Cramer-Rao Bound Study of Multiple Scattering

Effects in Target Separation Estimation

Edwin A. MarengoDepartment of Electrical and Computer Engineering

Northeastern University, Boston, MA 02115email:emarengo@ece.neu.edu

Paul BeresteskyDepartment of Electrical and Computer Engineering

Northeastern University, Boston, MA 02115email:berestesky.p@husky.neu.edu

January 16, 2013

Abstract

The information about the distance of separation between two pointtargets that is contained in scattering data is explored in the context ofthe scalar Helmholtz operator via the Fisher information and associatedCramer-Rao bound (CRB) relevant to unbiased target separation esti-mation. The CRB results are obtained for the exact multiple scatteringmodel and, for reference, also for the single scattering or Born approx-imation model applicable to weak scatterers. The effects of the sensingconfiguration and the scattering parameters in target separation estima-tion are analyzed. Conditions under which the targets’ separation can-not be estimated are discussed for both models. Conditions for multiplescattering to be useful or detrimental to target separation estimation arediscussed and illustrated.

1 Introduction

An important question in imaging and inverse scattering is the quantificationof theoretical limits in the information that can be extracted about parame-ters of a wave scatterer from given scattered field data. This question, withparticular interest in the quantification of limits related to target localizabilityand resolution, has been tackled in a number of papers (see [1, 2, 3] and thereferences therein) via the statistical signal processing framework of the Fisherinformation and the associated Cramer-Rao bound (CRB) [4]. This approachquantifies the best precision with which scattering parameters can be estimated

1

in the statistical framework of unbiased estimation under given signal corrup-tion or noise models. Importantly, this quantification is algorithm-independentand showcases the role of both scattering parameters and imaging or sensingconfiguration. Furthermore, this theoretical approach holds under nonnegligiblemultiple scattering conditions where the mapping from object function to datais nonlinear, which prevents the direct application of the standard diffractionlimits (λ/2 rule of thumb) of inverse scattering problems under the Born approx-imation as well as inverse source problems where the respective map is linear andtherefore tractable via bandlimitation considerations in spatial Fourier domain.

The present paper expands our research on CRB analysis of multiple scatter-ing effects in the estimation of target parameters initiated in a previous paper[1] co-authored by the current authors. In that paper we characterized an-alytically and computationally the roles of the sensing configuration and thescattering parameters in the task of localizing two point targets. Concrete con-ditions were derived under which localization is facilitated or obstructed. Theseparate roles of the sensing configuration and the scattering parameters wereisolated and interpreted. The results were obtained within the exact scatteringmodel including multiple scattering and, for reference purposes, also under theapproximate first Born approximation scattering model so as to obtain insighton the role of multiple scattering in either facilitating or obstructing target lo-calization relative to the baseline provided by the Born approximation. To easemathematical tractability and insight, in that paper the focus was the informa-tion about the targets’ positions under the assumption that the target scatteringstrengths and the separation of the two targets are known. In the present paperwe address the complementary question about the information on the targets’separation if the position of one of the targets is known or if the position of thecenter of the two targets is known, which is related to the question of how wellthe two targets can be resolved from the scattering data.

The original contributions of the present paper can be summarized as fol-lows. First, we provide the closed-form Fisher information and CRB expressionsapplicable to estimation problems relevant to target resolution and exploit theimplications of the resulting developments with the aid of computer illustra-tions. Second, the present work expands the current understanding of the roleof multiple scattering in imaging resolution by considering two different physicalsituations: one where prior knowledge of the position of one of the scatterersis available and another where prior knowledge of the center of the compositetwo-scatterer target is available. We derive for these two different scenarios thenecessary and sufficient conditions under which the estimation of the targets’separation is impeded within the exact multiple scattering model. In addition,we derive analytically and illustrate computationally the conditions under whichmultiple scattering outperforms the Born approximation predictions and viceversa where the Born approximation predictions are unrealistically optimistic.As explained in [1], the estimation performance depends on both conditions in-trinsic to the data, in particular the sensing configuration, as well as conditionsthat depend on the target, in the present case the scattering parameters of tar-get strengths and positions. The adoption of the canonical system of two point

2

scatterers allows us to gain insight into the role of both sensing configurationand scattering parameters and, in particular, the mathematical expressions forFisher information and CRB derived in the paper demonstrate factors that de-pend only on configuration or on parameters as well as more complex factorsthat depend on both.

The remainder of the paper is organized as follows. Section 2 reviews theforward scattering model. Section 3 presents the CRB results for target sep-aration estimation. Section 4 provides numerical illustrations of the derivedtheory. Section 5 provides concluding remarks. Appendix A summarizes thebasic Fisher information and CRB derivation.

