How are impact probability maps rigorously calculated?

Question

This is one example of an impact probability map. It was posted in this question and is from Aerospace Corporation. But My question is about the process, not this particular map, although it is a good working example to pose this question.

With a given spacecraft, its most recent TLE or the last few, and an arbitrary time window defined by $t_1$ and $t_2$, one can use SGP4 to propagate the orbit between those two times to make a ground-track plot, which would be a series of wavy, thin lines.

That's roughly the process outlined in this comment, but it is not enough. Probably $t_1$ and $t_2$ should be chosen to safely bracket any reasonable guess as to the time of impact. So at $t_1$ it's almost for sure still in orbit, and at $t_2$ it's almost for sure on the ground (or vaporized, or some of both). Ground tracks at those points should contribute minimally to the final 2D areal probability histogram.

A corollary to that might address what happens to the histogram below the ground track at the most likely time of impact. Image below: is from the ESA RocketScience blog post Tiangong-1 Reentry Updates. Click for full size.

Looking closer at the 2D probability histogram (see below), a few things stand out.

1. The inner borders between the wide, inner, green region (lower prob) and narrow yellow (higher prob) bands near the extremes of latitude are jagged! Since the atmosphere itself does not have this structure, this must be some kind of residual effect due to the finite number of orbits considered. They are no longer thin lines, but their effect is still seen.
2. Outside of the narrow high probability bands near the extremes are a second pair of even more narrow bands between high probability and zero probability. This is not what one would get by looking at the probability of ground track dwell time alone.

So there seems to be a lot more than just "what you need is the time it spends over a given unit area of surface over the course of many orbits" going on here.

Question: How are impact probability maps rigorously calculated?

above: Cropped from a larger version of the map, found here.

Answer 1

I don't fully agree with the explanation given so far. Here's my additional guesses:

2) The outer edges: This is not a binning artifact as stated by @PearsonArtPhoto. Debris doesn't only fall along the path of satellite, there is a region with a certain width around the path where we have to expect debris coming down. It's reasonable to assume the distribution within this region is shaped like a bell curve - the further away from the path the more unlikely to find debris. That means, there is not a sudden cutoff but a smooth distribution between high probability and zero.

1) I agree that this comes from the number of orbits that had to be considered given the length of the reentry window. But: Why are the waves not symmetric around the drawn orbit of the satellite? As the orbit decays, the orbital period gets shorter. On the other hand, as we don't know the exact rate of decay we can't precisely determine the orbital path relative to the surface of Earth.

I suspect the drawn curves are from a constant orbit, but the calculation of impact probability included the (unknown and random) change in orbital period. If the orbit gets shorter, the satellite will pass over areas further East with each revolution, but never further West than the "nominal" path. And this is what we see: On the right hand side of the path the probabilities are higher than on the left side.

In summary, this all looks like what you have to expect after you run a series of orbital predictions involving random fluctuations to the rate of decay and assuming a certain distribution of debris around the calculated point of decay.

Answer 2

My guess is the way that they did this is to take the orbital paths that might happen, and figured out where it would reenter if that was the case.

The artifact at the edges is most likely a binning size. Only the very middle part (Top and bottom) is a possibility , but the bin is still there, so the actual amount is small. A simpler way to think of this is to imagine that you had bins for a number every degree vertically from 0 to 90. Let's say the highest was 50.1. The small 0.1 bin would show it as a small chance, but it depends on which part of the bin you are at, the 50.0 might well be the most likely zone for impact. Also it might take in to account winds and other similar things that might blur the edge a bit.

The waves appear to have a period of about 12 degrees, or about 60 peaks. That corresponds to about 4 days worth of orbits, give or take a bit, which I suspect corresponds to the region of uncertainty.

The bottom line is, create a model of the path including uncertainties of key parameters. Vary that model with the different uncertainties varied. Have some kind of an image showing possible reentry points, and every time something strikes that point add a dot to it. Continue to iterate until you have a good feel for it. They might well have included a few things like having debris fall over an arc instead of just a single point as well.

Bottom line is, some kind of a simulation was done with random conditions being modeled, and the graph is what came out.