The solution of inverse problems may be improved by better estimating the noise correlations in the measured data. Minimum variance estimators for the solution of the inverse problem typically require knowledge of the measurement correlations. In certain situations, some of the measured noise may be attributed to small scale spatial variations in the coefficient one aims at recovering. These small scale variations then contribute to the noise in the measured data. We consider asymptotic models that allow us to approximate the correlation function of such a noise. We show on concrete examples how this physics-based modeling of the measurement correlations allows us to improve the reconstructions.