Elegant Manual
Digital terrain Modelling
ENGO 573
Digital Terrain Modelling
The University of calgary
Geomatics Engineering Department
Digital Terrain Modelling
By
Naser El-Sheimy
Department of Geomatics Engineering
The University of Calgary
Tel: (403) 220 7587
E-mail: [email protected]
September 1999
Table of Contents
11.Introduction
1.1Functional surfaces11.2Surface Continuity31.3Solid
models31.4Surface smoothness32.Interpolation: What and
Why52.1Methods for Interpolation62.2Exact and Inexact
Interpolators63.Global Fit Interpolation Methods73.1Trend surface
analysis73.1.1Problems with trend surfaces83.1.2Trend-Surface
Analysis Mathematics:83.1.3Analysis of Variance (ANOVAR)
Mathematics for Trend-Surface
Regressions:103.1.4Advantages/Disadvantages and When to use
TSA:124.Local Deterministic Methods for Interpolation134.1Tin
interpolators144.1.1LINEAR interpolation144.1.22nd Exact Fit
Surface164.1.3Quintic interpolation164.2Grid
Interpolation174.2.1Nearest Neighbor174.2.2Linear
Interpolation174.2.3Bilinear interpolation184.2.4Cubic
convolution194.2.5Inverse-Distance
Mathematics:195.Gridding205.1Moving averaging gridding215.2Linear
projection gridding226.Search algorithms237.Grid Resampling25
Chapter
2
Surface Representation from Point Data
This Chapter explains methods of creating discretized,
continuous surfaces for mapping the variation of elevation over
space. Emphasis will be given to gridding, spatial sampling
strategies and methods of spatial prediction including global
methods of classifications and methods of local deterministic
interpolation methods.
1. Introduction
There is no ideal DEM, because the DEM generation techniques can
not capture the full complexity of a surface; there is always a
sampling problem and a representation problem. Various sampling
schemes and representations may lead to very different results both
in the DEM itself and especially in its derivatives. The accuracy
of the individual elevations does not necessary ensure the accuracy
of derivatives such as slope maps.
There are two general ways to represent a surface: by a
mathematical function that expresses elevation as a function of the
horizontal coordinates, and by an image of the surface, explicitly
giving the elevation at some set of points, with no functional
dependence with horizontal coordinates.
Before we go into details, we will first introduces some of the
surface characteristics based on ARC/INFO definitions:
1.1 Functional surfaces
Functional surfaces have the characteristic that they store a
single z value, as opposed to multiple z values as for the
discontinuous surfaces, for any given X,Y location. Probably the
most common example of a functional surface is terrestrial surfaces
representing the Earths surface. Other examples of terrestrial
functional surfaces include bathymetric data and water table
depths. Functional surfaces can also be used to represent
statistical surfaces describing climatic and demographic data,
concentration of resources, and other biologic data. Functional
surfaces can also be used to represent mathematical surfaces based
on arithmetic expressions such as Z = a + bX + cY. Functional
surfaces are often referred to as 2.5-dimensional surfaces.
1.2 Surface Continuity
Functional surfaces are considered to be continuous. That is, if
you approach a given x,y location on a functional surface from any
direction, you will get the same z value at the location. This can
be contrasted with a discontinuous surface, where different z
values could be obtained depending on the approach direction. An
example of a discontinuous surface is a vertical fault across the
surface of the Earth.
Figure 1: Surface Continuity
A location at the top of a fault has one elevation, but
immediately below this point at the bottom of the fault you can
observe another elevation. As you can see, a model capable of
storing a discontinuous surface must be able to store more than one
Z value for a given (X, Y) location.
1.3 Solid models
Functional surface models can be contrasted with solid models
which are true 3D models capable of storing multiple Z values for
any given (X, Y) location. Solid models are common in Computer
Assisted Design (CAD), engineering, and other applications
representing solid objects.
Examples of objects suited to solid modeling are machine parts,
highway structures, buildings, and other objects placed on the
Earths surface. In some cases it is possible to represent some
three-dimensional objects such as faults and buildings on a
functional surface by slightly offsetting the duplicate (X,Y)
coordinates.
1.4 Surface smoothness
Smoothness can be described in terms of the perpendicular to the
slope of the surface. Mathematically, this vector is referred to as
the normal to the first derivative. In addition to being
continuous, a smooth surface has the additional property, that
regardless of the direction from which you approach a given point
on the surface, the normal is constant.
Two surfaces with different levels of smoothness are shown
below. The first surface, represented by flat planar facets, is not
smooth. The normal to the surface is constant throughout the extent
of an individual facet. However, as the normal crosses an edge
separating two adjacent facets, the normal abruptly changes.
Figure 2: Surface normals change abruptly when crossing
facets.
