An inverse source problem for linearly anisotropic radiative sources in absorbing and scattering medium

We consider in a two dimensional absorbing and scattering medium, an inverse source problem in the stationary radiative transport, where the source is linearly anisotropic. The medium has an anisotropic scattering property that is neither negligible nor large enough for the diffusion approximation to hold. The attenuating and scattering properties of the medium are assumed known. For scattering kernels of finite Fourier content in the angular variable, we show how to recover the anisotropic radiative sources from boundary measurements. The approach is based on the Cauchy problem for a Beltrami-like equation associated with $A$-analytic maps. As an application, we determine necessary and sufficient conditions for the data coming from two different sources to be mistaken for each other.


INTRODUCTION
In this work, we consider an inverse source problem for stationary radiative transfer (transport) [6,7], in a two-dimensional bounded, strictly convex domain Ω Ă R 2 , with boundary Γ.The stationary radiative transport models the linear transport of particles through a medium and includes absorption and scattering phenomena.In the steady state case, when generated solely by a linearly anisotropic source f pz, θq " f 0 pzq `θ ¨Fpzq inside Ω, the density upz, θq of particles at z traveling in the direction θ solves the stationary radiative transport boundary value problem (1) θ ¨∇upz, θq `apz, θqupz, θq ´żS 1 kpz, θ, θ 1 qupz, θ 1 qdθ 1 " f pz, θq, pz, θq P Ω ˆS1 , u| Γ ´" 0, where the function apz, θq is the medium capability of absorption per unit path-length at z moving in the direction θ called the attenuation coefficient, function kpz, θ, θ 1 q is the scattering coefficient which accounts for particles from an arbitrary direction θ 1 which scatter in the direction θ at a point z, and Γ ´:" tpζ, θq P Γ ˆS1 : νpζq ¨θ ă 0u is the incoming unit tangent sub-bundle of the boundary, with νpζq being the outer unit normal at ζ P Γ .The attenuation and scattering coefficients are assumed known real valued functions.The boundary condition in (1) indicates that no radiation is coming from outside the domain.Throughout, the measure dθ on the unit sphere S 1 is normalized to ş S 1 dθ " 1.
Under various assumptions, e.g., [9,8,2,17,4], the (forward) boundary value problem ( 1) is known to have a unique solution, with a general result in [27] showing that, for an open and dense set of coefficients a P C 2 pΩ ˆS1 q and k P C 2 pΩ ˆS1 ˆS1 q, the boundary value problem (1) has a unique solution u P L 2 pΩ ˆS1 q for any f P L 2 pΩ ˆS1 q.In [14], it is shown that for attenuation merely once differentiable, a P C 1 pΩ ˆS1 q and k P C 2 pΩ ˆS1 ˆS1 q, the boundary value problem (1) has a unique solution u P L p pΩ ˆS1 q for any f P L p pΩ ˆS1 q, p ą 1.Moreover, uniqueness result of the forward problem (1) are also establish in weighted L p spaces in [11].In our reconstruction method here, some of our arguments require solutions u P C 2,µ pΩ ˆS1 q, 1 2 ă µ ă 1.We revisit the arguments in [27,14] and show that such a regularity can be achieved for sources f P W 3,p pΩ ˆS1 q, p ą 4; see Theorem 2.2 (iii) below.
For a given medium, i.e., a and k both known, we consider the inverse problem of determining the scalar field f 0 , and the vector field F from measurements g f0,F of exiting radiation on Γ, where Γ `:" tpz, θq P ΓˆS 1 : νpzq¨θ ą 0u is the outgoing unit tangent sub-bundle of the boundary, with νpζq being the outer unit normal at ζ P Γ .
For anisotropic sources the problem has non-uniqueness [25,28].One of our main result, Theorem 4.1 shows that from boundary measurement data g f0,F , one can only recover the part of the linear anisotropic source f " f 0 `θ ¨F; in particular, only the solenoidal part F s of the vector field F is recovered inside the domain.However, in Theorem 4.2, if one know apriori that the source F is divergence-free, then from the data g f0,F , one can recover both isotropic field f 0 and the vector field F inside the domain.Moreover, instead of apriori information of the divergence-free source F, if one has the additional data g f0,0 information along with the data g f0,F , then in Theorem 4.3, one can recover both sources f 0 and F under subcritical assumption of the medium.One of the main crux in our reconstruction method is the observation that any finite Fourier content in the angular variable of the scattering kernel splits the problem into an infinite system of non-scattering case and a boundary value problem for a finite elliptic system.The role of the finite Fourier content has been independently recognized in [13] and [18].
