Topology rules in GIS PDF

	Objectives:	Understanding topology in vector data
Keywords:	Vector, topology, topology rules, topology errors, search radius, snapping distance, simple feature

Topology expresses the spatial relationships between connecting or adjacent vector features [points, polylines and polygons] in a GIS. Topological or topology-based data are useful for detecting and correcting digitising errors [e.g. two lines in a roads vector layer that do not meet perfectly at an intersection]. Topology is necessary for carrying out some types of spatial analysis, such as network analysis.

Imagine you travel to London. On a sightseeing tour you plan to visit St. Paul’s Cathedral first and in the afternoon Covent Garden Market for some souvenirs. Looking at the Underground map of London [see figure_topology_london] you have to find connecting trains to get from Covent Garden to St. Paul’s. This requires topological information [data] about where it is possible to change trains. Looking at a map of the underground, the topological relationships are illustrated by circles that show connectivity.

Topology of London Underground Network.

There are different types of topological errors and they can be grouped according to whether the vector feature types are polygons or polylines. Topological errors with polygon features can include unclosed polygons, gaps between polygon borders or overlapping polygon borders. A common topological error with polyline features is that they do not meet perfectly at a point [node]. This type of error is called an undershoot if a gap exists between the lines, and an overshoot if a line ends beyond the line it should connect to [see figure_topology_errors].

Undershoots [1] occur when digitised vector lines that should connect to each other don’t quite touch. Overshoots [2] happen if a line ends beyond the line it should connect to. Slivers [3] occur when the vertices of two polygons do not match up on their borders.

The result of overshoot and undershoot errors are so-called ‘dangling nodes’ at the end of the lines. Dangling nodes are acceptable in special cases, for example if they are attached to dead-end streets.

Topological errors break the relationship between features. These errors need to be fixed in order to be able to analyse vector data with procedures like network analysis [e.g. finding the best route across a road network] or measurement [e.g. finding out the length of a river]. In addition to topology being useful for network analysis and measurement, there are other reasons why it is important and useful to create or have vector data with correct topology. Just imagine you digitise a municipal boundaries map for your province and the polygons overlap or show slivers. If such errors were present, you would be able to use the measurement tools, but the results you get will be incorrect. You will not know the correct area for any municipality and you will not be able to define exactly, where the borders between the municipalities are.

It is not only important for your own analysis to create and have topologically correct data, but also for people who you pass data on to. They will be expecting your data and analysis results to be correct!

Fortunately, many common errors that can occur when digitising vector features can be prevented by topology rules that are implemented in many GIS applications.

Except for some special GIS data formats, topology is usually not enforced by default. Many common GIS, like QGIS, define topology as relationship rules and let the user choose the rules, if any, to be implemented in a vector layer.

The following list shows some examples of where topology rules can be defined for real world features in a vector map:

• Area edges of a municipality map must not overlap.
• Area edges of a municipality map must not have gaps [slivers].
• Polygons showing property boundaries must be closed. Undershoots or overshoots of the border lines are not allowed.
• Contour lines in a vector line layer must not intersect [cross each other].

Snapping distance is the distance a GIS uses to search for the closest vertex and / or segment you are trying to connect when you digitise. A segment is a straight line formed between two vertices in a polygon or polyline geometry. If you aren’t within the snapping distance, a GIS such as QGIS will leave the vertex where you release the mouse button, instead of snapping it to an existing vertex and / or segment [see figure_snapping_distance].

The snapping distance [black circle] is defined in map units [e.g. decimal degrees] for snapping to either vertices or segments.

Search radius is the distance a GIS uses to search for the closest vertex you are trying to move when you click on the map. If you aren’t within the search radius, the GIS won’t find and select any vertex of a feature for editing. In principle, it is quite similar to the snapping distance functionality.

Snapping distance and search radius are both set in map units so you may need to experiment to get the distance value set right. If you specify a value that is too big, the GIS may snap to a wrong vertex, especially if you are dealing with a large number of vertices close together. If you specify the search radius too small the GIS application won’t find any feature or vertex to move or edit.

Mainly designed for simplicity and for fast rendering but not for data analysis that require topology [such as finding routes across a network]. Many GIS applications are able to show topological and simple feature data together and some can also create, edit and analyse both.

Let’s wrap up what we covered in this worksheet:

• Topology shows the spatial relation of neighbouring vector features.
• Topology in GIS is provided by topological tools.
• Topology can be used to detect and correct digitizing errors.
• For some tools, such as network analysis, topological data is essential.
• Snapping distance and search radius help us to digitise topologically correct vector data.
• Simple feature data is not a true topological data format but it is commonly used by GIS applications.

Here are some ideas for you to try with your learners:

• Mark your local bus stops on a toposheet map and then task your learners to find the shortest route between two stops.
• Think of how you would create vector features in a GIS to represent a topological road network of your town. What topological rules are important and what tools can your learners use in QGIS to make sure that the new road layer is topologically correct?

If you don’t have a computer available, you can use a map of a bus or railway network and discuss the spatial relationships and topology with your learners.

In the section that follows we will take a closer look at Coordinate Reference Systems to understand how we relate data from our spherical earth onto flat maps!

Page 2

	Objectives:	Understanding of Coordinate Reference Systems.
Keywords:	Coordinate Reference System [CRS], Map Projection, On the Fly Projection, Latitude, Longitude, Northing, Easting

Map projections try to portray the surface of the earth or a portion of the earth on a flat piece of paper or computer screen. A coordinate reference system [CRS] then defines, with the help of coordinates, how the two-dimensional, projected map in your GIS is related to real places on the earth. The decision as to which map projection and coordinate reference system to use, depends on the regional extent of the area you want to work in, on the analysis you want to do and often on the availability of data.