Figure 3: Surface normals on a smooth surface do not change
abruptly.
Contrast the surface on the top with the smooth surface on the
bottom. The normal to the surface varies continuously across the
surface. The normal does not change abruptly as it crosses an edge
in the surface.
Terrain surfaces vary in smoothness. Geologically young terrain
typically have sharp ridges and valleys, in contrast to older
terrain which have been smoothed by prolonged exposure to erosion
forces. Statistical surfaces such as a rainfall or air temperature
surface are generally smooth.
2. Interpolation: What and Why
The word Interpolation comes from the Latin words "inter" means
between and "polire" means refine.
Interpolation = Refining by putting in between.
Interpolation is the process of predicting the value of
attributes at unsampled sites from measurements made at point
locations within the same area or region. Predicting the value of
an attribute at sites out side the are covered by existing
observations is called extrapolation.
Interpolation is used to convert data from point observations to
continuos fields so that the spatial patterns by these measurements
can be compared with the spatial pattern of other spatial entities.
Interpolation is necessary when (Burrough and McDonnell, 1998),
The discretized surface has a different level of resolution,
cell size or orientation from that required. For example, the
conversion of scanned data (satellite images, scanned aerial
photographs) from one gridded tessellation with one given size
and/or orientation to another. This process is known generally as
convolution.
A continuous surface is represented by data model that is
different from required. For example, the transformation of
continuous surface from one kind of tessellation to another (e.g.
Irregular data to grid).
The data we have do not cover the domain of interest completely.
For example, the conversion of discretized data to continuous
surfaces.
To construct the DTM, we need the estimation of elevation for
each point of the grid. To do this, we need to know first whether
the point is exactly at a point where the sampling data is
available, or between the sampling points. In the first case, the
elevation can be taken directly from the database (or original
observation), while in the second case, some method for estimating
elevation need to be used. One of such methods is called
interpolation.
In digital terrain modelling, interpolation is mainly used for
the following operations:
Computation of elevation (Z) at single point locations;
Computation of elevation (Z) of a rectangular grid from original
sampling points (so-called gridding);
Computation of locations (X,Y) of points along contours (in
contour interpolation); and
Densification or coarsening of rectangular grids (so-called
resampling).
Abundant literature exists on methods for interpolation of DTM,
some of these interpolation algorithms will be discussed in the
next sections. Some characteristics and peculiarities of DTM
interpolation from topographic samples can be listed as follows:
There is no 'best' interpolation algorithm that is clearly superior
to all others and appropriate for all applications;
The quality of the resulting DTM is determined by the
distribution and accuracy of the original data points, and the
adequacy of the underlying interpolation model (i.e. a hypothesis
about the behaviour of the terrain surface);
The most important criteria for selecting a DTM interpolation
method are the degree to which (1) structural features can be taken
into account, and (2) the interpolation function can be adapted to
the varying terrain character;
Suitable interpolation algorithms must adapt to the character of
data elements (type, accuracy, importance, etc.) as well as the
context (i.e. distribution of data elements).
Other criteria that may influence the selection of a particular
method are the degree of accuracy desired and the computational
effort involved.
The two general classes of techniques for estimating a regular
grid of points on a surface from scattered observations are methods
called "global fit" and "local fit." As the name suggests,
global-fit procedures calculate a single function describing a
surface that covers the entire map area. The function is evaluated
to obtain values at the grid nodes. In contrast, local-fit
procedures estimate the surface at successive nodes in the grid
using only a selection of the nearest data points.
2.1 Methods for Interpolation
The two general classes of techniques for estimating the
elevation of points on a surface from scattered and regular
observations are methods called "global fit" and "local fit." As
the name suggests, global-fit procedures calculate a single
function describing a surface that covers the entire map area. The
function is evaluated to obtain values at the grid nodes. In
contrast, local-fit procedures estimate the surface at successive
nodes in the grid using only a selection of the nearest data
points.
2.2 Exact and Inexact Interpolators
An interpolation method that estimates a value of an elevation
at a sample points which are identical to that measured is called
an exact interpolator. This is the ideal situation, because it is
only at the data we have direct knowledge of the elevation in
question. All other methods are inexact interpolators. The
statistics differences (absolute and squared) between measured and
estimated values at data points are often used an indicator of the
quality of an inexact interpolator.
Terminology: through this course we shall use the following:
: The measured elevation at point (Xi, Yi)
: The estimate elevation at point (Xi, Yi)
3. Global Fit Interpolation Methods
3.1 Trend surface analysis
Trend surface analysis is the most widely used global
surface-fitting procedure. The mapped data are approximated by a
polynomial expansion of the geographic coordinates of the control
points, and the coefficients of the polynomial function are found
by the method of least squares, insuring that the sum of the
squared deviations from the trend surface is a minimum. Each
original observation is considered to be the sum of a deterministic
polynomial function of the geographic coordinates plus a random
error.
