Picosecond accuracy VLBI of the two subsatellites of SELENE (KAGUYA) using multifrequency and same beam methods



[1] Same beam very long baseline interferometry (VLBI) observations of the two subsatellites of SELENE (KAGUYA) are demonstrated for purpose of the precise gravimetry of the Moon. Same beam VLBI contributes a great deal to cancel out the tropospheric and ionospheric delays and to determine the absolute value of the cycle ambiguity by using the multifrequency VLBI method. As a result, the differential phase delay of the X-band signal is estimated within an error of below 1 ps. This accuracy is more than 1 order of magnitude smaller than former VLBI results. The preliminary results for the orbit determination of the subsatellites show a decrease of the orbit error from a few hundreds of meters to around 10 m when the differential phase delay data are added to the conventional range and Doppler data. These results reveal the possibility of precise gravimetry.

1. Introduction

[2] One of the important questions still remaining about the Moon is the existence and state of the lunar core. The moment of inertia (MOI) of the Moon gives an important constraint for investigating the size and density of the lunar core [Konopliv et al., 1998, 2001]. The MOI can be estimated from the second degree coefficients C20 and C22 of the lunar gravity field amongst others. However, less accurate gravity field information especially for the limb region and the farside of the Moon restricted the accuracy of the second degree coefficients [Hanada et al., 2008].

[3] In the Japanese lunar exploration program SELENE (KAGUYA), the VRAD (the differential VLBI radio sources) mission aims to improve the lunar gravity field by employing a very long baseline interferometry (VLBI) technique [Hanada et al., 2008]. VLBI measures the difference in arrival time of a signal transmitted from a satellite to two ground stations. The differential VLBI (DVLBI) measurement consists of the differenced delay between the two spin-stabilized subsatellites of SELENE, Rstar (Ouna) and Vstar (Okina). The differential delays provide plane-of-sky position differences of R/Vstar in contrast to two-way Doppler measurements that provide line-of-sight position information. Thus DVLBI has the potential to improve the lunar gravity field especially for the limb region. Additionally, high-altitude orbits of R/Vstar are mainly sensitive to the low degree coefficients [Matsumoto et al., 2008]. After combination with farside gravimetry of four-way Doppler observations of the main orbiter in the RSAT (Relay Satellite Transponder) mission [Namiki et al., 2009], it is expected that the second degree coefficients can be derived with an improvement of 1 order of magnitude over existing models [Matsumoto et al., 2008].

2. VRAD System

2.1. Onboard System

[4] Rstar and Vstar were released from the main orbiter of SELENE on 9 and 11 October 2007 after the arrival to the Moon, and they entered polar elliptical orbits. Perilune and apolune heights of Vstar are 129 km and 792 km, and 120 km and 2395 km for Rstar. The VLBI radio sources are loaded on R/Vstar [Hanada et al., 2008], and these transmit three S-band and one X-band signals (S1 = 2212, S2 = 2218, S3 = 2287, X1 = 8456, all in [MHz]). X-band signal from Rstar is a white noise signal with a bandwidth of 120 kHz. The others are carrier wave signals.

2.2. Back-End System

[5] The signals transmitted from R/Vstar are recorded in four channels of a low-rate sampling system [Kikuchi et al., 2004]. The sampling rate is 200 kHz and the quantization is 6 bits. The sampling rate is an important specification. Because the bandwidth of the carrier wave signal used in VRAD is several tens of Hz, the amount of VLBI data can be reduced by using the low-rate sampling system. This also introduces the possibility of near-real-time data processing.

2.3. Data Processing

[6] Correlation of signals received at two VLBI stations is carried out by the software method. The software is composed of the correlation module which includes bit shifts, fringe stopping, fractional bit correction, and delay estimation module [Kono et al., 2003; Kikuchi et al., 2008]. A priori delay for the correlation process is calculated based on predicted orbits of R/Vstar. Because the plane wave assumption is not valid for spacecraft orbiting the moon, a finite distance delay model [Kono et al., 2003; Kikuchi et al., 2004] is used. In this model, starting from the receiving time at the reference station, transmitted time at the spacecraft and receiving time at the remote station are respectively calculated by light time equations. The propagation times from the spacecraft to the reference and remote stations are iteratively calculated in this way to get the a priori delay value. The cluster correlation system which is composed of 20 calculation nodes of CPU is used for data processing.

