Spatial Survey Data Modeling - Sampling Spatial Units for Agricultural Surveys - page 308

Spatial interpolation methods include any formal techniques for studying entities
using their topological, geometrical, or geographical properties. In geographic and
quantitative mapping, typical applications of spatial interpolation methods include
the construction of maps with contour lines, and isopleth maps (Tobler 1979 ).
Generally speaking, it is very important to examine the nature and main char-
acteristics of the different spatial interpolation methods, to identify the most
appropriate techniques for solving practical problems.
Spatial agricultural data can be generally collected as discrete points or areal
data (see Sect. 1.4.1 ) . As a consequence, spatial interpolation methods can be
classified according to the type of data involved:

￿ Point interpolation methods, where data are collected in each locality of the area
(i.e., points).
￿ Areal interpolation methods, where the values refer to an entire area of a
territorial partition object.

Point interpolation methods can be further divided into exact and approximate
methods, according to whether they preserve the original sample point values. Point
interpolation deals with data that can be collected at a point.
Several algorithms have been developed. Exact methods include interpolating
polynomials, most distance-weighting methods, kriging, spline interpolation, and
finite difference methods. Approximate methods include power-series trend
models, distance-weighted least squares, and least squares fitting with splines
(Lam 1983 ).
Kriging can be considered as synonymous with an optimal prediction. It is a
method of interpolation that predicts unknown values from data observed at known
locations. This method uses a semivariogram to express spatial variations, and
minimizes the error of predicted values that are estimated using their spatial
distributions.
As described in Sect. 1.4.3.1 , kriging computes the BLUP yˆ (z 0 ). It uses a
stochastic model of the spatial dependence defined by the expectation

(z) and
covariance function C (h) of a random field (Schabenberger and Gotway 2005 ). This
spatial prediction method is one possible example of a model-based approach to
spatial units sampling. This noteworthy interpretation of kriging has not been
studied in the literature. It represents another element that connects sampling theory
with spatial statistics, which is the main aim of this topic. We will leave the
development of this topic to an interested researcher.
Areal interpolation is the process of estimating the values of variables in a set of
target polygons, using known values from a set of sampled polygons (Goodchild
and Lam 1980 ).
Note that areal interpolation techniques are typically applied to data conversion
between different areal systems. Using the definition of Ford ( 1976 ), the geographic
areas for which data are available are called source areas, while those that we wish
to estimate are the target zones. In this case, the data related to the variable of
interest Y i are known for a set of areas, which constitute the source partition S .
However, they are unknown at the level of the zones that cover the target partition

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