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5. SPATIAL INTERPOLATION
5.1 Introduction
Spatial interpolation is the procedure for estimating the value of
properties at unsampled sites within an area covered by existing
observations (Waters, 1989). A wide variety of spatial interpolation
techniques exist. You may already be familiar with some of these
techniques, for example, contouring. Others, such as Theissen
polygons, Delaunay triangles, and spatial moving averages are likely to
be new. Many of the more complex interpolation techniques such as
kriging and Fourier analysis require a comprehensive understanding of
spatial statistics, mathematics and, in some cases, involve the solving
of linked simultaneous equations. Don't panic! The primary objective
of this section is to explain the concepts which underpin the
interpolation process. You will be introduced, in detail, only to those
interpolation techniques which you are likely to find in commercially
available GIS software. Where a more complex technique is
introduced, additional references will be provided to guide you to a
more detailed and practical discussion of the approach. A good
overview to spatial interpolation can be found in Burrough and
McDonnell (1998) and in Lam (1983).
5.2 What do we mean by spatial interpolation?
The definition by Waters (1989), provided in the first sentence of the
section above, is a good starting point from which to explore what is
meant by spatial interpolation. Another definition states the
geographical nature of the problem more explicitly:
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"Given a set of spatial data either in the form of
discrete points or for subareas, find the function
that will best represent the whole surface and that
will predict values at points or for other subareas."
(Lam, 1983)
From a GIS perspective (and for those of you who have never come
across the term before) it is useful to dissect these definitions and
translate them into the language of attributes and entities introduced in
Spatial Data Modelling and DatabaseTheory. Let's assume that you
have a GIS which contains the location of a set of weather stations
represented as a series of point entities. For each weather station a
range of attribute information is recorded on a regular basis. One
attribute observed for each station is the mean daily wind speed. Now,
let us assume that a new airport is planned for the region. You would
already have used a variety of the spatial operations outlined in
Sections 2 and 3 of this module to identify a series of potential sites for
the airport. The question that still remains is - which of these sites has
the most favourable wind conditions? The problem is that none of the
potential airport sites you have selected contain, or are adjacent to, a
weather station. Even if one of the weather stations was bang in the
middle of one of your preferred sites, the wind values available to you
are for a point source and may not be representative of the wind
conditions across the entire site.
With this airport example in mind, let us reconsider the definition of
interpolation given by Waters (1989):
"Spatial interpolation is the procedure of
estimating values of properties...."
Let us replace the word `properties' with `attributes'. In our example
the attribute we are interested in is mean daily wind speed
"at unsampled sites...."
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This indicates that we have a spatial reference which allows us to
locate the position of our airport and also infers that we have a
conceptual spatial model, in this case an area entity which represents
the potential airport site
"within the area...."
This infers that we have a geographical region of interest or study in
which we must perform our interpolation. The term `area' implies that
our area of study will have a set of boundaries. In certain cases these
boundaries may be physical, for example, the coastline of Britain. On
the other hand, our boundaries may be political, such as administrative
boundaries. One of the problems with all interpolation techniques is
what to do about boundaries because although we are rarely concerned
about what happens outside our area of interest we should be
concerned about how the properties of the `outside world' influence
our interpolation process. Unfortunately, in many cases we simply turn
a blind eye. We will revisit the boundary issue in relation to individual
interpolation techniques a little later in this section.
"covered by existing observations."
This implies that we have another set of spatial features (in the case of
our example weather stations) for which we have a known set of
attribute values, here mean daily wind speed.
Therefore, in the language of GIS, the problem is which spatial
interpolation procedures are designed to address how to estimate the
attribute values for one set of spatial entities from those for a second
set of spatial entities (which may or may not be of the same type) for
which a set of observed attribute values exist. Or, in the case of our
example how to make the `best guess' at what the wind speed is likely
to be at each of our potential airport sites.
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If we return to the definition by Lam (1983) two further concepts are
introduced. These are Function and Surface. Lam's statement of the
interpolation problem stresses that before we can estimate the values
for our airport sites we must first derive a spatial surface from our
known data points (weather stations). This surface should be defined
by the nature of the relationship (function) which exists between our
observed data points. In a spatial context this function is usually based
on the observation that:
" points (or spatial entities) close together in space
are more likely to have similar values than points
(entities) far apart"
(Tobler's Law of Geography; Tobler, 1976).
Other spatial relationships can be envisaged, for example repeating
sequences in which similar values are found for points only at regular
intervals and not for points in between. However, the vast majority of
phenomena are thought to conform to Tobler's `law'.
