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A method to calculate zero-signature satellite laser ranging normal points for millimeter geodesy - a case study with Ajisai
- Daniel Kucharski^{1}Email author,
- Georg Kirchner^{2},
- Toshimichi Otsubo^{3} and
- Franz Koidl^{2}
https://doi.org/10.1186/s40623-015-0204-4
© Kucharski et al.; licensee Springer. 2015
- Received: 13 October 2014
- Accepted: 6 February 2015
- Published: 1 March 2015
Abstract
High repetition-rate satellite laser ranging (SLR) offers new possibilities for the post-processing of the range measurements. We analyze 11 years of kHz SLR passes of the geodetic satellite Ajisai delivered by Graz SLR station (Austria) in order to improve the accuracy and precision of the principal SLR data product - normal points. The normal points are calculated by three different methods: 1) the range residuals accepted by the standard 2.5 sigma filter, 2) the range residuals accepted by the leading edge filter and 3) the range residuals given by the single corner cube reflector (CCR) panels of Ajisai.
A comparison of the statistical parameters of the obtained results shows that the selection of the range measurements from the leading edge of the SLR data distribution allows to minimize the satellite signature effect and to reduce the average single-shot RMS per normal point from 15.44 to 4.85 mm. The optical distance between the leading edge mean reflection point and the satellite’s center of mass is 1,023 mm, RMS = 1.7 mm.
Further, in addition, we utilize the complete attitude model of Ajisai during the post-processing which enables selection of the range measurements to the single CCR panels of the satellite and the formation of the normal points which most closely approximate the physical distance between the ground station and the center of mass of Ajisai. This method eliminates the satellite signature effect from the distribution of the post-fit range residuals and further improves the average single-shot RMS per normal point to 3.05 mm. The normal point RMS per pass is reduced from 2.97 to 0.06 mm - a value expected for a zero-signature satellite.
Keywords
- Satellite laser ranging
- Ajisai
- Satellite geodesy
- Zero-signature satellite
- Normal point
Background
Experimental geodetic satellite - Ajisai
Ajisai was launched with an initial spin period of 1.5 s (Hashimoto et al. 2012) and is currently the fastest spinning SLR satellite in orbit. The specific construction of the spherical body (laminated sheets of aluminum foil) minimizes the interaction with the Earth’s magnetic field and prevents loss of the rotational energy. The orientation of the spacecraft was set to be parallel to the Earth’s spin axis and is stabilized by the passive nutation dumper (Figure 2).
Satellite laser ranging
Satellite laser ranging (SLR) measures the distance to satellites equipped with retroreflectors. The laser pulses transmitted from the SLR station are reflected by the spacecraft back to the receiver telescope of the ground system. The time-of-flight measurements of the laser pulses are recalculated into an optical range to the satellite. The range measurements are used for precise orbit determination (POD), study of tectonic plate motion, as well as determination of Earth’s gravity field, Earth rotation parameters, and geocenter position (Moore and Wang 2003). The SLR data can be used for deriving the product GM of the universal gravitational constant G with the mass M of the Earth (Dunn et al. 1999).
During a pass of a satellite over an SLR station, the laser range measurements are collected. After the pass, the predicted range trend is fitted to the measured range values by adjusting the time bias and range bias estimations. As the next step, the range residuals are formed by calculating the difference between the observed and predicted (computed) range values, O-C. In the course of this process, fitting functions (orbital function or low degree polynomials) are used in order to remove the systematic trends from the distribution of the range residual data. The final step of the post-processing is the formation of the so-called normal points (NP).
Normal point formation with sigma filter
The development of the SLR technology and the use of the 10 to 15 Hz repetition rate lasers significantly increased the amount of the range measurements per pass compared to the early SLR systems. In order to reduce the computer time required for the orbital analysis of the large data sets, while retaining the information content of the individual range measurements, a data compression technique has been introduced to create normal points. At the 5th Laser Ranging Instrumentation Workshop held at Herstmonceux in 1984, a procedure was recommended for the formation of normal points from the full rate ranging data (Recommendation 84A: ‘On SLR normal-point generation and data exchange’). The calculated range residuals of the pass are clipped with an iterative sigma rejection filter. The retained data set is divided into short time slots (30 s for Ajisai); the arithmetic mean of the range residuals of each bin is calculated and used as the reference for the normal point.
