TY - JOUR
AU - Zhang, Shuangxi
AB - Abstract In this study, we propose an approach for determining the geopotential difference using high-frequency-stability microwave links between satellite and ground station based on Doppler cancellation system. Suppose a satellite and a ground station are equipped with precise optical-atomic 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. Satellite gravity, Geopotential theory, Ionosphere/atmosphere interactions 1 INTRODUCTION 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 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 GRACE-generated 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 for determining the geopotential difference, Bjerhammar (1985) put forward an idea to determine the gravitational potential using clock transportation approach (Shen et al.2009), which is based on the general relativity theory (Einstein 1915). The geopotential difference is determined by precise clocks, since clocks run at different rates at the positions with different geopotentials according to general theory of relativity. Equivalently, Shen et al. (1993) suggested that the gravitational potential could be determined by gravity frequency shift, because light signal's frequency will change as it travels between two positions with different geopotentials. In addition, Shen et al. (1993, 2011) proposed an approach, which stated that the geopotential could be determined by satellite frequency signal transmission, which may play a key role in directly determining the geopotential and unifying the world height datum system. This method is prospective and potential to be applied in the Global Navigation Satellite System (GNSS; Shen et al.1993, 2011), since GNSS satellites almost cover every corner of the Earth's surface. The principle of determining geopotential difference via GNSS frequency signals was introduced in our previous studies (Shen et al.1993, 2011). However, a key problem open is how to effectively extract gravitational frequency shift from various frequency shifts caused by other sources, such as Doppler frequency shift, ionospheric influence, etc. In this paper, we propose an approach to extract gravitational frequency shift from the propagation of light signals between a spacecraft (space station or satellite) and a ground station based on Doppler cancelling technique (DCT; Vessot & Levine 1979) which was proposed to test the general relativity theory. By extracting the gravitational frequency shift signals between the spacecraft and the ground station, we can determine the geopotential difference. Since optical-atomic clocks (OACs) with instability around 10−18 (s s−1 or Hz Hz−1) in several hours and inaccuracy of 10−18 level have been generated in laboratory (Hinkley et al.2013; Bloom et al.2014; Ushijima et al.2015), we may expect that in the very near future, portable or commercial clocks with inaccuracy of 10−18 level could be generated. For instance, the project Space Optical Clocks 2 of European Space Agency planned to install transportable lattice optical clocks with inaccuracy of 10−16–10−17 on board (Schiller et al.2012; Botter et al.2014). The Space-Time Explorer and QUantum Equivalence Space Test space mission was planned to be launched around 2024 which can compare clocks between satellite and ground down to the inaccuracy of 1 × 10−18 level (Altschul et al.2014; Hechenblaikner et al.2014). China planned to launch Experiment Space Station in the period 2021–2023, in which an OAC system with inaccuracy of 10−17 level or better will be installed on board (Private communication, 2015). The main purposes that ultrahigh precision clocks are installed in a Space Station include not only further testing general relativity theory but also investigating broad applications of ultrahigh precise time–frequency systems. Thus, the proposed approach in this study is prospective for precisely determining the geopotential globally. Further, based on this approach a unified world height system could be realized with high accuracy. In this study, we assume that the clocks with relative inaccuracy of 1 × 10−17 level are available in the near future. Although the gravitational potential changes as a function of time, due to mass migration for instance associated with tides, ice melting, sea level rise and so on, here we deal with static geopotential measurements. The corrections from temporal to