ArcGIS
®
9Using ArcGIS
®
Geostatistical Analyst
Copyright © 2001, 2003-2005 ESRI
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DATACREDITS
Carpathian Mountains data supplied by USDA Forest Service, Riverside, California, and is used here with permission.
Radioceasium data supplied by International Sakharov Environmental University, Minsk, Belarus, and is used here with permission. Copyright © 1996.
Air quality data for California supplied by California Environmental Protection Agency, Air Resource Board, and is used here with permission.
Copyright © 1997.
Radioceasium contamination in forest berries data supplied by the Institute of Radiation Safety “BELRAD”, Minsk, Belarus, and is used here with
permission. Copyright © 1996.
CONTRIBUTING WRITERS
Kevin Johnston, Jay M. Ver Hoef, Konstantin Krivoruchko, and Neil Lucas
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Attribution.pmd 11/25/2003, 4:46 PM1
IN THIS CHAPTER
11
With Geostatistical Analyst, you can easily create a continuous surface,
or map, from measured sample points stored in a point-feature layer, raster
layer, or by using polygon centroids. The sample points may be measurements
such as elevation, depth to the water table, or levels of pollution, as is the case
in this tutorial. When used in conjunction withArcMap, GeostatisticalAnalyst
provides a comprehensive set of tools for creating surfaces that can be used
to visualize, analyze, and understand spatial phenomena.
Tutorial scenario
The U.S. Environmental ProtectionAgency is responsible for monitoring
atmospheric ozone concentration in California. Ozone concentration is measured
at monitoring stations throughout the state.
The locations of the stations are shown here. The
concentration levels of ozone are known for
all of the stations, but we are also interested in
knowing the level for every location in California.
However, due to cost and practicality, monitoring
stations cannot be everywhere. Geostatistical
Analyst provides tools that make the best predictions
possible by examining the relationships
between all of the sample points and producing a
continuous surface of ozone concentration,
standard errors (uncertainty) of predictions, and
probabilities that critical values are exceeded.
Quick-start tutorial
2
• Exercise 1: Creating a surface
using default parameters
• Exercise 2: Exploring your data
• Exercise 3: Mapping ozone con-
centration
• Exercise 4: Comparing models
• Exercise 5: Mapping the probability
of ozone exceeding a critical
threshold
• Exercise 6: Producing the final
map
ch02_Tutorial.pmd 11/25/2003, 4:33 PM11
12 USING ARCGIS GEOSTATISTICAL ANALYST
The data you’ll need for this tutorial is included on the
Geostatistical Analyst installation disk. The datasets were
provided courtesy of the California Air Resources Board.
The datasets are:
Dataset Description
ca_outline Outline map of California
ca_ozone_pts Ozone point samples (ppm)
ca_cities Location of major California cities
ca_hillshade A hillshade map of California
The ozone dataset (ca_ozone_pts) represents the 1996
maximum eight-hour average concentration of ozone in
parts per million (ppm). (The measurements were taken
daily and grouped into eight-hour blocks.) The original data
has been modified for the purposes of the tutorial and
should not be taken to be accurate data.
From the ozone point samples (measurements), you will
produce two continuous surfaces (maps), predicting the
values of ozone concentration for every location in the State
of California based on the sample points that you have. The
first map that you create will simply use all default options
to show you how easy it is to create a surface from your
sample points. The second map that you produce will allow
you to incorporate more of the spatial relationships that are
discovered among the points. When creating this second
map, you will use the ESDA tools to examine your data.
You will also be introduced to some of the geostatistical
options that you can use to create a surface such as
removing trends and modeling spatial autocorrelation. By
Introduction to the tutorial
using the ESDA tools and working with the geostatistical
parameters, you will be able to create a more accurate
surface.
Many times it is not the actual values of some caustic
health risk that is of concern, but rather if it is above some
toxic level. If this is the case, immediate action must be
taken. The third surface you create will assess the probability
that a critical ozone threshold value has been exceeded.
For this tutorial, the critical threshold will be if the maximum
average of ozone goes above 0.12 ppm in any eight-hour
period during the year; then the location should be closely
monitored. You will use Geostatistical Analyst to predict the
probability of values complying with this standard.
This tutorial is divided into individual tasks that are designed
to let you explore the capabilities of Geostatistical Analyst
at your own pace. To get additional help, explore the
ArcMap online Help system or see Using ArcMap.
• Exercise 1 takes you through accessing Geostatistical
Analyst and through the process of creating a surface of
ozone concentration to show you how easy it is to create
a surface using the default parameters.
• Exercise 2 guides you through the process of exploring
your data before you create the surface in order to spot
outliers in the data and to recognize trends.
• Exercise 3 creates the second surface that considers
more of the spatial relationships discovered in Exercise 2
and improves on the surface you created in Exercise 1.
This exercise also introduces you to some of the basic
concepts of geostatistics.
Tutorial.p65 04/02/2001, 2:36 PM12
QUICK-START TUTORIAL 13
• Exercise 4 shows you how to compare the results of the
two surfaces that you created in Exercises 1 and 3 in
order to decide which provides the better predictions of
the unknown values.
• Exercise 5 takes you through the process of mapping the
probability that ozone exceeds a critical threshold, thus
creating the third surface.
• Exercise 6 shows you how to present the surfaces you
created in Exercises 3 and 5 for final display, using
ArcMap functionality.
You will need a few hours of focused time to complete the
tutorial. However, you can also perform the exercises one
at a time if you wish, saving your results after each
exercise.
ch02_Tutorial.pmd 03/02/2004, 1:16 PM13
14 USING ARCGIS GEOSTATISTICAL ANALYST
The ca_outline layer is now displayed transparently with
just the outline visible. This allows you to see the layers
that you will create in this tutorial underneath this layer.
Saving your map
It is recommended that you save your map after each
exercise.
7. Click the Save button on the Standard toolbar.
You will need to provide a name for the map because
this is the first time you have saved it (we suggest
Ozone Prediction Map.mxd). To save in the future, click
Save.
