Some more analysis of scattered light.
I take three synthetic lunar images —
1) a thin crescent with no ES, i.e. BS only
2) ES only
3) BS + ES
Raw images shown on the left — with scattered light on the right. (All images shown at the same display levels, making the halo hard to see!)
The albedo adopted was 0.30 for the ES on (2) and (3).
All three are generated with the correct flux scaling for the components
i.e. image (3) is the sum of (1) and (2).
These three images are convolved with the PSF with alpha=1.8.
No noise is included in the final output images.
I then compute the amount of light which has scattered off the
original synthetic images and onto new pixels. (A list of all pixels which
had non-zero counts in them in the original images was kept and the
counts in these pixels in the convolved images compared to the total
counts in the convolved images. The normalisation of the convolution is set
to the total counts in the original images).
While doing this, I checked how much light scatters off the edge of the 512×512
frames and into the larger (3*512)x(3*512) frames used for the FFT. This turns out
to be ~0.04%, a good deal less than our target accuracy (0.1%)) for alpha=1.8, so this is not a major source of error.
The amount of off-moon scattered light is as follows
1) BS only : 4.2 percent
2) ES only : 0.58 percent
3) BS and ES : 3.9 percent
let’s call this the “halo light”.
these numbers highlight why getting the BS to ES normalisation correct
when generating forward scattering fits to data is important!
We want to measure the DS to BS ratio to an accuracy of 0.1 percent, so the approximately 4 percent of halo light has to be gotten right when figuring out the amount of BS light. The difference between BS only halo light and BS+ES halo light is of order 0.3 percent, 3 times more than our desired accuracy. This shows that the ES makes a non-negligible contribution to the halo light.
Those numbers were for alpha=1.8. For alpha=1.6, I get
1) BS only : 6.9 percent
2) ES only : 1.58 percent
3) BS and ES : 6.0 percent
Now the ES is making a very significant contribution to the halo light — 16 times more than the accuracy with which we would like to know the BS light.
The light lost off the 512×512 frame is small at 0.04% – half of our target error.
However, for alpha = 1.6, this lost light increases to 0.3 percent of the total.
Lost light off 512×512 array for centered thin crescent Moon
2.0 0.004 %
1.8 0.04 %
1.7 0.11 %
1.6 0.32 %
It looks like we need to take into account lost light off the 512×512 array
when we are doing the fitting for the BS, especially if the images aren’t well centered, as is presently an issue!
In response to Peter’s comments:
We have both done tests on artificial data which suggest that we are able to measure alpha with an accuracy of +/- 0.01. I figured out the corresponding missing light off the edges of the image for this uncertainty in alpha:
1.80 +/- 0.01 0.038 -/+ 0.004 %
1.70 +/- 0.01 0.11 -/+ 0.01 %
1.60 +/- 0.01 0.32 -/+ 0.03 %
So if we really have alpha=1.8 and can measure it with an accuracy of 0.01, we are pretty safe — at least for this thin crescent which is well centered.
Peter asked: what if extinction and alpha are correlated? Great idea! We should certainly look into this!
I had another idea — what about if we made longer exposures of the moon which would saturate the moon but sample the halo light well all the way to the edge of the frame? Could we then measure alpha with appropriate accuracy where it counts — at the moment we are merely extrapolating a not very well sampled halo beyond the edges of the frame to estimate the missing light (in 30 ms exposures, the halo light is very close to the bias level at the edges of the frame, so is not well measured out there, and a bias error could lead to systematics in the estimate of alpha, perhaps larger than the error estimate adopted above of 0.01, which was obtained for rather idealised synthetic data.
The behaviour of the scattered light around the moon beyond the edges of the frame is still very much an open question — we didn’t acquire the data at the last full moon to settle this, but just got more questions!
More still: position of the Moon
Tested the effects of the positioning of the moon using the same setup as above
For 1 pixel shift of Moon in the x-axis, I got negligible changes in the amount of lost light.
For a 10 pixel (circa 70 arcseconds) shift (to the left along the X-axis for a crescent moon which is illuminated on the left side):
2.0 0.005 %
1.8 0.04 %
1.7 0.12 %
1.6 0.33 %
The changes to the fractional lost light are very small. So the good news here is that we are not very sensitive to the precise position of the moon if it’s near the center.