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Encyclopedia > Geodetic system

Examples of map data are:

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 World Geodetic System defines a reference frame for the earth, for use in geodesy and navigation. ... Part of an Ordnance Survey map at 1 inch to the mile scale from 1945 Ordnance Survey (OS) is an executive agency of the United Kingdom government. ... ED 50 (European Datum 1950) is a geodetic datum which was defined after World War II for the international connection of geodetic networks. ... For other uses, see North Pole (disambiguation). ... For other uses, see South Pole (disambiguation). ... World map showing the equator in red In tourist areas, the equator is often marked on the sides of roads The equator marked as it crosses IlhÃ©u das Rolas, in SÃ£o TomÃ© and PrÃ­ncipe. ... True Pizza is a navigational term referring to the direction of the North Pole relative to the navigators position. ... In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. ...

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. Surveyor at work with a leveling instrument. ... A scuba diver in usual sport diving gear SCUBA is an acronym for Self-Contained Underwater Breathing Apparatus. ...

The main reason that there are a number of datums is that before the advent of GPS positioning, national map making organizations did not have a common surveying reference point and only produced maps for their locality. Over fifty GPS satellites such as this NAVSTAR have been launched since 1978. ...

## 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. WGS 84 is the 1984 revision of the World Geodetic System. ...

## 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. For considerations of sea level change, in particular rise associated with possible global warming, see sea level rise. ... The GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earths gravity field. ...

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 has not stayed constant throughout geological time, so is less useful when measuring very long-term processes. For considerations of sea level change, in particular rise associated with possible global warming, see sea level rise. ... This article is about tides in the Earths oceans. ... Averages redirects here. ... A nautical chart is a graphic representation of a maritime area and adjacent coastal regions. ... The chart datum is the level of water that charted depths displayed on nautical charts are measured from. ...

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. The GOCE project will measure high-accuracy gravity gradients and provide an accurate geoid model based on the Earths gravity field. ... This article is about the Canadian province. ...

## 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 (φ), longitude (λ) and height (h). The ellipsoid is completely parameterised by the semi-major axis a and the flattening f.

### Geodetic versus geocentric latitude

The same position on a spheroid has a different angle for latitude depending on whether the angle is measured from the normal (angle α) or around the center (angle β). Note that the "flatness" of the spheroid in the image is greater than that of the Earth; therefore, the corresponding difference between the "geocentric" and "geodetic" latitudes is also greater.

It is important to note that geodetic latitude (φ) is different than geocentric latitude (φ'). 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. A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

### Geodetic defining parameters

Parameter Symbol
Semi-major axis a
Reciprocal of flattening 1/f

### Geodetic derived geometric constants

From a and f it is 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 e2 = 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)

ANS Defining Parameters
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)

### Geodetic Reference System 1980 (GRS80)

GRS80 Defining Parameters
Parameter Notation Value
semi-major axis a 6378137 m
Reciprocal of flattening 1/f 298.257222101

see GDA Technical Manual document for more details.

### World Geodetic System 1984 (WGS84)

The global positioning system (GPS) uses the world geodetic system 1984 (WGS84) to determine the location of a point near the surface of the Earth.

WGS84 Defining Parameters
Parameter Notation Value
semi-major axis a 6378137.0 m
Reciprocal of flattening 1/f 298.257223563
WGS84 derived geometric constants
Constant Notation Value
Semi-minor axis b 6356752.3142 m
First Eccentricity Squared e2 6.69437999014x10-3
Second Eccentricity Squared e2 6.73949674228x10-3

see The official World Geodetic System 1984 document for more details.

## Other Earth based coordinate systems

Local tangent plane

### Earth Centred Earth Fixed (ECEF) coordinates

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 X axis passes through the equator at the prime meridian. The Z axis passes through the north pole. The Y axis can be determined by the right hand rule to be passing through the equator at 90o longitude. ECEF stands for Earth-Centered, Earth-Fixed, and is a Cartesian coordinate system used for GPS, and is sometimes known as a conventional terrestrial system[1]. It represents positions as an X, Y, and Z coordinate. ...

