On finding the surface admittance of an obstacle via the time domain enclosure method

An inverse obstacle scattering problem for the electromagnetic wave governed by the Maxwell system over a finite time interval is considered. It is assumed that the wave satisfies the Leontovich boundary condition on the surface of an unknown obstacle. The condition is described by using an unknown positive function on the surface of the obstacle which is called the surface admittance. The wave is generated at the initial time by a volumetric current source supported on a very small ball placed outside the obstacle and only the electric component of the wave is observed on the same ball over a finite time interval. It is shown that from the observed data one can extract information about the value of the surface admittance and the curvatures at the points on the surface nearest to the center of the ball. This shows that a single shot contains a meaningful information about the quantitative state of the surface of the obstacle.


Introduction and statement of the results
In this paper, we pursue further the possibility of the time domain enclosure method [10] for the Maxwell system developed in [11,12]. We consider an inverse obstacle scattering problem for the wave governed by the Maxwell system in the time domain, in particular, over a finite time interval unlike the time harmonic reduced case, see [5,15,18].
Let us formulate the problem more precisely. We denote by D the unknown obstacle. We assume that D is a non empty bounded open set of R 3 with C 2 -boundary such that R 3 \ D is connected.
We assume that the electric field E = E(x, t) and magnetic field H = H(x, t) are generated only by the current density J = J (x, t) at the initial time located not far a way from the unknown obstacle. There should be several choices of current density J as a model of antenna [2,4]. In this paper, as considered in [11,12] we assume that J takes the form J (x, t) = f (t)χ B (x)a, (1.1) where a is an arbitrary unit vector; B is a (very small) open ball satisfying B ∩ D = ∅ and χ B denotes the characteristic function of B; f ∈ C 1 [0, T ] with f (0) = 0. Let 0 < T < ∞. In this paper, we assume that Assumption 1.
Note that ν denotes the unit outward normal to ∂D. The obstacle is embedded in a medium like air (free space) which has constant electric permittivity ǫ( > 0) and magnetic permeability µ(> 0). The boundary condition (1.3) is called the Leontovich boundary condition ( [1,5,15,18]) and see also [16] for the case when λ is constant. The quantity 1/λ is called the surface impedance, see [1] and thus λ is called the admittance. In what follows we use these equivalent forms without mentioning explicitly. The existence of the admittance λ causes the loss of the energy of the solution on the surface of the obstacle after stopping of the source supply.
In [14] it is stated that the existence of (E(t), H(t)) satisfying (i)-(iv) can be derived from the theory of C 0 contraction semigroups [20]. However, therein the detailed proof is not given as pointed out in [12]. To make the logical relation clear, here we assume that our pair (E(t), H(t)) satisfies (i)-(iv). This is our starting assumption. It should be pointed out that Assumption (λ) in [12] implies the existence of such (E(t), H(t)) which ensures that conditions (i)-(iv) has a sense.
We consider the following problem.
Problem. Fix a large (to be determined later) T < ∞. Observe E(t) on B over the time interval ]0, T [. Extract information about the geometry of D and the values of λ on ∂D from the observed data.
First of all let us recall the previous reslult on this problem. Denote the solution of the system (1.2) in the case when D = ∅ by (E 0 (t), H 0 (t)) with J given by (1.1). Note that in this case, the solvabilty has been ensured by applying theory of C 0 contraction semigroups [20].
Define the indicator function And also, to describe another assumtion, we introduce here Using the same argument as that of [12] under Assumption 1, we know the following fatcts.
• The pair (W e , W m ) belongs to (1.5) • The boundary condition (1.3) remains valid in the sense of the trace [14] as mentioned above if (E(t), H(t)) is replaced with (W e , W m ).