static fields are well known, an inspection beyond the scope of this study. In fact, we determine the difference between the geopotential WS(t) of the satellite-orbit position at some point in time t and the fixed geopotential WA at the ground station (see Fig. 1). Figure 1. Open in new tabDownload slide Ground station P emits a frequency signal fe at time t1. Satellite S transmits the received signal |$f_e^{\prime }$| and emits a frequency signal fs at time t2. The ground station receives signal |$f_e^{\prime \prime }$| and |$f_s^{\prime }$| at time t3 at position P′. ϕ is gravitational potential, |$\vec{r}$| is position vector, |$\vec{v}$| is velocity vector and |$\vec{a}$| is centrifugal acceleration vector. Figure 1. Open in new tabDownload slide Ground station P emits a frequency signal fe at time t1. Satellite S transmits the received signal |$f_e^{\prime }$| and emits a frequency signal fs at time t2. The ground station receives signal |$f_e^{\prime \prime }$| and |$f_s^{\prime }$| at time t3 at position P′. ϕ is gravitational potential, |$\vec{r}$| is position vector, |$\vec{v}$| is velocity vector and |$\vec{a}$| 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 |$f_e^{\prime }$| 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, and provide two-way Doppler frequency shift data as a beat frequency |$f_0^{\prime \prime }-f_0$| (Vessot & Levine 1979). Similarly, the microwave link 3 provides one-way frequency shift data as a beat frequency |$f_0^{\prime }-f_0$| (Vessot & Levine 1979). Dividing the two-way beat frequency by two and subtracting it from the one-way beat frequency, we obtain the equation of output frequency Δf (Vessot & Levine 1979): \begin{equation} \Delta f = f_0^{\prime }-f_0-\frac{f_0^{\prime \prime }-f_0}{2}. \end{equation} (1) If the clock errors and the errors introduced during the signals’ propagation (such as ionosphere influence, random noises, etc.) are neglected, the beat frequencies |$f_0^{\prime }-f_0$| and |$f_0^{\prime \prime }-f_0$| mainly consist of first-order Doppler effects, second-order Doppler effects and gravitational effect (conventionally referred to as gravitational red shift). After the subtraction of the two beat frequencies, the first-order Doppler effect is removed and the second-order Doppler effect can be calculated. Although this subtraction cannot totally cancel out the first-order Doppler effect because of Earth's rotation, we can manage to make additional correction for it. Then, we may obtain the gravitational red shift, and consequently, the gravitational potential difference. Figure 2. Open in new tabDownload slide Schematic concept of the Doppler cancelling system (modified after Vessot & Levine 1979). The ground oscillator emits a frequency signal f0 to the spacecraft, then the spacecraft transmits the received signal to ground, and emits a frequency signal f0 from spacecraft oscillator to the ground at the same time. The signals that are transmitted and emitted from satellite are received at ground station as |$f_0^{\prime }$| and |$f_0^{\prime \prime }$|⁠, respectively. Finally, the output signal Δf are calculated from f0, |$f_0^{\prime }$| and |$f_0^{\prime \prime }$|⁠, as described in the text (see Section 2). Figure 2. Open in new tabDownload slide Schematic concept of the Doppler cancelling system (modified after Vessot & Levine 1979). The ground oscillator emits a frequency signal f0 to the spacecraft, then the spacecraft transmits the received signal to ground, and emits a frequency signal f0 from spacecraft oscillator to the ground at the same time. The signals that are transmitted and emitted from satellite are received at ground station as |$f_0^{\prime }$| and |$f_0^{\prime \prime }$|⁠, respectively. Finally, the output signal Δf are calculated from f0, |$f_0^{\prime }$| and |$f_0^{\prime \prime }$|⁠, as described in the text (see Section 2). In an ideal case, in an Earth-centred inertial coordinate system the output frequency Δf can be expressed as following equation (Vessot & Levine 1979) \begin{equation} \frac{\Delta f}{f_0} = \frac{\phi _s - \phi _e}{c^2} - \frac{\left| \vec{v}_e - \vec{v}_s \right| ^2}{2c^2} - \frac{\vec{r}_{se}\cdot \vec{a}_{e}}{c^2}, \end{equation} (2) where ϕs − ϕe is Newtonian gravitational potential difference between ground station and spacecraft (satellite), |$\vec{v}_e$| and |$\vec{v}_s$| are velocities of ground station and