The inverse source problem above has applications in medical imaging: In a non-scattering (k " 0) and non-attenuating (a " 0) medium the problem is mathematically equivalent to the one occurring in classical computerized X-ray tomography (e.g., [5,20]).In the absorbing non-scattering medium, such a problem (with only isotropic source f " f 0 ), appears in Positron/Single Photon Emission Tomography (PET/SPECT) [20,21], and (with f 0 " 0 and f " θ ¨F), appears in Doppler Tomography [21,20,26].For applications in scattering media the inverse source problem formulated here is the two dimensional version of the corresponding three dimensional problem occurring in imaging techniques such as Bioluminescence tomography and Optical Molecular Imaging, see [29,15,16] and references therein.
In Section 2, we remark on the existence and regularity of the forward boundary value problem.The results in Section 2 consider both attenuation coefficient and scattering kernel in general setting.
In this work, except for the results in Section 2, the attenuation coefficient are assumed isotropic a " apzq, and that the scattering kernel kpz, θ, θ 1 q " kpz, θ ¨θ1 q depends polynomially on the angle between the directions.Moreover, the functions a, k and the source f are assumed real valued.
In Section 3, we recall some basic properties of A-analytic theory, and in Section 4 we provide the reconstruction method for the full (part) of the linearly anisotropic source.Our approach is based on the Cauchy problem for a Beltrami-like equation associated with A-analytic maps in the sense of Bukhgeim [5].The A-analytic approach developed in [5] treats the non-attenuating case, and the absorbing but non-scattering case is treated in [3].The original idea of Bukhgeim from the absorbing non-scattering media [5,3] to the absorbing and scattering media has been extended in [13,14].In here we extend the results in [13,14] to linear anisotropic sources.
In Section 5, the method used will explain when the data coming from two different linear anisotropic field sources can be mistaken for each other.

SOME REMARKS ON THE EXISTENCE AND REGULARITY OF THE FORWARD PROBLEM
In this section, we revisit the arguments in [27,14], and remark on the well posedness in L p pΩˆS 1 q of the boundary value problem (1).Adopting the notation in [27,14], we consider the operators rT ´1 1 gspx, θq " ż 0 ´8 e ´ş0 s apx`tθ,θqdt gpx `sθ, θqds, and rKgspx, θq " where the intervening functions are extended by 0 outside Ω.
Using the above operators, the problem (1) can be rewritten as If the operator I ´T ´1 1 K is invertible, then the problem (3) is uniquely solvable, and has the form u " pI ´T ´1 1 Kq ´1T ´1 1 f .By using the formal expansion (4) 1 f, the well posed-ness in L p pΩˆS 1 q of the (forward) boundary value problem (1) is reduced to showing that the operator I ´T We recall some results in [14].
Proposition 2.1.[14, Proposition 2.1] Let a P C 1 pΩ ˆS1 q and k P C 2 pΩ ˆS1 ˆS1 q.Then the operator The following simple result is useful.Lemma 2.1.[14, Lemma 2.2] Let X be a Banach space and A : X Ñ X be bounded.Then I ˘A have bounded inverses in X, if and only if I ´A2 has a bounded inverse in X.
For λ P C, we note that pT ´1 1 pλKqq 2 " λ 2 T ´1 1 pKT ´1 1 Kq.By Proposition 2.1, the operator pT ´1 1 pλKqq 2 is compact for any λ P C. By Lemma 2.1, if the operator I ´pT ´1 1 pλKqq 2 is invertible in L p pΩ ˆS1 q, then the operator I ´T ´1 1 pλKq is invertible in L p pΩ ˆS1 q.Since I ´pT ´1 1 pλKqq 2 is invertible for λ in a neighborhood of 0, an application of the analytic Fredholm alternative in Banach spaces, e.g., [10,Theorem VII.4.5], yields the following result.
