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:
ΔΣΔ
g-
gim/
gdhi=-1/
gΔΣΔ(
gim-γ
im)
dhi-1/
gΔΣΔγ
imdhi+ΔΣΔ
dhi-1/
gΔΣΔγ
imdhi≈1/
gΔΣΔ(γ
0i+1/2-γ
0i-1/2)
hi-ΔΣΔ
dhi ∴ΔΣΔ
g-
gim/
gdhi≈-1/
gΔΣΔ(
gim-γ
im)
dhi+1/
gΔΣΔ(γ
0i+1/2-γ
0i-1/2)
hi where
dhi =
hi-
hi-1 hi : measurement height at the point i in levelling circuit
gi : gravity at
i gim : 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
im : mean of standard gravity values in the interval [
i-1,
i]
Let
dhi be the sum of the true difference of height
dhi0 and the observation error δχ
i dhi=
dhi0 +δχ
i. Under suitable conditions,
ΔΣΔ
g-
gim/
gdhi=ΔΣΔ
dhi-1/
gΔΣΔ
gimdhi=ΔΣΔ(
dhi0+δχ
i)-1/
gΔΣΔ
gim+δχ
i∴∴1/
gΔΣΔ
gimδχ
i (∴ΔΣΔ
gimdhi0=0)
≈ΔΣΔ(
dhi0+δχ
i)-ΔΣΔδχ
iΔΣΔ
dhi0 The physical meaning of ΔΣΔ
g-
gim/
gdhi is that the theoretical value ΔΣΔ
dhi0 can be obtained by separating the observation error ΔΣΔδχ
i from the observation value ΔΣΔ
dhi using the gravity values in the levelling circuit.The theoretical closing error is defined as the simplest form,
T.C.E≡ΔΣΔ
dhi0≈Σ
g-
gim/
gdhi .
Here it is to be noticed that the equation is approximately derived from the definition mentioned above.
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