2.4. VLBI Network

[7] The VLBI network of VRAD consists of four domestic Japanese stations of the VERA network: Mizusawa, Ogasawara, Ishigaki, and Iriki [Kobayashi et al., 2003] and four foreign stations: Shanghai, Urumqi (China), Hobart (Australia) and Wettzell (Germany). The antenna time of about 100 hours per month on average for the four VERA stations has been assigned to the SELENE VLBI observations since December 2007 until October 2008, and a 4-month extended mission is also planned. Additionally, the foreign stations participated in intensive observations in January, May, and June 2008. These intensive observations are planned because a longer baseline is more sensitive to the motion of the spacecraft and is expected to contribute to more precise orbit determination and lunar gravity field estimation. The correlation and delay estimation for the data obtained in the intensive observations are just ongoing, so we concentrate on showing the results for the domestic VERA network in this paper.

3. Key Techniques

3.1. Multifrequency VLBI Method

[8] In order to execute the precise lunar gravimetry of SELENE, the differential delay must be estimated within an error of less than 3.3 picoseconds (ps) [Kono et al., 2003]. Two kinds of key techniques are applied in VRAD for this purpose. One is differential phase delay (DPD) estimation by a multifrequency VLBI method (MFV) [Kono et al., 2003]. DPD (ΔΔτ) is estimated from a difference of residual fringe phases (DRFP, ΔΔϕ) between R/Vstar. The residual fringe phase (RFP) is the phase of the cross correlation function of signals received at two VLBI stations. DRFP it has a cycle ambiguity of 2πN, where N is unknown integer.

equation image

Therefore, the cycle ambiguity must be estimated to derive the DPD without a bias. The VLBI technique has been applied to spacecraft tracking from the 1960s [Sagdeyev et al., 1992; Border et al., 1992; Antreasian et al., 2002; Thornton and Border, 2003]. Conventionally, the delay is estimated without deriving the cycle ambiguity in the group delay method [Kikuchi et al., 2004; Sagdeyev et al., 1992; Thornton and Border, 2003]. The delay is calculated from deviation of the RFP by using different frequency signals. However, the accuracy is limited to several tens of ps because it is inversely proportional to the bandwidth of the signals. This accuracy is not sufficient for precise gravimetry of the Moon and the phase delay method must be applied. The phase delay method has been used practically for precise orbit determination [Border et al., 1992]; however, the cycle ambiguity was not determined. In this case, the position of the spacecraft was well known and the DRFP did not contain the cycle ambiguity. This is not the common case: in most cases, the cycle ambiguity must be determined in order to use the DPD for precise orbit determination.

[9] The cycle ambiguity has been estimated by a phase referencing method [Lanyi et al., 2005; Martin-Mur et al., 2006]. By using the 10-station VLBA interferometer, imaging algorithms for astronomical sources are applied to derive the cycle ambiguity. In this experiment, the accuracy of the delay is demonstrated to be about 10 ps. The MFV method is a different approach than the imaging. The DRFP of the four different frequency signals are used to determine the cycle ambiguity NS1 of S-band signal S1 and NX1 of X-band signal X1. Once the cycle ambiguity is uniquely determined, the accuracy of the DPD is inversely proportional to the frequency and it is expected to be several ps [Kono et al., 2003; Kikuchi et al., 2008]. Advantage of the MFV method is that the cycle ambiguity can be derived directly from the DRFP. It can be done for any individual baseline without requiring a broad range of baseline lengths and orientations. This is an important aspect for a local network with small number of VLBI stations. Although the dedicated onboard system that transmits several carrier wave signals is needed, it has wide applications for a tracking and navigation of spacecraft.

[10] Three conditions must be satisfied to determine the cycle ambiguity according to the MFV method [Kono et al., 2003]. First, the root mean square (RMS) error of the DRFP must be smaller than 4.3 and 179 degrees for the S/X-bands. Second, the double differenced total electron content (DDTEC) between four propagation paths from R/Vstar to two ground stations must be compensated for within an error of 0.42 and 0.23 TECU (1 TECU is 1016 el/m2) for the S/X-bands. Third, the error of the a priori delay which is used for the correlation must be smaller than 83 ns. The third condition can be achieved by the orbit determination with the two-way range and Doppler observations of R/Vstar [Kikuchi et al., 2008]. However, the first and second conditions are especially strict due to ionospheric and tropospheric delays [Kikuchi et al., 2008].

