## 3 Results and Discussion
### Data Analysis
#### 3.3.5 Comparison with Previous Studies  
We compare our results to those expected by [31] by applying their method to calculate array trace responses involving position errors only, elevation errors only, and combined position and elevation errors using the parameters of our data. The head-wave arrival in our data has a dominant frequency of 65 Hz and is incident at 9.2\({}^{\circ}\) on a 12-element array that has an ideal spacing of 4 m laid on a dune with an average velocity of 313 m/s. \begin{table}
\begin{tabular}{c c c c c} \hline
**Trace** & **\(\boldsymbol{X}\) (m)** & **\(\boldsymbol{Y}\) (m)** & **Elevation Z (m)** & \(\boldsymbol{e_{\mathit{z}n}}\) (m)** \\ \hline Reference position & 2,915,129 & 383,141 & 9.41 & \\
13 & 2,915,126 & 383,189 & 13.44 & 4.03 \\
14 & 2,915,126 & 383,185 & 13.19 & 3.78 \\
15 & 2,915,127 & 383,181 & 12.97 & 3.56 \\
16 & 2,915,127 & 383,177 & 12.70 & 3.29 \\
17 & 2,915,127 & 383,173 & 12.41 & 3.00 \\
18 & 2,915,127 & 383,169 & 12.08 & 2.67 \\
19 & 2,915,127 & 383,165 & 11.68 & 2.27 \\
20 & 2,915,127 & 383,161 & 11.32 & 1.91 \\
21 & 2,915,128 & 383,157 & 10.90 & 1.50 \\
22 & 2,915,128 & 383,153 & 10.52 & 1.11 \\
23 & 2,915,128 & 383,149 & 10.19 & 0.78 \\
24 & 2,915,128 & 383,145 & 9.81 & 0.40 \\  & Minimum elevation error (m) & & 0.40 \\  & Maximum elevation error (m) & & 4.03 \\  & Mean elevation error (m) & & 2.36 \\  & Median elevation error (m) & & 2.47 \\  & Standard deviation of elevation errors (m) & & 1.22 \\ \hline \end{tabular}
\end{table} 
Table 4: Elevation values and errors of the 12 studied traces. Errors are measured with respect to the datum elevation at 9.41 m giving a relative mean and standard deviation of about 25% and 13%, respectively. Figure 6: Comparison between head-wave first arrival travel times of the 10 studied traces: (**a**) perturbed case; (**b**) elevation error removal case; and (**c**) position error removal case. Similar to [31], we first generated the ideal array response due to a 65 Hz zero phase [PERSON] wavelet incident at 9.2' on a 12-element equally weighted array and calculated the corresponding trace energy \(E_{c}=1060.57\). Then, we calculated array responses with various error types using the following procedure:  
Figure 7: Trace wavelet responses (G) for: (**a**) ideal case; (**c**) perturbed case; (**e**) elevation error removal case; and (**g**) position error removal case. The corresponding array wavelet response of these cases are: (**b**) ideal case; (**d**) perturbed case; (**f**) elevation error removal case; and (**h**) position error removal case. 1. Normalize the standard deviation of position errors (Tables 3 and 4) by the ideal geophone spacing. 2. Generate 12 random position errors from a normal distribution with zero mean and normalized standard deviation of position errors calculated in step 1. 3. Generate the array response with position errors only. 4. Calculate the corresponding trace energy. We repeat these steps for the cases of elevation errors only and combined position and elevation errors. For the case of combined position and elevation errors, we assume normally distributed independent position and elevation errors and calculate a combined standard deviation of 0.3052 [33]. Table 5 presents calculations using this procedure and compares them to their counterparts calculated using steps described in Sections 2.2 and 2.3 of this paper. Similar to the observed data, the array response due to position errors shows only a negligible degradation (0.009%) when calculated using the method of [31]. On the other hand, observed degradations due to elevation errors only (64.5%) and combined position and elevation errors (64.4%) are smaller than those expected by the method of [31] (88.0% and 87.9%, respectively). Nevertheless, observed degradations due to elevation errors only and combined position and elevation errors are comparable to each other and much higher than the degradation due to position errors only, which is also observed by the method of [31].