The polynomial can be expanded to any desired degree, although
there are computational limits because of rounding error. The
unknown coefficients are found by solving a set of simultaneous
linear equations which include the sums of powers and cross
products of the X, Y, and Z values. Once the coefficients have been
estimated, the polynomial function can be evaluated at any point
within the map area. It is a simple matter to create a grid matrix
of values by substituting the coordinates of the grid nodes into
the polynomial and calculating an estimate of the surface for each
node. Because of the least-squares fitting procedure, no other
polynomial equation of the same degree can provide a better
approximation of the data.
Figure 4: Principle of Trend Surface, A surface = trend +
residuals
3.1.1 Problems with trend surfaces
There are a number of disadvantages to a global fit procedure.
The most obvious of these is the extreme simplicity in form of a
polynomial surface as compared to most natural surfaces. A
first-degree polynomial trend surface is a plane. A second-degree
surface may have only one maximum or minimum. In general, the
number of possible inflections in a polynomial surface is one less
than the number of coefficients in the trend surface equation. As a
consequence, a trend surface generally cannot pass through the data
points, but rather has the characteristics of an average.
Polynomial trend surfaces also have an unfortunate tendency to
accelerate without limit to higher or lower values in areas where
there are no control points, such as along the edges of maps. All
surface estimation procedures have difficulty extrapolating beyond
the area of data control, but trend surfaces seem especially prone
to the generation of seriously exaggerated estimates.
Computational difficulties may be encountered if a very high
degree polynomial trend surface is fitted. This requires the
solution of a large number of simultaneous equations whose elements
may consist of extremely large numbers. The matrix solution may
become unstable, or rounding errors may result in erroneous trend
surface coefficients.
Nevertheless, trend surfaces are appropriate for estimating a
grid matrix in certain circumstances. If the raw data are
statistical in nature, with perhaps more than one value at an
observation point, trend surfaces provide statistically optimal
estimates of a linear model that describes their spatial
distribution.
3.1.2 Trend-Surface Analysis Mathematics:
Trend-surface analysis is different than local estimation
techniques which try to estimate "local" features; trend-surface
analysis is a mathematical method used to separate "regional" from
"local" fluctuations (Davis, 1973). What is defined as "local" and
"regional" is also often subjective and a function of scale, and
the regional trend may vary with scale. The use of trend analysis
allows observed data points to be divided into these two
components. A trend can be defined by three components (Davis,
1973):
1. It is based of geographic coordinates; i.e. the distribution
of material properties can be considered to be a function of
location.
2. The trend is a linear function. That is, it has the form:
Where Z(X,Y) is the data value at the described location, the
a's are coefficients, and X and Y are combinations of geographic
location.
3. The optimum trend, or linear function, must minimize the
squared deviations from the trend.
For the purpose of illustration, consider the set of points (X,
Z) in Figure 5. These may be separated into "regional" and "local"
components into a variety of ways, 1st order (Figure 5.b), 2nd
order (Figure 5.c), or 3rd order (Figure 5.d).
Figure 5: Example of trend surface in 2D (After Davis, 1986)
Trend-surface analysis is basically a linear regression
technique, but it is applied to two- and three-dimensions instead
of just fitting a line. A first order linear trend surface equation
has the form:
That is, an observation, with a value Z, can be described as a
linear function of a constant value (a0) related to the data set
mean, and east-west (a1), and north-south (a2) components (Davis,
1986). To solve for these three unknowns, three normal equations
are available:
Where n is the number of data points. Solving these equations
simultaneously will yield a "best-fit", defined by least-squares
regression, for a two-dimensional, first-order (a plane) trend
surface. This can be rewritten in matrix format:
For second-order polynomial trend surfaces, the general equation
is:
Third-order polynomial tend surfaces:
Forth-order, two-dimensional surfaces:
3.1.3 Analysis of Variance (ANOVAR) Mathematics for
Trend-Surface Regressions:
Having obtained the coefficients of trend surface, we can now
calculate the expected or trend values, , at any location (X, Y). A
trend-surface of a given order can be fit to any set of data, but
that does not mean that it is a meaningful or worthwhile model.
Each of the terms in the trend- surface polynomial equation has an
associated error, and the trend-surface itself is only used to
estimate the "regional" trend. The "local" fluctuations can be
considered to be errors in the trend-surface regression estimate.
