Below, we have investiagted the phenomenon of 'Mickey Mouse ears' - i.e. the tendency for theoretical lunar images to have too bright cusps so that the residual image ends up with a bipolar structure ressembling Mickey Mouse's ears. We have investigated what happens to the ears if a semi-empirical BS model image is used, based on scaling and clip-selecting the BS from the observed image. We showed that the ears went away.
Now we investigate what happens to the quality of the fit - for our goal is to derive terrestrial albedos with minimal error (both scatter and bias; bias we can live with as long as it is constant; scatter is a pain).
We selected 10 images that we fit with both semi-empirical BSs and theoretical BSs and tabulated the values for fit-parameters as well as some of their formal fit errors (output from LMFIT).
In this plot we show the changes, in percent, of some key parametrs and their errors, in the sense that the purely theoretical result is subtracted from the semi-empirical result:
Change, in %, of some key fitting parameters. (semi-empirical - theoretical)/(½*(semi+theor))*100.
A: albedo, is larger by several percent in the semi-empirical (S) fit, than in the purely theoretical (T) fit.
dA: error on albedo reported by the fitting routine, is mainly smaller in S compared to T.
alfa: the power to which the PSF is raised is less by several % in S compared to T. This implies a broader PSF in the final fit in S.
ped: the pedestal added to the whole image has both positive and negative changes.
dX: the shift of the model image in X direction needed to align model and observed image. Mainly smaller in S than in T.
cf: core factor on the PSF - has larger values in S than in T. So the PSF peak is 'pulled up' in S relative to the wings.
contrast: lunar albedo map contrast, has smaller values in S than in T. This is consistent with the 'pulled up' core which 'copies' detail better.
RMSE: the root mean square error in the area of the image used for minimization, is mainly smaller in S than in T.
So, what have we learned? The two formal estimates of errors - dA and RMSE are both smaller in S than in T, so S would appear to be a better way to go.
Taken together - the smaller errors (on A, and RMSE itself) in S compared to T is encouraging, but we really do not know how much closer to the 'true value' for A we are, just that we are probably closer - at least in terms of scatter, not necessarily in terms of bias.
Using the semi-empirical method compared to the fully synthetic method adds very little in terms of computational overhead, so we will use S henceforth.
We note that the width of the distribution of albedos in units of the mean is 8 percent smaller in S than in T. The S method gives tighter distributions of albedo. Of course, the intrinsic (geophysical) scatter in albedo is present in this number too.