The difference in co-ordinates between data is commonly referred to as datum shift. The datum shift between two particular datums can vary from one place to another within one country or region, and can be anything from zero to hundreds of metres (or several kilometres for some remote islands). The North Pole, South Pole and Equator may be assumed to be in different positions on different datums, so True North may be very slightly different. Different datums use different estimates for the precise shape and size of the Earth (reference ellipsoids).
The difference between WGS84 and OSGB36 is up to 140 metres (450 feet), which for some navigational purposes is an insignificant error. For most applications, such as surveying and dive site location for SCUBA divers, 140 metres is an unacceptably large error.
The main reason that there are a number of datums is that before the advent of GPS positioning, national map making organizations didn't have a common surveying reference point and only produced maps for their locality.
Datum
In surveying and geodesy, a datum is a reference point or surface against which position measurements are made, and an associated model of the shape of the earth for computing positions. Horizontal datums are used for describing a point on the earth's surface, in latitude and longitude or another coordinate system. Vertical datums are used to measure elevations or underwater depths.
Horizontal datums
The horizontal datum is the model used to measure positions on the earth. A specific point on the earth can have substantially different coordinates, depending on the datum used to make the measurement. There are hundreds of locally-developed horizontal datums around the world, usually referenced to some convenient local reference point. Contemporary datums, based on increasingly accurate measurements of the shape of the earth, are intended to cover larger areas. The WGS84 datum, which is almost identical to the NAD83 datum used in North America, is a common standard datum.
Vertical datum
A vertical datum is used for measuring the elevations of points on the earth's surface. Vertical data are either tidal, based on sea levels, gravimetric, based on a geoid, or geodetic, based on the same ellipsoid models of the earth used for computing horizontal datums.
In common usage, elevations are often cited in height above sea level; this is a widely used tidal datum. Because ocean tides cause water levels to change constantly, the sea level is generally taken to be some average of the tide heights. Mean lower low water — the average of the lowest points the tide reached on each day during a measuring period of several years — is the datum used for measuring water depths on some nautical charts, for example; this is called the chart datum. Whilst the use of sea-level as a datum is useful for geologically recent topographic features, sea level hasn't stayed constant throughout geological time, so is less useful when measuring very long-term processes.
A geodetic vertical datum takes some specific zero point, and computes elevations based on the geodetic model being used, without further reference to sea levels. Usually, the starting reference point is a tide gauge, so at that point the geodetic and tidal datums might match, but due to sea level variations, the two scales may not match elsewhere. One example of a geoid datum is NAVD88, used in North America, which is referenced to a point in Quebec, Canada.
Geodetic coordinates
In geodetic coordinates the Earth's surface is approximated by an ellipsoid and locations near the surface are described in terms of latitude (), longitude () and height (). The ellipsoid is completely parameterised by the semi-major axis and the flattening .
Geodetic versus geocentric latitude
It is important to note that geodetic latitude () is different than geocentric latitude (). Geodetic latitude is determined by the angle between the normal of the spheroid and the plane of the equator, whereas geocentric latitude is determined around the centre (see figure). Unless otherwise specified latitude is geodetic latitude.
Geodetic defining parameters
Parameter
Symbol
Semi-major axis
a
Reciprocal of flattening
1/f
Geodetic derived geometric constants
From a and f it's possible to derive the semi-minor axis b, first eccentricity e and second eccentricity e′ of the ellipsoid
Parameter
Value
semi-minor axis
b = a(1-f)
First eccentricity squared
e2 = 1-b2/a2 = 2f-f2
Second eccentricity
e′2 = a2/b2 - 1 = f(2-f)/(1-f)2
Parameters for some geodetic systems
A more comprehensive list of geodetic systems can be found here
Australian Geodetic Datum 1966 [AGD66] and Australian Geodetic Datum 1984 (AGD84)
AGD66 and AGD84 both use the parameters defined by Australian National Spheroid (see below)
Australian National Spheroid (ANS)
Parameter
Notation
Value
semi-major axis
a
6378160.000 m
Reciprocal of Flattening
1/f
298.25
Geocentric Datum of Australia 1994 (GDA94) and Geocentric Datum of Australia 2000 (GDA2000)
Both GDA94 and GDA2000 use the parameters defined by GRS80 (see below)
The Earth-centred Earth-fixed (ECEF) or conventional terrestrial coordinate system rotates with the Earth and has its origin at the centre of the Earth. The axis passes through the equator at the prime meridian. The axis passes through the north pole. The axis can be determined by the right hand rule to be passing through the equator at 90o longitude.