To evaluate the "worthiness" of the trend-surface several terms
must be calculated, in particular the total variation as sum of
squares of the dependent variable, Z:
Total Sum of Squares =
For the estimated values, , we can calculate the sum of squares
due to residuals or deviation from the trend:
Regression Sum of Squares =
The difference between these gives the sum of squares due to
residuals or deviations from the trend:
Error Sum of Squares =
Now, the percentage of goodness-of-fit of the trend is given
by:
and Multiple Correlation Coefficient is
The significance of a trend surface may be tested by performing
an Analysis of Variance (ANOVA), which is the process of separating
the total variation of a set of observations into components
associated with defined sources of variation. This has been done by
dividing the total variation of Z into two components, the trend
(regression) and the residuals (deviations). The degrees of freedom
associated with the total variation in a trend analysis is (n-1),
where n is the number of observations. The degrees of freedom
associated with the regression is m, where m is the number of
coefficients in the polynomial fit to the data not including the
constant term (a0). Degrees of freedom for deviations equal to
difference between both, i.e. (n-m-1).
The mean squares are found by dividing the various sums of
squares by the appropriate degrees of freedom, which is:
Variance of SSR = Variance of the regression function about its
mean =
Variance of SSD = = Variance about the regression function =
Variance of SST = = Variance of the total variation
A test statistic used to define the "worth" of the regression
=
In a general test of a trend-surface equation, the ratio of
interest is that between variance due to regression (trend) and
variance due to residuals (deviations). A test statistic used to
define the "worth" of the regression. If the regression is
significant, the deviation about the regression will be small
compared to the variance of the regression itself.
The Fstatistic is used with the F-test to determine if the group
of trend-surface coefficients are significantly different than
zero; i.e. the regression effect is not significantly different
from the random effect of the data. In formal statistical terms,
the F-test for significance of fit tests the hypothesis (H0) and
alternative (H1):
The hypothesis tested is that the partial regression
coefficients equal zero, i.e. there is no regression. If the
computed F value exceeds the table value of F (Critical values for
F for v1 = m and v2= m-n-1 Degrees of Freedom and Level of
Significance e.g. 5.0% ( = 0.05) ) the NULL hypothesis is rejected
and the alternative is accepted, i.e. all the coefficients in the
regression are significant and the regression is worthwhile.
In addition to the problems of getting a "good" fit for the
trend surface, there are a number of pit-falls to the technique
(Davis, 1986):
1. There must be adequate data control. The number of
observations should be much greater then the number of
coefficients.
2. The spacing of the observation points is important. It can
affect the size and resolution of features seen. Clustering can
cause problems or bias. The distribution can affect the shape of
the surface.
3. There are problems near boundaries. The surface can "blow-up"
in the corners. For this reason it is important to have a buffer
around the area of concern. It amounts to a problem of
interpolating between data to one of extrapolating beyond data
observations.
3.1.4 Advantages/Disadvantages and When to use TSA:
Advantages:
Unique surface generated
Easy to program
Same surface estimated regardless of orientation of reference
geographic
Calculation time for low order surfaces is low
Disadvantages:
Statistical assumptions of the model are rarely met in
practice
Local Anomalies can not be seen on contour maps of low order
polynomials (but can be seen on contour maps of residuals)
The surfaces are highly susceptible edge effects
Difficulty to ascribe a physical meaning to complex higher order
polynomial
When to use TSA:
As a pre-processor to remove regional trend prior to kriging or
IDW
Generating data in sparse data areas
4. Local Deterministic Methods for Interpolation
Global methods presented so far have imposed external, global
spatial structures on the interpolation. Local variations have been
considered as random, unstructured noise. Intuitively, this is not
sensible as one expects to the Z value of the interpolated points
to be similar to the Z values measured close by (Burrough and
McDonnell, 1998). Local methods of interpolation are those methods
which make use of the information from nearest data points. This
involves the following:
1. Defining a search area or neighborhood around the point to be
interpolated
2. Finding the data points within this neighborhood
3. Choosing a mathematical model to represent the variation over
this limited number of points
4. Evaluating it for the point of interest (most probably
regular grid)
The following issues need to be addressed:
The kind of interpolation function to use
The size, shape, and orientation of the neighborhood
The number of data points
The distribution of the data points: regular grid, irregularly
distributed or TIN
The possible incorporation of external information on trends or
different domains.
We will examine all these points in terms of different
interpolation functions. These functions will be subdivided as
follows:
1. Interpolation from TIN data
Linear Interpolation
2nd Exact Fitted Surface Interpolation
Quintic Interpolation
2. Interpolation from grid/irregular data
Nearest neighbor assignment
Linear Interpolation
Bilinear interpolation
Cubic convolution
Inverse distance weighting (IDW)
Optimal functions using geostatistics (Kriging)
All these methods smooth the data to some degree in that they
compute some kind of average value within a window. All the
methods, except Kriging which will be discussed in Chapter 3, are
examined below.
