waveform row (\(w_{r}\)):  
\[p_{j}=s_{j}+w_{r}, \tag{16}\]  
In Section 2.2.3 it is explained that the waveform row (\(w_{r}\)) can be obtained from the first element of the column vector (\(\mathbf{u}\)) and the predetermined length of the recorded waveform (\(l_{w}\)):  
\[w_{r}=u_{1}+\frac{l_{w}-1}{2}. \tag{17}\]  
Using expressions (16) and (17), the zero offset time (\(T_{0}\)) can be described with the row position of the wave in the first column (\(p_{1}\)) multiplied by the time difference between consecutive recordings (\(t_{d}\)):  
\[T_{0}\approx t_{d}(s_{1}+u_{1}+\frac{l_{w}-1}{2}). \tag{18}\]  
The expression (18) will be used to obtain the zero offset time from SMD results. If we assume that the relative arrival times were accurately recorded by the shift vector, we can make the following substitution:  
\[\frac{\partial^{2}T}{\partial x^{2}}=\frac{s^{\prime\prime}t_{d}}{x_{d}^{2}} \tag{19}\]where \(x_{d}\) is the distance between adjacent receivers. In the expression (19), \(s^{\prime\prime}\) is the second derivative of an element in the shift vector \(s\) with respect to the position (\(j\)) of the element in the shift vector. The value of \(s^{\prime\prime}\) is determined by fitting a parabola to first \(n_{r}\) elements of the shift vector. The parameter \(n_{r}\) need not have an exact value. While we want to use enough elements from the shift vector to confidently fit a parabola, we also want to only use data from the receivers close to the source in order to follow the \((x\ll z_{f})\) condition. Therefore, the parameter \(n_{r}\) is set to 50. Once we fit the parabola, the value of \(s^{\prime\prime}\) is estimated to be the quadratic coefficient \(c_{2}\), multiplied by 2:  
\[ \begin{split} j\in[1,n_{r}]&:\,s_{j}\approx c_{0}+ c_{1}j+c_{2}j^{2},\\ s^{\prime\prime}=2c_{2}.\end{split}  \tag{20}\]  
By plugging the expressions (18)-(20) into expression (15) we estimate the acoustic velocity. However, the values of a shift vector can be affected by interfering waves, or noise. To minimize the error from those sources, we estimate the velocity based on the first \(n_{s}\) pairs of shift vectors and column vectors. Similar to \(n_{r}\), the \(n_{s}\) parameter does not need to be set to any specific value. In this example, the parameter \(n_{s}\) is set to 5. The average acoustic velocity \(\alpha\) is estimated as the weighted average of the results from the first \(n_{s}\) extracted pairs of a of shift vector and a column vector. Each term is weighted by the quality of the parabolic fit, which is equal to the inverse of the error norm \(\varepsilon\):  
\[ \begin{split} s^{k}&\to\{c_{0}^{k},c_{1}^{k},c_{2}^ {k}\}\\ e^{k}&=\sum_{j=1}^{n_{s}}\big{(}s_{j}^{k}-(c_{0}^{k}+ c_{1}^{k}j+c_{2}^{k}j^{2})\big{)}^{2},\\ e_{sum}&=\sum_{k=1}^{n_{s}}\frac{1}{\varepsilon^{ k}},\\ \alpha&=\sum_{k=1}^{n_{s}}\Big{(}(2c_{2}^{k})(s_{1} +u_{1}^{k}+\frac{l_{w}}{2})\frac{t_{d}^{2}}{x_{d}^{2}}\Big{)}^{\frac{-1}{2}} \Big{(}\frac{1}{\varepsilon^{k}\varepsilon_{sum}}\Big{)}.\end{split}  \tag{21}\]  
The formula for \(\alpha\), defined in expression (21), was applied to 20 marine field data files, each recording a seismic response from a unique and controlled seismic source. For each file, we used the formula from expression (21) to estimate the velocity. The average estimated velocity was \(1601\frac{\overline{m}}{\overline{s}}\) and the standard deviation among the results was \(236\frac{\overline{m}}{\overline{s}}\). Considering the ocean depth being about 3 km, the correct average acoustic velocity was likely about \(1500\frac{\overline{m}}{\overline{s}}\). If we assume the correct average wave velocity experienced by the reflected waves was \(1500\frac{\overline{m}}{\overline{s}}\), then the average and the median error from the 20 velocity estimates were \(12.1\%\) and \(8.3\%\), respectively.