We compare the numerical behavior of the logarithmic barrier (LB), primal-dual interior-point (IPM), projected conjugate gradiend (PRCG), and primal-dual active sets (PDAS) methods for solving one-dimensional partial differential equations (PDE) constrained optimization problems arising on earthquake applications. Our main interest is to identify an effective strategy that allows to incorporate efficiently physical bound constraints into the problem. We run several test cases and identify that the strategy IPM works best. The inverse seismic wave problem that we present here requires the expertise of researchers from different areas (geologists, engineers, computational scientists), in order to develop the capability for estimating the geological structure and mechanical properties of the earth structure. Our contribution comes to help in the development of a new optimization approach to the new inversion methods based on PDE's.