System-dependent center-of-mass correction for spherical geodetic satellites

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B4, 2201, doi: /2002jb002209, 2003 System-dependent center-of-mass correction for spherical geodetic satellites Toshimichi Otsubo Communications Research Laboratory, Japan Graham M. Appleby NERC Space Geodesy Facility, United Kingdom Received 17 September 2002; revised 2 January 2003; accepted 28 January 2003; published 17 April [1] The temporal spread of optical pulse signals due to reflection from multiple onboard reflectors is now a critical problem in satellite laser ranging. The full rate residual profile of single-photon laser ranging can be used to model the response function of geodetic satellites, resolving the uncertainty of far-field diffraction. We constructed the response function model for three types of laser ranging targets already in Earth orbit, the LAGEOS, AJISAI, and ETALON satellites. The center-of-mass correction depends on the ranging system and observation policy at terrestrial stations and varies about 1 cm for LAGEOS and 5 cm for AJISAI and ETALON. INDEX TERMS: 1241 Geodesy and Gravity: Satellite orbits; 1294 Geodesy and Gravity: Instruments and techniques; 1247 Geodesy and Gravity: Terrestrial reference systems; 1214 Geodesy and Gravity: Geopotential theory and determination; KEYWORDS: satellite laser ranging, satellite signature effect, Earth s scale Citation: Otsubo, T., and G. M. Appleby, System-dependent center-of-mass correction for spherical geodetic satellites, J. Geophys. Res., 108(B4), 2201, doi: /2002jb002209, Introduction [2] Since laser tracking was first demonstrated in the 1960s, the measurement precision of satellite laser ranging has improved remarkably. State-of-the-art systems can now measure distances to satellites with a precision of 3 to 8 mm on a single-shot basis, equivalent to a precision of better than 1 mm on a normal point basis. [3] Satellite laser ranging, as an optical measurement technique, benefits from the well-modeled correction of signals propagating through the ionosphere and troposphere, compared with other geodetic technologies based on microwave radiation. Despite its relatively sparse data productivity, satellite laser ranging, especially to spherical geodetic satellites, is the primary means for determining the origin and scale of a terrestrial reference frame (i.e., the center of gravity of the Earth and the scale of the Earth, respectively) [Altamimi, 2001] (more details available at It is also useful for deriving the product GM of the universal gravitational constant G with the mass M of the Earth. Dunn et al. [1999] determined GM at an internal repeatability of 0.3 to 0.4 ppb using laser ranging data to the two LAGEOS satellites and concluded that the availability of more accurate center-of-mass corrections and a more accurate tropospheric delay correction would make it possible to determine GM even more precisely. Further, the analysis of laser ranging data to various other satellites should confirm the accuracy of the estimated GM value. Copyright 2003 by the American Geophysical Union /03/2002JB002209$09.00 [4] Thus along with the system noise and potential errors in the current tropospheric delay model, the spread of retroreflection due to multiple reflectors on satellites is now recognized as a key error factor. This satellite signature effect is illustrated in Figure 1. [5] This effect was theoretically predicted when the first LAGEOS satellite was launched in the 1970s [Fitzmaurice et al., 1974] and also when the TOPEX/Poseidon satellite was designed to carry a ring-shaped reflector array [Schwartz, 1990]. It was initially identified in actual laser ranging data in the early 1990s; the distribution of the residuals from full-rate data taken at the Herstmonceux station in the United Kingdom was found to depend on the satellite [Appleby, 1992]. [6] The center-of-mass correction is the one-way distance to be added to the observed range, so that one can obtain the point-to-point distance between the telescope reference point of a terrestrial station and the center of mass of a satellite. Historically, for spherical geodetic satellites, a constant value has been used for each satellite regardless of the laser ranging system and the detection energy level. Through prelaunch analyses and measurements, the centerof-mass corrections were derived as 251 mm for the LAGEOS-2 satellite [Minott et al., 1993], 1010 mm for the AJISAI satellite (also known as EGP) [Sasaki and Hashimoto, 1987], and 576 mm for the two ETALON satellites, ETALON-1 and ETALON-2 [Mironov et al., 1992]. These values have become standard and have been uniformly applied by analysts to all laser systems. Note that 251 mm is now widely used for the center-of-mass correction for the LAGEOS-1 satellite as well as for its twin, LAGEOS-2. ETG 9-1

2 ETG 9-2 OTSUBO AND APPLEBY: SATELLITE CENTER-OF-MASS CORRECTION Figure 1. Satellite signature effect for a spherical satellite. The retroreflected pulse becomes broader than the transmitted pulse. [7] However, to utilize the improvements in measurement precision achieved in the 1990s, we can no longer afford to ignore the system dependence of the center-of-mass corrections. Intuitively, it is clear that there will be a difference in the effective distance to a spherical satellite determined by systems working at very low and very high numbers of return photons; the system working at high numbers of photons will tend to measure a distance that is shorter on average than that measured by a low energy system, as photons in the leading edge of the return pulse are preferentially detected. Provided that calibration ranging is carried out at the same energy levels as obtained during satellite ranging, the effect will be reduced, but this study will show that this practice does not fully compensate for the effect. Thus, in general, to refer a