A traditional method of representing the earth’s shape is the use of globes. There is, however, a problem with this approach. Although globes preserve the majority of the earth’s shape and illustrate the spatial configuration of continent-sized features, they are very difficult to carry in one’s pocket. They are also only convenient to use at extremely small scales [e.g. 1:100 million].

Most of the thematic map data commonly used in GIS applications are of considerably larger scale. Typical GIS datasets have scales of 1:250 000 or greater, depending on the level of detail. A globe of this size would be difficult and expensive to produce and even more difficult to carry around. As a result, cartographers have developed a set of techniques called map projections designed to show, with reasonable accuracy, the spherical earth in two-dimensions.

When viewed at close range the earth appears to be relatively flat. However when viewed from space, we can see that the earth is relatively spherical. Maps, as we will see in the upcoming map production topic, are representations of reality. They are designed to not only represent features, but also their shape and spatial arrangement. Each map projection has advantages and disadvantages. The best projection for a map depends on the scale of the map, and on the purposes for which it will be used. For example, a projection may have unacceptable distortions if used to map the entire African continent, but may be an excellent choice for a large-scale [detailed] map of your country. The properties of a map projection may also influence some of the design features of the map. Some projections are good for small areas, some are good for mapping areas with a large East-West extent, and some are better for mapping areas with a large North-South extent.

The process of creating map projections can be visualised by positioning a light source inside a transparent globe on which opaque earth features are placed. Then project the feature outlines onto a two-dimensional flat piece of paper. Different ways of projecting can be produced by surrounding the globe in a cylindrical fashion, as a cone, or even as a flat surface. Each of these methods produces what is called a map projection family. Therefore, there is a family of planar projections, a family of cylindrical projections, and another called conical projections [see figure_projection_families]

The three families of map projections. They can be represented by a] cylindrical projections, b] conical projections or c] planar projections.

Today, of course, the process of projecting the spherical earth onto a flat piece of paper is done using the mathematical principles of geometry and trigonometry. This recreates the physical projection of light through the globe.

Map projections are never absolutely accurate representations of the spherical earth. As a result of the map projection process, every map shows distortions of angular conformity, distance and area. A map projection may combine several of these characteristics, or may be a compromise that distorts all the properties of area, distance and angular conformity, within some acceptable limit. Examples of compromise projections are the Winkel Tripel projection and the Robinson projection [see figure_robinson_projection], which are often used for world maps.

The Robinson projection is a compromise where distortions of area, angular conformity and distance are acceptable.

It is usually impossible to preserve all characteristics at the same time in a map projection. This means that when you want to carry out accurate analytical operations, you need to use a map projection that provides the best characteristics for your analyses. For example, if you need to measure distances on your map, you should try to use a map projection for your data that provides high accuracy for distances.

When working with a globe, the main directions of the compass rose [North, East, South and West] will always occur at 90 degrees to one another. In other words, East will always occur at a 90 degree angle to North. Maintaining correct angular properties can be preserved on a map projection as well. A map projection that retains this property of angular conformity is called a conformal or orthomorphic projection.

These projections are used when the preservation of angular relationships is important. They are commonly used for navigational or meteorological tasks. It is important to remember that maintaining true angles on a map is difficult for large areas and should be attempted only for small portions of the earth. The conformal type of projection results in distortions of areas, meaning that if area measurements are made on the map, they will be incorrect. The larger the area the less accurate the area measurements will be. Examples are the Mercator projection [as shown in figure_mercator_projection] and the Lambert Conformal Conic projection. The U.S. Geological Survey uses a conformal projection for many of its topographic maps.

The Mercator projection, for example, is used where angular relationships are important, but the relationship of areas are distorted.

If your goal in projecting a map is to accurately measure distances, you should select a projection that is designed to preserve distances well. Such projections, called equidistant projections, require that the scale of the map is kept constant. A map is equidistant when it correctly represents distances from the centre of the projection to any other place on the map. Equidistant projections maintain accurate distances from the centre of the projection or along given lines. These projections are used for radio and seismic mapping, and for navigation. The Plate Carree Equidistant Cylindrical [see figure_plate_caree_projection] and the Equirectangular projection are two good examples of equidistant projections. The Azimuthal Equidistant projection is the projection used for the emblem of the United Nations [see figure_azimuthal_equidistant_projection].

The Plate Carree Equidistant Cylindrical projection, for example, is used when accurate distance measurement is important.

The United Nations Logo uses the Azimuthal Equidistant projection.

When a map portrays areas over the entire map, so that all mapped areas have the same proportional relationship to the areas on the Earth that they represent, the map is an equal area map. In practice, general reference and educational maps most often require the use of equal area projections. As the name implies, these maps are best used when calculations of area are the dominant calculations you will perform. If, for example, you are trying to analyse a particular area in your town to find out whether it is large enough for a new shopping mall, equal area projections are the best choice. On the one hand, the larger the area you are analysing, the more precise your area measures will be, if you use an equal area projection rather than another type. On the other hand, an equal area projection results in** distortions of angular conformity** when dealing with large areas. Small areas will be far less prone to having their angles distorted when you use an equal area projection. Alber’s equal area, Lambert’s equal area and Mollweide Equal Area Cylindrical projections [shown in figure_mollweide_equal_area_projection] are types of equal area projections that are often encountered in GIS work.

The Mollweide Equal Area Cylindrical projection, for example, ensures that all mapped areas have the same proportional relationship to the areas on the Earth.

