Map scale is the relationship between a unit of length on a map and the
corresponding length on the ground. We will use concepts of map scale
throughout the course, so it will pay you to study this section carefully.
We can relate map and ground with three different types of scale.
Verbal scale expresses in words a relationship between a map
distance and a ground distance. Usually it is along the lines of:
One inch represents 16 miles. Here it is implied that the one inch is
on the map, and that one inch represents 16 miles on the ground.
Verbal scales are commonly found on popular atlases and maps.
The second type of scale is a graphic scale, or bar scale. This
shows directly on the map the corresponding ground distance.
For example:
The third type of scale is a representative fraction, or ratio
scale. Compared to the first two, it is the most abstract, but also
the most versatile. A representative fraction, or RF, shows the
relationship between one of any unit on the map and one of the same
unit on the ground. RFs may be shown as an actual fraction, for
example 1/24,000, but are usually written with a colon, as in
1:24,000. In this example, one unit of any length (one mm, one cm, one
inch, one foot, etc.) on the map represents 24,000 of those same units
on the ground (24,000 mm, 24,000 cm, 24,000", 24,000', etc.). The RF
is versatile because you are not tied to any specific units. You may
work in any unit you choose, either metric, English, or other.
The RF is a called a fraction because it is just that--a fraction that
shows how much the real world is reduced to fit on the map. A good
comparison is often made with scale models of automobiles or aircraft. A
1/32-model of an auto is 1/32nd as large as the actual auto. In the same
way, a 1:100,000-scale map is 1/100,000th as large as the ground area
shown on the map.
A related idea is that of small scale versus large scale. Geographers use
these terms differently than many people. A large scale map is where the
RF is relatively large. A 1:1200 map is therefore larger scale than a
1:1,000,000 map. The 1:1,000,000 map would usually be called a small scale
map. This is true even though the 1:1,000,000 map would show a much larger
area than the 1:1200 map.
Here is a rule of thumb for size of scale by RF:
Size of Scale
Representative Franction (RF)
Large Scale
1:25,000 or larger
Medium Scale
1:1,000,000 to 1:25,000
Small Scale
1:1,000,000 or smaller
Of course, what is small or large scale is relative. I noticed a surveying
text (Brinker & Wolf, 1984) that classed anything smaller than
1:12,000 as small scale -- surveyors rarely work with anything smaller
than this.
The large/small scale terminology can become confusing when talking about
large versus small areas. If you are talking about a phenomenon that
occurs across a large region, it is tempting to say it's a large-scale
phenomenon (e.g., "the forest blight is a large-scale disease"). But since
the map that would show this would be small-scale, it is better to use a
different term to avoid confusion. My favorite is "broad-scale."
Many maps include two or even all three types of scales. USGS topographic
maps have both bar scales and RFs.
2. Converting Between Scale
Types
If you are given one type of scale, you should be able to derive or
construct any of the other two. This takes some practice, and some
problems are included in your lab exercises. Some examples are given
below.
A vital step in doing any kind of conversion that involves differing units
is to include the units in the problem itself. You can then cancel the
units by multiplying or dividing. This way you avoid becoming confused
about which conversion factors to use and how to use them.
Verbal Scale to RF
The key here is to write the verbal scale as a fraction, then convert so
that both numerator and denominator have the same units, and the numerator
has a 1.
(a) Convert verbal scale of "1" to 18 miles" to RF
or 1:1,140,000.
Notice that the resulting fraction is rounded so that the RF does not
imply more accuracy than the original precision warranted.
(b) Convert verbal scale of "15 cm to 1 km" to RF
or 1:6700.
In many conversions you can save steps if you remember additional
equivalencies.
For example, in (a) above, we could have used the fact that 1 mile =
63,360 inches to skip a step.
Verbal Scale to Graphic Scale
Usually this is a relatively easy task if the map gives us reasonable
units in the verbal scale. We can use the verbal scale like a fraction to
transform the ground distance to map distance.
(c) Convert verbal scale of "1 cm to 14 km" to a graphic scale. One
centimeter is a relatively small distance, so we probably don't want our
bar scale to have major divisions much smaller than this. A centimeter
represents 14 km, so a division of 10 km is probably fine. Therefore we
want to find how many centimeters represent 10 km.
In other words, we can represent our 10 km increment on the bar scale by
measuring off 0.71 cm on the map. We'd draw the first tick at 0.71 cm, the
second at 1.42 cm, and so on:
RF to Graphic Scale
This adds an extra step to the example above. We can find the map-distance
equivalent of a ground distance, but we also need to be careful about
choosing which ground distance we want to portray on the map. Perhaps it's
easiest to choose a smaller ground distance that you can then multiply to
get a reasonable bar scale.
