## Abstract

In wave theory, the homogeneous Green’s function consists of the impulse response to a point source, minus its time-reversal. It can be represented by a closed boundary integral. In many practical situations, the closed boundary integral needs to be approximated by an open boundary integral because the medium of interest is often accessible from one side only. The inherent approximations are acceptable as long as the effects of multiple scattering are negligible. However, in case of strongly inhomogeneous media, the effects of multiple scattering can be severe. We derive double- and single-sided homogeneous Green’s function representations. The single-sided representation applies to situations where the medium can be accessed from one side only. It correctly handles multiple scattering. It employs a focusing function instead of the backward propagating Green’s function in the classical (double-sided) representation. When reflection measurements are available at the accessible boundary of the medium, the focusing function can be retrieved from these measurements. Throughout the paper, we use a unified notation which applies to acoustic, quantum-mechanical, electromagnetic and elastodynamic waves. We foresee many interesting applications of the unified single-sided homogeneous Green’s function representation in holographic imaging and inverse scattering, time-reversed wave field propagation and interferometric Green’s function retrieval.

## 1. Introduction

In wave theory, the homogeneous Green’s function consists of the impulse response to a point source, minus its time-reversal. When there are no losses, the impulse response and its time-reversal obey the same wave equation with a delta-singularity at the position of the point source. When the difference of these wave equations is taken, the delta-singularities cancel each other. Hence, the homogeneous Green’s function obeys a wave equation without a delta-singularity.

The homogeneous Green’s function can be represented by a closed boundary integral. This representation plays an important role in optical, acoustic and seismic holography [1–3], imaging and inverse scattering [4,5], time-reversal acoustics [6] and Green’s function retrieval from ambient noise [7,8]. In many practical situations, the closed boundary integral needs to be approximated by an open boundary integral because the medium of interest is often accessible from one side only. This can lead to unacceptable errors, particularly when multiple scattering cannot be ignored. To overcome this problem, we recently formulated a single-sided homogeneous Green’s function representation [9]. This representation was derived for the scalar wave equation. In this paper, we use a unified matrix-vector notation for acoustic, quantum-mechanical, electromagnetic and elastodynamic waves. Based on a unified wave equation, we derive double- and single-sided representations for the homogeneous Green’s function. In particular, the single-sided homogeneous Green’s function representation has many interesting potential applications in holographic imaging and inverse scattering, time-reversed wave field propagation and interferometric Green’s function retrieval, using scalar or vectorial wave fields.

## 2. Unified double-sided two-way representation

Throughout this paper, we define a ‘closed boundary’ as two infinite horizontal boundaries, one above and one below the medium of interest. For this configuration, we derive in this section a double-sided homogeneous Green’s function representation, expressed as integrals along these two boundaries. We start in §2a by defining a unified two-way wave equation, which relates the vertical derivative of a wave vector via an operator matrix to the derivatives of the same wave vector in the horizontal plane. In §2b, we discuss two-way reciprocity theorems, using specific symmetry properties of the two-way wave equation. In §2c, we introduce the homogeneous Green’s function of the two-way wave equation and discuss its symmetry properties. Finally, in §2d, we use the two-way reciprocity theorem to derive the double-sided two-way representation of the homogeneous Green’s function.

### (a) Two-way wave equation

We define the temporal Fourier transform of a space- and time-dependent quantity *f*(**x**,*t*) as
*t* denotes time, *ω* angular frequency, ‘i’ is the imaginary unit and **x**=(*x*_{1},*x*_{2},*x*_{3}) denotes the Cartesian coordinate vector; the *x*_{3}-axis is pointing downward. For notational convenience, we use the same symbol (here *f*) for quantities in the time domain and in the frequency domain. The starting point for our derivations is a unified wave equation in the space-frequency domain, in matrix-vector form given by
**q**=**q**(**x**,*ω*) is a *N*×1 vector containing a specific choice of wave field components, **d**=**d**(**x**,*ω*) is a *N*×1 vector containing the source functions, and *N*×*N* operator matrix containing the medium parameters and the differential operators ∂_{1} and ∂_{2} (∂_{k} for *k*=1,2,3 stands for differentiation in the *x*_{k}-direction). The value of *N* depends on the type of wave field considered. We call equation (2.2) the ‘two-way’ wave equation because, unlike the ‘one-way’ wave equation discussed in §3, equation (2.2) does not explicitly distinguish between ‘downward’ and ‘upward’ propagation. The wave field vector **q** implicitly contains the superposition of downgoing and upgoing waves.

