Maps and ...

"I have an existential map; it has 'you are here' written all over it." - Steven Wright

Map Scale.

"I have a map of the United States... actual size. It says, 'Scale: 1 mile = 1 mile.'

I spent last summer folding it.

I also have a full-size map of the world. I hardly ever unroll it." - Steven Wright

Outline:

1. Implications to Geographical Information Systems.

• Map Scale and Statements of Scale.
1. Representative Fraction (RF).
2. Verbal Statement.
3. Bar or Graphic Scale.
4. Area Scale
• Map Input.
• Scaleless GIS.
• Map Output.
• Influence on Analysis.
• Accuracy vs. Cost as a Function of Scale.

2. Vector Scale.

• Point.
• Line.
• Area.

3. Raster Scale.

• Point.
• Line.
• Area.

4. Questions:

Implications to Geographical Information Systems.

Many of us may ask the question "Why is scale important to GIS?" Once we become familiar with the GIS, we realize that our data can be displayed at many scales. So when do we need to consider scale in geographic information systems? As we will we see from this outline, there are many answers to that question, but perhaps they can all be summarized with one word: Accuracy.

Map Scale and Statements of Scale. Here is a quick refresher for those, who might be a little rusty on the subject of map scale. Most maps (except for Steven Wright's of course) are reduced and simplified representations of reality. The scale of the map lets the user know how much reality has been reduced. This gives the user a great deal more information about areas and distances shown on the map and allows for accurate calculations of both distances and areas. Scale is usually depicted in one of four ways:

1. Representative Fraction. The first we will consider is the RF or Representative Fraction. This scale can be shown as 1/100,000 or as 1:100,000. It simply means that 1 unit on the map is to 100,000 units on the earth's surface. In other words, one inch on the map is equal to one hundred thousand inches on the earth. The same RF means that one meter on the map equals one hundred thousand meters in reality. Any set of units can be placed into the representative fraction as long as the same unit is on both sides of the fraction.
2. Verbal Statement. This is very similar to the meaning behind the RF. If we have an RF of 1:100 and the units are in centimeters, we can use the verbal statement: 1 centimeter (on the map) equals 100 centimeters or 1 meter (on the ground). Simply stated as, "One centimeter equals one meter." Likewise (since there are 63,360 inches in one mile), the representative fraction 1/63,360 can be stated verbally as; "One inch represents one mile."
3. Bar or Graph Scale. This scale is a symbol that depicts a line of a certain length is equal to a certain distance. In this example, one inch as depicted by the graph scale is equal to ten miles.
4. 5. Area Scale. This is the ratio of an area on a map to that same area on the earth's surface. (This scale is only useful on equal-area map projections.) So, on a map with a RF of 1:24,000, 1 inch on the map represents 24,000 inches. We can think of this as 1 inch = 24,0000 inches. If we square both sides of this equation we get; 1 inch2 = 576,000,000 inches2. On this scale map, one square measuring an inch on both sides equals 576 million square inches on the earth.

Map Input. The first answer we considered involves map input. The vast majority of available GIS data is that which is derived from maps, whether scanned or digitized. It is very important to consider the scale of the source maps because the degree of generalization is directly proportional to the map's scale. That is, as scale moves from a large-scale map to a small-scale map, features must take on a more generalized (and essentially less-accurate) form. Ideally, we would know in advance the approximate scale that our results will be displayed, and we could acquire data at that scale, or a slightly larger scale. As we will see in the following sections, the potential exists for some serious problems if we try to integrate or overlay map layers that were derived from source maps of different scales. (As it is, there are plenty of sources of error for data that originated from the same or very similar scale.)

Scaleless GIS. This concept of a "scaleless GIS" comes from the fact that the GIS database is not a function of a scaled map. In other words, we can essentially reduce or increase the way that the data relate to each other and then display this change on the monitor or print it out on a "map". Even though the database is not a function of scale it is characterized by scale - the level of generalization. Herein lies the problem, even though the data may be manipulated in a scaleless environment (the database) when the data is displayed spatially, it must be assigned a scale. So, a GIS is still not truly scaleless unless you can bypass graphically displaying maps and instead produce a table of your results. An ideal database would allow us to display information at any scale with the appropriate amount of detail. (The first full paragraph on page 42 in the text discusses this aspect of GIS.)

Map Output. From above, we know that if GIS data is to be presented spatially, then it must have scale. An example involved overlaying the same object (the Missouri River) from two source maps of different scales. In this instance it is easy to see how the problem arises from generalization, since the river will not "match up" precisely at every location. However, it is quite possible to have a coverage of stream data that was captured at scale of 1:100,000 and overlay it with a landuse coverage that had an original scale of 1:24,000. The resulting output may make a 'nice-looking and apparently error-free' map, but what is the scale? In reality, the data is only as good as the smallest scale. Since in this case, the streams relate to each other and to the landuse at a scale of 1:100,000 that should be the maximum scale of your output. This is regardless of the fact that the data for landuse is capable of much more accurate presentation. The only remedy is to obtain data for the streams at a scale that is appropriate for your project.

Influence on Analysis. It would seem as though some aspects of this influence scale has on analysis has been alluded to throughout the notes. The example that was given in the class lecture involved using a raster based Geographical Information System, like Map Factory. Say you were given the problem of determining if the surface area of a certain property was a majority of open water. The answer you provide could depend largely on the scale of the data you used for your analysis. The calculation returned from analyzing data with 30 meter grid cells could be around 48%. If you ran the same analysis with data from 10-meter grid cells, the better resolution could return an answer of 51%. In this case, the obvious answer would be "Yes, a majority of this property is open water." This is not to say that the first answer was wrong, just less accurate. The first set of data provided an answer that was correct, when the scale of the data is taken into consideration.