This `law' leaves open a wealth of possibilities for the exact nature of
the function. If near points have similar values, then we need to
consider just how dissimilar they can be and how dissimilarity is a
function of distance. Our functions can be simple or complex, sensitive
or insensitive to distance and other factors. The mapping of these
functions produces surfaces in two dimensions. It is these surfaces
which have to be `fitted' to empirical observations for estimates to be
made at sites where values are unknown. It is worth remembering that
with such surfaces, as with map overlay, there are serious problems
with error in the fitting and estimation processes.
Therefore, to summarise, the spatial interpolation process involves:-
1. Establishing the spatial entity type for which observed attribute data
are available.
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2. Establishing the spatial entity type for which you wish to estimate
attribute values.
3. Establishing the scale of measurement of the observed information.
For example, is it ordinal, ratio etc.? (Refer to Section 3, Box 2 of
this module for a reminder of what these mean.).
4. Determining how much is known about the spatial behaviour of the
observed attribute you are trying to interpolate.
5. Establishing the relationship between the sites for which you have
observed attribute values and selecting an appropriate interpolation
technique. In virtually every case this will result in choosing a
mathematical function which fits a surface to the spatial entities for
which you have data.
6. Extracting from your interpolated surface, if required, the `best
guess value' for an entity of the same type or different type at a new
location where real world data are unavailable.
The interpolation process, described above, has also been summarised
in Figure 5.1 below.
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Figure 5.1 : A conceptual overview of the spatial interpolation process
5.3. Why is spatial interpolation important in GIS?
Spatial interpolation techniques have wide application in the design,
construction and application of a GIS. This is because we are rarely in
the fortunate position of having access to all the data required for every
location in our study area. It is much more likely that we will have
available partial, and very often patchy data. Figure 5.2 presents a
conceptual view of the more common situations in which GIS
designers will find themselves with respect to data availability (the
figure makes reference to a point data set by way of example).
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Figure 5.2 : Four different views of data availability in the real world
The basic role for interpolation in GIS is to fill in the missing data for
those areas where real world observations are not available. In the case
of Figure 5.2 this is the blank spaces or regions between data points.
The important issue about interpolation, from a design perspective, is
that any map layer we create through this process will contain
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information which is only a representation of what we believe to be
true. It is all too easy to forget this as we integrate and overlay data sets
in the GIS environment. Therefore, understanding the quality of a map,
or any data for that matter, derived from a set of GIS operations, relies
on your appreciation of the limitations of the techniques that you have
selected to generate the output data.
The field of spatial interpolation is an area of current research interest.
Many of the techniques that are described below have been adopted
from the one dimensional methods originally developed for the
analysis of time series data. Research is being undertaken into the
methods and problems of transferring these methods to two
dimensional data with different scales, sampling distributions and
temporal properties. At the present time few, if any, of the commercial
GIS in the market place will have even 50% of the techniques
described below as part of their GIS toolbox. Many of these missing
techniques will gradually appear in the next generation of GIS, so keep
your eyes and ears open!
Continuous Self Assessed Exercise 12
(time: 20 minutes)
Choose two examples of GIS applications with which you are familiar.
These could be from your work, examples provided with the Diploma or any
additional reading you may have done. Make a list of all the instances
where you think it has been necessary to use an interpolation technique to
generate a new data layer. Indicate what the original data layer consisted
of (for example, vegetation sample points), what the new data layer
produced was (for example, vegetation zones) and what the entity types
involved were (for example, point to area).
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The list of possible uses of interpolation techniques within GIS is vast.
Waters (1989) provides the following list of potential uses:
z to provide contours for displaying data graphically
z to calculate some property of the surface at a given point
z to change the unit of comparison when using different data
models in different layers
z to aid in the spatial decision making process both in physical and
human geography and in related disciplines such as mineral
prospecting and resource exploration
One of the most common methods used to classify the many different
methods of interpolation is to group them according to the spatial
entities on which they can operate; points, lines or areas. This approach
will be used to structure the discussion which follows.
5.4. Point interpolation methods
Point interpolation methods may be categorised under a number of
different headings:
z local or global
z exact or approximate
z gradual or abrupt
z deterministic or stochastic.
An individual interpolation method will fit several of these categories,
for instance it may be local, exact, abrupt and deterministic (such as
Theissen polygons). Before we can expect you to be able to recognise
what type of interpolation methods you are using, we had better look at
the ideas behind the terminology.
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5.4.1 Local or global
The terms local and global describe how an interpolation technique
uses the observed data collected for a set of geographically spread
sampling points. Global interpolation techniques apply a single
function to all the points in a study area. In such cases, as Waters
(1989) indicates, a change in one observed value would affect the
entire interpolation process. Global interpolators tend to produce a
surface with few abrupt changes, because, in general, they employ the
principle of averaging. This reduces the influence of extreme values.
Such methods are most appropriate when the surface being modelled is
known to have an overall trend (For example, a pollutant may be
known to decrease in occurrence in proportion to distance from a
source).
