32 ArcUser July–Se ptember 2004 www.esri.com
By Colin Childs, ESRI Education Services
Editorʼs note: In addition to supplying tools for spatial analysis (i.e.,
for modeling suitability, distance, or hydrology), the ArcGIS 9 Spatial
Analyst extension provides tools for spatial data analysis that apply sta-
tistical theory and techniques to the modeling of spatially referenced data.
Elevation, temperature, and contamination concentrations are types of
data that can be represented by surfaces. Each raster cell represents a mea-
surement such as a cellʼs relationship to a fixed point or specific concentra-
tion level. Because obtaining values for each cell in a raster is typically
not practical, sample points are used to derive the intervening values using
the interpolation tools in ArcGIS Spatial Analyst. This article provides an
introduction to the interpolation methods used by these tools.
GIS is all about spatial data and the tools for managing, compiling,
and analyzing that data. ArcGIS Spatial Analyst extension provides a
toolset for analyzing and modeling spatial data. A set of sample points
representing changes in landscape, population, or environment can be
used to visualize the continuity and variability of observed data across a
surface through the use of interpolation tools. These changes can be ex-
trapolated across geographic space. The morphology and characteristics
of these changes can be described. The ability to create surfaces from
sample data makes interpolation both powerful and useful.
Understanding Surfaces
Before discussing different interpolation techniques, the differences in
the methods used for surface representation need to be discussed. Each
representation is useful for specific situations. This discussion concen-
trates on the creation of a surface in the native ArcGIS grid format.
A grid representation of a surface is considered to be a functional sur-
face because for any given x,y location, it stores only a single Z value as
opposed to multiple Z values. Functional surfaces are continuous because
an x,y location has one and only one Z value regardless of the direction
from which the x,y point is approached. Functional surfaces are 2.5-di-
mensional surfaces, not true three-dimensional surfaces.
Functional surfaces can be used to represent terrestrial surfaces that
depict the earthʼs surface, statistical surfaces that describe demographic
and other types of data, and mathematical surfaces that are based on arith-
metic expressions. Surface representation in its simplest form is done
by storing x,y and Z values that define the location of a sample and the
change characteristic represented by the Z value.
Contours or isolines are used to define a common characteristic along
a line. Technically, contours join locations of equal value to each other.
In the case of a contour line representing height, it is a line drawn on a
map that connects points of equal elevation above a datum that usually
represents mean sea level.
A triangulated irregular network (TIN) is a vector data structure used
to store and display surface models. A TIN partitions geographic space
using a set of irregularly spaced data points, each of which has x-, y-,
and Z-values. These points are connected by edges that form contiguous,
nonoverlapping triangles and create a continuous surface that represents
the terrain.
A grid is a spatial data structure that defines space as an array of cells
of equal size that are arranged in rows and columns. In the case of a grid
that represents a surface, each cell contains an attribute value that repre-
sents a change in Z value. The location of the cell in geographic space is
obtained from its position relative to the gridʼs origin.
Interpolation Methods
To create a surface grid in ArcGIS, the Spatial Analyst extension employs
one of several interpolation tools. Interpolation is a procedure used to
predict the values of cells at locations that lack sampled points. It is based
on the principle of spatial autocorrelation or spatial dependence, which
measures degree of relationship/dependence between near and distant
objects.
Spatial autocorrelation determines if values are interrelated. If values
are interrelated, it determines if there is a spatial pattern. This correlation
is used to measure
n Similarity of objects within an area
n The degree to which a spatial phenomenon is correlated to itself in
space
n The level of interdependence between the variables
n Nature and strength of the interdependence
Different interpolation methods will almost always produce different results.
Points
Methods of Surface Representation
Contours
TINs Grids
Interpolation is the proce-
dure used to predict cell
values for locations that
lack sampled points.
Interpolating Surfaces
in ArcGIS Spatial Analyst
1.1 1.2 1.3 1.5 1.6 1.7 1.8 21
Interpolated Rainfall
Known Rainfall Values
1 Mile
0 0.2 0.5 0.7 1
