Waves in space-dependent and time-dependent materials: a systematic comparison

Waves in space-dependent and time-dependent materials obey similar wave equations, with interchanged time- and space-coordinates. However, since the causality conditions are the same in both types of material (i.e., without interchangement of coordinates), the solutions are dissimilar. We present a systematic treatment of wave propagation and scattering in 1D space-dependent and time-dependent materials. After formulating unified equations, we discuss Green's functions and simple wave field representations for both types of material. Next we discuss propagation invariants, i.e., quantities that are independent of the space coordinate in a space-dependent material (such as the net power-flux density) or of the time coordinate in a time-dependent material (such as the net field-momentum density). A discussion of reciprocity theorems leads to the well-known source-receiver reciprocity relation for the Green's function of a space-dependent material and a new source-receiver reciprocity relation for the Green's function of a time-dependent material. A discussion of general wave field representations leads to the well-known expression for Green's function retrieval from the correlation of passive measurements in a space-dependent material and a new expression for Green's function retrieval in a time-dependent material. After an introduction of a matrix-vector wave equation, we discuss propagator matrices for both types of material. Since the initial condition for a propagator matrix in a time-dependent material follows from the boundary condition for a propagator matrix in a space-dependent material by interchanging the time- and space-coordinates, the propagator matrices for both types of material are interrelated in the same way. This also applies to representations and reciprocity theorems involving propagator matrices.


INTRODUCTION
A wave that encounters a temporal change of material parameters (a so-called time boundary) undergoes reflection and transmission (Xiao et al. 2014), similar to a wave that is incident on a spatial change of material parameters (a space boundary).Although research on wave propagation and scattering in time-dependent materials has been around for several decades (Morgenthaler 1958;Jiang 1975), recent advances in the construction of dynamic metamaterials have given this field of research a significant boost (Caloz & Deck-Léger 2020a).Whereas most applications concern electromagnetic waves (Mounaix et al. 2020;Apffel & Fort 2022;Moussa et al. 2023), mechanical waves show a similar scattering behaviour when confronted with a temporal change of parameters (Bacot et al. 2016;Fink & Fort 2017;Peng et al. 2020;Hidalgo-Caballero et al. 2023).In particular, Fink and coworkers (Bacot et al. 2016;Fink & Fort 2017) show that water waves propagate back to their point of origin when the restoring force responsible for wave propagation (gravity), and hence the propagation velocity, is temporarily changed by a vertical acceleration.
Several authors have discussed the analogy between the underlying equations for spacedependent and for time-dependent materials (Xiao et al. 2014;Torrent et al. 2018;Mendonça & Shukla 2002;Salem & Caloz 2015;Caloz & Deck-Léger 2020b).For example, the roles of the time-and space-coordinates in the 1D wave equation for a space-dependent material are interchanged in the 1D wave equation for a time-dependent material.Despite the simple relations between the wave equations, the relation between the solutions of these equations (i.e., the wave fields in space-dependent and in time-dependent materials) is less straightforward.The reason for this is that the causality conditions are the same in both types of material.Only when the initial conditions and boundary conditions would be interchanged (along with the interchangement of time-and space-coordinates in the wave equations), the solutions would obey a simple relation as well.
The aim of this paper is to discuss a number of fundamental aspects of wave propagation and scattering in space-dependent and in time-dependent materials and compare these in a systematic way.Our discussion partly overlaps with earlier reviews, such as the excellent paper by Caloz & Deck-Léger (2020b), but we also discuss new results.We use a unified notation for different wave phenomena (electromagnetic, acoustic, elastodynamic), so that all relations discussed in this paper hold simultaneously for these phenomena.For simplicity, we restrict ourselves to 1D waves only.We discuss Green's functions, propagation invariants, reciprocity theorems, wave field representations and expressions for Green's function retrieval.
In most of these cases, the derived solutions for space-dependent and time-dependent materials are not exchangeable as a result of non-exchangeable causality conditions.We also discuss propagator matrices for space-dependent and time-dependent materials and show that they are completely exchangeable as a result of interchangeable boundary and initial conditions.Finally, we discuss Marchenko-type focusing functions for both types of material and show that they are also exchangeable.

UNIFIED BASIC EQUATIONS AND CONSTITUTIVE RELATIONS FOR 1D WAVE FIELDS
Throughout this paper, we consider 1D wave fields as a function of space (denoted by x) and time (denoted by t).We take x increasing towards the right.Using analogies between electromagnetic, acoustic and elastodynamic waves (Carcione & Cavallini 1995;de Hoop 1995;Wapenaar et al. 2001;Carcione & Robinson 2002;de Hoop & Lager 2014;Burns et al. 2020), the basic equations in a unified notation are where ∂ x and ∂ t denote partial derivatives with respect to space and time, respectively, U (x, t), V (x, t), P (x, t) and Q(x, t) are space-and time-dependent wave-field quantities and a(x, t) and b(x, t) are space-and time-dependent source quantities, see Table 1.The wave-field quantities are mutually related via the following constitutive equations where α(x, t) and β(x, t) are the parameters of space-and time-dependent materials, see Table 1.Rows 1 and 2 contain the quantities for electromagnetic wave propagation, with TE standing for transverse electric and TM for transverse magnetic.The quantities are electric flux densities D y (x, t) and D z (x, t), magnetic flux densities B y (x, t) and B z (x, t), electric field strengths E y (x, t) and E z (x, t), magnetic field strengths H y (x, t) and H z (x, t), permittivity ε(x, t), permeability µ(x, t), external electric current densities J e y (x, t) and J e z (x, t) and external magnetic current densities J m y (x, t) and J m z (x, t).The quantities in row 3, associated to acoustic wave propagation in a fluid material, are dilatation Θ(x, t), longitudinal mechanical momentum density m x (x, t), acoustic pressure p(x, t), longitudinal particle velocity v x (x, t), compressibility κ(x, t), mass density ρ(x, t), volume-injection rate density q(x, t) and external longitudinal force density f x (x, t).For horizontally polarised shear (SH) waves in a solid material, we have in row 4 transverse mechanical momentum density m y (x, t), shear strain e yx (x, t), transverse particle velocity v y (x, t), shear stress τ yx (x, t), mass density ρ(x, t), shear modulus µ(x, t), external transverse force density f y (x, t) and external shear deformation rate density h yx (x, t).

WAVE EQUATIONS AND GREEN'S FUNCTIONS
In this and subsequent sections, the first subsection reviews a specific subject for a spacedependent material.This serves as an introduction to the second subsection, which discusses the same subject for a time-dependent material, including the analogies and differences.