2 Review of the Multiple Scattering Model

Following [1], we consider scattering within the framework of the scalar Helmholtzoperator in three-dimensional free space,∇2+k2, where∇2 = ∂2/∂x2+∂2/∂y2+∂2/∂z2 is the Laplacian operator and k = 2π/λ is the wavenumber of the fieldcorresponding to wavelength λ. For analytical tractability and insight, atten-tion is restricted to a system of two point scatterers having complex scatteringstrengths τ1 and τ2 and positions R1 = (0, 0, d1) and R2 = (0, 0, d2) in the zaxis (see Figure 1 of [1]). We assume d2 > d1 so that the targets’ separationd = d2 − d1 > 0. Consider incident plane waves eiksi·r traveling in the directionof the unit vector si corresponding to incidence polar angle α (cosα = si · z)and far-zone sensing in the direction s corresponding to scattering polar angle β(cosβ = s · z). These (polar) angles lie in the range [0, π]. For this two-scatterersystem under plane wave excitation, the scattering amplitude including multiplescattering is given by [1]

f (α, β) =F (d, τ1, τ2)eikd1g(α,β)[τ1+

τ2eikdg(α,β) + τ1τ2G(d)Q(d, α, β)] (1)

where

G(d) = −eikd

4πd, (2)

F (d, τ1, τ2) =[1− τ1τ2G

2(d)]−1

, (3)

g(α, β) = cosα− cosβ, (4)

and

Q(d, α, β) = eikd cosα + e−ikd cos β

= 2eikd2 g(α,β) cos

(kd

2g′(α, β)

)(5)

whereg′(α, β) = cosα+ cosβ. (6)

3

Also, it is assumed that τ1τ2G2(d) �= 1 (nonresonance condition). For the special

case of weak scatterers where |τmG(d)| << 1,m = 1, 2, this takes the first Bornapproximation form f(α, β) � fBorn(α, β) where

fBorn (α, β) = eikd1g(α,β)[τ1 + τ2e

ikdg(α,β)]. (7)

3 Target Separation Estimation

This section describes the statistical information that is contained in scatteringdata about the separation distance d between two known point scatterers. Inimaging systems, the ability to estimate the separation distance d of two pointtargets is usually regarded as a metric for the resolution or ability of the imagingsystem to resolve target details. The smaller the separation distance d of twopoint targets that can be properly estimated, the higher the resolution. Thequestion is how the resolution is affected by the target parameters and the re-mote sensing configuration, particularly in the case when multiple scattering issignificant. In the following, we study this question in the exact multiple scat-tering framework. For reference and to facilitate interpretation of the results,we also derive the respective Born approximation results valid for weak scatter-ers. We derive and discuss the necessary and sufficient condition under whichit is theoretically impossible to resolve the targets (“no-resolution condition”).This, of course, implies as a corollary the contrary “resolution condition” underwhich the targets can be resolved. We also comparatively examine the predic-tions of the exact and Born approximation models, paying particular attentionto contrasting the degrees of freedom associated to the no-resolution conditionsof the two models, and investigate the conditions where multiple scattering isbeneficial or detrimental to resolution relative to the Born approximation modellimits. In the following analysis, we consider two different scenarios associatedto two different physical and informational situations. One is the case wherethe position of one of the targets, say target 1, is known (d1 is known). Theother case is where the position of the center of the two targets dc = (d1+d2)/2is known. The results for the two cases are different but have the same generalmathematical structure, and therefore lead to similar general conclusions aboutthe role of the scattering and configuration parameters in the task of resolvingthe targets and the differences between the exact and approximate models. Asummary of the basic Fisher information and CRB results needed to carry outthe computations of this section is given in Appendix A.

3.1 Calculation for Known-d1 Case

The following calculation assumes prior knowledge of the first scatterer’s posi-tion, d1, as well as of the scattering strengths, τ1 and τ2. Adopting the Bornapproximation model (Eq. (30)), the Fisher information I(n)(d) of the nth scat-tering experiment corresponding to given pairs (αn, βn) of incidence and scat-tering angles αn ∈ [0, π] and βn ∈ [0, π], respectively, is found from (34) with

4

ξ = d to be given by

I(n)Born(d) = 2σ−2k2|τ2|2g2(αn, βn). (8)

On the other hand, in the exact multiple scattering model we obtain fromEqs. (29,34)

I(n)(d) = 2σ−2|τ2|2|F (d, τ1, τ2)|2|C(αn, βn)|2 (9)

where

C(αn, βn) =ikg(αn, βn)eikdg(αn,βn) + τ1G(d)

{∂

∂dQ(d, αn, βn)+(

ik − 1

d

)[Q(d, αn, βn) + 2G(d)e−ikd1g(αn,βn)Kn(d)

]}. (10)

As in the localization problem considered in [1], where I(n)(d1) ∝ g2(αn, βn) and

I(n)Born(d1) ∝ g2(αnβn), the Fisher information I

(n)Born(d) in the Born approxima-

tion model (8) is also proportional to g2(αn, βn), which implies that under thecondition g(αn, βn) = 0 (which is equivalent to the line-of-sight (LOS) condition,αn = βn) no information is contained in the data about the target separationd. However, this is not the case under the multiple scattering model, where,according to Eqs. (9,10), I(n)(d) does not necessarily vanish if g(αn, βn) = 0 orαn = βn, i.e., the respective LOS data may carry information about d. Thisissue is illustrated and discussed further in the following.