Where there is little or no knowledge about the overall trend in the
surface you are trying to model then local interpolators are more
appropriate. Local interpolation techniques apply the same function
repeatedly to a small portion of the total set of sample points for which
data have been observed. Then a surface is constructed by linking
these regional observations together.
Figure 5.3 illustrates the distinction between local and global methods
of interpolation. A useful analogy is to think of global interpolators as
a telescope where everything you wish to observe is present in the field
of view and local interpolators as a microscope where you build up a
complete picture by observing each part in turn.
One question debated in the literature is when does a local
interpolation technique cease to be local and become global? This is an
interesting question since, in theory, the neighbourhood of points used
in a local interpolation procedure could be extended to cover the entire
study area. For our purposes we will consider a local interpolator to be
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any technique which makes more than a single pass over the data
before constructing a surface. Examples of local interpolation
techniques we will review in the following sections include Theissen
polygons, Delaunay triangles, moving averages and kriging. Global
interpolators to be considered include line threading, trend surface
analysis and Fourier analysis.
Figure 5.3 : Global and local interpolators
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5.4.2 Exact or approximate
Exact interpolation techniques must honour all the points for which
observed data are available. They are most appropriate when there is a
high degree of certainty attached to measurements. Altitude, for
example, is usually known with a high degree of certainty. You would
not want to use an interpolation technique which did not honour an
observed altitude value but instead replaced it with a smoother
approximation. Approximate interpolators, on the other hand, do not
have to honour all the points for which observed data are available.
These techniques are most appropriate when there is some uncertainty
about the ability to reproduce the observed value at a given sample
point. Figure 5.4 shows a surface which has been fitted to a set of
points using both techniques. Note how the surface produced by the
approximate interpolator does not honour all of the data points.
Figure 5.4 : Two surfaces produced by exact and approximate interpolators
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Examples of exact interpolation techniques discussed later include; line
threading, Theissen polygons and kriging. Examples of approximate
interpolators include spatial moving averages, trend surface analysis
and a variety of the filtering operations used in remote sensing.
5.4.3. Gradual or abrupt
Gradual or abrupt interpolators are distinguished by the continuity of
the surface they produce. Gradual interpolation techniques will
produce a smooth surface with gradual changes occurring between
observed data points. Abrupt interpolators will produce a surface
which is stepped in appearance. An example of an abrupt interpolation
method is buffering which you looked at in an earlier section. Another
technique to be introduced later is Theissen polygons. This defines
discrete territories associated with specific points. Waters (1989)
suggests that it may be necessary to include barriers into a gradual
interpolation process in which case an abrupt interpolator is
appropriate. In terrain modelling, for example, you may wish to
represent the presence of a cliff or in climatic modelling the presence
of a weather front. For such features there has to be a definable step
function which can be used with a gradual surface.
5.4.4. Deterministic or stochastic
The most desirable situation to be in when faced with interpolating
new values from a known data set is where there is sufficient
knowledge about the phenomena to allow its behaviour to be described
by a mathematical function. This is known as deterministic modelling.
Figure 5.5 shows how knowledge of the laws of physics and
measurements of actual velocity would allow the calculation, from
seven sample points, of the trajectory of a bouncing ball.
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Figure 5.5 : A deterministic model of the trajectory of a bouncing ball
In addition, deterministic modelling allows extrapolation beyond the
range of our sample data set. However, this is only possible if the
context of the data values is understood. Unfortunately, few
geographical phenomena are understood in sufficient detail or obey
such precise rules to permit a deterministic approach to interpolation or
extrapolation. There is a great deal of uncertainty about what happens
at unsampled locations. To handle this problem stochastic models are
frequently used. Stochastic models incorporate the concept of
randomness suggesting that the interpolated surface, point line or area
values are only one of an infinite number that might have been
produced from the known data points. Figure 5.6 shows a series of
random trajectories for our bouncing ball, all of which could in theory
occur if the behaviour of our ball was assumed to be random. Kriging,
trend surface analysis and Fourier analysis are all techniques which
assume some degree of random behaviour in the observed data.
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Figure 5.6 : Two possible random trajectories for the bouncing ball
The remainder of this discussion of interpolation techniques will focus
on methods that you are most likely to find in GIS. The discussion will
focus on the interpolation of attribute data associated with point, line
and area features. For the discussion of point interpolation methods, a
clear distinction has been made between exact and approximate
methods.
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Continuous Self Assessed Exercise 13
(time: 30 minutes)
In the above discussion we have introduced a range of concepts, terms
and techniques which we are certain will be new to many of you.