Keep in mind that map projection is a very complex topic. There are hundreds of different projections available world wide each trying to portray a certain portion of the earth’s surface as faithfully as possible on a flat piece of paper. In reality, the choice of which projection to use, will often be made for you. Most countries have commonly used projections and when data is exchanged people will follow the national trend.

With the help of coordinate reference systems [CRS] every place on the earth can be specified by a set of three numbers, called coordinates. In general CRS can be divided into projected coordinate reference systems [also called Cartesian or rectangular coordinate reference systems] and geographic coordinate reference systems.

The use of Geographic Coordinate Reference Systems is very common. They use degrees of latitude and longitude and sometimes also a height value to describe a location on the earth’s surface. The most popular is called WGS 84.

Lines of latitude run parallel to the equator and divide the earth into 180 equally spaced sections from North to South [or South to North]. The reference line for latitude is the equator and each hemisphere is divided into ninety sections, each representing one degree of latitude. In the northern hemisphere, degrees of latitude are measured from zero at the equator to ninety at the north pole. In the southern hemisphere, degrees of latitude are measured from zero at the equator to ninety degrees at the south pole. To simplify the digitisation of maps, degrees of latitude in the southern hemisphere are often assigned negative values [0 to -90°]. Wherever you are on the earth’s surface, the distance between the lines of latitude is the same [60 nautical miles]. See figure_geographic_crs for a pictorial view.

Geographic coordinate system with lines of latitude parallel to the equator and lines of longitude with the prime meridian through Greenwich.

Lines of longitude, on the other hand, do not stand up so well to the standard of uniformity. Lines of longitude run perpendicular to the equator and converge at the poles. The reference line for longitude [the prime meridian] runs from the North pole to the South pole through Greenwich, England. Subsequent lines of longitude are measured from zero to 180 degrees East or West of the prime meridian. Note that values West of the prime meridian are assigned negative values for use in digital mapping applications. See figure_geographic_crs for a pictorial view.

At the equator, and only at the equator, the distance represented by one line of longitude is equal to the distance represented by one degree of latitude. As you move towards the poles, the distance between lines of longitude becomes progressively less, until, at the exact location of the pole, all 360° of longitude are represented by a single point that you could put your finger on [you probably would want to wear gloves though]. Using the geographic coordinate system, we have a grid of lines dividing the earth into squares that cover approximately 12363.365 square kilometres at the equator — a good start, but not very useful for determining the location of anything within that square.

To be truly useful, a map grid must be divided into small enough sections so that they can be used to describe [with an acceptable level of accuracy] the location of a point on the map. To accomplish this, degrees are divided into minutes ['] and seconds ["]. There are sixty minutes in a degree, and sixty seconds in a minute [3600 seconds in a degree]. So, at the equator, one second of latitude or longitude = 30.87624 meters.

A two-dimensional coordinate reference system is commonly defined by two axes. At right angles to each other, they form a so called XY-plane [see figure_projected_crs on the left side]. The horizontal axis is normally labelled X, and the vertical axis is normally labelled Y. In a three-dimensional coordinate reference system, another axis, normally labelled Z, is added. It is also at right angles to the X and Y axes. The Z axis provides the third dimension of space [see figure_projected_crs on the right side]. Every point that is expressed in spherical coordinates can be expressed as an X Y Z coordinate.

Two and three dimensional coordinate reference systems.

A projected coordinate reference system in the southern hemisphere [south of the equator] normally has its origin on the equator at a specific Longitude. This means that the Y-values increase southwards and the X-values increase to the West. In the northern hemisphere [north of the equator] the origin is also the equator at a specific Longitude. However, now the Y-values increase northwards and the X-values increase to the East. In the following section, we describe a projected coordinate reference system, called Universal Transverse Mercator [UTM] often used for South Africa.

The Universal Transverse Mercator [UTM] coordinate reference system has its origin on the equator at a specific Longitude. Now the Y-values increase southwards and the X-values increase to the West. The UTM CRS is a global map projection. This means, it is generally used all over the world. But as already described in the section ‘accuracy of map projections’ above, the larger the area [for example South Africa] the more distortion of angular conformity, distance and area occur. To avoid too much distortion, the world is divided into 60 equal zones that are all 6 degrees wide in longitude from East to West. The UTM zones are numbered 1 to 60, starting at the international date line [zone 1 at 180 degrees West longitude] and progressing East back to the international date line [zone 60 at 180 degrees East longitude] as shown in figure_utm_zones.

The Universal Transverse Mercator zones. For South Africa UTM zones 33S, 34S, 35S, and 36S are used.

As you can see in figure_utm_zones and figure_utm_for_sa, South Africa is covered by four UTM zones to minimize distortion. The zones are called UTM 33S, UTM 34S, UTM 35S and UTM 36S. The S after the zone means that the UTM zones are located south of the equator.

UTM zones 33S, 34S, 35S, and 36S with their central longitudes [meridians] used to project South Africa with high accuracy. The red cross shows an Area of Interest [AOI].

Say, for example, that we want to define a two-dimensional coordinate within the Area of Interest [AOI] marked with a red cross in figure_utm_for_sa. You can see, that the area is located within the UTM zone 35S. This means, to minimize distortion and to get accurate analysis results, we should use UTM zone 35S as the coordinate reference system.