The two special cases of LOS data (αn = βn) and of backscattering data(αn = π−βn) allow further simplifications which facilitate visualization. Underthe LOS condition, the Fisher information is given by (9) where C(αn, αn)defined by (10) takes the particular form

C(αn, αn) =τ1G(d)

{− 2k cosαn sin(kd cosαn)+(

ik − 1

d

)[2 cos(kd cosαn)+

2G(d)

1− τ1τ2G2(d)

(τ1 + τ2+

2τ1τ2G(d) cos(kd cosαn))]}

. (11)

In this LOS condition, the Born approximation model applicable to weak targetsdoes not permit the estimation of the targets’ separation. This is a very specificsituation where categorically only non-weak scatterers exhibiting non-negligiblemultiple scattering can reveal the targets’ separation information. On the otherhand, for very specialized combinations of parameters, C(αn, αn) in (11) van-ishes so that the respective LOS Fisher information I(n)(d) vanishes as well.For example, if we let d > 0 be arbitrary while αn is chosen such that kd cosαn

5

takes one of the discrete values ±(2p+ 1)π2 , p = 0, 1, 2, . . . , pmax = �kdπ − 1/2�,

then the choice of scattering strengths τ1 = τ = τ2 where

τ = 4π(−1)pe−ikd

⎡⎣(id− 1

k

)secαn ∓

√(id− 1

k

)2

sec2 αn + 1

⎤⎦ (12)

gives I(n)(d) = 0.For backscattering data, the Fisher information about d is given by (9) with

βn = π − αn where the corresponding C(αn, π − αn) is given by

C(αn, π − αn) =2ik cosαne2ikd cosαn+

2τ1G(d)

{ik cosαne

ikd cosαn+(ik − 1

d

)[eikd cosαn +

G(d)

1− τ1τ2G2(d)

(τ1+

τ2e2ikd cosαn + 2τ1τ2G(d)eikd cosαn

)]}. (13)

Note that, according to (8), within the Born approximation for weak scatterers,the Fisher information is proportional to g2(αn, π − αn) = 2 cosαn so that itvanishes for αn = π/2. This situation contrasts with the exact model result(13) which gives

C(π2,π

2

)= 2τ1G(d)

(ik − 1

d

){1 +

G(d) [τ1 + τ2 + 2τ1τ2G(d)]

1− τ1τ2G2(d)

}(14)

which vanishes only for specialized values of the parameters d, τ1 and τ2.

3.2 Calculation for Known-dc Case

Alternatively, we may consider another scenario where the center of the twotargets, dc = (d1 + d2)/2, is known. Physically and informationally, this is adifferent situation from the one of the preceding results and, therefore, givesdifferent Fisher information and CRB results. In this case, the d-dependence ofthe data is conveniently highlighted by rewriting Eqs. (29) and (30) as

Kn(d) =F (d, τ1, τ2)eik(dc− d

2 )g(αn,βn)[τ1+

τ2eikdg(αn,βn) + τ1τ2G(d)Q(d, αn, βn)

](15)

and

KBornn (d) =eikdcg(αn,βn)

[τ1e

−i kd2 g(αn,βn) + τ2e

i kd2 g(αn,βn)

](16)

=eik(dc− d2 )g(αn,βn)

[τ1 + τ2e

ikdg(αn,βn)].

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The corresponding Fisher informations, to be denoted as I′(n)Born(d) and I ′(n)(d),

respectively, are found from Eqs. (15,16) and Eq. (34) to be given by

I′(n)Born(d) =

k2

2g2(αn, βn)DNRBorn(αn, βn) (17)

where

DNRBorn(α, β) =1

σ2

∣∣∣τ1 − τ2eikdg(α,β)

∣∣∣2 (18)

and

I ′(n)(d) =2

σ2|F (d, αn, βn)|2 |C′(αn, βn)|2 (19)

where

C′(αn, βn) =− i

2kg(αn, βn)

[τ1 − τ2e

ikdg(αn,βn)]+

τ1τ2G(d)

[(ik − 1

d

)Q(d, αn, βn)+

i

2kg′(αn, βn)Q

′(d, αn, βn)+

2G(d)