range measurement to the center of mass of the satellite, a larger correction must be applied to the data from a high energy system than to that working at low numbers of return photons. For example, an apparent +26-mm range bias was detected for single-photon laser ranging to the AJISAI satellite [Otsubo et al., 1999] when the standard center-of-mass correction was applied during the orbit determination process. This result is a clear demonstration that a smaller center-of-mass correction is more appropriate for single-photon systems. We will show in this paper that even the LAGEOS satellites can no longer be considered small targets and that a single standard center-of-mass correction should not be expected to be applicable at mm-levels of precision for all types of laser ranging stations. It is important to note that a 1-cm offset error in laser ranging data can cause a few ppb error in the determination both of the value of GM and in the scale of the terrestrial reference frame. [8] In this paper we investigate the optical response functions of spherical satellites to determine how much error we should expect from the satellite signature effect, taking into account both the satellite and the laser ranging system. In section 2, based on an optical response model of individual reflectors, we use the postfit full rate residual profile of single-photon data to empirically determine the width of the target response functions of the LAGEOS, AJISAI, and ETALON satellites. Single-photon data are especially suited to such studies, since during the course of many range measurements the entire response function of the satellite is sampled. In section 3, we describe our derivation of values of, or realistic limits for, the centerof-mass corrections of the three satellites for typical laser ranging systems. Note that in this paper, by photon we mean the state of a photoelectron after traveling through a detector. 2. Target Response Function 2.1. Response From a Single Reflector [9] The response of a single cube corner reflector is a function of the apparent area of the front face and the reflectance of the reflector itself. We analyzed these two contributing effects. [10] We used three types of geodetic satellites in our study. Their specifications are listed in Table 1. Their diameters range from 0.60 m (LAGEOS) to 2.15 m (AJI- SAI). Most of their cube corner reflectors were made of fused silica, with refraction indices of approximately 1.46 for the widely used 532-nm green light (the second harmonic of Nd:YAG lasers). A very small number of them were made of germanium for infrared laser ranging, which is not addressed in this paper. [11] The dimensions and shapes of the reflectors differ, as shown in Figure 2. The aperture of the AJISAI reflector is a triangle with cut vertices. That of ETALON is hexagonal, and that of LAGEOS is circular. It should be emphasized that the back faces of the ETALON reflectors are coated with aluminum, while those of the AJISAI and LAGEOS reflectors are not. The surface of each of the LAGEOS reflectors is recessed 1 mm below the surface of the reflector holder, so the front face can be partly shadowed. The holders of the AJISAI reflectors extend 2 mm above the level of the reflectors, therefore also partly obstructing the front face. Shadowing does not occur with the ETALON reflectors at any achievable angles of incidence because the side edges of the holders are cut obliquely. [12] This variety in the reflector properties results in different effective reflection areas. The area is a function of the viewing angle and is calculated as the overlapped area of the input and output apertures [Arnold, 1979]. After correcting for the shadowing effect, we numerically calculated a two-dimensional map of the effective reflection area for the three types of reflectors. Figure 3 displays the results as a function of the viewing angle (angles of azimuth and of incidence). Table 1. Specifications of Satellites LAGEOS-1/LAGEOS-2 AJISAI ETALON-1/ETALON-2 Launch 4 May 1976/22 Oct Aug Jan. 1989/31 May 1989 Mass, kg 411/ Approximate diameter, m Number of reflectors Material of reflectors fused silica (422) and germanium (4) fused silica fused silica (2140) and germanium (6)

3 OTSUBO AND APPLEBY: SATELLITE CENTER-OF-MASS CORRECTION ETG 9-3 Figure 2. Dimensions of cube corner reflectors on LAGEOS, AJISAI, and ETALON satellites. Unit is mm. [13] Reflectance, on the other hand, depends largely on whether the back faces of the reflectors are coated. It is modeled as a double refraction at the front face and triple reflection at the back face, where we have assumed unpolarized light in our study. For the uncoated reflectors of LAGEOS and AJISAI, most of the retroreflection is obtained as a result of the triple total reflection at the back face. As illustrated in the upper map in Figure 4, this causes a strong azimuthal pattern of the reflector s reflectance every 120 when the angle of incidence is wider than 17. By contrast, the aluminum-coated ETALON reflectors have a much wider acceptance angle with no azimuthal dependence. The resulting reflectance pattern is as shown in the lower map in Figure 4. [14] Given effective reflection area a and reflectance e as a function of the viewing angle, we can construct a function of the strength of the retroreflection. If we neglect the farfield diffraction of the reflectors (i.e., we assume they uniformly illuminate the footprint on the Earth), the strength of the retroreflection can be modeled as ðstrengthþ / ae: ð1þ In previous studies [Neubert, 1994; Otsubo et al., 1999] (some minor corrections to Neubert are available at it has been modeled as ðstrengthþ / a 2 e; ð2þ Figure 3. Effective reflection area of cube corner reflectors on LAGEOS, AJISAI, and ETALON satellites as a function of the viewing angle (angles of azimuth and of incidence). Normalized to be 1.0 when the angle of incidence is zero.

