A. INTRODUCTION

B. CLASSIFICATION OF INTERPOLATION PROCEDURES

C. POINT BASED INTERPOLATION - POPULAR METHODS

spatial interpolation is the procedure of estimating the value of properties at unsampled sites within the area covered by existing observations

in almost all cases the property must be interval or ratio scaled

can be thought of as the reverse of the process used to select the few points from a DEM which accurately represent the surface

rationale behind spatial interpolation is the observation that points close together in space are more likely to have similar values than points far apart (Tobler's Law of Geography)

spatial interpolation is a very important feature of many GISs

spatial interpolation may be used in GISs:

to provide contours for displaying data graphically

to calculate some property of the surface at a given point

to change the unit of comparison when using different data structures in different layers

frequently is used as an aid in the spatial decision making process both in physical and human geography and in related disciplines such as mineral prospecting and hydrocarbon exploration

many of the techniques of spatial interpolation are two-dimensional developments of the one-dimensional methods originally developed for time series analysis

this lecture introduces spatial interpolation and examines point based interpolation, while the next looks at areal procedures and some applications

surfaces are ways of conceiving of a wide range of phenomena

topographic elevation

variation in temperature

variation in population density

surfaces are instances of fields

one value of a variable at all points in space

surfaces are thought of as "2.5 D"

elevation is a function of elevation

continuity properties

do the values at x and x+dx converge as dx vanishes?

are there any cliffs?

no cliffs means zero-order continuity

do the slopes at x and x+dx converge as dx vanishes?

are there sharp breaks of slope?

no breaks of slope means first-order continuity

what about curvatures?

second-order continuity

invertibility

does the surface look similar the other way up?

are pits and peaks equally frequent?

self-similarity

does the surface look the same at all scales?

can you tell the scale by looking at the surface?

spatial autocorrelation

is the surface smooth?

white noise

topological properties

peaks, pits, passes

geomorphological properties

fluvial, glacial, lacustrine

there are several different ways to classify spatial interpolation procedures:

point-based

given a number of points whose locations and values are known, determine the values of other points at predetermined locations

point interpolation is used for data which can be collected at point locations e.g. weather station readings, spot heights, oil well readings, porosity measurements

interpolated grid points are often used as the data input to computer contouring algorithms

once the grid of points has been determined, isolines (e.g. contours) can be threaded between them using a linear interpolation on the straight line between each pair of grid points

point to point interpolation is the most frequently performed type of spatial interpolation in GIS

lines to points

e.g. contours to elevation grids

areal interpolation

given a set of data mapped on one set of source zones determine the values of the data for a different set of target zones

e.g. given population counts for census tracts, estimate populations for electoral districts

global interpolators determine a single function which is mapped across the whole region

a change in one input value affects the entire map

local interpolators apply an algorithm repeatedly to a small portion of the total set of points

a change in an input value only affects the result within the window

global algorithms tend to produce smoother surfaces with less abrupt changes

are used when there is an hypothesis about the form of the surface, e.g. a trend

some local interpolators may be extended to include a large proportion of the data points in the set, thus making them in a sense global

the distinction between global and local interpolators is thus a continuum and not a dichotomy

this has led to some confusion and controversy in the literature

exact interpolators honor the data points upon which the interpolation is based

the surface passes through all points whose values are known

honoring data points is seen as an important feature in many applications e.g. the oil industry

proximal interpolators, B-splines, and Kriging methods all honor the given data points

approximate interpolators are used when there is some uncertainty about the given surface values

this utilizes the belief that in many data sets there are global trends, which vary slowly, overlain by local fluctuations, which vary rapidly and produce uncertainty (error) in the recorded values

the effect of smoothing will therefore be to reduce the effects of error on the resulting surface

stochastic methods incorporate the concept of randomness

the interpolated surface is conceptualized as one of many that might have been observed, all of which could have produced the known data points

stochastic interpolators include trend surface analysis, Fourier analysis, and Kriging

