Overview of gridding and interpolation

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Overview of gridding and interpolation

There have been a many studies that compare the effectiveness of alternative interpolation techniques, using a wide range of different test datasets and conditions. Overall it has been found that a number of well-defined factors have a major influence on the quality of interpolation: data measurement accuracy; data density; data distribution; and spatial variability. These are fairly predictable findings, but prior examination of each of these elements may assist in choosing the most appropriate technique for the problem at hand and/or be used in guiding sampling of new or supplementary datasets. Interpolation quality can often be substantially improved through the use of ancillary information, such as remote sensing data or additional environmental information (e.g. location of stream networks).

Having obtained the best possible dataset, within budget and time constraints, achieving the maximum usage and value is very important — hence the need for interpolation procedures that assist in estimating values at unsampled locations. More generally, spatial interpolation is required:

to convert a sample of data points to a complete coverage (set of values) for the study region

to convert from one level of data resolution or orientation to another (resampling). Usually resolution is reduced to the coarsest in a set, but resolution can be increased using a suitable interpolator, such as an simple or incremental bicubic spline (stair interpolation)

to convert from one representation of a continuous surface to another, e.g. TIN to grid or contour to grid

Grid files may be pre-generated (as for example, with elevation grid datasets from national mapping agencies), or may be generated from an input dataset of {x,y,z} form. In the latter case values of z are estimated for unsampled points, typically a square grid of pre-specified resolution. This generated grid may then be analyzed in order to identify its statistical attributes, its approximation or fit to the input dataset and/or other information available for validation of its quality. It may then be mapped in a variety of forms, e.g. contour, filled contour, shaded relief, wireframe, perspective 3D surface. Grids may also be generated in a number of other ways: from remote sensing datasets (e.g. hyperspectral images); as a result of conversion of vector datasets (e.g. contour to grid conversion); as representations of mathematical functions (e.g. as a output from a mathematical expression, with or without random or fractal perturbations); or as a result of map algebra, resampling and/or overlay operations.

Grid generation using local interpolation functions, of whatever type, is typically a process based on weighted averages of values at nearby points. The assumption is that each grid cell or intersection is likely to be similar to other values in its neighborhood. Most such models are of the general form:

where zj is the z-value to be estimated for location j, the λi are a set of estimated weights (proportional contributions that sum to 1) and the zi are the known (measured) values at points (xi,yi). Assuming that λi>0 for all i, then

i.e. zj will always lie within the range of observed data values. If one imagines zj being a point on a small hilltop surrounded by observations that are below it such estimators will generate a flat surface in the neighborhood of zj at the height of or slightly below the maximum observed value.

The size and shape of the search neighborhood is usually selectable. The size is taken either as a static value (e.g. a specific radius) or is determined in an adaptive manner (e.g. requiring a minimum number of observations to be used in any calculation). The shape of the neighborhood or search area may be circular, elliptical, or segmented variants of these forms. A number of packages facilitate the inclusion of known breaklines, exclusion zones and/or missing values as part of the gridding process. Most GIS packages provide a wide range of facilities for gridding, many of which are described in Section 6.5.2, Gridding and interpolation methods. Surfer provides amongst the most extensive set of options and offers suggestions as to their selection and use. Recommendations based on those of Surfer and similar GIS-related facilities are summarized in Table 6‑2.

Generating a grid from a (global) function does not require separate interpolation, merely definition of the maximum and minimum values for the x and y components and either the grid increment (size) or number of rows and columns to be generated. An example of such a function might be:


where a, b and c are constants and rnd() is a random number generator providing numbers in the range [‑1,1]. Another example is the Gaussian surface plotted in Figure 6‑23.