Space-dependent material
We consider a space-dependent material that is constant over time, with parameters α(x) and β(x).Substituting the constitutive equations ( 3) and (4) into the basic equations ( 1) and (2), using the fact that α(x) and β(x) are independent of time, gives For a space-dependent material with piecewise continuous parameters, these equations are supplemented with boundary conditions at all points where α(x) and β(x) undergo a finite jump.The boundary conditions are that P (x, t) and Q(x, t) are continuous at those points (see A1 for a review of reflection and transmission coefficients at a space boundary).
We obtain a second order wave equation for the field P (x, t) by eliminating Q(x, t) from equations ( 5) and ( 6), according to with propagation velocity c(x) given by with space-dependent parameters α(x) and β(x).We also define where η stands for impedance in the case of TE and acoustic waves (rows 1 and 3 of Table 1) or admittance in the case of TM and SH waves (rows 2 and 4 of Table 1).
We define the Green's function G x (x, x 0 , t) as the response to an impulsive point source δ(x − x 0 )δ(t), hence with causality condition This condition implies that G x (x, x 0 , t) is outward propagating for |x| → ∞.The subscript x in G x denotes that this is the Green's function of a space-dependent material.
A simple representation for P (x, t) is obtained when P and G x are defined in the same material and both are outward propagating for |x| → ∞.Whereas P (x, t) is the response to an arbitrary source distribution ∂ t a(x, t) (equation ( 7), assuming b = 0), G x (x, x 0 , t) is the response to an impulsive point source at an arbitrary location x 0 at t = 0 (equation ( 10)).
Because equations ( 7) and ( 10) are linear, a representation for P (x, t) follows by applying Huygens' superposition principle.Assuming ∂ t a(x, t) is causal, i.e., ∂ t a(x, t) = 0 for t < 0, this gives (Morse & Feshbach 1953;Bleistein 1984) This representation is a special case of the more general representation for a space-dependent material, derived in section 6.1.We discuss a numerical example of an acoustic Green's function for a piecewise homogeneous material, consisting of five homogeneous slabs, each with a thickness of 40 mm.The propagation velocities are 1.0, 1.0, 2.0, 1.0 and 2.2 km/s, respectively.The half-spaces to the left and the right of the space-dependent material are homogeneous, with the same velocities causality condition < l a t e x i t s h a 1 _ b a s e 6 4 = " J J 0 B m 8 g j f r y X q x 3 q 2 P + W r F K m 8 O w R 9 Y n z 8 + N p f w < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " F 5 Green's function G x (x, x 0 , t) (convolved with a temporal wavelet) for a piecewise homogeneous space-dependent material.
as the first and last slab, respectively.The parameter β is constant throughout.The source is located between the first and the second slab, at x 0 = 40 mm.We use a recursive "layer-code" method (Kennett 1983) to model the response to this source.Figure 1 shows an x, t-diagram of G x (x, x 0 , t), convolved in time with a temporal wavelet with a central frequency ω 0 /2π = 300 kHz.The causality condition of equation ( 11) implies that the Green's function is zero above the green line at t = 0.The red arrows indicate the rightward propagating primary wave and the blue arrows the leftward propagating primary reflections.Multiply scattered waves are also clearly visible.Note that the field is outward propagating for x = 0 and x = 200 mm (and hence for |x| → ∞, since the left and right half-spaces are homogeneous).

Time-dependent material
We consider a time-dependent homogeneous material with parameters α(t) and β(t).Substituting the constitutive equations ( 3) and (4) into the basic equations ( 1) and (2), using the fact that α(t) and β(t) are independent of space, gives For a time-dependent material with piecewise continuous parameters, these equations are supplemented with boundary conditions at all time instants where α(t) and β(t) undergo a finite jump.The boundary conditions are that U (x, t) and V (x, t) are continuous at those time instants (see A2 for a review of reflection and transmission coefficients at a time boundary).
We obtain a second order wave equation for the field U (x, t) by eliminating V (x, t) from equations ( 13) and ( 14), according to with propagation velocity c(t) again given by equation ( 8), this time with time-dependent parameters α(t) and β(t).
Note that equations ( 5), ( 6) and ( 7) can be transformed into equations ( 14), ( 13) and ( 15) and vice-versa, by the following mapping We define the Green's function G t (x, t, t 0 ) as the response to an impulsive point source with causality condition This condition implies that G t (x, t, t 0 ) is outward propagating for |x| → ∞.The subscript t in G t denotes that this is the Green's function of a time-dependent material.
A simple representation for U (x, t) is obtained when U and G t are defined in the same material and both are outward propagating for |x| → ∞.Whereas U (x, t) is the response to a source distribution −∂ x b(x, t) (equation ( 15), assuming a = 0), G t (x, t, t 0 ) is the response to an impulsive point source at x = 0 at an arbitrary time t 0 (equation ( 17)).Because equations ( 15) and ( 17) are linear, a representation for U (x, t) follows by applying Huygens' superposition   principle.Assuming ∂ Using equation (3), a representation for P (x, t) follows from P (x, t) = 1 α(t) U (x, t), with U (x, t) given by equation ( 19).The representation of equation ( 19) is a special case of the more general representation for a time-dependent material, derived in section 6.2.We discuss a numerical example of an acoustic Green's function for a piecewise constant material, consisting of five time-independent slabs.Following the mapping of equation ( 16), we "construct" this material from the material used for the numerical example in section 3.1, with time and space interchanged and with the reciprocal propagation velocities.For convenience, we define 1 km as the unit of distance and 1 km/s as the unit of velocity.With this definition, the reciprocal propagation velocities are 1.0, 1.0, 0.5, 1.0 and 0.455 km/s, respectively.The half-times before and after the time-dependent material are constant, with the same velocities as the first and last slab, respectively.The parameter β is again constant throughout.Note that 1 mm, which is actually 1 µkm, maps to 1 µs and vice-versa.Hence, the slab thickness of 40 mm is mapped to a slab duration of 40 µs.The source is located between the first and second slab, at t 0 = 40 µs. Figure 2a shows an x, t-diagram of G t (x, t, t 0 ), convolved in space with a spatial wavelet with a central wavenumber k 0 /2π = 300 * 10 3 km −1 .The causality condition of equation ( 18) implies that the Green's function is zero left of the green line at t = t 0 .The red arrows indicate the rightward propagating primary wave (i.e., in the +x direction) and the blue arrows the leftward propagating primary reflections (in the −x direction).Multiply scattered waves are also clearly visible.
Since the causality conditions (equations ( 11) and ( 18)) do not follow the mapping of equation (16), the x, t-diagrams of the Green's functions for space-dependent and time-dependent materials (Figures 1 and 2a) are very different.For the specially designed case considered here (with reciprocal velocities), only the rightward propagating primary waves (indicated by the red arrows) exhibit interchangeable kinematical behaviour between the two cases (but they have different amplitudes).All other events are different in these figures.Whereas the multiply scattered waves in G x (x, x 0 , t) in Figure 1 consist of ongoing reverberations between space boundaries, the multiply scattered waves in G t (x, t, t 0 ) in Figure 2a are the result of "forward-in-time" reflections and transmissions at time boundaries (see also Figure A1 in the Appendix); their total number is finite.In section 9 we discuss propagator matrices for spacedependent and time-dependent materials and show that these follow the mapping of equation ( 16) for all events.Moreover, in section 9.2 we show how G t (x, t, t 0 ) is related to one of the elements of the propagator matrix for a time-dependent material.
Finally, Figure 2b is an example of an acoustic Green's function G t (x, t, t 0 ) (convolved with the same spatial wavelet as in Figure 2a) of a sinusoidally modulated time-dependent material, with propagation velocity c(t) = 1 + 1 10 sin(πt/2) km/s (with t in µs, ranging from t 0 = 0 µs to t N = 100 µs) and constant β.For the modeling we divided the velocity profile into 4000 constant velocity slabs with a duration of 0.025 µs each.Note the complex scattering behaviour and the increasing amplitudes of the scattered events with time.To compensate for the increasing amplitudes, Torrent et al. (2018) introduce a dissipative time-dependent material.

