##
``

### 2.5 Coordinate Systems in GeoTIFF

`Geotiff has been designed so that standard map coordinate system definitions can be readily stored in a single registered TIFF tag. It has also been designed to allow the description of coordinate system definitions which are non-standard, and for the description of transformations between coordinate systems, through the use of three or four additional TIFF tags.

However, in order for the information to be correctly exchanged between various clients and providers of GeoTIFF, it is important to establish a common system for describing map projections.

In the TIFF/GeoTIFF framework, there are essentially three different spaces upon which coordinate systems may be defined. The spaces are:

`

      1) The raster space (Image space) R, used to reference the pixel values
         in an image,
      2) The Device space D, and
      3) The Model space, M, used to reference points on the earth.

`

In the sections that follow we shall discuss the relevance and use of each of these spaces, and their corresponding coordinate systems, from the standpoint of GeoTIFF.

`

``

#### 2.5.1 Device Space and GeoTIFF

`

In standard TIFF 6.0 there are tags which relate raster space R with device space D, such as monitor, scanner or printer. The list of such tags consists of the following:

`

        ResolutionUnit (296)
        XResolution    (282)
        YResolution    (283)
        Orientation    (274)
        XPosition      (286)
        YPosition      (287)

``

`In Geotiff, provision is made to identify earth-referenced coordinate systems (model space M) and to relate M space with R space. This provision is independent of and can co-exist with the relationship between raster and device spaces. To emphasize the distinction, this spec shall not refer to "X" and "Y" raster coordinates, but rather to raster space "J" (row) and "I" (column) coordinate variables instead, as defined in section 2.5.2.2.

`

#### 2.5.2 Raster Coordinate Systems

``

#### 2.5.2.1 Raster Data

`

Raster data consists of spatially coherent, digitally stored numerical data, collected from sensors, scanners, or in other ways numerically derived. The manner in which this storage is implemented in a TIFF file is described in the standard TIFF specification.

Raster data values, as read in from a file, are organized by software into two dimensional arrays, the indices of the arrays being used as coordinates. There may also be additional indices for multispectral data, but these indices do not refer to spatial coordinates but spectral, and so of not of concern here.

Many different types of raster data may be georeferenced, and there may be subtle ways in which the nature of the data itself influences how the coordinate system (Raster Space) is defined for raster data. For example, pixel data derived from imaging devices and sensors represent aggregate values collected over a small, finite, geographic area, and so it is natural to define coordinate systems in which the pixel value is thought of as filling an area. On the other hand, digital elevations models may consist of discrete "postings", which may best be considered as point measurements at the vertices of a grid, and not in the interior of a cell.

`

#### 2.5.2.2 Raster Space

`

The choice of origin for raster space is not entirely arbitrary, and depends upon the nature of the data collected. Raster space coordinates shall be referred to by their pixel types, i.e., as "PixelIsArea" or "PixelIsPoint".

Note: For simplicity, both raster spaces documented below use a fixed pixel size and spacing of 1. Information regarding the visual representation of this data, such as pixels with non-unit aspect ratios, scales, orientations, etc, are best communicated with the TIFF 6.0 standard tags.

`

#### "PixelIsArea" Raster Space

`

The "PixelIsArea" raster grid space R, which is the default, uses coordinates I and J, with (0,0) denoting the upper-left corner of the image, and increasing I to the right, increasing J down. The first pixel-value fills the square grid cell with the bounds:

`

       top-left = (0,0), bottom-right = (1,1)

`

and so on; by extension this one-by-one grid cell is also referred to as a pixel. An N by M pixel image covers an are with the mathematically defined bounds (0,0),(N,M).

`

         (0,0)
          +---+---+-> I
          | * | * |
          +---+---+        Standard (PixelIsArea) TIFF Raster space R,
          | (1,1)  (2,1)      showing the areas (*) of several pixels.
          |
          J

`

`

#### "PixelIsPoint" Raster Space

`

The PixelIsPoint raster grid space R uses the same coordinate axis names as used in PixelIsArea Raster space, with increasing I to the right, increasing J down. The first pixel-value however, is realized as a point value located at (0,0). An N by M pixel image consists of points which fill the mathematically defined bounds (0,0),(N-1,M-1).

`

         (0,0)   (1,0)
          *-------*------> I
          |       |
          |       |       PixelIsPoint TIFF Raster space R,
          *-------*         showing the location (*) of several pixels.
          |     (1,1)
          J

`

If a point-pixel image were to be displayed on a display device with pixel cells having the same size as the raster spacing, then the upper-left corner of the displayed image would be located in raster space at (-0.5, -0.5).

