On two parameter identification problems arising from a special form of computerized tomography (Q1340791)

Let \(\Omega = \{P(x,z) \in \mathbb{R}^2 : (x,z) \in [0,L] \times (0,Z]\}\), \(\Gamma = \{P(x,0) \in \mathbb{R}^2 : x \in [0,L]\}\). The author considers the operator problem \[ (A(\sigma_s, \sigma_T) I_E) (P) = I_D(P),\;P\in \Omega \tag{1} \] where the operator \(A : L_\infty \to L_2 (\Omega, z^2)\) is defined by \[ \begin{multlined} (A(\sigma_s, \sigma_T) I_E) (P) = I_E(P) {\sigma_s (P) \over \text{dist} (P,D) 4\pi \sin (2\gamma)} \text{exp} \left( -\int_{K(P)} \sigma_T (P') ds\right),\\ \sigma_s \in L_2(\Omega),\quad \sigma_T \in W^1_2 (\Omega).\end{multlined} \] This problem describes mathematically a special problem in computerized tomography using backscatted photons: the photons are irradiated under a fixed angle \(\gamma\) with a fixed intensity \(I_E\) at an emiter point \(E\) on the surface \(\Gamma\). After scattering in the point \(P \in \Omega\) the stream of photons leaves the object at the point \(D \in \Gamma\) and an intensity \(I_D\) of this stream can be measured. Some photons can be absorbed along the straight lines connecting the points \(E\) and \(P\), and \(P\) and \(D\). The operator \(A\) in model (1), which is called Albedo operator, is dependent on the parameter of scattering \(\sigma_s\) and on the parameter of absorption \(\sigma_T\), and describes the connection between the conditions of irradiation (\(I_E\) and \(\gamma\)) and the intensity \(I_D\). According to the practical background of the model (1) the author restricts the domains of \(\sigma_T\), \(\sigma_s\) and \(I_E\) to the certain sets \(D_0 \subseteq W^1_2(\Omega)\), \(D_1 \subseteq L_2(\Omega)\) and \(D_2 \subseteq L_\infty (\Omega)\) respectively. The aim of this paper is to study the reconstruction problems for the parameter functions \(\sigma_s\) and \(\sigma_T\) in the model (1) from known values of \(I_E\), \(\gamma\) and \(I_D\). The author deals with two inverse problems. Problem (P1): given a function \(I_D \in R(F_1)\), find a function \(\sigma_s \in D_1\) which satisfies the linear equation \((F_1 (\sigma_T, I_E, \gamma) \sigma_s) (P) = I_D(P)\), \(P \in \Omega\), where \(\sigma_T \in D_0\), \(I_E \in D_2\) and \(\gamma \in (0, {\pi \over 2})\) are known, and the operator \(F_1 : D_1 \to L_2 (\Omega,z^2)\) is defined by \((F_1 \sigma_s) (P) = g_1 (P) \sigma_s(P)\) with \(g_1 (P) = I_E (P) {1\over 8\pi z \sin \gamma} \text{exp} \bigl( -\int_{K(P)} \sigma_T (P') ds\bigr)\). Problem (P2): given a function \(I_D \in R(F_2)\), find a function \(\sigma_T \in D_0\) which satisfies the nonlinear equation \((F_2 (\sigma_s, I_E, \gamma) \sigma_T) (P) = I_D(P)\), \(P \in \Omega\), where \(\sigma_s \in D_1\), \(I_E \in D_2\) and \(\gamma \in (0, {\pi \over 2})\) are known, and the operator \(F_2 : D_0 \to L_2 (\Omega, z^2)\) is defined by \((F_2 \sigma_T) (P) = g_2(P) \text{exp} \bigl( -\int_{K(P)} \sigma_T (P') ds\bigr)\) with \(g_2(P) = I_E(P) {\sigma_s(P)\over 8\pi z \sin \gamma}.\) It is proved that problem (P1) is well-posed in the Hadamard sense and problem (P2) is ill-posed (there is no stability of the solution).