Measuring geopotential difference between two points

Conventionally, the potential difference between two points P and Q located on the Earth’s surface are determined by gravimetry and levelling (Heiskanen and Moritz 1967), the drawback of which is that it is almost impossible to connect these two points in the case that they are located on two continents, because it is well known that the potential surface of the mean sea level (MSL) is not an equipotential surface. In another aspect, if given the gravity data on the Earth’s surface, one might determine the potential difference between two points by using the Stokes method or Molodensky method (ibid). In this case the potential field is determined and consequently the potential difference between two arbitrary points could be determined. However, Stokes method is approximate due to the requirement that the measured gravity data should be deduced on the geoid, which gives rise to obvious errors; Molodensky method is complicated, and the determined is the quasi-geoid. The transformation from quasi-geoid to geoid needs again levelling, which encounter difficulties in connecting two points located on different continents. Hence, applying the conventional approach, it is very dif?cult to establish a unified global datum system (with adequate precision). To avoid this difficulty, Bjerhammar (1985) suggested that the geopotential difference between two arbitrary points P and Q might be determined by using two precise clocks put at P and Q respectively, based on the theory of general relativity (e.g. Weinberg 1972). The basic principle is stated as follows. According to Einstein’s general relativity theory, the running rate of a clock is closely related with the geopotential at the point where the clock is located: the clock located at the position with higher geopotential runs faster than the clock located at the position with lower geopotential. Based on the above considerations, Bjerhammar defined a relativistic geoid as follows: the relativistic geoid is the equipotential surface nearest to MSL on which a precise clock runs with the same rate. This is referred to as the equi-time-rate geoid (Shen 1998).

Different from Bjerhammar, Shen et al(1993) argued that it is better to determine the geopotential difference by receiving the light signals emitted by an emitter (which could be located on the Earth’s surface or in space), based on the gravity frequency shift equation, which is not only the result derived from the general relativity but also the quantum mechanics and energy conservation law (Shen 1998). Further, it was shortly proposed that (Shen et al 1993; Shen 1998), it might be possible to directly determine the potential difference between two points P and Q on the surface of the Earth (even these two points are located on different continents) by receiving the light signals emitted by a satellite or a distant star.

Suppose an emitter is equipped with a fiying satellite, which can emit light signals with regular intervals. Then, by receiving the light signals emitted by the emitter simultaneously at two points P and Q, one could determine the geopotential difference between P and Q, based on the gravity frequency shift principle. Then, one can determine the potential field based on the truncated spherical harmonic expansion combining with the least squares adjustment (e.g., Rummel et al 1993) or the fictitious compress recovery approach (Shen 2004), the main idea of the latter is stated as follows. Compress the boundary value, given on the Earth’s surface, on the surface of an inner sphere located inside the Earth, and using Poisson integral one gets a harmonic field in the domain outside the inner sphere; compress again the residuals between the initial boundary value and the calculated one, provided on the Earth’s surface, on the surface of the inner sphere, and using Poisson integral one gets again a harmonic field in the domain outside the inner sphere; this procedure is repeated and a series result is obtained, which coincides with the real field in the domain outside the Earth.

The gravity frequency shift principle is stated as follows (Shen et al. 1993; Shen 1998).

Suppose a light signal with frequency f is emitted from point P by an emitter that is not necessarily located on a satellite, and the signal is received at point Q by a receiver (Cf. Fig.1). Because of the geopotential difference between these two points, the frequency of the received light signal is not f but f’. Using fP and fQ to denote f and f’ respectively, the following equation holds (Pound and Snider 1965; Shen et al. 1993):

where c is the velocity of light in vacuum, WP and WQ are the geopotentials at points P and Q respectively. Expression (1) is referred to as the gravity frequency shift equation (Pound and Snider 1965), or properly called the geopotential frequency shift equation due to the fact that the frequency shift is caused by the geopotential difference. Katila and Riski (1981) con?rmed Eq. (1) with accuracy 10-2. Vessot et al.

1) with accuracy 10-2. Vessot et al. (1980) proved that Eq. (1) is correct to the accuracy of 10-4. Scientists believe that Eq. (1) is correct, because it is a result derived from the theory of general relativity. As mentioned previously, Eq. (1) can be also derived out based on quantum theory and energy conservation law (Shen, 1998). Hence, suppose the geopotential at point P is given, then, from Eq. (1) the geopotential at an arbitrary point Q can be determined by measuring the geopotential frequency shift between P and Q.

