The linear sampling method and energy conservation

Transcription

1 The linear sampling method and energy conservation R Aramini, G Caviglia, A Massa and M Piana $ Dipartimento di Ingegneria e Scienza dell Informazione, Università di Trento, via Sommarive 4, 3823 Povo di Trento, Italy Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 646 Genova, Italy $ CNR - INFM LAMIA, via Dodecaneso 33, 646 Genova, Italy Abstract. In this paper we explain the linear sampling method and its performances in various scattering conditions by means of an analysis of the far-field equation based on the principle of energy conservation. Specifically, we consider the conservation of energy along the flow strips of the Poynting vector associated with the scattered field whose far-field pattern is one of the two terms in the far-field equation. The behavior of these flow lines is numerically investigated and theoretically described. Appropriate assumptions on the flow lines, based on the numerical results, allow characterizing a set of approximate solutions of the far-field equation which can be used to visualize the boundary of the scatterer in the framework of the linear sampling method. In particular, under the same assumptions, we can show that Tikhonov regularized solutions belong to this set of approximate solutions for appropriate choices of the regularization parameter.. Introduction The linear sampling method [6, 3, 4, 6] is a visualization algorithm for solving inverse scattering problems in the time-harmonic regime. It belongs to the family of noniterative qualitative methods [8, 9, 2, 9], aimed at determining the location and shape of the unknown scatterer (but not the pointlike values of its index of refraction) from the knowledge of the scattered field measured in either a near-field or a far-field region surrounding the target. From a computational viewpoint, all qualitative methods are based on the following procedure: the investigation domain is sampled by means of a computational grid; for each point of the grid a linear integral equation whose kernel is related to the scattering measurements is solved by means of a regularization method; To whom correspondence should be addressed.

2 2 the norm of the regularized solution is used to visualize the boundary of the scatterer. The linear sampling method is the qualitative method where the integral kernel is the far-field pattern of the scattered field. The most interesting features of these qualitative methods are well-known: they require a limited amount of a priori information on the scatterer, e.g. the linear sampling method only needs to assume that the target is contained in a known bounded region; their linearity is intrinsic, i.e. derives from no approximation, as Born or physical optics; they are not iterative and then do not suffer from the typical pathologies affecting iterative algorithms such as local minima, need for a sufficiently accurate initialization, or long computational times. Besides the impossibility of providing quantitative information on the scatterer, qualitative methods also show some drawbacks in their performances; moreover, they even pose some problems from the viewpoint of their theoretical foundation. In particular, as far as the linear sampling method is concerned, a satisfactory understanding of the reason why it should work at all is still an open issue [7]. This is due to a missing link between the general theorem inspiring the method and the method itself. Indeed, the general theorem shows that, for each sampling point z in the physical space, a far-field equation exists that admits approximate solutions whose L 2 - norm is bounded when z is inside the scatterer, tends to blow up when z approaches the boundary of the scatterer from inside and remains arbitrarily large when z is outside. On the other hand, there is no a priori guarantee that the regularized solution of the farfield equation, as computed by the algorithm and exploited to characterize the domain of the scatterer, should behave like one of those approximate solutions. However, many numerical simulations, performed under very different scattering conditions and with various noise levels, show that there is a good agreement between theory and practice, i.e. that the computed regularized solution behaves as indicated by the general theorem. So far, the attempts made to explain this agreement can be divided into two families: ) a first set of papers [2, 3, 8] focuses on the restrictive case in which, in addition to the linear sampling method, also the factorization method [9] can be used; 2) a second set of papers [, 22] uses physics-based arguments under restrictive hypotheses on the scattering conditions: in [] the scatterer is assumed to be a dielectric target, while in [22] only perfectly electrical conducting objects (in the resonance regime) are taken into account. The purpose of our paper is to conceive a physical interpretation of the farfield equation that does not depend on the penetrable or impenetrable nature of the scatterer. Our approach is based on the properties of the (time-averaged) Poynting vector associated with the field whose far-field pattern is the left-hand side of the farfield equation: this Poynting vector carries out electromagnetic energy from the scatterer to infinity, and we shall prove that energy conservation along its flow strips, together

3 with the constraint on energy fluxes expressed by the far-field equation, are sufficient conditions for the linear sampling method to work. We point out that this explanation is based on an a posteriori approach: more precisely, the performances of the linear sampling method are related to the behavior of the flow lines of the Poynting vector, but such a behavior is numerically observed and not theoretically predicted. This is also the reason why, at the current stage of advancement, our approach is not a mathematical justification of the linear sampling method: to this end, it would be necessary to deduce the geometric properties of these flow lines a priori, i.e. starting from the knowledge of the scattering conditions. Such an investigation could be made by using sophisticated tools of topological dynamics [2], which is however beyond the purposes of this paper. In order to keep our investigation as simple as possible, we shall focus on the 2D electromagnetic scattering problem for a penetrable and isotropic cylinder, by assuming that the measurements are taken in the far-field region of a lossless and homogeneous background. However, the key-ideas of our approach are still valid in different or more general situations: in particular, the physical properties of the scatterer are irrelevant, the acoustic case can be discussed in the same way, possible heterogeneities of the background could be simply accounted for, etc. The plan of the paper is as follows. In Section 2 we introduce both the direct and the (qualitative) inverse scattering problems: in particular, the general theorem [8], concerning the existence of ɛ-approximate solutions to the far-field equation and their qualitative behavior, is recalled and the logical gap between the theorem and the linear sampling method is highlighted. Section 3 introduces the Poynting vector of the scattered field and identifies some relevant features of its flow lines in the framework of energy (i.e. time-averaged power) conservation. This allows a physical interpretation of the far-field equation as a constraint on power fluxes. In Section 4 we perform a certain number of numerical simulations in order to visualize the behavior of the flow lines of the Poynting vector when the sampling point is inside the scatterer or on its boundary. In Section 5 we prove that such behavior, the energy conservation along the flow strips and the energy constraint induced by the far-field equation allow characterizing the ɛ- approximate solutions of the far-field equation in a fashion that is in agreement with the standard general theorem. Section 6 adapts the approach of Sections 4 and 5 to the case of a sampling point chosen outside the scatterer. Section 7 specifies the previous results to Tikhonov regularized solutions. Finally, our conclusions and suggestions for future developments are proposed in Section 8. 3