4 ETG 9-4 OTSUBO AND APPLEBY: SATELLITE CENTER-OF-MASS CORRECTION model 2 is narrower than that given by model 1. In other words, with model 2, the nearer reflectors give a more concentrated return. [17] To construct a function of the retroreflection intensity for an entire satellite using knowledge of the three-dimensional location of each of its reflectors, we need the time relations between the individual reflectors and the satellite s center of mass. The relative time delay, R i,oftheith reflector was calculated using this formula of Minott et al. [1993]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rðq i Þ ¼ R cos q i L n 2 sin 2 q i ; ð3þ where q i is the angle of incidence to the ith reflector s surface, R is the radius from the satellite center to the reflector s surface, and L is the distance from the reflector s vertex to the front face. [18] We can model the retroreflected pulse from an entire satellite by summing up the return intensities from all the reflectors, applying the delay given in equation (3). We do not consider the interference between the return signals from the different reflectors. Figure 4. Reflectance of cube corner reflectors on LAGEOS, AJISAI, and ETALON satellites as a function of the viewing angle (angles of azimuth and of incidence). implicitly assuming no relative motion between the satellite and terrestrial station and that the far-field diffraction pattern was projected symmetrically around the line of sight. Neither of these two models is theoretically correct, of course. [15] Note that previous studies [Minott et al., 1993; Sasaki and Hashimoto, 1987] had assumed a too narrow value for the acceptance angle of the reflectors. For instance, the LAGEOS-2 satellite was illuminated in the laboratory by laser only within a cone angle of 25, which is obviously well inside the effective acceptance angle in some azimuths (see Figures 3 and 4) Response From Entire Satellite [16] Let us now consider the response from an entire satellite. Model 2 (equation (2)) indicates that a reflector with a large effective reflection area gives stronger returns than with model 1 (equation (1)) and clearly the reflectors nearest the observer have the largest effective reflection area. Therefore the response of the entire satellite given by 2.3. Empirical Scaling [19] The remaining question is how we should model the return intensity, taking into account the diffraction effect, using models 1 and 2 or something different. The approach we will adopt is to compare the models with single-photon ranging data. [20] After computing the satellite response function as described above, we can convolve it with a function representing the response of the ranging system. When Otsubo et al. [1999] did this, they found that the approximation of equation (2) did not fit very well with the full rate residual histogram of the Herstmonceux single-photon station. The model was narrower than the actual residual histogram; this led us reconsider the basic assumptions of the model. [21] Because of velocity aberration, the far-field diffraction pattern does not fall symmetrically along the line of sight. The deviation amounts to 50 mrad (10 arc sec) at maximum [Degnan, 1993]. The cube corner reflectors on geodetic satellites are sometimes spoiled such that the angles between the reflective faces (dihedral angles) deviate slightly from 90 so that the diffraction effect partly compensates for the velocity aberration. For example, the dihedral angle of the LAGEOS-2 reflectors is 90 and 1.25 arc sec with a manufacturing error of ±0.5 arc sec [Minott et al., 1993]. The AJISAI reflectors were not intentionally spoiled, but the manufacturing error amounts to 2 arc sec. The degree or otherwise of spoiling is not known for the ETALON satellites. Thus for LAGEOS and AJISAI it would be possible to simulate the far-field diffraction pattern using the available information, but the manufacturing error prevents precise numerical computation. Moreover, we do not know the extent to which thermal deformation of a reflector disturbs the diffraction pattern. As a result, little is known about the actual diffraction effect for orbiting satellites. [22] Instead, we devised an empirical method for determining the profile of the response function without pre-

