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A large number of target board calibration ranges were obtained at low return rate (~10%). The range-distribution of these observations then represents the whole-system response at single-photon levels, and has previously been shown to be as expected a convolution of the responses of the SPAD, the Stanford counter and the distribution of the laser pulse. The calibration 'fiducial' value is then determined as usual from the raw data, by computing an iterated mean using a 2.5 sigma rejection level. The raw range data is then smoothed using Andrew Sinclair's Gaussian smoothing routine, to derive a smooth version of the system response. The process is illustrated in this system plot
The smooth system distribution can now be used to compute the expected range distribution for any satellite whose impulse response is known, for single-photon detection. The satellite impulses used here are expressed as a function of distance from the physical surface of the satellite. Then the crucial point is that provided the system response is convolved about its 'fiducial' value with the satellite response, the resulting distribution will be the expected distribution of calibrated range observations again with respect to the front surface of the satellite. In practice raw range observations from a given pass are filtered by forming residuals from a fitted orbit or arbitrary polynomial, and have zero mean. To compare the observations to the model the amplitude and range-offset can be adjusted, and in practice the amplitude of the model and the mean value of the range residuals are adjusted to give the best fit. The amount by which the mean value of the range residuals are changed with respect to their initial value of zero then gives the mean value of the range residuals with respect to the surface of the satellite. Subtraction of this mean from the radius of the satellite then gives the centre of mass correction that should be added to each range measurement. To illustrate this process, the smoothed system response was convolved about its mean value with the Lageos impulse function derived by Reinhart Neubert. Range-residuals from full-rate data from some fifty passes taken at Herstmonceux were used to form a histogram showing the distribution of range values. The model amplitude was then adjusted and the mean value of the range residuals altered from zero to give the best fit between the model and the data. This process is shown in this satellite plot
As can be seen, the data imply that the best-fit Centre-of-mass correction appropriate to the normal observing and processing methods at Herstmonceux is 240 mm. Work is continuing to make this fitting process give a simple estimate of goodness of fit, and lead to an error estimate for the CoM value. However, the method appears to be sensitive at about the one mm level. This is illustrated in some sense by the following plot, which shows the same data to which the 'standard' CoM value appropriate to high-energy MCP detection, namely 251 mm, has been applied. The plot clearly demonstrates that 251 is not appropriate for the Hx single-photon data
The same methods have been used to investigate CoM values for Ajisai, for which an impulse function developed by Toshi Otsubo has been used. Again, the range-distribution of full-rate data from some fifty Ajisai passes observed at single-photon level at Herstmonceux were compared with the model, formed by convolving the system response with Toshi's Ajisai impulse response. The appropriate CoM value was determined as for the Lageos data, and the results are shown in this satellite plot
There is currently no explanation for the presence of significant numbers of observations 'ahead' of the leading edge of the model, particularly evident in the Ajisai results, but the overall quality of fit of the model to the data does appear to 'tie down' the CoM value fairly well.
Several Team members have commented that the process of fitting the model to the distribution of range residuals should be equivalent to forming the mean directly from the model itself, and that conceptually this would be easier to understand. To check this, a 2.5-sigma iterated mean of the model distribution was formed. This is the essentially the same way that the mean of the residuals is determined operationally at Herstmonceux prior to the formation of normal points.
The mean value of the Lageos model distribution was found to be 58 mm, with sigma = 17mm. Thus the centre of mass correction determined from the model is 298-58 = 240, in agreement with the value found by fitting the model to the data residuals as described above. This result, coupled with the generally good agreement between the model distribution and the data residuals, leads to the conclusion that a value of 240 mm is close to the correct CoM value for the Herstmonceux Lageos data. The sigma value of 17mm from the model also agrees well with the single-shot precision of Lageos observations.
Also evident in the Lageos and Ajisai plots is the clipping that was applied to the raw data during the process to compute normal points. In both cases the models show that some 'real' data was removed from the tails of the distributions during this process. However, it does not matter at all what processing method was used, since the models developed here have been tailored to that process by using real calibration data and the derived system fiducial point. What does matter of course is that having derived appropriate CoM corrections in the way proposed here, that there are no subsequent changes to system procedures that would invalidate the results.
One interesting possibility to check the stability with time of the procedures carried out at a station is to look at pass-by-pass statistics which characterise the data distributions obtained from calibration and satellite ranging. A few years ago Andrew Sinclair recommended that Eurolas stations use his Gaussian smoothing routine on all their data to determine the distribution peak, mean, skewness and kurtosis. He has recently carried out an analysis of the results from Graz and Herstmonceux, and his summary of the results and further thoughts appear here
An example of a system change that would require a modification to the CoM value determined as proposed here would be if the return rate were significantly increased beyond the single-photon regime for which the models have been developed. The CSPAD in use at Herstmonceux has been tuned to give minimal time-walk as a function of return energy during calibration ranging, the tuning being a function of the laser pulse-length. It is therefore likely that there will be some residual time-walk in response to the 'stretched' return pulse from an extended laser array on a satellite. To test this expectation, an experimental pass of Lageos was observed at Herstmonceux during which, at carefully recorded intervals of time, return rates were allowed to increase to about 60%, equivalent to receipt of on average about 5 photons per laser shot. Calibration ranging was carried out at the standard single-photon level, and the observations processed in the normal way, except that range residuals for the high-rate observations were formed relative to an orbit fitted to the single-photon data in the pass. The mean value of the high-rate residuals was -2.8 mm, and the precision of the observations improved to about 12 mm, compared to the 15 mm normally achieved for Herstmonceux Lageos single-photon data. The high-rate observations were compared to the single-photon model, as shown in this satellite plot
The distribution of the observations is now much different than the single-photon model, being as expected much more 'Gaussian-like'. The quoted CoM value of 243 mm reflects the change of -3 mm in the residual mean value. The main point to note here is that the correction to the lageos high-rate range observations of -3mm would not normally be calibrated out, since for the CSPAD the range calibration value is near-constant for return rates of between zero and 100%.
More effort is required to model properly this effect, and to check that as expected the appropriate CoM correction will depend on return level. These energy dependent effects are treated in a very interesting theoretical study by Reinhart Neubert , in which he also develops a precise signature model for the TOPEX array, and discusses data clipping procedures.