4 4 2. The inverse scattering problem and the linear sampling method We first consider the following direct scattering problem: a plane, electromagnetic and time-harmonic wave, propagating in a homogeneous and non-conducting background medium, is scattered by an inhomogeneity consisting of a penetrable, isotropic and infinitely long cylinder. The geometrical and physical properties of the cylinder are invariant with respect to translations along its axis: in particular, its cross section is the closure of an open and C 2 -domain D IR 2. Moreover, we assume that the incident plane wave is TM-polarized. The related scattering problem [3, 5] consists in the determination of u = u( ; θ) C 2 (IR 2 \ D) C (IR 2 ) such that u(x) + k 2 n(x) u(x) = for x IR 2 \ D (a) u(x) = e ikx ˆd + u s (x) for x IR 2 (b) [ ( )] (2.) r u s lim r r ikus =, (c) where ˆd = ˆd(θ) = (cos θ, sin θ) is the incidence direction and k is the wavenumber; n(x) is the refractive index n(x) := [ ε(x) + i σ(x) ] x IR 2, (2.2) ε B ω where i = and ω denotes the angular frequency of the wave; ε(x) and σ(x) are the electrical permittivity and conductivity, respectively. We assume that ε(x) is uniform in IR 2 \ D and equal to the background value ε B >, while σ = in the same region. We consider a non-magnetic scatterer, i.e. we require that the magnetic permeability is a positive constant µ B everywhere in IR 2. For each incidence direction ˆd, there exists a unique solution to problem (2.) [5] and the corresponding scattered field u s = u s ( ; θ) has the following asymptotic behavior (holding uniformly in all directions ˆx := x/ x ): u s (x; θ) = eikr r u (ϕ; θ) + O ( r 3/2) as r = x, (2.3) where (r, ϕ) are the polar coordinates of the observation point x and the function u = u ( ; θ) L 2 [, 2π] is known as the far-field pattern of the scattered field u s. In this paper we consider the qualitative inverse problem of determining the support D of the scatterer under the assumption that the far-field pattern u (ϕ, θ) is known for all observation and incidence angles ϕ, θ [, 2π]. A procedure for its solution is provided by the linear sampling method. Define the linear and compact far-field operator F : L 2 [, 2π] L 2 [, 2π]

5 5 corresponding to the inhomogeneous scattering problem (2.) as [8] (F g)(ϕ) := 2π u (ϕ, θ)g(θ)dθ g L 2 [, 2π]. (2.4) The operator F is injective with dense range if k 2 is not a transmission eigenvalue [8]. By the superposition principle, (F g) is the far-field pattern of the scattered field u s g(x) := 2π u s (x, θ)g(θ)dθ x IR 2 \ D (2.5) corresponding to the incident field u i given by the Herglotz wave function v g with kernel g, i.e. u i (x) = v g (x) := 2π Next consider the outgoing scalar field e ikx ˆd(θ) g(θ)dθ for x IR 2. (2.6) Φ(x, z) = i 4 H() (k x z ) x z, (2.7) generated by a point source located at z IR 2, where H () ( ) denotes the Hankel function of the first kind and of order zero. The corresponding far-field pattern is given by Φ (ϕ, z) = eiπ/4 8πk e ikˆx(ϕ) z, with ˆx(ϕ) := (cos ϕ, sin ϕ) ϕ [, 2π]. (2.8) For each z IR 2, the far-field equation is defined as (F g z )(ϕ) = Φ (ϕ, z). (2.9) The linear sampling method depends on what we shall call the general theorem [8, 3], concerning the existence of ɛ-approximate solutions to the far-field equation and their qualitative behavior. Theorem 2.. (General theorem) Let D IR 2 be nonempty, open, bounded, with C 2 -boundary D, and such that IR 2 \ D is connected; let n : IR 2 IC be given by (2.2) and such that n D C( D); let k := ω ε B µ B > be such that k 2 is not a transmission eigenvalue and let F be the far-field operator corresponding to the inhomogeneous scattering problem (2.). Then: (i) if z D, it follows that for every ɛ > there exists a solution g ɛ z L 2 [, 2π] of the inequality F g ɛ z Φ (, z) L 2 [,2π] ɛ (2.) such that, for every z D, lim z z gɛ z L 2 [,2π] = (a) and lim vg L z z ɛ z =, (b) (2.) 2 (D) where v g ɛ z is the Herglotz wave function with kernel g ɛ z;

6 6 g ɛ,δ z (ii) if z / D, it follows that for every ɛ > and δ > there exists a solution L 2 [, 2π]of the inequality F gz ɛ,δ Φ (, z) ɛ + δ (2.2) L 2 [,2π] such that lim gz ɛ,δ δ L 2 [,2π] = (a) and lim δ v g ɛ,δ z =, (b) (2.3) L 2 (D) where v g ɛ,δ z is the Herglotz wave function with kernel g ɛ,δ z. On the basis of Theorem 2., the algorithm of the linear sampling method may be described as follows [6]. Consider a sampling grid that covers a region containing the scatterer. For each point z of the grid compute a regularized solution g α (z) of the (discretized) far-field equation (2.9) by applying Tikhonov regularization coupled with the generalized discrepancy principle [23]. The boundary of the scatterer is visualized as the set of grid points in which the (discretized) L 2 -norm of g α (z) becomes mostly large. Apart from noise and discretization issues, there is a logical gap between the content of Theorem 2. and the linear sampling method. Indeed, the proof of the former gives no evidence that the approximate solutions gz ɛ and gz ɛ,δ are just (or can be chosen as) the Tikhonov regularized solutions g α (z) of the far-field equation (for z D and z / D respectively) exploited by the latter. Even more, in Tikhonov regularization, the blowing up of is only possible for a vanishing regularization parameter L 2 [,2π] α (z); but, g α (z) owing to the denseness of the range of F and to the fact that, in general, Φ (, z) does not belong to this range, the discrepancy F gα (z) Φ (, z) L also vanishes when 2 [,2π] α (z) and gα (z) blows up even for almost all z inside D [23] (cf. also L 2 [,2π] Remark 3.2 of [8]). The main result of this paper consists in showing that the linear sampling method is a manifestation of the general principle of energy conservation. This approach will allow us to prove that every ɛ-approximate solution to the far-field equation blows up at the boundary of the scatterer; in particular, this holds for the Tikhonov regularized solution. From a technical viewpoint, we first need: i) some results concerning the energy transport along the flow strips of the Poynting vector and ii) some numerical tests concerning the behavior of its flow lines. 3. Power flux of the scattered field and far-field equation In our interpretation of the linear sampling method, a crucial role is played by the time-averaged Poynting vector S s [4] associated with a scattered field u s. In the present