In order to deliver an effective distance between the ground SLR system and the satellite’s center of mass (which represents its orbital motion), the normal points are corrected for the systematic errors such as the atmospheric refraction delay (Mendes and Pavlis 2004), the calibration system delay, and for the center-of-mass offset. The center-of-mass correction (CoM correction - Figure 3A) is the one-way distance to be added to a normal point at the precise orbit determination stage and can be determined from the prelaunch measurements or by the analysis of a theoretical satellite’s response function (Arnold 1979). The prelaunch tests give the center-of-mass correction for Ajisai of 1,010 mm (Sasaki and Hashimoto 1987). At the 9th International Laser Ranging Workshop in Canberra in 1994, papers were presented by J. J. Degnan and by R. Neubert which theoretically investigated the reflection signature of the LAGEOS-type satellites and calculated the correction from the mean reflection point to the center of mass of the spherical satellites.
Otsubo and Appleby (2003) analyzed the temporal spread of the optical pulse signals due to a reflection from the multiple onboard reflectors and modeled the response function of LAGEOS, Ajisai, and the Etalons. The obtained center-of-mass corrections depend on the ranging system configuration and observation policy at the terrestrial stations and vary by about 1 cm for LAGEOS and 5 cm for Ajisai and Etalons. Otsubo et al. (2000) demonstrated that the center-of-mass correction for Ajisai depends on the orientation of the satellite and can vary between 985 and 1,030 mm for a Gaussian profile of the incoming laser pulse with FWHM (full width at half maximum) of 200 ps.
The optical range measured by SLR is longer from the physical value due to the delay of a laser pulse passing through the glassy corner cube reflectors. The optical distance to a CCR refers to the optical reflection point and can be corrected by the range correction ΔR in order to represent the physical distance to the center of the CCR’s external surface. The range correction ΔR is defined as \( \varDelta R=L\sqrt{n^2-{ \sin}^2\theta } \), where L is the height of CCR (17.15 mm for Ajisai), n is the refractive index of the material (1.46 for fused silica and 532-nm wavelength) and θ is the incident angle between the laser vector (vector from the satellite’s center directed towards the ground SLR system) and the optical axis of a retroreflector (Fitzmaurice et al. 1977, Figure 3B).
In this paper, we propose a ‘reflector filter’ which allows us to calculate a physical distance to the single CCR panels of Ajisai. The physical range values are corrected by the center-of-mass vector (vector length along the laser range direction = R⋅cos(θ)) in order to determine the distance between the ground SLR system and the satellite’s center of mass (Figure 3A).
SLR measurements to Ajisai at Graz observatory
Since 8 October 2003, the Graz SLR station (Austria) operates with a 2-kHz repetition-rate Nd:Van laser of 10-ps FWHM pulse duration (Kirchner and Koidl 2004). The detection system is based on the Compensated Single Photoelectron Avalanche Detector (C-SPAD) (Kirchner and Koidl 1995) and responds to the first incoming photon. The energy-dependent range bias of C-SPAD is reduced to about 10 ps (time walk compensation), thus the measured range does not depend on the energy of the incoming signal.
The presented range residuals are determined during an iterative 2.5 sigma elimination process and give RMS of 8.81 mm. The statistical parameters of the range residual distribution can be obtained with the smoothing algorithm derived by Sinclair (1993). The distribution function is calculated with a strong smoothing coefficient of 15 mm which assures that all of the leading range residuals will be used during the process of normal point formation. The determined histogram function (Figure 5B) gives peak = −0.78 mm, full width at half maximum (FWHM) = 41.7 mm, and leading edge at half maximum (LEHM) = −21.2 mm; the determined leading edge corresponds to the front face of the spherical satellite.
Due to the fast spin of Ajisai, the distance between the nearest CCR panel and the satellite’s front face is quickly changing - this effect is represented as the V-shape distribution of the laser echoes on Figure 5C. The situation when the incident angle θ between the laser vector and the central axis of the panel (Figure 3B) is the smallest defines the minimum deviation of the observed CCR panel (MD). The epoch times of the minimum deviation events throughout the pass can be predicted by the complete attitude model of Ajisai. The spin axis orientation of the satellite (in J2000.0 celestial reference frame) during a single pass can be predicted with RMS of 0.128° (Kucharski et al. 2010b, 2013a). The spin period and the rotational phase angle of Ajisai can be obtained during the post-processing of a complete pass with RMS of 84 μs (42 ppm) for the spin period and with RMS of 0.48° for the phase angle (Kucharski et al. 2010a).