5 OTSUBO AND APPLEBY: SATELLITE CENTER-OF-MASS CORRECTION ETG 9-5 cisely calculating the far-field diffraction pattern. Rather than choosing either equation (1) or (2), we find the best fit value of n in: ðstrengthþ / a n e: ð4þ We found the full rate residual histogram of a single photon system useful for recovering the average response function from a satellite. Under ideal conditions with no system noise, the residual histogram made from a sufficient amount of full rate data would agree perfectly with the average response function. However, in actual data sets, the full rate residual histogram may be considered to be a convolution of the satellite response function and the system response. [23] We began our search for best fit n in equation (4) by collecting full rate residual histograms obtained at the NERC Herstmonceux station, United Kingdom, which adheres to a single-photon detection policy and uses a SPAD detector [Appleby et al., 1999]. We used a residual histogram of the small array ERS-2 satellite ranging data (Figure 5) as the system response. Including an error source due to transmission of the laser pulse through the atmosphere, this is a more realistic base to be convolved with the satellite response, than a residual histogram of terrestrial target ranging data. Because of the characteristics of a single-photon avalanche diode (SPAD), the residual histogram is skewed with a long tail that is cut significantly in the conventional data reduction process. However, for our use of this data as the system response profile, the full ERS- 2 residual profile was used unedited to represent the full system noise. [24] The response functions of the three satellites convolved with the ERS-2-based system response (Figure 5) were generated for n = 0.9 to 2.1 with a step size of 0.1. Then, the resulting distributions were compared with the Herstmonceux full rate residual histograms for LAGEOS, AJISAI, and ETALON. The residual histograms were constructed from data collected for a few tens of passes during June July 2000 and September October 2001 at the Herstmonceux station. Figure 6 shows those for the September October 2001 data. We used data already filtered on site (see section 3.1 for details), unlike the ERS-2 data. [25] Six selected cases for AJISAI are shown in Figure 7 for comparison. Two parameters, the vertical scale and horizontal offset, were adjusted to fit a convolved curve to a residual histogram. The fit with n = 2.0 that had been adopted in the previous studies [Neubert, 1994; Otsubo et al., 1999] is clearly not very good; the response function model is too narrow. The fit is better with n = 1.0. Similar results were obtained for LAGEOS and ETALON. We then used the differences between the convolved functions and the Herstmonceux residuals to estimate the best fit n. For range data obtained during June July 2000, the best fits were obtained for n = 1.2 for LAGEOS, 1.1 for AJISAI, and 1.3 for ETALON. For September October 2001 with more data, the results were 1.1 for LAGEOS, 1.2 for AJISAI (the Figure 7 case), and 1.3 for ETALON. A c 2 test applied for each fit indicated that two or three value of n including the best fit value agreed with the data at a 5% level of significance. For example, in the case shown in Figure 7, the convolved response functions with n = 1.2 and 1.3 both passed the test. Figure 5. Unedited residual histogram of ERS-2 ranging data tracked from Herstmonceux station during September October The mean after the on-site filtering is defined as zero. [26] By way of an independent confirmation of some of these results, we also obtained the full rate data of ETALON and CHAMP (the small laser array on CHAMP makes it another ideal target for determining system response) from the Graz laser station, Austria. The return energy in their CHAMP (low orbit) ranging was kept at a low level especially for this study, and most of the return from the high-orbiting ETALON was likely to be at single-photon levels. Nevertheless, as Graz does not strictly keep at a single-photon level of return, we additionally rejected data whenever the return rate was higher than 10% in order to treat only the single-photon data. Finally, applying the same procedure as for the Herstmonceux case, we obtained a best fit n of 1.3, in good agreement with the Herstmonceux results for ETALON. [27] Given these consistent results, we fixed the value of parameter n: nðlageosþ ¼ 1:1 nðajisaiþ ¼ 1:2 nðetalonþ ¼ 1:3: The resulting response functions for the three satellites using these values of n and convolved with negligibly narrow 1-ps full width half maximum (FWHM) Gaussian distributions to represent system noise are shown in Figure 8. The empirically adjusted values of n for each satellite were used to compute them and also shown for comparison are the functions obtained when n was 1.0 and 2.0. When these extreme values are used, there is clearly a significant variation in the response functions. However, it is clear that the realistic uncertainty of 0.1 in our determination of n would cause only a slight change in the functions and have only marginal effect on the computed center-of-mass corrections. [28] These results suggest that the diffraction effect must be taken into consideration to some extent because the values of n were all larger than 1.0, but they also suggest that the effect is much smaller than we had assumed in previous studies because the values of n are also significantly smaller than 2.0.