Exercise 1: Creating a surface using default parameters
Before you begin you must first start ArcMap and enable
GeostatisticalAnalyst.
Starting ArcMap and enable Geostatistical Analyst
Click the Start button on the Windows taskbar, point to
Programs, point to ArcGIS, and click ArcMap. In ArcMap,
click Tools, click Extensions, and check Geostatistical
Analyst. Click Close.
Adding the Geostatistical Analyst toolbar to
ArcMap
Click View, point to Toolbars, and click Geostatistical
Analyst.
Adding data layers to ArcMap
Once the data has been added, you can use ArcMap to
display the data and, if necessary, to change the properties
of each layer (symbology, and so on).
1. Click the Add Data button on the Standard toolbar.
2. Navigate to the folder where you installed the tutorial
data (the default installation path is
C:\ArcGIS\ArcTutor\Geostatistics), hold down the Ctrl
key, then click and highlight the ca_ozone_pts and
ca_outline datasets.
3. ClickAdd.
4. Click the ca_outline layer legend in the table of contents
to open the Symbol Selector dialog box.
5. Click the Fill Color dropdown arrow and click No Color.
6. Click OK on the Symbol Selector dialog box.
5
4
7
1
6
Tutorial.p65 03/07/2001, 2:38 PM14
QUICK-START TUTORIAL 15
Creating a surface using the defaults
Next you will create (interpolate) a surface of ozone
concentration using the default settings of the Geostatistical
Analyst. You will use the ozone point dataset
(ca_ozone_pts) as the input dataset and interpolate the
ozone values at the locations where values are not known
using ordinary kriging. You will click Next in many of the
dialog boxes, thus accepting the defaults. Do not worry
about the details of the dialog boxes in this exercise. Each
dialog box will be revisited in later exercises. The intent of
this exercise is to create a surface using the default options.
1. Click the Geostatistical Analyst toolbar, then click
Geostatistical Wizard.
2. Click the Input Data dropdown arrow and click
ca_ozone_pts.
3. Click the Attribute dropdown arrow and click the
OZONE attribute.
4. Click Kriging in the Methods dialog box.
5. Click Next.
By default, Ordinary Kriging and Prediction Map will be
selected in the Geostatistical Method Selection dialog
box.
Note that having selected the method to map the ozone
surface, you could click Finish here to create a surface
using the default parameters. However, steps 6 to 10 will
expose you to many of the different dialog boxes.
6. Click Next on the Geostatistical Method Selection dialog
box.1
2
3
4 5
6
Tutorial.p65 03/07/2001, 2:38 PM15
16 USING ARCGIS GEOSTATISTICAL ANALYST
The Semivariogram/Covariance Modeling dialog box allows
you to examine spatial relationships between measured
points. You assume things that are close are more alike.
The semivariogram allows you to explore this assumption.
The process of fitting a semivariogram model while capturing
the spatial relationships is known as variography.
7. Click Next.
The crosshairs show a location that has no measured value.
To predict a value at the crosshairs you can use the values
at the measured locations. You know that the values of the
closer measured locations are more like the value of the
unmeasured location that you are trying to predict. The red
points in the above image are going to be weighted (or
influence the unknown value) more than the green points
since they are closer to the location you are predicting.
Using the surrounding points, with the model fitted in the
Semivariogram Modeling dialog box, you can predict a
more accurate value for the unmeasured location.
8. Click Next.
7 8
Tutorial.p65 3/21/01, 8:02 AM16
QUICK-START TUTORIAL 17
W
The Cross Validation dialog box gives you some idea of
“how well” the model predicts the values at the unknown
locations. You will learn how to use the graph and understand
the statistics in Exercise 4.
9. Click Finish.
The Output Layer Information dialog box summarizes
information on the method (and its associated parameters)
that will be used to create the output surface.
10. Click OK.
The predicted ozone map will appear as the top layer in
the table of contents.
11. Click the layer in the table of contents to highlight it,
then click again, and change the layer name to
“Default”.
This name change will help you distinguish this layer
from the one you will create in Exercise 4.
12. Click save on the ArcMap Standard toolbar.
Notice that the interpolation continues into the ocean. You
will learn in Exercise 6 how to restrict the prediction
surface to stay within California.
9
Q
Tutorial.p65 3/21/01, 8:02 AM17
18 USING ARCGIS GEOSTATISTICAL ANALYST
Surface-fitting methodology
You have now created a map of ozone concentration and completed Exercise 1 of the tutorial. While it is a simple task to
create a map (surface) using the Geostatistical Analyst, it is important to follow a structured process as shown below:
You will follow this structured process in the following exercises of the tutorial. In addition, in Exercise 5, you will create a
surface of those locations that exceed a specified threshold and, in Exercise 6, you will create a final presentation layout of
the results of the analysis performed in the tutorial.
Note that you have already performed the first step of this process, representing the data, in Exercise 1. In Exercise 2, you
will explore the data.
Represent
the data
Explore
the data
Fit a
model
Perform
diagnostics
Compare
the models
Add layers and display them in the ArcMap data view.
Investigate the statistical properties of your dataset. These tools can be used
to investigate the data, whether or not the intention is to create a surface.
Select a model to create a surface. The exploratory data phase will help in
the selection of an appropriate model.
Assess the output surface. This will help you understand “how well”
the model predicts the unknown values.
If more than one surface is produced, the results can be
compared and a decision made as to which provides the better
predictions of unknown values.
Exercise 1
Exercise 2
Exercise 3
Exercise 3
Exercise 4
Tutorial.p65 03/07/2001, 2:38 PM18
QUICK-START TUTORIAL 19
In this exercise you will explore your data. As the structured
process on the previous page suggests, to make better
decisions when creating a surface you should first explore
your dataset to gain a better understanding of it. When
exploring your data you should look for obvious errors in the
input sample data that may drastically affect the output
prediction surface, examine how the data is distributed,
look for global trends, etc.
The Geostatistical Analyst provides many data-exploration
tools.
In this tutorial you will explore your data in three ways:
• Examine the distribution of your data.
• Identify the trends in your data, if any.
• Understand the spatial autocorrelation and directional
influences.
