Transcript Exploring Geography Cartographically Waldo Tobler Professor Emeritus of Geography University of California Santa Barbara, CA 93106-4060 http://www.geog.ucsb.edu/~tobler.

Exploring Geography
Cartographically
Waldo Tobler
Professor Emeritus of Geography
University of California
Santa Barbara, CA 93106-4060
http://www.geog.ucsb.edu/~tobler
1
Subjects To Be Covered
The talk will cover
several apparently
disjoint topics, but all
related to my interest
in using cartographic
methods to study
geography.
The emphasis is not
on the drawing of
maps, although this
is important, but
more on the way
cartographers think,
and on the methods
they use to study
geographic facts.
First presented at the Institute of Geography, Paris, 25 April 22000
To begin I stress
Location Location Location
3
There are many ways of
specifying location
For example, here is my address if you
wish to correspond with me
4
Another Way of Determining Locations
Telephone numbers
Wireless (cell) telephones
These allow you to be located within a few
meters.
5
Common Geographic Locational
Aliases and Conversions
6
Hand Held GPS with Map
Locations are estimated from distances
7
Wrist Watch GPS
Location is determined by measuring
distance to satellites.
8
The fact that locations can be
determined by specifying distances
can lead to some interesting results.
This is in fact how locations are determined
in surveying.
The technical name for this is trilateration.
It is also known as multidimensional scaling.
9
In the lower school grades one learns
the procedure for finding distance
from coordinates.
Dij = [(Xi - Xj)2 + (Yi - Yj)2]1/2
Why does one not learn the inverse?
Find the coordinates from the distances!
10
An Exercise Using Distances
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A Similar, But Simpler, Exercise
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Solution Procedure
Guess!
Then improve the guess.
This leads to an iterative procedure.
13
Step by Step Trilateration
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Trilateration Procedure Illustrated Graphically
15
The Beginning Steps
16
The Answer, with Coordinates and
Estimated Distances
17
Solution to the US Distance Exercise
with (enlarged) estimated standard errors
18
Locations increase linearly, distances quadraticaly thus
No Numbers are Needed!
Find points A through E, when
AE < BC < BE < CD < AB < AD < CE < BD < DE < AC
where < means the distance is less than
The geometric constraints override the need for
numbers.
D. Kendall, 1971, “Construction of Maps from ‘Odd’ bits of Information”, Nature,
231:158-159
W. Tobler, 1996, "A Graphical Introduction to Survey Adjustment",
Cartographica, 33-42.
19
I will now give you another
example where locations are
estimated from distances
20
Cappadocian Cuneiform Tablet
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The Cuneiform Tablets
Uncovered near Kultepe, Turkey in 1925.
From a commercial enterprise ~3500 years ago.
Containing the names of 119 towns
most of whose location is unknown.
This is not a “geographical tablet”.
(Geographical tablets identify locations, usually by giving an itinerary.)
22
Given the occurrence of town
names on tablets
Count the occurrences and set them
proportional to the size of the town.
Count co-occurrences and set these equal
to the total interaction between the
towns.
Then
23
Use the well known gravity model:
The interaction Iij between places i and j
is proportional to PiPj/dij
Invert the model so that distance is the
unknown variable
dij = PiPj/Iij
then predict the locations
24
Predicted Locations
of 33 Pre-Hittite Towns in Cappadocia using distances estimated from presumed interaction.
25
W. Tobler, Wineberg, S., 1971, “A Cappadocian Speculation”, Nature, 231, 5297(7 May):39-42.
USA Highway Distance Map
Here is another, perhaps less exciting, example
Using a road atlas the student took the values
from the table of distances between places.
These tables are common in such atlases.
Using these distances he then computed the
location of the places.
The US outline and latitude longitude grid were
then interpolated to complete the map.
26
Road Distance Map of the United States
Student drawing
27
This “distorted” map can be evaluated
by the methods of Tissot
The road system introduces distance distortions,
as is obvious,
but there are also areal and angular distortions
and these can be measured.
In other words, cartographic theory can be used
to evaluate several impacts of a road.
M. A. Tissot, 1881, Mémoire sur la représentation des surfaces…, Paris, Gauthier Villars
28
All maps are of course distorted
to some extent
And this brings me to the subject of
map projections.
29
The Surface of the Earth Is Two-dimensional
30
The Map Projection Problem is to
reduce the curved, closed, and bumpy
two dimensional surface of the earth
to a flat two dimensional surface
It is not, as erroneously suggested in many
books, to reduce three dimensions to two
dimensions. Map projections preserve the
dimensionality.
There are many choices; these depend on the
application.
31
The Transform - Solve - Invert
Paradigm
This is a classic way of solving problems.
