At MLO the aerosol load of the atmosphere is measured and a database of hourly, daily and monthly average values for the absorption cofficient, in three optical bands, exists. Here is the link:
The data cover the period in 2011+2012 where we observed..
Our fitting procedures have produced estimates of the power a PSF has to be raised to in order for the convolution of model images with the resulting PSF to produce a 'good fit' at the edge of the DS disc in lunar images.
We compare our 'alfa' values with AERONET absorption coefficients, and get this plot:
We used daily average AERONET values, which goes some way towards explaining why the scatter for our data (the 'Power' alfa) is large while AERONET seems less messy. The overplotted red and green lines are two different robust regression fits. The slope is negative implying that large absorbtion coefficients correspond to small 'alfa' values. This matches how we empirically undersatnd the behaviour - on bad foggy nights (presumably with lots of aerosols in the air) we got broad PSFs with small values of alfa.
Repeating the above with hourly-average data the situation does not improve,
So our first interpretation of the above is that 'alfa' is only a poor descriptor of the aerosol load - 'alfa' is really just a dummy fitting parameter for us - alfa is the value that causes a good fit at the edge of the DS disc. It describes the shape of the PSF. Apparently thisis only weakly coupled to the total amount of absorption.
We must understand the AERONET data better first.