For acoustic waves in fluids, we have *N*=2 [10–13], for quantum-mechanical waves *N*=2, for electromagnetic waves in matter *N*=4 [14–16], for elastodynamic waves in solids *N*=6 [17,18], for poroelastic waves in porous solids *N*=8, for coupled elastodynamic and electromagnetic waves in piezo-electric materials *N*=10, and for coupled elastodynamic and electromagnetic waves in porous solids *N*=12 [19]. From here onward, we consider lossless media, hence, we exclude the two situations that concern waves in porous solids.

We subdivide the vectors in equation (2.2) into *N*/2×1 subvectors and the matrix into *N*/2×*N*/2 submatrices, according to
*p*=*p*(**x**,*t*) and particle velocity *v*_{i}=*v*_{i}(**x**,*t*), the source functions are external force density *f*_{i}=*f*_{i}(**x**,*t*) and volume-injection rate density *q*=*q*(**x**,*t*) (not to be confused with the components **q**_{1} and **q**_{2} of wave field vector **q** in equation (2.3)), the medium parameters are compressibility *κ*=*κ*(**x**) and mass density *ρ*=*ρ*(**x**). Lower case Latin subscripts (except *t*) run from 1 to 3, whereas lower case Greek subscripts can take the values 1 and 2 to denote the horizontal components. The summation convention applies to repeated subscripts. We transform equations (2.4) and (2.5) to the space-frequency domain, using equation (2.1). The time derivatives are thus replaced by −i*ω*. Next, by eliminating the horizontal components of the particle velocity, *v*_{1} and *v*_{2}, we obtain
*p*=*p*(**x**,*ω*), etc. Equations (2.6) and (2.7) can be cast in the form of two-way wave equation (2.2) by defining the vectors and operator matrix as follows:

Schrödinger’s wave equation for a particle with mass *m* in a potential *V* =*V* (**x**) is given by [20,21]
*Ψ*=*Ψ*(**x**,*t*) is the wave function and *h* Planck’s constant. We transform this equation to the space-frequency domain, replace ∂_{t} by −i*ω*, and separate the vertical derivatives (∂_{3}∂_{3}) from the horizontal derivatives (∂_{α}∂_{α}). This gives
*Ψ*=*Ψ*(**x**,*ω*). This equation, together with the trivial equation

The vectors and matrices for the electromagnetic and elastodynamic situation can be found in several of the aforementioned references. Details of the definitions vary from author to author. In this paper, we employ the definitions of Appendix C in reference [22], albeit that here we only consider real-valued medium parameters (because we consider lossless media) and we replace −*j* by *i* (to be consistent with the use of *i* in Schrödinger’s equation).

For all considered cases, matrix **I** and **O** being identity and zero matrices of appropriate size. Superscript *t* denotes the transposed operator and † the adjoint (the complex conjugate transposed). Here transposed and adjoint operators are introduced via their integral properties
*x*_{3}=constant), **f**=**f**(**x**) and **g**=**g**(**x**) are vector functions with ‘sufficient decay’ at infinity, **f**^{t} is the transposed vector, **f**^{†} is the complex conjugate transposed vector and _{1} and ∂_{2}. Equation (2.14) implies that

For all considered cases, the submatrices of **q** for all considered cases is scaled such that the power-flux density in the *x*_{3}-direction (or, for quantum-mechanical waves, the probability current density) is given by
**N**, **K** and **J** that are frequently used throughout this paper without always explicitly mentioning this.

### (b) Two-way reciprocity theorems

We consider two wave field states *A* and *B*, characterized by independent wave vectors **q**_{A}(**x**,*ω*) and **q**_{B}(**x**,*ω*), obeying wave equation (2.2) with independent source vectors **d**_{A}(**x**,*ω*) and **d**_{B}(**x**,*ω*) and operator matrices *A* and *B*. We consider a spatial domain *n*_{3} is the vertical component of the outward pointing normal vector on *n*_{3}=−1 at the upper boundary *n*_{3}=+1 at the lower boundary

### (c) Two-way homogeneous Green’s function

We introduce the two-way Green’s function **G**(**x**,**x**_{B},*ω*) as a *N*×*N* matrix obeying wave equation (2.2), with the source vector replaced by a diagonal point-source matrix, according to
**x**_{B} denotes the position of the point source of Green’s function. As boundary condition we impose the physical radiation condition of outgoing waves at infinity, which corresponds to causality in the time domain, i.e. **G**(**x**,**x**_{B},*t*)=**O** for *t*<0. In other words, **G**(**x**,**x**_{B},*ω*) is the forward propagating Green’s function. Analogous to matrix *N*/2×*N*/2 submatrices, according to
**q**_{1} or **q**_{2}) observed at **x**; the second subscript refers to the type of source (**d**_{1} or **d**_{2}) at **x**_{B}. For example, for the acoustic situation we have
*p* and *v* stand for the observed wave quantities acoustic pressure (*p*) and particle velocity (*v*_{3}) at **x**, and subscripts *f* and *q* stand for the source types external force (*f*_{3}) and volume-injection rate (*q*) at **x**_{B}.