(ik − 1

d

)e−ik(dc−d

2 )g(αn,βn)Kn(d)

](20)

and

Q′(d, α, β) = eikd cosα − e−ikd cosβ

= 2ieikd2 g(α,β) sin

(kd

2g′(α, β)

). (21)

The behavior of I′(n)Born(d) in Eq. (17) applicable to the known-dc case is similar

to that in the counterpart result (8) for known d1 in that the Fisher informa-tion is proportional to the quantity g2(αn, βn). Again, the Fisher informationvanishes for g(αn, βn) = 0, i.e., αn = βn, so that under this LOS condition dcannot be estimated. However, instead of being proportional to |τ2|2 as in (8),the Fisher information in (17) is proportional to DNRBorn which is a functionof both scattering strengths and the incidence and sensing angles. Previously,because d1 was known, an estimation of d was effectively an estimation of d2and, therefore, the problem was essentially of locating the second scatterer.Hence, a stronger second scatterer strength leads directly to better resolutioninformation. In contrast, in the known-dc case, because it is the center pointthat is known, an estimation of d is effectively the estimation of both d1 and d2where it is known that the scatterers are equidistant from the known position dcon the z-axis. Thus, within the Born approximation, the two estimation prob-lems (known d1 and known dc) have a fundamental difference in that, while forknown d1 the no-resolution condition is the LOS condition, αn = βn, for knowndc, it is the LOS condition αn = βn or DNRBorn(αn, βn) = 0. Thus, unlike in

7

the known-d1 case, in the known-dc case, zero resolution information is possi-ble for certain values of the scattering parameters at non-LOS (NLOS) anglesαn �= βn. On the other hand, the exact multiple scattering expression (19)resembles its known-d1 counterpart (9). In this case, the angular dependenceis more complex and g(αn, βn) = 0 does not imply zero information about d.In the following, we elaborate the no-resolution conditions for both the exactand approximate models (whose disobeyance defines, of course, the contrary“resolution conditions” under which it is theoretically possible to resolve thetargets).

3.3 No-Resolution Condition: Known-d1 Case

3.3.1 Born Approximation

As explained in the preceding paragraph, it follows from (8) that, under theBorn approximation, the no-resolution condition for the known-d1 case is simplythe LOS requirement, αn = βn. Note that this condition depends only on thesensing angles and hence applies to any value of the scattering parameters d,τ1 and τ2. In particular, this no-resolution condition defines a plane whichhas 6 degrees of freedom in the 7-dimensional parameter space associated to(d, τ1, τ2, αn, βn).

3.3.2 Exact Model

Note that F (d, τ1, τ2) �= 0 for finite parameters d, τ1 and τ2. It follows fromEqs. (9,10) that I(n)(d) = 0 if and only if C(αn, βn) = 0, in particular,

ikg(αn, βn)eikdg(αn,βn) + τ1G(d)

{∂

∂dQ(d, αn, βn)+(

ik − 1

d

)[Q(d, αn, βn) + 2G(d)e−ikd1g(αn,βn)Kn(d)

]}= 0. (22)

By using expression (29) for Kn(d) in (22) and manipulating the resulting equa-tion for (d, τ1, τ2, αn, βn), it is not hard to show that this equation cannot beobeyed for arbitrary values of the scattering strengths τ1 and τ2 in clear contrastwith the corresponding Born approximation no-resolution condition (the LOScondition) which holds for any τ1 and τ2. The Born approximation conditioncorresponds to 6 degrees of freedom which is dimensionally less restrictive thanthe exact multiple scattering condition, suggesting that as the targets scattermore it also becomes less likely for the scattering and configuration parametersto be such that zero information is available about the targets’ separation d.In particular, it follows from (29) that if the values of d, αn, βn and τ1 (orτ2) are arbitrarily fixed, condition (22) reduces to a quadratic equation in τ2(or τ1) which gives either two different or one (double-root) solution for τ2 (orτ1). The number of degrees of freedom of this no-resolution condition is 5 in the7-dimensional parameter space of (d, τ1, τ2, αn, βn). On the other hand, while inthe Born approximation the angles for no-resolution are restricted by αn = βn,

8

under exact multiple scattering, the angles for no-resolution can take any valueas long as the other parameters are such that condition (22) holds.