Change to a more appropriate coordinate
system where the problem becomes simpler.
Solve the problem and then revert to the
original coordinates.
32
The conventional satellite
tracking chart
The satellite tracks are curves
The meridians and parallels are straight
33
Breckman Chart to Track Satellites
Bend the meridians and parallels so that
the satellite tracks are straight. It is
then easier to track the satellite. 34
Area Cartograms, Also Known As
Anamorphoses, Are A Form Of Map
Projection Designed to Solve Particular
Problems
Mercator’s projection is the most famous
anamorphose.
It is designed to solve a navigation problem.
Other anamorphoses (map projections) solve
other problems.
35
36
Another Map Projection To Solve
Another Problem
The next illustration shows the U.S. population
assembled into one degree quadrilaterals.
We would like to partition the U.S. into regions
containing the same number of people.
There follows a map projection (anamorphose)
that may be useful for this problem.
37
US Population By One Degree
Quadrilaterals
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The Lat/Lon Grid in the Two Spaces
Left, the usual grid. Right, transformed according to population.
39
US Map in the Two Spaces
Left, the usual map. Right, the transform.
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The Inversion
On the right are uniform hexagons in the transformed
space. On the left is the solution: The inverse
transformation partitions the US into cells of equal
population
W. Tobler, 1973, "A Continuous Transformation Useful for Districting", Annals,
41
New York Academy of Sciences, 219: 215-220
Most of you are familiar with linear
regression and correlation
Here is a bidimensional version of regression
applied to some 2000 year old data.
This is closely related to map projections.
42
Title page from Ptolemy’s Geography
43
A Page from Ptolemy’s Geography
A rather boring book.
Mostly a list of 8,000
latitudes and longitudes.
But it tells us something
about what was known
geographically ~2,000
years ago. We can
compare the coordinates
with known modern
locations, although this
is not easy.
44
Ptolemy’s World Map
45
Ptolemy’s Map of Gaul
(Modern France)
46
Modern and Ptolomaic Latitude and
Longitude for Gaul
47
The data in the foregoing table
were assembled by a student and
some of you may be more familiar
with the the ancient names than I.
Observe that we will be comparing two sets of
vectors (pairs of coordinates), not single
variables.
That is why we need a bidimensional regression.
48
W. Tobler, 1994, “Bidimensional Regression”, Geog. Analysis, 26:186-212
49
Modern Latitude versus
Ptolemy’s Latitude
50
Modern Longitude versus
Ptolemy’s Longitude
51
52
Ptolemaic Displacements
Amounts needed to move Ptolemy’s locations to modern locations.
The places needing further investigation can be seen.
53
Interpolated Vector Field
Amount needed to move the graticule to modern locations
54
Ptolemaic Grid Shifted to Fit Modern
‘Pushed’ by the interpolated vector field
55
There are many applications of vectors
(pairs of coordinates) in geography
The pairs of coordinates could, for example,
be considered vectors showing coordinates of
were where people moved (changed
addresses) during some period of time, so
that the same technique can be applied to
this situation. Or they could be displacements
from a “mental map”, or differences between
a satellite image and a map, thus requiring
“rubber sheeting”.
56
Geographical Interpolation
Sometimes one has numerical
observations given at point locations.
The objective is often to produce a
contour map.
There is a large literature on interpolation
from point data.
We mention only Kriging, inverse
distance, splining, and so on.
57
But often we have observations
assembled by statistical units
Census tracts, school districts, and the like
It is incorrect, in my opinion, to assign these
observations to points (centroids).
One criterion to be satisfied is that the resultant
maintain the data values within each unit.
This is why I invented pycnophylactic
reallocation.
58
Pycnophylactic Reallocation
Allows the production of density or contour maps
to be made from areal data.
It is reallocation - and somewhat of a
disaggregation operator. My assertion is that it
may actually improve the data.
It is also important for the conversion of data
from one set of statistical units to another, as
from census tracts to school districts.
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Population Density by County
Observe the discontinuities at the county
boundaries.
We would like a smooth map of
population density.
The usual interpolation procedure will
not work unless we use centroids and
this fiction could allow people to be
moved from one county to another.
60
Population Density in Kansas
By County
Courtesy of T. Slocum
A piecewise continuous surface
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Population Density in Kansas
by County
Each county still contains the same number of people
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A smooth continuous surface, with population pycnophylactically redistributed
How Pycnophylactic Reallocation
Works
Philosophically it is based on the notion that
people are gregarious, influence each other,
and tend to congregate.
This leads to neighboring and adjacent places
being similar.
Mathematically this translates into a
smoothness criterion.