spacecraft respectively, |$\vec{r}_{se}$| is vector from spacecraft to ground station, |$\vec{a}_e$| is centrifugal acceleration vector of ground station and c is the speed of light in vacuum. We note that, on the right-hand side of eq. (2), the first term denotes the gravitational red shift, the second term is identified as the second-order Doppler shift predicted by special relativity and the third term describes the effect of Earth's rotation during the propagation time |$\left| \vec{r}_{se} / c\right|$| of the light signal. It serves as an Earth's rotation correction term for the Doppler effect. Though this correction is quite small, around 10−16, it should be taken into account for the accuracy requirement of 10−17 level. Eq. (2) describes an ideal case for the DCT. However, a signals’ frequency will be influenced by ionospheric and tropospheric effects (Millman & Arabadjis 1984). Similar to Doppler effect, these influences cannot be totally cancelled out in eq. (1) because of Earth's rotation. Besides, various error sources should also be considered. Thus, the output frequency Δf should be expressed as \begin{eqnarray} \frac{\Delta f}{f_0} = \frac{\phi _s - \phi _e}{c^2} - \frac{\left| \vec{v}_e - \vec{v}_s \right| ^2}{2c^2} - \frac{\vec{r}_{se}\cdot \vec{a}_{e}}{c^2} + \frac{\Delta f_i}{f_0} + \frac{\Delta f_t}{f_0} + \frac{\Delta f_e}{f_0},\nonumber \\ \end{eqnarray} (3) where Δfi is ionospheric shift correction term (see Section 3.1), Δft is troposphere refraction effect correction term (see Section 3.2), Δfe is the error term which contain clock error, position error, velocity error, instrumental error, remained ionospheric shift error, remained tropospheric effect error and random errors. The clock error reflects the stability of oscillator, and the instrumental error contains a finite delay in spacecraft transponder. Since the DCT method only involves frequency measurement, it does not require time synchronization or calibration. Thus, the hardware delays in ground station can be neglected, if there is any. Here, we only focus on determining the geopotential difference between a satellite/spacecraft and a ground station, which means that the absolute value of the geopotential at the satellite/spacecraft's orbit is not considered in this study. For the determination of the absolute geopotential at a ground station via a satellite/spacecraft, the absolute value of the geopotential at the satellite/spacecraft's orbit should be a priori given. The determination of the geopotential difference between two ground stations can be realized via satellite frequency signal links. The determination of the (absolute) geopotential at a ground station and inversely that at satellite position at time t are topics of our future research. In the sequel, we will discuss the correction terms Δfi and Δft in details. 3 IONOSPHERE AND TROPOSPHERE CORRECTION 3.1 Ionospheric shift correction When a microwave signal travels through ionosphere, the Doppler and ionospheric frequency shift Δf is caused by the time variation of the phase path P (Namazov et al.1975): \begin{equation} \Delta f = -\frac{f}{c}\frac{{\rm d}P}{{\rm d}t} \end{equation} (4) and the phase path P in ionosphere can be expressed as: \begin{equation} P = \int _L n_{i}ds \end{equation} (5) where ni is the refractive index of ionosphere and L is the path of the signal's propagation. The refractive index ni in ionosphere is described as (Vessot & Levine 1979): \begin{equation} n_i = \left( {1 - \frac{{f_n^2}}{{{f^2}[1 \pm ({f_m}/f)]}}} \right)^{1/2}, \end{equation} (6) where fn is plasma frequency and fm is electron gyromagnetic frequency. The plasma frequency is described as \begin{equation} f_n = \frac{1}{2\pi }\left( \frac{\rho }{\varepsilon _0}\frac{e^2}{m} \right)^{1/2} \end{equation} (7) where e and m are electron's charge and mass, ρ is density (electron number per cubic metre) and ε0 is the permittivity of free space. When f = 2.0 GHz, the term fm/f ≈ 3 × 10−3 can be neglected. Combining eqs (4)–(7), we have \begin{eqnarray} \Delta f &=& -\frac{f}{c}\frac{{\rm d}}{{\rm d}t}\int _L {n_{i}ds} = -\frac{f}{c}\frac{{\rm d}}{{\rm d}t}\int _L {ds} + \frac{40.5}{cf}\frac{{\rm d}}{{\rm d}t}\int _L {\rho (t) ds} \end{eqnarray} (8) where in the right-hand side, the first term represents the conventional Doppler shift, and the second term represents the ionospheric influence. The first Doppler effect