The two-way homogeneous Green’s function **G**_{h}(**x**,**x**_{B},*ω*) is introduced as a solution of wave equation (2.20), but without the delta singularity on the r.h.s. We therefore search for a second solution of wave equation (2.20) (with the singularity), which will be subtracted from **G**(**x**,**x**_{B},*ω*) to form the homogeneous Green’s function. To find this second solution, take the complex conjugate of wave equation (2.20), and pre- and post-multiply all terms by **J**. This gives
**J****J**=**I**, this can be rewritten as
**J****G***(**x**,**x**_{B},*ω*)**J** is a second solution of wave equation (2.20). Subtracting wave equations (2.20) and (2.25), we obtain
**G**_{h}(**x**,**x**_{B},*ω*) defined as
**J****G***(**x**,**x**_{B},*ω*)**J** obeys the non-physical radiation condition of incoming waves at infinity, which corresponds to acausality in the time domain, i.e. **J****G**(**x**,**x**_{B},−*t*)**J**=**O** for *t*>0. In other words, **J****G***(**x**,**x**_{B},*ω*)**J** is the backward propagating Green’s function. Using equation (2.21) and the definition of **J** in equation (2.16), equation (2.27) can be written as

We conclude this section by deriving a reciprocity relation for Green’s function. To this end, we use the convolution-type reciprocity theorem (equation (2.18)). For state *A*, we replace the wave vector **q**_{A}(**x**,*ω*) by matrix **G**(**x**,**x**_{A},*ω*) and the source vector **d**_{A}(**x**,*ω*) by **I***δ*(**x**−**x**_{A}), which defines the source of Green’s function at **x**_{A}. Similarly, for state *B* we replace **q**_{B}(**x**,*ω*) by **G**(**x**,**x**_{B},*ω*) and **d**_{B}(**x**,*ω*) by **I***δ*(**x**−**x**_{B}). We choose **x**_{A} and **x**_{B} both in **N**^{−1}=−**N**,
**J****N**=−**N****J**, it follows that the same relation holds for the two-way homogeneous Green’s function defined in equation (2.27), i.e.

### (d) Double-sided two-way homogeneous Green’s function representation

We use the correlation-type reciprocity theorem (equation (2.19)) to derive a representation for the two-way homogeneous Green’s function **G**_{h}(**x**_{A},**x**_{B},*ω*). For states *A* and *B*, we make the same replacements as above. Taking also the medium parameters again the same in both states, the last integral in equation (2.19) vanishes. The remaining two integrals yield
**G**^{†}(**x**,**x**_{A},*ω*) obeys the non-physical radiation condition of incoming waves at infinity. Pre-multiplying all terms by **K**, using **K****K**=**I** and equations (2.30) and (2.27), gives
**G**_{h}(**x**_{A},**x**_{B},*ω*), with both **x**_{A} and **x**_{B} in

We conclude this section by considering a special case. First, note that for the upper-right submatrix of **G**_{h}(**x**_{A},**x**_{B},*ω*), which we will call **G**_{h,12}(**x**_{A},**x**_{B},*ω*), equation (2.33) gives
**G**_{h,12}(**x**_{A},**x**_{B},*ω*)=2ℜ{**G**_{12}(**x**_{A},**x**_{B},*ω*)}, see equation (2.28). For the acoustic Green’s matrix, defined in equation (2.22), we replace **G**_{12}(**x**,**x**_{B},*ω*) by *G*_{p,q}(**x**,**x**_{B},*ω*)≡*G*(**x**,**x**_{B},*ω*) and **G**_{22}(**x**,**x**_{B},*ω*) by **G**_{12}(**x**,**x**_{A},*ω*) etc). Equation (2.34) thus becomes
*G*_{h}(**x**_{A},**x**_{B},*ω*) defined as
*G*(**x**,**x**_{B},*ω*) is a unit point source of volume-injection rate density (hence, *q*(**x**,*ω*)=*δ*(**x**−**x**_{B})). On the other hand, Green’s function is often defined as the response to a unit source in the Helmholtz equation (for a medium with constant mass density *ρ*). Let us call this response **x**,**x**_{B},*ω*). The relationship between these two forms of Green’s function is *G*(**x**,**x**_{B},*ω*)=−i*ωρ***x**,**x**_{B},*ω*). Substituting this into equation (2.35), we obtain