If you closed the map after Exercise 1, click the File menu
and click Open. In the dialog box, click the Look in box
dropdown arrow and navigate to the folder where you
saved the map document (Ozone Prediction Map.mxd).
Click Open.
Examining the distribution of your data
Histogram
The interpolation methods that are used to generate a
surface give the best results if the data is normally distributed
(a bell-shaped curve). If your data is skewed (lopsided)
you may choose to transform the data to make it
normal. Thus, it is important to understand the distribution of
your data before creating a surface. The Histogram tool
Exercise 2: Exploring your data
plots frequency histograms for the attributes in the dataset,
enabling you to examine the univariate (one-variable)
distribution of the dataset for each attribute. Next, you will
explore the distribution of ozone for the ca_ozone_pts layer.
1. Click ca_ozone_pts, move it to the top of the table of
contents, then place ca_outline underneath
ca_ozone_pts.
2. Click the Geostatistical Analyst toolbar, point to Explore
Data, and click Histogram.
You may wish to resize the Histogram dialog box so you
can also see the map, as the following diagram shows.
1
2
Tutorial.p65 03/07/2001, 2:38 PM19
20 USING ARCGIS GEOSTATISTICAL ANALYST
3 4
5 6
1
3. Click the Layer dropdown arrow and click
ca_ozone_pts.
4. Click the Attribute dropdown arrow and click OZONE.
The distribution of the ozone attribute is depicted by a
histogram with the range of values separated into
10 classes. The relative proportion (density) of data within
each class is represented by the height of each bar.
Generally, the important features of the distribution are its
central value, its spread, and its symmetry. As a quick
check, if the mean and the median are approximately the
same value, you have one piece of evidence that the data
may be normally distributed.
The histogram shown above indicates that the data is
unimodal (one hump) and fairly symmetric. It appears to be
close to a normal distribution. The right tail of the distribution
indicates the presence of a relatively small number of
sample points with large ozone concentration values.
5. Click the histogram bar with ozone values ranging from
0.162 to 0.175 ppm.
The sample points within this range are highlighted on
the map. Note that these sample points are located
within the Los Angeles region.
6. Click to close the dialog box.
Normal QQPlot
The QQPlot is where you compare the distribution of the
data to a standard normal distribution, providing yet another
measure of the normality of the data. The closer the points
are to creating a straight line, the closer the distribution is to
being normally distributed.
1. Click the GeostatisticalAnalyst toolbar, point to Explore
Data, and click Normal QQPlot.
Tutorial.p65 03/07/2001, 2:38 PM20
QUICK-START TUTORIAL 21
4
32
2. Click the Layer dropdown arrow and click
ca_ozone_pts.
3. Click the Attribute dropdown arrow and click OZONE.
A General QQPlot is a graph on which the quantiles from
two distributions are plotted versus each other. For
two identical distributions, the QQPlot will be a straight line.
Therefore, it is possible to check the normality of the ozone
data by plotting the quantiles of that data versus the
quantiles of a standard normal distribution. From the
Normal QQPlot above you can see that the plot is very
close to a straight line. The main departure from this line
occurs at high values of ozone concentration (which were
highlighted in the histogram plot so they are highlighted here
also).
If the data did not exhibit a normal distribution in either the
Histogram or the Normal QQPlot, it may be necessary to
transform the data to make it comform to a normal distribution
before using certain kriging interpolation techniques.
4. Click to exit the dialog box.
Identifying global trends in your data
If a trend exists in your data, it is the nonrandom (deterministic)
component of a surface that can be represented by
some mathematical formula. For instance, a gently sloping
hillside can be represented by a plane. A valley would be
represented by a more complex formula (a second-order
polynomial) that creates a “U” shape. This formula may
produce the representation of the surface you desire.
However, many times the formula is too smooth to accurately
depict the surface because no hillside is a perfect
plane nor is a valley a perfect “U” shape. If the trend
surface does not adequately portray your surface for your
particular need, you may want to remove it and continue
with your analysis, modeling the residuals, which is what
remains after the trend is removed. When modeling the
residuals, you will be analyzing the short-range variation in
the surface. This is the part that isn’t captured by the
perfect plane or the perfect “U”.
The Trend Analysis tool enables you to identify the presence/absence
of trends in the input dataset.
1. Click the Geostatistical Analyst toolbar, point to Explore
Data, and click Trend Analysis.
2. Click the Layer dropdown arrow and click
ca_ozone_pts.
Tutorial.p65 03/07/2001, 2:38 PM21
22 USING ARCGIS GEOSTATISTICAL ANALYST
North–South
trend line
East–West
trend line
East–West
axis
North–South
axis
32
“U” shape trend
5
4
3. Click the Attribute dropdown arrow and click OZONE.
Each vertical stick in the trend analysis plot represents the
location and value (height) of each data point. The points
are projected onto the perpendicular planes, an east–west
and a north–south plane. A best-fit line (a polynomial) is
drawn through the projected points, which model trends in
specific directions. If the line were flat, this would indicate
that there would be no trend. However, if you look at the
light green line in the image above, you can see it starts out
with low values and increases as it moves east until it levels
out. This demonstrates that the data seems to exhibit a
strong trend in the east–west direction and a weaker one in
the north–south direction.
4. Click the Rotate Projection scroll bar and scroll left until
the rotation angle is 30o
.
This rotation enables you to see the shape of the east–west
trend more clearly. You can see that the projection actually
exhibits an upside-down “U” shape. Because the trend is
“U” shaped, a second-order polynomial is a good choice to
use for the global trend. Even though the trend is being
exhibited on the east–west projection plane, because we
rotated the points 30o
, the actual trend is northeast to
southwest. The trend seen is possibly caused by the fact
that the pollution is low at the coast, but moving inland there
are large human populations that taper off again at the
mountains. You will remove these trends in Exercise 4.
5. Click to exit the dialog box.
Tutorial.p65 03/07/2001, 2:39 PM22
QUICK-START TUTORIAL 23
1
2 3
Understanding spatial autocorrelation and
directional influences
1. Click the Geostatistical Analyst toolbar, point to Explore
Data, and click Semivariogram/Covariance Cloud.