6 ETG 9-6 OTSUBO AND APPLEBY: SATELLITE CENTER-OF-MASS CORRECTION 3.1. Single-Photon System [31] Only the Herstmonceux station currently has a policy of always maintaining single-photon detection for all satellites and terrestrial targets. The return rate is maintained below 15% so that only one photon at most is detected by the SPAD detector. Although this causes a reduction in the number of range measurements that are obtained and more scatter in the full rate data, the mean representing the normal point data is considered to be free from a systematic offset bias for all types of targets. There is a very small possibility that more than one photon is detected at the detector even with the return rate control, but this does not affect the overall profile of the residual distribution [Otsubo et al., 2001]. Figure 6. Residual histogram of ranging data to LA- GEOS, AJISAI, and ETALON satellites tracked from Herstmonceux station during September October The data were filtered on site. [29] Although AJISAI (2.15 m in diameter) is much larger than the ETALON satellites (1.294 m), the temporal spread in the reflected pulses is not very different. This is because the acceptance angle of the ETALON reflectors is much wider owing to their coated back faces (Figure 4). 3. Center-of-Mass Correction [30] We next derive the values of, or set realistic limits for, the center-of-mass corrections of the three satellites for three laser ranging systems typically used at present: a single-photon system, a C-SPAD system, and a photomultiplier system. We use the response functions shown by the thick curves in Figure 8 based on the values we obtained for n in each case. Figure 7. Search for best fit parameter n for response function of AJISAI using September October 2001 data; n was incremented from (top) 1.0 to (bottom) 2.0.

7 OTSUBO AND APPLEBY: SATELLITE CENTER-OF-MASS CORRECTION ETG 9-7 Figure 8. Response functions of LAGEOS, AJISAI, and ETALON satellites based on empirically adjusted value of n. [32] If there were no noise from a laser ranging system, the distribution of the range residuals would agree closely with the satellite response function as determined above. On this assumption, the center-of-mass correction would simply be the centroid of the response function. The corrections based on this assumption for the three satellites are listed in Table 2. [33] In the actual data processing, however, there are a number of outliers, and they have to be rejected. We assumed a simple iteratively applied filter; eliminating data points that deviate from the mean by more than the root mean squares (RMS) N (in most stations N = 3.0 or 2.5). Use of this filter assumes that the data points have a Gaussian distribution, whereas the satellite response function is significantly skewed. This discrepancy causes the filter to reject part of the tail, to an extent defined by N. After such iterative filtering, the centroid of the data set will be closer to the leading edge, meaning that a larger centerof-mass correction must be applied (Table 2). [34] We also have to consider the profile of the system response. As shown in the previous section, the full rate residual histogram even for terrestrial targets and small reflector array satellites is skewed with a long tail due to the responses of the SPAD detectors themselves. The tail is largely eliminated by the iterative N RMS filtering, but it is, to some extent, still included in the actual residual histogram when it comes to a large satellite whose response function is wider than the system response. This results in a slightly smaller center-of-mass correction than the ideal one from a symmetric system response. Using, as in the previous section, the ERS-2 ranging data obtained in September October 2001 to represent the system response, we generated a convolved function from which the center-ofmass correction for the Herstmonceux station is determined as the centroid after filtering. We used the same rejection procedure as is carried out by the Herstmonceux on-site data processing, in which data points beyond the 3.0 RMS region of the best fit Gaussian curve are iteratively rejected. For such a skewed distribution this filter is different from and actually tighter than the simple 3.0 RMS rejection. The center-of-mass correction values determined for Herstmonceux are also given in Table 2. [35] We note here that the newly developed SLR2000 prototype, with an eye-safe low-energy laser system [Degnan, 1999], is also very likely to operate at the single-photon level C-SPAD System [36] Since the first successful laser ranging with an avalanche photo diode at the Graz station, it has been used at a number of laser ranging stations thanks to its fast response and high sensitivity. Apart from the Herstmonceux station, the stations using it do not in general control the return energy at the single-photon level; they allow multiple photons to reach the detector. [37] However, a SPAD detector, a kind of avalanche photo diode, introduces time walk effects as a function of the return energy [Kirchner et al., 1998] due both to the energy-dependent transit time variation within the detector (electronic effect) and because with increasing numbers of photons reaching the detector, a photon near the leading edge will tend to be preferentially detected (optical effect). In total the effect can reach up to several hundred picoseconds, equivalent to a range bias of a few centimeters. Ways to compensate the time walk effect have been actively researched since the mid-1990s, utilizing the observed Table 2. Center-of-Mass Correction for Single-Photon Systems a LAGEOS AJISAI ETALON Standard No system noise (1-ps FWHM) No clipping Iterative 3.0-RMS clipping Iterative 2.5-RMS clipping Iterative 2.0-RMS clipping Herstmonceux noise profile (ERS-2) Gaussian fit and iterative 3.0-RMS clipping a In millimeters.

