GEODETIC REFERENCE SYSTEMS - GPS Satellite Surveying - page 35

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From (2.58) it follows that

H

W = W 0 − gdH (2.62)

0

or

H

C = W 0 − W = gdH (2.63)

0

or

W C

dW
g

dC
g

H =− = (2.64)

W 0 0

[35

Equation (2.63) shows how combining gravity observations and leveling yields po-
tential differences. The increment dH is obtained from spirit leveling, and the gravity
g is measured along the leveling path. Consider a leveling loop as an example. Be-
cause one returns to the same point when leveling a loop, i.e., one returns to the same
equipotential surface, (2.63) implies that the integral (or the sum) of the products gdH
adds up to zero. Because g varies along the loop, the sum over the leveled differences
dH does not necessarily add up to zero.
The difference between the orthometric heights and the leveled heights is called the
orthometric correction. Expressions for computing the orthometric correction from
gravity are available in the specialized geodetic literature. An excellent introduction
to height systems is Heiskanen and Moritz (1967, Chapter 4). Guidelines for accurate
leveling are available from the NGS (Schomaker and Berry, 1981).

Lin
—
4.9
——
No
PgE

[35

2.3.2 Ellipsoid of Revolution

Th e ellipsoid of revolution, called here simply the ellipsoid, is a relatively simple
m athematical figure that closely approximates the actual geoid. When using an ellip-
so id for geodetic purposes, we need to specify its shape, location, and orientation with
re spect to the earth. The size and shape of the ellipsoid is defined by two parameters:
th e semimajor axis a and the flattening f . The flattening is related to the semiminor
ax is b by

a − b

f = (2.65)

a

Appendix B contains the details of the mathematics of the ellipsoid and common val-
ues for a and b . The orientation and location of the ellipsoid often depends on when
and how it was established. In the presatellite era, the goal often was to establish a
local ellipsoid that best fitted the geoid in a well-defined region, i.e., the border of
a nation-state. The third axis, of course, always pointed toward the North Pole and
the first axis in the direction of the Greenwich meridian. Using local ellipsoids as