So far we have:-
local interpolator
global interpolator
exact interpolator
approximate interpolator
gradual interpolator
abrupt interpolator
deterministic interpolator
stochastic interpolator
Theissen polygons
kriging
Fourier analysis
trend surface analysis
moving average
line threading
Before you proceed we suggest that you cross off the list those that you
understand. Keep the list close to hand as you proceed through the
following sections and cross off terms and concepts as they are explained.
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5.5 Exact point interpolation methods
5.5.1 Line threading or eye balling
This is the original, and still considered by some to be the best, method
of interpolating a surface from a set of observed data. The process
involves threading a line, by eye, between data points with the
objective of enclosing points of similar value. In most cases, the person
performing the line threading will use their local knowledge about the
geographical area and their scientific knowledge about the phenomena
they are studying to enhance the interpolation process. For example, in
geological mapping, the presence or absence of a fault line may be
known to the geologist but not evident in the point sampled data set.
The geologist may be able to build this expert knowledge into the
interpolation process, where the non-specialist would be unable to do
so.
Three basic limitations of the manual line threading approach have led
to the computerisation of the method. These are:
z the problem of handling a large number of points,
z the subjective nature of the procedure, and
z the time consuming nature of the process.
Many automated line threading methods now exist which perform the
task with varying degrees of intelligence. Over the last few years
knowledge engineering techniques have been used to capture the
knowledge from `local experts' and build expert systems for this type
of interpolation.
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Continuous Self Assessed Exercise 14
(time: 45 minutes)
Use the manual line threading or eye balling technique to interpolate a
surface for the point data set in Figure 5.7. Use an interval of 10 units to
produce a contour map. Don't throw this diagram away as you will be
using it again later. To check how good your own eye balling is compare
your map with the figure under the 'SAE Answers' button. Do not worry if
your interpolated surface does not exactly match ours. The line threading
approach is very subjective and though the main trends of the lines you
have threaded should match with ours it is certain that the exact location of
the lines will differ.
Figure 5.7 : Observed values for a set of irregularly spaced points
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5.5.2 Theissen polygons
Theissen polygons are an exact method of interpolation that assumes
the unknown values of the points on a surface to be equal to the value
of the nearest known point. This can be considered as a local
interpolator because the characteristics of the global data set have no
influence on the interpolation process. The Theissen polygon technique
was originally used for computing areal estimates from rainfall data.
The approach is graphically represented in Figure 5.8 and involves
spreading the territory associated with a point until the boundary
associated with the territory of a neighbouring point is encountered. If
you like you can imagine a bubble being blown outwards from each
point simultaneously until they meet. Where they meet the boundaries
of the interpolated areas are drawn. If the data points are irregularly
spaced a surface made up of a mosaic of irregular polygons will result.
The attribute value of each point is then adopted by its' associated
polygon.
Figure 5.8 : Construction of Theissen polygons for a network of five points
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The method of Theissen polygons is a robust technique and will
always produce the same surface from the same set of data points. This
is a disadvantage in that the technique is unintelligent and unable to
respond to external knowledge about factors which may influence the
values recorded at the observed data points. One of the most common
uses for this technique is for the generation of area territories from a set
of points. For example, in the UK the smallest level of census
administrative regions is the Enumeration District (ED). For each ED a
geographical centroid (an (x,y) coordinate location) is recorded. The
precise boundary of each ED could be captured by digitising the
appropriate polygon boundary. However, an approximate boundary can
be generated by using the Theissen polygon method to interpolate the
area of influence associated with each ED centroid. The area polygons
produced will not accurately represent the true boundary of the
administrative area but may well be sufficient for the graphical
presentation of data.
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Continuous Self Assessed Exercise 15
(time: 2 hours)
For the point data set in Figure 5.7 generate a surface using the Theissen
polygon approach. This is not as easy as it may appear! Work with a pencil
and rubber so that you can change your mind about where the boundary
between pairs of points should be as you proceed. A hint - start with the
points on the boundary and work towards the points in the centre of the
data set. When you think you have solved the problem (or if you are having
problems) you can check your answer against ours under the 'SAE
Answers' button.
When you have completed your mosaic of Theissen polygons assign the
observed attribute associated with each data point to its surrounding
territory. Now, using the same classification interval of 10 units as in the
previous exercise produce a new map removing the boundary lines
between the territories of points which fall within the same classification
band. Then use either different colours or some form of graded shading to
distinguish between the areas (to check you are on the right lines you
should compare your map with our example under the 'SAE Answers'
button.). Compare this map with the one you produced using the threading
approach. What similarities and differences do you note between the two
images?
What your comparison of the two maps should reveal is that both maps
have a rough similarity. The greatest discrepancy occurs at the
boundaries between class intervals and at the edge of the study area.
