Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 155

design that avoids alignment problems that can often occur with a perfectly aligned
systematic design.
Stevens ( 1997 ) introduced the multiple density nested random tessellation strat-
ified (MD-NRTS) design to provide for non-constant spatial sampling intensity.
The geometric concept underlying the MD-NRTS design is the notion of coherent
intensification of a grid. That is, adding points to a regular grid to produce a finer
regular grid with similarly shaped, but smaller, tessellation cells.
Stevens and Olsen ( 2004 ) applied the same notion to the GRTS method by
extending the idea to a procedure that can potentially create an infinite series of
nested and coherent grids. When this hierarchical spatial coding process is juxta-
posed with a random ordering within each cell, it results in a function f that maps a
two-dimensional space onto a one-dimensional space and preserves some spatial
order.
Thus, at the heart of GRTS is a function f that maps the unit square onto the unit
interval. The function must preserve some proximity relationships, which implies
some additional restrictions on the class of functions to be considered.
When studying discrete two-to-one dimensional maps, Mark ( 1990 ) defined a
class of orderings called quadrant-recursive. In this class, when recursively
decomposing a rectangular region into sub-quadrants, the points of any
sub-quadrant always appear consecutively in the quadrant-recursive ordering. Dur-
ing the successive intensification of a grid a cell is divided into four sub-cells, each
of which is subsequently divided into four sub-sub-cells, and so on. In the final step,
the points have an address based on the order in which the divisions were carried
out, where each digit of the address represents a subdivision step. This spatially
referenced address induces a linear ordering of the sub-quadrants, with the property
that all successor cells of a cell have consecutive addresses.
This class of orderings constitutes the basis for the definition of GRTS. In this
context, the sample is selected in one dimension using systematic

π ps sampling (see

Sect. 6.4 ) and then mapped back into two dimensions.
Quadrant-recursive creates a line and systematic sampling with a random start. It
results in an equal probability sample that will be well-spread over the study area.
Unequal probability sampling is implemented by assigning each point a length
proportional to its inclusion probability. See Fig. 7.1 for an example of this
procedure.
The GRTS technique samples an area as follows:

1. The sampling units are assigned to an order according to a recursive, hierarchical
randomization process. This process preserves the spatial relationships of the
sample units.
2. The sampling units are arranged in order by creating a function that maps a
two-dimensional space to a one-dimensional space, so that it is defined an
ordered spatial address.
3. The one-dimensional space of sampling units is divided into a number of equal-
length segments depending on the requested sample size. A unit from each
segment is randomly selected.