Theorem 2.1.[14, Theorem 2.1] Let p ą 1, a P C 1 pΩ ˆS1 q, and k P C 2 pΩ ˆS1 ˆS1 q.At least one of the following statements is true.
(i) I ´T The regularity of the solution u of (1) increases with the regularity of f as follows.
Theorem 2.2.Consider the boundary value problem (1) with a P C 3 pΩ ˆS1 q.For p ą 1, let k P C 3 pΩ ˆS1 ˆS1 q be such that I ´T ´1 1 K is invertible in L p pΩ ˆS1 q, and let u P L p pΩ ˆS1 q in (4) be the solution of (1).
(i) If f P W 1,p pΩ ˆS1 q, then u P W 1,p pΩ ˆS1 q.
Proof.(i) We consider the regularity of the solution u of (1) term by term as in (4): It is easy to see that the operator T ´1 1 preserve the space W 1,p pΩ ˆS1 q, and also the operator K preserve the space W 1,p pΩ ˆS1 q, so that the first two terms, T ´1 1 f and T ´1 1 KT ´1 1 f , both belong to W 1,p pΩ ˆS1 q.Moreover, pI ´T ´1 1 Kq ´1T ´1 1 f P L p pΩ ˆS1 q, and now, by using Proposition 2.1, the last term is also belong in W 1,p pΩ ˆS1 q.
It is easy to see that T ´1 0 , r T ´1 0,j , K ξ j and K η i ξ j preserve W 1,p pΩ ˆS1 q.By evaluating the radiative transport equation in (1) at x `tθ and integrating in t from ´8 to 0, the boundary value problem (1) with zero incoming fluxes is equivalent to the integral equation: According to part (i), for f P W 1,p pΩˆS 1 q, the solution u P W 1,p pΩˆS 1 q, and so u x j P L p pΩˆS 1 q.In particular u x j solves the integral equation: Moreover, since a P C 2 pΩ ˆS1 q, k P C 2 pΩ ˆS1 ˆS1 q, and f P W 2,p pΩ ˆS1 q, the right-hand-side of (8) lies in W 1,p pΩ ˆS1 q.By applying part (i) above, we get that the unique solution to ( 8) For f P W 1,p pΩ ˆS1 q, also according to part (i), u θ j P L p pΩ ˆS1 q.In particular u θ j is the unique solution of the integral equation which is of the type (7) with K " 0.Moreover, since f P W 2,p pΩ ˆS1 q, and, according to (9), u x j P W 1,p pΩ ˆS1 q, j " 1, 2, the right-hand-side of (10) lies in W 1,p pΩ ˆS1 q.Again, by applying part (i), we get u θ j P W 1,p pΩ ˆS1 q, j " 1, 2.
(iii) For f P W 2,p pΩ ˆS1 q, according to part (ii), u x j , u θ j P W 1,p pΩ ˆS1 q, and u x i x j P L p pΩ ˆS1 q.In particular u x i x j is the unique solution of the integral equation ( 11) Moreover, since a P C 3 pΩ ˆS1 q, k P C 3 pΩ ˆS1 ˆS1 q, and f P W 3,p pΩ ˆS1 q, the right-hand-side of (11) lies in W 1,p pΩ ˆS1 q.By applying part (i) above, we get that the unique solution to (11) For f P W 2,p pΩ ˆS1 q, also according to part (ii), u x j , u θ j P W 1,p pΩ ˆS1 q, and u θ i θ j P L p pΩ ˆS1 q.In particular u θ i θ j is the unique solution of the integral equation which is of the type (7) with K " 0.
Moreover, since f P W 3,p pΩ ˆS1 q, and, according to (12), u x i x j P W 1,p pΩ ˆS1 q, j " 1, 2, the right-hand-side of ( 13) lies in W 1,p pΩ ˆS1 q.Again, by applying part (i), we get For f P W 2,p pΩ ˆS1 q, also according to part (ii), u x i u θ j P L p pΩ ˆS1 q.In particular u x i θ j is the unique solution of the integral equation which is of the type (7).Moreover, since a P C 3 pΩˆS 1 q, k P C 3 pΩˆS 1 ˆS1 q, and f P W 3,p pΩˆS 1 q, the right-hand-side of (15) lies in W 1,p pΩ ˆS1 q.Again, by applying part (i), we get From ( 12), (14), and ( 16), we get u P W 3,p pΩ ˆS1 q.