The position of a coordinate in UTM south of the equator must be indicated with the zone number [35] and with its northing [y] value and easting [x] value in meters. The northing value is the distance of the position from the equator in meters. The easting value is the distance from the central meridian [longitude] of the used UTM zone. For UTM zone 35S it is 27 degrees East as shown in figure_utm_for_sa. Furthermore, because we are south of the equator and negative values are not allowed in the UTM coordinate reference system, we have to add a so called false northing value of 10,000,000 m to the northing [y] value and a false easting value of 500,000 m to the easting [x] value. This sounds difficult, so, we will do an example that shows you how to find the correct UTM 35S coordinate for the Area of Interest.

The place we are looking for is 3,550,000 meters south of the equator, so the northing [y] value gets a negative sign and is -3,550,000 m. According to the UTM definitions we have to add a false northing value of 10,000,000 m. This means the northing [y] value of our coordinate is 6,450,000 m [-3,550,000 m + 10,000,000 m].

First we have to find the central meridian [longitude] for the UTM zone 35S. As we can see in figure_utm_for_sa it is 27 degrees East. The place we are looking for is 85,000 meters West from the central meridian. Just like the northing value, the easting [x] value gets a negative sign, giving a result of -85,000 m. According to the UTM definitions we have to add a false easting value of 500,000 m. This means the easting [x] value of our coordinate is 415,000 m [-85,000 m + 500,000 m]. Finally, we have to add the zone number to the easting value to get the correct value.

As a result, the coordinate for our Point of Interest, projected in UTM zone 35S would be written as: 35 415,000 m E / 6,450,000 m N. In some GIS, when the correct UTM zone 35S is defined and the units are set to meters within the system, the coordinate could also simply appear as 415,000 6,450,000.

As you can probably imagine, there might be a situation where the data you want to use in a GIS are projected in different coordinate reference systems. For example, you might get a vector layer showing the boundaries of South Africa projected in UTM 35S and another vector layer with point information about rainfall provided in the geographic coordinate system WGS 84. In GIS these two vector layers are placed in totally different areas of the map window, because they have different projections.

To solve this problem, many GIS include a functionality called on-the-fly projection. It means, that you can define a certain projection when you start the GIS and all layers that you then load, no matter what coordinate reference system they have, will be automatically displayed in the projection you defined. This functionality allows you to overlay layers within the map window of your GIS, even though they may be in different reference systems.

The topic map projection is very complex and even professionals who have studied geography, geodetics or any other GIS related science, often have problems with the correct definition of map projections and coordinate reference systems. Usually when you work with GIS, you already have projected data to start with. In most cases these data will be projected in a certain CRS, so you don’t have to create a new CRS or even re project the data from one CRS to another. That said, it is always useful to have an idea about what map projection and CRS means.

Let’s wrap up what we covered in this worksheet:

• Map projections portray the surface of the earth on a two-dimensional, flat piece of paper or computer screen.
• There are global map projections, but most map projections are created and optimized to project smaller areas of the earth’s surface.
• Map projections are never absolutely accurate representations of the spherical earth. They show distortions of angular conformity, distance and area. It is impossible to preserve all these characteristics at the same time in a map projection.
• A Coordinate reference system [CRS] defines, with the help of coordinates, how the two-dimensional, projected map is related to real locations on the earth.
• There are two different types of coordinate reference systems: Geographic Coordinate Systems and Projected Coordinate Systems.
• On the Fly projection is a functionality in GIS that allows us to overlay layers, even if they are projected in different coordinate reference systems.

Here are some ideas for you to try with your learners:

• Start QGIS and load two layers of the same area but with different projections and let your pupils find the coordinates of several places on the two layers. You can show them that it is not possible to overlay the two layers. Then define the coordinate reference system as Geographic/WGS 84 inside the Project Properties dialog and activate the checkbox
  Enable on-the-fly CRS transformation. Load the two layers of the same area again and let your pupils see how on-the-fly projection works.
• You can open the Project Properties dialog in QGIS and show your pupils the many different Coordinate Reference Systems so they get an idea of the complexity of this topic. With ‘on-the-fly’ CRS transformation enabled you can select different CRS to display the same layer in different projections.

If you don’t have a computer available, you can show your pupils the principles of the three map projection families. Get a globe and paper and demonstrate how cylindrical, conical and planar projections work in general. With the help of a transparency sheet you can draw a two-dimensional coordinate reference system showing X axes and Y axes. Then, let your pupils define coordinates [x and y values] for different places.

In the section that follows we will take a closer look at Map Production.

Page 3

	Objectives:	Understanding of map production for spatial data
Keywords:	Map production, map layout, scale bar, north arrow, legend, map body, map unit

Map production is the process of arranging map elements on a sheet of paper in a way that, even without many words, the average person can understand what it is all about. Maps are usually produced for presentations and reports where the audience or reader is a politician, citizen or a learner with no professional background in GIS. Because of this, a map has to be effective in communicating spatial information. Common elements of a map are the title, map body, legend, north arrow, scale bar, acknowledgement, and map border [see figure_map_elements].

Common map elements [labelled in red] are the title, map body, legend, north arrow, scale bar, acknowledgement and map border.

Other elements that might be added are e.g. a graticule, or name of the map projection [CRS]. Together, these elements help the map reader to interpret the information shown on the map. The map body is, of course, the most important part of the map because it contains the map information. The other elements support the communication process and help the map reader to orientate himself and understand the map topic. For example, the title describes the subject matter and the legend relates map symbols to the mapped data.

The map title is very important because it is usually the first thing a reader will look at on a map. It can be compared with a title in a newspaper. It should be short but give the reader a first idea of what the map is about.

The map border is a line that defines exactly the edges of the area shown on the map. When printing a map with a graticule [which we describe further down], you often find the coordinate information of the graticule lines along the border lines, as you can see in figure_map_legend.