## 3. Unified double-sided one-way representation

In this section, we follow a similar path as in §2, except that the ‘two-way’ wave fields will be replaced by ‘one-way’ (downgoing and upgoing) wave fields. In §3a, we define downgoing and upgoing wavefields and introduce a unified one-way wave equation which governs the coupled propagation of these fields. In §3b, we discuss one-way reciprocity theorems, using specific symmetries of the one-way wave equation. In §3c, we introduce the homogeneous Green’s function of the one-way wave equation and discuss its symmetry properties. In §3d, we use the one-way reciprocity theorem to derive the double-sided one-way representation of the homogeneous Green’s function.

### (a) One-way wave equation

We introduce a *N*×1 wave vector **p** and a *N*×1 source vector **s**, according to
**p** should not be confused with the acoustic pressure *p* in §2). Here **p**^{+}=**p**^{+}(**x**,*ω*) and **p**^{−}=**p**^{−}(**x**,*ω*) represent the downgoing (+) and upgoing (−) wavefield, respectively (recall that the *x*_{3}-axis is pointing downward). Similarly, **s**^{+}=**s**^{+}(**x**,*ω*) and **s**^{−}=**s**^{−}(**x**,*ω*) represent the source functions for downgoing and upgoing waves, respectively. We formally relate the vectors **p** and **s** to the vectors **q** and **d** in equation (2.2) as follows:
*N*×*N* matrix *N*×*N* matrices *N*/2×*N*/2 submatrices, according to
**p**^{+} and ‘upward’ propagation of subvector **p**^{−}. Note, however, that the one-way wave fields **p**^{+} and **p**^{−} are coupled via the matrix **J**=−**K****N**=**N****K**, we find the following additional symmetry property:
*x*_{3}-direction (or the probability current density) is given by

### (b) One-way reciprocity theorems

Following a similar derivation as for equations (2.18) and (2.19), but this time using the interaction quantities

### (c) One-way homogeneous Green’s function

We introduce the one-way Green’s function ** Γ**(

**x**,

**x**

_{B},

*ω*) as a

*N*×

*N*matrix obeying wave equation (3.3), with the source vector replaced by a diagonal point-source matrix, according to

**(**

*Γ***x**,

**x**

_{B},

*t*)=

**O**for

*t*<0. We subdivide the one-way Green’s matrix into

*N*/2×

*N*/2 submatrices, according to

**p**

^{+}or

**p**

^{−}) observed at

**x**; the second superscript refers to the type of source (

**s**

^{+}or

**s**

^{−}) at

**x**

_{B}(figure 2).

To find a second solution of wave equation (3.13), take the complex conjugate of this equation, and pre- and post-multiply all terms by **K**. Using the property **K****K**=**I** and equation (3.9), we obtain
**K**** Γ***(

**x**,

**x**

_{B},

*ω*)

**K**is a second solution of wave equation (3.13). Subtracting wave equations (3.13) and (3.15), we obtain

*Γ*_{h}(

**x**,

**x**

_{B},

*ω*) defined as

**K**in equation (2.13), equation (3.17) can be written as

**K**

**N**=−

**N**

**K**, it follows that the same relation holds for the one-way homogeneous Green’s function defined in equation (3.17), i.e.

### (d) Double-sided one-way homogeneous Green’s function representation

Following a similar derivation as for equation (2.33), starting with the one-way correlation-type reciprocity theorem (equation (3.12)), we find the following double-sided one-way homogeneous Green’s function representation:

## 4. Unified single-sided one-way representation

We derive a single-sided one-way representation of the homogeneous Green’s function from the one-way reciprocity theorems. In §4a, we briefly discuss the two states that will be used in the reciprocity theorems (a one-way Green’s function and a one-way focusing function). In §4b, we discuss the one-way focusing function in more detail, and in §4c, we use this focusing function and Green’s function in the one-way reciprocity theorems to derive the single-sided one-way representation of the homogeneous Green’s function.