7 framework, it is given by S s (x) = 4iω ε B µ B [ū s (x) u s (x) u s (x) ū s (x)]. (3.) In general, the time-averaged Poynting vector is related to the mean flow (over a period) of power per unit area. Accordingly, its flux over a given surface evaluates the timeaveraged amount of power crossing the surface. In our 2D framework, the flux of S s (x) over any curve γ in IR 2 \D equals the average power radiated through γ per unit length by the scatterer modeled as an equivalent source of electromagnetic waves [5]. Moreover, we consider only simple and (almost everywhere) regular curves. Then, we define the power flux of the scattered field u s across γ as the power flux of the associated Poynting vector, namely F γ (u s ) := S s (x) ν(x) dl(x), (3.2) γ where ν(x) denotes the unit normal to γ in x (chosen as outward when γ is closed) and dl(x) indicates the standard measure defined on γ. Since u s satisfies the Helmholtz equation in IR 2 \ D, the vector field S s (x) is divergence free in IR 2 \ D. Then, the Gauss divergence theorem implies that F γ (u s ) = (3.3) for any closed curve in IR 2 \ D not enclosing the scatterer (or any connected component of it). Furthermore F γ (u s ) = F γ2 (u s ) (3.4) for any pair of closed curves γ and γ 2 surrounding the whole scatterer. In order to determine any of the integrals in (3.4), we introduce the circle Ω R := {x IR 2 : x = R} and compute the power flux at infinity of u s as F (u s ) := lim F Ω R R (u s ). (3.5) This flux can be written in terms of the far-field pattern by observing that (2.)(c) and (2.3) imply [5] F (u s k ) = u 2 L 2ω ε B µ 2 [,2π]. (3.6) B When applied to specific scattered fields related to the far-field equation, relation (3.6) implies a technical consequence very helpful for a new formulation of the general theorem and a conceptual remark that naturally inspires a physical interpretation of the far-field equation and a near-field version of it. We first observe that if gz ɛ L 2 [, 2π] is such that F gz ɛ Φ (, z) L 2 [,2π] ɛ and ϕ, ϕ 2 [, 2π] are any two angles such that ϕ ϕ 2, simple algebraic manipulations lead to F gz ɛ 2 L 2 [ϕ,ϕ 2 ] Φ (, z) 2 L 2 [ϕ,ϕ 2 ] ɛ (ϕ, ϕ 2 ), (3.7) 7

8 8 where ɛ (ϕ, ϕ 2 ) := ɛ ( 2 Φ (, z) L 2 [ϕ,ϕ 2 ] + ɛ) (3.8) does not depend on z, since Φ (, z) L 2 [ϕ,ϕ 2 ] does not. Now, F gɛ z is the far-field pattern of the scattered field u s,ɛ z (x) := 2π u s (x, θ)g ɛ z(θ)dθ x IR 2 \ D. (3.9) Therefore, a comparison of (3.6) and (3.7) implies that, for any observation interval [ϕ, ϕ 2 ] in the far-field region, the flux of the scattered field u s,ɛ z can be made arbitrarily close to the flux of the field radiated by a pointlike source placed at the sampling point z IR 2. This result will play a crucial role in the proof of the new version of the general theorem based on energy conservation arguments, which will be discussed in Section 5. Equation (3.6) has also the important physical consequences described in the following Remark 3.. A physical interpretation of the far-field equation is a critical issue, since it is not a necessary consequence of physical laws. However, let us consider the radiating field wz s,ɛ (x) = u s,ɛ z (x) Φ(x, z), with far-field pattern wz, s,ɛ = F gz ɛ Φ (, z). Then equations (3.4) and (3.6) applied to wz s,ɛ, together with inequality (2.), imply F γ (wz s,ɛ ) = F (wz s,ɛ k ) = w s,ɛ z, 2 2ω ε B µ k ɛ2, (3.) B L 2 [,2π] 2ω ε B µ B for any z IR 2 and any γ enclosing D {z}. This may be regarded as the physical content of inequality (2.) in the statement of the general theorem; the power flux of the difference field u s,ɛ z (x) Φ(x, z) across any closed curve surrounding D {z} is bounded according to (3.). More importantly, equation (3.) naturally inspires the near-field equation F γ (wz s,ɛ ) =, which is characterized by a notable similarity with the integral equation at the basis of the reciprocity gap functional method [2]. As in that case, also here information on boundary values of both the field and its normal derivative is needed to qualitatively solve the problem. But here the physical interpretation is much more natural. In fact, it is known that, in L 2, a small far-field pattern does not necessarily correspond to a small scattered field [5]. Instead, equation (3.) shows that a small flux at infinity of wz s,ɛ remains small also close to the scatterer. Therefore the far-field equation (2.9) can be pulled back to a near-field region, provided that the gap between its two sides is estimated in terms of power fluxes instead of L 2 -norms. On the basis of equations (3.4) and (3.6) we have examined global conservation properties of the (time-averaged) power flux and their connection with the far-field equation. In view of the further developments we complete the discussion with an analysis of how power is radiated along flow strips, possibly emanating from parts of the boundary of the scatterer.

9 We introduce the flow lines of the time-averaged Poynting vector S s (x) by recalling that they are defined as the solutions to the initial value problem 9 dx dτ (τ) = Ss (x(τ)) x() = x (a) (b) (3.) where x is a point in IR 2 \ D. Since u s (and, consequently, S s (x)) is real-analytic in IR 2 \ D [8], for each x IR 2 \ D there exists a unique solution ζ x (τ) of the problem (3.), which is called the flow line of the scattered field u s starting from x. We are interested in considering the flow lines for τ. Henceforth, we assume that ζ x (τ) is defined for every τ and that there are no critical points x of S s, such that the flow line starting from x collapses into the point x itself []. Comparison with the definition in (3.2) shows that the average power crossing a flow line is zero, thus showing that power is carried out by the scattered field along flow strips whose boundaries are just flow lines. Now we can follow the power flux of u s from the near-field to the far-field region along its flow strips in the background medium. Accordingly, we consider a flow strip bounded by two (different) flow lines ζ x (τ) and ζ x (τ), and we assume that none of its transverse sections intersects the domain D: in particular, the two flow lines ζ x (τ), ζ x (τ) do not refold on the scatterer; moreover, we require that they are indefinitely outgoing toward the far-field region, in the sense that each of them approaches a definite direction. The last requirement is consistent with the Silver-Müller radiation condition [5], since the radiating electric and magnetic fields tend to be transverse in the far-field region, which implies that the corresponding Poynting vector becomes radial. More precisely: Definition 3.2. Let x IR 2 \ D. A flow line ζ x (τ) is regular if: (i) ζ x (τ) D = τ ; (ii) there exists R > such that R > R the flow line ζ x (τ) intersects Ω R in one and only one point P x (R) of polar coordinates (R, ϕ[p x (R)]); (iii) lim R ϕ[p x (R)] =: ϕ (x ). Moreover, a flow strip of u s is regular if it is bounded by two different regular flow lines and no one of its transverse sections intersects D. We point out in particular that no critical point of the Poynting vector is allowed to belong to a regular flow line. For an analysis of such points and their effects on the flow lines, see e.g. [2]. For future purpose, we also give the following Definition 3.3. Let ζ x (τ) and ζ x (τ) be two regular flow lines with x, x IR 2 \ D; let ϕ (x ) and ϕ (x ) be the corresponding asymptotic polar angles. Then, ψ (x, x ) := ϕ (x ) ϕ (x ) is called the asymptotic angular width of the flow strip bounded by ζ x (τ) and ζ x (τ).

