Page 72 - Geosystems An Introduction to Physical Geography 4th Canadian Edition

36 Chapter 1 essentials of Geography AQuantitativeSOlUTION Map Scales
A map does not have to be just something we look at qualita- tively, for example to see factual information and observe spatial relationships. If we apply the concept of scale, we can also use maps to make quantitative measurements of distances and areas. In this chapter we described map scale as being a ratio between map distance and ground distance. In the form of a representa- tive fraction, a map scale could be 1/100 000 or 1:100 000. This means that “any unit on the map represents 100 000 identical units on the ground.” The value 100 000 in this example is the scale factor. Before going on, review the concept of scale in the section titled The Scale of Maps on page 26.
It is important to remember that map scales are for linear measurements only. This statement is true: “On a 1:100 000 scale map, 1 cm on the map represents 100 000 cm on the ground,” but this statement is false: “On a 1:100 000 scale map, 1 cm2 on the map represents 100 000 cm2 on the ground.” To determine the size of a ground area represented on a map, we need to be careful about how we apply the linear scale. First we will see how to use map scale for measuring distances.
The general form of the solution is an equation with two ratios:
1 map distance scale factor = ground distance
To calculate the ground distance (x) represented by a distance of 1 cm on a 1:50 000 map, set up:
1
50 000 =
Calculating ground areas of features is more complicated. We will begin by illustrating the idea for a perfect square. To calculate the ground area of a square-shaped feature represented on a map, we must apply the linear scale to both sides of the square before calculating the area of the square.
On a 1:100 000 scale map, a square has sides 2 cm in length. What is the ground area of the square?
(2 cm × 100 000) × (2 cm × 100 000) =
200 000 cm × 200 000 cm = 4 × 1012 cm2
While the answer 4 × 1012 cm2 is numerically correct, we rarely use units of 1012 cm2 (or 1 000 000 000 000 cm2). To produce an answer with units that are easier to understand, we should convert 200 000 cm to units of metres or kilometres before the final multi- plication operation:
(2 cm × 100 000) × (2 cm × 100 000) = 2000 m × 2000 m = 4 000 000 m2
(2 cm × 100 000) × (2 cm × 100 000) = 2 km × 2 km = 4 km2
To convert from square metres to square kilometres divide by 1 000 000, so 4 000 000 m2 = 4 km2. Therefore both are correct answers, expressed with different units of area.
Features that are perfectly square or rectangular can be mea- sured with this technique.
What would have been the answer if we had applied the linear scale to the size of an area on the map? Calculating map area × scale factor gives an incorrect result for ground area:
(2 cm × 2 cm) × 100 000 = 4 cm2 × 100 000 = 400 000 cm2 On the map, 2 cm × 2 cm = 4 cm2. This is the correct map area,
but 400 000 cm2 is not the correct ground area. 1 cm2 = 0.0001 m2, so 400 000 cm2 is only 40 m2, far less (by a factor of 100 000) than the correct ground area which is 4 000 000 m2.
Irregularly shaped objects are more difficult to measure manu- ally from maps. One solution is to place a grid, consisting of squares smaller than the feature in question, on top of the map. Count the number of squares that are fully inside the feature. Around the edge, some squares will be partially inside the bound- ary of the feature. Count a full square for a square that is at least half inside the boundary, and do not count a square that is not at least half inside.
Then calculate the correct ground area represented by one square on the map, as above, and multiply by the number of squares counted for the feature. If the grid squares are small in relation to the feature, errors caused by counting partial squares around the boundary will be small and the method will give a reasonably accurate estimate of the area.
Mapping and GIS software programs that offer measuring tools eliminate the need to calculate distances and areas manu- ally from maps. However, it is a good idea to understand the fundamentals behind such automated tools.
Cross-multiplying yields
(1 cm × 50 000) x = 1
= 50 000 cm
1 cm x
Although the answer 50 000 cm is correct, it should be converted into equivalent units that are easier to understand. To convert
50 000 centimetres to metres, divide by 100 to give a final answer of 500 m.
Note that the general solution can be expressed as map dis- tance (units) × scale factor = ground distance (identical units).
The example above shows the basic approach to use in any situation, but be sure to use the appropriate scale factor in the ratio equation. What is the ground distance represented by a measurement of 14.8 mm on a 1:250 000 scale map?
250 000 = Cross-multiplying yields
(14.8 mm × 250 000) x = 1
x
= 3 700 000 mm
1 14.8 mm
Convert the answer into metres or kilometres:
3700 000 mm / 1000 = 3700 m, and 3700 m / 1000 = 3.7 km.
NASA’s World Wind software is another open-source browser with access to high-resolution satellite im- ages and multiple data layers suitable for scientific applications.
Geovisualization is the technique of adjusting geo- spatial data sets in real time, so that users can instantly
make changes to maps and other visual models. Geovisual tools are important for translating scientific knowledge into resources that nonscientists can use for decision making and planning. At East Carolina University, scientists are developing geovisual tools to assess the effects of sea level rise along the North Carolina coast, in
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