### (a) Introduction of the two states

For the derivation of the single-sided representation, we consider another configuration than for the double-sided representations. Instead of the domain *x*_{3}=*x*_{3,0}) and **x**_{A}, hence, it is defined by *x*_{3}=*x*_{3,A}. The boundary *a*,*b*, is obsolete in the following derivations. In §4c, we will apply the one-way reciprocity theorems to domain *B* we take again the one-way Green’s function ** Γ**(

**x**,

**x**

_{B},

*ω*) that we also used in the previous section. This Green’s function is illustrated (for the acoustic case) in figure 4

*b*, with more detail than in figure 3. The orange lines in figure 4

*b*indicate some interfaces between layers with different medium parameters, and the green rays between

**x**

_{B}(inside the medium) and

**x**(at the upper boundary

**(**

*Γ***x**,

**x**

_{B},

*ω*). The green rays between

**x**

_{B}and

**x**

_{A}represent Green’s function

**(**

*Γ***x**

_{A},

**x**

_{B},

*ω*). Note that

**x**

_{B}may lie above or below

**x**

_{A}, hence inside or outside domain

For state *A*, we introduce a one-way focusing function **F**(**x**,**x**_{A},*ω*). Although a Green’s function propagates from a source at **x**_{A} to **x**, a focusing function propagates from **x** to **x**_{A}, where it focuses. In the double-sided representations, discussed in §§2 and 3, the backpropagating Green’s functions **K****G**^{†}(**x**,**x**_{A},*ω*)**K** and **J***Γ*^{†}(**x**,**x**_{A},*ω*)**J** played this role. This ‘backpropagation approach’ breaks down when the medium is accessible from one side only (here *a*. This figure shows a reference medium, which is identical to the actual medium of figure 4*b* in the domain **F**^{+}(**x**,**x**_{A},*ω*) is incident from above (we consider the acoustic case, so the focusing function is a scalar field in this example). The outer red rays indicate the direct arrival of the downward propagating focusing function, converging to the focal point at **x**_{A} (these are only two of many direct rays converging to **x**_{A}). Before arriving at the focal point, these rays are reflected upward, indicated by the blue rays and denoted as **F**^{−}(**x**,**x**_{A},*ω*). If no further measures were taken, these rays would reflect downward again, giving rise to additional rays reaching the depth level *a*, additional red rays are launched from the upper half-space into the medium, which reach the interfaces at the same positions as the upgoing blue rays, in such a way that they annihilate the aforementioned downward reflected rays. As a consequence, only the direct arrival of the downward propagating focusing function reaches **F**^{+}(**x**,**x**_{A},*ω*) is known, the remainder of this function and the entire function **F**^{−}(**x**,**x**_{A},*ω*) (both for **x** at

### (b) One-way focusing function

Here we discuss the one-way focusing function in a more formal way. We introduce **F**(**x**,**x**_{A},*ω*) as a *N*×*N*/2 matrix obeying wave equation (3.3) without a source function on the r.h.s. and with an operator matrix *N*/2×*N*/2 submatrices, according to
**x**. The downgoing focusing function **F**^{+}(**x**,**x**_{A},*ω*) (variable **x**, fixed **x**_{A}) is incident to the medium from the reflection-free upper half-space (above **F**^{−}(**x**,**x**_{A},*ω*) and vice versa due to the inhomogeneities of the medium, after which it focuses at **x**_{A}. The focusing condition is denoted as
**x**_{H}=(*x*_{1},*x*_{2}) and **x**_{H,A}=(*x*_{1,A},*x*_{2,A}) denote the horizontal coordinates of **x** and **x**_{A}, respectively. The field at *x*_{3}=*x*_{3,A} can also be written as
**T**^{+}(**x**,**x**′,*ω*) is the transmission response of the medium between **F**^{+}(**x**′,**x**_{A},*ω*) for **x**′ at **T**^{+}(**x**,**x**′,*ω*) of the medium between *a*) and, in the elastodynamic situation, for wave conversion. Because there is no sink at **x**_{A}, the focused downgoing field at **x**_{A} acts as a virtual source for downgoing waves in the half-space below *h*(**x**_{H}) is also spatially band-limited (which is the case when evanescent waves are excluded).