8 ETG 9-8 OTSUBO AND APPLEBY: SATELLITE CENTER-OF-MASS CORRECTION Table 3. Center-of-Mass Correction for C-SPAD Systems a LAGEOS AJISAI ETALON Standard ps FWHM pulse width Iterative 3.0-RMS clipping 245/249/256/ /990/1020/ /573/608/613 Iterative 2.5-RMS clipping 247/250/256/ /996/1020/ /581/609/613 Iterative 2.0-RMS clipping 250/251/256/ /1004/1021/ /592/610/ ps FWHM pulse width Iterative 3.0-RMS clipping 245/247/251/ /989/1012/ /571/600/604 Iterative 2.5-RMS clipping 246/248/251/ /995/1013/ /578/600/604 Iterative 2.0-RMS clipping 249/249/251/ /1002/1013/ /590/601/604 a In millimeters. Four numbers separated by slashes are center-of-mass corrections for average number of detected photons of 0.1, 1, 10, and 100. correlation between input energy and the risetime of the SPAD output. As a result of these technological improvements, the intrinsic, energy-dependent range bias can be reduced to as little as 10 ps, or only 1-2 mm. This system is called a compensated-spad or simply C-SPAD [Kirchner et al., 1998]. [38] With this system, actual laser ranging to a terrestrial target (a point source target) could be used to generate and evaluate the time walk compensation profile. However, as discussed, since the SPAD detector responds to the first photon, the detection timing depends both on the laser pulse width and on the return energy. Thus the compensation profile generated by point source terrestrial ranging does not fully represent the optical portion of the time walk when ranging to the extended array on a real satellite. [39] Therefore, in order to determine this effect for the three satellites in this study, we have to carry out a numerical simulation of the situation using our satellite response functions. Taking a wide range of average numbers of photons, we numerically simulated the first photon s arrival time, both for terrestrial and satellite ranging, noting that the expected value of the first photon s arrival time can be determined statistically as a function of the average number of photons using Poission statistics [Neubert, 1994]. Then, by subtracting the first photon s arrival time for terrestrial ranging from that for satellite ranging, on the assumption that for a C-SPAD the terrestrial ranging results would not show a significant time walk, the center-of-mass corrections for the three satellites were calculated for a wide range of average photons assuming 1- and 100-ps FWHM laser pulse widths and no other noise sources. This computation was done for 10,267 angles of incidence with about a 2 two-dimensional interval. The average center-of-mass corrections derived in this way are listed in Table 3. It is obvious from the table that the intensity dependence cannot be fully compensated when carrying out satellite ranging using a C-SPAD, a stronger return increasing the center-ofmass correction. For the LAGEOS satellites, the effect is more than 1 cm, while for AJISAI and ETALON it reaches 5 cm or more. The largest jumps in correction values are seen between 1 and 10 photons in all cases. The values are almost constant below 0.1 photons and above 100 photons. Note that the single-photon detection discussed in section 3.1. corresponds to about 0.1 photons on average. [40] The filtering criteria (N RMS) affect the center-ofmass correction when the energy is low, but they do not significantly alter the correction when the energy is higher than 10 photons because with increasing energy the distribution tends to be more symmetric. Comparing the 1- and 100-ps cases, the spread of system noise reduces the maximum center-of-mass correction but barely changes the minimum one. In the extreme case of very large system noise, the center-of-mass correction reduces to the unedited centroid of the response function. [41] It should be noted that the standard corrections for LAGEOS and AJISAI agree well with the cases of highenergy return while for ETALON the standard value is close to that appropriate for a relatively low energy return. This fact somehow reflects reality; we cannot expect many photons from the high-orbiting ETALON satellites Photomultiplier System [42] Photomultiplier tubes (PMTs) have been used as a standard detector for laser ranging for decades. NASA stations as well as several other stations are currently using ones with microchannel plates (MCP). Although these systems are potentially sensitive to single-photon return, the threshold is usually set at some multiple-photon level. Their output signal is usually directed to a constant fraction discriminator to minimize the effect of signal strength variations. The response times of these electronic components, a few hundred ps for an MCP-PMT and probably slower inside a constant fraction discriminator, are typically slower than the pulse width of a modern laser, and most of the signature of the satellite is lost by convolution of the return pulse with the slow system response. In addition, the system components and configurations vary between stations, and it