Space-dependent material
We review propagation invariants for a space-dependent material with parameters α(x) and β(x).Given a space-and time-dependent function P (x, t), we define its temporal Fourier transform as with ω denoting the angular frequency and i the imaginary unit.With this definition, derivatives with respect to time transform to multiplications with −iω.Hence, equations ( 5) and (6) transform to In the following we consider two independent states, indicated by subscripts A and B, obeying equations ( 21) and ( 22).In the most general case, sources, material parameters and wave fields may be different in the two states.We derive relations between these states.First we consider the quantity ∂ x { PA QB − QA PB }.Applying the product rule for differentiation, using equations ( 21) and ( 22) for states A and B to get rid of the derivatives, we obtain This is the local reciprocity theorem of the time-convolution type (de Hoop 1995;Fokkema & van den Berg 1993), in which products like PA QB correspond to convolutions along the time coordinate in the x, t-domain.Next, we consider the quantity ∂ x { P * A QB + Q * A PB } (where the asterisk denotes complex conjugation) and apply the same operations, yielding This is the local reciprocity theorem of the time-correlation type (de Hoop 1995;Bojarski 1983), in which products like P * A QB correspond to correlations along the time coordinate in the x, t-domain.In section 5.1 we will use equations ( 23) and ( 24) as the basis for deriving global reciprocity theorems of the time-convolution and time-correlation type.Here we use these equations to derive propagation invariants for a space-dependent material.To this end we take identical material parameters in states A and B and we assume that there are no sources.With this, equations ( 23) and ( 24) simplify to hence, PA QB − QA PB and P * A QB + Q * A PB are space-propagation invariants, i.e., they are independent of the space coordinate x.This holds for continuously varying material parameters α(x) and β(x).For a material with piecewise continuous parameters, the boundary conditions state that P and Q are continuous at all points where α(x) and β(x) are discontinuous.This implies that the space-propagation invariants also hold for a space-dependent material with piecewise continuous parameters.Propagation invariants find applications in the analysis of symmetry properties of reflection and transmission responses and have been used for the design of efficient numerical modelling schemes (Haines 1988;Kennett et al. 1990;Koketsu et al. 1991;Takenaka et al. 1993).
For the special case that the wave fields in states A and B are identical, we may drop the subscripts A and B. The first space-propagation invariant then vanishes and is no longer useful.The second space-propagation invariant simplifies to 2ℜ{ P * Q}, where ℜ denotes the real part.We define ȷ(x, ω) = 1 2 ℜ{ P * Q} as the net power-flux density in the x-direction in the x, ω-domain.Hence, the net power-flux density is conserved (i.e., it is independent of x) in a space-dependent material with piecewise continuous parameters.

Time-dependent material
We derive propagation invariants for a time-dependent material with parameters α(t) and β(t).
Given a space-and time-dependent function U (x, t), we define its spatial Fourier transform as with k denoting the wavenumber.Following common conventions, we use opposite signs in the exponentials in the temporal and spatial Fourier transforms (equations ( 20) and ( 27)).
With this definition, derivatives with respect to space transform to multiplications with ik.
Hence, equations ( 13) and ( 14) transform to Note that equations ( 21) and ( 22) can be transformed into equations ( 29) and ( 28) and vice-versa, by the following modified mapping In the following we consider two independent states, indicated by subscripts A and B, obeying equations ( 28) and ( 29).In the most general case, sources, material parameters and wave fields may be different in the two states.Applying the mapping of equation (30) to equations ( 23) and (24) yields the local reciprocity theorems of the space-convolution and space-correlation type, in which products like ǓA VB and Ǔ * A VB correspond to convolutions and correlations, respectively, along the space coordinate in the x, t-domain.In section 5.2 we derive global reciprocity theorems of the space-convolution and space-correlation type.Here we derive propagation invariants for a time-dependent material.To this end we take identical material parameters in states A and B and we assume that there are no sources.Applying the mapping of equation (30) to equations ( 25) and ( 26) yields hence, ǓA VB − VA ǓB and Ǔ * A VB + V * A ǓB are time-propagation invariants, i.e., they are independent of the time coordinate t.This holds for continuously varying material parameters α(t) and β(t).For a material with piecewise continuous parameters, the boundary conditions state that Ǔ and V are continuous at all time instants where α(t) and β(t) are discontinuous.
This implies that the time-propagation invariants also hold for a time-dependent material with piecewise continuous parameters.
For the special case that the wave fields in states A and B are identical, we may drop the subscripts A and B. The first time-propagation invariant then vanishes and is no longer useful.
The second time-propagation invariant simplifies to 2ℜ{ Ǔ * V }.We define M (k, t) = 1 2 ℜ{ Ǔ * V } as the net field-momentum density (Burns et al. 2020;Feynman et al. 1963) in the x-direction in the k, t-domain (to be distinguished from the mechanical momentum densities m x and m y in Table 1).Hence, the net field-momentum density is conserved (i.e., it is independent of t) in a time-dependent material with piecewise continuous parameters.
Using equations (3), ( 4) and ( 8), we obtain for the net power-flux density ȷ(k, t) in the x-direction in the k, t-domain, defined as Hence, whereas the net field-momentum density M (k, t) is conserved, the net power-flux density ȷ(k, t) is not conserved (i.e., it is dependent on t) in a time-dependent material.This is explained as the result of energy being added to or extracted from the wave field by the external source that modulates the material parameters (Morgenthaler 1958;Mendonça & Shukla 2002; Caloz & Deck-Léger 2020b).

Space-dependent material
We review general reciprocity theorems for a space-dependent material with piecewise continuous parameters α(x) and β(x).Integrating both sides of equations ( 23) and ( 24) from x b to x e (with subscripts b and e standing for "begin" and "end"), taking into account that P and Q are continuous at points where α(x) and β(x) are discontinuous, yields These are the global reciprocity theorems of the time-convolution and time-correlation type, respectively, for a space-dependent material (de Hoop 1995; Fokkema & van den Berg 1993;Bojarski 1983;Rayleigh 1878;Lorentz 1895).We use equation ( 34) in section 6.1 to derive a general wave field representation and in section 7.1 we use equation ( 35) to derive an expression for Green's function retrieval, both for space-dependent materials.Here we discuss two special cases of equations ( 34) and ( 35).
First we derive an expression for source-receiver reciprocity of the Green's function of a space-dependent material from equation (34).To this end we take identical material parameters in both states, i.e., α A = α B = α and β A = β B = β and we assume that the material is homogeneous for x ≤ x b and for x ≥ x e .For state A we take a Green's state with a unit source at x A between x b and x e , hence, we substitute For state B we take a Green's state with a unit source at x B between x b and x e , and we substitute similar expressions.At x b and x e the field is leftward and rightward propagating, respectively (see for example Figure 1, with x b = 0 and x e = 200 mm), i.e., proportional to exp(−iωx/c(x b )) and exp(iωx/c(x e )), respectively.Hence with η defined in equation ( 9), and similar expressions for state B. With this, the left-hand side of equation ( 34) vanishes.From the remaining terms on the right-hand side we obtain Ĝx or, in the space-time domain, which formulates the classical source-receiver reciprocity relation for a space-dependent material (de Hoop 1995; Morse & Feshbach 1953;Hoenders 1979).These expressions remain valid for arbitrary x A and x B when taking x b → −∞ and x e → ∞.
Second, we derive a power balance for a space-dependent material from equation (35).
Taking (the right-hand side) is equal to the power leaving this region (the left-hand side).Hence, this equation formulates the power balance for a space-dependent material.

Time-dependent material
We derive general reciprocity theorems for a time-dependent material with piecewise continuous parameters α(t) and β(t).Applying the mapping of equation ( 30) to equations ( 34) and (35) yields These are the global reciprocity theorems of the space-convolution and space-correlation type, respectively, for a time-dependent material.We use equation ( 41) in section 6.2 to derive a general wave field representation and in section 7.2 we use equation ( 42) to derive an expression for Green's function retrieval, both for time-dependent materials.Here we discuss two special cases of equations ( 41) and ( 42).
First we derive an expression for source-receiver reciprocity of the Green's function of a time-dependent material from equation (41).Since the causality conditions for the Green's functions do not obey the mapping of equation ( 16), the derivation of source-receiver reciprocity is different from that in section 5.1.In particular, for the Green's function of a timedependent material there is not an equivalent of leftward and rightward propagating waves at t b and t e , respectively (see for example Figure 2a, with t b = 0 and t e = 200 µs).We take again ).With these definitions, the acausal Green's function is zero for t = t e and the causal Green's function is zero for t = t b (the latter is seen for example in Figure 2a, with t b = 0).With this, the left-hand side of equation ( 41) vanishes.
From the remaining terms on the right-hand side we obtain or, in the space-time domain, which formulates source-receiver reciprocity for a time-dependent material.These expressions remain valid for arbitrary t A and t B when taking t b → −∞ and t e → ∞.Note that for t A < t B these expressions reduce to the trivial relation 0 = 0.
Second, we derive a field-momentum balance for a time-dependent material from equation ( 42).Taking identical states A and B, we drop the subscripts A and B. Equation ( 42) thus yields This equation states that the field momentum generated by sources in the interval between t b and t e (the right-hand side) is equal to the field momentum leaving this interval (the lefthand side).Hence, this equation formulates the field-momentum balance for a time-dependent material.