2. Click the Layer dropdown arrow and click
ca_ozone_pts.
3. Click the Attribute dropdown arrow and click OZONE.
The Semivariogram/Covariance Cloud allows you to
examine the spatial autocorrelation between the measured
sample points. In spatial autocorrelation, it is assumed that
things that are close to one another are more alike. The
Semivariogram/Covariance Cloud lets you examine this
relationship. To do so, a semivariogram value, which is the
difference squared between the values of each pair of
locations, is plotted on the y-axis relative to the distance
separating each pair on the x-axis.
Each red dot in the Semivariogram/Covariance Cloud
represents a pair of locations. Since closer locations should
be more alike, in the semivariogram the close locations (far
left on the x-axis) should have small semivariogram values
(low on the y-axis). As the distance between the pairs of
locations increases (move right on the x-axis), the
semivariogram values should also increase (move up on the
y-axis). However, a certain distance is reached where the
cloud flattens out, indicating that the relationship between
the pairs of locations beyond this distance is no longer
correlated.
Looking at the semivariogram, if it appears that some data
locations that are close together (near zero on the x-axis)
have a higher semivariogram value (high on the y-axis) than
you would expect, you should investigate these pairs of
locations to see if there is the possibility that the data is
inaccurate.
4. Click and drag the Selection pointer over these points to
highlight them. (Use the following diagram as a guide. It
is not important to highlight the exact points the diagram
displays.)
Tutorial.p65 03/07/2001, 2:39 PM23
24 USING ARCGIS GEOSTATISTICAL ANALYST
4
7 8
56
The pairs of sample locations that are selected in the
semivariogram are highlighted on the map, and lines link the
locations, indicating the pairing.
There are many reasons why the data values differ more
among sample locations between the Los Angeles area and
other areas. One possibility is that there are more cars in
the Los Angeles area than in other areas, which will
invariably produce more pollution, contributing to a higher
ozone buildup in the Los Angeles area.
Besides global trends that were discussed in the previous
section, there may also be directional influences affecting
the data. The reasons for these directional influences may
not be known, but they can be statistically quantified. These
directional influences will affect the accuracy of the
surface you create in the next exercise. However, once you
know if one exists, the Geostatistical Analyst provides tools
to account for it in the surface-creation process. To explore
for a directional influence in the semivariogram cloud, you
use the Search Direction tools.
5. Check Show Search Direction.
6. Click and move the directional pointer to any angle.
The direction the pointer is facing determines which pairs of
data locations are plotted on the semivariogram. For
example, if the pointer is facing an east–west direction, only
the pairs of data locations that are east or west of one
another will be plotted on the semivariogram. This enables
you to eliminate pairs you are not interested in and to
explore the directional influences on the data.
7. Click and drag the Selection tool along the values with
the highest semivariogram values to highlight them on
the plot and in the map. (Use the following diagram as a
guide. It is not important to highlight the exact points in
the diagram or to use the same search direction.)
Tutorial.p65 03/07/2001, 2:39 PM24
QUICK-START TUTORIAL 25
9
You will notice that the majority of the linked locations
(representing pairs of points on the map), regardless of
distance, correspond to one of the sample points from the
Los Angeles region. Taking more pairs of points, at any
distance, into consideration, shows that it is not just pairs of
points from the Los Angeles region out to the coast that
have high semivariogram values. Many of the pairs of data
locations from the Los Angeles region to other inland areas
also have high semivariogram values. This is because the
values of ozone in the Los Angeles area are so much higher
than anywhere else in California.
8. Click to exit the dialog box.
9. Click Selection and click Clear Selected Features to
clear the highlighted points on the map.
In this exercise we learned:
1. The ozone data is close to a normal distribution. They
are unimodal and fairly symmetrical around the mean/
median line as seen in the histogram.
2. The Normal QQPlot reaffirmed that the data is normally
distributed since the points in the plot created a fairly
straight line, and transformation is not necessary.
3. Using the Trend Analysis tool you saw that the data
exhibited a trend and, once refined, identified that the
trend would be best fit by a second-order polynomial in
the southeast to northwest direction (330 degrees).
4. From the Semiovariogram/Covariance Cloud we found
that the high values of ozone concentration in Los
Angeles create high semivariance values with locations
nearby as well as far away.
5. The semivariogram surface indicates there is a spatial
autocorrelation in the data.
Knowing that there are no outlier (or erroneous) sample
points in the dataset and that the distribution is close to
normal, you can proceed with confidence to the surface
interpolation. Also, you will be able to create a more
accurate surface because you know that there is a trend in
the data that you can adjust for in the interpolation.
Tutorial.p65 03/07/2001, 2:39 PM25
26 USING ARCGIS GEOSTATISTICAL ANALYST
Exercise 3: Mapping ozone concentration
In Exercise 1, you used the default parameters to map
ozone concentration. However, you did not take into
account the statistical properties of the sample data. For
example, from exploring the data in Exercise 2, it appeared
that the data exhibited a trend. This can be incorporated
into the interpolation process.
In this exercise you will:
• Improve on the map of ozone concentration created in
Exercise 1.
• Be introduced to some basic geostatistical concepts.
Again you will use the ordinary kriging interpolation method
and will incorporate the trend in your model to create better
predictions.
1. Click the GeostatisticalAnalyst toolbar and click Geostatistical
Wizard.
2. Click the Input Data dropdown and click ca_ozone_pts.
3. Click the Attribute dropdown arrow and click the
OZONE attribute.
4. Click Kriging in the Methods box.
5. Click Next.
By default, Ordinary Kriging and Prediction are se-
lected.
From the exploration of your data in Exercise 2, you
discovered that there was a global trend in your data. After
refinement with the Trend Analysis tool, you discovered that
a second-order polynomial seemed reasonable and the trend
was from the southeast to the northwest. This trend can be
represented by a mathematical formula and removed from
the data. Once the trend is removed, the statistical analysis
will be performed on the residuals or the short-range
variation component of the surface. The trend will automatically
be added back before the final surface is created
so that the predictions will produce meaningful results. By
removing the trend, the analysis that is to follow will not be
influenced by the trend, and once it is added back a more
accurate surface will be produced.
