The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain

In this paper, a time domain enclosure method for an inverse obstacle scattering problem of electromagnetic wave is introduced. The wave as a solution of Maxwell's equations is generated by an applied volumetric current having an {\it orientation} and supported outside an unknown obstacle and observed on the same support over a finite time interval. It is assumed that the obstacle is a perfect conductor. Two types of analytical formulae which employ a {\it single} observed wave and explicitly contain information about the geometry of the obstacle are given. In particular, an effect of the orientation of the current is catched in one of two formulae. Two corollaries concerning with the detection of the points on the surface of the obstacle nearest to the centre of the current support and curvatures at the points are also given.


Introduction
In this paper, we consider an inverse obstacle scattering problem of a wave whose governing equation is given by Maxwell's equations. The wave is generated by a source at t = 0 which is not far a way from an unknown obstacle, and observe a single reflected wave from the obstacle over a finite time interval at the same place as the source. The inverse obstacle scattering problem is to: extract information about the geometry of the obstacle from the observed wave. This is a proto-type of so-called inverse obstacle problem [24] and the solution may have possible applications to radar imaging. Since we consider only the data over a finite time interval and thus, this is a time domain inverse problem. Our main interest is to find an analytical method or formula that extracts the geometry of the obstacle from the data by using the governing equation of the wave.
Let us describe the mathematical formulation of the problem. Let D be a nonempty bounded open subset of R 3 with C 2 -boundary such that R 3 \ D is connected. ν denotes the unit normal to ∂D, oriented towards the exterior of D.
Let 0 < T < ∞. We denote by E and H the electric field and the magnetic field, respectively. ǫ denotes the electric permittivity and µ the magnetic permeability assumed to be positive constant.
We assume that E and H are induced only by the current density J at t = 0 and that the obstacle is a perfect conductor. It is well known that the governing equations of E and H take the form (1.1) Now let us describe our problem. Fix a large (to be determined later) T < ∞. Let B be the open ball centred at p with very small radius η and satisfy B ∩ D = ∅. There are several choices of the current density J as a model of the antenna ( [3,6]). In this paper, we assume that J takes the form where a = 0 is a constant unit vector, χ B denote the characteristic function of B and f ∈ H 1 (0, T ) with f (0) = 0. Note that χ B (x) has discontinuity across the sphere ∂B.
Problem. Produce E and H by J and observe E on B over time interval ]0, T [. Extract information about the geometry of D from the observed data.
This may be the simplified model of the case when the reflected wave is observed at the same place where the source is located. Note that we consider the pair (E, H) is the solution of (1.1) in the sense as described on pages 433-435 in [9] which is based on Stone's theorem.
As far as the author knows there is no result for the problem mentioned above. The point is: the data is taken over a finite time interval and only a single (reflected) wave is employed.
In this paper, we employ the enclosure method for this problem. The origin goes back to a method developed for an inverse boundary value problem in two dimensions for the Laplace equation [11]. Since then this original version of the enclosure method in two dimensions has been applied to inverse obstacle scattering problems at a fixed wave number with a single incident wave [12,16] and an inverse boundary value problem for the Navier equation [21] and references therein.
The method consists of two parts: • constructing a special solution v of an elliptic partial differential equation which depends on a large parameter τ > 0 and is independent of unknown obstacles.
• constructing a so-called indicator function of independent variable τ by using observation data and v above and studying its asymptotic behaviour as τ −→ ∞.
New development of this method started in 2007. In [13] it was shown that the enclosure method is applicable to an inverse source problem for the heat equation in three-space dimensions [14] and inverse boundary value problems for the heat and wave equations in one-space dimension [13]. Now we have already some applications to inverse obstacle scattering problems whose governing equation is given by the classical wave equation in three-space dimensions [15,17,18,19]. The method enables us to extract information about the geometry of unknown obstacle from a single reflected wave over a finite time interval. However, the governing equation therein is a single partial differential equation and it is not clear that the method can cover also the very important case when the governing equation consists of a system of partial differential equations.
In the following subsection we describe our solution to Problem.

