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Geographic Information Systems (+/-)
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   Computer Resources GIS Manual  

An Overview of Geodesy and Geographic Referencing Systems

The magic of geographic information systems is that they bring together and associate representations from diverse sources and infer relationships based on spatial references. This ability depends on our data sources using well defined (coordinate) referencing systems. This is not to say that the coordinate systems need to be the same, only that they are well defined. Indeed, there are thousands of perfectly legitimate coordinate systems in active use. The notion of spatial referenicing systems is one of the most fundamental and interesting ideas that all users of GIS should understand. This document provides an overview of the basic ideas.

To tie this lecture together with the previous lecture on the context of geographic information systems, we will return to our framework for Modeling What's Important.

Big Ideas Covered in this Lecture

  • Spatial Datasets employ Coordinate Referencing Systems to charaterize the locations and shapes of entities.
  • Although Geographic Coordineate Systems (GCS) may be the best way to store coordinates in a broad-scale database, GCS is never an apropriate projection for portraying data on a map!
  • The choice of a CRS fror mapping and analysis involves many compromises and depends on the location and the purposes of the project.
  • It is very important to understand the Metadata requirements for identifying the CRS that is assumed by the geometry in a dataset. These are:
    • Earth Model Assumptions
    • Projection Method (or Geographic Coordinate System (GCS))
    • Projection Case
    • Coordinate Units
  • Problems with interoperbility in metadata will often lead to a failure of a GIS package to get layers to align correctly.

Understanding earth-based coordinate referencing systems and how they are documented is a critical skill for people who want to bring information together from many sources and to make useful maps and 3d models.

Deeper Reading

Technical Issues of Map Projections in ArcGIS

A Brief History and Theory of Latitude and Longitude

Almost everybody knows that Latitude and Longitude provide a framework for referencing places on the earth. This is very useful. Lat Lon coordinates are used in many worldwide datasets. The following paragraphs will describe a bit of history of this Geographic Coordinate System (GCS). We will also see how notwithstanding its usefulness, Latitude and Longitude are not a good coordinate system for interpreting important things like scale or plane geometric relationships amopng entities so measured.

In order to establish a system for referencing places on the earth someone first had to establish a means of measuring the earth so that the measurements could be subdivided into a referncing system. Eratosthenes Measured the Earth and its angle of rotation around 200 BC. He also figured out that a leap day would be necessary every 4 years in order to keep calendars consistent.

Around 50 years later, a greek, Hipparchus of Rhodes invented a world-wide referencing system of meridians and paralells that we use to describe earth locations: latitude and longitude.

It is actually very simple to figure out your latitude and logitude (geographic coordinates) from first principles, using a simple protractor and a clock. Here's How! (page from PBS)

Earth Models

Latitude and Longitude are angles, not distances. In order to use these angular displacements to understand distance relationships on the surface of the earth, you need to know the radius. If the earth was spherical, then the radius would be a simple constant; but it turns out that if you use this assumption, the maps that you make will not be very useful. Since about 1700 it has been postulated that the earth is not spherical (Isaac Newton). Therefore observations of latitude and longitude made on the surface of the earth need to be qualified with a radius which is derived from one of many possible models of the Size and shape of the earth (or Earth Model.)

See page 4 of the ESRI manual on Map Projections. for a discussion of the evolution of our concepts of the shape of the earth. When you gater together data for north america, you will find data registerd to the North American Datum of 1983, The World Geodetic Spheroid of 1994, and occaisionally the North American Datum of 1927. Daat collected in other parts of the world liukely use other local datums. Mis-specifying the earth model for a dataset can lead to displacements in your ground measurements of as much as 20 meters!

The Graphical Problem of Latitude and Longitude

Since Hipparchus described his system for referencing places on the earth why has humanity utilized so many diferent coordinate systems and variations on them? The problem with latitude and longitude is that it is impossible to represent accurately on a flat piece of paper, and it is extremely difficult to do the math required to analyze the relationships among places using sperical triginometry. Maps are an interesting instance of computing machines. The map maker arranges symbols on a piece of paper so that people can make useful (if not entirely correct) inferences about the relationships among things. The logic behind most maps (and GIS systems) is that of graph theory, which assumes a flat (orthogonal, planimetric) coordinate system. When topography is our tool, We will be misled if we use Latitude and Longitude as a coordinate system. Not only will our calculations be incorrect, but so will a map user's impression of the spatial relationships among places on our maps:

The map of Massachusetts in unprojected Decimal Degrees of Longitude and Latitude Note the proportion of the lines and the circle. The circle gives the impression that Worcester is approximately the same distance from Boston as New Hampshire, and that the state is nearly five times longer, east to west, as it is north-to-south.