4.1 Tin interpolators
Tin interpolators make use of the fact that the data is already
stored in a TIN structures (see Chapter 5). Three interpolators are
used for TIN surface models: Linear, 2nd exact fit surface and
bivariate quintic. All the three interpolation methods honor the Z
values at the triangle nodes (i.e., both surfaces pass through all
of the data points used to create the TIN). If you were to sample a
linear, 2nd exact fitted or quintic surface at exactly the same (x,
y) location as a point used to construct the TIN, the interpolated
Z value would be the same as the input value (ARC/INFO, 1998).
4.1.1 LINEAR interpolation
The liner interpolation method considers the surface as a
continuous faceted surface formed by triangles. The normal, or
perpendicular to the slope of the surface, is constant throughout
the extent of each triangle facet. However, as you cross over an
edge separating two adjacent triangles, the normal changes abruptly
to that of the next triangle.
In linear interpolation, the surface value to be interpolated is
calculated based solely on the Z values for the nodes of the
triangle within which the point lies. If breaklines are present in
the TIN, they do not influence interpolation because linear
interpolation is not affected by the surface behavior of adjacent
triangles.
The surface value at the certain point is obtained by
intersecting a vertical line with the plane defined by the three
nodes of the triangle. The generalized equation for linear
interpolation of a point (X, Y, Z) in a triangle facet is:
Ax + By + Cz + D = 0
where A, B, C, and D are constants determined by the coordinates
of the triangles three nodes.
Equation of plan using vector algebra:
Given: The position vectors of the three non-collinear nodes (A
,B,C) of a TIN triangle.
Required: the vector equation of the plane passing through the
three nodes.
This can be simply obtained from the mixed products of vectors ,
that is:
.Where is the position vector of a point P(X,Y,Z) which belong
to the plane ABC. The graphical presentation of the last formula is
shown in Figure 6.
Figure 6: Vector Representation of a Plane Through Three
PointsNote:
Mathematical Formulation
General formulation:
Computational Procedures:
1. Search for the triangle the intermediate point (X, Y) falls
into, i.e. get (X1, Y1, Z1), (X2, Y2, Z2), and (X3, Y3, Z3)
2. Solve the set of equations F = A x
3. Estimate (Z) for the location (X, Y)
Linear Interpolation produces continuous but not smooth
surface
4.1.2 2nd Exact Fit Surface
The assumption that the triangles represent tilted flat plates
obviously results in a very crude approximation. A better
approximation can be achieved using curved or bent triangle plates,
particularly if these can be these can be made to join smoothly
across the edges of the triangles. Several procedures can have been
used for this purpose (David, 1986). One of the earliest involved
finding the three neighbors closest to the faces of the triangle,
then fitting a second-degree polynomial trend surface to these and
to the points at the vertices of the triangle. A 2nd degree trend
surface is defined by six coefficients. This means that the fitted
surface is exactly passing through all six points. The equation can
be then used to estimate series of locations having a specified
elevation. Contour maps derived from this method will be curved
rather than straight lines as in the linear interpolation
method.
Even though adjacent plates are fitted using common points,
their trend surfaces will not coincide exactly along lines of
overlap. This means there may be abrupt changes in direction
crossing from one triangular plate to another.
4.1.3 Quintic interpolation
Similar to the linear and 2nd degree exact fit surface
interpolation methods, quintic interpolation considers the surface
model to be continuous. In addition, quintic interpolation also
considers the surface model to be smooth, that is, the normal to
the surface varies continuously within each triangle. In addition,
there are no abrupt changes in the normal as it crosses an edge
between triangles. This smooth characteristic is accomplished by
considering the geometry of the neighboring triangles when
interpolating the z value of a point in a tin triangle.
Quintic interpolation employs a bivariate 5th degree polynomial
in x and y. The generalized equation for quintic interpolation of
the z value of a point on a surface is (H. Akima, 1978):
There are 21 coefficients to be determined. The values of the
function and its first-order and second-order partial derivatives
are given at each node of the triangle, yielding 18 coefficients.
The three remaining coefficients are determined by considering the
surface to be both smooth and continuous in the direction
perpendicular to the three triangle edges.
Note:
The derivative of a polynomial of degree n is a polynomial of
degree n-1. The 2nd derivative of a polynomial of degree n is a
polynomial of degree n-2, and so on.
4.2 Grid Interpolation
4.2.1 Nearest Neighbor
Nearest neighbor assignment assigns the value of the nearest
mesh point in the input lattice or grid to the output mesh point or
grid cell. No actual interpolation is performed based on values of
neighboring mesh points. For example, a value 2 in the input grid
will always be the value 2; it will never be 2.2 or 2.3. Nearest
neighbor assignment is the preferred resampling technique for
categorical data since it does not alter the value of the input
cells. It is not usually used for surface interpolation.