Statement of the results
We denote by H(curl, R 3 ) the set of all vector valued-functions U ∈ L 2 (R 3 ) 3 such that ∇ × U ∈ L 2 (R 3 ) 3 . It is a Hilbert space with norm We call this V the weak solution of In this paper, unless otherwise stated, f ( · , τ ) has the form The following results give us some solutions to the problem raised above since W e on B can be computed from our observation data E on B over time interval ]0, T [ through (1.5).
Moreover, we have the following formula: A remarkable point in this theorem is: there is no restriction on direction a in (1.2).
we can find the sphere ∂B d ∂D (p) (p) via (1.7) regardless of the direction of a at any time. This sphere is the maximum one whose exterior encloses the unknown obstacle.
As is introduced in the author's previous papers [18,19,20] we denote by Λ ∂D (p) the set ∂D ∩ ∂B d ∂D (p) (p). We call this set the first reflector from p to ∂D and the points in the first reflector are called the first reflection points from p to ∂D. Using Theorem 1.1, one can also give a criterion for a given direction ω ∈ S 2 whether the point p + d ∂D (p)ω belongs to ∂D since as pointed out in [18,19,20,22] Here s ∈ ]0, 1[ and is fixed. Note that one can always compute d ∂D (p + sd ∂D (p)ω) via (1.7) using a suitable input current supported around p + sd ∂D (p)ω and the electronic wave observed at the same place as the support of the current.
The condition (1.6) is a restriction on the strength of the source at t = 0. Note that Let S q (∂D) and S q (∂B d ∂D (p) (p)) denote the shape operators at q ∈ Λ ∂D (p) of ∂D and ∂B d ∂D (p) (p) with respect to ν q and −ν q , respectively. These are symmetric linear operators on the common tangent space T q ∂D = T q ∂B d ∂D (p) (p). We have always S q (∂B d ∂D (p) (p)) − S q (∂D) ≥ 0 since q attains the minimum value of the function ∂D ∋ y −→ |y − p|. In general, given p the first reflector from p to ∂D can be an infinite set, even more, a continuum. For example, imagine the case when a part of ∂D coincides with that of ∂B d ∂D (p) (p). Note also that, in that case, we have S q (∂B d ∂D (p) (p)) = S q (∂D) the points q in that part. .
(1.10) (1.9) is a restriction on the direction of a in (1.2). Since |a · ν q | = 1 if and only if a = ± ν q , (1.9) means that there is no first reflection point from p on the straight line passing through p and parallel to a. It is clear that if Λ ∂D (p) consists of at least three points, then (1.9) is satisfied.
The denominator det (S q (∂B d ∂D (p) (p)) − S q (∂D)) in the right-hand side on (1.10) is independent of a and numerator 1 − (a · ν q ) 2 becomes maximum when a is perpendicular to ν q . Someone may think that this fact has similarity to a well known fact in the dipole antenna theory(e.g., [3]), that is, the maximum radiation from the antenna is directed along right angles to the dipole.
Note also that since we have from (1.5) we know that in (1.7) and (1.10) instead of all the components of E we need only a · E. It is a due course to deduce the following corollary from Theorem 1.2 (see [18,20]).
Let J j be the J given by (1.2) in which B = B j and f (t) = f j (t) satisfying (1.6) for a γ = γ j ∈ R; E j with j = 1, 2 denote the corresponding electric fields governed by (1.1). If T > 2 √ µǫ max j=1,2 dist (D, B j ) and a × (p − q) = 0, then one can extract the Gauss curvature K ∂D (q) of ∂D at q and mean curvature H ∂D (q) with respect to ν q from a · E j on B j with j = 1, 2 over time interval ]0, T [.
Briefly speaking, this corollary says that: one can completely know the Gauss and mean curvatures of the boundary of the obstacle at a known first reflection point q from a given point p outside the obstacle by observing two reflected electric fields produced by two sources whose centres are placed between p and q. Thus, one can know an approximate shape of the boundary of unknown obstacle at a known first reflection point by using two electromagnetic waves.
We think that Corollary 1.1 shows us an advantage of the near field measurement. Compare the results with those of [27] where the information about the mean curvature never appear explicitly in the scattering kernel which is the observation data in the context of the Lax-Phillips scattering theory [25].
The outline of this paper is as follows. In Section 2 we describe a representation formula of the indicator function To study the asymptotic behaviour of the indicator function as τ −→ ∞ we need some preliminary facts about V . In Section 3, using the mean value theorem for the modified Helmholtz equation, we give an explicit computation formula for V outside B. This formula is found in Subsection 3.1 and enables us to study the asymptotic behaviour of an energy integral of V over D as τ −→ ∞ in Subsection 3.2. However, unlike the previous applications to scalar wave equations, we need an upper bound of L 2 -norm of the Jacobian matrix V ′ over D in terms of the energy integral. This is not trivial and described in Subsection 3.2. Theorem 1.1 is proved in Section 4. The proof is based on a brief asymptotic formula of the indicator function and the resulted upper and lower bound in terms of the energy integral of V over D mentioned above. Theorem 1.2 is proved in Section 5. The proof of Theorem 2.1 is based on the precise asymptotic formula of the indicator function and the leading profile of the energy integral of V over D studied in Section 3. The precise asymptotic formula comes from a combination of the brief asymptotic formula of the indicator function established in Section 4 and the asymptotic coincidence of the energy of the so-called reflected solution over R 3 \ D with that of V over D as described in Lemma 5.1.
The proof of Lemma 5.1 is based on the reflection principle across ∂D for the Maxwell system as described in Propositions 5.1 and 5.2 and a representation of the difference of two energy integrals mentioned above in terms of the reflection as described in Proposition 5.3. Then, apply the Lax-Phillips reflection argument [25] to the difference. This story is parallel to the previous scalar wave equation cases [18,19,20], however, a proper problem for system of partial differential equations occurs in proving Lemma 5.1. In order to apply their argument, we need an upper bound of the L 2 -norm of the Jacobian matrix of the reflected solution in terms of the energy of the same reflected solution. However, it seems difficult to obtain such an estimate and instead, we give the upper bound in terms of the energy integral of V over D directly. This way is different from the original Lax-Phillips reflection argument and makes the argument for the proof of the asymptotic coincidence of two energies straightforward compared with the scalar wave equation case.
In Appendix we describe some differential identities for the vector fields obtained by the reflection across ∂D and the resulted reflection formula described in Proposition 5.2 is proved. Note that the regularity assumption that ∂D is C 4 in Theorem 1.2 is more restrictive compared with the scalar wave equation case [18,19,20] in which the corresponding theorems are valid for C 3 -smooth boundary. This is coming from the difference of the reflection principle used. Therein only a change of independent variables is used, however, for Maxwell's equations, the reflection principle involves also a change of dependent variables and this requires a higher regularity.

