The Cartesian coordinate frame of reference used in GPS/GLONASS is called

Earth-Centered, Earth-Fixed (ECEF). ECEF uses three-dimensional XYZ coordinates

(in meters) to describe the location of a GPS user or satellite. The term "Earth-

Centered" comes from the fact that the origin of the axis (0,0,0) is located at the mass

center of gravity (determined through years of tracking satellite trajectories). The term

"Earth-Fixed" implies that the axes are fixed with respect to the earth (that is, they

rotate with the earth). The Z-axis pierces the North Pole, and the XY-axis defines the

equatorial plane (Figure 1).

Figure 1 ECEF Coordinate Reference Frame

ECEF coordinates are expressed in a reference system that is related to mapping

representations. Because the earth has a complex shape, a simple, yet accurate,

method to approximate the earth's shape is required. The use of a reference ellipsoid

allows for the conversion of the ECEF coordinates to the more commonly used

geodetic-mapping coordinates of Latitude, Longitude, and Altitude (LLA). Geodetic

coordinates can then be converted to a second map reference known as Mercator

Projections, where smaller regions are projected onto a flat mapping surface (that is,

Universal Transverse Mercator ­ UTM or the USGS Grid system).

A reference ellipsoid can be described by a series of parameters that define its

shape and which include a semi-major axis (a), a semi-minor axis (b) and its first

eccentricity (e 2) and its second eccentricity (e '2) as shown in Figure 2. Depending on

the formulation used, ellipsoid "flattening" may be required.

    Figure 2 Ellipsoid Parameters

For global applications, the geodetic reference (datum) used for GPS is the World

Geodetic System 1984 (WGS84). This ellipsoid has its origin coincident with the

ECEF origin. The X-axis pierces the Greenwich meridian (where longitude = 0

degrees) and the XY plane make up the equatorial plane (latitude = 0 degrees).

Altitude is described as the perpendicular distance above the ellipsoid surface (which

not to be confused with the mean sea level datum).

Figure 3 ECEF and Reference Ellipsoid

LLA to ECEF

The conversion between the two reference coordinate systems can be performed

using closed formulas (although iteration methods also exist). The conversion from

LLA to ECEF (in meters) is shown below.

X = [N + H] cos ¢. cos £                    (1)

Y = [N + H] cos ¢. sin £                    (2)

Z = [(b 2 /a 2) N + h] sin ¢.                  (3)

Where:

¢ = latitude

£= longitude

h = height above ellipsoid (meters)

N = Prime Vertical of Curvature (meters) is defined as:

                a 2

N = ____________________                            (4)

        [a 2 cos 2 ¢. + b 2 sin 2 £ ] 1/2

ECEF to LLA

    The conversion between XYZ and LLA is slightly more involved but can be

achieved using the following closed formula set:

¢ = arctan [ _________________                ]      (5)

£ = arctan    [ ____    ]                            (6)

                        X

h = [  P

_______      ] ­ N                              (7)

        Cos ¢

Where auxiliary values are:

¥ = arctan [ _____      ]                              (8)

                    p b

p = [X 2 +Y 2 ] 1/2                                                      (9)

GPS Heights

The height determined by GPS measurements relates to the perpendicular distance

above the reference ellipsoid and should not be confused with the more well-known

height datum Mean Sea Level (MSL). The datum that defines the MSL (also called

the geoid) is a complex surface that requires dense and accurate gravity data to define

its shape. The WGS84 ellipsoid approximates the geoid on a worldwide basis with

deviations between the two datums never exceeding 100 meters. The transformation

between the two surfaces is illustrated in Figure 4.

Figure 4 Ellipsoid and MSL Reference Datums

p ­ e 2 a cos 3 ¥

Z + e '2 b sin 3 ¥

Y

Z a

The conversion between the two reference datums is shown by:

h = H + N                                                (10)

where:

h        ellipsoidal height (Geodetic)

H      orthometric height (MSL)

N      geoid separation (undulation)

Note ­ The ellipsoid/geoid separation ranges from a value of +100 meters to

­100 meters.

Although the conversion between the different height datums is straightforward, the

accuracy at which the undulation is known (N) varies greatly with gravity

measurement data density. It is even more difficult to determine in mountainous

regions where mass distribution can vary rapidly.