(with small partial derivatives)
63
Mass Preserving Reallocation Using Areal Data
W. Tobler, 1979, “Smooth Pycnophylactic Interpolation for Geographical Regions”, J. Am. Stat.
64
Assn., 74(367):519-536.
What the Mathematics Means
Imagine that each unit is built up of colored
clay, with a different color for each unit.
The volume of clay represents the number of
people, say, and the height represents the
density.
In order to obtain smooth densities a spatula
is used, but no clay is allowed to move from
one unit into another. Color mixing is not
allowed.
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Colored Clay Before Smoothing
66
Colored Clay After Smoothing
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Another Advantage of Mass-Preserving Reallocation
A frequent problem is the reassignment of
observations from one set of collection units to a
different set, when the two sets are not nested
nor compatible. For example converting the
number of children observed by census tract to a
count by school district. Boundaries also change
over time, requiring reallocation of information.
The density values obtained using the smooth
pycnophylactic method allow an estimate to be
made rather simply. A “cookie cutter” can cut the
continuous clay surface into the new zones with
subsequent addition to get the count.
68
The next important topic is
Movement Movement Movement
This is because most change in geography is
due to movement.
Movement of people, ideas, money, or materiel.
69
The Table is an Important Form of
Geographic Movement Data
Especially when the rows and columns
refer to known geographic locations.
The tables are then “square”, having the
same number of rows as columns.
Such tables can be decomposed into two
parts, a symmetric part and a skew
symmetric part.
For the statisticians in the audience the total variance
can also be partitioned into these two parts.
70
Let T represent the table, with i rows and j
columns. It can be decomposed into two
parts as follows:
Tij = T+ + T–
where
T+ = (Tij + Tji)/2 (symmetric)
T– = (Tij - Tji)/2 (skew symmetric)
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Both Parts Can Be Used
When geographic locations are not
known the symmetric part can be used
to estimate the positions using
trilateration (a.k.a. multidimensional
scaling).
The skew symmetric (asymmetric) part
can be used to infer movement.
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From B to A is Not the Same as A to B
(Gary Larson)
73
Asymmetries Are a Fact of Geography
Movement is generally not symmetric.
This gives rise to asymmetries which can
be exploited.
As one example consider this table of
mail delivery times.
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Table of Mail Delivery Times
Observe the asymmetry
Transit time for US mail, in days (1973)
To:
From: \
NYC
CHI
LAX
WDC
STL
HOU
NYC
CHI
LAX
WDC
STL
HOU
-------------------------------------| 0.9
1.8
2.5
2.0
2.3
2.3 |
|
|
| 2.6
0.8
3.1
2.2
1.9
2.3 |
|
|
| 2.5
2.2
1.1
2.2
2.3
2.6 |
|
|
| 1.8
2.3
2.6
1.3
2.4
2.5 |
|
|
| 2.4
2.1
3.1
2.4
0.9
2.5 |
|
|
| 2.3
1.9
2.8
2.2
2.2
1.1
|
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--------------------------------------
A Map of Wind Computed from Mail
Delivery Times
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From Wind to Pressure Field
An interesting property of vector fields, as on the
foregoing map, is that they may be inverted.
If you think of a vector field as having been derived
from the topography of some surface this assertion
is that the topography can be calculated when only
the slope is known.
At least up to a constant of integration (the
absolute elevation) and if the data are curl free.
In the particular instance here, this says that the
barometric pressure could be estimated from the
mail delivery times.
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In the United States the Currency
Indicates Where It Was Issued
For bills this is the Federal Reserve District.
Coins contain a mint abbreviation.
You can check your wallet to estimate your
interaction with the rest of the country.
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Dollar Bill
(Federal Reserve Note)
Issued by the 8th (St. Louis) Federal Reserve District.
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(H is the 8th letter of the alphabet)
The 12 Federal Reserve Districts
(Alaska and Hawaii omitted)
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Movement of One Dollar Notes
between Federal Reserve Districts, in hundreds, Feb. 1976
To: B
Boston
New York
Philadelphia
Cleveland
Richmond
Atlanta
Chicago
St. Louis
Minneapolis
Kansas City
Dallas
San Francisco
NY P
Cl
R
A Ch SL M
K
D
SF
From:
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The Table of Dollar Bill Movements
was obtained from MacDonalds outlets
throughout the United States.
Source: S. Pignatello, 1977, Mathematical Modeling for
Management of the Quality of Circulating Currency, Federal
Reserve Bank, Philadelphia
From the table we can compute a movement
map.
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Dollar Bill Movement in the U.S.
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The Map is Computed Using a
Continuous Version of the Gravity
Model
The result is a system of partial differential
equations solved by a finite difference iteration
to obtain the potential field.
This can be contoured and its gradient
computed and drawn on a map.
