In this paper, the identification of the elastic properties of a sphere is investigated. The inverse problem requires finding the deflection \(u(r,\theta,\varphi)\) together with the density \(\rho(r)> 0\) and the perturbation velocities of longitudinal waves \(c^2_0\ll c_1(r,\theta,\varphi)= c^2(r,\theta,\varphi)- c^2_0\), where \(c_0\) is known, and transverse \(a^2_0\ll a_1(r,\theta,\varphi)= a^2(r,\theta,\varphi)- a^2_0\), where \(a_0\) is known, \(c= (\lambda+ 2\mu)/\rho\), \(a= \mu/\rho\), \(\lambda\) and \(\mu\) are the Lamé parameters satisfying the Lamé equations of transient isotropic elsaticity
\[ \rho{\partial^2\underline u\over\partial t^2}= \mu\Delta\underline u+(\lambda+ \mu)\nabla(\nabla\cdot\underline u)+ (\nabla\cdot\underline u)\nabla\lambda+ (\nabla\underline u+(\nabla\underline u)^{\text{tr}})\nabla\mu \]
in \(B(0,1)\) together with the initial condition \(\underline u|_{t< 0}= 0\) on \(B(0,1)\) and the Cauchy boundary conditions on \(\partial B(0,1)\)
\[ \underline u|_{r= 1}=\underline h(t,\theta,\varphi)\quad\text{on }\partial B(0,1)\times [0,T], \]
\[ (\lambda\nabla\cdot\underline u+\mu(\nabla\underline u+(\nabla\underline u)^{\text{tr}})\underline e_r|_{r=1}= {1\over\sqrt{2\pi}} q(t)\underline i. \] Uniqueness and conditional stability estimates are provided.