(d) Convert an RF of 1:250,000 to a graphic scale If we aren't sure what
increments a bar scale would have for this scale, we could start out, say,
with finding the map equivalent of 1 mile:
This might work fine, with one mile marked off on the map every 0.25 inch;
or, we may want finer or broader increments, which we can find by dividing
or multiplying the .25" as needed.
RF to Verbal Scale
Again we have to choose appropriate units to convert into. Most verbal
scales are either "one inch represents ____ miles," or "one centimeter
represents ___ kilometers." These are relatively easy to do, since it
means only that we convert the denominator of our RF to the larger units.
(e) Convert from RF of 1:25,000 to a verbal scale, in metric
Therefore,
1 centimeter on this map represents 1/4 of a kilometer on the ground.
Graphic Scale to RF
Here we must take a measurement from the bar scale to determine the map
distance that corresponds to a ground distance.
(f) Find the RF scale for the following graphic scale
By measuring with a ruler, we find that 10 kilometers measures 2.4 cm. We
can use this relationship to find the RF for the bar scale:
3. Determining Scale from a Map or Photo
Some maps may come with no scale at all. Aerial photographs almost never
do (unless one was painted on the ground before the photo was taken!). How
can you derive a scale for use with the map or photo?
Actually the procedure is very similar to the last example above. But
instead of measuring along a bar scale, you must measure the length of an
object on the map or photo whose actual length you know. This might be a
football field, a city block, or the Equator (if it's a world map). Often
you can identify 1-mile-square sections in the US (see the account below,
under Survey Systems). You may even need to go out to the location mapped
or pictured and measure the distance between two identifiable objects.
Once you have the two distances, you can find the scale as above. For
another example, suppose you have a map where the distance between two
section-line roads is 3.5 inches on the map. We can usually assume this is
one mile on the ground (there are exceptions). The RF scale is then:
One caveat (exception) for air photos is that this method assumes the two
locations are at the same elevation--or that the terrain is flat. If you
are using air photos, the terrain may not be flat. If there are hills,
even moderate ones, the calculations can be thrown off. Keep this in mind
for later in the course.
Another way to calculate scale on an unknown map or photo is to compare it
to a map with a known scale. For example, suppose you have an air photo
where the distance between two hills is 7.2 centimeters.You have a map of
the same area at 1:24,000, and on the map the distance between the hills
is 2.4 centimeters.
The answer involves a little algebra. Since the ground distance is the
same on both photo and map, we can create an expression for this ground
distance for both, and then put them on either side of an equation. The
ground distance can be found by multiplying the map/photo distance by the
scale (in this case, by the inverse of the scale--notice how this makes
the units cancel correctly). We need to find, for the photo, how many
ground units are represented by one unit on the photo, so we use an x for
this unknown quantity and solve for it:
we can cancel the units on each side and divide by 7.2:
In other words, the RF scale for the
photo is 1:8,000.
4. Determining Distance and Area from Map
& Scale
Map scale isn't much use in and of itself. We can use a map's scale to
determine distances and areas on the map. Compared to converting between
scale types, calculating distance is simple. Area calculations are
trickier, since we have to square the numbers.
Finding distance from map and scale
As an example, suppose we have a map with a scale of 1:50,000. We measure
the distance along a property boundary as 1.7 cm. What is the length in
the real world?
To find ground distance, we must use the map scale to convert map distance
to ground distance. Notice that again we inverted the RF scale, so the
units will cancel properly. Once we multiply by the scale, we need to
convert the ground distance to units suitable for ground measurement--in
this case, from centimeters to kilometers.
We can also calculate distance from verbal and graphic scales. With verbal
scales, we use the same procedure as above with the RF. The only
difference is that we have to use the units given in the verbal scale
(e.g., 1 inch to 17 miles). We'd probably want to measure our map distance
in the same units (in this case, inches) to make our conversion easy.
Graphic scales are probably the scales most frequently used by laypersons.
You can mark off a distance on the map and compare it directly to the bar
scale. You need not know how many inches or centimeters the map distance
is. The main drawback of bar scales is that they are usually short
compared to the map itself, and hence measuring longer distances is
difficult.
A. Finding area measurement from map and
scale
Area must be expressed in areal units, which are usually distance units
squared -- cm^{2} , mi^{2}, and so on. We must therefore
used squared conversion factors when finding area from map measurements.
