## 3 Results and Discussion
### Data Analysis
#### 3.3.4 Responses Calculation  
After calculating the position and elevation errors, we calculate the array responses in various cases using the procedure described in Section 2.4 above. The ideal trace wavelet responses (\(G_{cm}(t)\)) are shown in Figure 7a and their corresponding array wavelet response (\(G_{c}(t)\)) is shown in Figure 7b. The array trace energy for the ideal case (i.e., after removal of both position and elevation errors) is \(E_{c}=2237\). The perturbed trace wavelet responses (\(G_{xzm}(t)\)) are shown in Figure 7c and their corresponding array wavelet response (\(G_{xz}(t)\)) is shown in Figure 7d. The array trace energy for the perturbed case is \(E_{xz}=797\). The normalized array trace energy of the perturbed case is calculated using Equation (23) to be \(E^{\prime}_{Nxz}=-9\) dB. These results demonstrate the considerable amount of degradation in the array trace energy due to position and elevation errors in this survey. Figure 5: Raw first arrival travel times of shot gathers 1 (circles) and 3 (crosses) picked manually for the 12 traces used in this study. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Trace** & **X (m)** & **Y (m)** &
\begin{tabular}{c} **Calculated** \\ **Distance (m)** \\ \end{tabular} & \(e_{\text{xm}}\) **(m)** \\ \hline Reference position & 2,915,129 & 383,141 & & \\
13 & 2,915,126 & 383,189 & 3.995 & \(-\)0.005 \\
14 & 2,915,126 & 383,185 & 3.945 & \(-\)0.055 \\
15 & 2,915,127 & 383,181 & 4.035 & 0.035 \\
16 & 2,915,127 & 383,177 & 3.939 & \(-\)0.061 \\
17 & 2,915,127 & 383,173 & 4.025 & 0.025 \\
18 & 2,915,127 & 383,169 & 3.972 & \(-\)0.028 \\
19 & 2,915,127 & 383,165 & 3.985 & \(-\)0.015 \\
20 & 2,915,127 & 383,161 & 3.938 & \(-\)0.062 \\
21 & 2,915,128 & 383,157 & 4.022 & 0.022 \\
22 & 2,915,128 & 383,153 & 3.940 & \(-\)0.060 \\
23 & 2,915,128 & 383,149 & 3.986 & \(-\)0.014 \\
24 & 2,915,128 & 383,145 & 4.056 & 0.056 \\  & Minimum position error (m) & & \(-\)0.062 \\  & Maximum position error (m) & & 0.056 \\  & Mean position error (m) & & \(-\)0.014 \\  & Median position error (m) & & \(-\)0.015 \\  & Standard deviation of position errors (m) & & 0.041 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Measured position coordinates using D-GPS and position errors of the 12 studied traces. Position errors are measured with respect to the ideal spacing of 4 m giving a relative mean and standard deviation of about 0.35% and 1%, respectively. To understand the separate effects of position and elevation errors on the seismic array response, the first arrival travel times after separately removing each of these errors is calculated. Starting with elevation error removal, Equation (13) is used to calculate the first arrival travel time for this case. Comparison between the travel times of the ideal case and the elevation error removal case shows that elevation errors have large contributions in all picked head-wave first arrival travel times (Figure 6b). Figure 7e shows the trace wavelet responses after application of elevation errors to the ideal trace responses (\(G_{\mathit{z}n}(t)\)) and Figure 7f shows the corresponding array response (\(G_{\mathit{z}}(t)\)). Now, the array trace energy is calculated to be \(E_{\mathit{z}}=795\), which is much smaller than the array trace energy of the ideal case (i.e., \(E_{\mathit{c}}=2237\)). The normalized array trace energy is \(E_{\mathit{Nz}}^{\prime}=-9\) dB, which is similar to that in the perturbed case, indicating that most degradation is attributed to elevation errors. The same steps are carried out for the removal of only the position errors using Equation (11) and, hence, update the first arrival travel times. Figure 6c shows a comparison of the head-wave travel times between the ideal and position error removal cases. This comparison indicates that position errors have a very small effect on the first arrival travel times. The trace wavelet responses after position error removal and their corresponding array response are shown in Figure 7g,h, respectively.