New method to resolve 2π ambiguity in NBV
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012
Received: 2 July 2010
Accepted: 5 August 2011
Published: 2 March 2012
The Narrow Bandwidth VLBI (NBV) technique is utilized to track spacecrafts in Japanese lunar exploration. However, the problem of 2π ambiguity affects the phase delay of the carrier waves transmitted by satellites. A probabilistic algorithm called coarse search and fine delay search, in combination with a simple ambiguity judgment procedure, is presented in the current paper to resolve the 2π ambiguity. This method is employed to estimate both the delay and the delay rate from the residual phases obtained using NBV. Compared with previous analytic methods, it does not require strict constraint conditions. The ambiguity can be resolved even when the phase variations in the residual phase are 0.2 rad (approximately 11.6 degrees) compared with the less than 4.3 degree phase variations in the analytic method. The method also has the advantage of giving short time variations in the orbital motion of a satellite without ambiguity using NBV.
In the Japanese lunar explorer SELenogical and ENgineering Explorer (SELENE) project, three satellites, including a main orbiter, a relay sub-satellite (Rstar), and a VLBI sub-satellite (Vstar), were launched into polar orbits around the Moon. Using four-way Doppler and differential VLBI techniques, the global map of the lunar gravity field was substantially improved. The measurement of the gravity field, obtained through the orbital motion of a spacecraft, is a powerful method to estimate the inner density structure of the Moon (Iwata et al.,2009).
The VLBI technique has been used in spacecraft tracking since the 1960s. However, one persisting problem is the ambiguity existing in the phases of the carrier waves transmitted by satellites. Therefore, group delay and delay rate have been primarily used (Border et al., 1992). However, the accuracy of group delay is limited to several hundred pico-seconds (ps), which is insufficient for a precise lunar gravity field estimation. To determine the low degree coefficients of the lunar gravity field, the estimation of phase delay with an accuracy of several ps is necessary.
Two radio sources are loaded on Rstar and Vstar, with each transmitting three carriers in the S-band and one carrier in the X-band. In the work of Kono et al. (2003), these sources help address cycle ambiguity using the multi-frequency VLBI (MFV) method (Kono et al., 2003), especially using the same-beam MFV method, when the elongation between two nearby spacecrafts is smaller than the beam width of the ground antenna (Liu et al., 2007; Kikuchi et al., 2009). In their methods, cycle ambiguity was derived analytically. The residual phase equation pairs were deduced individually, relying on strict error conditions, e.g. the error of the correlation phase must be less than 4.3°. Further, the ionospheric fluctuation must be less than 0.23 TECU (1 TECU equals 1016 el/m2). The TEC condition is more difficult for switching VLBI when a traveling ionospheric disturbance occurs in the ionosphere. However, it is addressed through the use of the same-beam method.
The derivation of cycle ambiguity is a critical aspect. Tropospheric fluctuation is a flicker noise (Liu et al., 2005). Therefore, the phase error cannot be reduced in inverse proportion to the integration time, and coherence decreases when the integration time is longer than a few tens of seconds. In the analytic method, the integration time is set to be approximately 1 min to decrease the number of phase variations. However, this duration is unsuitable for precise orbit determination and gravity field recovery.
In the current study, a new delay and delay rate estimation method is presented. The proposed method is based on probability theory and is called coarse search and fine delay search. It has been widely used in conventional VLBI. With this method, cycle ambiguity can be resolved with less stringent conditions in which the phase error of the residual phase is less than 0.2 rad. This condition indicates that the integration time becomes only 1/4 of that of the previous method, and shorter time information of the orbital motion can also be given. The improvement in time resolution in relative positioning is very important for the docking of two satellites and for understanding the dynamics of a medium-sized crater.
2. Main Concept
2.1 Main principle of coarse search and fine delay search in NBV
2.2 Problem of 2π ambiguity
However, the phase of each channel (e.g. Channel 4 in Fig. 1) can be validated in multiples of 2π, and this characteristic results in the ambiguity of delay, and fine delay search procedures are sometimes unsuccessful. In a previous study (Kono et al., 2003; Kikuchi et al., 2009), multi-frequency and same-beam differential VLBI technology are introduced to resolve ambiguity, and the correct delay values are obtained with an approximately 1 minute integration. In the current study, the 2π ambiguity problem can also be resolved with a shorter integration time by the coarse search and fine delay search method, and using the total delay and delay rate values predicted at the present epoch as references for judgment, which are derived from the values of the former epoch.