is difficult to measure what is going on inside specific devices. Therefore the development of an accurate model for all PMT systems is not a realistic proposition. [43] We instead applied a simple model in which the half maximum at the leading edge represents the detection timing. Various laser pulse widths, in the range from 1 ps to 3 ns, which we assumed includes the contribution of all the other pulse-broadening sources, were used to simulate the optical and electronic responses from the satellites. The responses were calculated for each of the 10,267 angles of incidence, and the half-maximum point was determined for each case. We also had to simulate the terrestrial ranging calibration in which the same halfmaximum detection was applied, finally defining the center-of-mass correction as the average half-maximum point for the satellite return less the half-maximum point from the terrestrial ranging simulation. Table 4 shows the center-of-mass corrections for the LAGEOS, AJISAI, and ETALON satellites for the half maximum detection sys-

9 OTSUBO AND APPLEBY: SATELLITE CENTER-OF-MASS CORRECTION ETG 9-9 Table 4. Center-of-Mass Correction for Leading Edge Half Maximum Systems a LAGEOS AJISAI ETALON Standard ps FWHM pulse width ps FWHM pulse width ps FWHM pulse width ns FWHM pulse width ns FWHM pulse width a In millimeters. tem. As the spread of the laser pulse increases, the center-of-mass correction gets smaller. [44] Actual systems have different system configurations such as the value of the pulse decay in the MCP-PMTs, different cables and variation in the tuned setup (delay, bias voltage, etc.) of discriminators. The center-of-mass corrections are very sensitive to these parameters. Although we also know empirically from orbit determination studies that the range bias of reliable MCP-PMT stations does not greatly exceed 1 cm for the LAGEOS satellites when applying the standard 251-mm correction, it is possible that there is an unpredictable variation of a few mm in the center-of-mass correction even between systems with the same type of photomultiplier and the same type of discriminator. 4. Discussions [45] Special care should be taken when one generates or analyses laser ranging data at its maximum precision and accuracy. If one uses the standard center-of-mass corrections without adjusting the range bias or applying any corrections, the estimated geodetic parameters are at risk of systematic errors. [46] As discussed in the previous section, a single photon system is the only type for which the center-of-mass correction can be given as a constant. Although Herstmonceux is now the only station with a strict single-photon policy, the newly developed SLR2000 system will also fall in this category. This means that more accurate links to geodetic satellites can be established when SLR2000 stations (or any other single-photon stations) join the worldwide network. [47] The center-of-mass correction for a C-SPAD system potentially has a significant energy dependence of about 1 cm (LAGEOS) to 5 cm (AJISAI and ETALON). Since we normally use normal point data that do not contain direct information on the return energy, we cannot fully compensate for the energy dependence in the data analysis stage. Furthermore, the return energy is likely on average to be strongest when the satellite is close to the local zenith, and weaker at low elevation. This means that the center-of-mass correction will depend on the elevation angle, which in turn is likely to bias a solution for station height. To achieve stable data quality, C-SPAD stations should strive to keep the return energy at a certain constant level, preferably at the single-photon level, which is probably the simplest to achieve. [48] The center-of-mass corrections for an MCP-PMT system were derived assuming half maximum detection; they are not based on the actual configuration of the various, mainly NASA, systems. However, the virtue of the MCP- PMT system is the small energy dependence of the range measurements. Thus, even though there is some ambiguity in the offset corrections (about 1 cm for LAGEOS and 5 cm for AJISAI and ETALON), assuming that the energy dependence is completely removed the correction itself can be treated as a constant for a given installation, unlike the situation for a C-SPAD system. [49] In this report, we have assumed an error-free laser ranging system without offset or systematic error sources (such as a timing instrument, a calibration system, a meteorological sensor, and so on). For the majority of the most productive laser ranging systems, this is a reasonable assumption at the 1 2 cm level or better, but is probably not true at the 1 2 mm level. Clearly, if a particular ranging station does have an undiscovered systematic error, careful analysis to derive a center-of-mass correction for the station would be to no avail because we would have no other choice than to estimate its range bias in the orbit analysis stage. Before discussing the satellite signature issues of a particular station, it is undoubtedly important for a station to minimize all sources of error. 