We consider the problem of reconstructing an audio signal from multiple observations, each of which is contaminated with time-varying noise. Assuming that the time-variation is different for each observation, we propose an estimation formulation that can adapt to these changes. Specifically, we postulate a parametric reconstruction and choose the parameters so that the reconstruction minimizes a cost function. The cost function is selected so that audio signals are penalized less compared to arbitrary signals with the same energy. As cost functions, we experiment with a recently proposed prior as well as mixed norms placed on the short time Fourier coefficients.