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The development of remote frequency transfer techniques, especially the appearance of optical clocks with unprecedented stability, has prompted geoscientists to study their applications in geodesy. Using remote frequency transfer technique, by frequency comparison of two optical clocks at two points P and Q connected by optical fibers, one can measure the signal's frequency shift between them, and the geopotential difference between them can be determined based on the gravity frequency shift equation. Given the orthometric height of P, the orthometric height of Q can be determined. Since the present stability of the optical clock has achieved 1 × 10−18 or better and comparing the frequency transfer via optical fiber provides stability at 10−19 level, the optical clock network enables determining the orthometric height at centimeter-level. This study provides a formulation to determine the height difference at one-centimeter level between two points on the ground based on the optical fiber frequency transfer technique.

General relativity theory (GRT) concludes that a precise clock ticks at different running rates if it is under the influence of different geopotentials. Therefore, by comparing the running rates of clocks at arbitrary two stations, the geopotential difference between them can be determined. In this study, with the help of two hydrogen atomic clocks (noted as H-masers), using the two-way satellite time and frequency transfer (TWSTFT) technique, we carried out experiments of the geopotential difference determination at the China Aerospace Science & Industry Corporation (CASIC), Beijing. Here the ensemble empirical mode decomposition (EEMD) method is adopted to remove periodic signals included in the original observations. Finally, the clock-comparison-determined geopotential difference in the experiments is determined. Results show that the difference between the geopotential difference determined by GRT and that determined by measuring tape is about 1316.1 ± 931.0 m2s-2, which is equivalent to 134.3 ± 95.0 m in height, and in consistence with the stability of the H-masers applied in the experiments (at the level of 10−15/day). With the rapid improvement of atomic clocks' accuracy, the geopotential determination by accurate clocks is prospective, and it is promising to realize the unification of the world vertical height system (WVHS).

According to the general theory of relativity, two clocks placed at two different positions with different geopotentials run at different rates. Thus one can determine the geopotential difference between these two points by comparing the running rates of the two clocks. Using the most precise optic-atomic clocks whose stability achieves 10−18 level and the time transfer technique with comparison accuracy higher than 10ps level by 100 m coaxial cable, the relativistic geodesy method for determining the geopotential may be realizable in the near future. In this paper, we propose, design and describe in detail an approach for determining the geopotential difference between two points based upon a simulation experiment. We select two stations A and B whose ground distance is within 100 m, height difference being about 30 m. Each station is equipped with an atomic clock whose instability is 3.2×10−16/τ (where τ is time in second). And the two stations are connected with a coaxial cable for time transfer. Our simulation experiment results show that the accuracy could reach 0.16 m2/s2 (equivalent to 1.6 cm in height) level after a four-hour period of observation.

According to the space-geodetic data recorded at globally distributed stations over solid land spanning a period of more than 20-years under the International Terrestrial Reference Frame 2008, our previous estimate of the average-weighted vertical variation of the Earth's solid surface suggests that the Earth's solid part is expanding at a rate of 0.24 ± 0.05 mm/a in recent two decades. In another aspect, the satellite altimetry observations spanning recent two decades demonstrate the sea level rise (SLR) rate 3.2 ± 0.4 mm/a, of which 1.8 ± 0.5 mm/a is contributed by the ice melting over land. This study shows that the oceanic thermal expansion is 1.0 ± 0.1 mm/a due to the temperature increase in recent half century, which coincides with the estimate provided by previous authors. The SLR observation by altimetry is not balanced by the ice melting and thermal expansion, which is an open problem before this study. However, in this study we infer that the oceanic part of the Earth is expanding at a rate about 0.4 mm/a. Combining the expansion rates of land part and oceanic part, we conclude that the Earth is expanding at a rate of 0.35 ± 0.47 mm/a in recent two decades. If the Earth expands at this rate, then the altimetry-observed SLR can be well explained. Keywords: ITRF2008 coordinates, Ice melting, Thermal expansion, Earth expansion, Sea level rise (SLR), Space-geodetic data, Velocities altimetry, Earth's solid surface