A representation formula of the indicator function
Define and where W e is given by (1.5).
The enclosure method starts with having the following representation formula of the indicator function.

Proposition 2.1. It holds that
Proof. We have Thus we obtain (2.5).
We have Thus we obtain (2.6). (2.7) is clear. ✷ Taking the rotation of (2.5) and (2.6), respectively, we obtain the following equation: Now we are ready to prove Proposition 2.1. Integration by parts gives (2.7) ensures that the first term on this right-hand side vanishes. And we have Thus Substituting (1.3) and (2.8) into this, we obtain integration by parts gives (2.13) Similarly we have (2.14) Now (2.3) follows from (2.1), (2.2), (2.9), (2.10), (2.13) and (2.14). Remark 2.1. From (2.5) and (2.6) we obtain also the following equation for W m : In this paper, we will not make use of this equation.

Preliminary facts about V
In this section first we give a detailed expression of V . Second using the expression we give an asymptotic behaviour of some integrals involving V .

An explicit form of V outside of B
Here we give an explicit computation formula of the weak solution of (1.
First, assume that V has the form where V 0 and V 1 are two vector-valued functions on the whole space.
From this we see that if and From this formal argument we have the following construction of the weak solution of Then, for each fixed τ by the interior regularity or from the expression we see that together with its all derivatives are exponentially decaying as |x| −→ ∞. Thus we have V 0 ∈ H 2 (R 3 ) 3 . Define V 1 is also smooth outside B and, for each fixed τ V 1 (x) together with all the derivatives are exponentially decaying as |x| −→ ∞; It is easy to see that this V satisfies (1.3) in the weak sense. Thus, by the uniqueness of the weak solution of (1.3) we conclude that the weak solution of (1.3) has the expression where V 0 and V 1 are given by (3.2) and (3.3), respectively. Note that this argument for the construction of V is based on the form of the fundamental solution for the operator (1/µǫ)∇ × ∇ × · − k 2 · with k > 0 (e.g., see [1]).
In what follows, for convenience we introducẽ By the mean value theorem for the modified Helmholtz equation [7], we know that Thus V 0 given by (3.2) takes the form where f (τ ) is given by (1.4) and A direct computation of V 1 via (3.3) and (3.5) yields the following explicit formula of V .
From these and (3.5), we have