### (c) Single-sided one-way homogeneous Green’s function representation

We use the convolution- and correlation-type one-way reciprocity theorems (equations (3.11) and (3.12)), with *Γ*_{h}(**x**_{A},**x**_{B},*ω*). For state *A*, we substitute the one-way focusing function defined in the reference medium, hence, we replace wave vector **p**_{A}(**x**,*ω*) by matrix **F**(**x**,**x**_{A},*ω*), source vector **s**_{A}(**x**,*ω*) by zero, and operator *B*, we substitute the one-way Green’s function defined in the actual medium, hence, we replace wave vector **p**_{B}(**x**,*ω*) by matrix ** Γ**(

**x**,

**x**

_{B},

*ω*), source vector

**s**

_{B}(

**x**,

*ω*) by

**I**

*δ*(

**x**−

**x**

_{B}) (with

**x**

_{B}below

*n*

_{3}=−1 at

*H*(

*x*

_{3}) is the Heaviside step function, hence, the second term on the l.h.s. of equations (4.6) and (4.7) only contributes when

**x**

_{B}lies above

**x**

_{A}. Post-multiplying all terms in equation (4.7) by

**K**, using

**J**=

**N**

**K**and

**J**

**K**=

**N**, and subtracting the resulting equation from equation (4.6), gives

*Γ*_{h}(

**x**,

**x**

_{B},

*ω*) defined in equation (3.17). Note that the matrix

*Γ*_{h}. We can recover the complete matrix

*Γ*_{h}as follows. First we define a matrix

*Γ*_{2}by pre-multiplying the l.h.s. of equation (4.8) by

**I**

_{2}(defined below). Using equation (3.18), this gives

*Γ*_{h}(

**x**

_{A},

**x**

_{B},

*ω*), as defined in equation (3.18), is thus recovered via

*Γ*_{2}(

**x**

_{A},

**x**

_{B},

*ω*) is given by

*a*. This representation states that the one-way homogeneous Green’s function

*Γ*_{h}(

**x**

_{A},

**x**

_{B},

*ω*), with both

**x**

_{A}and

**x**

_{B}below

It is interesting to note that *Γ*_{h}(**x**,**x**_{B},*ω*) in the r.h.s. of equation (4.11) can be represented in a similar way. To this end, in equation (4.8), we replace **x** by **x**′ at **x**_{B} by **x** at **x**_{A} by **x**_{B}. This gives
**N** and transposing the result (using equation (3.20)),
**I**_{1} on the l.h.s. selects the left two submatrices of *Γ*_{h}. Post-multiplying both sides of this equation by *Γ*_{h}(**x**,**x**_{B},*ω*) is recovered as follows:
*b*. Here *Γ*_{h}(**x**,**x**′,*ω*) is a one-way homogeneous Green’s function with its source at **x**′ and receiver at **x**, both at the upper boundary.

## 5. Applications of the single-sided representation

We briefly discuss applications of the single-sided homogeneous Green’s function representation in holographic imaging and inverse scattering, time-reversed wave field propagation, and interferometric Green’s function retrieval. The discussions are not meant to be exhaustive but only indicate some applications and possible new research directions.

### (a) Holographic imaging and inverse scattering

The central process in acoustic, electromagnetic and elastodynamic imaging and inverse scattering methods is the retrieval of the wave field inside the medium from measurements carried out at the boundary of that medium. The process to obtain the wave field inside the medium is in essence a form of holography. Here, we discuss the steps that are needed to obtain the homogeneous Green’s function **G**_{h}(**x**_{A},**x**_{B},*t*), with **x**_{A} and **x**_{B} inside the medium, from **G**(**x**,**x**′,*t*), with **x** and **x**′ at the upper boundary of the medium. First, the one-way Green’s function ** Γ**(

**x**,

**x**′,

*ω*) is obtained from the Fourier transform of

**G**(

**x**,

**x**′,

*t*), according to

Next, equations (4.14) and (4.15) are used to retrieve *Γ*_{h}(**x**,**x**_{B},*ω*). The one-way homogeneous Green’s function *Γ*_{h}(**x**,**x**′,*ω*) in equation (4.15) is obtained from ** Γ**(

**x**,

**x**′,

*ω*) via equation (3.17), with

**x**

_{B}replaced by

**x**′. The time-domain version of the focusing function,

**F**(

**x**′,

**x**

_{B},

*t*), can be derived from

**(**

*Γ***x**,

**x**′,

*t*) with the Marchenko method. For the one-dimensional acoustic situation, this does not require any additional information [37,40]. For the two- and three-dimensional acoustic situation, apart from

**(**

*Γ***x**,

**x**′,

*t*), an estimate of the direct arrival between

**x**

_{B}and

**x**′ is needed to obtain

**F**(

**x**′,

**x**

_{B},

*t*), but no other information about the medium is required [38,41]. For the elastodynamic situation, also the forward scattered response between

**x**

_{B}and

**x**′ is needed [39], although in moderately inhomogeneous media a reasonable estimate of

**F**(

**x**′,

**x**

_{B},

*t*) can be obtained when only the direct arrival is known [42]. Once

**F**(

**x**′,

**x**

_{B},

*t*) has been retrieved, its Fourier transform is, according to equations (4.14) and (4.15), applied to

*Γ*_{h}(

**x**,

**x**′,

*ω*) to obtain the one-way homogeneous Green’s function

*Γ*_{h}(

**x**,

**x**

_{B},

*ω*) (figure 5

*b*). Next, according to equations (4.10) and (4.11), the Fourier transform of

**F**(

**x**,

**x**

_{A},

*t*) is applied to

*Γ*_{h}(

**x**,

**x**

_{B},

*ω*) to obtain

*Γ*_{h}(

**x**

_{A},

**x**

_{B},

*ω*) (figure 5

*a*). The combination of these two steps, i.e.

