We conduct two experiments, to asses the abilities of the fit-a-model-image method.

Experiment 1: Same image, with differing noise.
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We pick an ideal image, generated from JD2455945, and to this one image we apply a PSF convolution, and 10 times generate images with varying Poisson noise. We then fit these 10 images, and look at the distribution of results.

We find

mean albedo: 0.28051
std. dev. of 10 fits: 0.00030
SD expressed as percentage of mean: 0.11%

The actual value for albedo used to generate the ideal image was 0.28083594, which is 1 SD above the mean value found.

So, we can (again) confirm that in principle, we can achieve fits with small bias and small errors, as per our original science goal.

Experiment 2: Same image with noise, different fit starting guesses.
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We pick now one of the noisy artificial images and refit it 10 times – each tiome starting at a random guess near the supposed best guess. We pick our vatiation froma normal distribution with amplitude 0.03 and multiply the fixed starting guesses by 1+0.03*N(0,1) for the 8 starting parameters (there are 8 – 7 before – we added shift om image along y-axis also). These guesses were then allowed to converge and we collected the results. We found:

mean albedo=0.280183
S.D.=1.4181365e-05
SD in % of mean=0.0051

We see that the sactter is very small, i.e. that convergence is almost always to the same value. However, there is a bias in that the known albedo (see above) was not found.

The bias is (0.280183-0.28083594)=0.0006529 which is 0.23% of the correct value.

From this we lkearn that
a) convergence may not always be dependent on starting point – the same best solution will be arrived at, and that
b) the noise added to the image itself will have enough influence, even if small (i.e. realistic for averaging 100 images in a stack, and coadding 20 ‘lines’ in each of 11 ‘bands’) to cause a bias in this perfect-case example, where noise is only Poissonian.

The bias is a mere 0.2% of the correct value which we could lkive with given our science goal of 0.1, I guess, but reality adds more problems to our images than just Poisson noise. These effects appear to give us a typical spread of 1% – about 10 times what we saw in experiment 1, above.

As it is, we are using 1/10 of the pixels in the image for our fits so we have at most a factor of 10 more pixels to work on. This could in principle give sus a factor f 3 reduction in the uncertainty, if all pixels are equally effective at driving the solution. If all went well, our 1% errors coul dbecome 0.3% errors which woul dbe acceptable (just), in the absence of a bias.

What can we do along this line?

1) Many of the DS sky pixels are not being put to use – they coul dhelp set the sky-level of the fitted model better.
2) Not all DS disc pixels are being put to use, but there is a limit how far onto the disc we can go before BS halo starts being a too importnat factor, but perhaps we could double the number of pixels.
3) We only use the BS disc and BS sky to calculate the total flux which is used to ensure same flux in model and observation. We might be able to also fit some of the edge near the BS or just try to match the level of the BS sky near the disc – but the levels are much higher and our cirrent use of errors based on differences between absolute levels would become dominated by the BS pixels in driving the fit.
4) We can use more images from the same night – at least inside some period of time where the geometry of Earth and Moon does not change significantly – perhaps half an hour or so. In that time some 10-30 images could be gathered in reality. If several filters are wanted then perhaps 10 images in each filter (in practise it seems we could reach that on some nights, inside a half hour) and then repeat for other filters.

In summary, if erros were due to Poisson noise onkly, we could hope to double the number of DS disc pixels, use a lot more of the DS sky pixels, and perhaps work towards 10 images in each filter inside each half hour, giving us, potentially, the required improvements in uncertainty.

We can work on this!

There is a “however”, however: All the noise is not Poissonian – there are issues having to do with:

a) Bias subtraction – and airglow – and zodical lights: letting the ‘pedestal’ solve for these may be a bad idea – perhaps we should fix the pedestal at predicted levels in each image? This needs some wor, since we are sensitive to errors there too!
b) The atmospheric turbulence causes ‘wiggle’ in the images,. described on this blog, and this is a factor in driving the error in the fits.
c) The model we fit is not faultless either – the BDRF is a simple one and we see hints of some azimuthal-angle problem ain fitting along the edge of the disc. Model couild be improved.
d) Other parameters in the model might be fixed too – such as the ‘core factor’. Experiments are still needed to see if this is a good idea.