Geodetic system

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Geodetic system
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Geodetic systems or geodetic data are used in geodesy, navigation, surveying by cartographers and satellite navigation systems to translate positions indicated on their products to their real position on earth.

The systems are needed because the earth is an imperfect sphere. Also the earth is an imperfect ellipsoid. This can be verified by differentiating the equation for an ellipsoid and solving for dy/dx. It is a constant multiplied by x/y. Then derive the force equation from the centrifugal force acting on an object on the earth's surface and the gravitational force. Switch the x and y components and multiply one of them by negative one. This is the differential equation which when solved will yield the equation for the earth's surface. This is not a constant multiplied by x/y. Note that the earth's surface is also not an equal-potential surface, as can be verified by calculating the potential at the equator and the potential at a pole. The earth is an equal force surface. A one kilogram frictionless object on the ideal earth's surface does not have any force acting upon it to cause it to move either north or south. There is no simple analytical solution to this differential equation. A power series solution using three terms when substituted into this differential equation bogs down a TI-89 calculator and yields about three hundred terms after about five minutes.

The USGS uses a spherical harmonic expansion to approximate the earth's surface. It has about one hundred thousand terms. This problem has applications to moving Apollo asteroids. Some of them are loose rock and spinning. Their surface will be determined by the solution to this differential equation.

An interesting experiment will be to spin a mass of water in the space station and accurately measure its surface and do this for various angular velocities. Also, we can accurately measure Jupiter's surface using our telescopes. We can accurately determine earth's surface by using GPS.

Examples of map datums 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 difference between WGS84 and OSGB36, for example, is up to 140 metres (450 feet), which for some navigational purposes is an insignificant error. For other applications, such as surveying, or dive site location for SCUBA divers, 140 metres is an unacceptably large error.

Because the Earth is an imperfect ellipsoid, localised datums can give a more accurate representation of the area of coverage than the global WGS 84 datum. OSGB36, for example, is a better approximation to the geoid covering the British Isles than the global WGS 84 ellipsoid. However, as the benefits of a global system outweigh the greater accuracy, the global WGS 84 datum is becoming increasingly adopted.

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 WGS 84 datum, which is almost identical to the NAD83 datum used in North America and the ETRS89 datum used in Europe, is a common standard datum.

Vertical datum

A vertical datum is used for measuring the elevations of points on the Earth's surface. Vertical datums 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 mean sea level; this is a widely used tidal datum. Mean Sea Level (MSL) is a tidal datum which is described as the arithmetic mean of the hourly water elevation taken over a specific 19 years cycle. This definition averages out tidal highs and lows due to the gravitational effects of the sun and the moon. MSL is defined as the zero elevation for a local area. However, zero elevation as defined by one country is not the same as zero elevation defined by another (because MSL is not the same everywhere). Which is why locally defined vertical datums differ from one another. 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.

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  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 (orange) in the image is greater than that of the Earth; as a result, the corresponding difference between the "geodetic" and "geocentric" latitudes is also exaggerated.

In geodetic coordinates the Earth's surface is approximated by an ellipsoid and locations near the surface are described in terms of latitude ( $\ \phi$), longitude ( $\ \lambda$) and height (h)[footnotes 1].

The ellipsoid is completely parameterised by the semi-major axis a and the flattening f.

Geodetic versus geocentric latitude

It is important to note that geodetic latitude ( $\ \phi$) is different from geocentric latitude ( $\ \phi^\prime$). 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.

Defining and derived parameters

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

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

Australian Geodetic Datum 1966 [AGD66] and Australian Geodetic Datum 1984 (GDA84)

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 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; the value given above for the flattening is not exact.

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.

A more comprehensive list of geodetic systems can be found here

Other Earth-based coordinate systems  Earth Centred Earth Fixed and East, North, Up coordinates.

Earth-centred earth-fixed (ECEF or ECF) coordinates

The Earth-centered earth-fixed (ECEF or ECF) 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 but it does not exactly coincide with the instantaneous Earth rotational axis. The Y 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 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 airplane most objects of interest are below you, so it is sensible to define down as a positive number. The NED coordinates allow you to do this as an alternative to 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 calculations

Geodetic to/from ECEF coordinates

From geodetic to ECEF

Geodetic coordinates (latitude $\ \phi$, longitude $\ \lambda$, height h) can be converted into ECEF coordinates using the following formulae: \begin{align} X & = \left( N(\phi) + h\right)\cos{\phi}\cos{\lambda} \\ Y & = \left( N(\phi) + h\right)\cos{\phi}\sin{\lambda} \\ Z & = \left( N(\phi) (1-e^2) + h\right)\sin{\phi} \end{align}

where $N(\phi) = \frac{a}{\sqrt{1-e^2\sin^2 \phi }},$

a and e2 are the semi-major axis and the square of the first numerical eccentricity of the ellipsoid respectively. $\, N(\phi)$ is called the Normal and is the distance from the surface to the Z-axis along the ellipsoid normal.