We remark that for Theorem 2.2 part (i) we only need a P C 1 pΩ ˆS1 q and k P C 2 pΩ ˆS1 ˆS1 q, and we only require a P C 2 pΩ ˆS1 q and k P C 2 pΩ ˆS1 ˆS1 q for Theorem 2.2 part (ii).We also refer to [14, Theorem 2.2] for part (i) and (ii) of Theorem 2.2.Moreover, in a similar fashion, one can show that under sufficiently increased regularity of a and k, the solution u of (1) belong to u P W m,p pΩ ˆS1 q for Z Q m ě 1, provided f P W m,p pΩ ˆS1 q.

INGREDIENTS FROM A-ANALYTIC THEORY
In this section we briefly introduce the properties of A-analytic maps needed later, and introduce notation.We recall some of the existing results and concepts used in our reconstruction method.
Bukhgeim's original theory [5] shows that solutions of ( 19), satisfy a Cauchy-like integral formula, where B is the Bukhgeim-Cauchy operator acting on v| Γ .We use the formula in [12], where B is defined component-wise for n ě 0 by ( 21) The theorems below comprise some results in [22,23].For the proof of the theorem below we refer to [23,Proposition 2.3].Theorem 3.1.Let 0 ă µ ă 1, and let B be the Bukhgeim-Cauchy operator in (21).
The following result recalls the necessary and sufficient conditions for a sufficiently regular map to be the boundary value of an L 2 -analytic function.
(i) If g is the boundary value of an L 2 -analytic function, then Hg P C µ pΓ ; l 1 q and satisfies pI `ıHqg " 0.
(ii) If g satisfies (24), then there exists an L 2 -analytic function v :" Bg P C 1,µ pΩ; l 1 q X C µ pΩ; l 1 q X C 2 pΩ; l 8 q, such that v| Γ " g. (25) For the proof of Theorem 3.2 we refer to [ Another ingredient, in addition to L 2 -analytic maps, consists in the one-to-one relation between solutions u :" xu 0 , u ´1, u ´2, ...y satisfying Bu `L2 Bu `aLu " 0, (26) and the L 2 -analytic map v " xv 0 , v ´1, v ´2, ...y satisfying (19), via a special function h, see [24, Lemma 4.2] for details.The function h is defined as where θ K is the counter-clockwise rotation of θ by π{2, Raps, θ K q " ż 8 ´8 a `sθ K `tθ ˘dt is the Radon transform in R 2 of the attenuation a, Dapz, θq " ż 8 0 apz `tθqdt is the divergent beam transform of the attenuation a, and Hhpsq " 1 π ´8 hptq s ´tdt is the classical Hilbert transform [19], taken in the first variable and evaluated at s " z¨θ K .The function h appeared first in [20] and enjoys the crucial property of having vanishing negative Fourier modes yielding the expansions e ´hpz,θq :" Using the Fourier coefficients of e ˘h, define the operators e ˘Gu component-wise for each n ď 0, by pe ´Guq n " pα ˚uq n " We refer [24,Lemma 4.1] for the properties of h, and we restate the following result [22, Proposition 5.2] to incorporate the operators e ˘G notation used in here.8 pΩq, and the operators maps piq e ˘G : C µ pΩ; l 8 q Ñ C µ pΩ; l 8 q; piiq e ˘G : C µ pΩ; l 1 q Ñ C µ pΩ; l 1 q; piiiq e ˘G : Y µ pΓ q Ñ Y µ pΓ q.Lemma 3.1.[23, Lemma 4.2] Let a P C 1,µ pΩq, µ ą 1{2, and e ˘G be operators as defined in (29).
