Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 148

Spatial units are also artificially defined over a domain partitioned into a number
of predetermined regularly, or irregularly, shaped sets of spatial objects. This may
happen, for example, when the original data generating process involves a contin-
uous spatial domain. Then, to simplify the problem, the researcher chooses to only
observe it over a possibly random selection of fixed points, or averaged over a
selection of predefined polygons (see Chap. 5 ).
In the remainder of this chapter, we will only focus our attention on this last
perspective, leading to the traditional sampling definition for finite populations.
However, it is useful to highlight that this approach covers an important, but not
necessarily exhaustive, part of all the possible sampling problems in natural
resource monitoring and estimation. In fact, there is a huge list of phenomena
that can be observed as a linear object (e.g., a river) or as a surface (e.g., meteo-
rological data). In these cases, the resulting sample is a set of points or polygons that
have possible positions chosen from an infinite set of possible sites, but which are
not predefined.
Another reason that we only consider finite populations is that the spatial
distribution of the population is a strong constraint that, we suspect, could have a
considerable impact on the performance of a random sampling method. For exam-
ple, the traditional solution for extending systematic sampling to multidimensional
data overlaying a grid of points onto a spatial domain is not reasonable if the
population cannot be considered distributed on a regular grid. This may be the case
if it is clustered, or if the units have different intensities over the domain.
To formalize the problem, let U ¼

f g be a finite population recorded
on a frame together with a set of q auxiliary variables X ¼

1 ; 2 ; ...;

x q ,
and a set of h coordinates obtained by the geo-coding of each unit
C ¼

x 1 ; x 2 ; ...; x j ; ...;

:

c 1 ; c 2 ; ...; c j ; ...; c h Here, the generic j- th auxiliary is

c Nj is the generic
j- th coordinate. From C we can always derive, according to any distance definition,
a matrixD U ¼

, and c j ¼

x j ¼ x 1 j ; x 2 j ; ...; x ij ; ...; x Nj c 1 j ; c 2 j ; ...; c ij ; ...;

f d kl ; k ¼ 1, ... , N , l ¼ 1, ... , N g that specifies the distance between

all the pairs of units in the population.
Typically, X and C play different roles in agricultural surveys, according to the
definition of the statistical unit:

1. When U is a list of agricultural households, C is rarely obtainable because it
depends on the availability of accurate cadastral maps. It should consist of a map
of polygons representing parcels of land used by each holding. X is usually filled
by administrative data sources, previous census data and, if C is available,
remotely sensed data obtained by overlaying the polygon map with a classified
image.
2. If U is a list of regularly (or irregularly) shaped polygons defined ad hoc for the
purpose of the agricultural survey, C is always available because it represents the
definition of each statistical unit. X can only consist of some geographical
coding and summaries of classifications that arise from the remotely sensed
data within each polygon, unless an overlay of C with a cadaster is possible.