procedures such as trend surface analysis allow the statistical significance of the surface and uncertainty of the predicted values to be calculated

deterministic methods do not use probability theory (e.g. proximal)

a typical example of a gradual interpolater is the distance weighted moving average

usually produces an interpolated surface with gradual changes

however, if the number of points used in the moving average is reduced, there would be abrupt changes in the surface

it may be necessary to include barriers in the interpolation process

semipermeable, e.g. weather fronts

will produce quickly changing but continuous values

impermeable barriers, e.g. geologic faults

will produce abrupt changes

sometimes interpolation can be aided by some additional variable

elevation helps in interpolating ground temperature, or pressure

land use might help in interpolating population density

several methods accept covariates

e.g. co-Kriging

Lam (1983) and Burrough (1986) describe a variety of quantitative interpolation methods suitable for computer contouring algorithms

all values are assumed to be equal to the nearest known point

is a local interpolator

computing load is relatively light

output data structure is Thiessen polygons with abrupt changes at boundaries

has ecological applications such as territories and influence zones

best for nominal data although originally used by Thiessen for computing areal estimates from rainfall data

is absolutely robust, always produces a result, but has no "intelligence" about the system being analyzed

available in very few mapping packages, SYMAP was a notable exception

uses a piecewise polynomial to provide a series of patches resulting in a surface that has continuous first and second derivatives

ensures continuity in:

elevation (zero-order continuity) - surface has no cliffs

slope (first-order continuity) - slopes do not change abruptly, there are no kinks in contours

curvature (second order continuity) - minimum curvature is achieved

produces a continuous surface with minimum curvature

output data structure is points on a raster

note that maxima and minima do not necessarily occur at the data points

is a local interpolator

can be exact or used to smooth surfaces

computing load is moderate

best for very smooth surfaces

poor for surfaces which show marked fluctuations, this can cause wild oscillations in the spline

are popular in general surface interpolation packages but are not common in GISs

can be approximated by smoothing contours drawn through a TIN model

see Burrough (1986), Davis (1986) and mathematical aspects in Lam (1983) and Hearn and Baker (1986)

also described in "numerical approximation theory"

estimates are averages of the values at n known points:

 
z = Sum over i (w_iz_i) / Sum over i (w_i)

where w is some function of distance, such as:

w = 1/d^k

w = e^-kd

an almost infinite variety of algorithms may be used, variations include:

 
the nature of the distance function

varying the number of points used

the direction from which they are selected


is the most widely used method

objections to this method arise from the fact that the range of interpolated values is limited by the range of the data

 
no interpolated value will be outside the observed range of z values

peaks and pits will be missed if they are not sampled

outside the area sampled the surface must flatten to the average value


other problems include:

 
how many points should be included in the averaging?

what to do about irregularly spaced points?

how to deal with edge effects?

developed by Georges Matheron, as the "theory of regionalized variables", and D.G. Krige as an optimal method of interpolation for use in the mining industry

the basis of this technique is the rate at which the variance between points changes over space

this is expressed in the variogram which shows how the average difference between values at points changes with distance between points

Kriging is based on an analysis of the data, then an application of the results of this analysis to interpolation

vertical axis is E(z_i - z_j)², i.e. "expectation" of the difference

i.e. the average difference in elevation of any two points distance d apart

d (horizontal axis) is distance between i and j

most variograms show behavior like the diagram

the upper limit (asymptote) is called the sill

the distance at which this limit is reached is called the range

the intersection with the y axis is called the nugget

a non-zero nugget indicates that repeated measurements at the same point yield different values

in developing the variogram it is necessary to make some assumptions about the nature of the observed variation on the surface:

simple Kriging assumes that the surface has a constant mean, no underlying trend and that all variation is statistical

universal Kriging assumes that there is a deterministic trend in the surface that underlies the statistical variation

in either case, once trends have been accounted for (or assumed not to exist), all other variation is assumed to be a function of distance

