Constraining the Parameter Space

 

Screenshot at Jul 25 12-18-01
Equation 7. Derived and discussed in my last post, this is the equation we are working with to find possible center points.

In my last post, I discussed how we can try to minimize equation (7) to ascertain what the radii and d_center could be. Using Sinlap, I know the profile distance (Dp) is ~77km, and the radii could range from 50% to 200% what literature assumed based on radar imagery. I used these to get a complete picture of the parameter space. By plugging in these values, I created a 3D field for f(rl,rr,dc). I show it in the video below. This gives a value for every point in 3D space, but it’s difficult to learn much from the full plot (left). I filter out values >0.05 to locate where the function approached 0 (right).

What we see is that there exists a curved plane of possible points for these values, but even in this plane, points seem to converge to a symmetric pattern of circles. The transition from higher to lower points is clearer when looking at points less than 1.

 

f(rl,rr,dc)1field
Figure 1. This shows a still view of the video above. We are looking at a curved plane of possible radii and dc values that would make this function approach to zero (<= 1).

 

 

These views have interesting implications. There are points of low dc or r where this function can be met, but the lower (blue) points cluster at higher values. Meaning, The center is more restricted when closer to the profile.

However, I we don’t know if this will translate to a unique range of center points. That is to say, lets visualize the field of center points this range of radii and dc values could produce.

 

 

There is no discernible pattern in the plot of all possible center points (Figure 3). All there is are lines of possible points using the defined ranges of each variable. However, we see a pattern when we only plot the center points produced using the range of values in the curved plain of f(‘x’) -> 0 for dc, rl, and rr (Figure 4)

 

 

soi crater center .05
Figure 4. Resulting center points after converting only the values for rc, rl, and dc where f(‘x’)->0 (<=.05) for Soi Crater.

I think the attempt to minimize the function is flawed because, as we see above, there is no one area were the function approaches 0. However in conjunction with radar imagery, we can better rule out unlikely points. And, I think this figure suggests that might be the trick. It isn’t obvious in this picture, but a tiny black dot indicates the literature assumed center.

CraterCenter_.0005fix
Figure 5. Possible (f<=.005) crater center points (blue) plotted over Soi Radar imagery. The literature center is shown with a black dot. The absolute minimum f is shown with a red dot.

The function is filtered to only plot points corresponding to function f <= 0.005. The red point represents the point with the lowest value for for function f(). Presumably, I should get a similar result if I ran this function using Matlab’s minimizing function over the same constraints (min and max rl, rr, and dc).

CraterCenter_.0005_crop
Figure 6. Figure 5 repeated looking only at points closer to the center.

If we repeat the same process, focusing closer to the center, the lowest point is still in the top left. It appears the absolute lowest point is a matter of its position to the topographic profile. It explains why changing the variable constraints had such a large impact on the final result. We can continue to filter out points, going as low as <=.0005. Even then we have a decent number of possible points if it is allowed to run with small enough increments. It constrains the possible points, but it will not give us a center value.

However, it is not useless. Up to now, we have relied on radar imagery to find the center, but with this we can adjust the center to better fit the topographic data. I ran the numbers for the literature values for Soi to see what value for function f it would produce and got ~0.8. Thats not far off, but it could be better.

The question becomes whether or not this is too drawn out or convoluted a process. The goal is to use the topo data to update the crater parameters, and in that sense this gives the crater parameters at least more small piece information to rely on.

Thoughts and comments appreciated.

 

 

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