Green's Function Representations in Wave Physics

• Green's Function Representations are mathematical tools that capture the impulse response of linear wave systems, enabling solution construction via superposition.
• They integrate concepts like Green's matrices, propagator matrices, and focusing functions to systematically model wavefield propagation in diverse media.
• These representations underpin advanced applications such as imaging, inversion, redatuming, and monitoring, offering efficient, data-driven approaches to complex wave physics problems.

A Green's function is the fundamental impulse response of a system described by linear or linearized equations, allowing the construction of general solutions through superposition. In wave physics and acoustics, Green's functions serve as the building blocks of wave-field representations, facilitating modeling, imaging, inversion, and monitoring in complex media. The unifying matrix-vector formalism extends these concepts to various domains—acoustics, elastodynamics, electromagnetics—enabling systematic construction of solution representations using Green's matrices, propagator matrices, and Marchenko-type focusing functions.

1. Matrix-Vector Wave Equation and Green’s Matrices

The matrix-vector wave equation encapsulates several classical wave phenomena in a unified operator form: $\partial_3 {\bf q} - {\cal A} {\bf q} = {\bf d}$ where

• ${\bf q}$: wavefield vector (may contain pressure, velocity, etc.)
• ${\cal A}$: medium/operator matrix
• ${\bf d}$: source vector.

The associated Green’s matrix ${\bf G}({\bf x}, {\bf x}_A, \omega)$ is defined as the solution of

$$\partial_3 {\bf G} - {\cal A} {\bf G} = {\bf I} \delta({\bf x} - {\bf x}_A)$$

where ${\bf I}$ is the identity matrix.

Green's matrices generalize the scalar Green's function, describing the full wavefield at ${\bf x}$ due to an impulsive source at ${\bf x}_A$, encompassing all field components.

Integral representations for the wavefield are then written as: $${\bf q}({\bf x}, \omega) = \int_{\mathbb{R}^3} {\bf G}({\bf x}, {\bf x}_A, \omega)\,{\bf d}({\bf x}_A, \omega)\,d^3{\bf x}_A$$ or, for boundary value problems (e.g., the Kirchhoff-Helmholtz theorem),

$${\bf q}({\bf x}_A,\omega) = \int_{\partial\mathbb{D}_0} {\bf G}({\bf x}_A, {\bf x}, \omega)\, {\bf q}({\bf x}, \omega)\,d^2{\bf x}$$

These representations stem from reciprocity theorems and apply across acoustics, elastodynamics, and electromagnetics.

2. Propagator Matrices in Wave-Field Representations

A propagator matrix ${\bf W}({\bf x},{\bf x}_A,\omega)$ provides a solution to the homogeneous (no source) wave equation,

$\partial_3 {\bf W} - {\cal A} {\bf W} = {\bf 0}$

with the boundary condition: $${\bf W}({\bf x}, {\bf x}_A, \omega)\,|_{x_3 = x_{3,A}} = {\bf I}\,\delta({\bf x}_H - {\bf x}_{H,A})$$ where $x_{3,A}$ is the reference depth (or plane) for propagation.

The propagator matrix describes the "propagation" of wavefields from one boundary or interface to another: $${\bf q}({\bf x}, \omega) = \int_{\partial\mathbb{D}_A} {\bf W}({\bf x}, {\bf x}_A, \omega) {\bf q}({\bf x}_A, \omega)\, d^2{\bf x}_A$$

Advantages:

• In contrast to Green's matrices (which require the full medium response), propagator matrices depend only on the portion of the medium between two boundaries.
• Modular structure allows "concatenation" or compositional building of solutions for layered or piecewise media.
• Enables representations built on single-sided boundary integrals rather than closed or two-sided boundaries, which is particularly advantageous for practical acquisition setups.

3. Marchenko-Type Focusing Functions and Their Integration

Marchenko-type focusing functions are specialized wavefields designed to focus energy at a specific interior point (or time), constructed so that their injection from a single accessible boundary results in a "virtual source" at an interior location.

Mathematically, for a focusing function ${\bf F}_1({\bf x}_A, {\bf x}, t)$: $${\bf F}_1({\bf x}_A, {\bf x}, t)\,|_{x_{3,A}=x_{3,0}} = {\bf I}\,\delta({\bf x}_H - {\bf x}_{H,A})\,\delta(t)$$

Connection to Propagator Matrices:

• Any single-sided propagator matrix can be expressed as a linear combination of Marchenko-type focusing functions (see Eq. 934ac for the acoustic case).
• This relation allows propagation operators in boundary-integral representations to be built entirely from focusing functions constructed using reflection response data.