A map is a simplified representation of the real world and map symbols are used to represent real objects. Without symbols, we wouldn’t understand maps. To ensure that a person can correctly read a map, a map legend is used to provide a key to all the symbols used on the map. It is like a dictionary that allows you to understand the meaning of what the map shows. A map legend is usually shown a a little box in a corner of the map. It contains icons, each of which will represent a type of feature. For example, a house icon will show you how to identify houses on the map [see figure_map_legend].

Two maps from the same area, both with a water body in the background but with different themes, map symbols and colours in the legend.

You can also use different symbols and icons in your legend to show different themes. In figure_map_legend you can see a map with a lake in light blue overlaid with contour lines and spot heights to show information about the terrain in that area. On the right side you see the same area with the lake in the background but this map is designed to show tourists the location of houses they can rent for their holidays. It uses brighter colours, a house icon and more descriptive and inviting words in the legend.

A north arrow [sometimes also called a compass rose] is a figure displaying the main directions, North, South, East and West. On a map it is used to indicate the direction of North.

For example, in GIS this means that a house that is located north from a lake can be found on top of the lake on a map. The road in the east will then be to the right of the water body on the map, a river in the south will be below the water body and if you are searching for a train station to the west of the lake you will find it on the left side on the map.

The scale of a map, is the value of a single unit of distance on the map, representing distance in the real world. The values are shown in map units [meters, feet or degrees]. The scale can be expressed in several ways, for example, in words, as a ratio or as a graphical scale bar [see figure_map_scale].

Expressing a scale in words is a commonly used method and has the advantage of being easily understood by most map users. You can see an example of a word based scale in a figure_map_scale [a]. Another option is the representative fraction [RF] method, where both the map distance and the ground distance in the real world are given in the same map units, as a ratio. For example, a RF value 1:25,000 means that any distance on the map is 1/25,000 th of the real distance on the ground [see figure_map_scale [b]]. The value 25,000 in the ratio is called the scale denominator. More experienced users often prefer the representative fraction method, because it reduces confusion.

When a representative fraction expresses a very small ratio, for example 1:1000 000, it is called a small scale map. On the other hand if the ratio is very large, for example a 1:50 000 map, it is called a large scale map. It is handy to remember that a small scale map covers a large area, and a large scale map covers a small area!

A scale expression as a graphic or bar scale is another basic method of expressing a scale. A bar scale shows measured distances on the map. The equivalent distance in the real world is placed above as you can see in figure_map_scale [c].

A map scale can be expressed in words [a], as a ratio [b], or as graphic or bar scale [c]

Maps are usually produced at standard scales of, for example, 1:10 000, 1:25 000, 1:50 000, 1:100 000, 1:250 000, 1:500 000. What does this mean to the map reader? It means that if you multiply the distance measured on the map by the scale denominator, you will know the distance in the real world.

For example, if we want to measure a distance of 100mm on a map with a scale of 1:25,000 we calculate the real world distance like this:

100 mm x 25,000 = 2,500,000 mm

This means that 100 mm on the map is equivalent to 2,500,000 mm [2500 m] in the real word.

Another interesting aspect of a map scale, is that the lower the map scale, the more detailed the feature information in the map will be. In figure_map_scale_compare, you can see an example of this. Both maps are the same size but have a different scale. The image on the left side shows more details, for example the houses south-west of the water body can be clearly identified as separate squares. In the image on the right you can only see a black clump of rectangles and you are not able to see each house clearly.

Maps showing an area in two different scales. The map scale on the left is 1:25,000. The map scale on the right is 1:50,000.

In the acknowledgment area of a map it is possible to add text with important information. For example information about the quality of the used data can be useful to give the reader an idea about details such as how, by whom and when a map was created. If you look at a topographical map of your town, it would be useful to know when the map was created and who did it. If the map is already 50 years old, you will probably find a lot of houses and roads that no longer exist or maybe never even existed. If you know that the map was created by an official institution, you could contact them and ask if they have a more current version of that map with updated information.

A graticule is a network of lines overlain on a map to make spatial orientation easier for the reader. The lines can be used as a reference. As an example, the lines of a graticule can represent the earth’s parallels of latitude and meridians of longitude. When you want to refer to a special area on a map during your presentation or in a report you could say: ‘the houses close to latitude 26.04 / longitude -32.11 are often exposed to flooding during January and February’ [see figure_map_graticule].

Graticules [red lines] representing the Earth’s parallels of latitude and meridians of longitude. The latitude and longitude values on the map border can be used for better orientation on the map.

A map projection tries to represent the 3-dimensional Earth with all its features like houses, roads or lakes on a flat sheet of paper. This is very difficult as you can imagine, and even after hundreds of years there is no single projection that is able to represent the Earth perfectly for any area in the world. Every projection has advantages and disadvantages.

To be able to create maps as precisely as possible, people have studied, modified, and produced many different kinds of projections. In the end almost every country has developed its own map projection with the goal of improving the map accuracy for their territorial area [see figure_map_projection].

The world in different projections. A Mollweide Equal Area projection left, a Plate Carree Equidistant Cylindrical projection on the right.

With this in mind, we can now understand why it makes sense to add the name of the projection on a map. It allows the reader to see quickly, if one map can be compared with another. For example, features on a map in a so-called Equal Area projection appear very different to features projected in a Cylindrical Equidistant projection [see figure_map_projection].

Map projection is a very complex topic and we cannot cover it completely here. You may want to take a look at our previous topic: Coordinate Reference Systems if you want to know more about it.