Space-dependent material
We review a general wave field representation for a space-dependent material with piecewise continuous parameters α(x) and β(x).Our starting point is the global reciprocity theorem of the time-convolution type for a space-dependent material, formulated by equation (34).For state A we take the Green's state, hence, we substitute For state B we take the actual field and drop the subscripts B. Substitution into equation (34), using reciprocity relation (38), this yields the following classical representation with and where χ(x A ) is the characteristic function, defined as Equation ( 46) is a generalisation of equation ( 12), transformed to the frequency domain.It expresses the wave field at any point x A between x b and x e (including these points).The first term on the right-hand side accounts for the contribution of the sources between x b and x e , the second term describes scattering due to the material contrast functions ∆α(x) and ∆β(x), and the last term describes contributions from the fields at x b and x e .The representation of equation ( 46) finds applications in the analysis of wave scattering problems in space-dependent materials (de Hoop 1995; Morse & Feshbach 1953;Bleistein 1984;Born & Wolf 1965;Oristaglio 1989).

Time-dependent material
We derive a general wave field representation for a time-dependent material with piecewise continuous parameters α(t) and β(t).Our starting point is the global reciprocity theorem of the space-convolution type for a time-dependent material, formulated by equation ( 41).In anticipation of using the reciprocity relation ( 43), for state A we take the acausal Green's state, hence, we substitute For state B we take the actual field and drop the subscripts B. Substitution into equation ( 41), using reciprocity relation ( 43), this yields the following representation and where χ(t A ) is the characteristic function, defined as Equation ( 50) is a generalisation of equation ( 19), transformed to the wavenumber domain.
It expresses the wave field at any time t A between t b and t e (including these time instants).
The first term on the right-hand side accounts for the contribution of the sources between t b and t e , the second term describes scattering due to the material contrast functions ∆α −1 (t) and ∆β −1 (t), and the last term describes contributions from the fields at t b and t e (with the contribution from the field at t e being zero when t A < t e ).The representation of equation ( 50) finds potential applications in the analysis of wave scattering problems in time-dependent materials.

GREEN'S FUNCTION RETRIEVAL
7.1 Space-dependent material Under specific circumstances, the correlation of passive wave measurements at two receivers yields the response to a virtual impulsive source at the position of one of these receivers, observed by the other receiver (i.e., the Green's function between the receivers).This concept has found numerous applications in ultrasonics (Weaver & Lobkis 2001, 2002;Malcolm et al. 2004), seismology (Campillo & Paul 2003;Wapenaar 2003;Snieder 2004;Schuster et al. 2004 Following the approach of Wapenaar & Fokkema (2006), we use the global reciprocity theorem of the time-correlation type (equation ( 35)) to derive an expression for Green's function retrieval for a space-dependent material with piecewise continuous parameters α(x) and β(x).To this end, we take identical material parameters in both states, i.e., α A = α B = α and β A = β B = β and we assume that the material is homogeneous for x ≤ x b and for x ≥ x e .For state A we take a Green's state with a unit source at x A between x b and x e , and we substitute −iωâ A (x, ω) = δ(x − x A ), bA (x, ω) = 0 and PA (x, ω) = Ĝx (x, x A , ω); for QA (x, ω) we use equations ( 36) and (37).For state B we take a Green's state with a unit source at x B between x b and x e , and we substitute similar expressions.Furthermore, we use the source-receiver reciprocity relation of equation ( 38).This yields where ℑ denotes the imaginary part.Applying an inverse temporal Fourier transform yields The right-hand side is the time derivative of the superposition of time correlations of mea-surements by receivers at positions x A and x B , in response to impulsive sources at x b and x e .
The left-hand side is the causal Green's function G x (x B , x A , t) between x A and x B , minus its time-reversed version.Hence, the Green's function G x (x B , x A , t) is retrieved by evaluating the right-hand side of this expression and taking the causal part.Note that this is independent of the positions x b and x e of the sources, as long as the receivers are located between these sources and the material left of x b and right of x e is homogeneous.When the impulsive sources at x b and x e are replaced by uncorrelated noise sources, the retrieved response is the Green's function G x (x B , x A , t), convolved with the autocorrelation of the noise.

Time-dependent material
Following a similar approach as in section 7.1, we use the global reciprocity theorem of the space-correlation type (equation ( 42)) to derive an expression for Green's function retrieval for a time-dependent material with piecewise continuous parameters α(t) and β(t).We take again α A = α B = α and β A = β B = β.For state A we take an acausal Green's state with a unit sink at t A between t b and t e and we substitute −ik bA (k, t) = δ(t − t A ), ǎA (k, t) = 0, ǓA (k, t) = Ǧa t (k, t, t A ) and VA (k, t) = − β(t) ik ∂ t Ǧa t (k, t, t A ).For state B we take an acausal Green's state with a unit sink at t B between t b and t e , and we substitute similar expressions.
Furthermore, we use the source-receiver reciprocity relation of equation ( 43).This yields Applying an inverse spatial Fourier transform yields The right-hand side is the superposition of space correlations of measurements by receivers at time instants t A and t B , in response to an impulsive source at t b and its time derivative and vice-versa.The left-hand side is the causal Green's function G t (x, t B , t A ) between t A and t B , minus its space-reversed acausal counterpart.Hence, the Green's function G t (x, t B , t A ) (when ) is retrieved by evaluating the right-hand side of this expression.Note that this is independent of the time instant t b of the source, as long t A and t B are both larger than t b .Unlike the two-sided representation of equation ( 55), which requires sources at x b and x e , this is a single-sided representation, which requires sources at t b only.Note that the time-derivatives in equation ( 57) act on a superposition of left-and right-going waves at t b (see for example Figure 2, with t b = t 0 ), hence, we cannot use an expression similar to equation ( 36) to simplify the right-hand side of equation ( 57) further.

Space-dependent material
For a space-dependent material with continuous parameters α(x) and β(x), equations ( 5) and ( 6) can be combined into the following matrix-vector wave equation in the x, t-domain with wave field vector q x (x, t), operator matrix A x (x, t) and source vector d x (x, t) defined as For a space-dependent material with piecewise continuous parameters, this equation is supplemented with boundary conditions at all points where α(x) and β(x) are discontinuous.The boundary condition is that q x (x, t) is continuous at those points.
Using the temporal Fourier transform defined in equation ( 20), we obtain the following matrix-vector wave equation in the x, ω-domain with wave field vector qx (x, ω), matrix Âx (x, ω) and source vector dx (x, ω) defined as Note that matrix Âx (x, ω) obeys the following symmetry properties where superscript t denotes transposition, superscript † denotes transposition and complex conjugation, and where

Time-dependent material
Applying the mapping of equation ( 16) to equations ( 58) and ( 59) yields the following matrixvector wave equation in the x, t-domain for a time-dependent material with continuous parameters α(t) and β(t) with wave field vector q t (x, t), operator matrix A t (x, t) and source vector d t (x, t) defined as For a time-dependent material with piecewise continuous parameters, this equation is supplemented with boundary conditions at all time instants where α(t) and β(t) are discontinuous.
The boundary condition is that q t (x, t) is continuous at those time instants.
Applying the mapping of equation ( 30) to equations ( 60) and ( 61), we obtain the following matrix-vector wave equation in the k, t-domain Matrix Ǎt obeys the same symmetries as Âx , as formulated by equations ( 62)-( 64).

