  We derive a method to reconstruct Gaussian signals from linear measurements
with Gaussian noise. This new algorithm is intended for applications in
astrophysics and other sciences. The starting point of our considerations is
the principle of minimum Gibbs free energy which was previously used to derive
a signal reconstruction algorithm handling uncertainties in the signal
covariance. We extend this algorithm to simultaneously uncertain noise and
signal covariances using the same principles in the derivation. The resulting
equations are general enough to be applied in many different contexts. We
demonstrate the performance of the algorithm by applying it to specific example
situations and compare it to algorithms not allowing for uncertainties in the
noise covariance. The results show that the method we suggest performs very
well under a variety of circumstances and is indeed qualitatively superior to
the other methods in cases where uncertainty in the noise covariance is
present.