Local east, north, up (ENU) coordinates
In many targeting and tracking applications the local East, North, Up (ENU) Cartesian coordinate system is far more intuitive and practical than ECEF or Geodetic coordinates. The local ENU coordinates are formed from a plane tangent to the Earth's surface fixed to a specific location and hence it's sometimes known as a "Local Tangent" or "local geodetic" plane. By convention the east axis is labeled , the north and the up .
Local north, east, down (NED) coordinates
In an aeroplane most objects of interest are below you, it's therefore sensible to define down as a positive number, the NED coordinates allow you to do this as an alternative the ENU local tangent plane. By convention the north axis is labeled , the east and the down . To avoid confusion between and , etc in this web page we'll restrict the local coordinate frame to ENU.
Conversion
From geodetic coordinates to local ENU coordinates
To convert from geodetic coordinates to local ENU up coordinates is a two stage process
Convert geodetic coordinates to ECEF coordinates
Convert ECEF coordinates to local ENU coordinates
From geodetic to ECEF coordinates
Geodetic coordinates (latitude , longitude , height ) can be converted into ECEF coordinates using the
following formulae: Note: Unambiguous determination of requires knowledge of the quadrant
From GPS measurements to ENU measurements: sample code
This code was written in MATLAB
Step 1: Convert GPS to ECEF
function [X,Y,Z] = llh2xyzTest(lat,long, h)
% Convert lat, long, height in WGS84 to ECEF X,Y,Z
%lat and long given in decimal degrees.
lat = lat/180*pi; %converting to radians
long = long/180*pi; %converting to radians
a = 6378137.0; % earth semimajor axis in meters
f = 1/298.257223563; % reciprocal flattening
e2 = 2*f -f^2; % eccentricity squared
chi = sqrt(1-e2*(sin(lat)).^2);
X = (a./chi +h).*cos(lat).*cos(long);
Y = (a./chi +h).*cos(lat).*sin(long);
Z = (a*(1-e2)./chi + h).*sin(lat);
Step 2: Convert ECEF to ENU
function [e,n,u] = xyz2enuTest(Xr, Yr, Zr, X, Y, Z)
% convert ECEF coordinates to local east, north, up
phiP = atan2(Zr,sqrt(Xr^2 + Yr^2));
lambda = atan2(Yr,Xr);
e = -sin(lambda).*(X-Xr) + cos(lambda).*(Y-Yr);
n = -sin(phiP).*cos(lambda).*(X-Xr) - sin(phiP).*sin(lambda).*(Y-Yr) + cos(phiP).*(Z-Zr);
u = cos(phiP).*cos(lambda).*(X-Xr) + cos(phiP).*sin(lambda).*(Y-Yr) + sin(phiP).*(Z-Zr);
From ENU measurements to GPS measurements: sample code
function [X,Y, Z] = enu2xyz(refLat, refLong, refH, e, n, u)
% Convert east, north, up coordinates (labelled e, n, u) to ECEF
% coordinates. The reference point (phi, lambda, h) must be given. All distances are in metres
[Xr,Yr,Zr] = llh2XYZ(refLat,refLong, refH); % location of reference point
