The standard physical model of the Earth used for GPS applications is the DOD’s World Geodetic System 1984 (WGS 84) [5]. One part of WGS 84 is a detailed model of the Earth’s gravitational irregularities. Such information is necessary to derive accurate satellite ephemeris information; however, we are concerned here with estimating the latitude, longitude, and height of a GPS receiver. For this pur- pose, WGS 84 provides an ellipsoidal model of the Earth’s shape, as shown in Fig- ure 2.7. In this model, cross-sections of the Earth parallel to the equatorial plane are circular. The equatorial cross-section of the Earth has radius 6,378.137 km, which is the mean equatorial radius of the Earth. In the WGS 84 Earth model, cross-sec- tions of the Earth normal to the equatorial plane are ellipsoidal. In an ellipsoidal cross-section containing the z-axis, the major axis coincides with the equatorial diameter of the Earth. Therefore, the semimajor axis,a, has the same value as the mean equatorial radius given previously. The minor axis of the ellipsoidal cross-sec- tion shown in Figure 2.7 corresponds to the polar diameter of the Earth, and the semiminor axis,b, in WGS 84 is taken to be 6,356.7523142 km. Thus, the eccen- tricity of the Earth ellipsoid,e, can be determined by
e b
= 1−a
2 2
WGS 84 takese2 =0.00669437999014. It should be noted that this figure is extremely close, but not identical, to the Geodetic Reference System 1980 (GRS 80) ellipsoid quantity of e2 =0.00669438002290. These two ellipsoids differ only by 0.1 mm in the semiminor axis, b.
Another parameter sometimes used to characterize the reference ellipsoid is the second eccentricity,e′, which is defined as follows:
′ = − =
e a
b
a be
2
2 1
WGS 84 takese′2=0.00673949674228.
Equatorial plane b
z
u
w n
N h S
φ P O
a
A
Figure 2.7 Ellipsoidal model of Earth (cross-section normal to equatorial plane).
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 calledgeodetic. Given a user receiver’s position vector ofu=(xu, yu, zu) 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 thexy-plane
λ=
≥
°+
<
arctan ,
arctan , y
x x
y
x x
u u
u
u u
u
0
180 0 and
and y y
x x y
u
u u
u u
≥
− °+
< <
0
180 arctan , 0 0
(2.1)
In (2.1), negative angles correspond to degrees west longitude. The geodetic parameters of latitude (ϕ) and height (h) are defined in terms of the ellipsoid normal at the user’s receiver. The ellipsoid normal is depicted by the unit vectornin 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 vectoru) 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 vectorn. Geodetic latitude,ϕ, is the angle between the ellipsoid nor- mal vectornand the projection ofninto the equatorial (xy) plane. Conventionally, ϕis taken to be positive ifzu> 0 (i.e., if the user is in the northern hemisphere), andϕ is taken to be negative ifzu< 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 ofnintersects 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 (ϕ,λ, 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, e2, ande′2are 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.
2.2.3.2 Conversion from Geodetic Coordinates to Cartesian Coordinates in ECEF Frame
For completeness, equations for transforming from geodetic coordinates back to Cartesian coordinates in the ECEF system are provided later. Given the geodetic parametersλ,ϕ, andh, we can computeu=(xu, yu, zu) in closed form as follows:
( )
( )
u=
+ − +
+ − +
a e
h a
e
h cos
tan
cos cos
sin tan
sin λ
φ λ φ
λ
φ λ
1 1
1 1
2 2
2 2
( )
cos
sin sin
sin
φ
φ
φ φ
a e
e 1 h 1
2
2 2
−
− +
2.2.3.3 WGS 84 Reference Frame Relationships
There have been four realizations of WGS 84 as of this edition. The original WGS 84 was used for the broadcast GPS orbit beginning January 23, 1987. WGS 84 (G730), where the “G730” denotes GPS week, was used beginning on June 29, 1994. WGS 84 (G873) started on January 29, 1997 [5]. And, the current frame, WGS 84 (G1150), was introduced on January 20, 2002. These reference frame real- izations have brought the WGS 84 into extremely close coincidence with the Inter-
Table 2.1 Determination of Geodetic Height and Latitude in Terms of ECEF Parameters
p= x2+y2 tanu z
p a
= b
Iteration Loop
cos tan
2
2
1 u 1
= u + sin2u= −1 cos2u
tan sin
ϕ= + ′ cos
−
z e b u
p e a u
2 3
2 3
tanu b tan
= a
ϕ
until tanuconverges, then
N a
= e
− 1 2sin2φ
h p
= −N ≠ ± °
cosφ ϕ 90
otherwise
h= z −N+e N ≠
sinφ 2 ϕ 0
national Terrestrial Reference Frame (ITRF), administered by the International Association of Geodesy. For example, the WGS 84 (G1150) matches the ITRF2000 frame to better than 1 cm, one sigma [7].
The fact that there have been four realizations of WGS 84 has led to some confu- sion regarding the relationship between WGS 84 and other reference frames. In par- ticular, care must be used when interpreting older references. For example, the original WGS 84 and NAD 83 were made coincident [8], leading to an assertion that WGS 84 and NAD 83 were identical. However, as stated above, WGS 84 (G1150) is coincident with ITRF2000. It is known that NAD 83 is offset from ITRF2000 by about 2.2m. Hence, the NAD 83 reference frame and the current realization of WGS 84 can no longer be considered identical.