Space-dependent material
For a space-dependent material, a propagator matrix "propagates" a wave field (represented as a vectorial quantity) from one plane in space to another (Thomson 1950;Haskell 1953;Gilbert & Backus 1966).It has found many applications, particularly in elastodynamic wave problems (Kennett 1983(Kennett , 1972;;Woodhouse 1974).
We define the propagator matrix W x (x, x 0 , t) for a space-dependent material with continuous parameters α(x) and β(x) as the solution of matrix-vector equation ( 58) without the source term, hence with operator matrix A x (x, t) defined in equation ( 59) and with boundary condition where I is the identity matrix.
A simple representation for the wave field vector q x (x, t) obeying equation ( 58) is obtained when q x and W x are defined in the same source-free material between x 0 and x (where x can be either larger or smaller than x 0 ).Whereas q x (x, t) can have any time-dependency at x = x 0 , W x (x, x 0 , t) collapses to Iδ(t) at x = x 0 .Because equations ( 58) and ( 70) are linear, a representation for q x (x, t) follows by applying Huygens' superposition principle, according to Note that W x (x, x 0 , t) propagates the wave field vector q x from x 0 to x, hence the name "propagator matrix".We partition W x (x, x 0 , t) as follows The first and second superscripts refer to the wave field quantities in vector q x , defined in equation ( 59), at x and x 0 , respectively.For more general representations with propagator matrices, including source terms and differences in material parameters (analogous to the representation with Green's functions discussed in section 6.1), see Wapenaar (2022).
Using the temporal Fourier transform defined in equation ( 20), we obtain the following space-frequency domain equation for Ŵx (x, x 0 , ω) with matrix Âx (x, ω) defined in equation ( 61) and with boundary condition The representation of equation ( 72) transforms to qx (x, ω) = Ŵx (x, x 0 , ω)q x (x 0 , ω). (76) By applying this equation recursively, it follows that Ŵx obeys the following recursive expression where are points where the material parameters may be discontinuous.As a special case of equation ( 77) we obtain from which it follows that Ŵx (x n−1 , x n , ω) is the inverse of Ŵx (x n , x n−1 , ω).For a homogeneous slab between x n−1 and x n , with parameters α n , β n , c n = 1/ √ α n β n , η n = β n /α n and thickness ∆x n = x n − x n−1 , we have From equation ( 77), we obtain a similar recursive expression in the space-time domain, according to where * t denotes a time convolution (more formally defined in equation ( 72)).For a homogeneous slab between x n−1 and x n , we find from equations ( 79)-( 82) For the same piecewise homogeneous material as used for the numerical example in section 3.1, the elements W P,P x (x, x 0 , t) and W P,Q x (x, x 0 , t) for x 0 = 40 mm (convolved with a temporal wavelet with a central frequency ω 0 /2π = 300 kHz) are shown as x, t-diagrams in Figures 3a and 3b.The green lines indicate the boundary conditions W P,P x (x 0 , x 0 , t) = δ(t) and W P,Q x (x 0 , x 0 , t) = 0 (equations ( 71) and ( 73)).Note that these figures clearly exhibit the recursive character, described by equation ( 83).
< l a t e x i t s h a 1 _ b a s e 6 4 = " F 5 < l a t e x i t s h a 1 _ b a s e 6 4 = " / X R C p 9 b p N F T B 5 y 5 B d m r x D / 8 w a p j p q + y Z e k m g g 8 f x S l D O o Y F m 3 A k E q C N Z s a g r A 0 0 T H E Y y Q R 1 q a z m i n B X Y y 8 T L o X D f e 6 c X V / W W 8 1 y z q q 4 B i c g D P g g h v Q A n e g D T o A g x w 8 g 1 f w Z j 1 Z L 9 a 7 9 T F f r V j l z S H 4 A + v z B 4 Z z l v 0 = < / l a t e x i t > initial condition     x (x, x 0 , t) (convolved with a temporal wavelet) for a piecewise homogeneous space-dependent material.The labels at the vertical axes denote time (in µs) and those at the horizontal axes denote space (in mm).With interchanged labels (and "boundary condition" replaced by "initial condition") these figures can be interpreted as W U,U t (x, t, t 0 ) (a) and W U,V t (x, t, t 0 ) (b) (convolved with a spatial wavelet) for a piecewise constant time-dependent material.

Time-dependent material
For a time-dependent material, a propagator matrix propagates a wave field from one instant in time to another.In the literature on time-dependent materials this matrix is usually called the transfer matrix (Torrent et al. 2018;Salem & Caloz 2015;Pacheco-Peña & Engheta 2020), but for consistency with section 9.1, we hold on to the name propagator matrix.
We define the propagator matrix W t (x, t, t 0 ) for a time-dependent material with continuous parameters α(t) and β(t) as the solution of matrix-vector equation ( 66) without the source term, hence with operator matrix A t (x, t) defined in equation ( 67) and with initial condition Note that the mapping of equation ( 16) not only applies to the wave equations (equations ( 70) and ( 88)), but also to the boundary and initial conditions (equations ( 71) and ( 89)).
Consequently, the mappings of equations ( 16) and ( 30) also apply to all expressions for the propagator matrix in the space-time and Fourier-domains, respectively.We discuss a few of these mappings explicitly.
The representation of equation ( 72) maps to Note that W t (x, t, t 0 ) propagates the wave field vector q t from t 0 to t.We partition W t (x, t, t 0 ) as follows The first and second superscripts refer to the wave field quantities in vector q t , defined in equation ( 67), at t and t 0 , respectively.The recursive expression of equation ( 83) maps to where * x denotes a space convolution (more formally defined in equation ( 90)).For a constant slab between t n−1 and t n with duration ∆t n = t n − t n−1 , we find from equations ( 84) -( 87) 10 MATRIX-VECTOR RECIPROCITY THEOREMS 10.1 Space-dependent material We review matrix-vector reciprocity theorems for a space-dependent material with piecewise continuous parameters α(x) and β(x).We consider two independent states, indicated by subscripts A and B, obeying equation ( 60), and we derive relations between these states.In the most general case, sources, material parameters and wave fields may be different in the two states.We consider the quantities ∂ x {q t x,A (x, ω)Nq x,B (x, ω)} and ∂ x {q † x,A (x, ω)Kq x,B (x, ω)}.Applying the product rule for differentiation, using wave equation ( 60) and symmetry relations ( 62) and ( 63) for states A and B, yields with ∆ Âx = Âx,B − Âx,A .Equations ( 100) and ( 101) are the matrix-vector forms of the local reciprocity theorems of the time-convolution and time-correlation type, respectively, as formulated by equations ( 23) and ( 24).Integration of both sides of equations ( 100) and ( 101 These are the matrix-vector forms of the global reciprocity theorems of the time-convolution and time-correlation type, respectively, as formulated by equations ( 34) and ( 35).When there are no sources and the material parameters are identical in both states, the right-hand sides of equations ( 100)-( 103) are zero.From equations ( 100) and ( 101) it then follows that qt x,A Nq x,B and q † x,A Kq x,B are space-propagation invariants (Haines 1988;Kennett et al. 1990;Koketsu et al. 1991;Takenaka et al. 1993).
We use equations ( 102) and ( 103) (with zeroes on the right-hand sides) to derive reciprocity relations for the propagator matrix.For state A we substitute qx,A (x, ω) = Ŵx (x, x A , ω) and, using equation ( 75 Using N −1 K = −J we find from equations ( 104) and ( 105) or, in the space-time domain 10.2 Time-dependent material The matrix-vector reciprocity theorems for a time-dependent material with piecewise continuous parameters α(t) and β(t) follow from those in section 10.1 by applying the mappings of equations ( 16) and (30).In particular, the reciprocity relations for the propagator matrix are 11 MARCHENKO-TYPE FOCUSING FUNCTIONS

