  The shallow water equations (SWE) model a variety of geophysical flows. Flows
in channels with rectangular cross sections may be modelled with a simplified
one-dimensional SWE with varying width. Among other model parameters,
information about the bathymetry and friction coefficient is needed for the
correct and precise prediction of the flow. Although synthetic values of the
model parameters may suffice for testing numerical schemes, approximations of
the bathymetry and other parameters may be required for applications.
Estimations may be obtained by experimental methods but some of those
techniques may be expensive, time consuming, and not always available. In this
work, we propose to solve the inverse problem to estimate the bathymetry and
the Manning's friction coefficient from transient velocity data. This is done
with the aid of a cost functional which includes the SWE through Lagrange
multipliers. The solution is obtained by solving the constrained optimization
problem by a continuous descent method. The direct and the adjoint problems are
both solved numerically using a second-order accurate Roe-type upwind scheme.
Numerical tests are included to show the merits of the algorithm.
