Waldo R. Tobler

A Computer Movie Simulating Urban Growth in the Detroit Region

(1970). A Computer Movie Simulating Urban Growth in the Detroit Region. Economic Geography: Vol. 46, PROCEEDINGS International Geographical Union Commission on Quantitative Methods, pp. 234-240.

Smooth Pycnophylactic Interpolation for Geographical Regions

Abstract Census enumerations are usually packaged in irregularly shaped geographical regions. Interior values can be interpolated for such regions, without specification of “control points,” by using an analogy to elliptical partial differential equations. A solution procedure is suggested, using finite difference methods with classical boundary conditions. In order to estimate densities, an additional nonnegativity condition is required. Smooth contour maps, which satisfy the volume preserving and nonnegativity constraints, illustrate the method using actual geographical data. It is suggested that the procedure may be used to convert observations from one bureaucratic partitioning of a geographical area to another.

Exploring the anchor-point hypothesis of spatial cognition*

Abstract The anchor-point hypothesis of spatial cognition, according to which primary nodes or reference points anchor distinct regions in cognitive space, brings together certain frequently reported apparent properties of mental maps: the regionalization and hierarchical organization of cognitive space, and the active role of salient cues in structuring spatial cognition. After a brief overview of the state of the art in cognitive mapping research, the anchor-point hypothesis is first explored conceptually, and then one particular version of it, the ‘tectonic plates’ hypothesis, is made operational. For that second part of the study, cognitive configurations derived from five subjects selected from a larger sample taken in Goleta, California are analyzed using three different methods, and features transcending any method-specific biases are identified. Although not entirely unambiguous, these first results seem encouraging and warrant further research in this direction.

Experiments In Migration Mapping By Computer

Migration maps represent patterns of geographical movement by arrows or bands between places, using information arriving in “from-to” tables. In the most interesting cases the tables are of large size, suggesting that computer assistance would be useful in the preparation of the maps. A computer program prepared for this purpose shows that graphical representation is feasible for tables as large as fifty by fifty, and possibly larger. The program contains options for alternate forms of movement depiction, and rules are suggested for the parsing of migration tables prior to the cartographic display, without loss of spatial resolution.

Thirty Five Years of Computer Cartograms

Abstract The notion of a cartogram is reviewed. Then, based on a presentation from the 1960s, a direct and simple introduction is given to the design of a computer algorithm for the construction of contiguous value-by-area cartograms. As an example, a table of latitude/longitude to rectangular plane coordinates is included for a cartogram of the United States, along with Tissots measures for this map projection. This is followed by a short review of the subsequent history of the subject and includes citation of algorithms proposed by others. In contrast to the usual geographic map, the most common use of cartograms is solely for the display and emphasis of a geographic distribution. A second use is in analysis, as a nomograph or problem-solving device similar in use to Mercators projection, or in the transform-solve-invert paradigm. Recent innovations by computer scientists modify the objective and suggest variation similar to Airys (1861) “balance of errors” idea for map projections.

On the First Law of Geography: A Reply

D aniel Sui has sent me a written version of comments presented by five geographers at a panel on the first law of geography organized by him at the 2003 AAG meeting in New Orleans. The comments seem to fall into two camps: some reject the idea of ‘‘laws’’ in geography, and others feel that my notion has been of some merit. Interestingly, several other laws are cited in the comments; two by Isaac Newton (gravity, motion), two additional ‘‘first’’ laws (ecology, social science), four additional laws applying to people (utility maximization, primate city, human behavior, and Gresham’s), and three ‘‘second’’ laws (thermodynamics, spatial heterogeneity, things are never equal—the last two being only suggestions). Curiously, they do not mention other well-known laws, such as those of Zipf (1949), or Ravenstein’s (1885) 10 laws, or a second law of geography by proposed by Arbia, Benedetti, and Espa (1996) and one by myself (Tobler 1999, 87). The comments then focus on three topics. One is a discussion of what constitutes a law, and whether ‘‘the first law of geography’’ fits into the appropriate definition. Much of the remaining material examines the concepts of ‘‘related’’ and ‘‘near.’’ Smith is quite correct in pointing out that this discussion would never have taken place if the specific word law had not been used. And Barnes is also correct when he puts it into context and observes that, by restricting myself to local effects, I used the notion to parse the possible complexities of simulating urban growth. I am a great believer in simplicity, when this is possible. For example, the point in science is to achieve as many results as possible with the fewest hypotheses. So, in order to simplify the problem of depicting the growth of population in the Detroit region, I tried to eliminate complicating factors. This is when I invoked ‘‘the first law of geography: everything is related to everything else but near things are more related than distant things.’’ Doing this allowed me to concentrate on local effects— using the idea of a change in the ‘‘unit inhabitant,’’ and ignoring many other possible influences. As a result, Miller calls my simulation effort ‘‘crude,’’ whereas Barnes considers that it invokes complex ideas. Miller would undoubtedly consider economics, transportation, geology, and other factors to produce a much richer, but also more involved, model. Curiously there is no discussion of other concepts from the 1970 paper.

Spatial Interaction Patterns

An algebraic examination of spatial models leads to the conclusion that a convenient description of the pattern of flows implicit in a geographical interaction table is obtained by displaying a field of vectors computed from the relative net exchanges. The vector field approximates the gradient of a scalar potential, and this may be invoked to explain the flows. The method can be applied to asymmetrical tables of a non-geographical nature.

A mathematical theory of migration

Section 1 considers growth of a population; section 2 treats geographical migration by analogy to the flow of heat; section 3 combines sections 1 and 2 and includes immigration; in section 4 are assembled empirical observations taken from the recent United States experience. These are designed to provide a test of the theory.

A Transformational View of Cartography

Cartographic transformations are applied to locative geographic data and to substantive geographic data. Conversion between locative aliases are between points, lines, and areas. Substantive transformations occur in map interpolation, filtering, and generalization, and in map reading. The theoretical importance of the inverses is in the study of error propagation effects.

MIGRATION: RAVENSTEIN, THORNTHWAITE, AND BEYOND

Over one hundred years have passed since Ernst Georg Ravenstein published his “Laws of Migration.” This paper addresses the question of how these laws have withstood the test of time. Ravenstein’s 1885 paper also includes a map of “Currents of Migration,” not mentioned in the text, which this article examines for its impacts on the study of migration. Warren Thornthwaite (1934) also compared migration to currents, but did not pursue this analogy in subsequent work. Others have used similar terminology, and examination of more recent migration studies suggests that they may yield new laws.