Formulation of geopotential difference determination using optical-atomic clocks onboard satellites and on ground based on Doppler cancellation system

Geophysical Journal International Geophys. J. Int. (2016) 206, 1162–1168 Advance Access publication 2016 June 9 GJI Gravity, geodesy and tides doi: 10.1093/gji/ggw198 Formulation of geopotential difference determination using optical-atomic clocks onboard satellites and on ground based on Doppler cancellation system Ziyu Shen,1 Wen-Bin Shen1,2 and Shuangxi Zhang1 1 Department of Geophysics, School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan 430079, China. E-mail: 2 State SUMMARY In this study, we propose an approach for determining the geopotential difference using highfrequency-stability microwave links between satellite and ground station based on Doppler cancellation system. Suppose a satellite and a ground station are equipped with precise opticalatomic clocks (OACs) and oscillators. The ground oscillator emits a signal with frequency fa towards the satellite and the satellite receiver (connected with the satellite oscillator) receives this signal with frequency fb which contains the gravitational frequency shift effect and other signals and noises. After receiving this signal, the satellite oscillator transmits and emits, respectively, two signals with frequencies fb and fc towards the ground station. Via Doppler cancellation technique, the geopotential difference between the satellite and the ground station can be determined based on gravitational frequency shift equation by a combination of these three frequencies. For arbitrary two stations on ground, based on similar procedures as described above, we may determine the geopotential difference between these two stations via a satellite. Our analysis shows that the accuracy can reach 1 m2 s−2 based on the clocks’ inaccuracy of about 10−17 (s s−1 ) level. Since OACs with instability around 10−18 in several hours and inaccuracy around 10−18 level have been generated in laboratory, the proposed approach may have prospective applications in geoscience, and especially, based on this approach a unified world height system could be realized with one-centimetre level accuracy in the near future. Key words: Satellite gravity; Geopotential theory; Ionosphere/atmosphere interactions. 1 I N T RO D U C T I O N One of the main objectives in geodesy is to accurately determine the geopotential as well as the orthometric height. If the geopotential can be precisely determined, then the orthometric height can be accordingly precisely determined (Hofmann-Wellenhof & Moritz 2006). Another objective is to unify the world height datum system with high accuracy. The conventional approach of determining the geopotential (as well as the orthometric height) by combining levelling and gravimetry has at least the following two drawbacks: (1) the error is accumulated with the increase of the length of the measurement line, and (2) it is difficult or impossible to transfer the orthometric height with high accuracy between two points located in mountainous areas or continents separated by sea. The point (2) of the drawbacks also means that it is very difficult to unify the 1162 world height datum system with high accuracy, which is an open problem in geodetic community. In recent decades, though gravity field models (such as GOCE/GRACE geopotential models and EGM2008 models) can be used for determining geopotential, there exist essential limitations. For instance, the main problems existing in the GRACEgenerated gravity field or GOCE-generated gravity field are that their resolution is low, achieving about 2◦ × 2◦ to 1◦ × 1◦ , equivalent to about 200–100 km resolution (Tapley et al. 2004; Pail et al. 2011). At present, though the gravity field model EGM2008 with degree/order 2160 (Pavlis et al. 2008) has the highest accuracy and resolution (about 10 km), its average accuracy is around 10–20 cm, which is not enough for high precision requirement (say several centimetres). In addition, it provides only ‘average’ results, not in situ. To overcome the difficulties existing in conventional approach  C The Authors 2016. Published by Oxford University Press on behalf of The Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Accepted 2016 May 19. Received 2016 May 18; in original form 2016 February 14 Formulation of geopotential difference Figure 1. Ground station P emits a frequency signal fe at time t1 . Satellite S transmits the received signal f e and emits a frequency signal fs at time t2 . The ground station receives signal f e and f s at time t3 at position P . φ is gravitational potential, r is position vector, v is velocity vector and a is centrifugal acceleration vector. 2 DOPPLER CANCELLING TECHNIQUE When a frequency signal is emitted from satellite to ground or from ground station to satellite, the first-order Doppler effect contributes the most amount of frequency shift. However, the first-order Doppler effect is hard to be precisely measured due to the fact that the velocity of satellite cannot be precisely enough determined. Thus, the gravity frequency shift cannot effectively be identified if the first-order Doppler effect is not cancelled. Fortunately, this problem could be solved by using the DCT (Vessot & Levine 1979). After the first-order Doppler effects are eliminated, the remained frequency shift effects caused by other factors are more easily to be distinguished. After subtracting the ionosphere frequency shift, troposphere frequency shift and other influences, we can obtain the target gravity frequency shift. In fact, the DCT not only cancels the first-order Doppler effect, but also almost eliminates the ionosphere and troposphere effects. The DCT (Vessot & Levine 1979) contains three micro-wave links as depicted in Fig. 1. Ground station P emits a frequency signal fe at time t1 . When the signal is received by satellite S at time t2 , it immediately transmits the received signal f e and emits a frequency signal fs at the same time. These two signals emitted from satellite are received by ground station P at time t3 , noting that during the time period from t1 to t3 the ground station has changed from position P to position P . As described in Fig. 1, we can extract the gravity frequency shift signals (or equivalently gravitational frequency shift signals) by combining the emitting and receiving frequencies. The simplest case is when fe = fs , the frequency shift signals can be determined (referring to Fig. 2). The frequencies of the signals emitted from ground oscillator and satellite oscillator are f0 . The microwave links 1 and 2 consist of a go-return link by a phase-coherent microwave transponder equipped at satellite, (...truncated)