Space-dependent material
Building on work by Rose (2001Rose ( , 2002)), geophysicists used the Marchenko equation to develop methods for retrieving the wave field inside a space-dependent material from reflection measurements at its boundary (Broggini & Snieder 2012;Broggini et al. 2012;Wapenaar et al. 2014;Ravasi et al. 2016;Staring et al. 2018;Jia et al. 2018).Focusing functions play a central role in this methodology.For a 1D space-dependent material, a Marchenko-type focusing function F x (x, x 0 , t) is defined as a specific solution of the wave equation, which focuses at the focal point x = x 0 (i.e., F x (x 0 , x 0 , t) ∝ δ(t)) and which propagates unidirectionally through the focal point.It has recently been shown that there exists a close relation between focusing functions and the propagator matrix (Wapenaar & de Ridder 2022;Wapenaar et al. 2023).
Here we briefly review this relation for a space-dependent material with parameters α(x) and β(x).We start by noting that the elements W P,P x (x, x 0 , t) and W Q,Q x (x, x 0 , t) are symmetric functions of time, whereas W P,Q x (x, x 0 , t) and W Q,P x (x, x 0 , t) are asymmetric functions of time.This follows simply from equation ( 107) and the expressions for matrices W x and J in equations ( 73) and (65).For elements W P,P x (x, x 0 , t) and W P,Q x (x, x 0 , t) these symmetry properties are also clearly seen in Figures 3a and 3b, respectively.Exploiting these symmetries, Marchenko-type focusing functions can be expressed in terms of the elements of the propagator matrix, according to (Wapenaar et al. 2023) From these expressions and equations ( 71) and ( 73) we obtain the following focusing conditions for x = x 0 Finally, note that the elements of the propagator matrix can be expressed in terms of the  x (x, x 0 , t) (convolved with a temporal wavelet) for a piecewise homogeneous space-dependent material.The label at the vertical axis denotes time (in µs) and that at the horizontal axis denotes space (in mm).With interchanged labels and reversed blue arrows this figure can be interpreted as F U t (x, t, t 0 ) (convolved with a spatial wavelet) for a piecewise constant time-dependent material.

Time-dependent material
The relations between the Marchenko-type focusing functions and the propagator matrix for a time-dependent material with parameters α(t) and β(t) follow from those in section 11.1 by applying the mapping of equation ( 16), hence F U t (x, t, t 0 ) = W U,U t (x, t, t 0 ) − η(t 0 )W U,V t (x, t, t 0 ), (120) with interchanged space coordinates.We have also derived a new representation for retrieving Green's functions from the space correlation of passive measurements in time-dependent materials.Unlike the corresponding representation for space-dependent materials it is singlesided, meaning that it suffices to correlate two responses (at two time instants) to sources at a single time instant.
Propagator matrices in space-dependent and in time-dependent materials obey boundary and initial conditions which can be transformed into one another in the same way as the underlying wave equations.Hence, these propagator matrices are interrelated in the same way.
This also applies to representations and reciprocity theorems involving propagator matrices, and to Marchenko-type focusing functions, which can be expressed as combinations of elements of the propagator matrix.
l e 4 c 2 R z o v z 7 n w s W g t O P n M M f + B 8 / g A E U o 2 b < / l a t e x i t > t 0 causality condition < l a t e x i t s h a 1 _ b a s e 6 4 = " J J 0 t L p v z G r O 7 r y U 6 m 3G i w p J p U w Y = " > A A A C A n i c b V D L S s N A F J 3 U V 6 2 v q C t x E y y C C y l J F e y y 4 M Z l B V s L T S i T y a Q d O o 8 w M x F C K G 7 8 F T c u F H H r V 7 j z b 5 y 0 W W j r g Q u H c + 7 l c k 6 Y U K K 0 6 3 5 b l Z X V t f W N 6 m Z t a 3 t n d 8 / e P + g p k U q E u 0 h Q I f s h V J g S j r u a a I r 7 i c S Q h R T f h 5 P r w r 9 / w F I R w e 9 0 l u C A w R E n M U F Q G 2 l o H / m S 5 Q i m C l K i M / 8 c C R 6 R w p o O 7 b r b c G d w l o l X k j o o 0 R n a X 3 4 k U M o w 1 4 h C p Q a e m + g g h 1 I T R P G 0 5 q c K J x B N 4 A g P D O W Q Y R X k s w h T 5 9 Q o k R M L a Y Z r Z 6 b + v s g h U y p j o d l k U I / V o l e I / 3 m D V M e t I C c 8 S T X m a P 4 o T q m j h V P 0 4 U R E Y q R p Z g h E 0 k R H D h p D C Z E 2 r d V M C d 5 i 5 G X S a z a 8 i 0 b z 9 r L e b p V 1 V M E x O A F n w A N X o A 1 u Q A d 0 A Q K P 4 Bm 8 g j f r y X q x 3 q 2 P + W r F K m 8 O w R 9 Y n z 8 + N p f w < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " F x L Y I O D 2 I Y + u k c z 7 h + B H 4 I y s R Y o = " > A A A B 8 n i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I v g q s y I j y 4 L b l x W s A + Y D i W T Z t r Q P I Y k I 5 S h n + H G h S J u / R p 3 / o 1 p O w u t H g g c z r n 3 5 t 4 T p 5 w 9 Y n z 8 + N p f w < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " F x L Y I O D 2 I Y + u k c z 7 h + B H 4 I y s R Y o = " > A A A B 8 n i c b V D L S g M x F L 1 T X 7 W + q i 7 d B I v g q s y I j y 4 L b l x W s A + Y D i W T Z t r Q P I Y k I 5 S h n + H G h S J u / R p 3 / o 1 p O w u t H g g c z r n 3 5 t 4 T p 5 w Z 6 / t f X m l t f W N z q 7 x d 2 d n d 2 z+ o H h 5 1 j M o 0 o W 2 i u N K 9 G B v K m a R t y y y n v V R T L G J O u / H k d u 5 3 H 6 k 2 T M k H O 0 1 p J P B I s o Q R b J 0 U 9 r X I l 7 N m g 2 r N r / s L o L 8 k K E g N C r Q G 1 c / + U J F M U G k J x 8 a E g Z / a K M f a M s L p r N L P D E 0 x m e A R D R 2 V W F A T 5 Y u V Z + j M K U O U K O 2 e t G i h / u z I s T B m K m J X K b A d m 1 V v L v 7 n h Z l N G l H O Z J p Z Ks n y o y T j y C o 0 v x 8 N m a b E 8 q k j m G j m d k V k j D U m 1 q V U c S E E q y f / J Z 2 L e n B d v 7 q / r D U b R R x l O I F T O I c A b q A J d 9 C C N h B Q 8 A Q v 8 O p Z 7 9 l 7 8 9 6 X p S W v 6 D m G X / A + v g H y 8 p G s < / l a t e x i t > source < l a t e x i t s h a 1 _ b a s e 6 4 = " f E V F p 3 Q d E 8 0 p H A p H N U U Z F T o n u K U = " > A A A B 8 n i c b V D L S g N B E O y N r x h f U Y 9 e B o M Q L 2 F X f O Q Y 8 O I x g n n A Z g m z k 9 l k y M z O M j M r h i W f 4 c W D I l 7 9 G m / + j Z N k D 5 p Y 0 F B U d d P d F S a c a e O 6 3 0 5 h b X 1 j c 6 u 4 X d r Z 3 d s / K B 8 e t b V M F a E t I r l U 3 R B r y l l M W 4 Y Z T r u J o l i E n H b C 8 e 3 M 7 z x S p Z m M H 8 w k

