10.2190/VAKC-3GRF-3XUG-WY4W VAKC3GRF3XUGWY4W 1 Spatial Interaction Patterns 271 301 20020509 20020509 20020509 20020509 VAKC3GRF3XUGWY4W.pdf http://baywood.metapress.com/link.asp?target=contribution&id=VAKC3GRF3XUGWY4W 4 Waldo Tobler Department of Geography, University of Michigan 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.

W. Tobler, H. Mielke and T. Detwyler, Geobotanical Distance Between New Zealand and Neighboring Islands, Bioscience, 20:9, pp. 537-542, 1970.

W. Tobler and S. Wineberg, A Cappadocian Speculation, Nature, 231:5297, pp. 39-42, May 7, 1971.

P. Wolf, Horizontal Position Adjustment, Surveying and Mapping, 29:4, pp. 635-644, 1969.

R. Shepard, The Analysis of Proximities, Psychometrika, 27, pp. 219-246, 1962.

J. Lingoes, The Guttman-Lingoes Non Metric Program Series, Mathesis Press, Ann Arbor, pp. 81-111, 1973.

P. Schwind, Migration and Regional Development in the United States: 1950-1960, University of Chicago, Department of Geography, Chicago, Research Paper 1933, 1971.

W. Borejko, Study of the Effectiveness of Migrations, Geographia Polonica, 14, pp. 305-312, 1968.

A. Wilson, Urban and Regional Models in Geography and Planning, J. Wiley, New York, 1974.

I. Lowry, Migration and Metropolitan Growth, Chandler, San Francisco, 1966.

M. Cordey-Hayes, On the Feasibility of Simulating the Relationship Between Regional Imbalance and City Growth, in Space-Time Concepts in Urban and Regional Models, E. Cripps, (ed.), Pion, London, p. 193, 1974.

J. H. Ross, A Measure of Site Attraction, Ph.D. thesis, Department of Geography, University of Western Ontario, London, Canada, p. 27 et seq, 1972.

F. Cesario, A Generalized Trip Distribution Model, Journal of Regional Science, 13:2, pp. 233-247, 1973.

W. Somermeijer, Multi-Polar Human Flow Models, Papers, Regional Science Assn., 26, pp. 131-144, 1971.

M. Dacey and J. Nystuen, A Graph Theory Interpretation of Nodal Regions, Papers, Regional Science Assn., 7, pp. 29-42, 1961.

M. Harvey and J. Auwerter, Derivation and Decomposition of Hierarchies From Interaction Data, Discussion Paper 31, Department of Geography, Ohio State University, Columbus, Ohio, 1973.

P. B. Slater, College Student Mobility in the United States in Fall 1968: Graph-Theoretic Structuring of Transaction Flows, Proceedings, Social Statistics Section, Am. Stat. Assn., 9:3, pp. 440-446, 1974.

S. Salas and E. Hille, Calculus, Toronto Xerox College Publishing, Part II, pp. 659-666, 1969.

H. N. Pollock, On the Separation of an Arbitrary Two Component Vector Into Divergence Free and Curl Free Parts, and the Determination of the Scalar and Vector Potentials, mimeo, Ann Arbor, 2 pp., 1973.

H. M. Schey, Div, Grad, Curl, and All That, Norton, New York, 1973.

B. Carnahan, H. Luther and J. Wilkes, Applied Numerical Methods, J. Wiley, New York, pp. 429-530, 1969.

R. McQuistan, Scalar and Vector Fields, J. Wiley, New York, pp. 292-305, 1965.

W. Leontief and A. Strout, Multiregional Input Output Analysis, in Structural Interdependence and Economic Development, T. Barna, (ed.), Macmillan, London, pp. 119-150, 1963.

T. Hägerstrand, A Monte Carlo Approach to Diffusion, in Spatial Analysis, B. Berry and D. Marble, (eds.), Prentice Hall, New York, 1968.

F. Pasquill, Atmospheric Diffusion: the Dispersion of Windborne Material From Industrial and Other Sources, Van Nostrand, New York, 1962.

National Center for Educational Statistics, Residence and Migration of College Students: Basic State-to-State Matrix Tables, United States Department of Health, Education, and Welfare, Government Printing Office, Washington, D. C., 1970.

W. Tobler, Local Map Projections, The American Cartographer, 1:1, pp. 51-62, 1974.

M. Beckmann, On the Equilibrium Distribution of Population in Space, Bulletin of Mathematical Biophysics, 19, pp. 81-90, 1957.

W. Warntz, Macro Geography and Income Fronts, Regional Science Research Institute, Philadelphia, 1965.

Department of Commerce, Statistical Abstract of the United States, 1972, G. P. O., Washington, D. C., pp. 130-132.

Provided by Professor W. Dekan of the University of Wisconsin, White Water, Wisconsin.

J. M. Rodgers, State Estimates of Interregional Commodity Trade, 1963, Lexington Heath Books, Cambridge, p. 447, 1973.

W. Tobler, Commodity Fields, Internal paper, International Institute for Applied Systems Analysis, Laxenburg, March 1975.

W. Tobler, Migration Fields, Internal paper, International Institute for Applied Systems Analysis, Laxenburg, March 1975.

D. Fliedner, Zyklonale Tendenzen bei Bevölkerungs und Verkehrsbewegungen in Städtischen Bereichen untersucht am Beispiel der Städte Göttingen, München, und Osnabrück, Neues Archiv für Niedersachsen, 10:15, Table 2, p. 281, April 4, 1965. Belgium census commuting data provided by Mr. Frans Willekens of IIASA.

OECD, Nights Spent by Foreign Tourists, Tourism Policy and International Tourism in OECD Member Countries, 1973, OECD, Paris, pp. 95-141, 1974.

Data provided by Professor A. Kilchenmann, Geographisches Institut, Technische Universität Karlsruhe, FRG.

G. Törnqvist, Contact Systems and Regional Development, Gleerup, Lund, p. 77, 1970.

C. Coombs, J. Dawes and A. Tversky, Mathematical Psychology, Prentice Hall, Englewood Cliffs, pp. 73-74, 1970.

D. Thompson, Spatial Interaction Data, Annals, Assn. Am. Geographies, 64:4, pp. 560-575.