3.4 No-Resolution Condition: Known-dc Case

3.4.1 Born Approximation

As outlined above, it follows from (17) that, for the Born approximation, theno-resolution condition for the known-dc case is

αn = βn, or DNRBorn(αn, βn) = 0. (23)

Thus, there are two alternative ways of having zero resolution information. Oneis the LOS condition, which involves 6 degrees of freedom in the 7-dimensionalparameter space. Another is the zero DNRBorn condition, which can be putexplicitly as

|τ1| = |τ2|kdg(αn, βn) = kd(cosαn − cosβn)

= θ1 − θ2 + lπ, l = ±2p, p = 0, 1, 2, . . . , pmax (24)

where pmax is constrained by

4p2max ±4

πpmax(θ1 − θ2) +

1

π2(θ1 − θ2)

2 ≤ 16d2

λ2. (25)

The condition involves 5 degrees of freedom in 7-dimensional space. This can bereadily visualized by noting that the first condition in (24) defines two planes(τ1 = ±τ2) in the parameter space, reducing the dimensionality from 7 to 6,while the second condition defines for any (θ1, θ2, αn, βn) a countably infiniteset of values of d. In particular, the values d±p > 0, p = 0, 1, 2, . . . ,∞ obeying

d±p =

[(θ1 − θ2)

2π± p

]λ

cosαn − cosβn, p = 0, 1, 2, . . . ,∞, (26)

implying the loss of another dimension for a total of 5 degrees of freedom. TheDNRBorn = 0 alternative has less degrees of freedom than the LOS condition.However, it offers greater flexibility to the no-resolution angles which are notrequired to be the same under this other condition.

3.4.2 Exact Model

It follows from Eqs. (19,20) that, within the exact model, the no-resolutioncondition is C′(αn, βn) = 0, i.e.,

− i

2kg(α, β)

[τ1 − τ2e

ikdg(α,β)]+

τ1τ2G(d)

[(ik − 1

d

)Q(d, α, β) +

i

2kg′(α, β)Q′(d, α, β)+

2G(d)

(ik − 1

d

)e−ik(dc− d

2 )g(α,β)Kn(d)

]= 0. (27)

9

This result has the same general form as the known-d1 counterpart (22) andyields, through the methodology outlined above, the same general conclusions.For instance, it is a constraint of 5 degrees of freedom. The comparative dis-cussion of the exact and approximate models given for the known-d1 case stillapplies. However, there is a minor difference which is that, in the known-dccase, the Born approximation model allows situations where estimation of thetargets’ separation is impossible at arbitrary sensing angles unlike in the known-d1 case where the LOS condition is the only way of impeding estimation of thisdistance.

4 Numerical Illustrations

4.1 Single Observation

Next we discuss a selection of single observation experiments which illustratehow variations in the system’s parameters (scatterer separation d, scattererstrengths τ1 and τ2) and observer configuration (incident and observation an-gles αn and βn) affect the estimation of the targets’ separation. In generatingthe following plots, we use σ2 = 1 so that the plotted CRB results are normal-ized by the noise variance σ2. In some of the plots, the CRB of the targets’separation d, CRB(d), is additionally normalized by d2 (equivalently

√CRB(d)

is normalized by σd). This highlights the estimation error relative to d whichfacilitates interpretation. Also, we consider unit value wavelength λ = 1 so thatthe wavenumber k = 2π/λ = 2π and all distances (e.g., d) can be given in theplots in terms of the wavelength.

Figure 1 shows plots of CRB(d) for τ1 = 1 = τ2 as a function of the obser-vation angle β while the incidence angle is held constant at α = 0. Both models(exact and approximate), as well as the two formulation cases (known-d1 andknown-dc cases), are plotted together for comparison. Two values of the sepa-ration d are considered: d = λ/4 in Figure 1(a) and d = λ/2 in Figure 1(b). Inboth the known-d1 case and the known-dc case, the Fisher information underthe Born approximation is proportional to g2 (see (8,17)) so that (as shown inthe plots) for β = 0, CRB(d) = ∞, since then g = 0. Also, in the known-dccase, the Fisher information under the Born approximation is proportional toDNRBorn (see (17)) which vanishes for (d = λ/2, β = π) so that CRB(d) = ∞in that case, as is shown in the respective plot in Figure 1(b). On the otherhand, the exact CRB is finite in all these cases: the information about thetargets’ separation contained in the data is thus greater than the Born approx-imation value. As a general trend, the exact CRB is seen to decrease with theobservation angle, reaching its lowest values for the near-backscattering angles,except for the known-dc case and d = λ/2 where the forward scattering andbackscattering values are comparable. This can be understood intuitively fromthe corresponding Born approximation result where the CRB goes to infinity forboth β = 0 and β = π. In addition to lacking blind spots, the exact CRB variesonly smoothly with limited regions of raised bound or peaks such as the ones

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10−2

100

102

104

106

β (π)

CR

B(d

)/(d

σ)2

d1 known, MSd1 known, Borndc known, MSdc known, Born

(a) d = λ/4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10−2

100

102

104

106

β (π)

CR

B(d

)/(d

σ)2

d1 known, MSd1 known, Borndc known, MSdc known, Born

(b) d = λ/2

Figure 1: CRB(d) as the observation angle β varies, normalized by (dσ)2, fortarget separations d = λ/4 and d = λ/2. The incidence angle is α = 0 and thescatterer strengths are τ1 = 1 = τ2.