• It holds that (1.6) Note that, at this stage, each term on (1.3) does not have a point-wise meaning. What we know is: the left-hand side on (1.3) just belongs to the dual space of H 1/2 (∂D) 3 . In this paper, we introduce another assumption which states a regularity up to boundary. This assumption makes us possible to treate vector-valued functions appeared in a dual paring pointwise. Note that Assumption 2 is a special version of Assumption (R) introduced in [12] by virtue of (1.6). However, for our purpose, it suffices to assume Assumption 2 instead of Assumption (R). We believe that Assumption 2 should be removed. Now, by Assumption 2, we have that both W e and W m belong to H 1 in the intersection of an open neighbourhood of ∂D with R 3 \ D. Then, we see that the boundary condition (1.3) is satisfied in the sense of the usual trace in H 1/2 (∂D) 3 : Note that this is equivalent to (1.7) Moreover, note also that: 3 , from the first equation on (1.6) and by applying Corollary 1.1 on page 212 and Remark 2 on page 213 in [6] one can conclude that W m ∈ H 1 (R 3 \ D) 3 . Set As done in [11], we introduce two conditions (A.I) and (A.II) on λ listed below: Roughly speaking, we can say that: the condition (A.I)/(A.II) means that the admittance λ is greater/less than the special value λ 0 which is the admittance of free space [1].
Under assumptions 1-2 we have already known that the following statement is true. 12]). Let a j , j = 1, 2 be two linearly independent unit vectors. Let Then, we have: In what follows, we denote by V 0 e the weak solution. Roughly speaking, the reason why such a replacement is possible is the following. Introduce another indicator function by the formulaĨ (1.10) Using the simple facts  [8]. See (15) in [12].
The main purpose of this paper is to go further beyond Theorem 1.1 under Assumptions 1 and 2. Especially, we consider how to extract quantitative information about the state of the surafce of an unknown obstacle using the time domain enclosure method. For the purpose, we clarify the leading profile of the indicator functions (1.4) or (1.10) as τ −→ ∞.
In what follows, we denote by B r (x) the open ball centered at x with radius r. Set d ∂D (p) = inf y∈∂D |y − p| and Λ ∂D (p) = {y ∈ ∂D | |y − p| = d ∂D (p)}. To describe the formula, we recall some notion in differential geometry. Let q ∈ Λ ∂D (p). Let S q (∂D) and S q (∂B d ∂D (p) (p)) denote the shape operators (or Weingarten maps) at q of ∂D and ∂B d ∂D (p) (p) with respect to ν q and −ν q , respectively (see [19] for the notion of the shape operator). Because q attains the minimum of the function: ∂D ∋ y −→ |y − p|, we have always S q (∂B d ∂D (p) (p)) − S q (∂D) ≥ 0 as the quadratic form on the common tangent space at q. Now we are ready to state the main result in this paper.
Theorem 1.2. Assume that ∂D is C 4 and λ ∈ C 1 (∂D). Assume that λ satisfies (A1) or (A2). Let f satisfy (1.8) and T > 2 √ µǫ dist (D, B). Assume that the set Λ ∂D (p) consists of finite points and det (S q (∂B d ∂D (p) (p)) − S q (∂D)) > 0 ∀q ∈ Λ ∂D (p). (1.12) And also assume that ν q × a = 0 for some q ∈ Λ ∂D (p). Then, we have Once we have the formula (1.13), as done in [13] for the scalar wave equation case, we immediately obtain the following corollary. To indicate the dependence of the indicator function on the surface admittance we writẽ Corollary 1.1. Assume that ∂D is C 4 . Let λ 1 and λ 2 belong to C 1 (∂D) and satisfy (A1) or (A2). Let f satisfy (1.8) and T > 2 √ µǫ dist (D, B). Assume that the set Λ ∂D (p) consists of finite points and satisfies (1.12). And also assume that ν q × a = 0 for some q ∈ Λ ∂D (p). Then, we have and its lower and upper estimates: (1.14) In particulr, if Λ ∂D (p) consists of a single point q ∈ ∂D, we have Estimates (1.14) and formula (1.15) are remarkable since they do not require information about the curvatures of the surface of the obstacle in advance. Note that if we know a point q ∈ Λ ∂D (p), then, all the intermediate points p ′ on the segment connecting q and p, satisfy Λ ∂D (p ′ ) = {q} and det (S q (∂B d ∂D (p ′ ) (p ′ )) − S q (∂D)) > 0. Thus, one gets immediately the following corollary in which the set Λ ∂D (p) can be an infinite one, even, continuum.
Corollary 1.2. Assume that ∂D is C 4 . Let λ 1 and λ 2 belong to C 1 (∂D) and satisfy (A1) or (A2). Le p be an arbitrary point in R 3 \ D and q ∈ Λ ∂D (p). Let p ′ be an arbitrary And also assume that ν q × a = 0. Then, we have Thus formula (1.16) can be used for monitoring of the quantitative state of the surface, that is, the change of λ 1 to λ 2 of the surface admittance at a given monitoting point q on the surface.
All the results mentioned above can be transplanted as follows.

