The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 152

g B dH

gdh ¼

g
g B dh,

ð 4 : 31 Þ

dH ¼

where g is the gravity at dh along the leveling line and g B is the gravity at dH along
the plumb line from point B. Substituting ( 4.31 ) into ( 4.30 ), we have:

ð ð

g
g B dh

H O ¼

dH ¼ : ð 4 : 32 Þ

CB OAB

The gravity along the plumb line, g B , varies with depth. Let their average be g m ;
then:

ð

1
g m

H O ¼

gdh, ð 4 : 33 Þ

OAB

where g m is a certain value relative to a certain surface point. Ð gdh, independent of
the leveling path, is the potential energy difference between the level surface
passing through point B and the geoid. So, the orthometric height is a unique
value. However, since g m is the average gravity at depth and we cannot know for
sure the subsurface mass density distribution, g m therefore can neither be measured
nor precisely calculated. Thus, the orthometric height of a point can only be
approximately evaluated.

4.3.4 Normal Height

The reason why orthometric height cannot be precisely obtained lies in that g m of
point B cannot be measured accurately. Replacing g m in ( 4.33 ) with the normal
gravity

m , one can get the normal height that belongs to another height system,
denoted by H N , namely:

ʳ

ð

1
ʳ

H N ¼ gdh, ð 4 : 34 Þ

m

OAB

where g can be measured through gravimetry along the leveling line, dh can be
measured by leveling, and

m can be calculated by the normal gravity formulae
( 4.24 ) and ( 4.25 ). Thus, we can get a precise normal height whose value is unique,
without varying with the changes in leveling routes. The concept of normal heights
was formulated by the Russian geodesist M.S. Molodensky in 1945. In China, the
normal height system is adopted as the unified system for computing the height of a
point on the Earth's surface.

ʳ