10 We can now describe how energy conservation is realized along regular flow strips. Indeed, let x, x IR 2 \ D, not belonging to the same flow line. Consider the flow lines ζ x (τ), ζ x (τ) and assume that they are regular. Next choose x 2 ζ x (τ) and x 3 ζ x (τ) and draw two non-intersecting curves, γ and γ 2, connecting x to x, and x 2 to x 3, respectively. We regard as a finite flow strip the subset of IR 2 \ D bounded by the closed curve γ resulting from the union of the transverse sections γ, γ 2 and the arcs of the flow lines with endpoints x, x 2 and x, x 3. According to (3.3), the power flux across γ is zero. As a consequence, if the unit normals to both γ and γ 2 are oriented towards infinity then F γ (u s ) = F γ2 (u s ). The last equation is the local counterpart of (3.4). If ζ x (τ) and ζ x (τ) are regular flow lines identifying a regular flow strip with asymptotic angular width ψ (x, x ) >, then a local version of (3.) holds in the form F γ (u s ) = F γ2 (u s k ) = u 2 L 2ω ε B µ 2 [ϕ (x ),ϕ (x )]. (3.2) B In particular, equation (3.2) shows that the power flux of a scattered field through an outwardly oriented arc is positive. Remark 3.4. The analysis performed so far is easily adapted to the acoustic case [5]. In particular, the role of the time-averaged Poynting vector (3.) is now played by the vector S s (x) = 4iω ρ [ p s (x) p s (x) p s (x) p s (x)], (3.3) where ρ is the constant equilibrium density of the background and p s (x) is the acoustic scattered pressure field in the inviscid background fluid. 4. Behavior of the flow lines In the present section we consider some numerical simulations showing the behavior of the flow lines for sampling points inside the scatterer or on its boundary. The implementation of the linear sampling method used for our simulations is the same as in [6]. Specifically, the regularized solution g α(z) (z) IC N of a noisy and discretized version 2π F N hg(z) = Φ (z) of the far-field equation is accomplished by means of Tikhonov method, where the regularization parameter α(z) is fixed to an optimal value α (z) by the generalized discrepancy principle. This regularized solution is used in (3.9) and in the associated Poynting vector (3.) to obtain a discretized version of u s,ɛ z and Sz s,ɛ. Our numerical simulations are performed by choosing a frequency ν = GHz, corresponding to a wavelength λ =.3 m in vacuum, which is the background medium (i.e. ε B = ε, σ B = ), and by using the same number N = 5 of incidence and observation angles. The far-field patterns are computed by a 2D TM direct code based

11 on the method of moments [2], then are corrupted by 3% Gaussian noise and used as entries of the noisy far-field matrix F h. The investigation domain T is a square of minimum side.5 m, i.e. 5. λ y/λ y/λ x/λ x/λ (a) (b) Figure. Implementation of the linear sampling method: (a) visualization of an elliptic scatterer with, superimposed, its true profile (solid black line); (b) values of the discretized discrepancy d(z). In the first numerical example we consider an ellipse centered at the origin, with semiaxes of length a =.4 λ and b =.2 λ: this scatterer is characterized by constant relative electric permittivity ε r = 2. and electric conductivity σ =.2 S m. In this case, the linear sampling method provides a satisfactory reconstruction, shown in Figure (a) together with the actual profile (solid black line). For future purpose, in Figure (b) we plot the point-values of the discretized discrepancy d(z) := 2π F N hg α (z)(z) Φ (z) IC, which is a numerical estimate of N ɛ F gz ɛ Φ (, z) L 2 [,2π]. Figure 2 shows the behavior of the flow lines of u s,ɛ z for a sampling point placed at the center of the ellipse: more precisely, the arrows represent the discretized and time-averaged Poynting vector field Sz s,ɛ, normalized in order to avoid scale effects impairing the visualization. The behavior of the flow lines is essentially radial with respect to z, in this resembling the field generated by a point source located at z: in other terms, the scattered field u s,ɛ z reproduces, at least qualitatively, the features of the fundamental solution Φ(, z). Such a radial behavior of the flow lines with respect to z is maintained for any z inside the scatterer or on its boundary, as shown e.g. in Figure 3 for z = (. λ,.2 λ): in particular, the flow lines are regular in the sense of Definition

12 y/λ x/λ Figure 2. Regular flow lines for a sampling point z (represented by a red bullet) placed at the origin. The second example is concerned with a case in which the reconstruction of the unknown scatterer is unsatisfactory. We consider two penetrable ellipses with the same dimensions as the previous one: the upper ellipse is centered at the point (. λ,.5 λ) and characterized by constant relative electric permittivity ε r, = and electric conductivity σ =.38 S m ; the lower ellipse is centered at (. λ,.5 λ) and its electric parameters are ε r,2 = 2 and σ 2 =.5 S m. Figure 4(a) shows the reconstruction provided by the linear sampling method, together with the true profile of the scatterer (solid black lines). In Figure 4(b) we plot the point-values of the discretized discrepancy d(z). As in the previous case, for a sampling point z inside the scatterer, or on its boundary, the behavior of the flow lines of u s,ɛ z resembles that of Φ(, z): see Figure

13 y/λ x/λ Figure 3. Regular flow lines for a sampling point z (represented by a red bullet) placed on the boundary of the scatterer. 5 (for z = (. λ,.5 λ)), Figure 6 (for z = (. λ,.7 λ)) and Figure 7 (for z = (. λ,.3 λ)). We point out that in the case of Figure 6 the flow lines starting from a neighborhood of z are regular in the sense of Definition 2.; instead, this is not true in the case of Figure 7, since these flow lines cannot reach the far-field region without crossing the lower ellipse: notably, in this case the visualization provided by the linear sampling method is bad in the region around z. From this first set of simulations, we can conclude that, if z D, the scattered field (3.9) resembles the field Φ(, z) radiated by a pointlike source placed in z: this fact, as we shall see in the next section, suffices to explain the growing of g ɛ z L 2 [,2π] as z approaches the boundary D. A second set of simulations will be performed in Section 6 to study the behavior of the flow lines when z is outside D.