It is sometimes difficult to create a map that is easy to understand and well laid out whilst still showing and explaining all the information that the reader needs to know. To achieve this, you need to create an ideal arrangement and composition of all the map elements. You should concentrate on what story you want to tell with your map and how the elements, such as the legend, scale bar and acknowledgements should be ordered. By doing this, you will have a well designed and educational map, that people would like to look at and be able to understand.

Let’s wrap up what we covered in this worksheet:

• Map production means arranging map elements on a sheet of paper.
• Map elements are the title, map body, map border, legend, scale, north arrow and the acknowledgement.
• Scale represents the ratio of a distance on the map to the actual distance in the real world.
• Scale is displayed in map units [meters, feet or degrees]
• A legend explains all the symbols on a map.
• A map should explain complex information as simply as possible.
• Maps are usually always displayed ‘North up‘.

Here are some ideas for you to try with your learners:

• Load some vector layers in your GIS for your local area. See if your learners can identify examples of different types of legend elements such as road types or buildings. Create a list of legend elements and define what the icons should look like, so a reader can most easily figure out their meaning in the map.
• Create a map layout with your learners on a sheet of paper. Decide on the title of the map, what GIS layers you want to show and what colours and icons to have on the map. Use the techniques you learned in Topics Vector Data and Vector Attribute Data to adjust the symbology accordingly. When you have a template, open the QGIS Map Composer and try to arrange a map layout as planned.

If you don’t have a computer available, you can use any topographical map and discuss the map design with your learners. Figure out if they understand what the map wants to tell. What can be improved? How accurately does the map represent the history of the area? How would a map from 100 years ago differ from the same map today?

In the section that follows we will take a closer look at vector analysis to see how we can use a GIS for more than just making good looking maps!

Page 4

Spatial analysis uses spatial information to extract new and additional meaning from GIS data. Usually spatial analysis is carried out using a GIS Application. GIS Applications normally have spatial analysis tools for feature statistics [e.g. how many vertices make up this polyline?] or geoprocessing such as feature buffering. The types of spatial analysis that are used vary according to subject areas. People working in water management and research [hydrology] will most likely be interested in analysing terrain and modelling water as it moves across it. In wildlife management users are interested in analytical functions that deal with wildlife point locations and their relationship to the environment. In this topic we will discuss buffering as an example of a useful spatial analysis that can be carried out with vector data.

Buffering usually creates two areas: one area that is within a specified distance to selected real world features and the other area that is beyond. The area that is within the specified distance is called the buffer zone.

A buffer zone is any area that serves the purpose of keeping real world features distant from one another. Buffer zones are often set up to protect the environment, protect residential and commercial zones from industrial accidents or natural disasters, or to prevent violence. Common types of buffer zones may be greenbelts between residential and commercial areas, border zones between countries [see figure_buffer_zone], noise protection zones around airports, or pollution protection zones along rivers.

The border between the United States of America and Mexico is separated by a buffer zone. [Photo taken by SGT Jim Greenhill 2006].

In a GIS Application, buffer zones are always represented as vector polygons enclosing other polygon, line or point features [see figure_point_buffer, figure_line_buffer, ].

A buffer zone around vector points.

A buffer zone around vector polylines.

A buffer zone around vector polylines.

There are several variations in buffering. The buffer distance or buffer size can vary according to numerical values provided in the vector layer attribute table for each feature. The numerical values have to be defined in map units according to the Coordinate Reference System [CRS] used with the data. For example, the width of a buffer zone along the banks of a river can vary depending on the intensity of the adjacent land use. For intensive cultivation the buffer distance may be bigger than for organic farming [see Figure figure_variable_buffer and Table table_buffer_attributes].

Buffering rivers with different buffer distances.

River Adjacent land use Buffer distance [meters]

Breede River	Intensive vegetable cultivation	100
Komati	Intensive cotton cultivation	150
Oranje	Organic farming	50
Telle river	Organic farming	50

Table Buffer Attributes 1: Attribute table with different buffer distances to rivers based on information about the adjacent land use.

Buffers around polyline features, such as rivers or roads, do not have to be on both sides of the lines. They can be on either the left side or the right side of the line feature. In these cases the left or right side is determined by the direction from the starting point to the end point of line during digitising.

A feature can also have more than one buffer zone. A nuclear power plant may be buffered with distances of 10, 15, 25 and 30 km, thus forming multiple rings around the plant as part of an evacuation plan [see figure_multiple_buffers].

Buffering a point feature with distances of 10, 15, 25 and 30 km.

Buffer zones often have dissolved boundaries so that there are no overlapping areas between the buffer zones. In some cases though, it may also be useful for boundaries of buffer zones to remain intact, so that each buffer zone is a separate polygon and you can identify the overlapping areas [see Figure figure_buffer_dissolve].

Buffer zones with dissolved [left] and with intact boundaries [right] showing overlapping areas.

Buffer zones around polygon features are usually extended outward from a polygon boundary but it is also possible to create a buffer zone inward from a polygon boundary. Say, for example, the Department of Tourism wants to plan a new road around Robben Island and environmental laws require that the road is at least 200 meters inward from the coast line. They could use an inward buffer to find the 200 m line inland and then plan their road not to go beyond that line.

Most GIS Applications offer buffer creation as an analysis tool, but the options for creating buffers can vary. For example, not all GIS Applications allow you to buffer on either the left side or the right side of a line feature, to dissolve the boundaries of buffer zones or to buffer inward from a polygon boundary.

A buffer distance always has to be defined as a whole number [integer] or a decimal number [floating point value]. This value is defined in map units [meters, feet, decimal degrees] according to the Coordinate Reference System [CRS] of the vector layer.

