Determining of the parameters of an elastic isotropic medium in a infinite cylinder

We consider an inverse problem for a system of an elastic isotropic equations in a cylinder infinite with respect to the variable $z$. The linearized problem of identification of three characteristics of elastic isotropic medium is investigated. It is supposed that the medium density $\rho(r,\theta,\varphi)$, the propagation velocities of longitudinal $c(r,\theta,\varphi)$ and transverse $a(r,\theta,\varphi)$ waves can be represented as $\rho(r,\theta,\varphi)\!=\!\rho_{0}+\rho_{1}(r,\theta,\!\varphi)$, $a^{2}(r,\theta,\varphi)=a_{0}^{2}+a_{1}(r,\theta,\varphi)$, $c^{2}(r,\theta,\varphi)=c_{0}^{2}+c_{1}(r,\theta,\varphi)$, where $\rho_{0}$, $a_{0}^{2}$, $c_{0}^{2}$ are some unknown constants, and unknown functions $\rho_{1}(r,\theta,\varphi)$, $a_{1}(r,\theta,\varphi)$, $c_{1}(r,\theta,\varphi)$ are small in comparison with the constants $\rho_{0}$, $a_{0}^{2}$ и $c_{0}^{2}$, correspondingly. The estimates of conditional stability of the inverse problem solution are obtained.