14 y/λ.5 y/λ x/λ x/λ (a) (b) Figure 4. Implementation of the linear sampling method: (a) visualization of a double-elliptic scatterer with, superimposed, its true profile (solid black lines); (b) values of the discretized discrepancy d(z). 5. A new version of the general theorem: z D In the present section we shall provide a new version of the general theorem for the linear sampling method when the sampling point is inside the scatterer. This new version utilizes both the considerations on the fluxes of the Poynting vector described in Section 3 and the numerical behavior of its flow lines shown in Section 4. In the next section, we shall consider the case of z outside the scatterer. According to the general theorem, for every ɛ > there exists a solution gz ɛ of inequality (2.) such that gz ɛ L 2 [,2π] if z z D. We prove that, under appropriate assumptions on the flow lines, the norm of any approximate solution of the far-field equation blows up for a non-vanishing (although small enough) bound ɛ on the discrepancy. Theorem 5.. Under the same hypotheses of Theorem 2., consider a point z D and a neighborhood U z of z. If z U z D and ɛ >, let gz ɛ L 2 [, 2π] be such that F gz ɛ Φ (, z) L 2 [,2π] ɛ. (5.) For each z U z D, denote by C z (z ) the circle of center z and radius r := z z, and by C z (z ) the intersection C z (z ) (IR 2 \ D). Suppose that U z is so small that C z (z ) is an arc with endpoints yz, yz 2 D for each z U z D. Moreover, assume that the flow lines ζ y z (τ) and ζ y 2 z (τ) of u s,ɛ z are regular and identify a regular flow strip with asymptotic angular width ψ (z) := ϕ (yz) ϕ (yz) ; 2 finally, assume that lim z z ϕ (yz) and

15 y/λ x/λ Figure 5. Flow lines for a sampling point z (represented by a red bullet) placed at the center of the upper ellipse. lim z z ϕ (y 2 z) exist finite and are different. Then, for any such g ɛ z L 2 [, 2π], lim z z gɛ z L 2 [,2π] = (5.2) if ɛ is small enough. Proof. We preliminarily note that the existence of gz ɛ L 2 [, 2π] satisfying inequality (5.) follows from the denseness of the range of F. Let z D. Since the boundary D is C 2, the condition that C z (z ) is an arc is satisfied provided z z is small enough (see Figure 8). Next we assume, by contradiction, that the limit (5.2) does not hold. Then, there exist a constant K > and a sequence { } n= U z D such that lim z = n and gz ɛ L n K n IN. (5.3) 2 [,2π]

16 y/λ x/λ Figure 6. Flow lines for a sampling point z (represented by a red bullet) placed at the top of the upper ellipse. Now, let B := {x IR 2 x < R } be the open disk centered at the origin and with radius R large enough, so that B D, and set G := B \ D. To show the contradiction we need a common bound for the fields u s,ɛ (x) (defined as in (3.9)) for any x Ḡ and n IN, and for their partial derivatives. Continuity in Ḡ of us (, θ) for each θ [, 2π] is obvious, while continuity in [, 2π] of u s (x, ), uniformly with respect to x Ḡ (i.e. lim u s (x, θ) u s (x, θ ) = ), follows max θ θ x Ḡ from the well-posedness of the direct scattering problem with respect to the maximum norm in C( B ) [5] and from the fact that lim e ikx ˆd(θ) e ikx ˆd(θ ) =. Then, u s is max θ θ x B continuous in A := Ḡ [, 2π]; since A is compact, we can define M := max (x,θ) A us (x, θ). As a consequence, by using the Cauchy-Schwarz inequality and comparing with (3.9)

17 y/λ x/λ Figure 7. Flow lines for a sampling point z (represented by a red bullet) placed at the bottom of the upper ellipse. and (5.3), we have u s,ɛ (x) 2π u s (x, θ)gz ɛ n (θ) g dθ M 2π ɛ L zn 2πM K =: Q, (5.4) 2 [,2π] for all n IN and for all x Ḡ. By a similar procedure, we can bound the incident field u i,ɛ (x) = v g ɛ z for all n IN n and for all x Ḡ: v g ɛ (x) 2π e ikx ˆd(θ) gz ɛ n (θ) dθ 2π gz ɛ L n 2πK. (5.5) 2 [,2π] From (5.4) and (5.5), we find an upper bound for the total field u ɛ = u i,ɛ + u s,ɛ, i.e. u ɛ (x) Q + 2πK x Ḡ, n IN. (5.6) To find a similar bound for the derivatives we recall [5] that the direct scattering

18 8 C zn (z ) D y z t y 2 C zn (z ) Figure 8. A point D approaching z D. problem for an incident field u i (x) is equivalent to the Lippmann-Schwinger equation u(x) = u i (x) k 2 Φ(x, y) m(y)u(y)dy, x B, (5.7) B where m := n, and B := {x IR 2 : x < R} is any open disk such that B D; in particular, by taking R > R, we can assume that B Ḡ. As a consequence of (5.7), we find that u s,ɛ Φ (x) = k2 xi B x (x, i y)m(y)uɛ (y)dy, x B Ḡ, (5.8) where the partial derivative with respect to x i (with i =, 2) can be brought inside the integral because of the boundedness of m(y)u ɛ (y) [7]. If we denote by M an upper bound for m(y), from (5.6) and (5.8) we get u s,ɛ x (x) i k2 M ( Q + 2πK ) Φ (x, y) dy x Ḡ, n IN. (5.9) B xi By the same arguments used in [7], it is possible to show that the integral at the righthand side of (5.9) is a continuous function of x: then it takes its maximum value, say M 2, on Ḡ. As a consequence, we find an upper bound for the derivatives of the scattered field, i.e. u s,ɛ x (x) i k2 M ( Q + 2πK ) M 2 =: Q 2 x Ḡ, n IN. (5.)

