**5. Conclusions**

Sommerfeld integral is included in the Green function for the seismic field in horizontal layered half-space. The numerical technique is used to compute the Sommerfeld integrals by deriving the integral into two parts, the infinite integral part and the finite integral part, and by applying DE quadrature rules to evaluate the finite part and DCIM to calculate the infinite part. Compared with the DE\_WA method, the new method can ge<sup>t</sup> an accurate result with a relative error less than 2.5 × 10−<sup>3</sup> and increase time saving by about 40%. The

ESPRIT method is introduced to extract DCIM for better accuracy and efficiency. Finally, the fast numerical method is applied to the calculation of the seismic field in horizontal layered half-space. The method in this paper takes both efficiency and accuracy into account, and theoretically, higher accuracy can be achieved by controlling the parameters.

**Author Contributions:** Conceptualization, Z.Z. and S.D.; methodology, S.L.; software, S.L.; validation, S.L., Y.Y. and Z.Z.; formal analysis, S.L.; writing—original draft preparation, S.L.; writing—review and editing, I.I. and Y.Y.; supervision, Z.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Fundamental Research Funds for the Central Universities of Central South University 2019ZZTS300.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author. The data are not publicly available due to some institutional reasons.

**Conflicts of Interest:** The authors declare no conflict of interest.
