GUIDE TO MAP SCALE
(note: This is a series of exerts from lots of various sites)
This guide has been prepared to help the users to
understand the concept of scale in cartography. Scale is defined as the ratio of the
distance on a map to the corresponding distance on the surface the map
represents.
I. THREE FORMS OF SCALE. Scale is shown on maps in three
ways:
A. Verbal Scale:
1 inch equals 16 miles
This example tells us that 1-inch
on the map represents 16 miles on the surface of the Earth. This is the easiest
scale to understand because it generally uses familiar units.
B. Graphic or Bar Scale:
______________________________________
_____16________0________16_________32_
miles
The Bar Scale is particularly
important when enlarging or reducing maps by photocopy techniques because it
changes with the map. If the Bar Scale is included in the photocopy, you will
have an indication of the new scale.
C. Representative Fraction (RF) or Natural
Scale:
1:1,000,000 (this is the same as 1/1,000,000)
The RF says that 1 of any
measurement on the map equals 1 million of the same measurement on the original
surface; for the example above 1-foot equals 1 million feet or 1 cm. equals
1,000,000 cm. This is the form of scale commonly used in the Map Collection. A
good quality map should have both the RF and Bar Scales.
II. LARGE AND SMALL SCALE:
when we speak of large-scale maps we are saying the RF is large, i.e. the RF's
denominator is small. 1:10,000 and 1:62,500 maps are large scale. Small-scale
maps have a small RF. 1:500,000 and 1:1,000,000 maps are small scale.
III. HOW TO CONVERT FROM ONE FORM OF SCALE TO ANOTHER. Often when using cartographic materials it is useful to convert from one form of scale to another. If you have a good understanding of the concept of scale, the techniques are fairly simple. Here is an example of converting from Verbal Scale to RF. Remember; the RF has the same unit of measurement on both sides of the colon.
1 inch equals 10 miles
1 inch = 10 miles
1 inch = 10 miles x 12 inches/foot x 5280 feet/mile
1 inch = 10 x 63360 inches = 633,600 inches
1:633,600
To convert from RF to Verbal Scale you convert the fraction to familiar units of measurements; for example:
1:250,000
1 inch = 250,000 inches
1 inch = 250,000 inches [d] 12 inches/foot = 20,833.3 feet
1 inch = 20,833.3 feet [d] 5280 feet/mile = 4 miles or
1 inch = 250,000 [d] 63360 inches/mile = 4 miles
1 inch equals 4 miles
[Note: [d] = divided
by]
IV. SOME COMMON SCALES. Here is a list of RF scales commonly used in the Map Collection and their equivalent Verbal Scales.
1:24,000 - 1 in. = .379 mi. 1:250,000 - 1 in. = 4 mi.
1:62,500 - 1 in. = .986 mi. 1:500,000 - 1 in. = 7.891 mi.
1:100,000 - 1 in. =
1.578 mi. 1:1,000,000 - 1 in. =
15.783 mi.
V. HOW TO DETERMINE WHICH
SCALE MAP WILL FIT ON A PARTICULAR SIZE OF PAPER. At times map users need to
calculate the scale of a map that will fit on a particular size of paper. To do
this you first need to determine the dimensions of the area you want on the map
and then relate that to the dimensions of the new sheet.
For example you want a map of Arizona on a 8 1/2 x 11 inch piece of paper. To allow for 1/2-inch margins the new sheet will then be 7 1/2 x 10 inches. Since Arizona's north-south dimension, 395 miles, is slightly longer than its east-west dimension, 340 miles, we will place the longer north-south dimension along the longer 10-inch dimension of the paper. The next step is to compute the scales for both dimensions of the State. The smaller of the two scales will be the one we need.
North-south East-west
10 in = 395 mi 7.5 in = 340 mi
10 in = 395 mi x 63360 in/mi 7.5 in = 340 mi x 63360 in/mi
10 in [d] 10 = 25027200 in [d] 10 7.5 in [d] 7.5 = 21542400 in [d] 7.5
1 in = 2502720 in 1 in = 2872320 in
1:2,502,720
1:2,872,320
[Note: [d] = divided by]
We therefore need a map of Arizona at a scale of 1:2,872,320 or less to place it on an 8 1/2 x 11 inch sheet of paper.