19 [If the scatterer is not penetrable, inequalities analogous to (5.4) and (5.) can be proved more easily, by exploiting the well-posedness of the direct problem: indeed, in this case the solution operator is bounded from C,α ( D) into C,α (IR 2 \ D), see [5].] Now, let us evaluate the flux F Czn (z ) (us,ɛ ) as n in two ways: a) near the boundary D, i.e. across C zn (z ). In view of (3.2), we have: [ ] F Czn (z ) (us,ɛ ) = 4iω ε B µ B C (z ) ū s,ɛ u s,ɛ z ū s,ɛ us,ɛ n ν ν 9 (x) dl(x). (5.) Then, by observing that C zn (z ) Ḡ for n large enough, and by applying inequalities (5.4), (5.), we have: F C(z ) (us,ɛ ) 2ω ε B µ B C (z ) ūs,ɛ (x) us,ɛ ν (x) dl(x) C zn (z ) Q Q 2, (5.2) 2ω ε B µ B where C zn (z ) denotes the length of the arc C zn (z ). Since C zn (z ) as n (i.e. as z ), we find that lim F C (z ) (us,ɛ ) = ; (5.3) n b) in the far-field region. To this end, we consider the regular flow strip delimited by the two regular flow lines ζ y zn (τ) and ζ y 2 zn (τ) of u s,ɛ, so that the power flux outgoing from C zn (z ) is preserved along the flow strip itself up to infinity. Hence, recalling (3.2), (3.5), (3.6), (3.7), following the notations introduced by Definition 2. and assuming (it is not restrictive) that ϕ (yz n ) ϕ (yz 2 n ), we have: [ ] F C(z ) (us,ɛ ) = 4iω ε B µ B = 4iω ε B µ B = ū s,ɛ u s,ɛ C (z ) ν ū s,ɛ us,ɛ ν [ ϕ[py 2 (R)] lim ū s,ɛ u s,ɛ z ū s,ɛ ] R us,ɛ n ϕ[p y (R)] r (x) dl(x) (5.4) r (R, ϕ) Rdϕ k F g ɛ 2 z 2ω ε B µ n B L 2 [ϕ (yzn ),ϕ (yzn 2 )] k [ Φ (, ) 2 L 2ω ε B µ 2 [ϕ (y B zn ),ϕ (yzn 2 )] ɛ (ϕ (yz n ), ϕ (yz 2 n )) ]. According to our assumptions, we have lim ϕ (y n ) =: ϕ and lim ϕ (y 2 n ) =: ϕ 2 ; then, from the chain of inequalities (5.4), we easily get lim inf n F C(z ) (us,ɛ ) k 2ω ε B µ B [ Φ (, z ) 2 L 2 [ϕ,ϕ 2 ] ɛ (ϕ, ϕ 2 ) ]. (5.5) Since ψ (z ) := ϕ ϕ 2 =, the inequality Φ (, z ) 2 L 2 [ϕ > holds; hence,,ϕ 2 ] by taking ɛ small enough, we can make ɛ (ϕ, ϕ 2 ) small enough too, so that the righthand side of (5.5) is strictly positive, and a contradiction between (5.3) and (5.5) is

20 2 obtained. More precisely, according to the definition (3.8), it suffices to take < ɛ < ( 2 ) Φ (, z ) L 2 [ϕ,ϕ2 ] = ( ψ (z 2 ) ) 8πk. (5.6) This concludes the proof. Remark 5.2. An explicit computation of the bound (5.6) requires the knowledge of the asymptotic angular width ψ (z ). Since numerical simulations show that, for z D, the behavior of the flow lines of u s,ɛ z is essentially radial with respect to z, we can identify ψ (z ) with the limit amplitude of the Euclidean angle yz n ẑ n yz 2 n of vertex and subtended by the arc C zn (z ) as n. Now, it is easily seen that yz n ẑ n yz 2 n is an angle at the circumference C zn (z ) whose corresponding angle at the center tends to π as n, owing to the existence of the tangent t in z to the C 2 -boundary D (see Figure 8). As a consequence, we find that ψ (z ) = π/2, and then bound (5.6) becomes < ɛ < ( ) 2 4 k. (5.7) 6. A new version of the general theorem: z / D Let us first observe that, if x D, uniqueness issues for the initial value problem (3.) may arise. Indeed, the scattered field u s is real-analytic in IR 2 \ D but is only in C (IR 2 \ D) (or in C,α (IR 2 \ D), with < α <, in the case of perfect conductors [5]). Accordingly, its first derivatives are only in C (IR 2 \ D) (or in C,α (IR 2 \ D)), i.e. they are not necessarily Lipschitz up to D. Hence the field S s defined in (3.) is not necessarily Lipschitz up to D: as a consequence, an initial point x taken on D may be a ramification point, i.e. a point whence several flow lines start. A clear example of this is provided by Figures 3 and 6. Indeed, as observed in Section 4, the radiality of flow lines is approximately verified for sampling points z in D and even on D: then, in the latter case, z is also a ramification point. A situation where no ramification point is allowed occurs when a penetrable scatterer stands out from the background in a smooth way, i.e. when n is in C (IR 2 ): indeed, in this case u s C 2 (IR 2 ) [5]. However, this smoothness property is seldom verified in practice; moreover, numerical simulations show that the support of such a smooth scatterer is significantly underestimated by the linear sampling method, as expected. Hence, we shall not explicitly investigate this situation in the following. In the case of sampling points z chosen in regions outside the scatterer where the visualization is good, ramification points on D systematically show up in our numerical experiments and the behavior of the flow lines, in general, is far from being radial with respect to z or any other point in the plane.

21 With reference to the same experiments of Section 4, in Figure 9 we consider a sampling point z = (.7 λ,.7 λ): notably, two ramification points (represented by red square boxes) are detectable on the boundary of the scatterer. In Figure the sampling point is z = (.7 λ,.7 λ): again, two ramification points show up on the boundary D. Finally, Figure shows the behavior of the flow lines for a sampling point z placed at the origin of the investigation domain: except for the region between the two ellipses, the field has a radial behavior with respect to the sampling point. However, we notice that no ramification point is detectable and the visualization of the scatterer around z is bad. The occurrence of ramification points is supported not only by numerics but also by theory, since assuming their existence allows proving that gz ɛ L 2 [,2π] must blow up for z / D, and therefore provides a coherent theoretical framework whereby the numerical simulations can be interpreted y/λ x/λ Figure 9. Flow lines for a sampling point z (represented by a red bullet) placed outside the scatterer. Two ramification points (square boxes) are detectable.