Corollary 1.3. Theorem 1.2 and Corollaries 1.1-1.2 remain valid ifĨ ⋆ is replaced with
This can be seen as follows. From (1.8) we have have the same leading profile as τ −→ ∞. Finally, we show that Theorem 1.2 suggests us a procedure for finding curvatures and λ at an arbitrary point q on Λ ∂D (p). It is a translation of the procedure described in [13] in which the scalar wave equation is considered.
Step 1. Choose three points p j , j = 1, 2, 3 on the segment connecting p and q. Denote by B j three open balls with very small radiuses centered at p j such that Step 2. Fix T > 2 max j √ µǫ dist (D, B j ) and generate E and H on B j by the source J j = f (t)χ B j a for a fixed unit vector a with a × ν q = 0 and observe E on B j over the time interval ]0, T [.
Step 3. ComputeĨ J j (τ, T ) from the observation data in Step 2.
Step 5. Use the expression where H ∂D (q) and K ∂D (q) denote the mean and Gauss curvatures at q of ∂D with respect to ν q . From F j we have Solving this linear system numerically, we may obtain H ∂D (q) and K ∂D (q).
Step 6. From F j one has Step 7. From the signature of one of F j one can know whether λ(q) > λ 0 or γ(q) < λ 0 .
This paper is organized as follows. In section 2, we give a proof of Theorem 1.2. The proof is based on a rough asymptotic formula of the indicator function as τ −→ ∞ as stated in Lemma 2.1 which has been established in [12]. The formula consists of two terms and remainder. The treatement of the remainder is not a problem. And the first term is explicitly given by (2.6) as a Laplace type surface integral of V 0 e and its rotation over ∂D. Thus the key point is the profile of the second term which is the energy integral of the so-called reflected solutions given by (2.7). Its asymptotic profile is stated as Theorem 2.1 which tells us that the leading profle is also given as a Laplace type surface integral of V 0 e and its rotation. Then, using the leading profile of those two terms which is described in Lemma 2.2 as an application of the Laplace method, we obtain the reading profile of the indicator function as stated in Theorem 1.2.
The proof of Theorem 2.1 is given in section 3. First we construct a candidate of the leading term of the reflected solutions. For the purpose we employ a combination of the reflection principle which has been established in [12] and a cut-off argument in a neighbourhood of ∂D with a cut-off parameter δ. Then the first and second terms of integral (2.7) is extracted as (3.7) in Lemma 3.1. To show that the first term is the reading profile we have to prove that the second term is small compared with first term. We see that it is true if δ is properly chosen according to the size of τ . It's essence is described as Lemma 3.2. The proof of Lemma 3.2 which is given in section 4, is a modification of the Lax-Phillips reflection argument [17] originally developed for the study of the leading singularity of the scattering kernel for the scalar wave equation in the context of the Lax-Phillips scattering theory, however, our version of the argument is rather straightforward.
2 Proof of Theorem 1.2 From this and (1.9) we have It is a due course to deduce that V 0 m ∈ H 1 (R 3 ) 3 and V 0 e belongs to H 1 in a neighbourhood of D.
Define 3 and R e belongs to H 1 in a neighbourhood of ∂D.
From (2.2) and (1.5) we see that R e and R m satisfy 3) It follows from (1.7) that and Thus, the essential part of the proof of Theorem 1.2 should be the study of the asymptotic behaviour of J(τ ) and E(τ ) as τ −→ ∞. The asymptotic behaviour of J(τ ) can be reduced to that of a Laplace-type integral [3]. See [12]. For that of E(τ ), we have the following result, which enables us to make a reduction of the study to a Laplace-type integral.
Theorem 2.1. Assume that ∂D is C 4 an λ ∈ C 2 (∂D). Assume that λ has a positive lower bound, the set Λ ∂D (p) consists of finite points, and (1.12) is satisfied; there exists a point q ∈ Λ ∂D (p) such that λ(q) = λ 0 and that Then, we have The proof of Theorem 2.1 is given in Section 3. Assumption (2.8) means that vector a is not parallel to ν q at q. Note that the factor 2 in the restriction T > 2 √ µǫ dist (D, B) in Theorem 1.2 is dropped in (2.9). The quantity √ µǫ dist (D, B) corresponds to the first arrival time of the wave generated at t = 0 on B and reached at ∂D firstly. The asymptotic formula (2.10) clarifies the effect on the leading profile of the energy of the reflected solutions R e and R m in terms of the deviation of th surface admittance from that of free-space admittance and the energy density of the incident wave.
To complete the proof of Theorem 1.2 we need the following asymptotic formulae of J(τ ) and J * (τ ) as τ −→ ∞.