phiP = atan2(Zr,sqrt(Xr^2+Yr^2)); % Geocentric latitude
X = -sin(refLong)*e - cos(refLong)*sin(phiP)*n + cos(refLong)*cos(phiP)*u + Xr;
Y = cos(refLong)*e - sin(refLong)*sin(phiP)*n + cos(phiP)*sin(refLong)*u + Yr;
Z = cos(phiP)*n + sin(phiP)*u + Zr;
Step 2: Convert ECEF to GPS
function [phi,lambda, h] = xyz2llh(X,Y,Z)
a = 6378137.0; % earth semimajor axis in meters
f = 1/298.257223563; % reciprocal flattening
b = a*(1-f);% semi-minor axis
e2 = 2*f-f^2;% first eccentricity squared
ep2 = f*(2-f)/((1-f)^2); % second eccentricity squared
r2 = X.^2+Y.^2;
r = sqrt(r2);
E2 = a^2 - b^2;
F = 54*b^2*Z.^2;
G = r2 + (1-e2)*Z.^2 - e2*E2;
c = (e2*e2*F.*r2)./(G.*G.*G);
s = (1 + c + sqrt(c.*c + 2*c) ).^(1/3);
P = F./(3*(s+1./s+1).^2.*G.*G);
Q = sqrt(1+2*e2*e2*P);
ro = -(e2*P.*r)./(1+Q) + sqrt((a*a/2)*(1+1./Q) - ((1-e2)*P.*Z.^2)./(Q.*(1+Q)) - P.*r2/2);
tmp = (r - e2*ro).^2;
U = sqrt(tmp + Z.^2 );
V = sqrt(tmp + (1-e2)*Z.^2 );
zo = (b^2*Z)./(a*V);
h = U.*(1 - b^2./(a*V));
phi = atan((Z + ep2*zo)./r );
lambda = atan2(Y,X);
Note: atan2(Y,X) uses quadrant information to return a value of lambda between and .
Sample Implementation Code
clear all
close all
clc
%% reference point
refLat = 39*pi/180;
refLong = -132*pi/180;
refH = 0;
%% Points of interest
lat = [39.5*pi/180;39.5*pi/180;39.5*pi/180];
long = [-132*pi/180;-131.5*pi/180;-131.5*pi/180];
h = [0;0;1000];
disp('lat long height')
for i = 1:length(lat)
disp([num2str(lat(i)*180/pi),'', num2str(long(i)*180/pi), ' ',num2str(h(i))])
end
% lat = [39.5*pi/180];
% long = [-132*pi/180];
% h = [0];
%% convering llh to enu
[Xr,Yr,Zr] = llh2xyz(refLat,refLong,refH);
[X,Y,Z] = llh2xyz(lat,long,h);
disp('X Y Z')
for i = 1:length(X)
disp([num2str(X(i)),'', num2str(Y(i)), ' ',num2str(Z(i))])
end
[e,n,u] = xyz2enu(Xr, Yr, Zr, X, Y, Z);
disp('e n u')
for i = 1:length(e)
disp([num2str(e(i)),'', num2str(n(i)), ' ',num2str(u(i))])
end
%% Converting enu to llh
[X,Y, Z] = enu2xyz(refLat, refLong, refH, e, n, u);
disp('X Y Z')
for i = 1:length(X)
disp([num2str(X(i)),'', num2str(Y(i)), ' ',num2str(Z(i))])
end
[phi,lambda, h] = xyz2llh(X,Y,Z);
disp('phi lambda h')
for i = 1:length(X)
disp([num2str(phi(i)*180/pi),'', num2str(lambda(i)*180/pi), ' ',num2str(h(i))])
end
P. Misra and P. Enge, Global Positioning System Signals, Measurements, and Performance. Lincoln, Massachusetts: Ganga-Jamuna Press, 2001.
J. Zhu, "Conversion of Earth-centered Earth-fixed coordinates to geodetic coordinates," Aerospace and Electronic Systems, IEEE Transactions on, vol. 30, pp. 957-961, 1994.
P. Misra and P. Enge, Global Positioning System Signals, Measurements, and Performance. Lincoln, Massachusetts: Ganga-Jamuna Press, 2001.
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