Analysis of some statistics of SLR data from RGO and Graz

1. Preamble

The SLR stations RGO and Graz have been routinely forming and archiving

statistics of their SLR tracking data for some time. Among these

statistics are the quantities Peak-Mean, Skewness and Kurtosis. In

this note we analyse these in order to try decide whether it would

be of value to recommend that the wider SLR network should also form

and archive these data. The RGO data analysed span about 790 days up

to mid May, 1999, and for Graz the data span about 130 days also up

to mid May.

If SLR data residuals were distributed symetrically then the peak would

coincide with the mean of the data. However it is now well recognised that

SLR data are not in general symetrically distributed. The effects of

satellite signature and some properties of first-photon detectors both

cause a skewness towards long ranges. This skewness is particulary well

seen by systems operating at low photon return rates. It is generally

agreed that the data should be clipped to remove the extremities of this

skewness, but there is not a concensus nor any good theory of what

exactly this clipping level should be. The clipping is carried out

relative to a 'Mean' of the data, and for tight clipping this Mean will

move close to the Peak. For Graz the clipping is at 2.2*rms from the

Mean. For RGO the Mean is actually achieved by a process of fitting a

Gaussian profile to the data, but extended tests have shown that this Mean

is very close to a Mean achieved by using a 2.5*rms clipping.

2. Results

This note gives a table of mean values of Peak-Mean, Skewness and

Kurtosis, plus the rms of these quantities from their mean, and the formal

standard error of the mean. The data in the table include a selection of

satellites and also calibration terrestrial ranging. Also attached are

plots of the data for Lageos-2 for both Herstmonceux

and Graz

It is seen from the plots and from the smallness of the formal standard

errors of the mean values that both stations have maintained a highly

mutually consistent behaviour over the period of the data. It is also

seen for the rms values that the scatter of the RGO data is considerably

greater than that of the Graz data, typically a factor of 2 to 3 larger

for Peak-Mean and Kurtosis, and a factor of 5 or more for Skewness. A

major cause of this must be the looser clipping of the RGO data, which

results in retaining more of the variable skew tail. The values of Kurtosis

are all close to 2.6 for RGO, and 2.3 for Graz, which is indicative of the

looser clipping of the RGO data.

For the Lageos satellites the RGO value of Peak-Mean is about -5.3 mm,

whereas the Graz value is about -2.2 mm. It would be nice to be able

to interpret this as a relative bias of the two stations for Lageos

ranging, as this is the sort of information that analysts are beginning

to determine from orbital analysis, and for which they would like

independent confirmation from engineering considerations. However in

order to be confident of this deduction it is necessary to be sure that

the Peaks of the RGO and Graz data actually refer to a mean reflection

point at the same distance from the CoM of the satellite, or at least to

know the relationship of the mean reflection points if they are different,

and to deduce this theoretically requires additional information of the

typical received energy level and the laser pulse width.

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Summary of statistics of RGO and Graz SLR data

for terrestrial ranging and satellite tracking.

HERS: 790 days up to 1999 May 15

Graz: 130 days up to 1999 May 15