Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 154

most interest is the ordering of the population. Surprisingly, this has been arranged
using some simplistic circular or linear one-dimensional ordering, without taking
advantage of developments in spatial statistics (see Sect. 1.4 ).
Attractive approaches that appear similar to that described in Sect. 7.6 are the
distance balanced sampling plans (DBSPs). In these methods, the

π k s are constant

π kl s depend on a non-decreasing function of the
distance between k and l (Mandal et al. 2009 ).

for all the population units, and the

7.4 Generalized Random Tessellation Sampling

An intuitive way to produce samples that are well-spread over the population is to
stratify the units of the population on the basis of their location. This technique is
widely used by practitioners. Problems arise when using this strategy because it
does not have a direct and substantial impact on the second-order inclusion prob-
abilities (particularly not within a given stratum), and a good partition of the study
area is frequently not obvious.
These drawbacks are in some way related, and for this reason they are usually
addressed together by defining a maximal stratification (see Sect. 6.5 ). In other
words, the study area is partitioned into as many strata as possible, and we select
one or two units per stratum. However, this simple and quick scheme for guaranty-
ing that the sample is well-spread over the population is somewhat arbitrary,
because it highly depends on the stratification criterion, which should be general
and efficient.
Another widely used basic option is to try to extend the use of systematic
sampling to two or more dimensions (Das 1950 ), overlapping a regular grid onto
the spatial population. The underlying idea is that it is always possible to collect the
units of the population by selecting them from a regular grid in a very similar way to
an indirect sample design (see Sect. 10.6 ). The concern is that it is very difficult to
obtain a design with the desired features within an indirect sampling framework
because the first-order inclusion probabilities are often unknown, making the
estimation process unfeasible.
However, these practices encouraged the development of the generalized ran-
dom tessellation stratified (GRTS) design (Stevens and Olsen 2004 ). It systemati-
cally selects the units, and maps the two-dimensional spatial population into one
dimension while trying to preserve some multi-dimensional order.
The preliminary developments that led to the GRTS design were made by
Stevens ( 1997 ) and by Stevens and Olsen ( 1999 ). They studied the properties of
several grid-based designs that were extensions of the random tessellation stratified
(RTS) design (Dalenius et al. 1961 ; Olea 1984 ; Overton and Stehman 1993 ).
The RTS design randomly selects from a spatial point frame using a two-step
procedure. First, a regular tessellation coherent with a regular grid is randomly
located over the domain to be sampled. Secondly, a random point is selected within
each random tessellation cell. The RTS design is a variation of the systematic