While Tobler may have fewer publications than some contemporary geographers, his publications covered a broad range of topics and are considered of exceptional quality.
Cartography Waldo Tobler described himself as a "geographical cartographer", and his research interests reflect this. He published the first paper on using computers for making maps, established the discipline of analytical cartography, and contributed to the literature around thematic maps. Geographer
Mark Monmonier described Tobler as "arguably the twentieth century's most innovative cartographer."
Map projections of deformation; α = 0, k = 3 One of Tobler's largest interests, especially early in his career, was map projections, with much of his dissertation focusing on them. He also invented a method for smooth two-dimensional mass-preserving areal data redistribution. In 1972, Tobler translated and published
Johann Heinrich Lambert's 1772 "Notes and comments on the Composition of Terrestrial and Celestial Maps."
Computer cartography Using his time and experience on the SAGE system, Tobler built upon the concepts and published his work in academic journals. Tobler's research in developing applications for computer cartography is described by
Mark Monmonier as occupying "a pivotal place in map history". Analytical cartography is the foundation for many of the developments in
Geographic information systems, and shapes how spatial analysis and cartography are taught today. Tobler argued that these maps would increase data density, and avoided many of the issues with
Data binning and
Statistical classification. There has been significant debate around the best approach to solve this issue with choropleth maps, and most choropleth maps today continue to make use of class breaks. Other approaches to creating classes in choropleth maps include using the
Jenks natural breaks optimization, quantile, or equal class intervals.
Cartograms Tobler's interest in cartograms stemmed from his interest in map projections. A chapter of his dissertation was developed for their creation, later adapted and published in the Geographical Review. Tobler was among the first to use computers to create cartogram maps, with the rubber sheet method being the first method he proposed for their creation. Tobler's methods for creating cartograms are still employed, however they have some practical problems in implementation that can sometimes ruin topology. Tobler's methods serve as the basis for many other methods to create them. While crude, the result of this research was that Tobler was the first to develop a software approach to creating flow maps in 1987. The first demonstration of this technology by Tobler involved mapping the flow of money through the US Federal Reserve to the various US states. Tobler's flow mapper software, and similar programs, continue to be built upon and applied to new topics. Tobler published several studies on different approaches to spatial interpolation, including an extension of bilinear weighted interpolation and other models. With regard to spatial resolution Tobler has formulated the following rule of thumb: "The usefulness of a GIS is constrained by its spatial resolution. The size of the smallest detectable feature is twice that of the resolution. The rule is: divide the denominator of the map scale by 1,000 to get the detectable size in meters. The resolution is one half of this amount." This text following the colon in his statement, which was not the main focus of the paper, is now known as "Tobler's First Law of Geography", and is probably what Tobler is most famous for. The first law of geography is widely cited and is relevant today, particularly within the sub-discipline of geographic information science. The
Geographic Information Science and Technology Body of Knowledge "Model Curricula" in particular emphasizes the importance of the first law in the section on "Metrical relationships: distance and direction." It is considered the theoretical basis of many statistics in spatial analysis, including those involved in
cluster analysis and spatial autocorrelation (such as
Moran's I). Spatial autocorrelation, and Tobler's 1970 paper, are considered central to modern approaches in
technical geography. Tobler's first law is included in the children's book "ABCs of Geography" under the letter "T" for "Tobler". In a 1999 paper titled "Linear pycnophylactic reallocation comment on a paper by D. Martin," Tobler stated "Philosophically, the phenomenon external to an area of interest affects what goes on in the inside; a sufficiently common occurrence as to warrant being called the second law of geography." In his 2004 paper "On the First law of Geography: A reply", he discussed this concept again. This has come to be known as Tobler's second law of geography. This has come to be known as
Arbia's law of geography. The laws of geography, particularly Tobler's first law of geography, have been debated heavily in literature, with their status as
scientific laws questioned, changes and amendments proposed, exceptions noted, and corresponding defenses by proponents of the laws. Tobler weighed in on this debate surrounding his law, and others, in a 2004 article titled "On the First law of Geography: A reply". While this project had serious limitations, largely due to data limitations, it was the finest scale population set produced to that point. The project was later supported by the
Columbia University's Center for International Earth Science Information Network. In this research, Tobler used himself as a subject, and published the results in a 1993 paper. Using the frequency of locations being noted in
Cuneiform tablets discussing commercial transactions, he estimated the distance between the towns in
Babylonia using a reverse gravity model. Many of these predictions were for unknown locations and were proven accurate for at least three known towns; however, more excavation is needed to confirm the remainder of his predictions. These ideas serve as the basis for numerous similar computer simulations to model ancient human migration, such as the settlement of Polynesia. ==Awards and honors==