### 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 plance tangent to the Earth's surface finxed to a specific location and hence it is sometimes known as a "Local Tangent" or "local geodetic" plane. By convention the east axis is labeled x, the north y and the up z.

### Local north, east, down (NED) coordinates

In an aeroplane most objects of interest are below you, it is 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 x', the east y' and the down z'. To avoid confusion between x and x', etc in this web page we will 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

1. Convert geodetic coordinates to ECEF coordinates
2. Convert ECEF coordinates to local ENU coordinates

#### From geodetic to ECEF coordinates

Geodetic coordinates (latitude φ, longitude λ, height h) can be converted into ECEF coordinates using the following formulae:

$begin{matrix} X & = & left(frac{a}{chi} + hright)cos{phi}cos{lambda} Y & = & left(frac{a}{chi} + hright)cos{phi}sin{lambda} Z & = & left(frac{a(1-e^2)}{chi} + hright)sin{phi} end{matrix}$

Where $chi = sqrt{1-e^2sin^2{phi}},$ a and e2 are the semi-major axis and the square of the first numerical eccentricity of the ellipsoid respectively

#### From ECEF to ENU Coordinates

To transform from ECEF coordinates to the local coordinates we need a local reference point, typically this might be the location of the radar. If a radar is located at {Xr,Yr,Zr} and an aircraft at {Xp,Yp,Zp} then the vector pointing from the radar to the aircraft in the ENU frame is

$begin{bmatrix} x y z end{bmatrix} = begin{bmatrix} -sinlambda & coslambda & 0 -sinphicoslambda & -sinphisinlambda & cosphi cosphicoslambda & cosphisinlambda& sinphi end{bmatrix} begin{bmatrix} X_p - X_r Y_p-Y_r Z_p - Z_r end{bmatrix}$

Note: φ is the geodetic latitude. A prior version of this page showed use of the geocentric latitude (φ'). The geocentric latitude is not the appropriate up direction for the local tangent plane. If the original geodetic latitude is available it should be used, otherwise, the relationship between geodetic and geocentric latitude has an altitude dependency, and is captured by:

$tanphi' = frac{Z_r}{sqrt{X_r^2 + Y_r^2}} = frac{frac{a}{chi}(1 - f)^2 + h}{frac{a}{chi} + h}tanphi$

Note that $frac{a}{chi}$ is also called the Normal, and is the length of the line segment, co-linear with the altitude vector and normal to the ellipsoid, which runs from the geodetic ellipsoid at the specified latitude/longitude to the intersection with the line connecting the north and south poles.

Obtaining geodetic latitude from geocentric coordinates from this relationship requires an iterative solution approach, otherwise the geodetic coordinates may be computed via the approach in the section below labeled "From ECEF to geodetic coordinates." Reference

for another example of computing ENU coordinates.

The geocentric and geodetic longitude have the same value

$tanlambda = frac{Y_r}{X_r}$

Note: Unambiguous determination of φ and λ requires knowledge of which quadrant the coordinates lie in.

### From local ENU coordinates to geodetic coordinates

As before it is done in two stages

1. Convert local ENU coordinates to ECEF coordinates
2. Convert ECEF coordinates to GPS coordinates

#### From ENU to ECEF

This is just the inversion of the ECEF to ENU transformation so

$begin{bmatrix} X Y Z end{bmatrix} = begin{bmatrix} -sinlambda & -sinphicoslambda & cosphicoslambda coslambda & -sinphisinlambda & cosphisinlambda 0 & cosphi& sinphi end{bmatrix} begin{bmatrix} x y z end{bmatrix} + begin{bmatrix} X_r Y_r Z_r end{bmatrix}$