4
3
2
5
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QUICK-START TUTORIAL 27
Trends should only be removed if there is justification for
doing so. The southwest to northeast trend in air quality can
be attributed to an ozone buildup between the mountains
and the coast. The elevation and the prevailing wind
direction are contributing factors to the relatively low
values in the mountains and at the coast. The high concentration
of humans also leads to high levels of pollution
between the mountains and coast. The northwest to
southeast trend varies much more slowly due to the higher
populations around Los Angeles and extending to lesser
numbers in San Francisco. Hence we can justifiably
remove these trends.
8. Click Next on the Detrending dialog box.
6. On the Geostatistical Method Selection dialog box, click
the Order of Trend Removal dropdown arrow and click
Second.
A second-order polynomial will be fitted because a
U-shaped curve was detected in the southwest to
northeast direction in the Trend Analysis dialog box in
Exercise 2.
7. Click Next on the Geostatistical Method Selection dialog
box.
By default, the Geostatistical Analyst maps the global
trend in the dataset. The surface indicates the most
rapid change in the southwest to northeast direction and
a more gradual change in the northwest–southeast
direction (causing the ellipse shape).
8
Southwest to
northeast global
trend
Northwest to
southeast global
trend
6
7
Tutorial.p65 03/07/2001, 2:39 PM27
28 USING ARCGIS GEOSTATISTICAL ANALYST
Semivariogram/Covariance modeling
In the Semivariogram/Covariance Cloud in Exercise 2, you
explored the overall spatial autocorrelation of the measured
points. To do so, you examined the semivariogram, which
showed the difference-squared of the values between each
pair of points at different distances. The goal of
Semivariance/Covariance modeling is to determine the best
fit for a model that will pass through the points in the
semivariogram (the yellow line in the diagram).
The semivariogram is a function that relates semivariance
(or dissimilarity) of data points to the distance that separates
them. Its graphical representation can be used to
provide a picture of the spatial correlation of data points
with their neighbors.
The Semivariogram/Covariance Modeling dialog box allows
you to model the spatial relationship in the dataset. By
default, optimal parameters for a spherical semivariogram
model are calculated. The Geostatistical Analyst first
determines a good lag size for grouping semivariogram
values. The lag size is the size of a distance class into
which pairs of locations are grouped in order to reduce the
large number of possible combinations. This is called
binning. As a result of the binning, notice that there are
fewer points in this semivariogram than the one in Exer-
cise 2. A good lag distance can also help reveal spatial
correlations. The dialog box displays the semivariogram
values as a surface and as a scatterplot related to distance.
Then it fits a spherical semivariogram model (best fit for all
directions) and its associated parameter values, which are
typically called the nugget, range, and partial sill.
Try to fit the semivariogram at small lags (distances). It is
possible to use different bin sizes and refit the default
spherical model by changing the lag size and number of
lags.
9. Type a new Lag Size value of 12000.
10. Click in the input box and type 10 for the Number of
Lags.
Reducing the lag size means that you are effectively
zooming in to model the details of the local variation between
neighboring sample points. You will notice that with a
smaller lag size, the fitted semivariogram (the yellow line)
rises sharply and then levels off. The range is the distance
where it levels off. This flattening out of the semivariogram
indicates that there is little autocorrelation beyond the
range.
By removing the trend, the semivariogram will model the
spatial autocorrelation among data points without having to
consider the trend in the data. The trend will be automatically
added back to the calculations before the final surface
is produced.
Q
9
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QUICK-START TUTORIAL 29
The color scale, which represents the calculated semivariogram
value, provides a direct link between the empirical
semivariogram values on the graph and those on the
semivariogram surface. The value of each “cell” in the
semivariogram surface is color coded, with lower values
blue and green and higher values orange and red. The
average value for each cell of the semivariogram surface is
plotted on the semivariogram graph. The x-axis on the
semivariogram graph is the distance from the center of the
cell to the center of the semivariogram surface. The
semivariogram values represent dissimilarity. For our
example, the semivariogram starts low at small distances
(things close together are more similar) and increases as
distance increases (things get more dissimilar farther apart).
Notice from the semivariogram surface that dissimilarity
increases more rapidly in the southwest to northeast
direction than in the southeast to northwest direction.
Earlier, you removed a coarse-scale trend. Now it appears
that there are directional components to the autocorrelation
at finer scales, so we will model that next.
Semivariogram
surface
Semivariogram value
Fitted semivariogram
model
Available
semivariogram
models
Associated
parameter
values
Empirical
semivariogram
values
Color scale
Tutorial.p65 03/07/2001, 2:39 PM29
30 USING ARCGIS GEOSTATISTICAL ANALYST
Directional semivariograms
A directional influence will affect the points of the semivariogram
and the model that will be fit. In certain directions
closer things may be more alike than in other directions.
Directional influences are called anisotropy, and the Geostatistical
Analyst can account for them. Anisotropy can be
caused by wind, runoff, a geological structure, or a wide
variety of other processes. The directional influence can be
statistically quantified and accounted for when making your
map.
You can explore the dissimilarity in data points for a certain
direction with the Search Direction tool. This allows you to
examine directional influences on the semivariogram chart.
It does not affect the output surface. The following steps
show you how to achieve this.
11. Check Show Search Direction. Note the reduction in the
number of semivariogram values. Only those points in
the direction of the search are displayed.
12. Click and hold the cursor on the center line in the
Search Direction. Move the direction of the search tool.
As you change the direction of the search, note how the
semivariogram changes. Only the semivariogram
surface values within the direction of the search are
plotted on the semivariogram chart above.
To actually account for the directional influences on the
semivariogram model for the surface calculations, you must
calculate the anisotropical semivariogram or covariance
model.
13. Check Anisotropy.
The blue ellipse on the semivariogram surface indicates the
range of the semivariogram in different directions. In this
case the major axis lies approximately in the NNW–SSE
direction.
