UC Santa Barbara
Core Curriculum-Geographic Information Systems (1990)
Title
Unit 41 - Spatial Interpolation II
Permalink
https://escholarship.org/uc/item/32z017q8
Authors
Unit 41, CC in GIS
Waters, Nigel M.
Publication Date
1990
Peer reviewed
eScholarship.org
Powered by the California Digital Library
University of California
UNIT 41 - SPATIAL INTERPOLATION II
UNIT 41 - SPATIAL INTERPOLATION II
Compiled with assistance from Nigel M. Waters, University of Calgary
A. INTRODUCTION
B. AREAL INTERPOLATION - NON-VOLUME PRESERVING
Procedure
C. AREAL INTERPOLATION - VOLUME-PRESERVING
1. Overlay
2. Pycnophylactic
Boundary conditions
D. SPECIAL CASES OF SPATIAL INTERPOLATION
1. Mapping populated areas
2. Estimating trade areas
E. A GIS PERSPECTIVE ON INTERPOLATION
Expert systems for spatial interpolation algorithms
Conclusion
REFERENCES
DISCUSSION AND EXAM QUESTIONS
NOTES
UNIT 41 - SPATIAL INTERPOLATION II
Compiled with assistance from Nigel M. Waters, University of Calgary
A. INTRODUCTION
this unit continues the examination of spatial interpolation by looking at areal
interpolation techniques and some applications
areal interpolation is the problem of transferring data from one set of areas (source
reporting zones) to another (target reporting zones)
this is easy if the target set is an aggregation of the source set, but more difficult if
the boundaries of the target set are independent of the source set
NCGIA Core Curriculum in GIS - 1990
Page 1
Unit 41 - Spatial Interpolation II
later we look at applications that do not fall easily into either point or areal interpolation
categories
B. AREAL INTERPOLATION - NON-VOLUME PRESERVING
e.g. interpolating population counts from census tracts to school districts
Procedure
overhead - Non-volume preserving areal interpolation
calculate the population density for each source census tract by dividing population by
area
identify a centroid for each region
assign to the point located at each centroid, the population density value
determined for its enclosing area
using this set of points, interpolate a gridded population density surface using any of the
methods described previously
convert each grid cell's value to a population by multiplying the estimated density by the
cell's area
overlay the interpolated grid on the target map and assign each grid value to each its
target region (school district)
calculate the total population in each target region
this method is criticized because:
choosing the center point is ill-defined
inadequacy of point based interpolation methods
most importantly, the total value of each zone is not conserved
e.g. if a source zone is divided into two target zones, the total population of
the target zones after interpolation need not equal the population of the
source zone
C. AREAL INTERPOLATION - VOLUME-PRESERVING
1. Overlay
discussed by MacDougall (1976) and Goodchild and Lam (1980)
procedure involves:
overlay of target and source zones
determining the proportion of each source zone that is assigned to each target
zone
apportioning the total value of the attribute for each source zone to target zones
NCGIA Core Curriculum in GIS - 1990
Page 2
Unit 41 - Spatial Interpolation II
according to the areal proportions
assumes uniform density of the attribute within each zone
e.g. uniform population density if the attribute is total zone population
2. Pycnophylactic
see Tobler (1979) for the original algorithm
the technique has two objectives: 1. create a smooth surface, no steps
attribute values should not change suddenly at zone boundaries 2. the total
value of the attribute within each zone must be correct
procedure: 1. overlay a dense raster on a choropleth map 2. divide each zone's total
value equally among the raster cells that overlap the zone
3. smooth the values by replacing each cell's value with the average of its neighbors 4.
sum the values of the cells in each zone 5. adjust the values of all cells within each zone
proportionally so that the zone's total is the same as the original total
e.g., if the total is 10% low, increase the value of each cell by 10% 6. repeat
steps 3, 4 and 5 until no more changes occur
does not require an assumption of homogeneity within zones but rapid variation within
zones may affect the quality of interpolation
output is a contour or continuously shaded map
Boundary conditions
at the boundary of the reporting zones, pixels will have neighbors outside the study area
and therefore without values
some decision must be made about the behavior of the surface outside the study
area
e.g. population density equals zero (a lake or rural area)
e.g. population density unknown, assumed equal to the values of the
outermost pixels of the study area
D. SPECIAL CASES OF SPATIAL INTERPOLATION
1. Mapping populated areas
objective is to create a map showing "populated areas", given point population values
for a number of cities and towns
this problem arises frequently when populated areas are represented as points
it arises for small reporting zones when boundary files are unavailable, but data includes
centroid locations e.g. US or UK census data
NCGIA Core Curriculum in GIS - 1990
Page 3
Unit 41 - Spatial Interpolation II
are several methods that could be used
a simple approach would be to estimate the populated area using an empirical
relationship like:
A is proportional to p0.84
and draw a circle around the point, of radius:
/ (A / p)
Bracken and Martin (1989) have developed methods for replacing ED centroids by
disks, the radius of each disk being estimated from the distances to neighboring
centroids
the method works very well with UK ED data
an alternative approach might proceed as follows:
establish a critical population density for defining an urban area
spread the population over each urban area so that population density is highest in
the center and decreases gradually outwards
e.g. use a normal distribution function
interpolate densities to a raster, accumulating values where the population spread
from two urban areas overlap
draw contours at the critical value to define the boundaries of the populated areas
both of these methods fall within the general heading of density estimation
a density is being estimated from a collection of points
see Silverman
2. Estimating trade areas
in marketing, it is often desirable to plot the boundary of a trade area for e.g. a store,
given information on the home locations of customers
simplest case is when the location of all customers and non-customers is known
