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Exploring Geography Cartographically

Waldo Tobler

Professor Emeritus of Geography

University of California

Santa Barbara, CA 93106-4060

http://www.geog.ucsb.edu/~tobler

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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 2000

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To begin I stress

Location Location Location

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There are many ways of specifying location

For example, here is my address if you wish to correspond with me

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Another Way of Determining Locations

Telephone numbersWireless (cell) telephones

These allow you to be located within a few meters.

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Common Geographic Locational Aliases and

Conversions

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Hand Held GPS with MapLocations are estimated from distances

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Wrist Watch GPS

Location is determined by measuring distance to satellites.

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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.

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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!

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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.

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Step by Step Trilateration

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Trilateration Procedure Illustrated Graphically

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The Beginning Steps

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The Answer, with Coordinates and Estimated Distances

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Solution to the US Distance Exercise

with (enlarged) estimated standard errors

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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 thanThe 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.

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I will now give you another example where locations are estimated from distances

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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.)

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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

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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

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Predicted Locationsof 33 Pre-Hittite Towns in Cappadocia using distances estimated from presumed

interaction.

W. Tobler, Wineberg, S., 1971, “A Cappadocian Speculation”, Nature, 231, 5297(7 May):39-42.

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USA Highway Distance MapHere 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.

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Road Distance Map of the United States

Student drawing

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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

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All maps are of course distorted to some extent

And this brings me to the subject of map projections.

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The Surface of the Earth Is Two-dimensional

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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.

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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.

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The conventional satellite tracking chart

The satellite tracks are curvesThe meridians and parallels are straight

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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.

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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.

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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.

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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.

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US Map in the Two SpacesLeft, the usual map. Right, the transform.

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The InversionOn 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, New York Academy of Sciences, 219: 215-220

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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.

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Title page from Ptolemy’s Geography

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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.

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Ptolemy’s World Map

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Ptolemy’s Map of Gaul(Modern France)

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Modern and Ptolomaic Latitude and Longitude for Gaul

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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.

W. Tobler, 1994, “Bidimensional Regression”, Geog. Analysis, 26:186-212

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Modern Latitude versus Ptolemy’s Latitude

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Modern Longitude versus Ptolemy’s Longitude

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Ptolemaic DisplacementsAmounts needed to move Ptolemy’s locations to modern locations. The places needing further investigation can be

seen.

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Interpolated Vector Field Amount needed to move the graticule to modern

locations

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Ptolemaic Grid Shifted to Fit Modern

‘Pushed’ by the interpolated vector field

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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”.

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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.

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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.

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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.

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Population Density in KansasBy County

Courtesy of T. Slocum

A piecewise continuous surface

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Population Density in Kansasby County

Each county still contains the same number of people

A smooth continuous surface, with population pycnophylactically redistributed

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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)

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Mass Preserving Reallocation Using Areal Data

W. Tobler, 1979, “Smooth Pycnophylactic Interpolation for Geographical Regions”, J. Am. Stat. Assn., 74(367):519-536.

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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

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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.

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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.

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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.

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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)

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Movement is generally not symmetric.

This gives rise to asymmetries which can be exploited.

As one example consider this table of mail delivery times.

Asymmetries Are a Fact of Geography

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Table of Mail Delivery TimesObserve the asymmetry

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A Map of Wind Computed from Mail Delivery Times

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From Wind to Pressure FieldAn 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.(H is the 8th letter of the alphabet)

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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 NY P Cl R A Ch SL M K D SF

From: BostonNew York

PhiladelphiaClevelandRichmond

AtlantaChicagoSt. Louis

MinneapolisKansas City

DallasSan Francisco

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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

G. Dorigo, & Tobler, W., 1983, “Push Pull Migration Laws”, Annals, AAG, 73(1):1-17.

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First the Federal Reserve Districts Are “Rasterized”

There will be one finite difference equation for each node on this raster

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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 vectors.

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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.

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.

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Nine Region Migration TableUS 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.

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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 magnitude of the change, as on the next map

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Gaining and Losing StatesBased on the marginals of a 48 by 48 migration

table

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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 map.

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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.

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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.

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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

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Major Flux Across State Boundaries Predicted from the model and

table marginals

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The Previous Maps Have Used Observations based on States

Patterns within urban areas could not be seen.

Average resolution 409 km. Detectable objects > 800 km in size.

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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!

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County Units

Average resolution ~55 km. Patterns >110 km detectable.Still not sufficient to see movement within cities.

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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!

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36,545 Communes of France

This many communes could lead to a migration table with 1,335,537,025 entries.

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For a world table of international migration refugee movementscommodity 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.

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Migration in SwitzerlandAt 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.

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3090 Communities. 3.6 km. Average resolution

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Net Migration In SwitzerlandBy 3090 Communities, 3.5 km resolution

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184 Districts. 14.7 km average resolution

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26 Cantons. 39.2 km average resolution

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Swiss Migration at Reduced Resolution

To emphasize the filtering effect of resolution

184 Districts, 14.7 km resolution 26 Cantons, 39.2 km resolution

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In this presentation I have emphasized

Location and Movement

As represented by various kinds of cartographic

modeling.

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I appreciate your attention and thank you.