Impact of Turbulence (Seeing) to an Image

The ``quality'' of an image can be described in many different ways. The overall shape of the distribution of light from a point source is specified by the point spread function. Diffraction gives a basic limit to the quality of the PSF, but any aberrations or image motion add to structure/broadening of the PSF. Another way of describing the quality of an image is to specify it's modulation transfer function (MTF). The MTF and PSF are a Fourier transform pair. Turbulence theory gives:

$$MTF(\nu) = \exp \bigg[-3.44\bigg(\frac{\lambda\cdot\nu}{r_o}\bigg)^{5/3}\bigg]$$ (2.14)

where $\nu$ is the spatial frequency. Note that a gaussian goes as $\nu^2$, so this is close to a gaussian. The shape of seeing-limited images is roughly Gaussian in core but has more extended wings This is relevant because the seeing is often described by fitting a Gaussian to a stellar profile. A potentially better empirical fitting function is a Moffat function :

$$I = p_1 (1 + (x-p_2)^2/p_4^2 + (y-p_3)^2/p_5^2) ^{-p_6}$$ (2.15)

A final way of characterizing the image quality, more commonly used in adaptive optics applications, is the Strehl ratio. The Strehl ratio is the ratio between the peak amplitude of the PSF and the peak amplitude expected in the presence of diffraction only.