The range coordinate of the minimum deviation (MD) can be calculated as an arithmetic mean of the range residuals located around the predicted MD epoch (data bin width of 20 ms) and clipped by an iterative 2.2 sigma filter. In the example case (Figure 5C), the range coordinate of the MD event is calculated as the mean of 23 selected points (return rate of 57.5%) and is equal to −11.7 mm, RMS = 2.29 mm.
Methods
Normal point formation with leading edge filter
The standard normal point algorithm was prepared before the effects of the satellite signature were fully recognized. Since the position of the mean reflection point (its distance to the satellite’s center of mass) can vary depending on the optical response of the retroreflector array, the further theoretical investigation was done (Appleby 1993, 1995, Neubert 1995, Sinclair et al. 1995) to find the center-of-mass corrections for a variety of the reference points: mean, peak, leading edge at half maximum (LEHM) (Figure 5).
The large amount of data delivered by the high repetition-rate SLR system allows to find a stable leading edge level of the range residual distribution (Figure 5 - LEHM) and to use only the leading data points located between the peak and LEHM for the normal point formation. It was demonstrated by Kirchner et al. (2008) that selecting only the leading range measurements significantly improves the stability of the normal points.
In order to compare different post-processing methods presented in this paper, we modify the initial definition of the leading edge filter given by Kirchner et al. (2008). The initial algorithm defines the reference for the normal points as a polynomial function fitted to the 10% of the nearest range residuals. The normal points are formed from the range residuals which lay between the fixed and arbitrary set limits of −3 to 17 mm from the polynomial function. The modification of the procedure introduced here sets the acceptance levels at the peak and LEHM of the data distribution given by the smoothing algorithm (with the smoothing coefficient of 15 mm, Figure 5B). Thus, the mean reflection point calculated with the leading edge filter (Figure 5A: mean_{LE} level) is located on the leading slope of the data distribution.
Normal point formation with reflector filter
This process is applied to the predicted minimum deviation events if the incident angle θ between the laser beam vector and the central axis of a panel is ≤3°. This arbitrarily set limit is related to the complex structure of the Ajisai CCR panel (Figure 2 - right: CCR panel) and assures that during the minimum deviation event, the SLR system receives the laser echoes in equal proportions from the two sides of the panel. Identification of the laser echoes given by the single CCR panels allows to refer the particular range measurements to the satellite’s center of mass and eliminates the satellite signature effect from the normal points.
The zero-signature SLR data of Ajisai can be compared with the range residuals of the spherical glass retroreflector satellite BLITS (Vasiliev et al. 2007). The satellite represents the concept of a Luneburg lens and consists of an inner glass sphere covered by the concentric glass half-shells of which one is aluminum coated. BLITS acts as a single retroreflector and thus it is the zero-signature satellite (Kucharski et al. 2011a, b). The mission started on 17 September 2009 and ended after the possible collision with a space debris on 22 January 2013 (Parkhomenko et al. 2013).
Figure 6 presents the zero-signature range residuals of Ajisai (Figure 6A) and BLITS (Figure 6D) measured by Graz SLR system on 13 March 2014 and 24 January 2012, respectively. The selected time span of 210 s corresponds to the duration of seven normal points. The range residuals are processed by an iterative 2.5 sigma filter and give RMS of 2.8 mm for Ajisai and 2.67 mm for BLITS. The specific construction of BLITS makes the range measurements possible only when the transparent hemisphere of the lens is visible from the SLR station. Due to the spin of the satellite (Kucharski et al. 2013b), the laser beam of the SLR system is alternately pointing to the transparent and the coated hemisphere, thus series of measurement/no measurement intervals during a pass are visible (Figure 6D). Figure 6C presents the selected range residuals during one rotation of Ajisai (approximately 2.2 s).
