# July Update

## Crater Topography (depths and roughness)

I read through Catherine’s paper on Titan crater topography, and realized that roughness wasn’t the way to go about it. I’ve since realized she left a comment saying as much on my last post. I’ve changed the roughness parameter to relative depth, the measurement that Catherine uses to see how much a crater has changed from its state (as compared to Ganymede craters).

I calculate depths using the same technique that Catherine does finding the lowest point on each side of the center of the crater. I used the depths for Ganymede (Bray and Schenck papers) I found a while back when I worked to create meshes for the tekton code. I’m still faced with the same issues in regards to the very large craters, but that isn’t the main priority at the moment. I know Catherine included Menrva in her data, so I am curious how she determined what a Ganymede depth for a 400+km crater should be.

## Best Fit Circle

The circle fit that I discussed before is a function developed by Izhak Bucher in 1991. It works by using a little matrix algebra that I am having a little trouble follow. I had hoped to decipher exactly how it works before I updated you all but wasn’t successful. We start with a matrix A consisting of three columns: x, y, ones(length of x). This is divided into a vector = -x.^2+-y.^2. The result is a vector of the x and y center points and can calculate. My knowledge of matrix division is a little rusty, and from what I could find ab = inv(a)*b but inv only works with square matrices. Without really understanding this step I can’t really explain the method exactly, merely that it tries to best fit the circle to all the given points.

I just reread it again, and I think I may have misunderstood what you were getting at. It doesn’t use RADAR imagery if that is what you are worried about. It uses the position of the rim from multiple flyovers.

If we look back at one of my figures from my last post, we can see how the rim is identified in each sartopo strip. It’s these sartopo points that the circle is being fitted to. The radar image is merely there as reference.

I thought this was a rather good way to incorporate all the data when we have multiple points available. It allows us to fit the shape to all the data and back out exactly where the center should be. Its when we have less than two fly bys, like in the example I gave last post, where it gets difficult to solve. I don’t like having to make the assumptions, but I’m finding it difficult to find a better way when we have such little data. A circle requires 3 or more points.

## Single Sartopo Profile Approach

The use of the Chi-squared test is an interesting approach I didn’t think of. I haven’t done it yet because I hadn’t realized I had a response on my blog. However, I have used it before in a statistical analysis class back at Tech. I still have all my notes for this class, so I think I can do this. However, from what I can ascertain from an initial overview, it compares the modeled to the observed. I assume I would input a range of possible values for each variable to get different fits. Then the chi-squared test would be done on the fit and the actual data. It seems that I am to just compare two numbers.

r_left = sqrt(d_c^2 + (D_p/x)^2)
r_right = sqrt(d_c^2 + (D_p/y)^2)

The variables x and y are not exactly two unless the two radii are the same. So, we have two equations with five unknowns: r_l, r_r, d_c, x, and y. You could try a minimization process to get the ‘best guess’ for these values. Start with reasonable assumptions and see which combination produces the smallest chi-squared value when compared to the known value of D_p.

D_p = x * sqrt(r_left^2 – d_c^2) = y * sqrt(r_right^2 – d_c^2)

-Catherine

Here I am just calculating what the profile length (two calculated values) would be and comparing it to the single value it actually is. Why would I try to do a chi analysis when I could just find the closest value? I don’t understand what I would be performing it on.

## Error and Uncertainty

I still have a ways to go with uncertainty. I updated the code to pull the random and systematic errors, but I don’t entirely understand how they fit together. Does one incorporate the other? I’ve added the two together for now and can add them to the outputted result of the code.