Heights - Physical Geodesy - page 159

or

g = g +0 . 0424 H ( g in gal, H in km) . (4-31)
The factor 0.0424 refers to the normal density =2 . 67 g/cm 3 . The corre-
sponding formula for arbitrary constant density is, by (3-43),

1
2 ∂h +2 πG H.

∂γ

g = g − (4-32)

If we use g according to (4-31) or (4-32) in the basic formula (4-27), we
obtain the so-called Helmert height:

C
g +0 . 0424 H

H = (4-33)

with C in g.p.u., g in gal and H in km.
As we have seen in Sect. 3.5, this approximation replaces the terrain
with an infinite Bouguer plate of constant density and of height H .Thisis
often sucient. Sometimes, in high mountains and for highest precision, it
is necessary to apply to g a more rigorous Prey reduction, such as the three
steps described in Sect. 3.5. A practical and very accurate method for this
purpose has been given by Niethammer in 1932. It takes the topography into
account, assuming only that the free-air gradient is normal and the density
is constant down to the geoid.
It is also sucient to calculate g as the arithmetic mean of gravity g mea-
sured at the surface point P and of gravity g 0 computed at the corresponding
geoidal point P 0 by the Prey reduction:

g = 2 ( g + g 0 ) . (4-34)

This presupposes that gravity g varies linearly along the plumb line. This
can usually be assumed with sucient accuracy, even in extreme cases, as
shown by Mader (1954) and by Ledersteger (1955).

Orthometric correction
The orthometric correction is added to the measured height difference, in
order to convert it into a difference in orthometric height.
We let the leveling line connect two points A and B (Fig. 4.5) and apply
a simple trick first:

H dyn + H dyn +( H dyn H dyn ∆ H AB = H B − H A = H B − H A −

B −

)

B A A

(4-35)

=∆ H dyn H dyn H dyn +( H B − ) − ( H A − ) .

AB B A