From ECEF to geodetic

The conversion of ECEF coordinates to geodetic coordinates (such WGS84) is a much harder problem, except for longitude, $\,\lambda$.

There exist two kinds of methods in order to solve the equation.

Newton-Raphson method

The following Bowring's irrational geodetic equation is efficient to be solved by Newton–Raphson iteration method: $\kappa - 1 - \frac{e^2 a \kappa}{\sqrt{p^2+(1-e^2) z^2 \kappa^2 }} = 0,$

where $\kappa = \frac{p}{z} \tan \phi$ and $p=\sqrt{x^2+y^2} .$ $h=(1-(1-\kappa^{-1}) e^{-2}) \sqrt{p^2+ z^2 \kappa^2 }. \,$

The iteration can be transformed into the following calculation: $\kappa_{i+1} = \frac{c_i+\left(1- e^2\right) z^2 \kappa_i ^3 }{c_i- p^2} = 1 + \frac{p^2+\left(1- e^2\right) z^2 \kappa_i ^3 }{c_i- p^2} ,$

where $c_i = \frac{\left(p^2+\left(1-e^2\right) z^2 \kappa_i ^2\right)^{3/2}}{a e^2} .$ $\kappa_0=\frac{1}{1-e^2}$ is a good starter for the iteration when $h \approx 0$. Bowring showed that the single iteration produces the sufficiently accurate solution. He used trigonometric functions in his original formulation.

Ferrari's solution

The following solve the above equation: \begin{align} \zeta &= (1 - e^2) z^2 / a^2 ,\\ \rho &= (p^2 / a^2 + \zeta - e^4) / 6 ,\\ s &= e^4 \zeta p^2 / ( 4 a^2) ,\\ t &= \sqrt{s + \rho^3 + \sqrt{s (s + 2 \rho^3)}} ,\\ u &= \rho + t + \rho^2 / t ,\\ v &= \sqrt{u^2 + e^4 \zeta} ,\\ w &= e^2 (u + v - \zeta) / (2 v) ,\\ \kappa &= 1 + e^2 (\sqrt{u + v + w^2} + w) / (u + v). \end{align}

ECEF to/from 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

To convert from local ENU up coordinates to geodetic coordinates is a two stage process

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

From ECEF to ENU

To transform from ECEF coordinates to the local coordinates we need a local reference point, typically this might be the location of a 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 $\begin{bmatrix} x \\ y \\ z\\ \end{bmatrix} = \begin{bmatrix} -\sin\lambda_r & \cos\lambda_r & 0 \\ -\sin\phi_r\cos\lambda_r & -\sin\phi_r\sin\lambda_r & \cos\phi_r \\ \cos\phi_r\cos\lambda_r & \cos\phi_r\sin\lambda_r& \sin\phi_r \end{bmatrix} \begin{bmatrix} X_p - X_r \\ Y_p-Y_r \\ Z_p - Z_r \end{bmatrix}$

Note: $\ \phi$ is the geodetic latitude. A prior version of this page showed use of the geocentric latitude ( $\ \phi^\prime$). 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: $\tan\phi^\prime = \frac{Z_r}{\sqrt{X_r^2 + Y_r^2}} = \frac{ N(\phi) (1 - f)^2 + h}{ N(\phi) + h}\tan\phi$

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."

The geocentric and geodetic longitude have the same value. This is true for the Earth and other similar shaped planets because their latitude lines (parallels) can be considered in much more degree perfect circles when compared to their longitude lines (meridians). $\tan\lambda = \frac{Y_r}{X_r}$

Note: Unambiguous determination of $\ \phi$ and $\ \lambda$ requires knowledge of which quadrant the coordinates lie in.

From ENU to ECEF

This is just the inversion of the ECEF to ENU transformation so $\begin{bmatrix} X\\ Y\\ Z\\ \end{bmatrix} = \begin{bmatrix} -\sin\lambda & -\sin\phi\cos\lambda & \cos\phi\cos\lambda \\ \cos\lambda & -\sin\phi\sin\lambda & \cos\phi\sin\lambda \\ 0 & \cos\phi& \sin\phi \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} + \begin{bmatrix} X_r \\ Y_r \\ Z_r \end{bmatrix}$

Footnotes

1. ^ About the right/left-handed order of the coordinates, i.e., $\ (\lambda, \phi)$ or $\ (\phi, \lambda)$, see Spherical_coordinate_system#Conventions.

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