There used to be two obstacles for progress of geography. One is that it is hard to be mathematically modeled in an efficient way because of nonlinearity of geographical processes and regularity of geographical phenomena, the other is that it is impossible to perform controlled experiments on geographical systems because of irreversibility and uncontrollability of geographical evolvements. Fortunately, as the development of the postmodern mathematics of fractals and chaos and others, certain geographical systems can be modeled with the aid of mathematical equations today. As the development of cellular automata (CA) model, geographical information system (GIS) based computer simulation can be employed to make geographical analysis as an experiment approach to revealing causalities (cause and effect). In this case, a new study procedure termed "three-step method" for theoretical geographical research is propounded as follows. Step 1: building mathematical models empirically based on observational data and estimating the values of parameters. Step 2: constructing theoretical equation based on certain postulates. The solution to the theoretical equation should be found in some way, and the solution must be identical in form and structure to the empirical model made in the first step. Or else repeat step 2 by giving new postulates and new theoretical model until the solution is satisfying. Step 3: carrying out experiments of computer simulation upon the studied object based on the postulates raised in the second steps. The result of simulation should be identical in pattern or structure to the observed phenomena in the first step.The first step relates to the quantification of geographical research, the second step to the theorization of geographic discipline, and the third step to the demonstration and experimentation of geography. The method used in the first step is mature at the present time. Building of empirical models with mathematical theory has been developing since the well-known "quantitative revolution" of geography from the 1950s to the 1970s. Of course, "the purpose of models is not to fit the data, but to sharpen the questions." The means employed in the third step is rapidly developed in recent years and in its full flourish now. Simulation of a geographical system may be very helpful practically, but geo-simulation doesn’t help us conceptually, in understanding the rules of behavior at the higher level. Comparatively speaking, the process utilized in the second step is still a difficult problem to be solved despite the fact that the theorization of geography began long ago. The three-step method is proposed for the purpose of theoretical exploration of geography, but it can be extended to the domain of applied geography. In application research, the second step can be simplified dramatically by giving up mathematical derivation. However, the third step will be more difficult when it is designed for use in practice, as the simulation device should be drastically altered before it becomes even vaguely realistic representations of geographical systems.
[1] Henry J. The Scientific Revolution and the Origins of Modern Science (2nd Edition)[M]. New York: Palgrave, 2000.
[2] Trefill J. The Edge of the Unknown[M]. Boston: Houghton Mifflin Co., 1996.
[3] 陈彦光,刘继生.地理学的主要任务与研究方法——从整个科学体系的视角看地理科学[J].地理科学,2004,24(3):257~263.
[4] Waldrop M. Complexity: The Emerging of Science at the Edge of Order and Chaos [M]. NY: Simon and Schuster, 1992.
[5] Bak P. How Nature Works: The Science of Self-organized Criticality [M]. New York: Springer-Verlag, 1996.
[6] 高泳源.竺可桢的关于地理学性质、任务与方法的论述[J].地理科学,1999,10(1):20~27.
[7] 曹桂发.地理分析方法库及其应用研究[J].地理科学,1992,12(2):172~178.
[8] Helbing D, Keltsch J, Molnar P. Modelling the evolution of human trail systems. Nature, 1997, 388: 47-50.
[9] Makse H A, Andrade Jr J S, Batty M, Havlin S, Stanley HE. Modeling urban growth patterns with correlated percolation [J]. Physical Review E, 1998, 58(6): 7054-7062.
[10] Zanette D, Manrubia S. Role of intermittency in urban development: a model of large-scale city formation[J]. Physical Review Letters, 1997, 79(3): 523-526.
[11] Manrubia S, Zanette D. Intermittency model for urban development[J]. Physical Review E, 1998, 58(1): 295-302.
[12] Gordon K. The mysteries of mass[J]. Scientific American, 2005, 293(1): 41-48.
[13] Couclelis H. From cellular automata to urban models: new principles for model development and implementation[J]. Environment and Planning B: Planning and Design, 1997, 24: 165-174.
[14] Philo C, Mitchell R, More A. Reconsidering quantitative geography: things that count (Guest editorial) [J]. Environment and Planning A, 1998,30(2): 191-201.
