The non-parallelism of the geopotential surfaces at different heights may cause some amounts of the closing discrepancy of a levelling circuit having no connection with the observation errors. In order to evaluate the closing error, the well-known formula after Helmert; ΔΣΔg
0-g
i/g
0 Δh
i has been widely used, where gi, Δhi and g
0 are the observed gravity, levelling increment and a constant respectivelly. Starting from the equation of the astronomical levelling after Molodenski, a new formula for computing the closing error is given as a line integral along the circuit, as follows; ∫dhw=-∫Δθdl
h =-1/gΔΣΔ(Δg
i′Δg
i′+1/2Δh
i+1i+γ
i′+γ
i′+1/2Delta;h
i+1i)
where Δθ denotes the difference θ'-θ between the components of the vertical deviation to the direction of the route θ at a bench mark and θ' at the point which is located above the bench mark and on one external geopotential surface named Exogeoid, Δgi' and γi' being the station free air anomaly and the normal gravity, both reduced to the values on the Exogeoid above i-th bench mark respectivelly. For the purpose of calculating Δg'(P') from the distribution of observed gravity on an undulated physical surface of the earth, an iterative approximation for computing the anomaly of the vertical gradient of gravity was proposed, where the first approximation for Δg'(P') on the Exogeoid is the station free air anomaly Δg(P) itself on the ground, and its first correction Δg
1(P) is obtained from the value of ∂g/∂z anomaly computed from the distribution of the first approximated value Δg (χ, y, h) insted of Δg'(χ, y) on the Exogeoid surface, An example of the closing error was computed on the levelling circuit; Tokyo-Takasaki-Suwa-Yamanashi-Tokyo, the closing discrepancy amounting to +21.45 mm to the direction of clock wise, using the result of levelling with the normal orthometric correction.
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