**x**′ at the boundary are focused to

**x**

_{B}, whereas in the second step the receivers at all

**x**at the boundary are focused to

**x**

_{A}. The resulting one-way homogeneous Green’s function

*Γ*_{h}(

**x**

_{A},

**x**

_{B},

*ω*) can be seen as the response to a virtual source at

**x**

_{B}, observed by a virtual receiver at

**x**

_{A}(note that

**x**

_{A}and

**x**

_{B}can be chosen anywhere inside the medium). This two-step procedure resembles a method called ‘source–receiver redatuming’ in exploration seismology [43,44], except that in that method only primary waves are accounted for, whereas equation (5.2) accounts for primaries and all orders of multiple scattering and wave conversion.

Next, the two-way homogeneous Green’s function **G**_{h}(**x**_{A},**x**_{B},*ω*) can be obtained from *Γ*_{h}(**x**_{A},**x**_{B},*ω*) via
**G**_{h}(**x**_{A},**x**_{B},*t*) is obtained by an inverse Fourier transformation. This homogeneous Green’s function can be used for imaging the internal structures or finding the local medium parameters via inverse scattering [45–47]. Note, however, that **G**_{h}(**x**_{A},**x**_{B},*t*) (for variable **x**_{A}, **x**_{B} and *t*) contains a wealth of additional information about the interior of the medium, of which the advantages need to be further explored.

An example of obtaining *G*_{h}(**x**_{A},**x**_{B},*t*) from *G*(**x**,**x**′,*t*) for the acoustic situation can be found in [9]. This example shows the evolution of *G*_{h}(**x**_{A},**x**_{B},*t*) through space and time, with multiple scattering occurring at interfaces between layers with different material parameters. Apart from imaging and inverse scattering, the elastodynamic version could be very useful to predict the propagation of microseismic signals through an unknown complex subsurface.

### (b) Time-reversed wave field propagation

In the field of time-reversal acoustics, the response to a source inside a medium is recorded at the boundary, reversed in time and emitted back from the boundary into the medium [48]. The back-propagated field focuses at the position of the source and after that the focal point acts as a virtual source. The generalization of time-reversal acoustics for other types of waves we call time-reversed wave field propagation. The back-propagated field can be quantified with the homogeneous Green’s function representation, as follows. Transposing both sides of equation (2.33), using symmetry properties (2.30) and (2.31), we obtain
**G**(**x**,**x**_{A},*t*) represents the response to a source at **x**_{A} inside the medium, observed at **x** at the boundary. According to the r.h.s. of equation (5.5), this field is reversed in time and used as a source function which is injected from all **x** at the boundary **G**(**x**_{B},**x**,*t*) propagates the field from the boundary to **x**_{B} inside the medium. The homogeneous Green’s function **G**_{h}(**x**_{B},**x**_{A},*t*) on the l.h.s. (fixed **x**_{A}, variable **x**_{B} and *t*) describes the time-dependent evolution of the injected field through the medium. Note that, according to equation (5.5), standard time-reversed wave field propagation requires that the medium is accessible from two sides. The single-sided homogeneous Green’s function representations developed in this paper provide an alternative. Because time-reversed wave field propagation is a physical process, we cannot use equation (4.11) which contains the non-physical homogeneous Green’s function under the integral on the r.h.s. Therefore, we start with equation (4.6). Post-multiplying all terms by **N** in the r.h.s. by **F**^{c}(**x**,**x**_{A},*ω*) and the modified Green’s function *Γ*^{c}(**x**,**x**_{A},*ω*) are defined as
**x** and one-way fields at **x**_{A}. In the time domain, equation (5.8) becomes
**F**^{c}(**x**,**x**_{A},*t*) should be injected into the medium, instead of the time-reversed Green’s function (as in equation (5.5)). When reflection measurements are available at the boundary, this focusing function can be obtained in a similar way as described in §5a, except that now the required direct arrival between **x**_{A} and **x** can be obtained directly from the measured response to the source at **x**_{A}.