4.2.2 Linear Interpolation
In 1-D case two adjacent reference points values are connected
by a straight line which is used for interpolation. In case of a
2-D space, three reference points are used to define a plane (as in
the TIN linear interpolation), which serves as interpolation
function on the respective grid. The whole interpolation domain is
subdivided into triangles, the corners at the reference points.
Computational Procedures, See Figure 7:
1. Search for the grid where the intermediate point (X, Y) falls
into, i.e. get (X1, Y1, Z1), (X2, Y2, Z2), and (X3, Y3, Z3), and
(X3, Y3, Z3).
2. Affine transformation of the intermediate point
EMBED Equation.2 and
3. Determination of :
4. Evaluate the formula
Figure 7: Linear Interpolation from Grids
Linear Interpolation produces continuous but not smooth
surface
4.2.3 Bilinear interpolation
Instead of dividing the girds into triangles, as in the linear
interpolation, bilinear interpolation take the grid mesh as basic
unit to define a piecewise polynomial function. A bilinear
polynomial of the form;
defined on every mesh will yield a surface which is continuous
on the whole domain, but not smooth (discontinuous first
derivative)
Computational Procedures:
1. Search for the grid where the intermediate point (X, Y) falls
into, i.e. get (X1, Y1, Z1), (X2, Y2, Z2), and (X3, Y3, Z3), and
(X4, Y4, Z4).
2. Affine transformation of the intermediate point
and
3. Evaluate the formula
Figure 8: Bilinear Interpolation
4.2.4 Cubic convolution
Cubic convolution calculates the output Z value using the values
of the sixteen nearest input mesh points. A cubic spline curve is
fit horizontally through each column of four points, and a final
curve is fit vertically to the four horizontal curves, through the
data point location. Using a smooth curve and a larger neighborhood
gives cubic convolution a tendency to smooth the data, and can in
some cases result in output mesh point values which are outside the
range of values in the input lattice.
4.2.5 Inverse-Distance Mathematics:
The technique estimates the Z value at a point by weighting the
influence of nearby data the most, and more distant data the least.
This can be described mathematically by:
or
EMBED Equation.3 where Z(X,Y) is the estimated Z value at the
location (X,Y), Zi is the Z value at the grid location(Xi, Yi), di
is the distance between the grid location (Xi, Yi) and point (X,Y),
and p is the power to which the distance is raised. The basis of
this technique is that nearby data are most similar to the actual
field conditions at the grid location. Depending on the site
conditions the distance may be weighted in different ways. If p =
1, this is a simple linear interpolation between points. Many
people have found that p = 2 produce better results. In this case,
close points are heavily weighted, and more distant point are
lightly weighted (points are weighted by 1 / d2). At other sites, p
has been set to other powers and yielded reasonable results.
Inverse-distance is a simple and effective way to estimate
parameter values at grid locations of unknown value. Beside the
directness and simplicity, however, inverse-distance techniques
have a number of shortcomings. Some of these are:
If a data point is coincident with a grid location, d = 0, a
division by 0 occurs, unless it is treated specially with the
following: condition:
if di = 0, Z(X,Y) = Zi.
4.2.5.1 Weighting functionsThe data points (or the slopes at the
points) used in the estimation procedure are weighted according to
the distances between the grid node being estimated and the points.
Figure 9shows four different weighting functions. The four
weighting functions decrease at increasing rates with distance.
The weights that are assigned to the control points according to
the weighting function are adjusted to sum to 1.0. Therefore, the
weighting function actually assigns proportional weights and
expresses the relative influence of each control point. A widely
used version of the weighting process assigns a function whose
exact form depends upon the distance from the location being
estimated and the most distant point used in the estimation. This
inverse distance-squared weighting function is then scaled so that
it extends from one to zero over this distance.
Figure 9: Distance Weighting Functions
5. Gridding
Gridding is the estimation of values of the surface at a set of
locations that are arranged in a regular pattern which completely
covers the mapped area. In general, values of the surface are not
known at these uniformly spaced locations, and so must be estimated
from the irregularly located control points where values of the
surface are known. The locations where estimates are made are
referred to as "grid points" or "grid nodes."
The grid nodes usually are arranged in a square pattern so the
distance between nodes in one direction is the same as the distance
between them in the perpendicular direction. In most commercial
software, the spacing is under user control, and is one of many
parameters that must be chosen before a surface can be gridded and
mapped. The area enclosed by four grid nodes is called a "grid
cell." If a large size is chosen for the grid cells, the resulting
map will have low resolution and a coarse appearance, but can be
computed quickly. Conversely, if the grid cells are small in size,
the contour map will have a finer appearance, but will be more time
consuming and expensive to produce.