simply draw a boundary contour between them
if the location of non-customers is not known: 1. calculate the average distance to all
customers and draw a circle or 2. divide the area into sectors, average the distance to
customers within the sectors and draw a distance arc for each sector (see Huff and
Batsell, 1977)
these techniques do not pick up islands or holes in the trade area
or 3. give each customer a small probability surface
accumulate values as in the populated areas example
NCGIA Core Curriculum in GIS - 1990
Page 4
Unit 41 - Spatial Interpolation II
set critical value for delimiting trade area
E. A GIS PERSPECTIVE ON INTERPOLATION
both point and areal interpolation try to estimate a continuous surface
in the point case, the surface has been measured at sample points
in the areal case, the surface of population density is estimated from total
population counts in each reporting zone
in other cases it is impossible to conceive of a continuous surface
e.g. if each point is a city and the attribute is city population
if city A has population 1 million and city B 100 km away has population 2
million, there is no reason to believe in the existence of a city half way
between A and B with population 1.5 million
in this case, the variable population exists only at the points, not as a continuous
surface
in other cases the variable might exist only along lines e.g. traffic density on a
street network
we must distinguish here between layer and object views of the world
a continuous surface of elevations is a layer view of the world - there is one value
of elevation at an infinite number of possible places in the space
the point map of cities is an object view of the world - the space in between points
is empty, and has no value of the population variable
the street map is an object view of the world - the world is empty except where
there are streets - only along streets is traffic density defined
spatial interpolation implies a layer view of the world, and it requires special techniques
(e.g. density estimation) to apply it to objects such as store customers
Expert systems for spatial interpolation algorithms
a good GIS should include a range of spatial interpolation routines so that the user can
choose the most appropriate method for the data and the task
ideally, these routines should provide a natural language interface which would lead the
user through an appropriate series of questions about the intentions, goals and aims of
the user and about the nature of the data
a number of prototype expert systems for guiding the choice of a spatial interpolation
algorithm have been developed
these may be written in the form of:
an expert system shell (Waters, 1988)
in one of the artificial intelligence languages such as Prolog or LISP (see DuttonMarion, 1988)
or in a high level language such as Pascal (Maslyn, 1987)
Conclusion
NCGIA Core Curriculum in GIS - 1990
Page 5
Unit 41 - Spatial Interpolation II
if computer contouring and surface generation techniques are to be incorporated
successfully into GIS, they must be easy to use and effective
"easy to use" implies that those without a detailed knowledge of the mathematical
and statistical characteristics of the procedure should be able to choose the
correct technique for displaying a particular data set for a particular purpose
note: statisticians argue that this is not an ideal goal as people may use
techniques without a proper understanding of the underlying assumptions
"effective" means that these techniques should be informative, highlighting the
essential nature of the data and/or surface and serving the purpose of the
researcher/analyst
the researcher's measure of success will be largely subjective and visual does the result look right?
this purpose may vary from an attempt to model all the "real" intricacies of the surface
to simply trying to highlight the general, spatial trend of the data in order to aid in the
decision-making process
REFERENCES
Bracken, I. and D. Martin, 1989. "The generation of spatial population distributions from
census centroid data," Environment and Planning A 21:537-44.
Dutton-Marion, K.E., 1988. Principles of Interpolation Procedures in the Display and
Analysis of Spatial Data: A Comparative Analysis of Conceptual and Computer Contouring,
unpublished Ph.D. Thesis, Department of Geography, University of Calgary, Calgary,
Alberta.
Goodchild, M.F., and Lam, N., 1980. "Areal Interpolation: A Variant of the Traditional
Spatial Problem," Geo- Processing 1: 297-312.
Huff, D.L. and R.R. Batsell, 1977. "Delimiting the areal extent of a market area," Journal of
Marketing Research 14:581-5.
MacDougall, E.B., 1976. Computer Programming for Spatial Problems, Arnold, London.
Maslyn, R.M., 1987. "Gridding Advisor: An Expert System for Selecting Gridding
Algorithms," Geobyte 2(4):42-43.
Silverman, B.W., 1986. Density Estimation for Statistics and Data Analysis, Chapman and
Hall, London.
Tobler, W.R., 1979. "Smooth pycnophylactic interpolation for geographical regions," Journal
of the American Statistical Association 74:519-30.
Waters, N.M., 1988. "Expert Systems and Systems of Experts," Chapter 12 in W.J. Coffey,
ed., Geographical Systems and Systems of Geography: Essays in Honour of William Warntz,
Department of Geography, University of Western Ontario, London, Ontario.
NCGIA Core Curriculum in GIS - 1990
Page 6
Unit 41 - Spatial Interpolation II
DISCUSSION AND EXAM QUESTIONS
1. What are the main considerations to be aware of in computer contouring? What are the key
aspects for the design of an expert system to aid in choosing a computer contouring algorithm
within a GIS? How long do you think it will be before such expert systems become widely
available?
2. Describe how Tobler's pycnophylactic method differs from volume-preserving overlay.
What model of the underlying spatial distribution is assumed by each? Give examples of
phenomena and application which fit each method's assumptions.
3. Describe the application of areal interpolation in political districting.
4. One test of a spatial interpolation method is that its results would be judged as equal or
better to hand contouring by a specialist, e.g. a field geologist, with detailed knowledge of the
phenomenon being mapped. How well do the methods discussed in these two units do against
this criterion?
Last Updated: August 30, 1997.
NCGIA Core Curriculum in GIS - 1990
Page 7
Unit 41 - Spatial Interpolation II
UNIT 41 IMAGES
NCGIA Core Curriculum in GIS - 1990
Page 8