One of the implications of Tobler’s First Law and spatial dependence is that it is possible to capture a reasonably accurate description of the Earth’s surface with a few well-placed samples. Meteorologists capture weather patterns by sampling at measuring stations, knowing that conditions between these sample locations will vary systematically and smoothly. Geographers describe the contents of entire regions with a few well-chosen statements, provided that the regions have been defined to enclose approximately uniform conditions. Census data are published in spatially aggregated form to protect the confidentiality of individual returns, and Tobler’s First Law ensures that the variance within each aggregate unit is not so large as to render the aggregated data meaningless.
Numerous approaches to spatial sampling exist, depending on the nature of the questions being asked. Perhaps the simplest occurs when sample observations are made in order to obtain a best estimate of average conditions within an area. For example, to show that average temperatures across the globe are increasing due to global warming, one would want to ensure that every location on the globe had an equal chance of being sampled (in reality weather stations are distributed far from randomly, often co-located with airports, and more common on land and in heavily populated areas). In other cases it might be more important to characterize various regions, by placing an equal number of sample points in each region, or to characterize each of a number of vegetation or soil types. If the objective is to characterize the pattern of spatial dependence in an area, then it may be more important to place samples so as to obtain equally reliable estimates of similarity in every range of distances. A discussion of spatial sampling can be found in the book by Longley et al. (2015, Ch2.4) and in Section 5.1.2, Spatial sampling, of this Guide.