can be completely cancelled after application of DCT. Hence, the interested ionospheric shift can be expressed as \begin{equation} \Delta f_i = \frac{40.5}{cf}\frac{{\rm d}}{{\rm d}t}\int _L {\rho (t) ds} \end{equation} (9) where ∫Lρ(t)ds is the columnar electron density, and from eq. (9) we can see that the ionosphere frequency shift Δfi may be caused by the variation of electron density ρ along the trajectory and by the variation of the geometric path of the signal. Eq. (9) describes the ionosphere frequency shift for a single wave link. However, our DCT contains a two-way go-return link and a single downlink as shown in Fig. 2. The uplink and downlink paths are different due to the Earth's rotation, thus the uplink wave path li1 in ionosphere is slightly different from the downlink wave path li2 in ionosphere, as shown in Fig. 3. Based on eq. (9), the ionosphere frequency shift Δfud of go-return (i.e. up- and down-) link is: \begin{equation} \Delta f_{ud} = \frac{40.5}{cf}\frac{{\rm d}}{{\rm d}t}\left( \int _{l_{i1}} \rho ds + \int _{l_{i2}} \rho ds \right) \end{equation} (10) while the ionosphere frequency shift Δfd of downlink is: \begin{equation} \Delta f_d = \frac{40.5}{cf}\frac{{\rm d}}{{\rm d}t}\int _{l_{i2}} \rho ds. \end{equation} (11) Considering the output frequency defined by eq. (1), the frequency shift Δfi caused by ionosphere frequency effect in output frequency is: \begin{equation} \Delta f_i = \Delta f_d - \frac{\Delta f_{ud}}{2} = \frac{40.5}{2cf}\frac{{\rm d}}{{\rm d}t}\left( \int _{l_{i2}} \rho ds - \int _{l_{i1}} \rho ds \right). \end{equation} (12) Figure 3. Open in new tabDownload slide The path difference between uplink and downlink. The oscillator at ground station emits a frequency signal at point P1, then the spacecraft (satellite) at S receives and transmits the signal toward the ground station. Finally, the receiver at ground station receives the transmitted signal at point P2 because of Earth's rotation. li1 and li2 are the uplink and downlink wave paths in ionosphere respectively, lt1 and lt2 are the uplink and downlink wave paths in troposphere, respectively. Figure 3. Open in new tabDownload slide The path difference between uplink and downlink. The oscillator at ground station emits a frequency signal at point P1, then the spacecraft (satellite) at S receives and transmits the signal toward the ground station. Finally, the receiver at ground station receives the transmitted signal at point P2 because of Earth's rotation. li1 and li2 are the uplink and downlink wave paths in ionosphere respectively, lt1 and lt2 are the uplink and downlink wave paths in troposphere, respectively. 3.2 Tropospheric refraction effect correction When a microwave signal travels through troposphere, similar to the ionospheric case, we have the tropospheric frequency shift equation (Millman & Arabadjis 1984): \begin{equation} \Delta f = -\frac{f}{c}\frac{{\rm d}}{{\rm d}t}\int _L n_{t}ds \end{equation} (13) where nt is the refractive index of troposphere, which is expressed as (Millman & Arabadjis 1984): \begin{equation} n_t = 1 + \frac{a}{T}\left( p + \frac{b\epsilon }{T}\right) \times 10^{-6}, \end{equation} (14) where T is absolute temperature (°K), p is total pressure (mbar) and ϵ is the partial pressure of water vapour (mbar), a and b are constants, with their values being 77.6 K mbar−1 and 4810 K, respectively (Smith & Weintraub 1953). According to eqs (13) and (14), similar to the ionospheric case (omitting the first Doppler term), we obtain the troposphere shift, expressed as \begin{equation} \Delta f_t = -\frac{f}{c}\frac{{\rm d}}{{\rm d}t}\int _L (M_1 + M_2) ds \end{equation} (15) where M1 = 77.6 × 10−6p/T and M2 = 0.373ϵ/T2. From eq. (15), we can see that the troposphere frequency shift Δft may be caused by the variation of temperature T, total pressure p and partial pressure of water vapour ϵ along the trajectory, or by the variation of the geometric path of the signal. Similar to eq. (12) that describes the frequency shift caused by the ionosphere effect, the frequency shift Δft caused by troposphere refraction effect in output frequency is \begin{equation} \Delta f_t = -\frac{f}{2c}\frac{{\rm d}}{{\rm d}t}\left( \int _{l_{t2}}(M_1 + M_2)ds - \int _{l_{t1}}(M_1 + M_2)ds \right). \end{equation} (16) 3.3 Simplified expressions for ionosphere and troposphere correction In previous subsections, we derived eqs (12) and (16) as the expressions for ionospheric and tropospheric frequency shift corrections, which can be used in practice. However, for convenience in practical usage in calculating the relevant quantity and estimating the accuracy, we can further simplify eqs (12) and (16) as described in the sequel. The signal propagation paths li1 and li2 (in ionosphere) are relatively close to each other (Fig. 3), thus it is safe to ignore the difference of the electron density ρ between the paths li1 and li2. Then, the ionospheric frequency correction Δfi is only caused by variations of the geometric path of the wave, and eq. (12) can be expressed as: \begin{equation} \Delta f_i = \frac{40.5\bar{\rho }}{2cf}\left( \frac{{\rm d}l_{i2}}{{\rm d}t}-\frac{{\rm d}l_{i1}}{{\rm d}t}\right) = \frac{40.5\bar{\rho }}{2cf}\left( \left| \vec{v}_{i2}\right| - \left| \vec{v}_{i1}\right| \right), \end{equation} (17) where |$\vec{v}_{i1}$| and |$\vec{v}_{i2}$| denote the variation velocities of |$\vec{l}_{i1}$| and |$\vec{l}_{i2}$| (see Fig. 3) (here |$\vec{l}_{ij}$| denote the vector from ground station to satellite at time tj, j = 1, 2), |$\bar{\rho }$| is the average electron density along the signals’ propagation paths li1 and li2. The signal propagation paths li1 and li2 (in troposphere) are very similar to the case in ionosphere, hence we can also ignore the difference of M1 and M2 between the paths lt1 and lt2. Then, eq. (16) can be expressed as: \begin{equation} \Delta f_t = -\frac{f(\bar{M}_1 + \bar{M}_2)}{2c}\left( \left| \vec{v}_{t2}\right| - \left| \vec{v}_{t1}\right| \right), \end{equation} (18) where |$\vec{v}_{t1}$| and |$\vec{v}_{t2}$| denote the variation velocities of |$\vec{l}_{t1}$| and |$\vec{l}_{t2}$| (here |$\vec{l}_{tj}$| denote the vector from ground station to satellite at time tj, j = 1, 2), |$\bar{M}_1$| and |$\bar{M}_2$| are the average value of M1 and M2 along the signals’ propagation paths lt1 and lt2. The uplink and downlink paths between ground station and spacecraft are, respectively, denoted as P1S and P2S, as shown by Fig. 3. We define the variation velocities of paths P1S and P2S as |$\vec{V}_1$| and |$\vec{V}_2$|⁠, respectively, then we have the following relationship (note that |$\left| \vec{a}_e \right| \cdot \Delta t$| is much smaller than |$\left| \vec{v}_s - \vec{v}_e\right|$|⁠): \begin{eqnarray} | \vec{V}_2| - | \vec{V}_1| &=& \left| \vec{v}_s - \vec{v}_e - \vec{a}_e\cdot \Delta t\right| - \left| \vec{v}_s - \vec{v}_e\right| \nonumber \\ &=& \left| \vec{a}_e \right| \cdot \Delta t \cdot {\rm cos}\alpha \nonumber \\ &=& \vec{a}_e \cdot \frac{2\left| \vec{r}_{se}\right|}{c} \cdot \frac{\left( \vec{v}_s - \vec{v}_e \right) \cdot \vec{a}_e}{\left| \vec{v}_s - \vec{v}_e \right| \left| \vec{a}_e\right|} \nonumber \\ &=& \frac{2\left| \vec{r}_{se}\right| \cdot \left( \vec{v}_s - \vec{v}_e \right) \cdot \vec{a}_e}{c\cdot \left| \vec{v}_s - \vec{v}_e \right|}, \end{eqnarray} (19) where |$\vec{v}_s$| and |$\vec{v}_e$| are velocities of spacecraft and ground station, respectively, |$\vec{a}_e$| is centripetal acceleration of ground station, |$\vec{r}_{se}$| is vector from spacecraft to ground station, α is the angle between relative velocity |$\vec{v}_s - \vec{v}_e$| and acceleration |$\vec{a}_e$|⁠, Δt is the time duration of go–return microwave link. Suppose the height of the spacecraft is H (km) from the ground, then we have the following approximations: \begin{eqnarray} \left| \vec{v}_{i2}\right| - \left| \vec{v}_{i1}\right| &\approx &\frac{1940}{H}(| \vec{V}_{2}| - | \vec{V}_{1}|) \nonumber \\ \left| \vec{v}_{t2}\right| - \left| \vec{v}_{t1}\right| &\approx & \frac{60}{H}(| \vec{V}_{2}| - | \vec{V}_{1}| ), \end{eqnarray} (20) where 1940 (km) and 60 (km) denote the thicknesses of ionosphere and troposphere layers, respectively. According to eqs (17) and (18), the ionospheric and tropospheric frequency correction terms Δfi and Δft can be, respectively, written as \begin{equation} \Delta f_i = \frac{78570\bar{\rho }\left| \vec{r}_{se}\right| \left( \vec{v}_s - \vec{v}_e \right) \cdot \vec{a}_e}{c^2fH\left| \vec{v}_s - \vec{v}_e \right|} \end{equation} (21) and \begin{equation} \Delta f_t = -\frac{60f\left( \bar{M}_1 + \bar{M}_2 \right) \left| \vec{r}_{se}\right| \left( \vec{v}_s - \vec{v}_e \right) \cdot \vec{a}_e}{c^2H\left| \vec{v}_s - \vec{v}_e \right|} \;{.