  Inverse problems use physical measurements along with a computational model
to estimate the parameters or state of a system of interest. Errors in
measurements and uncertainties in the computational model lead to inaccurate
estimates. This work develops a methodology to estimate the impact of different
errors on the variational solutions of inverse problems. The focus is on time
evolving systems described by ordinary differential equations, and on a
particular class of inverse problems, namely, data assimilation. The
computational algorithm uses first-order and second-order adjoint models. In a
deterministic setting the methodology provides a posteriori error estimates for
the inverse solution. In a probabilistic setting it provides an a posteriori
quantification of uncertainty in the inverse solution, given the uncertainties
in the model and data. Numerical experiments with the shallow water equations
in spherical coordinates illustrate the use of the proposed error estimation
machinery in both deterministic and probabilistic settings.