5.5.3 The Triangulated Irregular Network (TIN)
You have already been introduced to the Triangulated Irregular
Network (TIN) Model in Spatial Data Modelling. Developed in the
early 1970's, the TIN is a simple way to construct a surface from a set
of known points. It is a particularly useful technique for irregularly
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spaced points. The TIN approach is an exact interpolation method. In
the TIN model, the known data points are connected by lines to form a
series of triangles. Figure 5.9 shows a TIN model for a set of 10 points.
Because the value at each node of the triangle is known and the
distance between nodes can be calculated, a simple linear equation can
be used to calculate an interpolated value for any position within the
boundary of the TIN.
Figure 5.9 : A simple TIN for a set of 10 points
To construct contour isolines from a TIN model you must first choose
a contour interval. Then, identify all the arcs which would be dissected
by a contour of a given value. To do this the computer must compare
the values of the points which make up the nodes of a particular
triangle. For example, in Figure 5.9 above, 9 out of the 19 arcs could
be dissected by a value of 100. The next stage would be for the
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computer to calculate the (x, y) coordinates of the point of intersection.
Finally, the intersections are joined (see Figure 5.10). As you can see
from Figure 5.10 one of the problems with producing contours from
the TIN model is the angular character of the isolines produced. To
overcome this some TIN models represent the surface in each triangle
using a non-linear mathematical function chosen to ensure that the
isoline changes continuously, not abruptly, at the edge of the triangle.
The TIN model is also used to generate digital terrain models, which
we consider in the next section of this module.
Figure 5.10 : Contouring a TIN
What should be apparent is that the isolines produced from the TIN
method are much more angular than those produced by threading a line
through the data points by hand. In addition, in the TIN model it is
usual to interpolate a surface only within what is know as the `hull' of
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the data set. The hull is defined as the area enclosed by the arcs joining
the sampling points which are on the periphery of the data set.
5.5.4 Kriging
Kriging, also known as the "Theory of regionalized variables", was
developed by G. Matherson and D. G. Krigge as an optimal method of
interpolation for use in the mining industry. Burrough (1986) notes that
the origins of the method lie in the recognition that the spatial variation
of many geographical properties is too irregular to be modelled by a
smooth mathematical function.
The basis of kriging lies in estimating the average rate at which the
difference between values at points change with distance between
points. This is the most complex of all the exact interpolation methods
and until recently was absent from virtually all commercially available
GIS software. It has found wide application in the earth and
environmental sciences, such as geology, where it is essential to both
honour the known data points and take into consideration local
variability. The complex nature of the technique and the prerequisites
in terms of statistical knowledge which are necessary to understand its
application are too detailed for us to pursue here. If you are interested
in learning more about kriging a comprehensive and readable account
is provided in Oliver and Webster (1990).
5.6 Approximate point interpolation methods
Approximate interpolation techniques are applicable when there is
uncertainty associated with reproducing the measured values for the
observed data points but where it is known that there are global trends
overlain by local fluctuations (Waters, 1989). Approximate
interpolation techniques, therefore, employ smoothing techniques
which reduce the effects of local variability on the interpolated surface.
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An important characteristic of all approximate point interpolation
techniques is that the surface being interpolated does not have to pass
through the known data points.
5.6.1 Spatial moving average
The most common method used in GIS is the Spatial Moving Average.
This process involves calculating a new value for each known location
based upon a range of values associated with neighbouring points. The
new value will usually be either an average or weighted average of all
the points within a predefined neighbourhood. In some textbooks you
will find that they use the term `moving window'.
Applying a moving average involves making a number of decisions
about the shape, size and character of the neighbourhood about a point.
The most common shape for a neighbourhood is a circle, since points
in all directions have an equal chance of falling within the radius of a
given point. However, in the raster world a square or rectangular
window of cells is often used. The size of the neighbourhood is
determined by the user and is usually based upon assumptions about
the distance over which local variability in the data is important.
One weighting function which may be used to enhance the technique is
the distance of points away from the centre of the neighbourhood. For
example, a point which is only 100m away may be considered to be
twice as important as another point 200m away. This is described as a
linear weighting. It is also possible to have a non linear weight where
points closer to the centre of neighbourhood are considered to be one
or two orders of magnitude more important than those towards the
edge of the neighbourhood. The number of values used in the
calculation is also important. Burrough (1986), for example, suggests a
spatial moving average should use between 4 and 12 data points with
the norm being between 6 and 8. Including too many points creates a
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very smooth surface, ultimately resulting in a single mean value for the
entire study area. Conversely, too few points can result in individual
points with extreme values dominating the interpolated surface.
Once a mathematical model has been constructed to describe the
character of the neighbourhood the next stage is to calculate a new
value for each of the known data points. Figure 5.11 summarises this
process for a neighbourhood in which each point is equally weighted.
