If you fit each image in a 100-image stack with our model, and plot the histogram of albedos, you get this:

The dashed line shows the fitted V-band albedo of the 101st image, which is the average image made up of the aligned and averaged 100 images in the stack.

In fitting the above 100 images we found residuals that were ‘wavy’ – probably the waves of ‘something’ that is seen in the movie here. The residuals look like this:

If you fit the 100-image average image, you get this:

The residuals are much smaller – but also ‘wiggly’. The RMSE of the 100-image average fit is about 3.7 times smaller than the single-image fit’s. Can this, and the width of the histogram (1.8%), be used to form a scaling argument so that the uncertainty on the 100-image fit can be deduced?

I’d like to think that

error_fit_100_average = SD_histogram*(RMSE_100/RMSE_1)

which would give us an uncertainty on the fit of the 100-image average of 0.5%whichis quite a lot. In contrast, the standard deviation of the mean, SD_m=SD/sqrt(N-1), is 1/10th of the 1.8%, or 0.18%. This is more in keeping with the experience we have had with fitting ideal images with added noise.

The advantage of the fit of the 100 individual images is that you get the histogram and thus a measure of the actual uncertainty – when you fit the 100-average image you just get the fit – formal uncertainty estimates depend on knowing (independently) the noise on each pixel. One way around is to consider Monte Carlo Markov Chain fitting using the Metropolis-Hastings algorithm – this iterates the fit in a Bayesian framework and generates histograms of all fitted parameters, but is very very slow because MANY iterations are needed. Another way would be to bootstrap the single images, somehow.


Here are the similar histograms for the same night from V, B and VE2. Again, the dashed line is the albedo found in the 100-image average image, while the histogram is made up of 100 single-image determinations:

We now have 3 stacks for which the mean of the histogram and the albedo of the mean image are very close – this encourages us to think that the uncertainty of the determination based on the mean image is similar to the standard deviation of the mean for the histogram. Let us test this by fitting even more stacks and their component images along with the mean image. It would be a way towards using the mean-image determinations as the ‘real thing’ and designate the variations we see from stack to stack as geophysical and not based in analysis method bias.