Anisotropy will now be incorporated into the model to adjust
for the directional influence of autocorrelation in the output
surface.
E
W
R
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14. Type the following parameters for the Search direction
to make the directional pointer coincide with the minor
axis of the anisotropical ellipse:
Angle Direction: 236.0
Angle Tolerance: 45.0
Bandwidth (lags): 3.0
Note that the shape of the semivariogram curve increases
more rapidly to its sill value. The x- and y-coordinates are
in meters, so the range in this direction is approximately
74 km.
15. Type the following parameters for the Search direction
to make the directional pointer coincide with the major
axis of the anisotropical ellipse:
Angle Direction: 340.0
Angle Tolerance: 45.0
Bandwidth (lags): 3.0
The semivariogram model increases more gradually, then
flattens out. The range in this direction is 114 km.
The plateau that the semivariogram models reach in both
steps 14 and 15 is the same and is known as the sill. The
range is the distance at which the semivariogram model
reaches its limiting value (the sill). Beyond the range, the
dissimilarity between points becomes constant with
increased lag distance. The lag is defined by the distance
Y
Range
T
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32 USING ARCGIS GEOSTATISTICAL ANALYST
between pairs of points. Points separated by a lag distance
greater than the range are spatially uncorrelated. The
nugget represents measurement error and/or microscale
variation (variation at spatial scales too fine to detect). It is
possible to estimate the measurement error if you have
multiple observations per location, or you can decompose
the nugget into measurement error and microscale variation
by checking the Nugget Error Modeling check box.
16. Click Next.
Now you have a fitted model to describe the spatial autocorrelation,
taking into account detrending and directional
influences in the data. This information, along with the
configuration and measurements of locations around the
prediction location, is used to make a prediction. But how
should man-measured locations be used for the calcula-
tions?
U
Range
Nugget
Sill
Lag distance
Anisotropical
ellipse
Tutorial.p65 03/07/2001, 2:39 PM32
QUICK-START TUTORIAL 33
Searching neighborhood
It is common practice to limit the data used by defining a
circle (or ellipse) to enclose the points that are used to
predict values at unmeasured locations.
Additionally, to avoid bias in a particular direction, the circle
(or ellipse) can be divided into sectors from which an equal
number of points are selected. By using the Searching
Neighborhood dialog box, you can specify the number of
points (a maximum of 200), the radius (or major/minor axis),
and the number of sectors of the circle (or ellipse) to be
used for prediction.
The points highlighted in the data view window give an
indication of the weights that will be associated with each
location in the prediction of unknown values. In this
example, four locations (red) have weights of more than
10 percent. The larger the weight, the more impact that
location will have on the prediction of unknown values.
17. Click inside the graph view to select a prediction
location (where the crosshairs meet). Note the change
in the selection of data location (together with their
associated weights) that will be used for calculating the
value at the prediction location.
18. For the purpose of this tutorial, type the following
coordinates in the Test Location input boxes:
X = -2044968 and Y = 208630.37.
19. Check the Shape check box and type 90 in the Angle
input box. Notice how the shape changes. However, to
account for the directional influences, change the angle
back to 338.1.
Crosshairs define the
location prediction
Perimeter of search
neighborhood
Sector of search
neighborhood
Locations used and
associated weights
Preview surface or
neighbors
I
S
In each sector of the search
neighborhood, the number
of points used to predict a
value at an unmeasured
location
In each sector of the search
neighborhood, the minimum
number of points to be used
Geometry and number of
sectors used in the search
O
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34 USING ARCGIS GEOSTATISTICAL ANALYST
20. Uncheck the Shape check box—the Geostatistical
Analyst will use the default values (calculated in the
Semivariogram/Covariance dialog earlier).
21. Click Next on the Searching Neighborhood dialog box.
Before you actually create the surface, you next use the
Cross-validation dialog to perform diagnostics on the
parameters to determine “how good” your model will be.
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QUICK-START TUTORIAL 35
Cross-validation
Cross-validation gives you an idea of “how well” the model
predicts the unknown values.
For all points, cross-validation sequentially omits a point,
predicts its value using the rest of the data, and then
compares the measured and predicted values. The calculated
statistics serve as diagnostics that indicate whether
the model is reasonable for map production.
In addition to visualizing the scatter of points around this
1:1 line, a number of statistical measures can be used to
assess the model’s performance. The objective of crossvalidation
is to help you make an informed decision about
which model provides the most accurate predictions. For a
model that provides accurate predictions, the mean error
should be close to 0, the root-mean-square error and
average standard error should be as small as possible (this
is useful when comparing models), and the root-meansquare
stardardized error should be close to 1.
Here the term “prediction error” is used for the difference
between the prediction and the actual measured value. For
a model that provides accurate predictions, the mean
prediction error should be close to 0 if the predictions are
unbiased, the root-mean-square standardized prediction
error should be close to 1 if the standard errors are accurate,
and the root-mean-square prediction error should be
small if the predictions are close to the measured values.
The Cross Validation dialog box also allows you to display
scatterplots that show the Error, Standardized Error, and
QQPlot for each data point.
Line of best fit 1:1 Line
Results from
cross-validation
exercise
Summary
statistics
Cross-validation
scatter plot
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36 USING ARCGIS GEOSTATISTICAL ANALYST
22. Click the QQPlot tab to display the QQPlot.
From the QQPlot you can see that some values fall slightly
above the line and some slightly below the line, but most
points fall very close to the straight dashed line, indicating
that prediction errors are close to being normally distributed.
23. To highlight the location for a particular point, click on
the row that relates to the point of interest in the table.
The selected point is highlighted in green on the
scattergram.
24. Optionally, click Save Cross Validation to save the table
for further analysis of the results.
25. Click Finish.
The Output Layer Information dialog box provides a
summary of the model that will be used to create a
surface.
26. Click OK.
The predicted ozone map will appear as the top layer in
ArcMap.
J
H
F
G
D
Selected point
Tutorial.p65 03/07/2001, 2:39 PM36
QUICK-START TUTORIAL 37
By default, the layer assumes the name of the kriging
method used to produce the surface (e.g., Ordinary Krig-
ing).