3.2. Same Beam VLBI Method

[11] To satisfy the conditions of the MFV method, same beam VLBI observations (SBV) are applied [Border et al., 1992; Liu et al., 2007a, 2007b; Kikuchi et al., 2008]. As shown in Figure 1, when the separation angle of R/Vstar (Δθ) is smaller than the beam width of the ground antenna, the signals from R/Vstar are simultaneously received by tracking the midpoint of R/Vstar. By using SBV, ionospheric and tropospheric delays are almost the same for R/Vstar, and they are thus canceled. Although these delays can also be canceled out by a conventional switching VLBI method (SWV), fluctuations whose periods are shorter than the switching interval still remain. Therefore, the schedule of DVLBI in VRAD is such that SBV is carried out as long as possible.

Figure 1.

Configuration of the same beam and switching VLBI methods.

[12] The DVLBI observations consist of three types: SBV of the X-band signal when Δθ is smaller than 0.1 degrees (A), SBV of the S-band signal when Δθ is between 0.1 to 0.56 degrees (B), and SWV when Δθ is larger than 0.56 degrees. These conditions are decided according to the beam width of the ground antenna for each frequency band [Liu et al., 2007a, 2007b] and the tracking error of R/Vstar. Percentages of the periods for three type observations are about 10, 60, and 30% for A, B, and C types respectively.

4. Results

4.1. Results of Correlation

[13] Results of the DPD estimation for the SBV acquired on 15 January 2008 are shown. The VLBI stations are Iriki, Ogasawara and Mizusawa and Δθ is between 0.02 to 0.06 degrees. The RFP of the signals S1 and X1 from R/Vstar, and the DRFP between R/Vstar are shown in Figure 2. The baseline is Mizusawa-Ogasawara. The integration interval is set to 5 s. Long-term trends included in the RFP and the DRFP are removed by fitting a fourth-order polynomial to reveal the contribution of SBV. These trends are considered to be attributed mainly to the error of the a priori orbit. In Figure 2, it is shown that the fluctuations of the RFP which are mainly caused by tropospheric delay are similar between R/Vstar because the propagation paths from R/Vstar to the ground station are nearly the same. These fluctuations are canceled out by differencing the RFP of R/Vstar. Generally, tropospheric fluctuations are a bottleneck in the improvement of the RMS error by time integration because they are considered to be a flicker noise [Kikuchi et al., 2008]. However, SBV cancels out most of the tropospheric fluctuations and allows the long time integration. This effect is the same for the frequencies S2 and S3. When the integration interval is changed from 1 s to 60 s, the RMS error of the DRFP decreases from 7 degrees to 1.8 degrees for the three S-band signals and from 10 degrees to 1.3 degrees for the X-band. These RMS values satisfy the condition for the RMS error of the MFV method. The RMS error still depends on the climate condition even if most of the tropospheric fluctuations are canceled out. However, in most cases of the SBV, the RMS error satisfies the condition of the MFV method.

Figure 2.

The RFP and DRFP of the S-band signal S1 and the X-band signal X1. “Rstar” and “Vstar” represent the RFP of the signal from Rstar and Vstar, respectively. “Rstar-Vstar” represents the DRFP between Rstar and Vstar. Integration intervals of RFP and DRFP are 5 s.

[14] According to the simulation of Kikuchi et al. [2008], the paths from R/Vstar to each ground station are close in the period of SBV and DDTEC is expected to be less than 0.23 TECU. In this case, the condition for the TEC error of the MFV method is satisfied. However, this is not the case when a traveling ionospheric disturbance [Afraimovich et al., 2000] and/or other ionospheric fluctuation occur. Therefore, the DDTEC are estimated from the DRFP of the S/X-band signals in order to satisfy the condition with certainty. Because the ionospheric delay is inversely proportional to the square of frequency [Kikuchi et al., 2008], DDTEC can be estimated from the DRFP of different frequency signals. As a result, the DDTEC is between −0.07 to −0.02 TECU and the error evaluated from the RMS error of the DRFP is about 0.01 TECU in this observation period. This result satisfies the condition for the TEC error of the MFV method.

4.2. Results of Differential Phase Delay Estimation

[15] Figure 3 shows the result for the estimation of the cycle ambiguities of the S-band signal NS1 and the X-band signal NX1 according to the MFV method. When the integration interval of the DRFP is 1 s, the cycle ambiguity does not converge because the condition for the RMS error is not satisfied in this case. However, it converges to a unique value in 60-s integration interval. Once the cycle ambiguity is uniquely determined, the DPD can be derived from the DRFP without any bias. Figure 4 shows the residual of the DPD for S/X-band signals for three baselines. The trends of the DPD are removed by fitting a fourth-order polynomial. This is because these trends are considered to be attributed to the error of the a priori orbit and are smaller than 83 ns which is the condition for the error of the a priori delay of the MFV method. The RMS error is calculated from the residual of the DPD. The average of the RMS error for three baselines is 2.27 ps for S-band and 0.29 ps for X-band in a 60-s integration interval.