Figure 5.11 : Calculating a spatial moving average in the raster vector world
27
Continuous Self Assessed Exercise 16
(time: 1 and a half hours)
Use the spatial moving averaging technique described above to calculate
new values for the sampling locations in Figure 5.7. For this exercise you
will need to hunt out, borrow or buy a pair of compasses. Or, if you are
clever, invent one from a piece of string and a drawing pin! Set your
compasses to a radius of 3cm and centre them on each of the sample
points in turn and use them to identify the neighbourhood of a given point.
Before you begin, make sure you have a separate copy of Figure 5.7 to
hand. For each point in turn calculate a new value from the average of all
the points within the neighbourhood. Use this value to replace the original
observed value in Figure 5.8. You can check how you are doing by
comparing your new values for the sample data points with the answers
under the 'SAE Answers' button.
Now use the TIN method described in Section 5.5.3 to produce a contour
map for the new surface. Compare this surface with the one produced
using the TIN and line threading approaches. What differences do you
note?
The basic problem with spatial moving average techniques is the
number of variations that exist. It is, therefore, highly probable that the
spatial moving average functions available in one proprietary GIS will
produce significantly different results from those in another GIS. This
is because different systems will allow you to select different distance
decay models, select the number of points used in the model, and apply
different procedures for handling those points which fall on the edge of
a neighbourhood. In addition, in the raster world, it is more common
for the neighbourhood to be defined as a group of cells (usually 9)
compared with the circular neighbourhood used in the vector world.
These problems do not arise if you have a complete understanding of
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the phenomena which you are trying to interpolate because you can
design a `perfect' moving window.
The spatial moving average approach is best suited to those situations
where:-
1 - observations are not considered to be exact representations
of a true surface, but rather singular observations within a
statistical sample
2 - where the measurement techniques used to measure values
at discrete locations on the true surface are known to be
subject to error
3 - where it is known that the true surface exhibits a general
pattern within which there is local variability
Examples include demographic data, survey or questionnaire responses
and pollution monitoring observations. A spatial moving average will
produce a surface which suppresses individual values and reveals
broad structures in data. The nature of those structures depends to
some extent on the filter size and the weightings, which you determine
based on known objectives of the analysis.
5.6.2 Trend surface analysis
To introduce the concept of trend surface analysis let us take a look at
what it says next to the appropriate command from the IDRISI help:
`TREND calculates linear, quadratic and cubic
trend surface polynominal equations for spatial
data sets and interpolates surfaces based on these
equations'.
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If this sounds like a statistical nightmare do not worry, the ideas are
simple. All we have time to do here is introduce the basic principles. In
addition, because the statistical assumptions of the model are rarely
met in practice the technique is not a very robust interpolation method.
Therefore, it is much safer to use one of the simpler methods described
elsewhere in this section. For those of you interested in getting to grips
with trend surface analysis in detail Unwin (1981) provides an
excellent introduction which includes a worked example based around
a set of five points.
For those of you who have done some basic statistics then the concepts
which underpin trend surface analysis should be immediately apparent.
If you have not done any statistics then the following discussion should
help. The trend surface method fits a surface to the known observation
points by a method known as least squares. The shape of the surface is
determined by the mathematical equation, or polynomial, which is used
to describe it. A polynomial surface has some very important
characteristics, most importantly it changes smoothly and predictably
from one map edge to the other. The polynomial equation may be
linear, quadratic or even cubic. A linear equation would be used to
describe a tilted plane. A quadratic equation would define a simple hill
or hollow and a cubic equation would define a more complex surface
(Figure 5.11). These concepts should be familiar to anyone who has
calculated regression equations (`lines of best fit').
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Figure 5.11
31
To fit a simple tilted plane to a surface of observed points all you need
to do is solve a simple linear regression equation in which the spatial
coordinates (x, y) are the independent variables, and the z value, the
attribute of interest, the dependent variable. For those of you who are
not familiar with regression what this statistical procedure does is fit
what is often termed a ` best fit line' through observed data values. It is
easiest to understand by way of a two dimensional example. Graph A
in Figure 5.12 is known as a scatter plot and shows how height varies
with distance along a transect. If we were certain that these height
values were exact then we might join the points together as in Graph B
to interpolate the height value at unsampled locations.
However, if we are uncertain of the quality of our data, or we are more
interested in the general trend, then it might be safer to place a line of
best fit through our points (see Graph C).Conceptually, all trend
surface analysis does is fit a similar `best-fit surface' to the known data
points based on a mathematical definition of fit, i.e. least squares.
Trend surfaces are, therefore, smoothing functions, which do not
necessarily pass through the original data points.
The main limitation of trend surface analysis is that it makes
adjustments for all points simultaneously. I would suggest that these
techniques are more appropriate for exploratory data analysis than for
interpolation.