3. Simulation Analysis
In the simulations, the RF frequency components (n = 1, 2, 3, 4) consisted of three carrier wave signals in the S-band (2.212, 2.218, and 2.287 GHz) and one in the X-band frequency (8.456 GHz). The differential residual delay was fitted with a five-order polynomial based on a real orbit model. Differential phase noise for same-beam observation was derived from the observed tropospheric phase variations (Liu et al., 2005) assuming different traveling times (4, 5, 8, and 11s). Traveling time is proportional to the elongation between Rstar and Vstar. The differential frequency-dependent ionospheric delay was less than 5 ps in the S-band and less than 0.3 ps in the X-band (Liu et al., 2007), which can be disregarded. The simulation signal used as a received signal was produced by adding random noise generated by random numbers in a computerand the observed tropospheric phase variations. The root mean square (RMS) of the fringe phase errors with a pp integration were set to 18.0 degrees for the S-band and 26.6 degrees for the X-band, which can diminish to 3.1 degrees and 5.2 degrees with 50 s integration, respectively. Coarse search and fine delay search of the simulation signals were carried out to acquire the residual delay and the residual delay rate.
First, the total integration time must be roughly determined. Based on the capability to acquire delay rate correctly, the threshold of b is set at approximately several tens of ps/s. Second, the threshold of a is expected to be between 0.118 and 0.437 ns, with the former corresponding to 1 ambiguity for the X-band carrier and the latter to 1 ambiguity for the S-band carrier. If a wrong ambiguity is detected, the quantity of the ambiguity will be counted with 1 ambiguity corresponding to the carrier as a unit. The delay value is then shifted with the ambiguities, and the results are considered as a posteriori estimates. The a priori estimates for the next epoch are revised using Eq. (6), and the process is repeated.
However, the estimation method is dependent on the accuracy of the initial a priori estimates. When the first a priori estimate values significantly deviate from the true values over the threshold, such as 0.2 ns for delay and several tens of ps/s for the delay rate, the results are no longer valid. The first correct delay needs to be estimated using long-term integration so that the phase error will be sufficiently small to resolve 2π ambiguity. However, such an integration time is extremely long so the phase variation exceeds π in some cases.
In this case, the estimated delay is incorrect, so the carrier signal cannot be correctly detected. Assuming that both the phase variations attributable to the atmosphere, and the random noise from receivers, are distributed as Gaussian functions, a frequency distribution table with respect to the ambiguity can be generated by applying the search procedures several times in a given period. The frequency at the correct ambiguity would then be at a maximum in the table. If the first correct delay value is evaluated in this method, and the same procedure is repeated during the observation period, appropriate delay times through the whole observation period can be estimated accordingly.
4. Simulation Results
In Fig. 4, when the phase errors are larger than 0.2 rad, the possibility of resolving the ambiguity using an ambiguity judgment must be set as a conditional probability, which is equal to the product of the possibilities of searching a correct ambiguity and resolving ambiguity using an ambiguity judgment, assuming a correct a priori delay and delay rate. A value of 0.2 rad is the upper limit of the phase error in a conservative estimate to resolve ambiguity, with a corresponding integration time 1/4 of that needed in the previous method. Therefore, a greater amount of information on orbital motion can be obtained by applying this condition.
5. Discussion and Conclusion
To resolve 2π ambiguity, and to estimate the phase delay of an RF carrier, phase errors resulting from the atmosphere and receiver noise, among others, should be controlled to a low level. Switching VLBI, especially same-beam VLBI, can decrease a large portion of the errors and can finally provide the conditions for completely resolving ambiguity. Still, the integration time should be adequately long to enable the same-beam method to reduce phase errors and to resolve the ambiguity using the analytic solutions employed in previous work.
In the current study, a probabilistic algorithm, which can be used to obtain correct results under comparatively less stringent conditions, is presented. To realize this goal, there is a need to ensure first the estimation accuracy of the initial values. This estimation accuracy can also be solved using probabilistic algorithms to obtain the frequency distribution of ambiguity and to determine the first correct delay. The current study proposes another method for resolving ambiguity, in which shorter period information on orbital motion can be obtained because of the shorter integration time. Further, the computations, including coarse search and fine delay search and the ambiguity judgment procedure, can be easily performed using a personal computer. The application of a Kalman filter for delay and delay rate estimation, and the use of real data from SELENE, are expected to be done in our future work.
The authors wish to thank the editor and two anonymous referees for their comments. The current work was financially supported by the National Natural Science Foundation of China (No. 10973033) and the Shanghai Natural Science Foundation (09ZR1437300).
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