#### From ECEF to geodetic coordinates

The conversion of ECEF coordinates to geodetic coordinates (such WGS84) is a much harder problem. A number of techniques are available but the most accurate according to Zhu (Ref 6), is the following 15 step procedure summarised by Kaplan. It is assumed that geodetic parameters {a,b,e,e'} are known

$begin{matrix} r &=& sqrt{X^2+Y^2} E^2 &=& a^2 - b^2 F &=& 54b^2Z^2 G &=& r^2 + (1-e^2)Z^2 - e^2E^2 C &=& frac{e^4Fr^2}{G^3} S &=& sqrt[3]{1+C+sqrt{C^2 + 2C}} P &=& frac{F}{3left(S+frac{1}{S}+1right)^2G^2} Q &=& sqrt{1+2e^4P} r_0 & =& frac{-(Pe^2r)}{1+Q} + sqrt{frac12 a^2left(1+1/Qright) - frac{P(1-e^2)Z^2}{Q(1+Q)} - frac12 Pr^2} U &=& sqrt{(r - e^2r_0)^2 + Z^2} V &=& sqrt{(r-e^2r_0)^2 + (1-e^2)Z^2} Z_0 &=& frac{b^2Z}{aV} h &=& Uleft(1-frac{b^2}{aV}right) phi & = & arctanleft[ frac{Z+e'^2Z_0}{r}right] lambda &=& arctan[frac{Y}{X}] end{matrix}$

Note: Unambiguous determination of λ 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

This code was written in MATLAB Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ...

#### Step 1: Convert ENU to ECEF

` 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 − π and π.

### 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 `

## Reference material

1. List of geodetic parameters for many systems
2. Kaplan, Understanding GPS: principles and applications, 1 ed. Norwood, MA 02062, USA: Artech House, Inc, 1996.
3. GPS Notes
4. Introduction to GPS Applications
5. P. Misra and P. Enge, Global Positioning System Signals, Measurements, and Performance. Lincoln, Massachusetts: Ganga-Jamuna Press, 2001.
6. 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.
7. P. Misra and P. Enge, Global Positioning System Signals, Measurements, and Performance. Lincoln, Massachusetts: Ganga-Jamuna Press, 2001.
8. Peter H. Dana: Geodetic Datum Overview - Large amount of technical information and discussion.
9. UK Ordnance Survey
10. US National Geodetic Survey
11. Borkowski, Kazimiers (1987), "Transformation of Geocentric to Geodetic Coordinates Without Approximations", Astrophysics and Space Science 139: 1-4

The World Geodetic System defines a reference frame for the earth, for use in geodesy and navigation. ... In the British Isles, an Ordnance Datum or OD is a vertical datum used by an ordnance survey as the basis for deriving altitudes on maps. ...

Results from FactBites:

 Geodetic Datum Overview (1532 words) Geodetic datums define the size and shape of the earth and the origin and orientation of the coordinate systems used to map the earth. Plane and geodetic surveying uses the idea of a plane perpendicular to the gravity surface of the earth, the direction perpendicular to a plumb bob pointing toward the center of mass of the earth. The geodetic longitude of a point is the angle between a reference plane and a plane passing through the point, both planes being perpendicular to the equatorial plane.
 Ordnance Survey Ireland :: Geodetic services :: Overview :: Irish Grid Reference System (574 words) Map positions expressed in this system are based on a co-ordinate reference frame observed by two primary triangulations during the 1950's and 60's, and combined in one adjustment in 1975 to produce geographic positions (latitude and longitude) for the primary stations in the reference frame. The Geodetic Datum is known as the 1965 Datum, and is defined by the positions of the ten Northern Ireland primaries (as defined by the 1952 adjustment) and the positions of two primary stations in the Republic (as defined by the 1965 adjustment). The Geodetic Datum of the Irish Grid is a derived one based on the positions of ten OSNI primary triangulation stations (1952 adjustment values), and the positions of three OSI primary triangulation stations fixed to their 1965 adjustment values.
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