11

for the known-d1 formulation near (d = λ/2, β = 0.2π) and for the known-dcformulation near (d = λ/4, β = 0.45π).

Figures 2 and 3 show plots of CRB(d) for backscattering experiments asa function of (d, α) where d ∈ [0.1λ, 10λ] and α ∈ [0, π]. The results are foran equal scatterer system having τ1 = 1 = τ2. Figure 2 shows the results forthe known-d1 case while Figure 3 shows the results for the known-dc case. Thedark areas highlight regions where the CRB is high. These do not explicitlycorrespond to blind conditions but the black regions do follow loci of infiniteCRB conditions in general. The simplest plot is the Born approximation known-d1 case result shown in Figure 2(b) which shows a single high CRB ridge alongα = π/2 where g = 0 so that, according to (8), the CRB is infinite. The Bornapproximation known-dc case plot shown in Figure 3(b) contains the same ridgewith the addition of a widening region of increasing bound in the vicinity ofα = π/2 for small separations d as well as a number of blind zones per givenvalue of d which increases as d grows larger. All of this is in agreement withthe analysis presented in (17) and (24,25). The exact CRB results shown inthese figures exhibit similar general trends with differences between the exactand approximate results becoming more prominent for smaller d. This is alsoas expected since this is where the multiple scattering becomes stronger. Theseeffects appear to be favorable toward facilitating estimation of d as the blindconditions predicted by the Born approximation break up into sparser individualpeaks or disappear entirely as d decreases.

Figures 4 and 5 illustrate further the results in Figures 2 and 3, showing therelation between the Fisher information for the multiple scattering and Bornapproximation models. The most noticeable difference between the known-d1and known-dc cases is the shape of the alternating beneficial and harmful in-fluence regions of the multiple scattering. In the known-d1 case, the patternresembles stripes rather than a checker-board as in the known-dc case. As inthe localization case studied in [1], the alternating pattern ends below somevalue of d (d � 0.05λ for known-d1 and d � 0.08λ for known-dc). Notably, thebehavior in this small d region is opposite from the localization case in thathere CRB(d) of the multiple scatter is consistently higher. This result is alsoconsistent with the multiple observation results discussed later and illustrated inFigure 7(a) where the error bounds of the multiple scattering and Born approx-imation models averaged over all observation and incidence angles converge forhigher values of d but, for lower values of d, the CRB of the multiple scatteringmodel is consistently lower.

We consider next the LOS or forward scattering configuration, α = β, forτ1 = 1 = τ2. In this configuration, the exact expressions for CRB(d) in terms ofd1 and dc become equal. On the other hand, the Born approximation predictsCRB(d) = ∞ due to the vanishing of g for this configuration. Figure 6 showsa plot of the CRB for the multiple scattering model. As expected, the CRBincreases as d grows because the multiple scattering becomes less pronouncedwith greater scatter separation and the results approach the Born approximationprediction. High bound ridges appear for separations above one wavelength andincrease in number as d increases above this size. This is as expected from the

12

α (π)

d(λ

)

0 1/4 1/2 3/4 110

−1

100

101

−20dB

−15dB

−10dB

−5dB

0dB

5dB

10dB

(a) Multiple Scattering Model

α (π)

d(λ

)

0 1/4 1/2 3/4 110

−1

100

101

−20dB

−15dB

−10dB

−5dB

0dB

5dB

10dB

(b) Born Approximation Model

Figure 2: Plots of CRB(d) as a function of incidence angle α and scattererseparation d, normalized by σ2, for the case where d1 is known. The observationangle β = π − α and τ1 = 1 = τ2.

13

α (π)

d(λ

)

0 1/4 1/2 3/4 110

−1

100

101

−20dB

−15dB

−10dB

−5dB

0dB

5dB

10dB

(a) Multiple Scattering Model

α (π)

d(λ

)

0 1/4 1/2 3/4 110

−1

100

101

−20dB

−15dB

−10dB

−5dB

0dB

5dB

10dB

(b) Born Approximation Model

Figure 3: Plots of CRB(d) as a function of incidence angle α and scattererseparation d, normalized by σ2, for the case where dc is known. The observationangle β = π − α and τ1 = 1 = τ2.

14

α (π)

d(λ

)

0 1/4 1/2 3/4 1

0.1

0.25

0.5

1

10

IBorn(d) = 0 I(d) ≤ IBorn(d)

Figure 4: Map of the regions where multiple scattering effects do not aid esti-mation of target separation in relation to blind conditions of the Born approx-imation model. Black lines mark conditions where IBorn(d) = 0. Gray areasmark regions where I(d) ≤ IBorn(d). The same parameters are used here as inFigure 2 (τ1 = τ2 = 1, β = π − α, d1 is known).