Two basic lemmas about
The following two lemmas concerned with the asymptotic behaviour of some integrals involving V and its derivatives over D is one of key points in this paper.
it follows from (3.6) and (3.7) that, for all and also from (3.10) that These yield Choose an arbitrary point x 0 ∈ Λ ∂D (p). First consider the case when Then, from (3.10) and (3.13), we know that there exists positive constants δ and τ 0 such that, for all Next consider the case when Then Thus, if µǫ = 1, then, from (3.7) we see that there exist positive constants C, δ and τ 0 such that, for all If µǫ = 1, then, from (3.7) again we see that there exist positive constants C, δ and τ 0 such that, for all x ∈ D ∩ B δ (x 0 ) and τ ≥ τ 0 Thus, anyway, at least, from these and (3.6) we obtain and hence Since (3.14) yields (3.15), in any case we have (3.15). A standard technique [22] yields Thus rewriting (3.15) as we obtain (3.12). ✷ For establishing Theorem 1.2 we need a more accurate information about the asymptotic behaviour of J(τ ) as τ −→ ∞. (3.17) Proof. First of all we note that a combination of (2.14) and the identity We have Let ν = ν x . We consider the map a −→ (ω × a) × (a − µǫ(ω · a)ω) · ν a quadratic form on R 3 . From (3.19) we see that this quadratic form has the expression where Let q ∈ Λ ∂D (p). We have ν = ν q = −ω| x=q . Then Therefore, |a · ν| = 1 if and only if the following condition is satisfied: From (3.6) and (3.10) we have
And also d ∂D (p) − η = dist (D, B). These together with (3.24) yield (3.16). Next we prove (3.17). Since we have it suffices to study the asymptotic behaviour of the second integral on this right-hand side. Thus, for this purpose we compute V ′ . A direct computation yields In particular, we have as τ −→ ∞, uniformly for x ∈ ∂D.
On the other hand, we have In particular, we have as τ −→ ∞ uniformly for x ∈ ∂D. Now from (3.6), (3.26) and (3.27), as τ −→ ∞ we obtain A combination of this and (3.9) gives (3.28) Since we have
As a direct corollary of this and (3.24), we conclude that: there exist positive constants C and τ 0 such that, for all τ ≥ τ 0  Once we have these lemmas, then from Lemma 3.1 we conclude that lim sup . From these we immediately obtain Theorem 1.1.
Thus, the following subsections are devoted to the proofs of Lemmas 4.1 and 4.2.

Proof of Lemma 4.1
We claim that, as τ −→ ∞ and First we prove (4.2). From (1.3) we obtain Using the completion of the square, one can rewrite this as Dropping the second term on this left-hand side, we obtain Moreover, using the inequality Now (4.2) follows from this, (4.5) and the following trivial estimate: Next we prove (4.3). We make use of (2.3). Write Substituting this into (2.3), we obtain (4.8) Dropping the second and third terms on the left-hand side of (4.8), we obtain Applying these and (4.7) to the right-hand side of (4.9), we obtain On the other hand, dropping the first and third terms on the left-hand side of (4.8) and using (4.6), we obtain Thus, by the same reason above we obtain This completes the proof of (4.3). From (4.2), (4.3) and (4.10) we have Now a combination of (2.3) and (4.11) yields (4.1).

Proof of Lemma 4.2
Set R = W e − V . Taking the scalar product of equation (2.11) with R, integrating over R 3 \ D and using boundary condition (2.12) on ∂D, we have Note that C is a positive constant and independent of V .
Again the trace theorem tells us that where C ′ is a positive constant and independent of V . Thus, we have Moreover, from equation (2.11) one gets Substituting this into (4.12), we obtain This gives (4.14) Here we make use of the following trivial estimates and Applying (4.10), (4.15) and (4.16) to the right-hand side on (4.14), we obtain Thus, a standard technique to the first and last terms on this right-hand side, we obtain Now applying (4.13) to this, we obtain Now from this and trivial inequality V 2 H(curl,D) ≤ C(1+τ −2 )J(τ ) we obtain the desired estimate.
The estimate in Lemma 4.2 is not sharp, however, for Theorem 1.1 it is enough. For Theorem 1.2 we need more accurate estimate like E(τ ) ∼ J(τ ).

