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I have state vectors describing the satellite’s orbit under the influence of moon, sun, radiation, and the spherical Earth with zonal and tesseral terms. I want to generate a time series of mean orbital elements from these state vectors so that they resemble TLE data in some way. The purpose is to generate data for model training.

I’ve read many similar answers on this forum (this this, this), but they mostly deal with converting from TLE to osculating elements.

For converting from osculating (state vectors) to TLE, the general consensus seems to be using SGP4 to iteratively guess TLE elements that return the closest state vector. However, I believe this solution is not computationally feasible for my application.

I’ve also looked at the Brouver and Kozai methods described in Vallado and this article that discusses osculating to mean conversion, but the theory behind it seems to be quite complex and I’m still quite confused about these methods and not sure if I can even use them for my application.

My understanding is that TLE are essentially the average of observed state vectors. So, when given TLE is fed to SGP4, the predicted trajectory fits the observations as best as possible. However, based on this and the fact that I have essentially very closely spaced observations, why isn't it possible to numerically integrate (average) the elements over several periods to produce mean elements? I think this averaging should remove the periodic oscillations from perturbations, but I haven’t found any resources about it. Also unsure about the time window for the averaging(one period, days, etc.)

The question is, what’s the best way to go from state vectors to mean elements in my case?

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    $\begingroup$ To be honest, I haven’t tried the iterative guessing method. Apart from the computational difficulties, which you rightly mentioned can be overcome with more computational power, I have another concern about this process. I may be mistaken, but I believe there might be an issue with degeneracy. The approaches I’ve seen attempt to iteratively guess a single state vector. However, from my understanding, you need to guess the TLE data point, which fits the best the evolution of satellite, therefore several state vectors. I’m not sure how to proceed with that. $\endgroup$ Commented Jan 29 at 23:27
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    $\begingroup$ Instead of guessing look up "gradient descent". My bet is the real algorithm is more of a least squares method. Since it dates back to the 80's when non deterministic algorithms were really expensive. $\endgroup$ Commented Jan 30 at 1:43
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    $\begingroup$ Years ago, I did this by wrapping a general-purpose numerical nonlinear equation solver around an SGP4 propagator. $\endgroup$ Commented Jan 31 at 0:41
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    $\begingroup$ Would you tell us more about your data? "I have state vectors describing the satellite’s orbit ..." does this mean you have an ephemeris with time steps every few minutes/seconds or do you have a set of SV in different phases of a mission? Are there manoeuvres included? Depending on this details you would need to begin slightly different but in the end you should end up in doing a (weighted) least squares run (I think thats what you described as "guessing") over the ephemeris to get a TLE. Analytical conversion aproaches mostly end up with quite big errors. $\endgroup$ Commented Jan 31 at 11:36
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    $\begingroup$ You would need an Least Squares Solver including a SPG4-Propagator. (I wrote one a couple of years ago, but I am not allowed to share the code). Seggregate the orininal Ephemeris in maneuovre-free blocks between the maneuovre (or shorter) and do a LS run for every segment. That should be the least bad possibility I know. If you have to performe specific tasks like track the object with a sensor, just use the time span with the specific estimated access to generate the TLE. $\endgroup$ Commented Mar 4 at 9:53

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