On Theoretical Closing Error of Levelling Circuit

Abstract

The equations of theoretical closing error in levelling circuit due to the non-parallelism of geopotential surfaces have been proposed.
These equations are proved to be equivalent in principle as follows:
ΔΣΔg-g_i^m/gdh_i=-1/gΔΣΔ(g_i^m-γ_i^m)dh_i-1/gΔΣΔγ_i^mdh_i+ΔΣΔdh_i-1/gΔΣΔγ_i^mdh_i≈1/gΔΣΔ(γ⁰_i+1/2-γ⁰_i-1/2)h_i-ΔΣΔdh_i
∴ΔΣΔg-g_i^m/gdh_i≈-1/gΔΣΔ(g_i^m-γ_i^m)dh_i+1/gΔΣΔ(γ⁰_i+1/2-γ⁰_i-1/2)h_i
where dh_i = h_i-h_i-1
h_i : measurement height at the point i in levelling circuit
g_i : gravity at i
g_i^m : mean of gravity values in the interval [i-1, i]
g : mean of gravity values in the levelling circuit
γ_i : standard gravity at i
γ_i⁰: standard gravity on the reference ellipsoid below i
γ_i^m : mean of standard gravity values in the interval [i-1, i]
Let dh_i be the sum of the true difference of height dh_i⁰ and the observation error δχ_i dh_i=dh_i⁰ +δχ_i. Under suitable conditions,
ΔΣΔg-g_i^m/gdh_i=ΔΣΔdh_i-1/gΔΣΔg_i^mdh_i
=ΔΣΔ(dh_i⁰+δχ_i)-1/gΔΣΔg_i^m+δχ_i∴∴1/gΔΣΔg_i^mδχ_i (∴ΔΣΔg_i^mdh_i⁰=0)
≈ΔΣΔ(dh_i⁰+δχ_i)-ΔΣΔδχ_i
ΔΣΔdh_i⁰
The physical meaning of ΔΣΔg-g_i^m/gdh_i is that the theoretical value ΔΣΔdh_i⁰ can be obtained by separating the observation error ΔΣΔδχ_i from the observation value ΔΣΔdh_i using the gravity values in the levelling circuit.The theoretical closing error is defined as the simplest form,
T.C.E≡ΔΣΔdh_i⁰≈Σg-g_i^m/gdh_i .
Here it is to be noticed that the equation is approximately derived from the definition mentioned above.