Let’s wrap up what we covered in this worksheet:

• Buffer zones describe areas around real world features.
• Buffer zones are always vector polygons.
• A feature can have multiple buffer zones.
• The size of a buffer zone is defined by a buffer distance.
• A buffer distance has to be an integer or floating point value.
• A buffer distance can be different for each feature within a vector layer.
• Polygons can be buffered inward or outward from the polygon boundary.
• Buffer zones can be created with intact or dissolved boundaries.
• Besides buffering, a GIS usually provides a variety of vector analysis tools to solve spatial tasks.

Here are some ideas for you to try with your learners:

• Because of dramatic traffic increase, the town planners want to widen the main road and add a second lane. Create a buffer around the road to find properties that fall within the buffer zone [see figure_buffer_road].
• For controlling protesting groups, the police want to establish a neutral zone to keep protesters at least 100 meters from a building. Create a buffer around a building and colour it so that event planners can see where the buffer area is.
• A truck factory plans to expand. The siting criteria stipulate that a potential site must be within 1 km of a heavy-duty road. Create a buffer along a main road so that you can see where potential sites are.
• Imagine that the city wants to introduce a law stipulating that no bottle stores may be within a 1000 meter buffer zone of a school or a church. Create a 1 km buffer around your school and then go and see if there would be any bottle stores too close to your school.

Buffer zone [green] around a roads map [brown]. You can see which houses fall within the buffer zone, so now you could contact the owner and talk to him about the situation.

If you don’t have a computer available, you can use a toposheet and a compass to create buffer zones around buildings. Make small pencil marks at equal distance all along your feature using the compass, then connect the marks using a ruler!

Books:

• Galati, Stephen R. [2006]. Geographic Information Systems Demystified. Artech House Inc. ISBN: 158053533X
• Chang, Kang-Tsung [2006]. Introduction to Geographic Information Systems. 3rd Edition. McGraw Hill. ISBN: 0070658986
• DeMers, Michael N. [2005]. Fundamentals of Geographic Information Systems. 3rd Edition. Wiley. ISBN: 9814126195

Websites:

• //www.manifold.net/doc/transform_border_buffers.htm

The QGIS User Guide also has more detailed information on analysing vector data in QGIS.

In the section that follows we will take a closer look at interpolation as an example of spatial analysis you can do with raster data.

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	Objectives:	Understanding of interpolation as part of spatial analysis
Keywords:	Point data, interpolation method, Inverse Distance Weighted, Triangulated Irregular Network

Spatial analysis is the process of manipulating spatial information to extract new information and meaning from the original data. Usually spatial analysis is carried out with a Geographic Information System [GIS]. A GIS usually provides spatial analysis tools for calculating feature statistics and carrying out geoprocessing activities as data interpolation. In hydrology, users will likely emphasize the importance of terrain analysis and hydrological modelling [modelling the movement of water over and in the earth]. In wildlife management, users are interested in analytical functions dealing with wildlife point locations and their relationship to the environment. Each user will have different things they are interested in depending on the kind of work they do.

Spatial interpolation is the process of using points with known values to estimate values at other unknown points. For example, to make a precipitation [rainfall] map for your country, you will not find enough evenly spread weather stations to cover the entire region. Spatial interpolation can estimate the temperatures at locations without recorded data by using known temperature readings at nearby weather stations [see figure_temperature_map]. This type of interpolated surface is often called a statistical surface. Elevation data, precipitation, snow accumulation, water table and population density are other types of data that can be computed using interpolation.

Temperature map interpolated from South African Weather Stations.

Because of high cost and limited resources, data collection is usually conducted only in a limited number of selected point locations. In GIS, spatial interpolation of these points can be applied to create a raster surface with estimates made for all raster cells.

In order to generate a continuous map, for example, a digital elevation map from elevation points measured with a GPS device, a suitable interpolation method has to be used to optimally estimate the values at those locations where no samples or measurements were taken. The results of the interpolation analysis can then be used for analyses that cover the whole area and for modelling.

There are many interpolation methods. In this introduction we will present two widely used interpolation methods called Inverse Distance Weighting [IDW] and Triangulated Irregular Networks [TIN]. If you are looking for additional interpolation methods, please refer to the ‘Further Reading’ section at the end of this topic.

In the IDW interpolation method, the sample points are weighted during interpolation such that the influence of one point relative to another declines with distance from the unknown point you want to create [see figure_idw_interpolation].

Inverse Distance Weighted interpolation based on weighted sample point distance [left]. Interpolated IDW surface from elevation vector points [right]. Image Source: Mitas, L., Mitasova, H. [1999].

Weighting is assigned to sample points through the use of a weighting coefficient that controls how the weighting influence will drop off as the distance from new point increases. The greater the weighting coefficient, the less the effect points will have if they are far from the unknown point during the interpolation process. As the coefficient increases, the value of the unknown point approaches the value of the nearest observational point.

It is important to notice that the IDW interpolation method also has some disadvantages: the quality of the interpolation result can decrease, if the distribution of sample data points is uneven. Furthermore, maximum and minimum values in the interpolated surface can only occur at sample data points. This often results in small peaks and pits around the sample data points as shown in figure_idw_interpolation.

In GIS, interpolation results are usually shown as a 2 dimensional raster layer. In figure_idw_result, you can see a typical IDW interpolation result, based on elevation sample points collected in the field with a GPS device.

IDW interpolation result from irregularly collected elevation sample points [shown as black crosses].