`

#### 2.5.3 Model Coordinate Systems

`

The following methods of describing spatial model locations (as opposed to raster) are recognized in Geotiff:

`

    	Geographic coordinates
    	Geocentric coordinates
    	Projected coordinates
    	Vertical coordinates

`Geographic, geocentric and projected coordinates are all imposed on models of the earth. To describe a location uniquely, a coordinate set must be referenced to an adequately defined coordinate system. If a coordinate system is from the Geotiff standard definitions, the only reference required is the standard coordinate system code/name. If the coordinate system is non-standard, it must be defined. The required definitions are described below.

Projected coordinates, local grid coordinates, and (usually) geographical coordinates, form two dimensional horizontal coordinate systems (i.e., horizontal with respect to the earth's surface). Height is not part of these systems. To describe a position in three dimensions it is necessary to consider height as a second one dimensional vertical coordinate system.

To georeference an image in GeoTIFF, you must specify a Raster Space coordinate system, choose a horizontal model coordinate system, and a transformation between these two, as will be described in section 2.6

`

#### 2.5.3.1 Geographic Coordinate Systems

`

Geographic Coordinate Systems are those that relate angular latitude and longitude (and optionally geodetic height) to an actual point on the earth. The process by which this is accomplished is rather complex, and so we describe the components of the process in detail here.

`

#### Ellipsoidal Models of the Earth

`

The geoid - the earth stripped of all topography - forms a reference surface for the earth. However, because it is related to the earth's gravity field, the geoid is a very complex surface; indeed, at a detailed level its description is not well known. The geoid is therefore not used in practical mapping.

It has been found that an oblate ellipsoid (an ellipse rotated about its minor axis) is a good approximation to the geoid and therefore a good model of the earth. Many approximations exist: several hundred ellipsoids have been defined for scientific purposes and about 30 are in day to day use for mapping. The size and shape of these ellipsoids can be defined through two parameters. Geotiff requires one of these to be

`

              the semi-major axis (a),

`and the second to be either`

              the inverse flattening (1/f)

`or`

              the semi-minor axis (b).

`Historical models exist which use a spherical approximation; such models are not recommended for modern applications, but if needed the size of a model sphere may be defined by specifying identical values for the semimajor and semiminor axes; the inverse flattening cannot be used as it becomes infinite for perfect spheres.

Other ellipsoid parameters needed for mapping applications, for example the square of the eccentricity, can easily be calculated by an application from the two defining parameters. Note that Geotiff uses the modern geodesy convention for the symbol (b) for the semi-minor axis. No provision is made for mapping other planets in which a tri-dimensional (triaxial) ellipsoid might be required, where (b) would represent the semi-median axis and (c) the semi-minor axis.

Numeric codes for ellipsoids regularly used for earth-mapping are included in the Geotiff reference lists.

`

#### Latitude and Longitude

`

The coordinate axes of the system referencing points on an ellipsoid are called latitude and longitude. More precisely, geodetic latitude and longitude are required in this Geotiff standard. A discussion of the several other types of latitude and longitude is beyond the scope of this document as they are not required for conventional mapping.

Latitude is defined to be the angle subtended with the ellipsoid's equatorial plane by a perpendicular through the surface of the ellipsoid from a point. Latitude is positive if north of the equator, negative if south.

Longitude is defined to be the angle measured about the minor (polar) axis of the ellipsoid from a prime meridian (see below) to the meridian through a point, positive if east of the prime meridian and negative if west. Unlike latitude which has a natural origin at the equator, there is no feature on the ellipsoid which forms a natural origin for the measurement of longitude. The zero longitude can be any defined meridian. Historically, nations have used the meridian through their national astronomical observatories, giving rise to several prime meridians. By international convention, the meridian through Greenwich, England is the standard prime meridian. Longitude is only unambiguous if the longitude of its prime meridian relative to Greenwich is given. Prime meridians other than Greenwich which are sometimes used for earth mapping are included in the Geotiff reference lists.

`

#### Geodetic Datums

`

As well as there being several ellipsoids in use to model the earth, any one particular ellipsoid can have its location and orientation relative to the earth defined in different ways. If the relationship between the ellipsoid and the earth is changed, then the geographical coordinates of a point will change.

Conversely, for geographical coordinates to uniquely describe a location the relationship between the earth and the ellipsoid must be defined. This relationship is described by a geodetic datum. An exact geodetic definition of geodetic datums is beyond the current scope of Geotiff. However the Geotiff standard requires that the geodetic datum being utilized be identified by numerical code. If required, defining parameters for the geodetic datum can be included as a citation.

