Inverse problems are known to be very intolerant to both data errors
and errors in the forward model.
With several inverse problems the adequately accurate forward model can turn out to be
computationally excessively complex.
The Bayesian framework for inverse problems has recently been shown to enable
the adoption of highly approximate forward models.
This approach is based on the modelling of the associated approximation errors
that are incorporated in the construction of the computational model.
In this paper we investigate the extension of the approximation error theory to
nonstationary inverse problems.
We develop the basic framework for linear nonstationary inverse problems
that allows one to use both highly reduced states and extended time steps.
As an example we study the one dimensional heat equation.