Set at point P an emitter which emits a light signal with frequency f and a receiver at point Q receives the light signal emitted by the emitter at point P. Suppose the received signal’s frequency is f’. Then, it could be compared the frequency f’ of the received light signal with it self’s standard frequency f (this is not only the emitting frequency at point P but also the standard innate frequency of the receiver at point Q), and the frequency shift fif = f’ – f can be determined. Consequently, according to Eq. (1) the geopotential difference fiWPQ between P and Q
can be determined. Applying the same principle it will be found the geopotential difference fiWOP = WP – WO between the point O on the geoid and an arbitrarily point P, where WO = C0 is the geoid geopotential constant. If C0 is a known constant, WP as well as WQ can be determined. Hence, based on the geopotential frequency shift equation (1), the relativistic geoid could be defined as follows (Shen et al 1993): the relativistic geoid is the equi-potential surface nearest to MSL on which there does not exist light signal’s

frequency shift. This is referred to as the equi-frequency geoid (Shen 1998). Now, suppose the light signal emitter E is located in a satellite, and two light signal receivers at P and Q receive the light signals coming from E corresponding to an emitting time t (Cf. Fig.2). Further suppose the received signals’ frequencies corresponding to time t are recorded by receivers at P and Q in some way, respectively, i.e., fP and fQ corresponding to time t are recorded by receivers at P and Q, respectively. Note that the time at which the signal is received by P is generally different from that by Q. By comparing the received frequencies fP and fQ it could be determined the geopotential difference ?WPQ = WQ – WP (Shen et al., 1993), which is just given by Eq. (1).

By the above mentioned way, the geopotential on the Earth’s whole surface could be determined based on the geopotential frequency shift approach by receiving the light signals emitted by satellites. Once the geopotential W on the Earth’s surface is determined, the potential field V outside the Earth could be determined, and as a result, the geoid could be more precisely determined.

The emitter could be a distant stable star. The receivers at P and Q could be designed in such a way that the received light signals might be recorded on diskettes at P and Q in details. Then, comparing the recorded diskettes at a centre process system. In this way, the frequency shift information between P and Q might be drawn out.

In the case that the emitter is located on the Earth’s surface, the potential difference between the two points P and Q could be also directly determined based on the same principle, as long as the the receivers at these two points could simultaneously receive the light signals emitted by the emitter.

It is noted that the accuracy for determining the potential difference by using the geopotential frequency shift approach is becoming more and more prospective for the goal of determining the centimeter-geoid, which is mainly depending on the frequency stability of the receiver. More than 10 years ago, the frequency stability was around 10-14, which is very poor for determining the geoid with an adequate accuracy. At present however, the frequency stability is around 10-15 – 10-16 (HMC Project 2005), which corresponds to the height variation about 1 m. In the next ten years, it is prospective that the frequency stability 10-17 – 10-18 could be achieved, which corresponds to the height variation 1cm.

The great advantage by using the geopotential frequency shift approach lies in that a uni?ed global datum system could be established: two receivers located at two datum points A and B which belong to two separated continents could simultaneously receive the signals emitted by a satellite source emitter, and consequently the frequency shift between A and B is determined; then, based on the geopotential frequency shift equation the geopotential difference between A and B is determined.

With satellite technique (e.g. CHAMP mission, Cf. Gerlach 2003), the potential on the satellite surface could be determined, and consequently it could be determined the Earth’s could be determined the Earth’s external potential field, which is quite accurate for the long and middle wave-length of the field, but not for the short wave-length of the field.

The will-be launched GOCE system could provide a 10 x 10 global gravity model with accuracy level around 1cm. However, concerning a single datum point A, the accuracy of the potential at A determined based on the global gravity model is far from centimeter level. To get more precise result, local measurements (gravimetry and leveling) are needed. Hence, with the goal of determining the global centimeter-geoid, the connections between different datum points located on different continents should be precisely established, which might be completed by geopotential frequency shift approach.

Acknowledgements

This study is funded in parts by Natural Science Foundation China (No.40574004), and the Foundation of the Key Lab. of Geospace Environment and Geodesy, Ministry of Education China (03-04-13).

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