Proof of Theorem 1.2
Since under the assumption (1.6) for a γ ∈ R it holds that it is clear that Theorem 1.2 is a direct consequence of (3.16) in Lemma 3.2, Lemma 4.1 and the following lemma.
The proof of Lemma 5.1 employs the Lax-Phillips reflection argument in [25], however, some technical parts are different. Anyway that is based on: a representation formula of E(τ ) − J(τ ) via a reflection. Thus, the following subsection starts with describing a reflection principle across ∂D from inside to outside.

Reflection principle
One can choose a positive number δ 0 in such a way that: given Lemma 14.16 in [10] for this. For The reflection principle what we say in this paper consists of two parts summarized as the following propositions.
Proposition 5.1. Assume that ∂D is C 4 . Let V be a vector field over D and C 2 in D.
then, V * defined as (5.1) satisfies and all the coefficients in this right-hand side are independent of τ and continuous, in particular, the coefficients come from the second order terms are C 1 in a tubular neighbourhood of ∂D.
Remark 5.1. If ∂D is a plane, then n ′ (x) ≡ 0 and the third term in the right-hand side on (5.1) vanishes. Then, Propositions 5.1 and 5.2 becomes the reflection principle used in [26] for inverse obstacle scattering for Maxwell's equations in a frequency domain (replaced τ 2 with −k 2 ) . They employed this principle for a different purpose from us, more precisely, establishing a uniqueness theorem for polygonal obstacles in a single frequency domain. In the curved boundary case, n ′ = 0 and we need the correction term 2d ∂D (x)n ′ (x)A(x r ). and First, we claim thatṼ satisfies the following boundary conditions. Claim 1.Ṽ satisfies the following boundary conditions where S x (∂D) denotes the shape operator of ∂D at x ∈ ∂D with respect to ν x . Proof of Claim 1. Clearly thisṼ satisfies (5.7). To check (5.8) we have to compute ∇×Ṽ .
On the other hand, we have q ′ (y)v = v for all vectors with ν y · v = 0. Thus (5.9) gives From these we obtain Here we note that n( This gives ∇ × n = 0. Thus, whereB(x) = B(x r ). Thus, from (5.10) we obtain Let y ∈ ∂D. Applying (5.10) for B replaced with −A i for each i = 1, 2, 3, we have Note that and the same for ∇ × A. Thus, one gets Thus, we obtain (∇ ×Ã)(y) = −(∇ × A)(y) + 2ν y × {A ′ (y)ν y }. (5.12) Taking the vector product of both sides on (5.12) with ν y , we obtain Note that, in the derivation of this, we have made use of the identity the equation A ′ (x r )n(x) · n(x) = 0 which is an easy consequence of the property A(x) · n(x) = 0. It is easy to see that, for any vector v, we have Rewrite the right-hand side on (5.13) as Then applying (5.15) to the second term of this, we know that (5.13) becomes Let y(σ) be an arbitrary curve on ∂D with y(0) = y. We have A(y(σ)) · ν y(σ) = 0. Differentiating this both sides with respect to σ j , we obtain, Recalling the definition and symmetry of the shape operator for ∂D at y ∈ ∂D with respect to ν y , we have for all tangent vectors v at y of ∂D. Since S y (∂D)A(x) is a tangent vector at y of ∂D and A ′ (y) T ν y · ν y = 0, we know that (5.17) is valid for all vectors of R 3 . Thus we obtain A ′ (y) T ν y = S y (∂D)A(y). Now from this and (5.16) we obtain We have Now from this and (5.11) we obtain (5.8). ✷ Next we claim Proof of Claim 2. Since ∇(d ∂D (x)) = n(x), we have n ′ = (n ′ ) T and thus n ′ (x)n(x) = 0. These yield C(x) = 2d ∂D (x)n ′ A(x r ).

(5.22)
Since R satisfies R × ν = −V × ν on ∂D (see (2.12)), we have Thus, applying (5.3) to this, we obtain Substituting this into (3.18), we obtain Substituting this into (2.13), we obtain Substituting (5.23) and (5.24) into the right-hand side on (5.22), we obtain where C ijkl are of class C 1 ; D ijkl and E ijk are of class C 1 and C 0 in a neighbourhood of ∂D; F ijkl are constants.
Substituting this into the first term on the right-hand side of (5.19) and making a change of variables x = y r , we obtain 25) where J(y) denotes the Jacobian of the map: y −→ y r . A routine involving an integration by parts and d ∂D (y r ) = d ∂D (y) yields where R r (y) = R(y r ) and D δ = {y ∈ D | d ∂D (y) < 2δ}. Just simply estimating other terms on the right-hand of (5.25) which are coming from the lower order terms and combining the results with (5.26), we obtain