5.6.3 Fourier analysis
Fourier analysis is another global interpolation technique which uses
applied geostatistical methods to approximate a surface by overlaying a
series of sine and cosine waves. The technique was developed for
signal analysis and has been widely applied in climatic change
modelling. At present it has received little attention as a tool for
interpolating geographical phenomena. Burrough (1986) suggests this
32
is probably because it is best suited to data sets which exhibit
periodicity (such as ocean waves and sand dunes) which are rare in
most geographical phenomenon. Therefore, because of the lack of
practical applications using this technique and the complex
mathematical nature of the approach we will only mention it here. If
your inquisitive nature gets the better of you then both Burrough
(1986) and Waters (1989) suggest Davis (1986) as a readable
introduction.
Figure 5.12 : The best fit concept
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5.7 Line interpolation methods
You have already examined the most common method used to
interpolate observed data associated with a linear feature. The
technique known as buffering or corridor analysis was described in
Section 2.4.1 of this module. Buffering is an exact, abrupt and
deterministic interpolation technique which produces a surface to
which an attribute can be applied which is some derivative of that
associated with the linear feature.
Buffering is an exact interpolation technique because the procedure
honours the observed values associated with the linear entity from
which it has been derived. It is abrupt because there are exact
boundaries between the surface territories associated with the line
features used to produce the buffer (though these will be less marked if
a graded buffering approach is used to generate a surface). The
technique is deterministic in that it assumes a linear feature will have a
known fixed influence on its neighbourhood. The interpolated surface
and subsequent values produced using buffering hold true only when
the character of the linear feature is known to be the dominant feature
in the area of interest.
Another common interpolation operation found in many GIS is
estimating point values from isolines. The most common instance of
this application is the conversion of height contour lines into an
elevation grid for the purpose of constructing a digital terrain module.
Since digital terrain models are now an integral part of most GIS we
will leave our discussion on the interpolation of point data from linear
features until later in this module.
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5.8 Area based interpolation methods
Most socio-economic and environmental data are reported by reference
to an areal unit. Examples include: population density by country; the
number of employed or unemployed by administrative district; crime
statistics by police beat and water quality measures by water districts.
Within the UK alone there are some 70 different areal units which are
used for reporting socio-economic data. Area based interpolation
techniques are necessary when we wish to allocate the attributes
associated with one set of area features to another. For example, in the
UK we might wish to report crime statistics for police beat at electoral
ward level. As Laurini and Thompson (1992) indicate, assigning a
value is easy if either:-
1. the area units are an identical match, or
2. the source areas are a nested subset of the target area.
It is when the areas do not match that interpolation is necessary and
problematic.
Waters (1989) summarises the areal interpolation problem as follows:
"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."
Burrough (1986), Waters (1989) and Lam (1983) identify two different
approaches to solving the problem:-
z non-volume preserving
z volume preserving
which will be considered in turn.
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5.8.1 Non-volume preserving
The non-volume preserving method is best described by way of
example. The following example is based loosely around that
described by Waters (1989). Consider the two maps in Figure 5.13.
Map A shows the total population for four administrative districts on
the planet Mars. Map B shows the flood corridor which has been
established for one of the Martian built canals on the planet. What is
the population density within the flood zone?
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Figure 5.13 : Non-volume areal interpolation method
37
The first step is to calculate the population density for each district by
dividing population by area. Assuming we know the area of each
district to be 100 m units. This gives us a population density of 2, 4, 6
and 8 Martians/m unit for zones A, B, C and D respectively.
The next stage is to identify a centroid for each region (this is the point
at the geometric centre of the area) and then assign to this location the
population density value associated with its enclosing area. This would
give us Map C in Figure 5.13. The next stage would be to interpolate a
gridded population density surface using any of the methods described
earlier. Map D provides one possible surface which may result from
such a process.
Each grid cell value can then be adjusted to a population by
multiplying the estimated density by the cell area (see Map E). Now,
overlay the target map. In this case our flood corridor goes on top of
the interpolated grid and then each grid value can be assigned to the
flood and non-flood zones. From this map an estimated value of the
total number of Martians at risk in the flood zone can be calculated
(see Map F).
Continuous Self Assessed Exercise 17
(time: 20 minutes)
Taking a careful look at the above example can you identify the
weaknesses in this approach? Hint: how does the total population across
all zones compare between Map A and Map F?
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The most important point which we hope you will have identified from
the exercise above is that the total population of the whole region is not
preserved. In our example, the total population has dropped from 2000
in Figure 5.13A to 1950 in Figure 5.13F. Another criticism of this
method is that the choice of the centroid as the point location on which
to base an interpolation technique may be a poor assumption. In our
example there may well in fact be nobody living at this central location
and all the Martians might live at the edge of zone A. A further
problem is that this approach suffers from all the inadequacies
associated with point interpolation methods as described earlier.