For example, suppose we measure a rectangular piece of property that is 3
cm by 4 cm on a map. The map is at a scale of 1:24,000. What is the area
of the parcel?
The area of the parcel on the map is
on the ground.
Since this is a large number, we might want to translate to other units.
There are 10,000 square meters per hectare, so the area is 69 hectares
(ha) (a hectare is about 2.5 acres). Or, there are (1,000)^{2} =
1,000,000 square meters per square kilometer, so the area is also 0.69 km^{2}.
Notice that by writing the units as part of the problem, and squaring them
along with the numbers, our units cancel properly and we end up with a
sensible answer.
There is another way to tackle area problems if you have distance
dimensions like 3 x 4 cm to start out the problem. You can convert the
distance dimensions to real-world distances first, and then multiply them
to find the area. This makes the problem longer but perhaps simpler.
B. Measurement Scales
1. Definition
Any type of information on a map (or in a table, a list, a survey, etc.)
can be described in terms of how pieces of the information can be related
to each other. Examples of mapped information include land-use classes,
road classes, city populations, and county areas.
We can think of these types of information as either categorical (or
qualitative) or numerical (or quantitative). Categorical information can
only be described qualitatively, whereas items of numerical information
can be compared quantitatively. That is, numerical information is on a
real scale, such as distance (e.g., kilometers), area (e.g., hectares), or
temperature (e.g., Celsius).
Example of categorical information:
Land-Use categories of :
1= forest, 2 = grassland, 3 = urban;
We cannot say that based on the classes, 1 + 2 = 3 !
We can only use the numbering to describe the items qualitatively.
Example of Numerical information, city populations:
Springfield = 100, Harmony = 200, Centerville = 300;
We can say that Centerville has as many people as Springfield and
Harmony.
2. Four Measurement Levels
The categorical and numerical types have traditionally been further broken
down, each into two levels of measurement:
Categorical information can either be nominal level or ordinal level . At
the nominal level, categories simply describe different types of things,
and cannot be compared to each other. Ordinal-level categories can, on the
other hand, be ranked against each other.
An example of ordinal level would be road classes of interstate, primary,
secondary, and primitive; although two primary roads don't make an
interstate, you could say in a sense that interstates are at a higher
level than the others, at least in terms of expense to build!
The land-use classes would be an example of nominal-level measurement. You
probably wouldn't rank the classes against each other (unless you prefer
one over the other, in which case they could be ordinal!).
Numerical information can either be interval level or ratio level . This
distinction is subtle, and borders on artificial. The only difference is
that ratio-level information is on a scale that includes a true zero, that
is, a zero that truly represents a lack of whatever it is you're
measuring.
One of the few common examples of an interval scale is temperature. You
can compare two temperature readings quantitatively. For example, 20 C is
15 degrees warmer than 5 C. But the Celsius scale has an arbitrary zero
point (the freezing temperature of water, or 32 F). 0 C doesn't mean a
lack of temperature. For this reason, 20 C is not four times as warm as 5
C! (One temperature scale, Kelvin, does feature a true zero -- 0 K is -273
C, the temperature at which all molecular motion ceases). Fortunately,
there are few examples of this on maps. You might want to include an
exception for elevation, which can be below sea level, or longitude, where
the 0° line (Prime Meridian) is arbitrarily drawn on the globe.
Table 2: Measurement Levels
Measurement Level
Description
Examples
Categorical
Nominal
Simple categories; cannot rank categories
Colors; land use types; computer components
Categorical
Ordinal
Distinct categories; can
rank categores in order
Road classes; flavor preferences
Numerical
Interval
Continuous scale; measure with numbers; real
zero
Temperature; longitude, compass direction
Numerical
Ratio
Continuous scale; measure
with numbers; real zero
Distance; weight; bank balance
Most numerical scales are ratio-level. We can compare 20 km with 10 km in
any number of ways, including saying the first is twice as far as the
second.
3. So What?
We will use this terminology frequently to describe the kinds of
information on maps and in other sources. You will also encounter it
elsewhere, I guarantee it!
Information often is translated from one level to another. The general
rule is that it's easier to translate to a lower level (e.g., from ratio
to ordinal) than to a higher level. In most cases, it's impossible to go
legitimately to a higher level. Why bother using a lower-level scale?
Often it saves space and energy to use a lower scale.
For example, if you have a table of city populations in Sonoma County
(i.e., ratio level) and want to show population on a map, you could print
a number next to each city name. This would preserve your ratio-level data
on the map itself. A more common technique is to have a set of
increasingly large symbols for the city location, each symbol indicating a
range of population. These symbols would give ordinal-level information
about population. This conveniently reduces the clutter on the map, but
you cannot recover the original ratio-level numbers from the map.