Compare with the same data layers transformed according to the Massachusetts State Plane coordinate system system This shows an image that is much truer relative to distances you would actually experience on the ground. It is actually much closer to Worcester, and the state is only 3 times longer than it is tall. The grid on each map is one degree on a side. Using degrees of longitude and latitude as a coordinate system makes the grid square. But actually at the lattitude of Massachusetts, the degrees of longitude are much shorter. The concept of scale and measurement makes no sense when your coordinates are not orthogonal!

Transformation of Geographical Coordinates to Cartesian Coordinate Systems

While the system of latitude and longitude provides a consistent referencing system for anywhere on the earth, and it is therefore used in geographic databases that are not specific to a particular place. However, in order to portray our information on maps or for making calculations, we need to transform these angular measures to orthogonal coordinates. These transformations are known as projections. Methods for projecting the spherical coordinates of lattitude and longitude to orthoganal coordinates, may be thought of as metaphorical projections of light through a transparent globe onto a developable surface such as a flat piece of paper.

It turns out that any way you try to do this, you must unavoidably incorporate some distortion into the picture. Any projection has its area of least distortion. Projections can be shifted around in order to put this area of least distortion over the topographer's area of interest. Thus any projection can have an unlimited number of variations or cases that determined by standard paralells or meridians that adjust the location of the high-accuracy part of the projection. See ESRI's Handbook on Map Projections, chapters 2 and 4

Projection Cases

In the case of the orthographic projection above, the area of least distortion is occurs where the figurative projection plane touches the model of the earth. To create a projection that works well for a particular area, we can create a case of the orthographic projection that has its point of tangency wherever we want:

Other projection methods are based on more complicated flattenable projection surfaces, and instead of points of tangency, spacial cases of these projections can be made by adjusting their Standard Paralells or Central Meridians

These images are by Erwin Raisz, who worked at the Harvard Institute for Geographical Exploration between 1931 and 1951 and wrote many fine books on cartography and topography.

The Mercator Projection

Presentation of uniform scale is not always the goal when using maps as graphical calculators. A case in point is the Mercator's Cylyndrical Projection, which is unique among projections in terms of portraying lines of constant compass direction (rhumb lines) as straight lines on the map, as shown on this site by Carlos A Furuti. In a zone along its standard paralell, the Mercator projection has good scale and shape and direction presentation along with its property that makes a a terriffic tool for aiming missles and artillery! This is why the Transverse Case of the Mercator projection was invented and is in such common use in broad-scale series of national mapping projects.

Projection Systems

Over the years, cartographers have adopted several systems of standard cases for projections in certain areas. The most common is the Universal Transverse Mercator (UTM) system that divides the world into 60 longitudinal zones. see map by Alan Morton. This system is the basis for most national and global map series. In the US, all states have adopted special cases of map projections that suit the location shape of their territory. These conventional cases are called State Plane Coordinate Systems. Most states are divided into several different state plane zones.

Metadata Issues with Map Coordinate Systems

Given that geographic datasets are primarly concerned with spatial references, we really must have metadata (data about the data) describing the coordinate system that is embedded in the files. Here is an example of the FGDC Content Standards for Geospatial Metadata regarding Coordinate Systems But in actuality, because most datasets make use of special cases of projections, much of this metadata can be generated automatically if we know the following facts about the coordinate system of a dataset:

  • The projection method, or 'Geographic.'
  • Special projection parameters (e.g. standard paralells or meridians)
  • Assumed Earth Model
  • The coordinate units

On-the-Fly Projection

With all of this potential confusion about coordinate systems, it is truly a blessing that we have mathematics and software that can transform geographic coordinates (latitude and longitude) to projected systems (Forward projection) and projected systems to geographic (backward projection) on the fly without our having to worry about it. Unfortunately, things are not always this simple. Automatic transformation of coordinate systems requires that datasets include machine-readable metadata. In about 2002, the makers of ArcMap added one more file to the schema of a shape file. The .prj file contains the description of the projection of a shape file, and if it exists, it is always copied with the shape file or dlements that are exported from it. This is the machine-readable metadata that allows ArcMap to know how to handle the dataset if any transformation (reprojection) is required. There are plenty of datasets that do not include such machine readable metadata. This includes data that are not created with ArcMap since 2002 and even some that are. So we should get used to understanding map projections and their properties. If you need to learn to set the coordinate system for a dataset, use ArcCatalog - as explained in the The ArcMap Projections Tutorial.

Parting Thoughts

Aside from being a very important and interesting chapter in the history of how people understand their surroundings, the evolving understanding and application of Geodesy and Topography offer a very important lesson for anyone who would attempt to understand the world through representations or models. Understading the logic of one's referencing systems is crucial to understanding the utility of your conclusions. Choosing a projection means choosing one sort of error over another. This sort of choice comes into play with almost every decision that an educated person makes when creating and evaluating maps and GIS.

Additional References