Relationship between turbulent image variance and average image gradient

Optical turbulence can cause substantial distortions in imaging over long horizontal paths. For Lambertian objects, these distortions are only seen where there is a gradient in the object's radiance. It is possible to establish a relationship between the intensity variance of a turbulent image and the average image's gradient squared. We test the validity of a linear relationship between these quantities using turbulent imaging data. We find that it performs reasonably well for weak and intermediate optical turbulence regimes, but that some discrepancies remain to be explained.