22 y/λ x/λ Figure. Flow lines for a sampling point z (represented by a red bullet) placed outside the scatterer. Two ramification points (square boxes) are detectable. To this aim we introduce a definition describing the behavior of the flow lines starting from a ramification point and reaching the far-field region. Of course, this definition is inspired by the radial behavior of the flow lines of the Green function with respect to the point source and generalizes some of its relevant features by using the concepts of regularity and asymptotic angular width introduced in Definitions 3.2 and 3.3. Definition 6.. A radiating solution u of the Helmholtz equation in IR 2 \ D is said to be partially pseudo-radial with respect to a ramification point z D if there exist at least two regular flow lines ζz (τ) and ζz 2 (τ) starting from z such that their asymptotic polar angles ϕ (z ) and ϕ 2 (z ) are different, i.e. ϕ (z ) ϕ 2 (z ), and the flow strip delimited by ζz (τ) and ζz 2 (τ) is regular. If { ζz i (τ) } denotes the set of all i I such flow lines and {ϕ i (z )} i I is the set of their asymptotic polar angles, the quantity ψ (z ) := sup i,j I ϕ i (z ) ϕ j (z ) > is called the asymptotic angular width of the beam of flow lines outgoing from z.

23 y/λ x/λ Figure. Flow lines for a sampling point z (represented by a red bullet) placed outside the scatterer. Theorem 6.2. Under the same hypotheses of Theorem 2., consider a point z IR 2 \D. If ɛ is small enough then there cannot exist g ɛ z L 2 [, 2π] such that F g ɛ z Φ (, z) L 2 [,2π] ɛ (6.) and the field u s,ɛ z is partially pseudo-radial with respect to some point z D. Proof. Let z / D. Assume by contradiction that there exist ɛ > and gz ɛ L 2 [, 2π] such that inequality (6.) holds and the field u s,ɛ z is partially pseudo-radial with respect to a ramification point z D. Let ψ (z ) > be the asymptotic angular width of the beam of flow lines outgoing from z. According to Definition 6., we can find two regular flow lines ζz (τ), ζz 2 (τ) of asymptotic polar angles ϕ (z ), ϕ 2 (z ) respectively, such that the corresponding regular flow strip has asymptotic angular width ψ (z ) := ϕ (z ) ϕ 2 (z ) >, which can be made arbitrarily close to (but not greater than) ψ (z ).

24 24 Consider a family of circles C n (z ) of center z and radius r n, with lim r n =, and n let yn := C n (z ) { ζz (τ) }, τ [,+ ) y2 n := C n (z ) { ζz 2 (τ) }. Finally, denote by τ [,+ ) C n (z ) be the arc with end points yn and yn, 2 resulting from the intersection between the circle C n (z ) and the flow strip identified by the flow lines ζz (τ) and ζz 2 (τ). Following the procedure of the previous theorem, we evaluate the flux F Cn (z ) (us,ɛ z ) as n in two ways: a) close to the point z, across C n (z ): F Cn (z ) (us,ɛ z ) = 4iω ε B µ B C n (z ) [ ū s,ɛ z u s,ɛ z ν us,ɛ z ū s,ɛ ] z (x) dl(x). (6.2) ν Then, by applying the mean-value theorem for integration, we have: F Cn (z ) (us,ɛ z ) 2ω ε B µ B C n(z ) ūs,ɛ z (x) us,ɛ z ν (x) dl(x) C n (z ) = ū s,ɛ u s,ɛ z z ( x n ) 2ω ε B µ B ν ( x n), (6.3) where x n is an appropriate point of C n (z ). Since n lim Cn (z ) = and n lim x n z =, by the continuity of u s,ɛ z lim n ūs,ɛ z and us,ɛ z ν we have n ( x n ) = ū s,ɛ z (z ), lim ν ( x n) = u s,ɛ z ν (z ). (6.4) u s,ɛ z As a result, from (6.3) and (6.4), we find lim F Cn (z ) (us,ɛ = ; (6.5) n z ) b) in the far-field region. Following the proof of Theorem 5., we consider the regular flow strip bounded by the flow lines ζz (τ) and ζz 2 (τ) of u s,ɛ z, outgoing from z, and track the power flux outgoing from C n (z ) up to the far-field region. As in the proof of (5.4), we obtain, for each n IN: [ F Cn (z ) (us,ɛ z ) = ū s,ɛ u s,ɛ z z 4iω ε B µ B C n (z ) ν ū s,ɛ ] us,ɛ z z (x) dl(x) (6.6) ν [ ϕ[p 2 z (R)] = lim ū s,ɛ u s,ɛ z 4iω ε B µ B R ϕ[pz z (R)] r ū s,ɛ ] us,ɛ z z (R, ϕ) Rdϕ r k = F g 2ω ε B µ z ɛ 2 L 2 [ϕ (z ),ϕ 2 (z )] B k [ Φ (, z) 2 L 2ω ε B µ 2 [ϕ (z ),ϕ 2 (z )] ɛ (ϕ (z ), ϕ 2 (z )) ]. B Since ψ (z ) = ϕ (z ) ϕ 2 (z ) >, we have Φ (, z) 2 L 2 [ϕ (z ),ϕ 2 (z )] > ; this

25 25 leads to a contradiction between (6.5) and (6.6) for ɛ such that < ɛ < ( ψ 2 ) Φ (, z) L 2 [ϕ,ϕ2 ] = ( (z ) 2 ) 8πk. (6.7) This concludes the proof. Remark 6.3. Analogously to Remark 5.2, we note that an explicit computation of (6.7) is only possible by knowing the asymptotic angular width ψ (z ). In our framework, this knowledge can only be obtained a posteriori, by looking in each case at the flow lines provided by the numerical simulations, as we shall do in the next section, in order to compare bounds (5.6) and (6.7) with the values of the discrepancy. However, as observed in Section 4, for sampling points z on D the flow lines still have a radial behavior: then, bound (5.7) is even an underestimate of the maximum ɛ allowed by (6.7) for such points, at least in those cases (like in Figures 3 and 6) where the flow lines can reach the far-field region without intersecting the scatterer. Remark 6.4. The proof of Theorem 6.2 is obtained by considering a single ramification point z, but it is easily adapted to account for two or more such points. For example, if we assume the existence of two ramification points z and z, we can consider two flow strips with asymptotic angular widths ψ (z ) = ϕ (z ) ϕ 2 (z ) and ψ (z ) = ϕ (z ) ϕ 2 (z ). Then, our argument still holds by considering two vanishing sequences {r n } n=, {r n} n= and two corresponding families of collapsing circles C n (z ), C n(z ). Accordingly, the integral on C n (z ) appearing in (6.2), (6.3) and (6.6) should be replaced by an integral on C n (z ) C n(z ), i.e. by the sum of two integrals, each of which behaves as shown in the proof. As a result, in bound (6.7) the norm of Φ (, z) in L 2 [ϕ (z ), ϕ 2 (z )] must be replaced by the norm of Φ (, z) in L ( 2 [ϕ (z ), ϕ 2 (z )] [ϕ (z ), ϕ 2 (z )] ). Accordingly, in the same (6.7), ψ (z ) should be replaced by the total asymptotic angular width, defined as ψ T := ψ (z ) + ψ (z ). So far we have only dealt with the full-view configuration of probes. Then, we conclude this section by sketching the extension of our approach to the aspect-limited case. Let Γ i, Γ o [, 2π] be the sets of incidence and observation angles θ, ϕ respectively (for brevity, we consider Γ o independent of θ). An ɛ-approximate solution of the modified far-field equation [8] is a function gz ɛ L 2 (Γ i ) such that u (, θ)g ɛ z(θ)dθ Φ (, z) ɛ. (6.8) Γ i L 2 (Γ o ) In any case, the corresponding field u s,ɛ z (x) := Γ i u s (x, θ)g ɛ z(θ)dθ and its flow lines are defined for all x IR 2 \D; moreover, its far-field pattern, defined for ϕ [, 2π], satisfies condition (6.8) at least for ϕ Γ o, and even in a larger set Γ o Γ o if the left-hand side