VI. HOW TO FIND MAPS AT A
PARTICULAR SCALE IN THE MAP COLLECTION. Maps cannot be located in the online
catalog directly by scale. You need to look under the geographic area or under a
thematic subject heading to see what maps are available. Map scales are given in
the catalog in the RF form. A map series for a larger area may include the area
you are interested in; so be sure to check for maps of larger areas such as
countries or continents. For example, the entry for the 1:100,000 series
topographic maps for Arizona may be found only under the heading "United
States-maps, topographic". If you need assistance in locating a map at a
particular scale please ask a staff member at the reference desk.
VII. TOPOGRAPHIC MAPS FOR
ARIZONA. Here is a list, from the largest scale to the smallest, of the
various series of topographic maps available for Arizona.
What is Map Scale?
Map scale is the relationship between a unit of length on a map and the corresponding length on the ground. It's also an expression of how much the area represented has been reduced on the map. Map scale is important for understanding maps both in paper and computer form, so it will pay you to understand the types and uses of scales.
Types of Map Scales
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:
Bar scales is probably the most common kind of
scale found on maps, perhaps because their graphical nature makes them easily
understood? Another great thing about bar scales is that they remain correct
if the map is reduced or enlarged photographically. This is not true of the
other two types of scales.
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 units 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, 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.
Many people get confused by the
abstract nature of RF scales. They ask, "one what is equal to 24,000 of what?"
The key is that you can put in any unit you want instead of "what" in the
question. That's what makes it the most flexible type of scale.
RF scales are common among
geographers as shorthand for types of maps. For instance, the common local-area
quadrangle maps produced by the US Geological Survey, which each cover 7.5
minutes of latitude and longitude, are printed at 1:24,000. Hence these are
often referred to as 1:24,000 maps, or even "24K quads." Many maps include two or even all
three types of scales. USGS topographic maps have both bar scales and RFs.
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
1:10,000,000 maps. The 1:10,000,000 maps would usually be called a
small-scale map. This is true even though the 1:10,000,000 maps
would show a much larger area than the 1:1200 maps. Here is a rule of thumb for
size of scale by 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."
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.
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.
Area must be expressed in areal units, which are usually distance units squared -- cm2, mi2, 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 3 cm * 4 cm = 12 cm2.
|
|
|
|
|
|
The area is 691,200 square meters
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
km2.
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.
If you are given one type of
scale, you may need to derive or construct any of the other two. This takes some
practice. 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.
The key here is to write the
verbal scale as a fraction, and then convert so that both numerator and
denominator have the same units, and the numerator has a 1. Convert verbal scale of "1 inch
represents 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.
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.
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.
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 (note: the scale below
will vary in size depending on your screen resolution and size):
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.
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. We would draw the
scale the same way as in the previous example.
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. Convert from RF of 1:25,000 to a verbal scale, in metric.
Therefore, one centimeter on this map represents
1/4 of a kilometer on the ground (or, 10 centimeters represents 2.5 kilometers,
etc.).
Here we must take a measurement
from the bar scale to determine the map distance that corresponds to a ground
distance. Find the RF scale for the
following graphic scale:
By measuring with a ruler (your results may
differ depending on screen resolution and size!), we find that 10 kilometers
measures 2.4 cm. We can use this relationship to find the RF for the bar
scale:
,
Or 1:420,000.
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 whole Equator (if
it's a world map). Often you can identify 1-mile-square sections in the
US (in areas under the Public Land Survey System, which is most of the US other
than the East Coast, Texas and parts of coastal California). 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 in the above example for graphic-to-RF scale.
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 distance is one mile on the ground (there
are exceptions). The RF scale is then
,
Or 1:18,000.
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 due to relief displacement. See references on aerial
photography and photogrammetry for more information.