Figure 2 .
Figure 2. Green's function G t (x, t, t 0 ) (convolved with a spatial wavelet) for (a) a piecewise constant and (b) a sinusoidally modulated time-dependent material.
identical states A and B (hence, identical sources, material parameters and wave fields), we drop the subscripts A and B. Equation (35) thus yields (de Hoop 1995that the power generated by sources in the region between x b and x e 7 p a 9 u P 2 Z p h N I w Q b X u e G 5 i / I w q w 5 nA S b G b a k w o G 9 E B d i y V N E L t Z 7 N T J + T U K n 0 S x s q W N G S m / p 7 I a K T 1 O A p s Z 0 T N U C 9 6 U / E / r 5 O a s O p n X C a p Q c n m i 8 J U E B O T 6 d + k z x U y I 8 a W U K a 4 v Z W w I V W U G Z t O 0 Y b g L b 6 8 T J r n F e + q c n l 3 U a 5 V 8 z g K c A w n c A Y e X E M N b q E O D W A w g G d 4 h T d H O C / O u / Mx b 1 1 x 8 p k j + A P n 8 w c K a o 2 f < / l a t e x i t > x 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " O Y z E 1 + C / g E X D D s n O F 7 M / 7 J v t R Y E = " > A A A B 6 n i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e y K j x w D X j x G N A 9 I l j A 7 m U 2 G z M 4 u M 7 1 C l e 4 c 2 R z o v z 7 n w s W g t O P n M M f + B 8 / g A E U o 2 b < / l a t e x i t > t 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " D m j R k 9 V 3 S x c 8 y L 0 + O 6 T w l c o p q M 0 = " > A A A C A X i c b V D L S s N A F J 3 4 r P U V d S O 4 G S y C C y m J + O i y 4 M Z l B f u A J p T J Z N I O n c y s r P p w k m 8 M Q o I Y y E N M M 1 n K q / L 3 I U K 5 X F g d m M k R 6 p e a 8 Q / / P 6 q Y 4 a f k 5 5 k m r C 8 e x R l D K o B S z q g C G V B G u W G Y K w N N E x x C M k E d a m t K o p w Z 2 P v E g 6 5 3 X 3 q n 5 5 d 1 F r N s o 6 K u A I H I N T 4 I J r 0 A S 3 o A X a A I N H 8 A x e w Z v 1 Z L 1 Y 7 9 b H b H X J K m 8 O w B 9 Y n z 9 u 4 Z e B < / l a t e x i t > boundary condition < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 v 7 C s F a I v r D + q G K a + z f 5 b 2 d 5 x r k = " > A A A C H 3 i c b Z B N S 8 N A E I Y 3 f l u / o h 6 9 B I t Q L y W R W j 0 K X j w q 2 A 9 o S t l s p n F x s w m 7 E 7 G E / h M v / h U v H h Q R b / 0 3 b t s g 2 j q w 8 P C + M + z M G 6 S C a 3 T d k b W w u L S 8 s r q 2 X t r Y 3 N r e s X f 3 m j r J F I M G S 0 S i 2 g H V I L i E B n I U 0 E 4 V 0 D g Q 0 A r u L 8 d + 6 w G U 5 o m 8 x U E K 3 Z h G k v c 5 o 2 i k n l 3 3 A 4 i 4 z G O K i j 8 O / R A E 0 g o e + 3 6 B j 8 c + y P D H D 3 g U K X / Y s 8 t u 1 Z 2 U M w 9 e A W V S 1 H X P / v L D h G U x S G S C a t 3 x 3 B S 7 O V X I m Y B h y c 8 0 p J T d 0 w g 6 B i W N Q X f z y X 1 D 5 8 g o o d N P l H k S n Y n 6 e y K n s d a D O D C d Z s 8 7 P e u N x f + 8 T o b 9 8 2 7 O Z Z o h S D b 9 q J 8 J B x N n H J Y T c g U M x c A A Z Y q b X R 1 2 R x V l a C I t m R C 8 2 Z P n o X l S 9 e r V 0 5 t a + a J W x L F G D s g h q R C P n J E L c k W u S Y M w 8 k R e y B t 5 t 5 6 t V + v D + p y 2 L l j F z D 7 5 U 9 b o G / W 2 p C M = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " r a + U d 2 s b W 3 v 7 O 7 Z + w d d F a c S k w 6 O W S z 7 A V K E U U E 6 m m p G + o k k i A e M 9 I L J b e H 3 H o l U N B Y P e p o Q n 6 O R o B H F S B t p a B 9 5 k m d U U E 0 R 8 8 5 x L E J a G P n Q r j s N Z w a 4 T N y S 1 E G J 9 t D + 8 s I Y p 5 w I j R l S a u A 6 r z P S k t G 0 b O P / s D 4 + A Z 2 S 5 2 J < / l a t e x i t > 0 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " f E V F p 3 Q d E 8 0 p H A p H N U U Z F T o n u K U = " > A A A B 8 n i c b V D L S g N B E O y N r x h f U Y 9 e B o M Q L 2 F X f O Q Y 8 O I x g n n A Z g m z k 9 l k y M z O M j M r h i W f 4 c W D I l 7 9 G m / + j Z N k D 5 p Y 0 F B U d d P d F S a c a e O 6 3 0 5 h b X 1 j c 6 u 4 X d r Z 3 d s / K B 8 e t b V M F a E t I r l U 3 R B r y l l M W 4 Y Z T r u J o l i E n H b C 8 e 3 M 7 z x S p Z m M H 8 w k o Y H A w 5 h F j G B j J f + p m v W U Q E J M z / v l i l t z 5 0 C r x M t J B X I 0 + + W v 3 k C S V N D Y E I 6 1 9 j 0 3 M U G G l W G E 0 2 m p l 2 q a Y D L G Q + p b G m N B d Z D N T 5 6 i M 6 s M U C S V r d i g u f p 7 I s N C 6 4 k I b af A Z q S X v Z n 4 n + e n J q o H G Y u T 1 N C Y L B Z F K U d G o t n / a M A U J Y Z P L M F E M X s r I i O s M D E 2 p Z I N w V t + e Z W 0 L 2 r e d e 3 q / r L S q O d x F O E E T q E K H t x A A + 6 g C S 0 g I O E Z X u H N M c 6 L 8 + 5 8 L F o L T j 5 z D H / g f P 4 A t 3 m Q 3 g = = < / l a t e x i t > x(mm) < l a t e x i t s h a 1 _ b a s e 6 4 = " O u M 6 6 V M 6 u Y R Q w n h a / x P V C T B L A S A = " > A A A B 9 X i c b V D L S g N B E O z 1 G e M r 6 t H L Y B D i J e y K j x w D X j x G M A / I r m F 2 M p s M m Z l d Z m a V s O Q / v H h Q x K v / 4 s 2 / c Z L s Q R M L G o q q b r q 7 w o Q z b V z 3 2 1 l Z X V v f 2 C x s F b d 3 d v f 2 S w e H L R 2 n i t A m i X m s O i H W l D N J m 4 Y Z T j u J o l i E n L b D 0 c 3 U b z 9 S p V k s 7 8 0 4 o Y H A A 8 k i R r C x 0 o O p + C J F m a 8 E 0 p O z X q n s V t 0 Z 0 D L x c l K G H I 1 e 6 c v v x y Q V V B r C s d Z d z 0 1 M k G F l G O F 0 U v R T T R N M R n h A u 5 Z K L K g O s t n V E 3 R q l T 6 K Y m V L G j R T f 0 9 k W G g 9 F q H t F N g M 9 a I 3 F f / z u q m J a k H G Z J I a K s l 8 U Z R y Z G I 0 j Q D 1 m a L E 8 L E l m C h m b 0 V k i B U m x g Z V t CF 4 i y 8 v k 9 Z 5 1 b u q X t 5 d l O u 1 P I 4 C H M M J V M C D a 6 j D L T S g C Q Q U P M M r v D l P z o v z 7 n z M W 1 e c f O Y I / s D 5 / A G i v 5 H v < / l a t e x i t > t(µs) < l a t e x i t s h a 1 _ b a s e 6 4 = "f E V F p 3 Q d E 8 0 p H A p H N U U Z F T o n u K U = " > A A A B 8 n i c b V D L S g N B E O y N r x h f U Y 9 e B o M Q L 2 F X f O Q Y 8 O I x g n n A Z g m z k 9 l k y M z O M j M r h i W f 4 c W D I l 7 9 G m / + j Z N k D 5 p Y 0 F B U d d P d F S a c a e O 6 3 0 5 h b X 1 j c 6 u 4 X d r Z 3 d s / K B 8 e t b V M F a E t I r l U 3 R B r y l l M W 4 Y Z T r u J o l i E n H b C 8 e 3 M 7 z x S p Z m M H 8 w k o Y H A w 5 h F j G B j J f + p m v W U Q E J M z / v l i l t z 5 0 C r x M t J B X I 0 + + W v 3 k C S V N D Y E I 6 1 9 j 0 3 M U G G l W G E 0 2 m p l 2 q a Y D L G Q + p b G m N B d Z D N T 5 6 i M 6 s M U C S V r d i g u f p 7 I s N C 6 4 k I b a f A Z q S X v Z n 4 n + e n J q o H G Y u T 1 N C Y L B Z F K U d G o t n / a M A U J Y Z P L M F E M X s r I i O s M D E 2 p Z I N w V t + e Z W 0 L 2 r e d e 3 q / r L S q O d x F O E E T q E K H t x A A + 6 g C S 0 g I O E Z X uH N M c 6 L 8 + 5 8 L F o L T j 5 z D H / g f P 4 A t 3 m Q 3 g = = < / l a t e x i t > x(mm) < l a t e x i t s h a 1 _ b a s e 6 4 = " O u M 6 6 V M 6 u Y R Q w n h a / x P V C T B L A S A = " > A A A B 9 X i c b V D L S g N B E O z 1 G e M r 6 t H L Y B D i J e y K j x w D X j x G M A / I r m F 2 M p s M m Z l d Z m a F 4 i y 8 v k 9 Z 5 1 b u q X t 5 d l O u 1 P I 4 C H M M J V M C D a 6 j D L T S g C Q Q U P M M r v D l P z o v z 7 n z M W 1 e c f O Y I / s D 5 / A G i v 5 H v < / l a t e x i t > t(µs) < l a t e x i t s h a 1 _ b a s e 6 4 = " F 5F / j H v U V v 8 8 Y k x x t 8 v E M L j / l n 4 = " > A A A B 6 n i c b V D L S g N B E O z 1 G e M r 6 t H L Y B A 8 h V 3 x k W P A i 8 e I 5 g H J E m Y n v c m Q 2 d l l Z l Y M S z 7 B i w d F v P p F 3 v w b J 8 k e N L G g o a j q p r s r S A T X x n W / n Z X V t f W N z c J W c X t n d 2 + / d H D Y 1 H G q G D Z Y L G L V D q h G w S U 2 D D c C 2 4 l C G g U C W 8 H o Z u q 3 H l F p H s s H M 0 7 Q j + h A 8 p A z a q x 0 / 9 R z e 6 W y W 3 F n I M v E y 0 k Z c t R 7 p a 9 u P 2 Z p h N I w Q b X u e G 5 i / I w q w 5 n A S b G b a k w o G 9 E B d i y V N E L t Z 7 N T J + T U K n 0 S x s q W N G S m / p 7 I a K T 1 O A p s Z 0 T N U C 9 6 U / E / r 5 O a s O p n X C a p Q c n m i 8 J U E B O T 6 d + k z x U y I 8 a W U K a 4 v Z W w I V W U G Z t O 0 Y b g L b 6 8 T J r n F e + q c n l 3 U a 5 V 8 z g K c A w n c A Y e X E M N b q E O D W A w g G d 4 h T d H O C / O u / M x b 1 1 x 8 p k j + A P n 8 w c K a o 2 f < / l a t e x i t > x 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " O Y z E 1 + C / g E X D D s n O F 7 M / 7 J v t R Y E = " > A A A B 6 n i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e y K j x w D X j x G N A 9 I l j A 7 m U 2 G z M 4 u M 7 1 C W P I J X j w o 4 t U v 8 u b f O E n 2 o I k F D U V V N 9 1 d Q S K F Q d f 9 d g p r 6x u b W 8 X t 0 s 7 u 3 v 5 B + f C o Z e J U M 9 5 k s Y x 1 J 6 C G S 6 F 4 E w V K 3 k k 0 p 1 E g e T s Y 3 8 7 8 9 h P X R s T q E S c J 9 y M 6 l e 4 c 2 R z o v z 7 n w s W g t O P n M M f + B 8 / g A E U o 2 b < / l a t e x i t > t 0 (b) < l a t e x i t s h a 1 _ b a s e 6 4 = " / X R C p 9 b p N F T B 5 y 5 B d m F r a + U d 2 s b W 3 v 7 O 7 Z + w d d F a c S k w 6 O W S z 7 A V K E U U E 6 m m p G + o k k i A e M 9 I L J b e H 3 H o l U N B Y P e p o Q n 6 O R o B H F S B t p a B 9 5 k m d U U E 0 R 8 8 5 x L E J a G P n Q r j s N Z w a 4 T N y S 1 E G J 9 t D + 8 s I Y p 5 w I j R l S a u A 6