[15] 陈彦光.地理数学方法:从计量地理到地理计算[J].华中师范大学学报(自然科学版),2005,39(1):113-119/125.
[16] Fotheringham A S. Trends in quantitative methods I: Stressing the Local[J]. Progress in Human Geography, 1997, 21: 88-96.
[17] Fotheringham A S. Trends in quantitative method Ⅱ: Stressing the computational[J]. Progress in Human Geography, 1998, 22: 283-292.
[18] Fotheringham A S. Trends in quantitative methods III: Stressing the visual[J]. Progress in Human Geography, 1999, 23(4): 597-606.
[19] Batty M. Cities as fractals: Simulating growth and form. In: Fractals and Chaos. Eds A J Crilly, R A Earnshaw, and H Jones. New York: Springer–Verlag, 1991,43-69.
[20] Allen P M. Cities and Regions as Self-Organizing Systems: Models of Complexity [M]. Amsterdam: Gordon and Breach Science Pub., 1997.
[21] Prigogine I, Stengers I. Order Out of Chaos: Man's New Dialogue with Nature [M]. New York: Bantam Book, Inc., 1984, 196-203.
[22] Batty M, Longley P A. Fractal Cities: A Geometry of Form and Function [M]. London: Academic Press, 1994.
[23] Batty M, Couclelis H, Eichen M. Urban systems as cellular automata (Editorial) [J]. Environment and Planning B: Planning and Design, 1997, 24: 159-164.
[24] White R, Engelen G. Cellular automata and fractal urban form: a cellular modeling approach to the evolution of urban land-use patterns[J]. Environment and Planning A, 1993, 25(8): 1175-1199.
[25] White R, Engelen G. Urban systems dynamics and cellular automata: fractal structures between order and chaos[J]. Chaos, Solitons & Fractals, 1994, 4(4): 563-583.
[26] Benenson I, Torrens P M. Geosimulation: Automata-based Modeling of Urban Phenomena [M]. Chichester: John Wiley & Sons, Ltd., 2004.
[27] Batty M. Cities and Complexity: Understanding Cities with Cellular Automata, Agent-Based Models, and Fractals[M]. London, England: The MIT Press, 2005.
[28] Portugali J. Ed. Complex Artificial Environments: Simulation, Cognition and VR in the Study and Planning of Cities [M]. Berlin: Springer-Verlag, 2006.
[29] 黎夏,叶嘉安,刘小平,杨青生.地理模拟系统:元胞自动机与多智能体[M].北京:科学出版社,2007.
[30] Albeverio S, Andrey D. Giordano P, Vancheri A. The Dynamics of Complex Urban Systems: An Interdisciplinary Approach [M]. Heidelberg: Physica-Verlag, 2008.
[31] Fischer M M, Leung L. Eds. Geocomputational Modelling:?Techniques and Applications [M]. Berlin:?Springer, 2001.
[32] Diappi L. Evolving Cities: Geocomputation in Territorial Planning [M]. Aldershot, Hants: Ashgate, 2004.
[33] Bossomaier T, Green D. Patterns in the Sand: Computers, Complexity and Life [M]. Reading, Massachusetts: Perseus Books, 1998.
[34] 郝柏林.复杂性的刻画与"复杂性科学"[J].科学,1999,51(3):3~8.
[35] Torrens PM, O'Sullivan D. Cellular automata and urban simulation: where do we go from here (Editorial)? [J] Environment and Planning B: Planning and Design, 2001, 28: 163-168.
[36] Armstrong MP. Geography and computational science [J]. Annals of the Association of American Geographers, 2000, 90(1): 146-156.
[37] Clark C. Urban population densities [J]. Journal of Royal Statistical Society, 1951, 114: 490-496.
[38] Bussiere R, Snickers F. Derivation of the negative exponential model by an entropy maximizing method [J]. Environment and Planning A, 1970, 2: 295-301.
[39] 陈彦光.城市人口空间分布函数的理论基础与修正形式——利用最大熵方法推导关于城市人口密度衰减的Clark模型[J].华中师范大学学报(自然科学版),2000, 34(4):489~492.
[40] 冯健.杭州市人口密度空间分布及其演化的模型研究 [J].地理研究,2002,21(5):635~646.
[41] 冯健.转型期中国城市内部空间重构[M].北京:北京科学出版社,2004.