5. Conclusions [50] We have demonstrated a new method to retrieve the average response function of the spherical geodetic satellites, LAGEOS, AJISAI, and ETALON. Matching full rate residual histograms with theoretical response functions, we found the model used in previous studies to be too narrow. [51] Using the newly constructed response functions, we derived center-of-mass corrections for a single-photon system, a C-SPAD system, and a photomultiplier system. The center-of-mass corrections depend on the ranging system and the detection policy, and varied about 1 cm for the LAGEOS satellites and about 5 cm for the AJISAI and ETALON satellites. We also demonstrated the advantageous stability of the single-photon ranging policy. [52] Through this paper, we would like to encourage the worldwide laser ranging network to provide more accurate data and also to see the satellite signature problem properly treated in the orbit determination process. This study would help the satellite laser ranging technology to make full use of its potential accuracy in the determination of geodetic parameters, especially the terrestrial reference frame and the gravitational scale GM. [53] Acknowledgments. We thank Roger Wood of the NERC Space Geodesy Facility and Georg Kirchner of the Austrian Academy of Sciences for providing the full rate laser ranging data of the Herstmonceux and Graz stations, respectively. We also thank Vladimir P. Vasiliev of the Russian Institute of Space Device Engineering and Hidekazu Hashimoto of the National Space Development Agency of Japan, for providing the ETALON and AJISAI satellite specification data, respectively. We also thank Howard Donovan of Honeywell Technology Solutions Inc. for useful discussions on aspects of the detector circuits of the NASA laser ranging systems. References Altamimi, Z., The accuracy of ITRF2000, paper presented at XXVI General Assembly, Eur. Geophys. Soc., Nice, France, Appleby, G. M., Satellite signatures in SLR observations, paper presented at the 8th International Workshop on Laser Ranging Instrumentation, Annapolis, Md., Appleby, G. M., P. Gibbs, R. A. Sherwood, and R. Wood, Achieving and maintaining sub-centimetre accuracy for the Herstmonceux single-photon

10 ETG 9-10 OTSUBO AND APPLEBY: SATELLITE CENTER-OF-MASS CORRECTION SLR facility, in Proceedings of Laser Radar Ranging and Atmospheric Lidar Techniques II, Proc. SPIE Int. Soc. Opt. Eng., 3865, 52 63, Arnold, D. A., Method of calculating retroreflector-array transfer functions, Spec. Rep. 382, Smithson. Astrophys. Obs., Cambridge, Mass., Degnan, J. J., Millimeter accuracy satellite laser ranging: A review, in Contributions of Space Geodesy to Geodynamics: Technology, Geodyn. Ser., vol. 25, edited by D. E. Smith and D. L. Turcotte, pp , AGU, Washington, D. C., Degnan, J. J., Engineering progress on the fully automated photon-counting SLR2000 satellite laser ranging station, Proceedings of Laser Radar Ranging and Atmospheric Lidar Techniques II, Proc. SPIE Int. Soc. Opt. Eng., 3865, 76 82, Dunn, P., M. Torrence, R. Kolenkiewicz, and D. Smith, Earth scale defined by modern satellite ranging observations, Geophys. Res. Lett., 26, , Fitzmaurice, M. W., P. O. Minott, J. B. Abshire, and H. E. Rowe, Prelaunch testing of the laser geodynamic satellite (LAGEOS), NASA Tech. Pap., TP-1062, Kirchner, G., F. Koidl, I. Prochazka, and K. Hamal, SPAD time walk compensation and return energy dependent ranging, paper presented at the 11th International Workshop on Laser Ranging Instrumentation, Deggendorf, Germany, Minott, P. O., T. W. Zagwodzki, T. Varghese, and M. Selden, Prelaunch optical characterization of the Laser Geodynamic Satellite (LAGEOS 2), NASA Tech. Paper, TP-3400, Mironov, N. T., A. I. Emetz, A. N. Zaharov, and V. E. Tchebotarev, ETALON-1, -2 center of mass correction and array reflectivity, paper presented at the 8th International Workshop on Laser Ranging Instrumentation, Annapolis, Md., Neubert, R., An analytical model of satellite signature effects, paper presented at the 9th International Workshop on Laser Ranging Instrumentation, Canberra, Australia, Otsubo, T., J. Amagai, and H. Kunimori, The center-of-mass correction of the geodetic satellite AJISAI for single-photon laser ranging, IEEE Trans Geosci. Remote Sens., 37, , Otsubo, T., G. M. Appleby, and P. Gibbs, GLONASS laser ranging accuracy with satellite signature effect, Surv. Geophys., 22, , Sasaki, M., and H. Hashimoto, Launch and observation program of the experimental geodetic satellite of Japan, IEEE Trans. Geosci. Remote Sens., 25, , Schwartz, J. A., Pulse spreading and range correction analysis for satellite laser ranging, Appl. Opt., 29, , G. M. Appleby, NERC Space Geodesy Facility, Monks Wood, Abbot Ripton, Huntingdon, PE28 2LS, UK. T. Otsubo, Communications Research Laboratory, Hirai, Kashima, Ibaraki, , Japan. (otsubo@crl.go.jp)