27. Click the layer name to highlight it, then click it again
and change it to “Trend removed”.
You can also create a Prediction Standard Error surface to
examine the quality of the predictions.
28. Right-click on the “Trend removed” layer that you
created and click on Create Prediction Standard Error
Map.
29. Click Save on the Standard toolbar.
The Prediction Standard Errors quantify the uncertainty for
each location in the surface that you created. A simple rule
of thumb is that 95 percent of the time, the true value of the
surface will be within the interval formed by the predicted
LKZ
value ± 2 times the prediction standard error if data are
normally distributed. Notice in the Prediction Standard Error
surface that locations near sample points generally have
lower error.
The surface you created in Exercise 1 simply used the
defaults of the Geostatistical Analyst, with no consideration
of trends in the surface, of using smaller lag sizes, or of
using an anisotropic semivariogram model. The prediction
surface you created in this exercise took into consideration
the global trends in the data, adjusted the lag size, and
adjusted for the local directional influence (anisotropy) in
the semivariogram.
In Exercise 4, you will compare the two models to see
which one provides a better prediction of unknown values.
Note: Once again, you see that the interpolation continues
into the ocean. You will learn in Exercise 6 how to restrict
the prediction surface to stay within California.
Tutorial.p65 3/21/01, 8:04 AM37
38 USING ARCGIS GEOSTATISTICAL ANALYST
Exercise 4: Comparing models
Using the Geostatistical Analyst, you can compare the
results of two mapped surfaces. This allows you to make
an informed decision as to which provides more accurate
predictions of ozone concentration based on cross-validation
statistics.
1. Right-click the “Trend removed” layer, point to Compare....
You will be comparing the “Trend removed”
layer with the Default layer you created in Exercise 2.
Because the root-mean-square prediction error is smaller
for the Trend removed layer, the root-mean-square standardized
prediction error is closer to one for the Trend
removed layer, and the mean prediction error is also closer
to zero for the Trend removed layer, you can state with
some evidence that the Trend removed model is better and
more valid. Thus, you can remove the default layer since
you no longer need it.
2. Click Close on the Cross Validation Comparison dialog
box.
1
3. Right-click the Default layer and click Remove.
4. Click the Trend removed layer and move it to the bottom
of the table of contents so that you can see the sample
points and outline of California.
5. Click Save on the Standard toolbar.
You have now identified the best prediction surface, but
there may be other types of surfaces that you might wish to
create.
3
2
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QUICK-START TUTORIAL 39
Exercise 5: Mapping the probability of ozone exceeding a critical threshold
In Exercises 1 and 3 you used ordinary kriging to map
ozone concentration in California using different parameters.
In the decision making process, care must be taken in
using a map of predicted ozone for identifying unsafe areas
because it is necessary to understand the uncertainty of the
predictions. For example, suppose the critical threshold
ozone value is 0.12 ppm for an eight-hour period, and you
would like to decide if any locations exceed this value. To
aid the decision making process, you can use the Geostatistical
Analyst to map the probability that ozone values
exceed the threshold.
While the Geostatistical Analyst provides a number of
methods that can perform this task, for this exercise you
will use the indicator kriging technique. This technique does
not require the dataset to conform to a particular distribution.
The data values are transformed to a series of 0s and
1s according to whether the values of the data are below or
above a threshold. If a threshold above 0.12 ppm is used,
any value below this threshold will be assigned a value of 0,
whereas the values above the threshold will be assigned a
value of 1. Indicator kriging then uses a semivariogram
model that is calculated from the transformed 0–1 dataset.
1. Click the Geostatistical Analyst toolbar and click Geostatistical
Wizard.
2. Click the Layer dropdown arrow and click
ca_ozone_pts.
3. Click the Attribute dropdown arrow and click the
OZONE attribute.
4. Click Kriging in the method box.
5. Click Next on the Choose Input Data and Method dialog
box.
6. Click Indicator Kriging; notice that Probability Map is
selected.
7. Set the Primary Threshold Value to 0.12.
8. Click the Exceed radial button to select it.
9. Click Next on the Geostatistical Method Selection dialog
box.
10. Click Next on theAdditional Cutoffs Selection dialog
box.
11. Click Anisotropy to account for the directional nature of
the data.
12. Type 25000 for the lag size and 10 for the number of
lags.
6
7
9
8
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40 USING ARCGIS GEOSTATISTICAL ANALYST
13. Click Next on the Semivariogram/Covariance Modeling
dialog box.
14. Click Next on the Searching Neighborhood dialog box.
The blue line represents the threshold value (0.12 ppm).
Points to the left have an indicator-transform value of 0,
whereas points to the right have an indicator-transform
value of 1.
15. Click and scroll right until the Measured, Indicator, and
Indicator Prediction columns are displayed.
16. Click and highlight a row in the table with an indicator
value of 0. That point will be highlighted in green on the
scattergraph, to the left of the blue threshold line.
The measured and indicator columns display the actual and
transformed values for each sample location. The indicator
prediction values can be interpreted as the probability of
exeeding the threshold. The indicator prediction values are
calculated using the semivariogram modeled from the
binary (0,1) data, created as indicator transformations of
your original data. Cross-validation sequentially omits a
point and then calculates indicator prediction values for
each.
For example, the highest measured value is 0.1736. If this
location had not actually been measured, a prediction of
about an 85 percent chance that it was above the threshold
based on the indicator kriging model would have been
made.
17. Click Finish on the Cross Validation dialog box.
18. Click OK on the Output Layer Information dialog box.
The probability map will appear as the top layer in the
ArcMap data view.
The map displays the indicator prediction values,
interpreted as the probability that the threshold value of
0.12 ppm was exceeded on one or more days in the
year 1996.U
I
Y
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QUICK-START TUTORIAL 41
P
It is clear from the map that near Los Angeles the probability
of values exceeding our threshold (staying, on average,
below 0.12 ppm for every eight-hour period during the year)
is likely.
19. Click and hold the Indicator Kriging layer. Drag the
layer and reposition between the ca_outline and trend
removed layers.
