Inexplicable Offset in Satellite Laser Ranging (SLR) observations

Question

I am struggling with the implementation of a Satellite Laser Ranging (SLR) validation tool.

The situation is as follows: I am provided with a framework which gives me the position of a SLR observation station and the position of a satellite at a specified GPS time. Those positions can be assumed to be sufficiently correct for the given task. From those positions I calculate a reference distance which I will use for comparison with given SLR observations. The observation files I use as input are in CRD format and provided by data centers like http://edc.dgfi.tum.de/en/. From those files I extract, for each observation record, the GPS time stamp and time of flight in seconds. According to the simplest observation equation for SLR given by $$d = c * \frac{\Delta t}{2}$$ I calculate the distance between station and satellite and compare it to the distance provided by my framework.

The problem is the distances deviate seemingly randomly with deviations ranging from 1 to 300 meters in both directions. I am aware that the simple observation equation is missing all correction terms but even without those an accuracy of up to 10 meters should be feasible.

So far I thought I was dealing with a station clock or observation clock bias but adding offsets in positive as well as negative direction to the observation times always resulted in greater deviations of the two distances.

The only thing I am constantly experiencing in all observations is (surprisingly) a greater deviation at steeper elevation angles, which is in contrast to my expectations, since this results in far shorter absolute distances.

One example of my results is the following observation from Graz Lustbühel of the 22 June 2017 monitoring the GRACE A satellite.

 | standard Deviation 156.32185
 |  Time               | Ref Distance| Obs Distance| Deviation   | Elevation Angle
 | ------------------------------------------------------------------------------
 | 2017-06-22_13-57-47 | 573630.53   | 573510.16   | -124.37518  | 34.1093
 | 2017-06-22_13-57-51 | 550922.4    | 550796.16   | -130.05672  | 35.989845
 | 2017-06-22_13-57-56 | 528348.25   | 528215.69   | -136.19948  | 38.060675
 | 2017-06-22_13-58-00 | 507464.43   | 507325.52   | -142.39058  | 40.19135
 | 2017-06-22_13-58-06 | 484607.77   | 484461.27   | -149.80929  | 42.81356
 | 2017-06-22_13-58-11 | 465610.45   | 465457.04   | -156.57817  | 45.279134
 | 2017-06-22_13-58-16 | 449075.77   | 448915.81   | -162.99763  | 47.689646
 | 2017-06-22_13-58-20 | 435233      | 435067.11   | -168.8336   | 49.94278
 | 2017-06-22_13-58-25 | 424230.88   | 424059.89   | -173.84746  | 51.921769
 | 2017-06-22_13-58-30 | 415972.62   | 415797.47   | -177.94548  | 53.540971
 | 2017-06-22_13-58-35 | 411432.82   | 411255.13   | -180.45426  | 54.491375
 | 2017-06-22_13-58-40 | 409363.89   | 409184.61   | -182.0285   | 54.951053
 | 2017-06-22_13-58-46 | 411421.31   | 411242.31   | -181.76171  | 54.532429
 | 2017-06-22_13-58-50 | 416021.81   | 415844.36   | -180.23949  | 53.598194
 | 2017-06-22_13-58-56 | 427050      | 426876.64   | -176.23183  | 51.499173
 | 2017-06-22_13-58-59 | 433701.19   | 433530.31   | -173.80953  | 50.323281
 | 2017-06-22_13-59-06 | 452168.09   | 452003.85   | -167.28999  | 47.357755
 | 2017-06-22_13-59-11 | 468417.38   | 468258.68   | -161.8747   | 45.049964
 | 2017-06-22_13-59-15 | 487200.93   | 487048.25   | -155.9925   | 42.663639
 | 2017-06-22_13-59-21 | 509791.27   | 509645.32   | -149.43676  | 40.116813
 | 2017-06-22_13-59-26 | 532186.88   | 532047.1    | -143.43855  | 37.873845
 | 2017-06-22_13-59-30 | 555171.16   | 555037.25   | -137.7555   | 35.809979
 | 2017-06-22_13-59-35 | 580787.28   | 580659.4    | -131.91771  | 33.744171
 | 2017-06-22_13-59-40 | 607376.04   | 607253.94   | -126.36003  | 31.816812
 | 2017-06-22_14-00-11 | 792767.05   | 792675.37   | -97.569278  | 22.258777

Is there any phenomenon explaining this varying offsets?

Answer 1

A few things to consider:

1. Refraction- The atmosphere will bend the light somewhat, which might cause it to take longer.
2. Speed of light changes- Light moves a bit slower when going through more of the atmosphere
3. Timing accuracy- The position of the satellite might be off in time slightly, which would result in a deviation.
4. Positional accuracy- This results if you don't know the location of your source point exactly.
5. Floating point errors. Floating points are accurate to about 7 decimal digits. That means it is accurate to about a meter, give or take. Not likely to be the source of your error, but it could crop up in the time estimates.

To narrow this down a bit, try the following:

1. See if there is a correlation between the error and the line of sight angle. If the distance is most accurate when straight overhead, it could be a refraction or speed of light error.
2. Make sure your GPS location and time are accurate.
3. Try plotting various of your values vs others.
4. Determine the distance to a known source. I recommend using the Lunar laser reflectors on the moon. With these, you can get great results. If you are able to do this with satellites, consider using one of the GPS satellites, as their exact positions can easily be calculated.

Looking at your data, I found a few things of some interest.

1. The higher the elevation, the higher the error. This would indicate an overcorrection.
2. The relative error is highest at the highest angles.

Thinking about this, that indicates that when the overhead motion of the satellite is the most, the error is the most. Also when the Doppler change is the most, the error is the most. I rather strongly suspect the issue is some kind of a timing issue, where the exact position of an object isn't as well determined as you think it is. Try some of the tests I indicated (Moon and GPS satellites) if you can to further validate this.