Kriging is an interpolation method that makes predictions at unsampled locations using a linear combination of observations at nearby sampled locations. The influence of each observation on the kriging prediction is based on several factors: 1) its geographical proximity to the unsampled location, 2) the spatial arrangement of all observations (i.e., data configuration, such as clustering of observations in oversampled areas), and 3) the pattern of spatial correlation of the data. The development of kriging models is meaningful only when data are spatially correlated.. Kriging has several advantages over traditional interpolation techniques, such as inverse distance weighting or nearest neighbor: 1) it provides a measure of uncertainty attached to the results (i.e., kriging variance); 2) it accounts for direction-dependent relationships (i.e., spatial anisotropy); 3) weights are assigned to observations based on the spatial correlation of data instead of assumptions made by the analyst for IDW; 4) kriging predictions are not constrained to the range of observations used for interpolation, and 5) data measured over different spatial supports can be combined and change of support, such as downscaling or upscaling, can be conducted.
Areal interpolation is the process of transforming spatial data from source zones with known values or attributes to target zones with unknown attributes. It generates estimates of source zone attributes over target zone areas. It aligns areal spatial data attributes over a single spatial framework (target zones) to overcome differences in areal reporting units due to historical boundary changes of reporting areas, integrating data from domains with different reporting conventions or in situations when spatially detailed information is not available. Fundamentally, it requires assumptions about how the target zone attribute relates to the source zones. Areal interpolation approaches can be grouped into two broad categories: methods that link target and source zones by their spatial properties (area to point, pycnophylactic and areal weighed interpolation) and methods that use ancillary or auxiliary information to control, inform, guide, and constrain the interpolation process (dasymetric, statistical, streetweighted and point-based interpolation). Additionally, there are new opportunities to use novel data sources to inform areal interpolation arising from the many new forms of spatial data supported by ubiquitous web- and GPS-enabled technologies including social media, PoI check-ins, spatial data portals (e.g for crime, house sales, microblogging sites) and collaborative mapping activities (e.g. OpenStreetMap).
Spatial data can be represented in vector or raster form. The vector spatial data model is coordinate-based and represents geographic features as points, lines, and polygons. The raster spatial data model is pixel-based and represents geographic phenomena as an organized matrix of cells. Each model possesses advantages, disadvantages, and tradeoffs in how data can be manipulated, analyzed, and rendered. As a result, GIS professionals often need to work between data models to achieve their analytical goals. Vector-to-raster and raster-to-vector conversions are fundamental spatial data manipulation processes used to transform one model of spatial data representation into the other to extend the utility of a spatial dataset. Vector-to-raster conversion, also known as rasterization, is the process of converting vector points, lines, and polygons into a surface of gridded cells or pixels. Advanced rasterization techniques, such as spatial interpolation and density mapping, can be used to predict raster surfaces at unsampled locations based on known values of nearby vector spatial data inputs. Raster-to-vector conversion, also known as vectorization, is the process of converting gridded cell- or pixel-based data into vector points, lines, and polygons. While powerful, these conversion processes also have implications for geographic accuracy and potential feature loss.
Gridding is the act of taking a field of measurements and discretizing it into a regular tessellation, often either a lattice of squares or hexagons. Gridding can either discretize continuous phenomena or aggregate discrete instances; in either case, gridding serves conceptually to assist analysis, for example in finding local minima or maxima (i.e., "hotspots"). The process of gridding often involves interpolation, which is the rational estimation of unknown data values within the bounds of known values. Contouring refers to the creation of isolines throughout a data surface, often one represented by a grid. This section describes gridding, interpolation, and contouring, highlighting a few example methods by which interpolation is frequently done in the geospatial analysis.
Spatial interpolation methods use the measured values at given locations to estimate the values at unsampled locations, for example, in computing raster digital elevation models from scattered measured elevations. Since this problem does not have a unique solution, many approaches have been developed to accomplish this task. Methods based on linear superposition of radial basis functions (RBF) centered at the data points include multivariate splines that simultaneously minimize the sum of the deviations from the measured points and the smoothness seminorm referred to also as a roughness penalty. The thin plate spline minimizes the 2D surface curvature and mimics a thin steel plate forced to pass through the data points: its equilibrium shape minimizes the bending energy which is closely related to the surface curvature. There are many generalizations such as spline with tension that controls the plate stiffness, while regularized spline enables direct calculations of surface gradients and curvatures making it effective for terrain modeling with simultaneous topographic analysis. Trivariate splines are used to interpolate meteorological variables with influence of topography. The RBF splines are related to universal kriging with the choice of the covariance function determined by the smoothness seminorm. Multiquadric RBF methods are similar in formulation and performance to splines.