TIN interpolation is another popular tool in GIS. A common TIN algorithm is called Delaunay triangulation. It tries to create a surface formed by triangles of nearest neighbour points. To do this, circumcircles around selected sample points are created and their intersections are connected to a network of non overlapping and as compact as possible triangles [see figure_tin_interpolation].

Delaunay triangulation with circumcircles around the red sample data. The resulting interpolated TIN surface created from elevation vector points is shown on the right. Image Source: Mitas, L., Mitasova, H. [1999].

The main disadvantage of the TIN interpolation is that the surfaces are not smooth and may give a jagged appearance. This is caused by discontinuous slopes at the triangle edges and sample data points. In addition, triangulation is generally not suitable for extrapolation beyond the area with collected sample data points [see figure_tin_result ].

Delaunay TIN interpolation result from irregularly collected rainfall sample points [blue circles]

It is important to remember that there is no single interpolation method that can be applied to all situations. Some are more exact and useful than others but take longer to calculate. They all have advantages and disadvantages. In practice, selection of a particular interpolation method should depend upon the sample data, the type of surfaces to be generated and tolerance of estimation errors. Generally, a three step procedure is recommended:

1. Evaluate the sample data. Do this to get an idea on how data are distributed in the area, as this may provide hints on which interpolation method to use.
2. Apply an interpolation method which is most suitable to both the sample data and the study objectives. When you are in doubt, try several methods, if available.
3. Compare the results and find the best result and the most suitable method. This may look like a time consuming process at the beginning. However, as you gain experience and knowledge of different interpolation methods, the time required for generating the most suitable surface will be greatly reduced.

Although we concentrated on IDW and TIN interpolation methods in this worksheet, there are more spatial interpolation methods provided in GIS, such as Regularized Splines with Tension [RST], Kriging or Trend Surface interpolation. See the additional reading section below for a web link.

Let’s wrap up what we covered in this worksheet:

• Interpolation uses vector points with known values to estimate values at unknown locations to create a raster surface covering an entire area.
• The interpolation result is typically a raster layer.
• It is important to find a suitable interpolation method to optimally estimate values for unknown locations.
• IDW interpolation gives weights to sample points, such that the influence of one point on another declines with distance from the new point being estimated.
• TIN interpolation uses sample points to create a surface formed by triangles based on nearest neighbour point information.

Here are some ideas for you to try with your learners:

• The Department of Agriculture plans to cultivate new land in your area but apart from the character of the soils, they want to know if the rainfall is sufficient for a good harvest. All the information they have available comes from a few weather stations around the area. Create an interpolated surface with your learners that shows which areas are likely to receive the highest rainfall.
• The tourist office wants to publish information about the weather conditions in January and February. They have temperature, rainfall and wind strength data and ask you to interpolate their data to estimate places where tourists will probably have optimal weather conditions with mild temperatures, no rainfall and little wind strength. Can you identify the areas in your region that meet these criteria?

If you don’t have a computer available, you can use a toposheet and a ruler to estimate elevation values between contour lines or rainfall values between fictional weather stations. For example, if rainfall at weather station A is 50 mm per month and at weather station B it is 90 mm, you can estimate, that the rainfall at half the distance between weather station A and B is 70 mm.

Books:

• Chang, Kang-Tsung [2006]. Introduction to Geographic Information Systems. 3rd Edition. McGraw Hill. ISBN: 0070658986
• DeMers, Michael N. [2005]: Fundamentals of Geographic Information Systems. 3rd Edition. Wiley. ISBN: 9814126195
• Mitas, L., Mitasova, H. [1999]. Spatial Interpolation. In: P.Longley, M.F. Goodchild, D.J. Maguire, D.W.Rhind [Eds.], Geographical Information Systems: Principles, Techniques, Management and Applications, Wiley.

Websites:

The QGIS User Guide also has more detailed information on interpolation tools provided in QGIS.

This is the final worksheet in this series. We encourage you to explore QGIS and use the accompanying QGIS manual to discover all the other things you can do with GIS software!

Page 6

Select graphics from The Noun Project collection

Untranslated page? Or you spot a translation error: fix me
Textual error, missing text or you know better: fix me

Page 7

Version 1.3, 3 November 2008

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However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated [a] provisionally, unless and until the copyright holder explicitly and finally terminates your license, and [b] permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.

Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License [for any work] from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.

Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it.

10. FUTURE REVISIONS OF THIS LICENSE

The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See //www.gnu.org/copyleft/.

Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License “or any later version” applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published [not as a draft] by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published [not as a draft] by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy’s public statement of acceptance of a version permanently authorizes you to choose that version for the Document.

11. RELICENSING

“Massive Multiauthor Collaboration Site” [or “MMC Site”] means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A “Massive Multiauthor Collaboration” [or “MMC”] contained in the site means any set of copyrightable works thus published on the MMC site.

“CC-BY-SA” means the Creative Commons Attribution-Share Alike 3.0 license published by Creative Commons Corporation, a not-for-profit corporation with a principal place of business in San Francisco, California, as well as future copyleft versions of that license published by that same organization.

“Incorporate” means to publish or republish a Document, in whole or in part, as part of another Document.

An MMC is “eligible for relicensing” if it is licensed under this License, and if all works that were first published under this License somewhere other than this MMC, and subsequently incorporated in whole or in part into the MMC, [1] had no cover texts or invariant sections, and [2] were thus incorporated prior to November 1, 2008.

The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing.

ADDENDUM: How to use this License for your documents

To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:

Copyright © YEAR YOUR NAME. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”.

If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the “with ... Texts.” line with this:

with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.

If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.

If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.

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