} \end{equation} (22) In the above, we have discussed in detail a microwave signal's frequency shift caused by ionosphere and troposphere. It should be noted that, according to our DCT concept described in Section 2, what we need to measure are frequencies, not time duration. Besides, the delay of a microwave signal do not influence its frequency shift [see eqs (4) and (13), there are no terms describing signal's delay]. Thus, the influences caused by the ionosphere and troposphere delays are not our interests. The main frequency shift error sources caused by ionosphere and troposphere are ionospheric and tropospheric Doppler effects, just as discussed in details in Sections 3.1 and 3.2. Since we have made corrections for the ionospheric and tropospheric Doppler effects [see eqs (21) and (22)], and the remained errors are at the level of 10−19 after corrections [see eqs (29) and (30)]. Thus, after corrections as given by eqs (21) and (22), the frequency shift errors caused by ionosphere and troposphere can be neglected. 4 ACCURACY ESTIMATION AND ERROR ANALYSIS After the ionospheric and tropospheric frequency corrections [see eqs (21) and (22)], the remained error term Δfe in eq. (3) is a sum of various error sources, expressed as \begin{equation} \Delta f_e = \Delta f_{{\rm osc}} + \Delta f_{{\rm pos}} + \Delta f_{{\rm vel}} + \Delta f_{{\rm rion}} + \Delta f_{{\rm rtro}} + \Delta f_{{\rm ran}} \end{equation} (23) where Δfosc is oscillator error, Δfpos is position error, Δfvel is velocity error, Δfrion is the residual ionosphere frequency shift error, Δfrtro is the residual troposphere frequency shift error, Δfran is the sum of instrumental errors, other high-order errors and random errors. To estimate the quantity of Δfe in a realistic DCT, we assume that the spacecraft is a typical GNSS satellite whose average height is about 20 000 km above the ground (Cohenour & Graas 2011), and the satellite and ground station are both equipped with OACs with inaccuracy of 10−17 (Westergaard et al.2011). The frequency f0 of the signals used is assumed as 2.0 GHz (Levine 2008). In this study we assume that OACs with their inaccuracy about 1 × 10−17 are available. Hence, we have Δfosc/f0 = 1 × 10−17. The distance between satellite and ground station, |$\left| \vec{r}_{se}\right|$|⁠, is around 22 000 km (the angle between observation sight and zenith is within 35° range at the ground station), and the position error of satellite is around 10−2 m (Kang et al.2006; Guo et al.2015), which is the precision of current GPS satellite precise ephemeris. Satellite's velocity relative to ground is about 3000 m s−1, with its accuracy better than 10−3 m s−1 (Remondi 2004; Zhang et al.2006). The centrifugal acceleration of ground station |$\left| \vec{a}_e \right|$| is at most 3.4 × 10−2 m s− 2 (at the equator) (Pavlis et al.2008). According to eq. (2), the error terms caused by position plus acceleration and velocity uncertainties are, respectively, expresses as \begin{equation} \frac{\Delta f_{{\rm pos}}}{f_0} = \sqrt{\left( \frac{\Delta \vec{r}_{se}\cdot \vec{a}_e}{c^2}\right) ^2 + \left( \frac{\vec{r}_{se}\cdot \Delta \vec{a}_e}{c^2}\right) ^2} \end{equation} (24) \begin{equation} \frac{\Delta f_{{\rm vel}}}{f_0} = \frac{\Delta \vec{v}_{es}\cdot \vec{v}_{es}}{c^2} \end{equation} (25) where |$\Delta \vec{r}_{se}$| and |$\Delta \vec{v}_{es}$| are the position error and velocity error between ground station and spacecraft, respectively, |$\Delta \vec{a}_e$| is the centrifugal acceleration error of ground station. The Earth's angular velocity ω is 7.2921150 × 10−5 rad s−1, with its relative uncertainty Δω = 1.4 × 10−8 (Groten 2000). According to the centrifugal acceleration expression a = ω2Re (on equator), we can estimate the uncertainty of |$\vec{a}_e$| as \begin{equation} \left| \Delta \vec{a}_e \right| < 2\Delta \omega \cdot \omega \cdot R_e + \omega ^2 \cdot \Delta R_e \end{equation} (26) where Re and ΔRe are the Earth's equatorial average radius (6378136.6 m) and its uncertainty (1.5 × 10−8) (Groten 2000). Then, from eq. (26) we have |$| \Delta \vec{a}_e | \le 1.46\times 10^{-9} \, \rm {m\, s^{-2}}$|⁠, and based on eqs (24) and (25), we have \begin{equation} \Delta f_{{\rm pos}}/f_0 \le 3.57\times 10^{-19},\quad \Delta f_{{\rm vel}}/f_0 = 3.33\times 10^{-17}. \end{equation} (27) Eq. (21) provides the ionosphere shift effect, and the residual ionosphere error Δfrion mainly comes from the error of electron density ρ in ionospheric electron