Figure 3 .
Figure 3. Propagator matrix elements W P,P x (x, x 0 , (a) and W P,Q ) from x b to x e yields (Haines & de Hoop 1996; Wapenaar & Grimbergen 1996N∆ Âx qx,B + dt x,A Nq x,B + qt x,A N dx,B }dx, ), qx,A (x A , ω) = I.Similarly, For state B we substitute qx,B (x, ω) = Ŵx (x, x B , ω) and qx,B (x B , ω) = I.Taking x A and x B equal to x b and x e (in arbitrary order) yields Ŵt

Figure 4
Figure 4 shows an x, t-diagram of F P x (x, x 0 , t) (convolved with a temporal wavelet with a central frequency ω 0 /2π = 300 kHz) for a focal point at x 0 = 40 mm.The interpretation is as follows.The four blue arrows in the right-most slab indicate leftward propagating waves that are emitted into the material from the right, at x = x N = 200 mm.After interaction with the boundaries between the homogeneous slabs, a single leftward propagating wave arrives at x = x 0 = 40 mm, where it obeys the focusing condition of equation (114).The red arrows indicate the rightward propagating scattered part of the focusing function F P x (x, x 0 , t).
< l a t e x i t s h a 1 _ b a s e 6 4 = " f EV F p 3 Q d E 8 0 p H A p H N U U Z F T o n u K U = " > A A A B 8 n i c b V D L S g N B E O y N r x h f U Y 9 e B o M Q L 2 F X f O Q Y 8 O I x g n n A Z g m z k 9 l k y M z O M j Mr h i W f 4 c W D I l 7 9 G m / + j Z N k D 5 p Y 0 F B U d d P d F S a c a e O 6 3 0 5 h b X 1 j c 6 u 4 X d r Z 3 d s / K B 8 e t b V M F a E t I r l U 3 R B r y l l M W 4 Y Z T r u J o l i E n H b C 8 e 3 M 7 z x S p Z m M H 8 w k

0 Figure 4 .
Figure 4. Focusing function F Px (x, x 0 , t) (convolved with a temporal wavelet) for a piecewise homogeneous space-dependent material.The label at the vertical axis denotes time (in µs) and that at the horizontal axis denotes space (in mm).With interchanged labels and reversed blue arrows this figure can be interpreted as F U t (x, t, t 0 ) (convolved with a spatial wavelet) for a piecewise constant time-dependent material.
A ) is the acausal Green's function, i.e., Ǧa t (k, t, t A ) = 0 for t > t A (hence, the impulse at t A is actually a sink).Using equation (28) Ǧa t (k, t, t A ).For state B we take a Green's state with an impulsive source at t B between t b and t e , according to −ik bB (k, t) = δ(t−t For state A we take a Green's state with an impulse at t A between t b and t e , according to−ik bA (k, t) = δ(t − t A ) and ǎA (k, t) = 0.However, we define ǓA (k, t) = Ǧa t (k, t, t A ),where Ǧa t (k, t, t B ) and ǎB (k, t) = 0. We define ǓB (k, t) = Ǧt (k, t, t B ), where Ǧt (k, t, t B ) is the causal Green's function, i.e., Ǧt (k, t, t B ) = 0 for t < t B .Moreover, VB (k, t) = − β(t) ik ∂ t Ǧt (k, t, t B