MAP PROJECTIONS AND SPATIAL REFERENCING
FOR GLOBAL DATA SETS
1. Abstract
The three-to-two dimensional model that has been applied to map data through the use of map projections has fulfilled the needs of map makers and map users for about 2 millennia. In the current digital era of cartography and geographic information science, this model continues to thwart the efforts of researchers who study the globe or very large regions. The distortion that map projections introduce into global, digital map data is significant due to the size of the area being studied, the resampling of raster data during transformation, and multiple reprojections of data. There are two general approaches to resolving this problem. The first approach examined is the minimization of distortion effects through judicious use of map projections. The second approach is to avoid the use of map projections by either performing analysis in the spherical realm, or by using hierarchal referencing methods.
keywords: map projection, global data, spherical analysis, tessellation, polyhedral globe
2. Introduction
Researchers have always been interested in the world in a global sense, but they never had the tools to study it all at once. Now earth scientists have the option of studying the entire Earth as a single entity. Most detailed analysis and mapping of the Earth has occurred at very large scales that take into account small regions. At first modern geo-techniques also focused on fairly small areas. The paradigm for referencing spatial data sets was the same as was being used for paper maps. The standard, familiar map projections, datums, and coordinate systems were implemented without alteration. How can spatial data for very large expanses, continents, regions, and the entire globe be managed effectively?
A map projection transformation converts three dimensional information to a more manageable two dimensional plane. This three-to-two dimensional model of transforming the globe to a flat map has served well and reliably for over two thousand years. Map projections are very efficient for managing spatial data for small areas, in part because only a known amount of distortion will be introduced via the map projection transformation. For very large areas however, significant amounts of distortion are introduced. The magnitude of the distortion, or error, introduced when transforming world maps is one indicator that the traditional map projection is less successful for global applications. Digital formats for spatial data create additional sources of error.
There are two major sources of error related to map projection to transformations of global data sets. The first is related to the simple fact that a spherical, three dimensional object cannot be squashed flat by any means without compression in some areas and expansion in others. Changes in shapes, distances and areas occur as a result. The second source of error relates to the difficulty of transforming a digital data set. Most of the global data sets being currently used, particularly for environmental applications, use a raster or grid data structure. A raster data structure cannot adjust to geometric changes without some combination of changing pixel values, elimination of pixels or duplication of pixels.
A spatial data set is repeatedly altered when it is projected several times. It is not unusual for a global data set to be built up from smaller sections. Each time data sets are combined, the extent of the data set changes, as does the choice of map projection. Multiple re-projection of data sets may also occur because global studies often require the use of several data sets that must be transformed to a common map projection. A number of researchers have been addressing these issues and a special task force has been developed to explore these issues.
The Task Team for Global Mapping is a part of The Committee on Earth Observation Satellites, CEOS, an international organization composed of representative of nearly all major space organization (NASA, NOAA, ESA, RSA, NASDA). Major international and scientific programs (IGBP, FAO, GCOS, WCRP, etc.) are CEOS affiliates. This task force describes itself as follows:
An overview of several approaches that are used to manage global spatial data sets is presented. First, the topic of map projections is explored including the properties of map projections and current recommendations for world map projections. Second, alternatives to using world map projections are explored. Analysis may be performed in the spherical domain or through alternative referencing systems which employ various combinations of tessellations of the globe, polyhedral globes, hierarchical referencing, and regional map projections.
3. Map Projections
Maling (1992) defined a map projection as:
Snyder (1987) wrote that:
Maling's definition and others that include statements similar to, "any systematic arrangement of meridians and parallels", originate from the days of manual cartography when a change in map projection meant a very slow painstaking and expensive manual process. First a new graticule was created and then map features were transferred in relation to the established graticule. Snyder updates this phrase to a systematic representation of all or part of the surface of a round body on a plane. We employ map projections and coordinate systems for nearly all planar representations of map data or paper maps.
Map projections are often categorized by the spatial property, or characteristic that is retained in the final product. Three significant characteristics of map projections include; conformality, equidistance, and equivalence. Conformality is a characteristic of map projections where the local angles or shapes are correct in the final map. Furthermore, a map is proven to be conformal when scale deformation is the same in all directions from all points on the map. Conformal projections, such as the Universal Transverse Mercator (UTM), are used for nearly all large scale mapping throughout the world. The Mercator projection is used for world maps and data sets but it is only suitable for a very limited set of navigational purposes. It produces very large distortions in area, particularly in the higher latitudes.
Another class of projections, the equidistant, provide true scale only between one or two points as it is impossible to show all distances correctly on a flat map. For example, the Lambert Azimuthal Equidistant projection provides accurate distant measurements from the center of the map to any other point on the map. This projection distorts both area and shapes.
For global studies the area of various features types, such as the amount of ice or clouds or the expanse of forests or deserts, needs to be calculated. Equivalent projections, also known as equal area projections are recommended because they ensure that a region on the surface of the earth is represented by the same amount of area on a map. Equal area projections are the focus of current research for determining the optimal choices of map projections for global data sets.