density model, which is generally less than 20 per cent (Kang et al.1997; Nava et al.2008). Since the frequency f is set as f0 = 2.0 GHz, satellite's typical height above ground is H = 20 000 km, assuming the electron density |$\bar{\rho }$| being critical value 5 × 1011 m−3 as a peak case (Bilitza et al.2014), then we have \begin{equation} \frac{\Delta f_{i}}{f_0} = \frac{78570\bar{\rho }}{2cf_0^2}\cdot \left| \vec{a}_e\right| \cdot \Delta t\cdot {\rm cos}\alpha < 2.75\times 10^{-18} \end{equation} (28) and the remaining ionospheric frequency shift error Δfrion is smaller than the 20 per cent of Δfi (Kang et al.1997; Nava et al.2008), namely \begin{equation} \frac{\Delta f_{\rm rion}}{f_0} < 5.5\times 10^{-19}. \end{equation} (29) We have also given the troposphere shift correction by eq. (22). According to Earth Global Reference Atmospheric Model (Leslie & Justus 2011), the values of |$\bar{M}_1$| and |$\bar{M}_2$| are 1.8 × 10−5 and 2.0 × 10−5, respectively. Then, we have: \begin{equation} \frac{\Delta f_t}{f_0} = -\frac{60\left( \bar{M}_1 + \bar{M}_2 \right) \left| \vec{r}_{se}\right| \left( \vec{v}_s - \vec{v}_e \right) \cdot \vec{a}_e}{c^2H\left| \vec{v}_s - \vec{v}_e \right|} < 9.47\times 10^{-19}. \end{equation} (30) According to eq. (30), the troposphere refraction effect in DCT is tiny (below the order of 10−18) compared to other error sources, and it can be safely neglected without needing consideration. The final term, random error Δfran comes from other high-order error sources, instrumental errors or system errors. For example, there would be a finite delay in the spacecraft transponder. Since the satellite is in motion, its position when receives signals are different from that when emits signals. Suppose we adopt a transponder (Pierno & Varasi 2013) that is designed for radio frequency (RF) signals, and the frequency f0 = 2.0 GHz is covered by the transponder's frequency range. The delay of the transponder is 800 ns (Pierno & Varasi 2013), thus the satellite moves only 0.24 mm between receiving and emitting signals and the satellite's direction of velocity changes 2.53 × 10−14 rad. The absolute value of the satellite's velocity can be assumed to be unchanged during the 800 ns. Then, according to eqs (24) and (25), the errors caused by this position and velocity change are on the order of 10−19, which can be neglected. As explained in Section 2, the DCT method only involves frequency measurements; time delay does not need to be measured. Thus, the hardware time delays in both ground station and satellite can be neglected. In the short period of 800 ns, the signal's frequency shift does not change, and since the stability of atomic clocks are assumed to be on the level of 10−17, we expect that the noises of frequency measurements are also on the level of 10−17. Thus, we only consider the satellite's position and velocity change in the error budget. Other error sources, such as measurement and data processing errors, can be reduced by multiple measurements and adjustment. We may expect this error term being controlled below 1 × 10−17. Hence, the total error Δfe/f0 is within the order of 10−17, namely \begin{eqnarray} \frac{\Delta f_e}{f_0} &=& \sqrt{\frac{\Delta f_{{\rm osc}}^2 + \Delta f_{{\rm pos}}^2 + \Delta f_{{\rm vel}}^2 + \Delta f_{{\rm rion}}^2 + \Delta f_{{\rm rtro}}^2 + \Delta f_{\rm ran}^2}{f_0^2}} \nonumber \\ &&\approx 10^{-17}. \end{eqnarray} (31) This study focuses on the general idea of determining the geopotential difference between a satellite/spacecraft and a ground station via radio links, closely related to the optical frequency of the OACs. The RF signal links are influenced by signal emission, transmission, reception and inevitable background noises (Rubiola 2005; Dawkins et al.2007), which are referred to as instrument-induced errors. The influences due to the signal's propagation in free space are addressed as signal propagation errors. Concerning the error sources, here we concentrate on errors introduced by signal propagation in free space, leaving the instrument-induced errors to be dealt with in a separate paper, dedicated to instrumental techniques. 