5.8.2 Volume preserving
Volume preserving procedures involve overlaying the target zones on
top of the source zones and then determining the proportion of the
target zone which falls within each source zone. The total value of the
attribute for each source zone is then given to the target zone according
to the areal proportions. Figure 5.14 summarises this approach for
estimating the total Martian population in our flood zone. The main
problem is that the method assumes uniform density of the attribute
within each zone. An enhanced version of this approach is the
Pycnophylactic method developed by Tobler in 1979.
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Figure 5.14 :Volume preserving areal interpolation methods
40
Continuous Self Assessed Exercise 18
(time: 45 minutes)
The procedure for the Pycnophylactic method is summarised below:
1. overlay a dense grid of cells on the source map.
2. divide each zone's total value equally among the raster cells that overlap
the zone.
3. smooth the values by replacing the value of each with the average of its
neighbours.
4. sum the values of the cells in each zone.
5. adjust the values of all the cells within each zone proportionally so that
the zone's total is the same as the original. For example if the zone is 10%
low increase the value of each cell by 10%.
Follow this procedure to calculate the total population for the flood zone.
Use a raster size of 16 cells. To check you are on the right track the
answer and procedure are summarised under the 'SAE Answers' button.
This particular interpolation procedure raises an interesting question.
What do you do about the cells whose neighbours are external to the
study area and therefore unknown. Do you:-
z assume the population is equal to zero, or
z assume the population density is equal to the last
neighbour?
z simply ignore them as we did above?
There is no simple answer to this question. In this case it is necessary
to rely on your own judgement or external information to help you
assess the situation and determine the appropriate action. Here another
interesting problem arises, since when we are performing this
41
operation by hand it is possible for us to assign boundary values as we
see fit. For example, if we know there is a desert to the top of our study
area we can build zero values into the cell averaging procedure.
However, if this process was being performed by a computer algorithm
this information would have to come from another source? What other
source? Artificial intelligence or another map layer perhaps? At the
present time most commercial GIS have no simple answer to this
problem.
5.9 Discussion
In this section we have provided an overview of the objectives,
importance and applications of a range of point, line and area
interpolation techniques. As with so many other topics in this diploma,
we could quite easily have written a whole module on this topic.
However, our purpose is to provide you with a flavour of the subject
and a rational for why, where and when you might use some of the
more common methods.
You probably now feel that selecting and applying the appropriate
interpolation technique to a given GIS problem is much more complex
than you had first considered. Unfortunately, there are few hard and
fast rules that can be applied to selecting the appropriate interpolation
procedure. Where possible we have tried to identify and isolate these in
the above discussion (these are also summarised in Table 4 which is
reproduced from Burrough (1986)).
Perhaps the best piece of advice we can give is that you should try to
ensure that you understand the data you are using and that you are
aware of its characteristics and limitations. Remember though that
even the most rigorous procedure is still only providing you with a
`best guess' - the only exact method is to go out there and sample each
and every location!
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Table 4 : Methods of interpolation (Burrough, 1986)
43
If you are intrigued and enthralled by the complexities surrounding
spatial interpolation procedures and wish to delve further then the
bibliography at the end of this module should provide you with an
appropriate starting point. If you have access to a GIS other than the
one you are using as part of this course an interesting exercise would
be to undertake a review of the spatial interpolation procedures which
it contains. At the present time it will probably make use of one or a
few of the methods described in this section. The more analytically
intensive techniques are often absent from proprietary GIS because
there has been no demand for them. It may be that until somebody is
able to develop a usable interface to these more sophisticated
procedures it is likely they will remain in the domain of the spatial
scientists.
So, after all that, why have we spent a large part of this module
discussing interpolation techniques ? The answer lies in the fact that
even if you aren't required to interpolate data in your job then it is still
highly likely that you will be using data that have been interpolated by
someone else. If this section has achieved nothing else then it should
have warned you of the need to understand the methods which have
been applied to those data.
Before completing the final exercise I would like to draw your
attention to the difference between interpolation and extrapolation.
All of the interpolation techniques that we have described have been
developed to estimate values at locations within the area covered by
the source data. Estimating values beyond those boundaries
(extrapolating) is more difficult and prone to greater uncertainty
because there is less `control' on the interpolation technique. You will
also appreciate that not all techniques can be used for extrapolation.
Strictly (statistically) speaking none of them should be used but we live
in a world where sometimes we are asked to perform analysis against
our better judgement.
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We will return to some of these issues in the Data Quality module
when we look at error propagation.
Periodic Self Assessed Exercise 5
(time: 30 minutes)
To help you summarise this section try to define and describe what is
meant by the following terms in the context of interpolation:-
1. Local/Global
2. Deterministic
3. Theissen polygons
4. Volume preserving area interpolation
5. Exact/Approximate
7. TIN
8. Eye balling
9. Neighbourhood