Inverse problems of 3-dimensional wave equations

In $\IR\sp3$ with a scattering medium in the upper half-space $x0\sb3 \u3e 0,$ we place a point source on the negative $x\sb3$ axis generating an incident spherical wave. Inverse problems associated with two types of linear wave equations are considered,$$u\sb{tt} - \Delta u = q(x)u = 0,\ x\sb3 \u3e 0,\ t \u3e 0$$and$${1\over c(x\sb3)\sp2} u\sb{tt} - \Delta u = 0,\ x\sb3 \u3e 0,\ t \u3e 0.$$For the problem associated with the first equation, by the application of wave splitting techniques and asymptotic analysis in the neighborhood of the wave front, we show that $q(x)$ can be derived from the knowledge of the reflected field on the surface $x\sb3 = 0$. The precise form of the incident wave is known. The existence of wave splitting of the solution u of the first equation is proved. We use the leading coefficients of the asymptotic expansion of the up-going wave component to yield $q(x)$. And, in fact, it is a necessary condition for the solution of the ill-posed problem which involves the inverse Dirichlet operator. This ill-posed problem of recovering the reflective split wave component is also examined in detail. The inverse problem for the second equation is treated by a similar method with some modifications. As a result of the analysis for this problem, a possible approach to more general problems. (For example, c is a function of $x\sb1,\ x\sb2$ and $x\sb3$) is indicated