Figure 3.

The absolute value of the cycle ambiguity estimated by the MFV method. The cycle ambiguity of the S-band signal S1 and that of the X-band signal X1 are shown. The green and red points represent the results for 1- and 60-s integration intervals, respectively.

Figure 4.

The residual of the differential phase delay of the S-band signal S1 and X-band signal X1. The baselines are Iriki-Mizusawa, Mizusawa-Ogasawara, and Ogasawara-Iriki.

[16] Closure delay which is signed sum of delays around a three-baseline triangle is also shown. If the cycle ambiguity is misestimated, the closure delay contains the bias which corresponds to the cycle ambiguity. However, the closure delays converge to near zero and the RMS error is less than 1 ps. This result confirms the successful estimation of the absolute value of the cycle ambiguity.

[17] The accuracy of the DPD is evaluated from the contribution of conceivable error sources: thermal noise, ionospheric delay, static component of tropospheric delay, clock offset/rate, instrumental delay, and phase characteristics of the ground antenna. The error of ionospheric delay is evaluated from the error of the DDTEC which are estimated from the DRFP in this paper. The error is 2.7 ps for the S-band and 0.2 ps for the X-band. The error of the DPD caused by the thermal noise of the VRAD system is evaluated from the phase error of the DRFP according to the result in the work of Kikuchi et al. [2008]. The phase errors are 0.19 degrees for S-band and 0.23 degrees for X-band, corresponding to 0.24 ps and 0.1 ps. Most of the static component of the tropospheric delay can be canceled out in the period of SBV. However, a residual signal still remains because the propagation paths from R/Vstar to the ground station are not entirely the same. The tropospheric delays in the line-of-sight directions of Rstar and Vsar are estimated and compensated for. The zenith hydrostatic delay (ZHD) is calculated from surface pressure and zenith wet delay (ZWD) is estimated by using GPS method [Behrend et al., 2000]. The GPS receivers are placed near the telescopes of VERA. The tropospheric delays in the line-of-sight directions are calculated from ZHD, ZWD and Herring-Niell Mapping function of hydrostatic and wet delay [Niell, 1996]. The error is predicted to be smaller than 0.02 ps both for S/X-bands [Kikuchi et al., 2008]. The offset and drift of the time stamps recorded at each station completely cancel out because these are the same for R/Vstar. The phase-frequency characteristics of the receiving system are linear in the video bandwidth from 0 to 100 kHz. The residual phase error is ±0.1 degrees in the case of the SBV in which the difference of the frequencies of the signals from R/Vstar is less than 5 kHz [Liu et al., 2007a, 2007b]. This error corresponds to 0.3 ps for S-band and 0.1 ps for X-band. Effects of the phase characteristics of the VERA antenna are less than 2.1 ps for S-band and 0.6 ps for X-band [Liu et al., 2007a, 2007b]. Consequently, the error of the DPD is 3.44 ps and 0.64 ps for S/X-band from the root-sum-square of each error component. The difference from the RMS error of the derived DPD is considered to result from the ionospheric delay, the static component of tropospheric delay, and the phase-frequency characteristics. These are partly included in the DPD as bias.

[18] Finally, the achievement of the accuracy below one ps for the X-band is unprecedented and it is accurate more than 1 order of magnitude compared to previous VLBI results. Both of the DPD of S/X-bands almost fulfill the desired accuracy of the DVLBI observations of VRAD which is 3.3 ps [Kono et al., 2003].