C. Accuracy, Precision, and Significant Digits
1. Accuracy versus Precision
Some people make a distinction between accuracy and precision.
Accuracy would be how close a measured value is to the actual value.
Precision would be the number of digits reported with the value. For
example, I added a thermometer to my car that shows the outside
temperature to tenths of a degree (e.g., 65.2° F). The thermometer's
precision is to tenths of a degree Fahrenheit. But it often seems to give
readings that are too high, perhaps due to the engine's heat nearby. Its
accuracy, then, might only be to within 5° F.
2. Significant Digits
Significant digits are the number of digits in a value that have real
meaning and reflect the accuracy of the value or measurement. For example,
a distance measurement of 45.8 km would have three significant digits. The
value implies that the measurement was made to an accuracy of tenths of a
kilometer. If the instrument, such as an odometer, is not accurate, the
value should not include as many significant digits. Of course, if the
instrument measures more accurately than this, the value should be
reported with the appropriate precision.
The main point is to beware measurements with highly precise values when
the accuracy could be called into question. This is especially true when
converting values from one kind of scale to another. For example, if I
measure a map distance as 11 inches with an ordinary ruler, then want to
convert to centimeters, I find on my calculator:
(11 in.) * (2.54 cm/in.) = 27.94 cm But 27.94 cm implies high accuracy,
which I didn't have with the original measurement. The solution here is to
round the answer to about the same number of significant digits as the
original measurement. In this case, I should probably say the distance is
28 cm.
More example of significant digits:
Two significant digits: 3700, 37, 3.7, 0.37, 0.0037
Four significant digits: 19120, 19.12, 0.001912, 19.00, 0.1900
Notice that zeroes added after the decimal point imply precision equal to
any other digits. Normally, zeroes to the right of other digits (e.g.,
3700) are not considered significant, unless a decimal point is added and
more zeroes occur to the right of it (e.g., 3700.0 implies 5 significant
digits). What if we have a measurement of 1900 where the zeroes are
significant? Traditionally, a bar is placed over the zeroes to indicate
significance.
Another solution is to use scientific notation.
For example, 3700 with the zeroes actually being significant can be
written as 3.700 x 103. This breaks the number down into the mantissa (the
significant digits, here 3.700) and the exponent on the ten (here 3). In
scientific notation, only one digit is placed before the decimal point in
the mantissa. The exponent on the 10 indicates how many places the decimal
point should be moved to restore the number to ordinary notation (in this
case, three places to the right; this is equivalent to multiplying the
mantissa by 1000). Negative exponents mean smaller numbers, and we shift
the decimal point to the left. For example, 3.700 x 10-4 is the same as
0.0003700. Scientific notation is handy with very large or small numbers.
Numbers in scientific notation can be used just like other numbers, as
long as you follow the rules of exponents -- for multiplying, dividing,
etc.
D. Classification, Simplification and
Symbolization of Data
This is not a course in cartography, so we will not deal in detail with
how data is classified and simplified for mapping, and how symbols are
created to suit those data. But you should be aware that any map,
including topographic maps, is the result of a lot of processing of the
data behind the map. Here is a brief discussion of some elements that go
into data processing for maps.
1. Classification
Maps must take the relevant data and assign it to a given number of
classes, each with its own number of members. Examples are maps of land
uses and of temperature. For categorical data like land uses, it is easy
to overload the map reader with too many classes. Cartographers recommend
a maximum of five or six different classes (for land use, these might be
forest, rangeland, cropland, water, and urban). Numerical data shown on
isarithmic maps (which show zones of equal values) can be somewhat more
complex. For example, temperature zones can trend from 80's F to -20's F.
But the data should be shown with logical symbolization (see below).
Data can be broken down into categories in three ways:
(a) Each category can have an equal number of members. For example, in a
map of US population by state, we could have five categories of
population, each with ten states.
(b) Each category could span an equal interval of values. In the
population map, we would take the range of populations (highest state
minus lowest state), and divide the range by five. Each category would be
one-fifth of the overall range. In this map, categories may have different
numbers of states. The highest category may only have a few states (e.g.,
California, New York and Texas), and other categories may have a large
number of states.
(c) Finally, we could subjectively assign states to classes, perhaps by
looking for "natural breaks" in population rankings. Often map-makers
start out with one of the first two methods, then refine it by looking at
the results and changing classes slightly. All of this should reflect the
purposes of the map. Different methods can produce radically
different-looking maps (see any cartography text for examples).