Click Save on the Standard toolbar to save your map.
Exercise 6 will show you how you can use the functionality
within ArcMap to produce a cartographically pleasing map
of the prediction surface that you created in Exercise 3 and
the probability surface that you created in this exercise.
Tutorial.p65 3/21/01, 8:04 AM41
42 USING ARCGIS GEOSTATISTICAL ANALYST
You will now produce a final map for presentation. You will
use ArcMap to produce a final output map in which the
prediction and the probability surfaces will be displayed.
Displaying both surfaces
You can change the display of the probability map so you
will be able to see both the prediction and the probability
maps at the same time. The probability levels will be
displayed as a contour map.
1. Right-click the Indicator Kriging layer. Click Properties.
2. Click the Symbology tab.
3. Uncheck the Filled Contours check box, then check the
Contours check box.
4. Click the Color Ramp dropdown arrow and choose an
alternative color ramp.
5. Click OK.
You now see both the probability map (the contours) and
the prediction map as the diagram to the right shows.
Exercise 6: Producing the final map
2
3
4
5
Tutorial.p65 03/07/2001, 2:39 PM42
QUICK-START TUTORIAL 43
Extrapolating ozone values
By default, the Geostatistical Analyst interpolates the value
of the selected variable at any location that lies within the
area defined by the north–south and east–west limits of the
sample point data. However, the map of predicted ozone
does not cover the geographical extent of California (the
ca_outline layer). To overcome this problem you will
extrapolate values (predict values outside the default
bounding box) for both surfaces.
1. Right-click the Indicator Kriging layer in the table of
contents and click Properties. Click the Extent tab. In
Set the extent to: select a custom extent entered below
and type the following values for the Visible Extent, then
click OK:
Left: -2400000 Right: -1600000
Top: 860000 Bottom: -400000
Repeat this step for the Trend removed layer.
Clipping the layers to the California State outline
You will now clip the layers to the ca_outline layer as you
are only interested in mapping the ozone levels within the
State of California and this will produce a more appealing
map.
1. Right-click Layers and click Properties.
2. Click the Data Frame tab.
3. Check the Enable Clip to Shape check box.
4. Click Specify Shape.
5. Click Outline of Features.
6. Click the Layer dropdown and click ca_outline.
7. Click OK.
8. Click OK to close the Data Frame Properties dialog box.
1
Tutorial.p65 03/07/2001, 2:39 PM43
44 USING ARCGIS GEOSTATISTICAL ANALYST
2
3
4
8
7
6
5
The clipped map should look like the following diagram.
Locating the City of Los Angeles
1. Click the Add Data button on the Standard toolbar.
2. Navigate to the folder where you installed the tutorial
data (the default installation path is
C:\ArcGIS\ArcTutor\Geostatistics), then click ca_cities.
3. ClickAdd.
A map of the location of cities in California will be
displayed.
Tutorial.p65 03/07/2001, 2:39 PM44
QUICK-START TUTORIAL 45
4. Right-click the ca_cities layer and click Open Attribute
Table.
5. Scroll through the table and find the AreaName called
Los Angeles. Click this row.
The City of Los Angeles is highlighted on the map.
6. Click to close the attribute table.
7. Click the Zoom In tool on the Tools toolbar and zoom in
on the City of Los Angeles.
Notice that the area with the highest ozone concentration is
actually located just to the east of Los Angeles.
Create a layout
1. Click View on the Main menu and click Layout View.
2. Click the map to highlight it.
3. Click and drag the bottom-left corner of the Data Frame
to resize the map.
5
6
7
3
Tutorial.p65 03/07/2001, 2:39 PM45
46 USING ARCGIS GEOSTATISTICAL ANALYST
4. Click Insert on the Main menu and click Data Frame.
A new data frame is inserted on the map. You can now
copy all the layers in the first data frame into this one in
order to display a map of ozone values for the whole of
California alongside the ozone map, which zooms in on
the Los Angeles area.
5. Right-click the Trend removed layer and click Copy.
6. Right-click the New Data Frame in the table of contents
and click Paste Layer(s).
Follow steps 5 and 6 for all the other layers.
7. Click and drag the New Data Frame to fit the whole
page.
5 6
7
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QUICK-START TUTORIAL 47
8. Click the Full Extent button on the Tools toolbar to view
the full extent of the map in the New Data Frame.
9. Right-click the New Data Frame and click Properties.
10. Click the Data Frame tab and, as you did for the first
Data Frame, check Enable Clip to Shape and click the
Specify Shape button. Choose ca_outline as the layer to
clip to, then click OK.
Adding a hillshade and transparency
1. Right-click the New Data Frame and click Add Data.
2. Navigate to the folder where you installed the tutorial
data (the default installation path is
C:\ArcGIS\ArcTutor\Geostatistics), then click
ca_hillshade.
3. ClickAdd.
A hillshade map of California will be displayed.
4. Click ca_hillshade and move it to the bottom of the table
of contents.
5. Right-click the Trend removed layer in the New Data
Frame table of contents and click Properties.
6. Click the Display tab.
7. Type 30 for the percentage of transparency.
8. Click OK.
The hillshade should now partially display underneath the
Trend removed layer.
Adding map elements
1. Click Insert on the Main menu and click Legend.
2. Move the legend to the bottom-left corner of the layout.
3. Optionally, click Insert and add a North arrow, a Scale
bar, and Text.
8
6
7
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48 USING ARCGIS GEOSTATISTICAL ANALYST
The following diagram shows the type of finished map you
could produce using the functionality of ArcMap. Refer to
Using ArcMap if necessary to learn about inserting
elements into the layout.
The map shows that the area east of Los Angeles has the
highest predicted levels of ozone and the highest probability
of exceeding the critical average threshold (0.12 ppm) on at
least one eight-hour period during 1996. Since this is the
case in the analysis (but remember the original data has
been altered), you may wish to focus on these areas and
analyze time series measurements of ozone to accurately
identify the areas at potential risk.
Tutorial.p65 03/07/2001, 2:39 PM48