`

#### Defining Geographic Coordinate Systems

`

In summary, geographic coordinates are only unique if qualified by the code of the geographic coordinate system to which they belong. A geographic coordinate system has two axes, latitude and longitude, which are only unambiguous when both of the related prime meridian and geodetic datum are given, and in turn the geodetic datum definition includes the definition of an ellipsoid. The Geotiff standard includes a list of frequently used geographic coordinate systems and their component ellipsoids, geodetic datums and prime meridians. Within the Geotiff standard a geographic coordinate system can be identified either by

`

               the code of a standard geographic coordinate system

`or by`

                a user-defined system.

`

The user is expected to provide geographic coordinate system code/name, geodetic datum code/name, ellipsoid code (if in standard) or ellipsoid name and two defining parameters (a) and either (1/f) or (b), and prime meridian code (if in standard) or name and longitude relative to Greenwich.

`

#### 2.5.3.2 Geocentric Coordinate Systems

`

A geocentric coordinate system is a 3-dimensional coordinate system with its origin at or near the center of the earth and with 3 orthogonal axes. The Z-axis is in or parallel to the earth's axis of rotation (or to the axis around which the rotational axis precesses). The X-axis is in or parallel to the plane of the equator and passes through its intersection with the Greenwich meridian, and the Y-axis is in the plane of the equator forming a right-handed coordinate system with the X and Z axes.

Geocentric coordinate systems are not frequently used for describing locations, but they are often utilized as an intermediate step when transforming between geographic coordinate systems. (Coordinate system transformations are described in section 2.6 below).

In the Geotiff standard, a geocentric coordinate system can be identified, either

`

    	through the geographic code (which in turn implies a datum),

`or`

    	through a user-defined name.

`

`

#### 2.5.3.3 Projected Coordinate Systems

`

Although a geographical coordinate system is mathematically two dimensional, it describes a three dimensional object and cannot be represented on a plane surface without distortion. Map projections are transformations of geographical coordinates to plane coordinates in which the characteristics of the distortions are controlled. A map projection consists of a coordinate system transformation method and a set of defining parameters. A projected coordinate system (PCS) is a two dimensional (horizontal) coordinate set which, for a specific map projection, has a single and unambiguous transformation to a geographic coordinate system.

In GeoTIFF PCS's are defined using the POSC/EPSG system, in which the PCS planar coordinate system, the Geographic coordinate system, and the transformation between them, are broken down into simpler logical components. Here are schematic formulas showing how the Projected Coordinate Systems and Geographic Coordinates Systems are encoded:

`

         Projected_CS  =  Geographic_CS + Projection
         Geographic_CS =  Angular_Unit + Geodetic_Datum + Prime_Meridian
         Projection    =  Linear Unit + Coord_Transf_Method + CT_Parameters
         Coord_Transf_Method   = { TransverseMercator | LambertCC | ...}
         CT_Parameters = {OriginLatitude + StandardParallel+...}

`

(See also the Reference Parameters documentation in section 2.5.4).

Notice that "Transverse Mercator" is not referred to as a "Projection", but rather as a "Coordinate Transformation Method"; in GeoTIFF, as in EPSG/POSC, the word "Projection" is reserved for particular, well-defined systems in which both the coordinate transformation method, its defining parameters, and their linear units are established.

Several tens of coordinate transformation methods have been developed. Many are very similar and for practical purposes can be considered to give identical results. For example in the Geotiff standard Gauss-Kruger and Gauss-Boaga projection types are considered to be of the type Transverse Mercator. Geotiff includes a listing of commonly used projection defining parameters.

Different algorithms require different defining parameters. A future version of Geotiff will include formulas for specific map projection algorithms recommended for use with listed projection parameters.

To limit the magnitude of distortions of projected coordinate systems, the boundaries of usage are sometimes restricted. To cover more extensive areas, two or more projected coordinate systems may be required. In some cases many of the defining parameters of a set of projected coordinate systems will be held constant.

The Geotiff standard does not impose a strict hierarchy onto such zoned systems such as US State Plane or UTM, but considers each zone to be a discrete projected coordinate system; the ProjectedCSTypeGeoKey code value alone is sufficient to identify the standard coordinate systems.

Within the Geotiff standard a projected coordinate system can be identified either by

`

            the code of a standard projected coordinate system

`or by`

            a user-defined system.

`

User-define projected coordinate systems may be defined by defining the Geographic Coordinate System, the coordinate transformation method and its associated parameters, as well as the planar system's linear units.

