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. 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.

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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.

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. In standard TIFF 6. 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. 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.

Raster space coordinates shall be referred to by their pixel types, i. 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. An N by M pixel image covers an are with the mathematically defined bounds 0,0 , N,M.

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 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 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. 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.

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. 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. 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 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 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. In the Geotiff standard, a geocentric coordinate system can be identified, either through the geographic code which in turn implies a datum , or 2.

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. 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. 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.

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## Cloud optimized GeoTIFF

This is generally accomplished through tying raster space coordinates to a model space coordinate system, when no further information is required. In the GeoTIFF nomenclature, "georeferencing" refers to tying raster space to a model space M, while "geocoding" refers to defining how the model space M assigns coordinates to points on the earth. In most cases the model space is only two-dimensional, in which case both K and Z should be set to zero; this third dimension is provided in anticipation of future support for 3D digital elevation models and vertical coordinate systems. A raster image may be georeferenced simply by specifying its location, size and orientation in the model coordinate space M. This may be done by specifying the location of three of the four bounding corner points. However, tiepoints are only to be considered exact at the points specified; thus defining such a set of bounding tiepoints does not imply that the model space locations of the interior of the image may be exactly computed by a linear interpolation of these tiepoints.

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