α (π)

d(λ

)

0 1/4 1/2 3/4 1

0.1

0.25

0.5

1

10

I′Born(d) = 0 I′(d) ≤ I′Born(d)

Figure 5: Map of the regions where multiple scattering effects do not aid esti-mation of target separation in relation to blind conditions of the Born approx-imation model. Black lines mark conditions where I ′Born(d) = 0. Gray areasmark regions where I ′(d) ≤ I ′Born(d). The same parameters are used here as inFigure 3 (τ1 = τ2 = 1, β = π − α, dc is known).

15

α (π)

d(λ

)

0 1/4 1/2 3/4 110

−1

100

101

−20dB

−15dB

−10dB

−5dB

0dB

5dB

10dB

15dB

20dB

25dB

30dB

Figure 6: Plot of CRB(d) as a function of incidence angle α and scattererseparation d, normalized by σ2, for the multiple scattering model. The resultsapply for τ1 = 1 = τ2 in the LOS or forward scattering configuration, α = β.

discussion in (24,25).

4.2 Multiple Observations

We consider next the case of multiple observations data. The correspondingcalculations provide values of the CRB averaged over the sensing configurationparameters which effectively reduces the parameter dimensionality and allowsus to decipher general patterns that are not obvious from the single observationanalysis and associated computer results.

Figure 7 shows the behavior of the average CRB(d) normalized by (dσ)2. Itis calculated from the average Fisher information corresponding to 104 scatteringexperiments involving 100 uniformly-spaced values of incidence and scatteringangles in the full interval [0, π]. Two scatterer strength cases are shown: (a)identical scatterers, τ1 = 1 = τ2, and (b) unequal scatterers, τ1 = 1, τ2 = 1/4. Itis worth noting that the exact formulations in terms of d1 and dc approach eachother below d = λ/10. In addition, the relation between the exact and Bornapproximation models is similar to the one in the localization context consideredin [1] in that the two models match reasonably above d = λ/2 but divergebelow this separation. However, the small d behavior is the opposite of whatwe obtained for the localization problem in [1]. Thus, unlike in the localizationproblem, where the Born approximation is unrealistically optmistic for smalld � λ/10, in the target separation problem, the Born approximation CRB issignificantly higher than the exact CRB for d � λ/10 so that the approximatepredictions are in reality quite pessimistic. It is important to note that, inclear contrast with the target localization problem where the strong multiplescattering present for small d has a destructive effect in SNR which in turnreduces target localizability for small d as we discussed earlier, in the related

16

10−2

10−1

100

101

10−4

10−2

100

102

104

d (λ)

CR

B(d

)/(d

σ)2

dc known, MSdc known, Bornd1 known, MSd1 known, Born

(a) τ1 = 1, τ2 = 1

10−2

10−1

100

101

10−4

10−2

100

102

104

d (λ)

CR

B(d

)/(d

σ)2

dc known, MSdc known, Bornd1 known, MSd1 known, Born

(b) τ1 = 1, τ2 = 1/4

Figure 7: CRB(d) as the scatterer separation d varies, normalized by (dσ)2. TheFisher information on which the bound is based is averaged over all combinationsof 100 samples each of incidence angles αn ∈ [0, π] and observation angles βn ∈[0, π] taken at regular intervals. Two variations are shown: (a) equal (τ1 = 1 =τ2) and (b) unequal (τ1 = 1, τ2 = 1/4) scatterer strengths.

17

problem of estimating the target separation, the strong multiple scattering forsmall d plays a beneficial role since the information about d does not rely onSNR. In particular, in the targets’ separation problem, d is the sought-afterparameter and estimation is based on the rate at which the data signal changesdue to variations in d which is significant for small values of d due to multiplescattering. Therefore, the associated information that can be extracted aboutd is also significant for small values of d, as is shown in these plots.

5 Conclusions

This paper has expanded the research on CRB for scattering parameter esti-mation initiated in the previous paper [1], co-authored by the current authors,where we studied, via the CRB approach, the effects of multiple scattering inthe localization of a two-target scattering system. In the present paper, we haveaddressed the pending question of the effects of multiple scattering in the esti-mation of the separation distance between two targets which is related to theresolvability of the two targets from scattering data. As in [1], we have assumedthat the target strengths are known which simplifies the formulation and theinterpretation of the derived findings. Two possible situations were considered:one where the position of one of the scatterers is known, which essentially mod-els localization of the other target in a (non-free-space) medium including theother target which acts as a multipathing agent, and another where the center ofthe two targets system is known but the targets’ separation is unknown, whichessentially simulates estimation of the size of the total two-target scatterer. Thepresented results complete the discussion initiated in [1], clarifying further therole of multiple scattering in the localization and resolution of targets.