Accuracy vs. Cost as a Function of Scale. The example given in the previous section provides an excellent introduction to this portion of the notes. In that example, two different data sets were used to provide an answer to a question. One set of data had a thirty-meter resolution, while the other had 10-meter resolution. Undoubtedly, the more precise data was the most costly data. When faced with this kind of choice, one must answer this compound question: How much accuracy is worth what cost? A large part of the answer comes from the scale of the project. For the more general and small scale project, extreme accuracy is not necessary, so cost of data should be relatively small. However, as the project becomes more specialized and moves to a larger scale, the issue of cost versus accuracy becomes an increasingly important and perplexing question. Perhaps a good approach would be to try to find other projects of similar size and scope. They might be able to provide either a good example or a not-so-good example of the balance between cost and accuracy.

Vector Scale.

Point. Point locations are very easily displayed in a vector system. A point is simply located by its coordinate pair. In other words, a point has a unique set of values in the x and y directions from the origin point of X = 0 and Y = 0. In the Cartesian coordinate system it is easiest to work in the first quadrant since this gives both coordinates positive values.

Line. A vector representation of a line is simply that which connects two points. If we know the coordinates of the two points then we can this method to calculate the length of the line. From the diagram below, we see the two end points of the arc as denoted by X2, Y2 and X1,Y1. These points have values of 3, 5 and 1, 1. If we subtract X1 from X2 and Y1 from Y2, then we get the lengths of lines a = 2 and b = 4. From the Pythagorean theorem we know that c2 = a2 + b2 or that c = Ö a2 + b2. In this case if a = 2 and b = 4 then c = Ö 4 + 16 = Ö 20 @ 4.47. Now that we know this length, we could use it to calculate scale if we knew the actual length the line represented. On the other hand, if we knew the scale of the map we could calculate the length of the actual line from the length we just calculated. Area. Areas can be calculated in the vector domain through use of the following formula.

Area = (-1/2) å (xi yi+1 - yi xi+1) (Where the 'I' values are integrated from 1 to n.)

This procedure works because the software breaks the area in question down into numerous triangles. The areas of the smaller, overlapping triangles are calculated and recalculated until the sum of these areas (and the area of the polygon) start to agree. In the following diagram the area of polygon ABCD equals the areas of triangles ABO and BCO minus the areas of triangles ADO and DCO. The program would calculate these areas using differences in something called the vector crossproduct and assigning the various triangles either a positive or negative area. Finally, the areas are added and subtracted to calculate the area for the polygon in question. Raster Scale.

Point. Raster systems record point data slightly differently than do vector systems. In this case, if a point is contained within a grid-cell, the whole cell is counted as being "on" and only that point is displayed within its cell. In this example the "point" actually has an area. So, the spacing between adjacent points is controlled by the size of the unit cell.

Line. The length of a line can be calculated by counting up the number of grid cells that contain the line and multiplying that number by the height, width or some diagonal of the unit cell. If the line runs north-south or east-west, then multiplying the number of cells containing the line by the width of an individual cell will give the total length of the line. If the line is running northeast-southwest or northwest-southeast, then the length of the line can be determined by multiplying the width of the cell by the Ö 2 = 1.41 which is the diagonal length of a square. This result can then be multiplied by the number of cells that the line passes through to get the overall length of the line.

Area. The raster data structure is ideally suited for area calculations. It can easily calculate the total number of grid-cells contained within the area of concern and then approximate the areas of the cells that are partially contained within the area being calculated. Simply addition of these two values provides the answer to question of calculating an area. We can use this rule of thumb to quickly estimate the area of a closed figure. We count up the total number of cells that are completely contained inside the area. Then we count up the total number of partially contained cells and divide this second by two. Once we have added this second quantity to the first, we multiply the result by the area of an individual cell to calculate a rough estimate of the actual area. Try it out on this figure of a lake. Questions:

# 1) If a line that is 10.24 inches long on a map is equal to a real world distance of 4.3 kilometers, what is the scale of the map? (Remember that there are 2.54 centimeters in an inch.) If we multiply 10.24 inches by 2.54 centimeters/inch we calculate that the line is 26.01 cm long. (Also remember that there are 100,000 centimeters per kilometer.) The actual ground distance of the line must be 4.3 kilometers X 100,000 centimeters/kilometer = 4,300,000 cm for the distance that the line represents. We know have both the map length and the 'real' length in the same units - centimeters. From this we now know that 26.01 cm = 4,300,000 cm. If we divide both sides by 26.01 cm, we end up with this RF 1:165,321.03 for the map. So, 1 map unit is equal to 165,321.03 real world units.

# 2) For this question we need to convert 5,500 square meters into square kilometers. The first step is to take the square route of 5,500 meters, which is close 74.16 meters. This means that we can now picture a square with sides equal to 74.16 meters. Since there are 1000 meters in a kilometer we need to divide 74.16 by one thousand in order to convert the length into kilometers. From this calculation, we know have a square with sides equal to 0.07416 kilometers. If we square this number we obtain a value of approximately 0.0055 square kilometers. This sounds right because we could have just moved the decimal point over six places since there are 1,000,000 square meters in a square kilometer. (Then you wouldn't have seen the process to use if the areas are in different systems.)