3.1 Recommended Projections for Digital Regional and Global Data
In an effort to choose the most effective map projections for the distribution of global data sets an analysis of the amount and types of changes that occur when data sets are projected was undertaken by Steinwand, Hutchinson, and Snyder (1995). They provided a set of equal area map projections specifically for continental, regional and global data sets. Recommendations were based on traditional map projection distortion analyses and they created two new methods for examining distortion in raster data sets. The four equal area map projections recommended for global data sets include; the Goodes Homolosine projection (Figure 1), the Wagner IV projection (Figure 2), the Wagner VII projection (Figure 3), and the interrupted Mollweide projection
The recommendations provided by these researchers for continental sized digital data sets are summarized below. The equal area projection chosen for all continents except the U.S. is the Lambert Azimuthal Equal Area projection. The results of their suggestions with optimized parameters appears in Table 1. No other research makes specific recommendations about map projection use for global data sets.
| Continent | Recommended Projection and Projection Parameters |
| North America | Oblated Equal Area with center 48N., 95W with shape constants m = 1.33, n = 2.27, rotation = 13.95 --or -- Lambert Azimuthal Equal Area w/center 50N, 100W |
| South America | Lambert Azimuthal Equal Area w/center 15N, 60W |
|
Europe |
Lambert Azimuthal Equal Area w/center 55N, 20E |
|
Africa |
Lambert Azimuthal Equal Area w/center 5N, 20E |
|
Asia |
Lambert Azimuthal Equal Area w/center 45N, 100E |
|
Australia |
Lambert Azimuthal Equal Area w/center 15S, 135E |
|
Antarctica |
Lambert Azimuthal Equal Area centered on the South Pole |
4. Spherical Analysis
Although most geographical analysis has been performed on a two dimensional plane the possibility of performing analysis on the sphere has received attention from a variety of different fields. Robert Raskin, in his review of spatial analysis on the sphere (1994), lists geology, astronomy, meteorology, statistics, operations research, and computer science as having researched this spherical analysis.
Raskin reports on three conceptual modeling approaches. The first is via a projection where a conventional map projection transformation is performed, analysis is performed on the plane, and the results are projected back to the sphere. The second approach is termed intrinsic. In this model the sphere surface is "considered an intrinsic space in its own right," and analysis is performed in non-Euclidean space. The third model is described as embedding. This model treats the sphere as a subset of an overall three dimensional space. Analysis is performed in this space with a constraint imposed to limit the solutions to the surface of the sphere.
Raskin reports that he found only one example of working GIS that operates in the spherical domain. After completing his report he went on to produce the Sphere Kit, now available from the NCGIA ftp site. At the time of writing in 1994, he reports that the Hipparchus GIS was the one working spherical based GIS. Raskin reports that Spherekit was developed at the National Center for Geographic Information and Analysis at the University of California, Santa Barbara and that developers included Cort Willmott, Rob Raskin, Chris Funk, Scott Webber, and Mike Goodchild. He describes Spherekit as an integrated toolkit for spatial interpolation and comparison of spatial interpolation algorithms. It permits interpolation over continental or global scales and computations are based upon spherical distances and orientations. (Raskin 1996)
A different representation of data for spherical analysis is through the use of spherical harmonics. Climatology data, topography and even world population have been represented with spherical harmonics (Raskin, 1994). A much more common method used to represent and manage global data in the geographic literature are suggestions for tessellations, polyhedral globes, and hierarchical referencing of the Earth as a basis for global analysis.
5. Hierarchal Referencing & Tessellated Globes
Managing spatial data sets of large extent is possible, and under development, through the use of various referencing schema. Several researchers from diverse points of view, have been exploring this possibility. The CEOS group (1997) recommends, ``Further support for investigation into alternative gridding schemes, such as spherical tessellation, which might allow near-distortion-free aggregation of global data.'' Raskin (1994) defines a tessellation of the sphere as "a spatial decomposition that is exhaustive and mutually exclusive (except perhaps along boundaries)'' and goes on to indicate that these tessellations may be classified as either polyhedral or empirical.
The most common spatial form selected by these investigators is of one of the five Platonic solids. See Figure 4 for examples of both the solids themselves and the flattened forms of the shapes. The tetrahedron has 4 equilateral triangles, the hexahedron, or cube, has 6 squares, the octahedron has 8 triangles, the dodecahedron has 12 pentagons, and the icosahedron has 20 equilateral triangles. This is far from a new concept however. The artist Albrecht Dürer considered several of the Platonic solids for the purpose of creating a world globe in 1538. The use of global tessellations and hierarchal data structures for managing data about the surface of the Earth map provide a means for more accurate and richer spatial representations.
A series of publications by Dutton (1983, 1988, 1989, 1996) have focused upon his development and implementation of a polyhedral based hierarchical data structure. Dutton needed an efficient way of modeling global terrain data including its assembly and management. He found that the traditional models used for modeling terrain data, gridded terrain data and triangular irregular networks, fell short of ideal for global data. He cites poor documentation, and variable accuracy through the gridded surface, both in horizontal and vertical dimensions, as just two of the problems he encountered.
