## Laplacian revisited: A Brain-Wave

From flux to AlbedoPosted by Peter Thejll Dec 27, 2012 09:52AMOn this blog we have previously investigated the laplacian method of estimating earthshine intensity. [See here: http://iloapp.thejll.com/blog/earthshine?Home&post=273

and here http://iloapp.thejll.com/blog/earthshine?Home&post=272 .]

One of the limitations we realized then was that image resolution influenced the results from the Laplacian edge signature - the more the edges of the image are smeared the lower the Laplacian signature estimate will be.

We now revisit the method after realizing that the whole image is affected in the same way by resolution issues - the intensity estimate derived from the DS and BS edges will both depend on image resolution in the same way. The DS estimate is proportional to earthshine intensity, while the BS estimate is proportional to 'moonshine' intensity (plus earthshine - a vanishing contribution). The ratio of the two estimates will be independent of resolution - i.e. the ratio has the 'common mode rejecting' property.

We convolved ideal images of the Moon with the alfa=1.8 PSF and estimated both the 'step size' at the edges (on DS as well as BS) and the Laplacian signature. We plot them below:

*The*

The

**top panel**shows the step size estimate against the Laplacian signature estimate on the DS only. We see an offset, with Laplacian estimate being smaller than the step size estimate. This is caused by the resolution issue - the edge is fuzzy and the Laplacian is degraded. The step size is estimated robustly from the original ideal image (i.e. before the convolution is performed) and is thus not dependent on image resolution.The

**bottom panel**shows the ratio of DS and BS estimates - x-axis is the step size estimator and the y-axis is the Laplacian estimator. We now see that the dependence on image resolution is gone from the Laplacian estimate. The correlation between the two estimates is good, at R=0.92, but not perfect. Noise was not added to the images. Estimates of Laplacian were performed on the line through the disc centre. The step size estimator is based on the average of 9 lines through the disc center.[Added later:] Since we are working with synthetic models we know the actual earthshine intensity in each image, so we can compare what the actual intensity is to what is found with the 'step' and the 'Laplacian signature' methods:

*In the*

Since the Laplacian signature method (and the step size method) literally express the DS/BS intensities ratio (which is only proportional to the earthshine intensity) we have different values along the x- and y-axes. The geometric factors having to do with distances and Earth's radius are not compensated for.

**first panel**we see what we saw above - that the step and Laplacian methods are rather consistent. In the**panel below**we compare to the actual earthshine intensity.*The main result is that the relationship between actual earthshine intensity and the Laplacian signature method (and by extension, the step size method) are not quite proportional - there is a slight curve. In interpreting the above we should keep in mind that the Laplacian estimate of DS illumination from the edge derivatives is in a different role than the estimate of BS illumination - the former is not very geometry dependent while the latter is: That is, earthshine is due to light from a source we are sitting on doing our observations from, while the BS illumination is more angle-dependent in that the reflectance properties of thr Moon come into play to a larger degree - at the moment I am not sure whether there is an angle-dependence 'along the BS edge of the disc' that can cause a problem in interpretation.*

*Will need to look at this.*

Another thought: is it possible to make a calibration relation between actual earthshine intensity and the Laplacian DS/BS signature estimate?

Another thought: is it possible to make a calibration relation between actual earthshine intensity and the Laplacian DS/BS signature estimate?

## Posted by Chris Flynn Dec 30, 2012 12:11AM

yep -- I did wonder about that!

## Posted by Peter Dec 29, 2012 08:01AM

... not 'reluctance' ... 'reflectance'!

## Posted by Peter Dec 29, 2012 07:51AM

Yes indeed! The BS is still reluctance dependent, and we need models to interpret. The edge is just an easier place to find because craters and mare have to be navigated to. The edge signature in the Laplacian can be robustly found without any fitting.

On the DS I wonder if we do not have a separate problem: Since the Earth is an extended source which illuminates the DS we could be located at positions where parts of the bright Earth does not 'reach around the corner' on the DS limb. This will affect a few degrees in lunar longitude and I wonder if we can model this - or use it for something? through the observing session the position of the observer wrt to the source is changing - more student projects!

## Posted by Chris Flynn Dec 29, 2012 03:34AM

Hi Peter, that looks really interesting -- I think I had misunderstood parts of the Laplacian procedure until now, we may need to discuss this in more depth. In any case, apart from the phase dependent properties of the Laplacian signature on the BS, which I am guessing we can estimate from artificial images (or even real images) and calibrate out, this is looking very promising!