  We develop an adjoint approach for recovering the topographical function
included in the source term of one-dimensional hyperbolic balance laws. We
focus on a specific system, namely the shallow water equations, in an effort to
recover the riverbed topography. The novelty of this work is the ability to
robustly recover the bottom topography using only noisy boundary data from one
measurement event and the inclusion of two regularization terms in the
iterative update scheme. The adjoint scheme is determined from a linearization
of the forward system and is used to compute the gradient of a cost function.
The bottom topography function is recovered through an iterative process given
by a three-operator splitting method which allows the feasibility to include
two regularization terms. Numerous numerical tests demonstrate the robustness
of the method regardless of the choice of initial guess and in the presence of
discontinuities in the solution of the forward problem.