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:
,
And 7.2 * x cm ground = 2.4 * 24,000 cm
ground; we can cancel the units on each side and divide by 7.2: x = (2.4 * 24,000)/7.2 = 8000. In other words, the RF scale for
the photo is 1:8,000.
Note: This document was
created in 1995 as a general introduction. A section on precision was added in
1998. See Error,
Accuracy, and Precision at The Geographer's
Craft Project for a much more comprehensive treatment.
Because GIS data is stored in a
very different way than paper map data, the relationships between map scale,
data accuracy, resolution, and density are very different between GIS and paper
maps.
Map scale specifies the amount of
reduction between the real world and its graphic representation (usually a paper
map). It is usually expressed as a ratio (e.g. 1:20,000), or equivalence (e.g. 1
mm = 20 m). Since a paper map is always the same size, its scale is fixed when
it is printed, and cannot change.
However, a map in a GIS can be
shrunk or enlarged at will on the screen or on paper. You can zoom in until the
screen displays a square meter or less, or zoom out until the screen displays
all of BC. This means that geographic data in a GIS doesn't really have a 'map
scale'.
The display scale of a map is the
scale at which it 'looks right'. Because a paper map is created at certain
scale, its 'map scale' and 'display scale' are the same. The display scale
influences two things about a map: The amount of detail. The map must not be
overwhelmed with detail, and become too crowded. The size and placement of text
and symbols. These must be sized to be readable at the display scale, and placed
so that they do not overlap each other.
If you put a 1:20,000 scale paper map on a reducing photocopier, you can make it into a 1:100,000 map (i.e. reduce it by a factor of 5). However, probably areas of detail will be merged into big black blobs, and most of the text on the map will be too small to read.
A GIS map's annotation (i.e. text
and symbols) must be designed with a display scale, just like a paper map. There
is a range of scale in which it will 'look right', even though it is possible to
display it at other scales with the GIS software.
Absolute accuracy: How
close is the location on the map or data representation to its real location on
the earth? For example, '95% of the well locations are within 50 meters of their
surveyed locations'.
Relative accuracy: How
similar is a shape on the map or data representation to the shape of the object
on the earth. For example, 'cutblock boundaries do not vary by more than 10
meters from their actual shape'. These are separated because a map object may
have a very accurate shape, but not be registered (located) correctly.
A rigorous statement of accuracy will include statistical
measures of uncertainty and variation, as well as how and when the information
was collected. Spatial data accuracy is independent of map scale and display
scale, and should be stated in ground measurement units.
Data precision is the smallest
difference between adjacent positions that can be recorded and stored. Most GIS
store locations in ground units (e.g. UTM coordinates, or Longitude/Latitude)
with a precision of a meter, centimeter or less.
Resolution is the degree to which
closely related entities can be discriminated. Since a paper map is always the
same size, its data resolution is tied to its scale. Resolution also limits the
minimum size of feature that can be stored. Generally, a line cannot be drawn
much narrower than about 1/2 a millimeter. Therefore, on a 1:20,000 scale paper
map, the minimum distance which can be represented (resolution) is about 10
meters. On a 1:250,000-scale paper map, the resolution is 125 meters.
Usually, it is desirable to
specify the resolution of a dataset as a minimum feature size. For example, 'no
lakes of less than 5 hectares surface area should be captured'. In a GIS, this
is the most important reason for having the same data represented at different
'scales'.
Raster data is stored as (usually
square) pixels, which form a grid or mesh over an area of the earth. The size of
these pixels determines the resolution of the raster, because it is impossible
to store anything, which falls 'between' the pixels. A GIS allows raster pixels
to be any size, although they should not be smaller than the uncertainty of the
data. If raster coverage is derived from vector line work, its pixels should not
be smaller than the uncertainty in the line work. If it comes from an air-photo
or satellite image, its pixels should not be smaller than the resolution of the
camera that recorded it.
Data density is a measure of how
many features per area are stored, and may imply a minimum feature size. Greater
density implies more features in a given area, and therefore the features may be
smaller. The density of paper map's
data is limited by its scale (and therefore its resolution). Areas (polygons)
cannot be shown if they are smaller than the lines, which draw them. For
example, a polygon less than 250 meters wide cannot be drawn on a
1:250,000-scale map. This minimum size also limits the number of polygons that
can be represented in a given area of a paper map.
