Green's functions for geophysics: a review.

Area of Science:

• Geophysics
• Applied Mathematics
• Physical Sciences

Background:

• Green's functions (GFs) are fundamental solutions to differential equations, crucial for boundary-value problems in physical and engineering fields.
• GFs model phenomena like heat transfer, fluid dynamics, and wave propagation, with applications ranging from Earth sciences to quantum nanostructures.

Purpose of the Study:

• To provide a comprehensive review of Green's functions (GFs) specifically for geophysical applications.
• To consolidate and present various GF derivations and solution methods relevant to geophysics.

Main Methods:

• Review of George Green's potential functions, GF definitions, theorems, and basic relations.
• Analysis of boundary-value problems for elastic and viscoelastic materials.
• Derivation and presentation of GFs in full- and half-spaces, including layered systems and spherical Earth models.

Main Results:

• Detailed derivations of GFs for concentrated forces and dislocations in layered media using Cartesian and cylindrical coordinates.
• Presentation of GFs for self-gravitating, layered spherical Earth models in spherical coordinates.
• Analysis of singularities and infinities associated with GFs in layered systems, including a review of layer matrix methods.

Conclusions:

• The review offers a thorough treatment of GFs for geophysics, synthesizing existing knowledge and methods.
• The presented GF methods and derivations are broadly applicable to various engineering and scientific disciplines beyond geophysics.
• This work serves as a valuable resource for researchers and practitioners utilizing Green's functions in Earth sciences and related fields.