the input data for Kriging is usually an irregularly spaced sample of points

to compute a variogram we need to determine how variance increases with distance

begin by dividing the range of distance into a set of discrete intervals, e.g. 10 intervals between distance 0 and the maximum distance in the study area

for every pair of points, compute distance and the squared difference in z values

assign each pair to one of the distance ranges, and accumulate total variance in each range

after every pair has been used (or a sample of pairs in a large dataset) compute the average variance in each distance range

plot this value at the midpoint distance of each range

fit one of a standard set of curve shapes to the points

"model" the variogram

once the variogram has been developed, it is used to estimate distance weights for interpolation

interpolated values are the sum of the weighted values of some number of known points where weights depend on the distance between the interpolated and known points

weights are selected so that the estimates are:

unbiased (if used repeatedly, Kriging would give the correct result on average)

minimum variance (variation between repeated estimates is minimum)

problems with this method:

when the number of data points is large this technique is computationally very intensive

the estimation of the variogram is not simple, no one technique is best

since there are several crucial assumptions that must be made about the statistical nature of the variation, results from this technique can never be absolute

simple Kriging routines are available in the Surface II package (Kansas Geological Survey) and Surfer (Golden Software), and in the GEOEAS package for the PC developed by the US Environmental Protection Agency, and in ArcInfo 8 as an add-on

surface is approximated by a polynomial function

output data structure is a polynomial function which can be used to estimate values of grid points on a raster or the value at any location

the elevation z at any point (x,y) on the surface is given by an equation in powers of x and y

e.g. a linear equation (degree 1) describes a tilted plane surface:

z = a + bx + cy

e.g. a quadratic equation (degree 2) describes a simple hill or valley:

z = a + bx + cy + dx² + exy + fy²


in general, any cross-section of a surface of degree n can have at most n-1 alternating maxima and minima

 
e.g. a cubic surface can have one maximum and one minimum in any cross-section

equation for the cubic surface:

z = a + bx + cy + dx² + exy + fy² + gx³ + hx²y + ixy² + jy³


a trend surface is a global interpolator

 
assumes the general trend of the surface is independent of random errors found at each sampled point


computing load is relatively light

problems

 
statistical assumptions of the model are rarely met in practice

edge effects may be severe

a polynomial model produces a rounded surface

 
this is rarely the case in many human and physical applications


available in a great many mapping packages

 
see Davis (1973) and Sampson (1978) for non- orthogonal polynomials; Mather (1976) for orthogonal polynomials

Burrough, P.A., 1986. Principles of Geographical Information Systems for Land Resources Assessment, Clarendon, Oxford. See Chapter 8.

Davis, J.C., 1986. Statistics and Data Analysis in Geology, 2nd edition, Wiley, New York. (Also see the first, 1973, edition for program listings.)

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.

Hearn, D., and Baker, M.P., 1986. Computer Graphics, Prentice-Hall Inc, Englewood Cliffs, N.J.

Jones, T.A., Hamilton, D.E. and Johnson, C.R., 1986. Contouring Geologic Surfaces with the Computer, Van Nostrand Reinhold, New York

Lam, N., 1983. "Spatial Interpolation Methods: A Review," The American Cartographer 10(2):129-149.

Mather, P.M., 1976. Computational Methods of Multivariate Analysis in Physical Geography, Wigley, New York.

Sampson, R.J., 1978. Surface II, revised edition, Kansas Geological Survey, Lawrence, Kansas.

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.

1. Are there other techniques for surface generation? How many of the above procedures are commonly used? How would they be ranked in terms of popularity? Give examples from the literature of where they have been used.

2. How does hand contouring rate as an alternative? What did you think of it and have you changed your mind? What are the key features and processes involved in hand contouring?

3. Explain the advantages and disadvantages of manual interpolation as used in hand contouring over computer based interpolation as used in a computer contouring package.

4. Describe the different ways in which spatial interpolation algorithms can be classified.