Data detail is a measure of how
much information is stored for each feature. A GIS stores lines (e.g., a lake
shoreline) as a sequence of point locations, and draws it with the edges that
join them. There is no limit to how many points can be stored, or how close
together they may be. The amount of detail on line features should be limited
just like data density. It does not make sense to store points at intervals,
which are shorter than the accuracy of their locations.
In a GIS, analysis is done at the precision of the data, not at any display scale. For example, the area of a habitat polygon is calculated to the nearest square centimeter. The GIS will carry much more precision through its calculations than are justified by the data's accuracy. The results of these calculations should be rounded to a value appropriate to the uncertainty of the data for reporting. Some operations may result in features, which are smaller than the data uncertainty. For example, overlaying rivers and forest polygons may create 'slivers' along the riverbanks, which are 10 meters wide, when the uncertainty of the data is 20 meters. These slivers should be ignored, or included with their neighbors before the results of the overlay are used for further analysis.
Separation of data and annotation
In a GIS, it is common to display the same data (e.g. wildlife management unit boundaries) at several different scales for different purposes. It is also possible to create symbols and text that 'look right' at several different scales, and store them apart from the data they label. For example, management unit boundaries could be stored in one provincial coverage, and annotation layers could be developed for labeling them at display scales of 1:20,000, 1:250,000, and 1:2,000,000. If done carefully, this avoids duplication of the same data for display at different scales.
Generalization
In a GIS, it is possible to create a new coverage by reducing the amount of detail in existing coverage. This 'generalizing' may or may not reduce the number of objects in the coverage. For example, a detailed forest cover map may be generalized by combining polygons with similar characteristics. This reduces the number of objects in the coverage.
Conversely, a detailed ecosystem classification map may be generalized by reducing the amount of detail in the boundaries between regions, without reducing the number of regions. Generalizing a raster image usually reduces both the number of objects, and the amount of detail.
Map series
It is convenient to identify a series of paper maps by their scale (the 1:50,000 water atlas), or the amount of earth they cover (e.g. NTS 2-degree letter blocks). Neither of these is well suited to GIS data. GIS data can be displayed at any scale, and can be manipulated as a seamless coverage for either analysis or display. GIS coverage should be identified by its accuracy (or uncertainty) and data resolution or density (or minimum feature size).
|
Scale |
Miles/inch |
Line Width
on Ground* |
Examples |
|
1:2,000,000 |
~32 |
2000’ |
USGS Nation-wide maps |
|
1:1,000,000 |
~16 |
1000’ |
National and state maps |
|
1:500,000 |
~8 |
500’ |
State or regional maps |
|
1:250,000 |
~4 |
250’ |
US Army Map Series |
|
1:100,000 |
~1.6 |
100’ |
USGS 30' quads |
|
1:62,500 |
5208 feet |
62.5’ |
USGS 15' quads |
|
1:24,000 |
2000 feet |
24’ |
USGS 7' quads |
|
1:15,840 |
1320 feet |
15.84’ |
Soils |
|
1:9,600 |
800 feet |
|
Aerial photos |
*Approximate real width on ground of a pencil line on a map - harder pencils give a finer line
1:2,000,000 to about 1:250,000 are SMALL-scale maps. Items on these maps appear small (e.g. a county on a 1:2,000,000 map is much smaller than on a 1:100,000 map).
1:24,000 on towards 1:9,600 are LARGE-scale maps. Items on these maps appear larger. 1:100,000 are pretty much in the middle - intermediate scale.
Other Reference
Sites:
http://idaho.usgs.gov/reference/map_scales.html
http://mac.usgs.gov/mac/isb/pubs/factsheets/fs03800.html
http://www.usgs.gov/education/learnweb/MA/MAlesson7.html
http://www.epa.gov/ceisweb1/ceishome/atlas/learngeog/understandingmaps.html