Concrete examples of conditions under which multiple scattering aids or im-pedes target separation estimation relative to the baseline of the Born approxi-mation have been derived and illustrated. We have derived for both the exactand approximate scattering models the conditions under which the targets’ sep-aration cannot be estimated and concluded that these conditions are generallymore restrictive under the Born approximation than for the exact model. Thus,in general, multiple scattering enhances target resolvability. The provided nu-merical results of CRB(d) corresponding to multiple observations show that,for d � λ/2, the Born approximation gives results which are, on average, verysimilar to those of the exact model. On the other hand, when considering singleobservations, specific cases exist where one model or the other exhibits a largerCRB. The multiple observation results for the two models diverge for smallervalues of d (d � λ/2). The boundary depends on whether the center of the twotargets or the position of one of the targets is known. However, in contrast tothe localization problem of the previous paper [1], the error bound of the mul-tiple scattering model is consistently smaller than the error bound of the Bornapproximation model and, for very small target separations, multiple scatteringsignificantly enhances the estimability of the two targets’ separation.

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A CRB Derivation

Following [1], we express the signal model as

K(ξ) = K(ξ) +W (28)

where K is the noise-free data vector, K is the collected noisy data vector, ξ isthe parameter vector to be estimated and W is complex Gaussian noise withknown variance σ2. In this paper, ξ = d, assuming prior knowledge of thestrengths and a reference position such as the position of the first target, d1, orthe position of the center of the two targets (dc = (d1 + d2)/2).

The data entries Kn, n = 1, 2, . . . , N , of the N × 1 data vector K are thevalues of the scattering amplitudes f(αn, βn) measurable in scattering experi-ments corresponding to given pairs (αn, βn) of incidence and scattering anglesαn ∈ [0, π] and βn ∈ [0, π], respectively. From (1), the entries Kn are given by

Kn(ξ) =F (d, τ1, τ2)eikd1g(αn,βn)[τ1+

τ2eikdg(αn,βn) + τ1τ2G(d)Q(d, αn, βn)]. (29)

The corresponding Born approximation is given from (7) by Kn(ξ) � KBornn (ξ)

whereKBorn

n (ξ) = eikd1g(αn,βn)[τ1+τ2e

ikdg(αn,βn)]. (30)

The CRB of the parameter ξi, CRB(ξi), constitutes a lower bound, achiev-

able under mild conditions, for the variance var(ξi) = E[(ξi − ξi)2] (where E

denotes the expected value) of any unbiased estimate ξi of the parameter ξi. Itis given by the diagonal elements of the Fisher information matrix (FIM) [4,Eq. (3.20)]. In particular,

var(ξi) ≥ [I−1(ξ)]i,i = CRB(ξi) (31)

where the FIM, I(ξ), is given by [4, Eq. (15.52)]

I(ξ)i,j = 2 [∂KH (ξ)

∂ξiC−1

˜K(ξ)

∂K (ξ)

∂ξj

](32)

where (·)H denotes the conjugate transpose and C˜K is the covariance matrix

which, in our case, is C˜K = σ2I. I denotes the N×N identity matrix. Therefore,

(32) reduces to

I(ξ)i,j =

N∑n=1

I(n)i,j (ξ) (33)

where the entry I(n)i,j of the FIM of the nth scattering experiment, I(n)(ξ), is

given by

I(n)i,j (ξ) = 2σ−2

[∂K ∗

n(ξ)

∂ξi

∂Kn(ξ)

∂ξj

]. (34)

19

Acknowledgments

This research was supported by the Air Force Office of Scientific Research undergrant FA9550-12-1-0285.

References

[1] E.A. Marengo, M. Zambrano-Nunez, and P. Berestesky, “Cramer-Raobound study of multiple scattering effects in target localization”, Interna-tional Journal of Antennas and Propagation, Vol. 2012, Article ID 390312,13 pages, 2012.

[2] A. Sentenac, C.A. Guerin, P.C. Chaumet, F. Drsek, H. Giovannini, N.Bertaux, and M. Holschneider, “Influence of multiple scattering on theresolution of an imaging system: A Cramer-Rao analysis”, Opt. Express,Vol. 15, pp. 1340-1347, 2007.

[3] F. Simonetti, M. Fleming, and E.A. Marengo, “An illustration of the role ofmultiple scattering in subwavelength imaging from far-field measurements”,J. Opt. Soc. Am. A, Vol. 25, pp. 292-303, 2008.

[4] S. Kay, Fundamentals of Signal Processing: Estimation Theory, WoodCliffs, NJ: Prentice Hall, 1993.

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