Figure
6. The Pattern of Subdivision used for the QTM Facets
Dutton's data structure is known as the quaternary triangular mesh (QTM) and he describes its characteristics as storing elevation alone explicitly, being global in scope, uses hierarchical sampling, having an economic data structure, having verifiable accuracy and is based upon subdivision of the octahedron (Figure 6). Each triangular facet of the original octahedron is subdivided into four triangles. Each of these four new triangles is then subdivided into four triangles. This Dutton has continued examining this QTM structure to include information regarding both qualitative and quantitative aspects of the features being encoded.
The implementation of a hierarchical data structure to take advantage of data compression and fast access was the objective of Goodchild and Shiren (1992). They define three criteria for the development of this data structure; at any level of the hierarchy the cells should be of equal size and likewise, they should be of the same shape at any level of the hierarchy, and the data structure must correctly describe the relationships between neighboring cells. They propose a system similar to Dutton's. The data structuring begins with the use of an octahedron and subdivides each of the eight triangular faces recursively using the same pattern seen in (Figure 6). Ultimately these researchers would like to apply this data structure to a global GIS. They report a partial success in terms of their three criteria, the data structure does accurately incorporate relationships, but it only provides an approximate solution in terms of maintaining recursively divided cells of equal area and shape.
Lugo (1994) implements a data structure based on triangular quad trees and two dimensional run length encoding to represent global relief. Based on Dutton's QTM system, his objective was ``to store a digital data set in a triangular hierarchical structure and link it directly onto a a three dimensional data structure.'' He calls his data structure the TQS, or triangulated quadtree sequencing. A map projection, the Modified Interrupted Collignon, (Clarke's Butterfly, Figure 8), was developed to support this mapping effort. Lugo successfully linked his Triangulated Quadtree Structure to a portion of the ETOPO5 global data set to the Modified Interrupted Collignon projection.
Figure
7. Truncated Icosahedron. (Weisstein 1997)
A polyhedral based solution was also reached by a group of researchers with a very different objective. Kimerling, Overton and White (1992) needed to develop a statistically sound sampling program for global ecological data. They described a number of criteria including; a randomly positioned sampling grid, an equal area map projection to allow for equal probabilities for sampling locations, compact areas, regions that did not align with cultural or physical features, and to be able to densify the sampling grid in a hierarchical pattern. A sampling grid was devised by the group based upon the Lambert Azimuthal Equal Area map projection on the faces of a 20 sided truncated icosahedron (Figure 7).
The determination of how to map the Earth onto a polyhedral globe has usually been approached by using a gnomonic map projection, recentered for each face of the figure. The characteristics of the gnomonic projection introduce significant area and scale distortion (Snyder 1987). There are several projections that have been developed specifically for the purpose of mapping onto polyhedral globes. In 1965 Lee presented the conformal projection of the sphere on a regular tetrahedron. It was regarded as interesting novelty at first due to its attractive presentation of repetitions of the sphere. One critic sarcastically suggested that it would be "suitable as a design for a curtain'' (Lee 1973). Today polyhedral based projections are receiving much more interest and the conformal tetrahedric projection would be a suitable choice when the preservation of shapes is desired.
Figure 8. Modified Interrupted Collignon, or Clarke's Butterfly
If an equal area projection is required there are two choices available. Snyder (1992) presents a modified Lambert Azimuthal Equal Area projection that was derived from an approach by Irving Fisher. Although the Snyder projection may be applied to any polyhedral globe, it provides the best (lowest) distortion characteristics for the 12 sided dodecahedron (Figure 4) and the 20 sided truncated icosahedron (Figure 7). He reports the angular deformation as not exceeding 3.75 degrees for the truncated icosahedron. Clarke and Mulcahy (1995) describe the development of a projection that was motivated by the implementation of a global relief data structure based upon Dutton's QTM system (Lugo and Clarke 1995). Called the Modified Interrupted Collignon, or Clarke's Butterfly, the projection is derived from the Collignon projection and is a near equal area projection that maps the surface of the Earth onto eight equilateral triangles of an octahedron. See Figure 8 for an example of the projection constructed as a polyhedral globe. In its flat form the triangular octants are arranged in a butterfly pattern reminiscent of Cahill's Butterfly (see Figure 4).
New challenges have been created by the development of new computational capabilities that allow analysis of the entire Earth at once. One of these challenges is the efficient referencing of global spatial data sets. A limited, but focused examination of the current methods for dealing with map projection distortion or alternatives to using world map projections was presented. No attempt has been made to recommend one solution over another. Instead they have been outlined to show that there is a serious issue that needs to be addressed and that there is more than one possible solution to the problem. Furthermore, research should be continued in all of these areas. The most frequently implemented model, that of using a world map projection, still lacks standardization and detailed information regarding the changes these transformations cause. The use of spherical analysis is limited by the paucity of algorithms and software that have been developed to date. Likewise, the various polyhedral based, hierarchal referencing systems are either not fully developed or have not been implemented in commercial software. Any combination of these approaches could hold the key to global geographic information science.
Ironically, even after more than two millennia, geographers still face the problem of how best to represent the Earth.