Lemma 2.2. We have
(2.14) Proof. Using (2.1), (2.12) and a simple computation in vector analysis, one can rewrite the right-hand side on (2.6) as By (18) in [11] we have already shown that V 0 e has the form This yields and thus (2.1) gives

17)
A combination of (2.16) and (2.17) gives where O(τ −1 ) means uniformly with respect to x ∈ ∂D and Thus we obtain and (2.15) gives Note that if x ∈ Λ ∂D (p), then ν at x coincides with −ω x . Thus we have It is well known that the Laplace method under the assumption that Λ ∂D (p) is finite and satisfies (1.12), yields where A ∈ C 1 (∂D). See [3], for example. The point is that the Hessian matrix of the function ∂D ∋ x −→ |x − p| at q ∈ Λ ∂D (p) is given by the operator S q (∂B d ∂D (p) (p)) − S q (∂D). See, for example, [9] for this point. Replacing τ above with τ √ µǫ, we obtain (2.22) Note also that K(τ ) ∼ τ −1 ηe τ η √ µǫ 2ǫ and thus λ 2 This is nothing but (2.13 provided (1.8) is satisfied, we obtain (1.13). This completes the proof of Theorem 1.1.

Proof of Theorem 2.1
We denote by x r the reflection across ∂D of the point x ∈ R 3 \ D with d ∂D (x) < 2δ 0 for a sufficiently small δ 0 > 0. It is given by [7]). Defineλ(x) = λ(q(x)) for x ∈ R 3 \ D with d ∂D (x) < 2δ 0 . The functionλ is C 2 therein and coincides with λ(x) for x ∈ ∂D. Choose a cutoff function φ δ ∈ C 2 (R 3 ) with 0 < δ < δ 0 which satisfies 0 ≤ φ δ (x) ≤ 1; Using the reflection across the boundary ∂D, in [11] we have already constructed from V 0 e in D the vector field (V 0 e ) * for x ∈ R 3 \ D with d ∂D (x) < 2δ 0 and another one Since R e and R m satisfiy (2.4), we obtain It follows from (2.3) and (3.3) that R 1 e and R 1 m satisfy Now we are ready to state an asymptotic formula of E(τ ) − J * (τ ) as τ −→ ∞ which extracts the main term involving ν × R 1 m on ∂D. Lemma 3.1. We have, as τ −→ ∞ Proof. Recall (40) in [12]: It follows from this and (2.7) that (3.8) This gives Substituting R m = R 0 m + R 1 m into the second term on this right-hand side and using (3.1) and (3.2), we obtain Thus (3.8) becomes (3.9) By Lemma 3.2 in [12] we have Here we make use of the following asymptotic formula which can be shown similarily as formulae in Lemma 2.2 by using (2.18): (3.11) Applying this to the right-hand side on (3.10), we obtain (3.12) Now a combination of (3.9) and (3.12) yields (3.7). ✷ Thus, the problem is: clarify the asymptotic behaviour of ν × R 1 m on ∂D as τ −→ ∞. The point is the choice of δ.
The proof of Leema 3.2 is given in Section 4. Now choose δ in the pair (R 0 e , R 0 m ) as that of Lemma 3.2. Write Applying (2.14), (3.11) and (3.13) to this right-hand side, we obtain (3.15) Note that, if f and T satisfiy (1.8) and (2.9), respectively, then we have, as τ −→ ∞ Now, applying this to (3.15) with the help of (2.14), we see that the left-hand side on (3.15) converges to 0 as τ −→ ∞. Applying this and (3.14) to the right-hand side on (3.7), we obtain (2.10).

Proof of Lemma 3.2
In this section, we denote by C several positive constants independen of δ and τ .
Lemma 4.1. We have

(4.2)
Proof. Taking the inner product of the both sides of the first equation on (3.5) with R 1 m , we obtain Taking the inner product of the both sides of the second equation on (3.5) with R 1 e , we obtain By virtue of the fact that R 1 m ∈ H 1 (R 3 \ D) 3 and R 1 e ∈ H(curl, R 3 \ D), we have where this right-hand side denotes the value of the bounded linear functional R 1 e × ν on H 1/2 (∂D) 3 of ν ×(R 1 m ×ν) ∈ H 1/2 (∂D) 3 . However, R 1 e belongs to H 1 in a neighbourhood of ∂D this coincides with the integral Note also that From these, (4.2), (4.3) and (4.4) we obtain (4.5) Sine we have

Thus (4.5) becomes
where Rewrite (4.6) further as This immediately yields Then, the boundary condition (3.6) yields (4.1). ✷ In order to make use of the right-hand side on (4.1), we prepare the following two lemmas.
✷ Lemma 4.3. We have and Proof. This is an application of a reflection argument developed in [17]. First of all, we compute both ∇ × (V 0 e ) * and ∇ × (V 0 m ) * . From the definition we have We have ∇ × R 0 e =φ δ ∇ × (V 0 e ) * + ∇φ δ × (V 0 e ) * and from (4.11) one gets the expression