Moreover, for f s 1 " pF s 1 `ıF s 2 q {2, the Hodge decomposition (38) can be rewritten as (40) f 1 " Bϕ `f s 1 , ϕ| BΩ " 0, RepBf s 1 q " 0. Theorem 4.1.Let Ω Ă R 2 be a strictly convex bounded domain, and Γ be its boundary.Consider the boundary value problem (30) for some known real valued a, k 0 , k ´1, ..., k ´M P C 3 pΩq such that (30) is well-posed.If scalar and vector field sources f 0 and F are real valued, W 3,p pΩ; Rq and W 3,p pΩ; R 2 q-regular, respectively, with p ą 4, then u| Γ `uniquely determine the solenoidal part F s in Ω and u ´u0 in Ω, where u 0 is the zeroth Fourier mode of u in the angular variable.
We note from (37) that the shifted sequence valued map L M u " xu ´M , u ´M ´1, u ´M ´2, ...y solves BL M upzq `L2 BL M upzq `apzqL M `1upzq " 0, z P Ω. (41) Let v :" e ´GL M u, then by Lemma 3.1, and the fact that the operators e ˘G commute with the left translation, re ˘G, Ls " 0, the sequence v " L M e ´Gu solves Bv `L2 Bv " 0, i.e v is L 2 analytic.
By (2), the data u| Γ `" g determines L M u on Γ.By Proposition 29 (iii), and the convolution formula, traces L M u| Γ determines the traces v P Y µ pΓ q on Γ.
Since v| Γ P Y µ pΓ q is the boundary value of an L 2 -analytic function in Ω, then Theorem 3.
where H is the Bukhgeim-Hilbert transform in (23).
From v on Γ , we use the Bukhgeim-Cauchy Integral formula (21) to construct the sequence valued map v inside Ω.By Theorem 3.1 and Theorem 3.2 (ii), the constructed sequence valued v P C 1,µ pΩ; l 1 q X C µ pΩ; l 1 q X C 2 pΩ; l 8 q is L 2 -analytic in Ω.
We use again the convolution formula L M u " e G v, and determine modes u ´n now inside Ω, for n ě M. In particular, we recover modes u ´M ´1, u ´M P C 2 pΩq.
Recall that the modes u ´1, u ´2, ¨¨¨, u ´M , u ´M ´1 satisfy Bu ´M `j " ´Bu ´M `j´2 ´rpa ´k´M`j´1 qu ´M `j´1 s , 1 ď j ď M ´1, (43a) u ´M `j| Γ " g ´M `j .(43b) By applying 4B to (43a), the mode u ´M `1 (for j " 1) is then the solution to the Dirichlet problem for the Poisson equation where the right hand side of (44) is known.
Since u 0 , u ´1, u ´2 P C 2 pΩq, we can take 4B on both sides of the equation ( 35) to get Moreover, since u 0 is real valued and div F " ∆ϕ, by equating the real part in (46) we get the boundary value problem: where the right hand side of (47) is known.Thus, real valued pu 0 ´ϕq is recovered in Ω, by solving the Dirichlet problem for the above Poisson equation (47).
From (35) and using f s 1 " f 1 ´Bϕ from (40), we get f s 1 :" Bpu 0 pzq ´ϕpzqq `Bu ´2pzq `rapzq ´k´1 pzqsu ´1pzq, z P Ω, (48) with f s 1 satisfying RepBf s 1 q " 0. Thus, the solenoidal part F s " x2 Re f s 1 , 2 Im f s 1 y, of the vector field F is recovered in Ω.If we know apriori that the vector field F is incompressible (i.e divergenceless), then we can reconstruct both scalar field source f 0 and vector field source F in Ω.
Theorem 4.2.Let Ω Ă R 2 be a strictly convex bounded domain, and Γ be its boundary.Consider the boundary value problem (30) for some known real valued a, k 0 , k ´1, ..., k ´M P C 3 pΩq such that (30) is well-posed.If the unknown scalar field source f 0 and divergenceless vector field sources F are real valued, W 3,p pΩ; Rq and W 3,p pΩ; R 2 q-regular, respectively, with p ą 4, then the data g f0,F defined in (2) uniquely determine both f 0 and F in Ω.