### (c) Interferometric Green’s function retrieval

In the field of interferometric Green’s function retrieval, wave field observations at two points **x**_{A} and **x**_{B} are cross-correlated. Under specific conditions, the time-dependent cross-correlation function is proportional to the homogeneous Green’s function between **x**_{A} and **x**_{B} [8,49–51]. For general wave fields, this principle is quantified by the homogeneous Green’s function representation. Using symmetry property (2.30) (which also holds in the time domain), we obtain from equation (5.5)
**x** on **x**_{A} and **x**_{B} (actually the inline asterisk denotes a convolution, but because of the time-reversal of the second Green’s function, the integrand can be interpreted as a correlation). The l.h.s. represents the response to a virtual source at **x**_{A}, observed at **x**_{B}. When the primary sources are present only on a single boundary, the single-sided homogeneous Green’s function representations provide an alternative. Following a similar procedure as in §4b, but this time starting with equation (4.8), using equation (6.10) and symmetry relations (2.31) and (3.20), we obtain

Finally, note that instead of *Γ*_{h}(**x**_{B},**x**_{A},*t*) can be obtained, following a similar procedure as outlined at the end of §4, according to
**G**_{h}(**x**_{B},**x**_{A},*t*) can be obtained via equation (6.10).

## 6. Conclusion

We have captured double- and single-sided representations of the homogeneous Green’s function in a unified matrix notation. This notation accounts for acoustic, quantum-mechanical, electromagnetic and elastodynamic wave fields. A double-sided homogeneous Green’s function representation can only be used in situations where the medium of interest is accessible from two sides. Nevertheless, it is often used in an approximate sense in situations where the medium can be accessed from one side only. The inherent approximations are acceptable as long as the effects of multiple scattering (and wave conversion) are negligible. However, in the case of strongly inhomogeneous media (or inhomogeneous potentials in the quantum-mechanical case), the effects of multiple scattering can be quite severe. In this case, approximating a double-sided Green’s function representation by a single-sided representation leads to unacceptable errors. For example, in holographic imaging these errors manifest themselves as artefacts in the image of the interior of the medium. The single-sided homogeneous Green’s function representation, on the other hand, correctly handles multiple scattering (and wave conversion) in situations where the medium can be accessed from one side only. It employs a focusing function instead of a backward propagating Green’s function. Evanescent waves are ignored in the single-sided representation. When reflection measurements are available at the accessible boundary of the medium, the focusing function can be retrieved from these measurements and an estimate of the direct arrival between the boundary and the focal point. By employing the single-sided homogeneous Green’s function representation in a two-step procedure, sources and receivers at the boundary are focused to virtual sources and virtual receivers inside the medium. The response between these virtual sources and receivers, i.e. the homogeneous Green’s function, includes multiple scattering and wave conversion. We foresee many interesting applications of the unified single-sided homogeneous Green’s function representation in holographic imaging and inverse scattering, time-reversed wave field propagation and interferometric Green’s function retrieval.

## Authors' contributions

All authors contributed to the development of the theory and methodology presented in this paper. K.W. wrote the manuscript. All authors critically read the manuscript, provided revisions, approve the final version and agree to be accountable for all aspects of the work.

## Competing interests

We have no competing interests.

## Funding

This research is supported by the Dutch Technology Foundation STW (the applied science division of NWO and the Dutch Technology Program of the Ministry of Economic Affairs; grant nos. 13939 and 13078).

## Acknowledgements

We thank the reviewers and editor for their constructive remarks, which helped us to improve the clarity of the paper.

## Appendix A. Relationship between two- and one-way Green’s functions

Two- and one-way wave vectors obey the simple relation **q** and **p**. Consider again the two-way convolution-type reciprocity theorem (equation (2.18)). For state *A*, we replace the wave vector **q**_{A}(**x**,*ω*) by matrix **G**(**x**,**x**_{A},*ω*) and the source vector **d**_{A}(**x**,*ω*) by **I***δ*(**x**−**x**_{A}), with **x**_{A} in **x**_{A} lies between these boundaries. For state *B*, we replace **q**_{B}(**x**,*ω*) by **q**(**x**,*ω*) and assume that this wave field obeys a source-free wave equation. We take the medium parameters the same for both states, hence, the last integral in equation (2.18) vanishes. Making the mentioned substitutions in the remaining integrals and using symmetry relation (2.30), we obtain
**q**(**x**,*ω*) and any integration boundary **x**_{A}, we find

- Received March 2, 2016.
- Accepted May 17, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.