---
title: Green's Function Representations in Wave Physics
url: https://www.emergentmind.com/topics/green-s-function-representations
type: topic
---

# Green's Function Representations in Wave Physics

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.

Source: https://www.emergentmind.com/topics/green-s-function-representations