## 6 Examples
### Love-Wave Case
#### 6.2.2 A Six-Layer Model  
All of the above-mentioned models are aimed at the scale of crustal seismology. In this example, we chose a near-surface six-layer model (Table 11) to demonstrate the applicability and effectiveness of our code at the scale of near-surface geophysics. This model was also chosen because some researchers have reported their results [1; 35]. As shown in Figure 5, the analytical PDs of the fundamental-mode Rayleigh-wave phase velocity of the six-layer model are quite consistent with [35] (Figures 3-5 of his paper). In order to show more details, as an example, the PDs with respect to the S-wave velocities at five different frequencies, along with the outcomes from other researchers, are presented in Table 12 (modified from [35]). Individual column of these matrices corresponds to a layer, whereas individual row corresponds to a frequency. Obviously, only the numerical results from [PERSON] and [PERSON] [6] show a certain lack of accuracy due to the adopted numerical integration in their code [35], whereas other results (including our result) are quite similar. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**T (s)** & \multicolumn{2}{c}{_U (m-s\({}^{-1}\))_} & \multicolumn{2}{c}{\(\partial\)_U/\(\partial\)\(\theta_{1}\)_} & \multicolumn{2}{c}{\(\partial\)_U/\(\partial\)\(\theta_{2}\)_} \\ \hline  & **Novotnj** & **This paper** & **Novotnj** & **This paper** & **Novotnj** & **This paper** \\ \hline
20 & 2864.63 & 2864.6298 & 1.09993 & 1.0999266 & \(-\)0.28108 & \(-\)0.2810790 \\
30 & 3191.45 & 3191.4546 & 1.05173 & 1.0517331 & \(-\)0.01968 & \(-\)0.0196826 \\
40 & 3585.94 & 3585.9400 & 0.45706 & 0.4570642 & 0.36729 & 0.3672883 \\ \hline  & & & \(\partial\)_U/\(\partial\)\(\alpha_{1}\)_ & \multicolumn{2}{c}{\(\partial\)_U/\(\partial\)\(\alpha_{2}\)_} \\ \cline{3-6}  & & **Novotnj** & **This paper** & **Novotnj** & **This paper** \\ \hline
20 & & 0.02799 & 0.0279941 & \(-\)0.00754 & \(-\)0.0075417 \\
30 & & 0.21034 & 0.2103442 & \(-\)0.01419 & \(-\)0.0141852 \\
40 & & 0.23521 & 0.2352072 & \(-\)0.00267 & \(-\)0.0026709 \\ \hline  & & & \(\partial\)_U/\(\partial\)\(h\)_ & \multicolumn{2}{c}{\(\partial\)_U/\(\partial\)\(\rho\)_} \\ \cline{3-6}  & & **Novotnj** & **This paper** & **Novotnj** & **This paper** \\ \hline
20 & & 0.00492 & 0.0049175 & \(-\)0.27858 & \(-\)0.2785814 \\
30 & & \(-\)0.04428 & \(-\)0.0442750 & 0.35718 & 0.3571744 \\
40 & & \(-\)0.03018 & \(-\)0.0301845 & 0.60146 & 0.6014577 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Computational results relevant to group velocities for a two-layer model. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**T (s)** & \multicolumn{2}{c}{_c (m-s\({}^{-1}\))_} & \multicolumn{2}{c}{\(\partial\)_c/\(\partial\)\(\theta_{1}\)_} & \multicolumn{2}{c}{\(\partial\)_c/\(\partial\)\(\theta_{2}\)_} \\ \hline  & **Novotnj** & **This paper** & **Novotnj** & **This paper** & **Novotnj** & **This paper** \\ \hline
20 & & 0.14041 & 0.1404094 & 0.00231 & 0.0023095 \\
30 & & 0.16901 & 0.1690098 & 0.01454 & 0.0145360 \\
40 & & 0.13987 & 0.1398705 & 0.02492 & 0.0249162 \\ \hline  & & & \(\partial\)_c/\(\partial\)\(h\)_ & \multicolumn{2}{c}{\(\partial\)_c/\(\partial\)\(\rho\)_} \\ \cline{3-6}  & & **Novotnj** & **This paper** & **Novotnj** & **This paper** \\ \hline
20 & & 0.14041 & 0.1404094 & 0.00231 & 0.0023095 \\
30 & & 0.16901 & 0.1690098 & 0.01454 & 0.0145360 \\
40 & & 0.13987 & 0.1398705 & 0.02492 & 0.0249162 \\ \hline  & & & \(\partial\)_c/\(\partial\)\(h\)_ & \multicolumn{2}{c}{\(\partial\)_c/\(\partial\)\(\rho\)_} \\ \cline{3-6}  & & **Novotnj** & **This paper** & **Novotnj** & **This paper** \\ \hline
20 & & 0.00492 & 0.0049175 & \(-\)0.27858 & \(-\)0.2785814 \\
30 & & \(-\)0.04428 & \(-\)0.0442750 & 0.35718 & 0.3571744 \\
40 & & \(-\)0.03018 & \(-\)0.0301845 & 0.60146 & 0.6014577 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Computational results relevant to phase velocities for a two-layer model.