Proof.Let u be the solution of the boundary value problem (30) and let u " xu 0 , u ´1, u ´2, ...y be the sequence valued map of its non-positive Fourier modes, Since the isotropic scalar and vector field f 0 P W 3,p pΩ; Rq, and F P W 3,p pΩ; R 2 q respectively for p ą 4, then the anistropic source f " f 0 `θ ¨F P W 3,p pΩ ˆS1 q and by applying Theorem 2.2 (iii), we have u P W 3,p pΩ ˆS1 q.By the Sobolev embedding [1], W 3,p pΩ ˆS1 q Ă C 2,µ pΩ ˆS1 q with µ " 1 ´2 p ą 1 2 , we have u P C 2,µ pΩ ˆS1 q, and thus, by [22, Proposition 4.1 (ii)], u P Y µ pΓ q.
By Theorem 4.1, the data u| Γ `" g f0,F uniquely determine the solenoidal field F s " F in Ω by equation (48) with ϕ " 0, and the sequence valued map Lu " xu ´1, u ´2, ...y in Ω.Moreover, the real valued mode u 0 is also then recovered (with ϕ " 0 ) in Ω, by solving the Dirichlet problem for the Poisson equation (47).
Thus, from modes u ´1 and u 0 , the scalar field f 0 is recovered in Ω by f 0 :" 2 RerBu ´1s `ra ´k0 su 0 .(49) In the radiative transport literature, the attenuation coefficient a " σ a `σs , where σ a represents pure loss due to absorption and σ s pzq " 1 2π ş 2π 0 kpz, θqdθ " k 0 pzq is the isotropic part of scattering kernel.We consider the subcritical region: σ a :" a ´k0 ě δ ą 0, for some positive constant δ. (50) Remark 4.1.In addition to the hypothesis to Theorem 4.1, if we assume that coefficients a, k 0 satisfies (50), then in the region tz P Ω : f 0 pzq " 0u, one can recover explicitly the entire vector field F " x2 Re f 1 , 2 Im f 1 y.Indeed, the equation ( 34) gives u 0 " ´2 RepBu ´1q{σ a and, following (35), the vector field F can be recovered by the formula Next, we show that one can also determine both scalar field f 0 and vector field F, if one has the additional data g f0,0 (or g 0,F ) information, instead of F being incompressible as in Theorem 4.2.
Theorem 4.3.Let Ω Ă R 2 be a strictly convex bounded domain, and Γ be its boundary.Consider the boundary value problem (30) for some known real valued a, k 0 , k ´1, ..., k ´M P C 3 pΩq such that (30) is well-posed.If the unknown scalar field source f 0 and vector field source F are real valued, W 3,p pΩ; Rq and W 3,p pΩ; R 2 q-regular, respectively, with p ą 4, and coefficients a, k 0 satisfying (50), then the data g f0,F and g f0,0 defined in (2) uniquely determine both f 0 and F in Ω.
We consider first the boundary value problem (52), and will reconstruct the scalar field f 0 from the given boundary data g f0,0 . If v n pzqe ınθ is the Fourier series expansion in the angular variable θ of a solution v of boundary value problem (52), then, by identifying the Fourier modes of the same order, the equation in (52) reduces to the system: Bv ´1pzq `Bv ´1pzq `rapzq ´k0 pzqsv 0 pzq " f 0 pzq, (54) Bv ´npzq `Bv ´n´2 pzq `rapzq ´k´n´1 pzqsv ´n´1 pzq " 0, 0 ď n ď M ´1, (55) Bv ´npzq `Bv ´n´2 pzq `apzqv ´n´1 pzq " 0, n ě M, (56) and let v " xv 0 , v ´1, v ´2, ...y be the sequence valued map of its non-positive Fourier modes.
By Theorem 4.1, from data g f0,0 , the sequence Lv " xv ´1, v ´2, ...y is determined in Ω.Moreover, as (55) holds also for n " 0 (f 1 " 0 in this case), the mode v 0 is also determined in Ω by solving the Dirichlet problem for the Poisson equation where the right hand side of (57) is known.

WHEN CAN THE DATA COMING FROM TWO SOURCES BE MISTAKEN FOR EACH OTHER ?