2. Simplification
A second major task in map-making is simplifying the information behind
the map. Classification is one way of simplifying the data we use going
into the map. Often this involves lowering the measurement level of the
data (such as transforming ratio-level data to ordinal-level data, as in
our example of city populations indicated by various circle sizes).
Another simplification task relates to the spatial aspects of the data.
For example, mountainous roads may take many twists and turns. A map that
shows all these curves would be cluttered and messy. We may want to
generalize the road spatially to clarify things. Beware, of course, those
map users who curse you for those curves they didn't expect! The
cartographer's job is to simplify without distorting and misleading the
map user.
Sometimes the task may necessitate eliminating features altogether. For
example, a small-scale US map can show only selected cities; how do you
choose which cities to include without offending too many people? I saw a
US map recently that included moderate-sized cities but omitted my
hometown, San Diego (now the 6th largest city in the US).
3. Symbolization
Symbolization of information is another crucial part of communicating
through maps. What symbols are chosen will influence whether will people
will understand, or even use, a map. Symbols should be chosen so that the
intended audience will understand them. Different symbols should be used
for general-purpose maps than those intended for specialists in a field.
Symbols can be placed on a continuum between intuitive and abstract.
Intuitive symbols suggest to the reader what they represent. Pictographs
are recognizable representations of the actual objects. For example,
marshes are often shown with swampy grass symbols, or airports as planes.
On the other extreme are abstract symbols with no obvious connection to
what they represent. Examples are circles as cities, or streams as dotted
lines. Some abstract symbols have been used so widely to represent a
particular feature that, although abstract, they do connect with the
object in most people's minds. A star within a circle, for instance,
usually means a capital city.
Symbols can also be classed according to their dimension: point
(0-dimension), line (1-dimension) or area (2-dimension) symbols. Like the
other two, point symbols can convey categorical or numerical information.
Examples of categorical point symbols are airports, campgrounds, springs,
and quarries. Variable-sized circles for city population would be examples
of numerical point symbols.
Line symbols can convey information about linear features, such as roads,
railroads, and streams. Another important use of lines on maps are
isolines. Isolines (also called isarithms) connect points of equal value.
You'll often see these on a map of temperatures on the weather page of the
newspaper. Every place along an isoline on this map has the same
temperature, at least theoretically. This implies also that places between
two isolines have values somewhere between the values of the surrounding
lines. Without additional information, we cannot say with confidence much
more about intervening values. We can estimate, or interpolate, the
intervening values, for example by measuring the distance of a point from
each line. There is no guarantee, of course, that our estimate is really
correct, so beware such estimates from isoline maps.
An important type of isoline for this course is the elevation contour .
These isolines show equal elevations, and are included on all USGS
topographic maps. The contour interval is the height difference between
each contour. Contour intervals on USGS maps vary depending upon the
terrain. Common intervals on 1:24,000-scale maps are 20, 40 and 80 feet. A
20-foot interval would show, for example, contours at 0 (sea level), 20,
40, 60, 80, and so on.
It takes some practice to feel comfortable working with elevation contours
and other isoline maps. We will have lab exercises to give you some
practice with them, including visualizing terrain and constructing
profiles, or two-dimensional views of how elevation changes on a line
across the topographic map.
The zones between isolines are often shaded or colored to portray the
information more effectively. Temperature maps in most newspapers are now
in color, with warmer zones getting "warmer" colors, colder areas "cooler"
colors. These shaded-isoline zones are a blend of line and area symbols.
Speaking of color and shading, let us cover one final point about map
symbolization. The rules about using color and shading are different for
nominal versus higher-level data (ordinal, interval, and ratio). If you
are mapping nominal-level data, the categories have no particular order,
and you probably want to maximize the contrast between categories. Land
uses, for example, should be portrayed so that you can separate urban,
suburban, forest, and pasture, or whatever categories you include on the
map.
Ordinal, interval or ratio data should be mapped differently. Here there
is a definite ordering of categories. To use the temperature map again, we
have a trend from warm to cold in the zones we portray. We want the map
reader to immediately appreciate this gradual trend in the data. Therefore
we select an ordering of colors or symbols that conveys this continuous
trend in the data, for example: red-orange-yellow-green-blue-violet.
Many people make the mistake of thinking they should select colors that
maximize contrast between categories, even with ordinal or higher-level
data. Beware this mistake! You may be able to pick out categories easily
with this approach, but you destroy the impression of a trend in the data.