`

#### 2.5.3.4 Vertical Coordinate Systems

`

Many uses of Geotiff will be limited to a two-dimensional, horizontal, description of location for which geographic coordinate systems and projected coordinate systems are adequate. If a three-dimensional description of location is required Geotiff allows this either through the use of a geocentric coordinate system or by defining a vertical coordinate system and using this together with a geographic or projected coordinate system.

In general usage, elevations and depths are referenced to a surface at or close to the geoid. Through increasing use of satellite positioning systems the ellipsoid is increasingly being used as a vertical reference surface. The relationship between the geoid and an ellipsoid is in general not well known, but is required when coordinate system transformations are to be executed.

`

#### 2.5.4 Reference Parameters

`

Most of the numerical coding systems and coordinate system definitions are based on the hierarchical system developed by EPSG/POSC. The complete set of EPSG tables used in GeoTIFF is available at:

ftp://ftp.remotesensing.org/pub/geotiff/tables

` Appended below is the README.TXT file that accompanies the tables of defining parameters for those codes:

                        +-----------------------------------+
                        |     EPSG Geodesy Parameters       |
                        |    version 2.1, 2nd June 1995.    |
                        +-----------------------------------+

     The European Petroleum Survey Group (EPSG) has compiled and is
     distributing this set of parameters defining various geodetic
     and cartographic coordinate systems to encourage
     standardisation across the Exploration and Production segment
     of the oil industry.  The data is included as reference data
     in the Geotiff data exchange specification, in Iris21 the
     Petroconsultants data model, and in Epicentre, the POSC data
     model.  Parameters map directly to the POSC Epicentre model
     v2.0, except for data item codes which are included in the
     files for data management purposes.  Geodetic datum parameters
     are embedded within the geographic coordinate system file.
     This has been done to ease parameter maintenance as there is a
     high correlation between geodetic datum names and geographic
     coordinate system names.  The Projected Coordinate System v2.0
     tabulation consists of systems associated with locally used
     projections.  Systems utilising the popular UTM grid system
     have also been included.

     Criteria used for material in these lists include:
       - information must be in the public domain: "private" data
         is not included.
       - data must be in current use.
       - parameters are given to a precision consistent with
         coordinates being to a precision of one centimetre.

     The user assumes the entire risk as to the accuracy and the
     use of this data.  The data may be copied and distributed
     subject to the following conditions:

          1)   All data must then be copied without modification
     and all pages must be included;

          2)   All components of this data set must be distributed
     together;

          3)   The data may not be distributed for profit by any
     third party; and

          4)   Acknowledgement to the original source must be
     given.

     INFORMATION  PROVIDED IN THIS DOCUMENT IS PROVIDED "AS IS"
     WITHOUT WARRANTY  OF  ANY  KIND,  EITHER  EXPRESSED OR
     IMPLIED, INCLUDING  BUT  NOT LIMITED TO THE IMPLIED WARRANTIES
     OF MERCHANTABILITY AND/OR FITNESS FOR A PARTICULAR PURPOSE.

     Data is distributed on MS-DOS formatted diskette in comma-
     separated record format.  Additional copies may be obtained
     from Jean-Patrick Girbig at the address below at a cost of
     US$100 to cover media and shipping, payment to be made in
     favour of Petroconsultants S.A at Union Banque Suisses,
     1211 Geneve 11, Switzerland (compte number 403 458 60 K).

     The data is to be made available on a bulletin board shortly.

     Shipping List
     -------------

     This data set consists of 8 files:

     PROJCS.CSV  Tabulation of Projected Coordinate Systems to
                 which map grid coordinates may be referenced.

     GEOGCS.CSV  Tabulation of Geographic Coordinate Systems to
                 which latitude and longitude coordinates may be
                 referenced.  This table includes the equivalent
                 geocentric coordinate systems and also the
                 geodetic datum, reference to which allows latitude
                 and longitude or geocentric XYZ to uniquely
                 describe a location on the earth.

     VERTCS.CSV  Tabulation of Vertical Coordinate Systems to
                 which heights or depths may be referenced. This
                 table is currently in an early form.

     PROJ.CSV    Tabulation of transformation methods and
                 parameters through which Projected Coordinate
                 Systems are defined and related to Geographic
                 Coordinate Systems.

     ELLIPS.CSV  Tabulation of reference ellipsoids upon which
                 geodetic datums are based.

     PMERID.CSV  Tabulation of prime meridians upon which geodetic
                 datums are based.

     UNITS.CSV   Tabulation of length units used in Projected and
                 Vertical Coordinate Systems and angle units used
                 in Geographic Coordinate Systems.

     README.TXT  This file.

``