The estimation process involves three essential steps. First,
the control points must be sorted according to their geographic
coordinates. Second, from the sorted coordinates, the control
points surrounding a grid node to be estimated must be searched
out. Third, the program must estimate the value of that grid node
by some mathematical function, as described in the previous
sections, of the values at these neighboring control points.
Sorting greatly affects the speed of operation, and hence the cost
of using a contouring program. However, this step has no effect on
the accuracy of the estimates. Both the search procedure and the
mathematical function do have significant effects on the form of
the final map.
5.1 Moving averaging gridding
The most obvious function that can be used to estimate the value
of a surface at a specific grid node is simply to calculate an
average of the known values of the surface at nearby control
points. In effect, this projects all of the surrounding known
values horizontally to the grid node location. Then, a composite
estimate is made by averaging these, usually weighting the closest
points more heavily than distant points. The resulting mathematical
model of the surface, and the corresponding contour map, will have
certain characteristics. The highest and lowest areas on the
surface will contain control points, and most grid nodes will have
intermediate values, since an average cannot be outside the range
of the numbers from which it is calculated. At grid nodes beyond
the outermost control points, the grid nodes must be estimated by
extrapolation; their values will be close to those of the nearest
control points.
Figure 10: Steps in Computation of Grid Value using Moving
Average
Figure 10 shows a series of observations; each point is
characterized by its X coordinate (east-west or across the page),
its Y coordinate (north-south or down the page), and its Z
coordinate (the value to be mapped). On the illustration, values of
Z are noted beside each point. The observations may be identified
by numbering them sequentially as they are read by the program,
from 1 to i. Therefore, an original data point i has coordinates
X(i) in the east-west direction, Y(i) in the north-south direction,
and value Z(i). In (b), a regular grid of nodes has been
superimposed on the map. These grid nodes also are numbered
sequentially from 1 to k. Grid node k has coordinates X(k) and Y(k)
and has an estimated elevation Z(k). To estimate the grid node
value Z(k) from the nearest n data points, these points must be
found and the distances between them and the node calculated. The
search procedure may be simple or elaborate. Assume that by some
method the n data points nearest to grid node k have been located.
The distance D(k) from observation i to grid node k is found by the
Pythagorean equation
Having found the distances D(ik) to the nearest data points, the
grid point elevation Z is estimated from these. The completed grid
with all values of Z is shown in (d).
This type of algorithm is sometimes called a "moving average,"
because each node in the grid is estimated as the average of values
at control points within a neighborhood that is "moved" from grid
node to grid node. In effect, values at the control points are
projected horizontally to the location of the grid node, where they
are weighted and averaged.
5.2 Linear projection gridding
The linear projection gridding is a two-part procedure, in which
a weighted average of slopes projected from the nearest neighboring
data points around each grid node are used to estimate the value at
the node.
Initially, the slope of the surface at every data point must be
estimated. The nearest n neighboring observations around a data
point are found and each is weighted inversely to its distance from
the data point. A linear trend surface is then fitted to these
weighted observations. The constant term of the fitted regression
equation is adjusted so the plane passes exactly through the data
point. The slope of the trend surface is used as the local dip. If
at least five points cannot be found around the data point or if
the set of simultaneous equations for the fitted plane cannot be
solved, the coefficients of a global linear trend are used to
estimate the local slope. The slope coefficients are saved for each
data point.
The second part of the algorithm estimates the value of the
surface at the grid nodes. A search procedure finds n nearest
neighboring data points around the node to be estimated. The X,Y
coordinates of the grid node are substituted into each of the local
trend surface equations associated with these data points, in
effect projecting these local dipping planes to the location of the
node. An average of these estimates is then calculated, weighting
each slope by the inverse of the distance between the grid node and
the data point associated with the slope. If a data point lies at
or very near a grid intersection, the value of the data point is
used directly as the value of the grid node.
The projection of slopes may be disadvantageous in some
circumstances. For example, the method may tend to create spurious
highs or lows both in areas of limited density of control points
and along the edges of maps if surface dips are projected from
areas where there are tight clusters of observations. Also, the
two-phase algorithm obviously requires more time for computation
than simpler procedures.
Variants of the linear projection algorithm are among the most
popular of those used for constructing the mathematical model used
in contouring. These algorithms are especially good within areas
that are densely controlled by uniformly spaced data points.
However, like the piecewise linear least squares methods, they have
the distressing habit of creating extreme projections when used to
estimate grid nodes beyond the geographic limits of the data.