W. Tobler, 1981,"A Model of Geographic Movement", Geogr. Analysis, 13 (1): 1-20
84
G. Dorigo, & Tobler, W., 1983, “Push Pull Migration Laws”, Annals, AAG, 73(1):1-17.
First the Federal Reserve Districts
Are “Rasterized”
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There will be one finite difference equation for each node on this raster
Solving the Equations Yields the Potential
Shown here by contours
The raster is indicated by the tick marks. The arrows are the gradients to the
potentials. The earlier streakline map is obtained by connecting the gradient 86
vectors.
Population movement in the
United States
Is recorded in migration tables using
census collection units and published at
several levels of resolution. The tables
are of course asymmetric.
Average resolution is here defined by
(Area of domain/ Number of units)1/2.
The resolution level is important because it tells one the size of pattern
which can be detected.
87
Nine Region Migration Table
US Census 1973
(Note asymmetry)
This is an example of a census migration table. There are also
(50 by 50) state tables and county by county tables.
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The Population Change Information Can Be
Positioned Locationally using centroids
Observe the spatial autocorrelation and how this is brought out more clearly by
omitting the collection unit boundaries, as on the next map. 89
Population Change at State Centroids
Mentally sketch a boundary around the losing region
The map is even better if the symbol size is made proportional to the
90
magnitude of the change, as on the next map
Gaining and Losing States
Based on the marginals of a 48 by 48 migration table
91
The Conventional Net Movement Map
Based on movement between state centroids
(Computer sketch. Optimum deletion: values below mean ignored)
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Using a model, this information can be
converted to a potential field and its
gradient
The model is, in essence, again a continuous
version of the familiar gravity model.
The gradient vectors can also be connected to
give a streakline map.
Next next two maps are based on the same observations as the previous
93 map.
The Migration Potentials and Gradients
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Migration Potentials and Streaklines
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Recall that tens of millions of people are
moving in the five year interval
The model, and map, thus illustrate aggregate
movement, and not the tracks of single
individuals.
On the other hand it is generally true that people living
to the East of Detroit move to Florida, Minnesotans to
the Northwest, and the rest to the Southwest.
96
That these migration maps resemble maps
of wind or ocean currents is not surprising
given that we in fact speak of migration
flows and backwaters, and use many such
hydrodynamic terms when discussing
movement phenomena.
97
By the insertion of arbitrary areal boundaries,
and by measuring the amount of flux across
these boundaries, one can obtain information
not contained in the original data, i.e., make a
prediction.
It’s like using a cookie cutter pressed into the
continuous flow model to look at an arbitrary piece
and computing the flow across its borders.
The next map is an example, using state boundaries.
The US Census Bureau does not provide this information. The model is used to
make the prediction
98
Major Flux Across State Boundaries
Predicted from the model and table marginals
99
The Previous Maps Have Used
Observations based on States
Patterns within urban areas could not be seen.
100
Average resolution 409 km. Detectable objects > 800 km in size.
If we used the 3,141 counties of the United
States the migration table could contain
9,862,740 numbers
This is not a lot for a computer, but for humans?
We need models and visualization techniques!
101
County Units
Average resolution ~55 km. Patterns >110 km detectable.
Still not sufficient to see movement within cities.
102
The 9x106 numbers in a county to county table
could not be comprehended without some
vizualization techniques or without a model.
Of course we know that most of the cells
in the county to county table would be
empty.
In the 3141 by 3141 US county migration
table only 5% of the cells have non-zero
entries.
Still, that is almost half a million numbers!
103
36,545 Communes of France
104
This many communes could lead to a migration table with 1,335,537,025 entries.
For a world table of
international migration
refugee movements
commodity trade
one would have a table of nearly 40,000 entries.
It is thus no surprise that few such tables exist.
Have you noticed that almost no statistical volumes contain from-to tables.
105
Migration in Switzerland
At several levels of resolution
(Guido Dorigo, University of Zürich)
The administrative units form a 3 level hierarchy.
Notice that one can pick out the Alps from the
administrative units pattern.
At the finest level of resolution the migration flow
field is “turbulent”.
Importantly we see that reducing the
resolution has the effect of a high frequency
spatial filter.
106
3090 Communities. 3.6 km. Average resolution
107
Net Migration In Switzerland
By 3090 Communities, 3.5 km resolution
108
184 Districts. 14.7 km average resolution
109
26 Cantons. 39.2 km average resolution
110
Swiss Migration at Reduced Resolution
To emphasize the filtering effect of resolution
184 Districts, 14.7 km resolution
26 Cantons, 39.2 km resolution
111
In this presentation I have
emphasized
Location and Movement
As represented by various kinds of
cartographic modeling.
112
I appreciate your attention and thank you.
113