Implications:

• Marchenko representations move the construction of internal Green's functions from model-based to data-driven, facilitating virtual source creation, internal imaging, redatuming, and accounting for multiples without explicit modeling of the full internal structure.

Example: In the plane-wave domain for acoustics,

$$F^p(s_1, x_3, x_{3,0}, \tau) = W^{p,p}(s_1, x_3, x_{3,0}, \tau) - \frac{s_{3,0}}{\rho_0} W^{p,v}(s_1, x_3, x_{3,0}, \tau)$$

focuses energy at $x_{3,0}$, and all non-focused contributions cancel by interference.

4. Applications in Imaging, Inversion, and Redatuming

These unified representations have broad practical implications:

• Advanced Inverse Scattering and Imaging: Knowledge of Green’s, propagator, or focusing function representations enables the recovery or prediction of fields anywhere in the medium, accounting for all orders of multiples and interactions.
• Source/Receiver Redatuming: The ability to "move" sources and receivers virtually within the medium using only single-sided data at the boundary allows for flexible acquisition geometries and post-survey repositioning of virtual sources/receivers for optimal imaging.
• Holography and Wavefield Retrieval: Homogeneous Green’s function representations can be constructed using only reflection data and focusing functions (see Eqs. eqhomgts and eqrepgenwghomag), which are foundational for time-reversal and holographic imaging.
• Monitoring: These representations are used for monitoring the evolution of the medium, such as in 4D seismics or process monitoring in engineering.

5. Mathematical and Algorithmic Structure

Key equations from the matrix-vector formalism include:

Quantity	Representation (acoustic)
Green’s matrix	$$\partial_3 {\bf G} - {\cal A}{\bf G} = {\bf I} \delta({\bf x} - {\bf x}_A)$$
Propagator matrix	$\partial_3 {\bf W} - {\cal A}{\bf W} = {\bf 0}$, with boundary condition at $x_3 = x_{3,A}$
Wavefield representation	$${\bf q}({\bf x}, \omega) = \int_{\mathbb{D}_A} {\bf W}({\bf x}, {\bf x}_A, \omega) {\bf d}({\bf x}_A, \omega) d^3{\bf x}_A + \int_{\partial\mathbb{D}_0} {\bf W}({\bf x}, {\bf x}_A, \omega) {\bf q}({\bf x}_A, \omega) d^2 {\bf x}_A + \ldots$$
Marchenko focusing func.	$${\bf F}_1({\bf x}_A, {\bf x}, t)|_{x_{3,A} = x_{3,0}} = {\bf I} \delta({\bf x}_H - {\bf x}_{H,A}) \delta(t)$$
Green’s function via propagator	$${\bf q}({\bf x}_A, \omega) = \int_{\partial\mathbb{D}_0} {\bf W}({\bf x}_A, {\bf x}, \omega) {\bf q}({\bf x}, \omega) d^2 {\bf x}$$
Propagator via focusing functions	For acoustics, via Eq. 934acsf

Algorithmic considerations:

• The modular structure facilitates stable, efficient computation, especially for stratified or layered media where propagator matrices can be expressed recursively.
• For single-sided data, Marchenko-type algorithms (recursive schemes based on reflection responses and imposing focusing conditions) are employed to construct focusing functions, which can then be used for all representation needs.

6. Limitations and Implementation Challenges

• Data Requirements: Marchenko-type focusing function construction requires high-quality, broadband, densely-sampled reflection data.
• Evanescent Components: The handling of evanescent (non-propagating) wave components can pose numerical challenges and is a limiting factor for some representation techniques.
• Multiphysics Complexity: While the matrix-vector formalism is general, practical inversion and computation in complex anisotropic or dissipative media require further algorithmic development.
• Iterative Construction: While focusing functions move much of the work to data, their retrieval (from single-sided data) is typically iterative and sensitive to acquisition artifacts or model inaccuracies.

7. Implications for Unified Wave Physics

By reformulating classical and modern wave-field representations within a unified, matrix-vector, and operator-theoretic framework—anchored by Green's matrices, propagator matrices, and Marchenko-type focusing functions—the methodology supports the development of advanced, general, and efficient algorithms for imaging, inversion, and monitoring across diverse wave phenomena. The approach bridges model-based and data-driven paradigms, lending itself to the synthesis of fields within the medium using only single-sided data, thereby aligning mathematical theory with practical acquisition constraints in modern wave physics.