Fundamentals of Satellite Navigation - Understanding GPS: Principles and Applications - page 30

2.2.3.1 Determination of User Geodetic Coordinates: Latitude, Longitude, and

Height
The ECEF coordinate system is affixed to the WGS 84 reference ellipsoid, as shown
in Figure 2.7, with the point O corresponding to the center of the Earth. We can now
define the parameters of latitude, longitude, and height with respect to the reference
ellipsoid. When defined in this manner, these parameters are called geodetic . Given
a user receiver's position vector of u

= ( x u ,y u ,z u ) in the ECEF system, we can com-

pute the geodetic longitude ( λ ) as the angle between the user and the x -axis, mea-

sured in the xy -plane

y
x

u

arctan , x ≥ 0

u

u

y
x

λ = 180 °+ arctan

u

, x < 0 and y ≥ 0 (2.1)

u u

u

y
x

u

−°+ 180 arctan , x < 0 and y < 0

u u

u

In (2.1), negative angles correspond to degrees west longitude. The geodetic
parameters of latitude ( ) and height ( h ) are defined in terms of the ellipsoid normal

at the user's receiver. The ellipsoid normal is depicted by the unit vector n in Figure
2.7. Notice that unless the user is on the poles or the equator, the ellipsoid normal
does not point exactly toward the center of the Earth. A GPS receiver computes
height relative to the WGS 84 ellipsoid. However, the height above sea level given on
a map can be quite different from GPS-derived height due to the difference, in some
places, between the WGS 84 ellipsoid and the geoid (local mean sea level). In the
horizontal plane, differences between the local datum, such as North American
Datum 1983 (NAD 83) and European Datum 1950 (ED 50), and WGS 84 can also
be significant.
Geodetic height is simply the minimum distance between the user (at the end-
point of the vector u ) and the reference ellipsoid. Notice that the direction of mini-
mum distance from the user to the surface of the reference ellipsoid will be in the
direction of the vector n . Geodetic latitude,

ϕ

, is the angle between the ellipsoid nor-
mal vector n and the projection of n into the equatorial ( xy ) plane. Conventionally,
ϕ

ϕ

ϕ
is taken to be negative if z u < 0. With respect to Figure 2.7, geodetic latitude is the
angle NPA, where N is the closest point on the reference ellipsoid to the user, P is the
point where a line in the direction of n intersects the equatorial plane, and A is the
closest point on the equator to P. Numerous solutions, both closed-form and itera-
tive, have been devised for the computation of geodetic curvilinear coordinates (

is taken to be positive if z u > 0 (i.e., if the user is in the northern hemisphere), and

,
h ) from Cartesian coordinates ( x, y, z ). A popular and highly convergent iterative
method by Bowring [6] is described in Table 2.1. For the computations shown in
Table 2.1, a , b , e 2 , and

ϕ , λ

′

2

are the geodetic parameters described previously. Note
that the use of “N” in Table 2.1 follows Bowring [6] and does not refer to geoid
height described in Section 2.2.4.

e