6. Search algorithms
One critical difference between various local-fit algorithms for
interpolating to a regular grid is the way in which "nearest
neighbors" are defined and found. Because some search techniques
may be superior to others in certain situations (primarily
reflecting the arrangement of the data points), most commercial
software provides a variety of search procedures that may be
selected by the user. The simplest method finds the n nearest
neighboring data points, in a Euclidean distance sense, regardless
of their angular distribution around the grid node being estimated.
This method is fast and satisfactory if observations are
distributed in a comparatively uniform manner, but provides poor
estimates if the data are closely spaced along widely separated
traverses.
An objection to a simple nearest neighbor search is that all the
nearest points may lie in a narrow wedge on one side of the grid
node. The resulting estimate of the node is essentially
unconstrained, except in one direction. This may be avoided by
restricting the search in some way which ensures that the control
points are equitably distributed about the grid node being
estimated.
Figure 11: Search Techniques those Locates N Nearest Neighbors
around a Grid Node
The simplest method that introduces a measure of radial is a
quadrant search. Some minimum number of control points must be
taken from each of the four quadrants around the grid node being
calculated. An elaboration on the quadrant search is an octant
search, which introduces a further constraint on the radial
distribution of the points used in the estimating equation. A
specified number of control points must be found in each of the 45
degrees segments surrounding the grid node being estimated. This
search method is one of the more elegant procedures and is widely
used in commercial contouring programs.
These constrained search procedures require finding and testing
more neighboring control points than in a simple search, which
increases the time required. Most commercial software provides a
number of different types of search procedure which finds all
points within a radius r of the point being estimated. Estimates
made using the other search procedures are based on a fixed number
of points collected at variable distances from the grid node; this
search algorithm uses a variable number of points found within a
fixed distance of the node.
Any constraints on the search for the nearest control points,
such as a quadrant or octant requirement, will obviously expand the
size of the neighborhood around the grid node being estimated. This
occurs because some nearby control points are likely to be passed
over in favor of more distant points in order to satisfy the
requirement that only a few points may be taken from a single
sector. Unfortunately, the autocorrelation of a surface decreases
with increasing distance, so these more remote control points are
less closely related to the location being estimated. This means
the estimate may be poorer than if a simple nearest neighbor search
procedure were used.
7. Grid Resampling
Resampling is the process of determining new values for grid
cells in an output grid that result from applying some geometric
transformation to an input grid. The input grid may be in a
different coordinate system, at a different resolution, or may be
rotated with respect to the output grid. Interpolation techniques,
nearest neighbor assignment, bilinear interpolation, cubic
convolution, and IDW, discussed in previous Sections, are usually
used in the resampling process.
Resampling grids of different resolutions should be done when
using grids of different resolutions as input. The default
resampling is to the coarsest resolution of the input grids. This
resampling on the fly uses the nearest neighbor assignment as the
resampling method. The following Figures shows typical examples of
when resampling is needed.
Figure 12: (a) Resampling when changing the grid size (b)
Resampling when rotating the grid plus changing the grid size.
Figure 13: Example of Nearest Neighbor Resampling
References
1. Akima, H. (1978), " A Method of Bivariate Interpolation and
Smooth Surface Fitting For Irregularly Distributed Data Points",
ACM Transactions on Mathematical Software, Vol. 4, No.2, June 1978,
pp. 148-159.
2. Davis, John, 1986, "Statistics and data analysis in geology",
New York, Wiley, 2nd ed , 1986, (UofC Call # QE48.8 .D38 1986 )
3. ESRI, Inc. 1998, "Arc/Info v7.0.2 Manual", Redlands, CA:
Environmental Systems Research Institute
4. P.A. Burrough and McDonnell R, 1998, "Principles of
geographical information systems for land resources assessment",
Oxford, Clarendon Press, 1998. (UofC Call # HD108.15 .B87 1986)
Trend
+
p-a
(b-a) x (c-a)
b-a
p
b
c
a
Z
X
Y
P
B
Z value ?
Z2
Z1
Residuals
Surface
A. Original Surface
D. 3rd Order Trend
C. 2nd Order Trend
B.1st Order Trend
A
C
1
Weight
Distance
1/D6
1/D2
1/D
1/D4
(d)
(c)
(b)
(a)
17.6
63.3
12.3
11.4
16.2
14.3
12.6
10.8
15.7
13.8
13.1
10.9
15.1
13.3
12.3
11.1
14
15
11
10
12
11
13
17
15
14
15
18
16
11
10
12
11
13
12
17
15
14
15
18
16
11
10
12
11
13
12
Octant Search
Quadrant Search
12
12
17
15
14
15
18
16
11
10
12
11
13
12
17
15
14
15
18
16
11
10
12
11
13
12
Input Grid
Output Grid
Output Grid
Input Grid
= 0
= 1
Y n
Y 1
X 1
j
i
Y
X
X n
4
3
2
1
Y
X
=1
=0
Y
X
4
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