In this section we will address when the data coming from two different linear anisotropic field sources can be mistaken.
In the theorem below the data are assuming the same attenuation a and scattering coefficient k.
Theorem 5.1.(i) Let a P C 3 pΩq, k P C 3 pΩ ˆS1 q be real valued, with σ a " a ´k0 ą 0, and f 0 , r f P W 3,p pΩq, p ą 4 be real valued with pf 0 ´r f q{σ a P C 0 pΩq.Then F :" r F `∇ ˜f0 ´r f σ a ¸is a real valued vector field such that the data g f 0 ,F coming from the linear anisotropic source f 0 `θ ¨F, is the same as data g r f , F coming from a different linear anisotropic source r f `θ ¨r F : (ii) Let a, k 0 , k ´1, ..., k ´M P C 3 pΩq be real valued with σ a " a ´k0 ą 0. Assume that there are real valued linear anisotropic sources f 0 `θ ¨F and r f `θ ¨r F, with isotropic fields f 0 , r f P W 3,p pΩq, p ą 4, and vector fields F, r F P W 3,p pΩ; R 2 q, p ą 4. If the data g f 0 ,F of the linear anisotropic source F is the data of some real valued anisotropic source r f `θ ¨r F, i.e., it is the trace on Γ ˆS1 of solutions w to the stationary transport boundary value problem: (65) θ ¨∇w `aw ´Kw " r f `θ ¨r F, where the operator rKwspz, θq :" kpz, θ ¨θ1 qwpz, θ 1 qdθ 1 , for z P Ω and θ P S 1 .
For σ a " a ´k0 with σ a ą 0, and isotropic real valued functions ψ and σ a , we note: (66) kpz, θ ¨θ1 qdθ where the second equality uses the linearty of K and (66), the last equality uses (65), and the definition of F.Moreover, since f 0 ´r f {σ a vanishes on Γ , we get (ii) Let f 0 `θ ¨F, be the real valued linear anisotropic source with isotropic field f 0 P W 3,p pΩq, p ą 4, and vector field F P W 3,p pΩ; R 2 q, p ą 4. If the data g f 0 ,F of the linear anisotropic source f 0 `θ ¨F equals data g r f , r F of some real valued r f , r F P W 3,p pΩq, p ą 4, i.e. g r f , r F " g " g f 0 ,F .Then by Theorem 2.2 (iii), there exist u, w P W 3,p pΩ ˆS1 q solutions to the corresponding transport equations θ ¨∇u `au ´Ku " f 0 `θ ¨F, and θ ¨∇w `aw ´Kw " r f `θ ¨r F (67) respectively, subject to u| Γ ˆS1" g " w| Γ ˆS1.
In particular, L M v and L M ρ are L 2 -analytic, and coincide at the boundary Γ.By uniqueness of L 2 -analytic functions with a given trace, they coincide inside: L M vpzq " L M ρpzq, for z P Ω.For the mode ψ ´M `1 (when j " 1), the right hand side of (83a) contains modes ψ ´M ´1 and ψ ´M which are both zero by (82).Thus, the mode ψ ´M `1 " 0 is the unique solution to the Cauchy problem for the B-equation, BΨ " 0, in Ω, (84a) Ψ " 0, on Γ. (84b) We then solve repeatedly (83) starting for j " 2, ..., M ´1, where the right hand side of (83a) in each step is zero, yielding the Cauchy problem (84) for each subsequent modes, and thus, resulting in the recovering of the modes ψ ´M `1 " ψ ´M `2 " ¨¨¨ψ ´2 " ψ ´1 " 0 in Ω.Therefore, establishing (80).
Moreover, by subtracting (73) from (69) and using (77) and (80), yields 2Bpu 0 ´w0 q " pF 1 ´Ă F 1 q ìpF 2 ´Ă F 2 q.Since both u 0 and w 0 are real valued we see that F ´r F " ∇pu 0 ´w0 q, and we have Remark 5.1.Note that in Theorem 5.1(i), the assumption on scattering kernels of finite Fourier content in the angular variable is not assumed, and the result holds for a general scattering kernels which depends polynomially on the angle between the directions.