## 7 Discussion and Conclusions  
The accurate and reliable calculation of PDs of surface-wave DCs is an essential component of any linearized inversion strategy based on surface-wave DCs. In this paper, we have rederived a series of algorithms and the related expressions based on the reduced delta matrix theory to compute analytically the PDs of surface-wave (both Love and Rayleigh waves) phase-velocity DCs. Moreover, we have also applied this analytical method to compute the PDs of Love-wave group velocity. Although Equations (9)-(15) constitute an abstract framework applicable to both waves, the derivation of all the second-order mixed PDs for Rayleigh waves could be rather difficult. Therefore, we have implemented a high-precision hemi-analytical method, which computes all the first-order partial differentiation analytically (Section 3) and approximates the only mixed second-order partial differentiation term with a central finite-difference scheme. For the fully analytical computation of Rayleigh-wave group-velocity PDs, [PERSON] et al. [32] is the only work we could find, despite the fact that only the case of two-layer model was studied due to the complexity of this problem. The comparison (Section 6.2.1) demonstrates that this mixed computational strategy is satisfying, with a comparable accuracy to five decimal places, as shown in Table 10.  
An open-source MATLAB package named SWPD has been developed accompanying this paper. All examples in this paper have been validated by comparing with the results of previous studies, and all the results computed by our package are satisfactory with rather high precision. Therefore, it has been demonstrated that the analytical expressions and the algorithms derived in this paper are effective. Also, these examples can be reproduced by other users with the help of four demo scripts in the package. Upon becoming familiar with this package, people could modify and extend it without any difficulty, including any further development of the surface-wave linearized inversion strategy. We think this package could be a useful tool for people who are engaged in the field of near-surface geophysics, especially for those who are interested in surface-wave exploration methods. Figure 5: Analytical PDs of the fundamental-mode phase velocity with respect to \(\beta\) (a), \(\alpha\) (b), \(\rho\) (c), and \(h\) (d) for the six-layer model in Table 10.  
**Author Contributions:** Conceptualization, D.W., X.W. and Q.S. ; Methodology, D.W., X.W. and Q.S. ; Software, D.W.; Formal analysis, D.W., X.W., Q.S. and T.Z. ; Funding acquisition, X.W. and Q.S. ; Investigation, D.W., Q.S. and T.Z. ; Project administration, X.W., Q.S. and T.Z. ; Resources, X.W. and Q.S. ; Validation, D.W., X.W., Q.S. and T.Z. ; Writing--original draft, D.W.; Writing--review & editing D.W., X.W., Q.S. and T.Z. **Funding:** This research is funded by the National Science and Technology Major Project of the Ministry of Science and Technology of China \"the study of key geophysical exploration technologies for Lower Palaeozoic Erathem-Precambrian basement tectonics (2016 ZX05004-003)\"  
**Acknowledgments:** We are grateful to the valuable comments by the anonymous reviewers, which have helped improve this manuscript. **Conflicts of Interest:** The authors declare no conflict of interest. **Appendix A**  
The expressions presented below are adapted from Buchen & Ben-Hador [23] just for convenience of reference. Basic physical quantities employed:  
\(\omega=\) angular frequency  
\(c=\) phase velocity  
\(k=\omega/c=\) angular wavenumber  
\(i=\sqrt{-1}=\) imaginary unit  
Basic derived quantities employed for the \(m\)-th layer (subscript \(m\) is omitted for simplicity):  
\[r=\left\{ \begin{array}{ll}\sqrt{1-c^{2}/\alpha^{2}},&c<\alpha\\ i\sqrt{c^{2}/\alpha^{2}-1},&c>\alpha\end{array} \right.\] (A1)  
\[r=\left\{ \begin{array}{ll}\sqrt{1-c^{2}/\alpha^{2}},&c<\alpha\\ i\sqrt{c^{2}/\alpha^{2}-1},&c>\alpha\end{array} \right.\] (A2)  
\[C_{\alpha}=\left\{ \begin{array}{ll}\cosh(krh),&c<\alpha\\ \cos(ikrh),&c>\alpha\end{array}