26 26 of (6.8) is strictly smaller than ɛ. Now, if Γ o = [, 2π], the situation is analogous to the full-view case; otherwise, Theorems 5. and 6.2 ensure that g ɛ z L 2 (Γ i ) blows up for each z approaching z D (or placed in IR 2 \ D), such that the flow lines starting from a neighborhood of z (or from a ramification point z ) reach, under the usual assumptions, the far-field region inside Γ o with a non-vanishing asymptotic angular width. Again, this is in qualitative agreement with numerical simulations, which show [, 2] that, when the emitters and receivers are placed in the same region, the scatterer is, in general, best reconstructed in its illuminated part. 7. Tikhonov regularization and numerical validation Theorems 5. and 6.2 deal with generic ɛ-approximate solutions gz ɛ of the far-field equation. However, it is of interest to focus on Tikhonov regularized solutions, from both the theoretical and the numerical viewpoint, since they play a major role in the implementation of the linear sampling method. First of all, we observe that Theorem 5. does not state that gz ɛ L 2 [,2π] is bounded for z D: rather, the existence of gz ɛ L 2 [, 2π], ensured by the denseness of the range of F, is a starting point of our argument. However, when Tikhonov regularized solutions g α (z) of the far-field equation are considered, the boundedness of g α (z) L 2 [,2π] for z D simply follows from the fact that α (z) >. What Theorem 5. can establish is that, in spite of regularization, appropriate conditions on the flow lines as well as bound (5.6), which reads F g α (z) Φ (, z) L 2 [,2π] < ( ) 2 4 if z D (7.) k in consequence of (5.7), force the norm g α (z) L 2 [,2π] to blow up when z tends to a point z D, i.e. α (z) must vanish as z z. In other terms, inequality (7.) suggests a criterion for choosing the regularization parameter α (z) for z D in such a way that g α (z) behaves as one of the approximate solutions of the far-field equation satisfying the condition described by limit (2.)(a) of Theorem 2.. By a similar argument, Theorem 6.2 shows that, if z / D, the partial pseudoradiality of u s,ɛ z with asymptotic angular width ψ (z ) and bound (6.7), i.e. F g α (z) Φ (, z) L 2 [,2π] < ( ψ (z ) 2 ) if z / D, (7.2) 8πk cannot be simultaneously verified for positive α (z). Accordingly, taking α (z) makes g α (z) L 2 [,2π] blow up, which corresponds to limit (2.3)(a) of Theorem 2.. Up to discretization and noise issues not explicitly addressed here, this behavior of α (z) is in qualitative agreement with numerical simulations, which show, as well known [6], that the values of α (z) are generally much smaller for z outside than for z inside the scatterer.

27 Moreover, although discretization and noise prevent an exact correspondence between numerical simulations and theoretical results, it is anyway interesting to compare, as a check of internal consistency for our framework, the theoretical bounds on ɛ, as given by (5.6) (i.e. (5.7)) and (6.7), with the values of the discretized discrepancy d(z) (plotted in Figures (b) and 4(b)) at the sampling points z considered in the simulations of Sections 4 and 6. This comparison is non-trivial since, on the one hand, the physical interpretation formalized by Theorems 5. and 6.2 can be applied only if d(z) ɛ and, on the other hand, the values of d(z) are fixed by using the generalized discrepancy principle, while the bounds on ɛ are estimated in a completely different way, i.e. on the basis of the (total) asymptotic angular width of the flow lines. For both the scattering experiments considered in Sections 4 and 6, the wavenumber k is the same, i.e. k = 2π = 2.9 m. As a consequence, bound (5.7) for λ z z D reads ɛ < Figures (b) and 4(b), which plot the values of d(z) for each z in the investigation domain, clearly show that, for both numerical experiments, d(z) < for any z inside D and even on D. However, while in the first experiment also the assumptions on the flow lines required by Theorem 5. are verified (see Figure 3), in the second one this only happens for a sampling point placed as in Figure 6, but not as in Figure 7: coherently with our approach, in the latter case the sampling point is placed in a region where the visualization of the scatterer is bad. Let us now turn to the case of sampling points external to the scatterer. For the single ellipse, the behavior of the flow lines is shown in Figure 9. If we refer the square investigation domain T to the usual polar coordinates (r, ϕ), from Figure 9 we can see that the asymptotic angular width ψ (z ) of the beam of flow lines outgoing from the left-upper ramification point z is approximately given by ϕ (z ) ϕ 2 (z ) π, since ϕ (z ) and ϕ 2 (z ) π. For the right-lower ramification point z, the asymptotic angular width ψ (z ) can be estimated from ϕ (z ) 5π/4 and ϕ 2 (z ) 2π, i.e. ψ (z ) 3π/4. As a result, the total asymptotic angular width (see Remark 6.4) is ψ T 7π/4 and then bound (6.7) becomes ɛ < This bound is approximately fulfilled by the value of the discrepancy d(z) = at the sampling point z chosen for Figure 9. In the experiment with two ellipses, only the visualization of Figure is worth discussing for the case of an external sampling point z: indeed, in Figure no ramification point appears and the visualization around z is bad. For the upper ramification point z of Figure, we can estimate ϕ (z ) π/4 and ϕ 2 (z ) π, and then ψ (z ) 3π/4; for the right-lower ramification point z, Figure suggests the values ϕ (z ) 3π/2 and ϕ 2 (z ) 2π, i.e. ψ (z ) π/2. Accordingly, the total asymptotic angular width is ψ T 5π/4 and then bound (6.7) reads ɛ < 3.6 2, which is fulfilled by the value of the discrepancy d(z) = at the sampling point of Figure. 27