The mean values of the parameters calculated from 2,731 passes of Ajisai and 450 passes of BLITS
Unit | Ajisai | BLITS | |
---|---|---|---|
Pass RMS of range residuals (after 2.5 sigma filtering) | mm | 3.04 ± 0.29 | 3.17 ± 0.31 |
Normal point RMS per pass | mm | 0.06 ± 0.02 | 0.09 ± 0.05 |
Single-shot RMS per normal point | mm | 3.05 ± 0.36 | 3.21 ± 0.32 |
Return rate per normal point | %, points | 1.6%, 960 | 5.8%, 3,480 |
The mean pass RMS of the range residuals of Ajisai and BLITS is at the level of 3 mm (obtained with 2.5 sigma filter) and coincides with the RMS of the SLR system calibration to the ground target of 2.25 mm (obtained with 2.2 sigma filter). The RMS of BLITS range residuals is slightly larger than the value of Ajisai; the complex structure of the spherical lens reflector (two kinds of glass with different refractive indices) can affect the shape of the retroreflected laser pulse. The symmetrical distribution of the range residuals around the 0 level (Figure 6A,D) gives a very low normal point RMS per pass: 0.06 mm for Ajisai and 0.09 mm for BLITS.
The low, average return rate of 1.6% for Ajisai normal points is caused by the 3° limit of the maximum incident angle θ between the laser beam vector and the central axis of a CCR panel. The average number of the range measurements per normal point can be increased from the present 960 by widening the θ incident angle limit or by increasing the repetition rate of the laser from present 2 to 10 kHz (while maintaining the return rate at the high level).
Results
The minimum deviation level (MD on Figure 5C) indicates the position of the specific CCR panel, thus it can be used as the reference to calculate the distance between the satellite’s center of mass and the significant levels of the range residuals distribution: peak, LEHM, NP_{S} (refers to mean on Figure 5), and NP_{LE} (refers to mean_{LE} on Figure 5).
Distance between the significant levels of the range residual distribution (Figure 5 ) and Ajisai center of mass
The optical distance to the satellite’s center of mass presented in Table 2 is shorter than the physical distance by the range correction value (ΔR = 25.04 mm) calculated for the incident angle θ = 0° (Figure 3B). The optical distance between the mean (reference level for NP_{S}) and CoM is 1,012 mm and coincides with the standard center of mass correction of 1,010 mm (Otsubo and Appleby 2003). The RMS of the mean_{LE}-CoM distance (1.7 mm) is significantly lower than the RMS of the mean-CoM (4.7 mm), thus the mean_{LE} can serve as a more stable reference level for the normal points.
Discussion
The leading edge post-processing method reduces the satellite signature effect and gives a more stable reference level (mean_{LE}) for the normal points. This method is based on the standard routines and can be easily implemented in the on-site post-processing software. Graz SLR station uses the leading edge filter to compute the normal points of Ajisai since 5 March 2008 and of Etalon-1, Etalon-2, LAGEOS-1, and LAGEOS-2 since 5 February 2008. It should be noted that the use of the mean_{LE} as the reference for the normal points must be followed by a proper center-of-mass correction in the course of the POD analysis.
The reflector filter eliminates the satellite signature effect from the distribution of the post-fit range residuals and enables the computation of the normal points which represent the physical distance to the satellite’s center of mass at the ultimate accuracy level.
It is important to note that approximately 10 mm offset error in the laser range measurements can cause a few ppb error in the determination both of the value of GM and in the scale of the terrestrial reference frame. The zero-signature normal points of the geodetic satellites delivered by the globally distributed SLR systems (Pearlman et al. 2002) could improve the accuracy of the precise orbit determination and help to achieve the GGOS geodetic reference frame requirements (Plag and Pearlman 2009).
Conclusions
The high repetition-rate SLR technology offers new possibilities for the post-processing of the range observations. The standard clipping process can be improved by the leading edge filter described here which minimizes the satellite signature effect and reduces the average single-shot RMS per normal point from 15.44 to 4.85 mm (Figure 7). The stable, optical distance between the leading edge mean reflection point (Figure 5: mean_{LE}) and the center of mass of Ajisai is 1,023 mm, RMS = 1.7 mm.
High repetition-rate satellite laser ranging efficiently measures the attitude of fully passive, geodetic satellites during day and night. The attitude model of Ajisai applied during post-processing allows to select the range measurements to the single CCR panels and to form normal points which are accurately referenced to the physical distance between the ground station and the satellite’s center of mass. This process eliminates completely the satellite signature effect from the distribution of the post-fit range residuals and further improves the average single-shot RMS per normal point to 3.05 mm (Figure 7). The normal point RMS per pass is reduced from 2.97 to 0.06 mm - a value expected for a zero-signature satellite.
Declarations
Authors’ Affiliations
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