5 CONCLUSIONS We formulated and discussed the gravitational frequency shift method for determining gravitational potential difference between a spacecraft (satellite) and a ground station. Based on this formulation, we may draw out gravitational frequency shift signals between a spacecraft and a ground station based on DCT. According to the general theory of relativity, the frequency shift of a light signal between a spacecraft and a ground station reflects the difference of the gravitational potential between them. Suppose atomic clocks with their inaccuracy of 10−17 are available, our analysis shows that the uncertainty of the determined frequency shift by the approach proposed in this study can reach 10−17, which is equivalent to about 1 m2 s− 2 in geopotential or 0.1 m in height determination. In the accuracy requirement in the level of 10−17 as assumed in this study, we need not to consider the ionosphere and troposphere corrections, which are very small, around 10−18. However, for theoretical consideration and future ultrahigh precision requirement, we should take into account these corrections. We used the characteristics of a GNSS satellite for error analysis. In fact any satellite/spacecraft is suitable for DCT, as long as it is visible at the ground station. Low-orbit satellites bear limitations when intended to serve as ‘bridges’ for the determination of geopotential differences between two ground sites that are separated by a long distance. For this purpose, a satellite ought to have a height of around 20 000 km (or larger) above the Earth's surface (like a GNSS satellite or a communication satellite). In practice, the worst case of two ground stations located on the opposite sides of the Earth, more ‘bridges’ (satellite network) might be needed. Alternatively, we may determine the geopotential difference between the mentioned two opposite ground stations via an intermediate ground station. With quick development of time and frequency science and technology, the approach proposed in this study for determining the geopotential difference is prospective. The advantage of this approach lies in that it is a very direct and convenient method to determine the orthometric height compared to the conventional levelling method. Based on this approach, besides the geometric position (coordinates) that can be precisely determined by well-known GNSS technique, the geopotential as well as orthometric height could be determined, realizing the determination of geometric position and orthometric height simultaneously. One of most prospective benefits from this approach is the realization of the world height system unification with centimetre level accuracy, once clocks with inaccuracy of 10−18 are available. We sincerely thank four anonymous reviewers for their valuable comments, corrections and suggestions on the original manuscript, which greatly improved the manuscript. We also thank Editor Prof Joerg Renner for his valuable comments, suggestions and corrections to improve the final manuscript. This study is supported by National 973 Project China (grant nos 2013CB733301 and 2013CB733305), NSFC (grant nos 41210006, 41374022 and 41429401), DAAD (grant no. 57173947) and NASG Special Project Public Interest (grant no. 201512001). REFERENCES Altschul B. et al. 2014 Quantum tests of the Einstein Equivalence Principle with the STE-QUEST space mission Adv. Space Res. 55 1 501 524 Google Scholar Crossref Search ADS WorldCat Bilitza D. Altadill D. Zhang Y. Mertens C. Truhlik V. Richards P. McKinnell L. Reinisch B. The International Reference Ionosphere 2012—a model of international collaboration 2014 J. Space Weather Space Clim. 4 A07 doi:10.1051/swsc/2014004 Bjerhammar A. 1985 On a relativistic geodesy Bull. Géod. 59 3 207 220 Google Scholar Crossref Search ADS WorldCat Bloom B.J. et al. 2014 An optical lattice clock with accuracy and stability at the 10–18 level Nature 506 7486 71 75 Google Scholar Crossref Search ADS PubMed WorldCat Botter T. Williams J. Chiow S.-W. Kellogg J. 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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. © The Authors 2016. Published by Oxford University Press on behalf of The Royal Astronomical Society.
TI - Formulation of geopotential difference determination using optical-atomic clocks onboard satellites and on ground based on Doppler cancellation system
JF - Geophysical Journal International
DO - 10.1093/gji/ggw198
DA - 2016-08-01
UR - https://www.deepdyve.com/lp/oxford-university-press/formulation-of-geopotential-difference-determination-using-optical-mj0Ncl9CUg
SP - 1162
EP - 1168
VL - 206
IS - 2
DP - DeepDyve
ER -