The geometry of the earth - Physical Geodesy - page 227

These relations may be used to find computational formulas for the cur-
vature reductions δϕ and δλ .Wehave

= d C
g

dC
g

dC
g

d (OC) = dH − −

(5-136)

= dC

C
g 2 dg −

dC
g g 2 dg + g

C − g dC

g g −

= −

g

or

H
g

dg + g

− g

d (OC) = − dn . (5-137)

g

By substituting this into (5-131) and (5-132), we obtain

H
g

∂g

∂x + g

− g

δϕ = − tan β 1 ,

g

(5-138)

H
g

∂g

∂y + g

− g

δλ cos ϕ = −

tan β 2 ,

g

where we have set

tan β 1 = ∂n tan β 2 = ∂n

∂x , ∂y ,

(5-139)

so that β 1 and β 2 are the angles of inclination of the north-south and east-
west profiles with respect to the local horizon; g is the mean value of gravity
between the geoid and the ground. In these formulas, we need only this
mean value g , together with its horizontal derivatives, and the ground value
g , whereas in (5-128) and (5-129), we must know the horizontal derivatives
of gravity all along the plumb line. The detailed shape of the plumb lines
does not directly enter into (5-138) as it does into (5-128) and (5-129).
The mean value g is found by a Prey reduction of the measured gravity
g . In order that the numerical differentiations ∂g/∂x and ∂g/∂y give reliable
results, a dense gravity net around the station is necessary, and the Prey
reduction must be performed carefully. The inclination angles β 1 and β 2 are
taken from a topographical map.
The sign of these corrections may be found in the following way. If g
decreases in the x -direction, then formulas (5-128) and (5-138) give δϕ > 0
and Fig. 5.18 shows that Φ at P 0 is then greater than at P . The same holds
for Λ, so that we have

Φ geoid =Φ ground + δϕ ,

(5-140)

Λ geoid =Λ ground + δλ .