(5.27)
And also, using a change of variables again, we can easily obtain From (2.1) and (2.2) we have also the following trivial inequality: And also, under the same assumption as Lemma 5.1, we have (3.17), that is, This is one of different points from the scalar case. Another non trivial different point from a previous scalar case is to give estimates of R ′ L 2 ((R 3 \D) δ ) in terms of E(τ ). It seems that one can not obtain such type of the estimate unlike the scalar case. Here we show that for the proof of Lemma 5.1 it suffices to have the following estimates. and . Applying (4.2), (5.31), (5.32) and (5.33) to this right-hand side, we obtain Let θ > 0 and set δ = τ −θ . Then (5.34) becomes Choosing θ in such a way that θ = 1 − θ, that is, θ = 1/2, we obtain 2dist (D,B)) ), This completes the proof of Lemma 5.1.

Proof of Lemma 5.2
Let ϕ be a smooth function on the whole space and satisfy 0 ≤ ϕ ≤ 1; ϕ(x) = 1 for x with d ∂D (x) ≤ δ 0 /2 and ϕ(x) = 0 for d ∂D (x) ≥ δ 0 . Here δ 0 is chosen in such a way that given x with d ∂D (x) < 2δ 0 there exists a unique q = q(x) ∈ ∂D that attains the minimum of the function ∂D ∋ y : −→ |y − x|. We assume that ∂D is C 2 . Then one may think that both d ∂D (x) and q(x) are C 2 for x ∈ R 3 \ D with d ∂D (x) < 2δ 0 (see [10], p.355, Lemma 14.16).
Taking the scalar product of (2.11) with ϕV * and integrating over R 3 \ D, we obtain Since V * satisfies (5.2), from this and (4.12) we obtain the expression This yields However we need a special care to obtain a similar estimate for ∇ × (ϕV * ). The point is the estimate of the second term on the right-hand side of the following estimate: It seems to be difficult to give an estimate of ∇ × V * in terms of only ∇ × V . Instead we simply estimate and thus However, the estimate is not easy since V does not satisfy any standard boundary condition on ∂D. For this we employ (3.17) in Lemma 3.2. This is the reason why we need some additional assumptions in Lemma 5.2. Once we have these estimates, from (5.36) together with (4.10), (4.15), (4.16) and (5.29), we obtain 6 Concluding remarks In this paper, we employed a simple form (1.2) as a model of the current density, however, in principle, it may be possible to cover more complicated model of the current density, at least, in the frame work of the solution as constructed in [9]. The method presented here can be applied also to an interior problem similar to that considered in [18]. The problem therein aims at extracting information about the geometry of an unknown cavity from the wave which is produced by the initial data localized inside the cavity and propagates therein. See also [30] and [31] for similar problems in frequency domain.
Single measurement version of the time domain enclosure method also finds an application to an inverse initial boundary value problem for the heat equation in three-space dimensions. For this see Theorem 1.1 in [23] and consult Section 3 in [22] for an open problem in the visco elasticity.
Some of open problems are in order.
• How about the case when the electromagnetic wave satisfies a more general boundary condition like the Leontovich condition (see, e.g., [2])? In the acoustic wave case we have [17] for Theorem 1.1 and [20] for Theorem 1.2.
• How about the case when the reflected electromagnetic wave is observed at a different place from the support of the source ? The observed data is so-called the bistatic data. We expect that the data give us different information about the geometry of unknown obstacle as seen for an acoustic wave case in [19]. It would be interstersting to read [4] for a comparison of monostatic and bistatic radar images.
• There are several other inverse obstacle scattering problems in time domain whose governing equations are systems of partial differential equations. Extend the range of the applications of the method presented here to such systems. For example, it would be intersting to consider the inverse fluid-solid interaction problem in time domain. See [28] and references therein for the problem in the frequency domain.
The point is the first term on this right-hand side. We have And by (A.7) we know that this right-hand side consists of at most first order terms. Summing up, we have obtained the following lemma.
Lemma A.2. Assume that ∂D is C 4 . Let C be given by (5.6). Then, we have the expression (A.10) All the coefficients are independent of τ and continuous in a tubular neighbourhood of ∂D, in particular, R ijkl (x) which come from the second order terms in (A.9) is C 1 . Now Proposition 5.2 is a direct consequence of (A.8) and (A.10).