4.3. Preliminary Results of Orbit Determination of Two Subsatellites

[19] The DVLBI data have not yet been used for the new lunar gravity model because the amount of data is not large enough. Here, we show the contribution of the DVLBI data to orbit determination of R/Vstar by using the GEODYN II software [Pavlis et al., 2001]. The lunar gravity model is based on SGM90d [Namiki et al., 2009]. Data used are two-way Doppler and range, as well as DVLBI of the S/X-bands signal. Data weights are 0.2 mm/s, 0.7 m and 1 cm, respectively. Data are distributed throughout the arc, with total time spans of 157 min for two-way Doppler and range for Rstar, 162 min for Vstar, and 116 min for DVLBI, of which 56 min are X-band data. The orbit errors are evaluated by overlap analysis, where orbit differences are computed between two overlapping arcs. The arcs of R/Vstar are listed in Table 1. Since the orbit is nearly edge-on in this example here, it is expected that the DVLBI data contribute mostly to improvements in the cross track direction of orbit differences. Here, edge-on orbit means that the normal of the orbital plane is perpendicular to the line-of-sight vector from the Earth to the Moon. On the other hand, face-on orbit means that the normal of the orbital plane is parallel to the line-of-sight vector. Overlaps are computed with and without DVLBI data to assess the influence. The results in Table 1 show the drastic improvement in orbit consistency that can be obtained from including the DVLBI data. Without DVLBI data, orbit consistency for Rstar is about 103 m, and for Vstar it is about 230 m. With DVLBI data and proper data weighting, consistency levels of 14.8 m for Rstar and 9.9 m for Vstar can be obtained, with the largest contribution being in the cross track direction, as expected.

Table 1. Arcs of Overlap Analysis for Rstar and Vstar and Overlap Results for Different Data Typesa
Satellite/DataRadial (m)Along Track (m)Cross Track (m)Total (m)
  • a

    All times in UT. RSTAR: arc 1: 14 January 2008 06:00:00 to 15 January 2008 14:00:00; arc 2: 15 January 2008 04:10:00 to 16 January 2008 09:20:00. VSTAR: arc 1: 14 January 2008 03:40:00 to 15 January 2008 14:00:00; arc 2: 15 January 2008 04:10:00 to 16 January 2008 13:00:00.

Rstar (Doppler/range)3.968.78103.14103.59
Rstar (Doppler/range/DVLBI)2.735.6113.4414.82
Vstar (Doppler/range)2.5523.00229.56230.72
Vstar (Doppler/range/DVLBI)1.305.748.019.94

[20] The RMS of postfit residuals for the orbit determination is 0.7 m for range data, 0.1 mm/s for Doppler data, and around 7 mm (23.3 ps) for DVLBI data. The RMS of 23 ps is larger than the error of the DPD which is estimated in section 4.2. The difference could be due to unmodeled lunar gravity signatures or satellite orbit errors. This is because the VLBI data have not been included in our gravity model estimation yet. The RMS of the postfit residuals are expected to approach their intrinsic accuracy after the accumulation of VLBI data and the improvement of the current gravity field model. A similar overlapping result can also be obtained for a face-on orbit including DVLBI data, although the Doppler/range only is already in the order of a total accuracy of 20 m. It should be noted that this is a preliminary result, because the amount of VLBI data is not yet enough for orbit determination on a routine basis. Nevertheless, the results confirm the expected contribution of DVLBI.

5. Conclusions

[21] In conclusion, the DPD estimation is demonstrated by using the MFV and SBV methods. SBV ensures satisfaction of the severe conditions for the MFV method. As a result, the absolute value of the cycle ambiguity is uniquely determined. The accuracy of the DPD of X-band signal is below one ps. Additionally, the desired accuracy of 3.3 ps for precise gravimetry of SELENE has been achieved for most of the SBV of the S/X-bands until now. The results of overlap analysis confirm the contribution of DVLBI for the orbit determination of R/Vstar, and subsequently this will lead to further improvements of the lower degrees of the lunar gravity field in the future. Although the RMS of the postfit residuals does not reach one ps, this could be due to lunar gravity signatures which are not modeled in our gravity model. The RMS of the postfit residuals are expected to approach their intrinsic accuracy after gravity models are improved.

[22] Additionally, the combination of the MFV and SBV methods is anticipated to be applied for precise positioning in future planetary missions. Because the accuracy of the phase delay is inversely proportional to the frequency of radio signal, the accuracy would be improved by expanding observation frequency to higher band. Although the requirement for the separation angle between two radio sources become stricter, the precise phase delay measurement would be useful for gravity field estimation of the Moon and planets, and for determining their rotations.


[23] The authors appreciate the contribution of all engineers of NEC/Toshiba Space Systems Ltd. (NTS), Nippon Antenna Co. Ltd., and Nippi Corporation, the entire staff of the Kaguya mission, and VERA project. The authors also wish to express their gratitude to the chair of the IVS committee Harald Schuh, Wolfgang Schlüter, Dirk Behrend, Simon Ellingsen, Xiaoyu Hong, Yusufu Aili, and Yasuhiro Koyama. The VRAD mission would never have been achieved without a prominent technique and the profound knowledge of Fumio Fuke who was an engineer of NTS and passed away 2 months after the launch. We express sincere thanks to him.