Lab Exercise #3: Spatial Interpolation: Inverse Distance Weighting and Kriging of Several
Datasets including Ozone concentration.
Background
In this lab you will use Arc/INFO (or ArcView if you can find the Avenue scripts to do it) to
perform some spatial interpolations on various datasets. You will use two kinds of spatial
interpolation: Inverse Distance Weighting (IDW) and Kriging. To get an idea of how IDW and
Kriging works simply read the info in Arc/INFO’s on-line help system: Arc Doc. Both IDW and
Kriging are functions in Arc’s Grid module that produce an interpolated surface from a set of point
measurements. For example, let’s say you have a single rainfall measure at the centroid of each of
the 48 conterminous United States; the IDW and Kriging functions can ‘estimate’ the rainfall
anywhere in the conterminous U.S. and produce a grid of those estimates. Using 50 points for the
whole conterminous U.S. is not such a great idea. The points will be too dispersed relative to the
spatial variability of rainfall. Your estimates will probably be better on the east coast where the
states are smaller. Getting a sense of the potential, drawbacks, and ‘rules of thumb’ for using these
spatial interpolation methods is the primary purpose of this lab.
Outline of your tasks/objectives
1) Do a single IDW calculation by hand and describe how the Grid function IDW does a
whole lot more.
2) Your second task will be to simply take a point coverage of ozone measurements made
throughout the United States and spatially interpolate a grid from these points using both IDW and
Kriging. You will then display these two different interpolations and comment on the differences
between them. You will also look at the error grid or variance output of the Kriging command.
3) One problem with the IDW and Kriging exercises in task #2 is that you can’t really assess
how good or bad your interpolations were. So in this third task you will take a complete grid of three
datasets (a DEM of the U.S., a landcover grid of the U.S., a population density grid of the U.S) and
sample them at the points of the Ozone monitoring stations used in task #2. (You will need to add ‘z’
values from those grid to the usozonesites coverage using ‘summarize zones’ command in spatial
analysty of ArcView). You will then produce four kriged and four IDW’d estimates of Elevation,
Landcover, popualtion density, and rainfall. You will then look at the errors of these estimates and
discuss.
4) Build a ‘detrended’ kriged estimate of ozone in Great Smoky NP and compare to U.S.
estimate and discuss method.
Specific Instructions for Tasks 1-4
Task #1: Inverse Distance Weighting by hand
The table below represents x and y coordinates of something called ‘Value’. Estimate the value of
the fourth record (indicated by a ‘?’) using inverse distance weighting. Draw a crude scatter plot of
the data in the table to the right of the table.
X coord
1
3
2
3
7
Y coord
3
5
7
3
4
Value
11
7
3
?
9
Now, enter this table into Excel or some text editor and read it in as an ‘event theme’ to ArcView.
Convert that shapefile to an Arc/INFO point coverage and do an IDW on that point coverage. How
did the output of Grid’s IDW command differ from your simple hand calculation? Explain.
Task #2: Kriging and IDWing without a ‘truth’ or reference grid
If you look at the Point Attribute table for the point coverage ‘usozonesites’ (usozonesites.pat) you
will find that there are numerous attributes. We will work with the attribute: ‘O3apr2oct_’ (don’t
forget the underscore ‘_‘at the end). First we will create an interpolated grid of the ‘O3apr2oct_’
values using the Inverse Distance Weighting command. Get into the right workspace and get into
Grid. At the Grid prompt type:
Grid: idwozone = IDW(usozonesites, o3apr2oct_,. #, 2,
Sample, 11, 1000000, 10000)
Take a look at the grid you created ‘idwozone’ in ArcView. Now create another interpolated filed
using the ‘kriging’ command in grid. At the grid prompt type:
Grid: krigozone = kriging(usozonesites, o3apr2oct_,. #, both, varozone, linear, sample, 11, 1000000,
10000)
This command produced three things: 1) a kriged estimate of the ‘3apr2oct_’ attribute of the
usozonesites point coverage, 2) A semi-variogram of that attribute of the point coverage, and 3) a
variance of the kriged estimate grid. To look at these outputs read the interpolated grid ‘krigozone’
into ArcView. Also read the interpolated grid ‘varozone’ into ArcView. To see the semi-variogram
run Arcplot, do a disp 9999, then type semivariogram krigozone.svg. This should
display a semi-variogram in the ArcPlot display. Use Snag-it to save this image somewhere.
A little bit on Variogram analysis
Some of this material was cribbed from the following web site:
http://www.bioss.sari.ac.uk/smart/unix/mvariog/slides/frames.htm (you might want to read it)
The book An Introduction to Applied Geostatistics by Isaaks and Srivastava is best at explaining this
and it is still pretty abstruse. If you look at the book look see Ch 12 Ordinary Kriging
Variogram : summarises the relationship between the variance of the difference between
measurements and the distance of the corresponding points from each other.
Kriging : uses the information from a variogram to find an optimal set of weights that are used in
estimating a surface at unsampled locations.
Sill (c1 + c0): describes where the variogram develops a flat region, i.e. where the variance no longer
increases.
Range (a): the distance between locations beyond which observations appear independent i.e. the
variance no longer increases.
Nugget variance (C0): when the variogram appears not to go through the origin.
The gist of simple ordinary kriging (this is for a single point. To do a whole field; you must do
this many times e.g. lather rinse repeat NxM times):
1) Characterize spatial autocorrelation with variogram (determine sill, nugget, etc.)
2) Create function to describe variance as a function of distance (linear, gaussian, etc.)
3) Get covariance matrix of all points used in specific interpolation C
4) Get covariance matrix for distance between point being estimated and all points with data D
5) Do magic math: C-1 x D = w (w is a weights matrix that sums to one just like in IDW but
differenct values result, for example two points close to each other can ‘cancel’ each other a
bit giving a greater weight to a lone point a little farther away)
6) Multiply the Z-values of the points used to interpolate times appropriate weights for estimate at
single location.
7) Kriging is more ‘rigorous’ than Inverse Distance Weighting because it supposedly provides
unbiased estimates and characterizes the magnitude of your error variance at all locations.
However, empirical validation exercises like ‘Cross-Validation, Bootstrapping, and JackKnifing’ often show IDW to outperform kriging because the assumptions needed fro kriging
are not met.
A key assumption is that of ‘Stationarity’ which essentially means that the auto-correlation function
is the same throughout the space being interpolated. You will see how this fails soon.
Anyway, BTW, you should have obtained a variogram for the ozone data that looked like this:
We chose to fit a linear model to this data. The nugget was estimated ab about 33.
By putting ‘Spherical, Gaussian, Exponential’, etc. into the kriging command instead of ‘Linear’
Arc/INFO would have calculated a different function to characterize the autocorrelation.
Questions for Task #2:
1) Read the three grids (idwozone, krigozone, and varozone) and a co-registered state boundary
coverage into ArcView. Compare and contrast the appearance of these three grids.
2) Where are the areas of high ‘ozone’ concentration in idwozone and krigozone? Do the two
images agree in general?
3) What is the IDW and Kriged estimate of o3apr2oct_ at the Four Corners location? What is the
value of varozone at Four Corners? If the varozone value is the standard error of the estimate
of the kriged ozone estimate, what is your 95% confidence interval for the o3apr2oct_ for
Four Corners?
Task #3: Spatial interpolation with a known full valid coverage to validate estimates against
In this task you will be IDWing and Kriging interpolated grids from data you already have ‘truth’
for. To do this you will artificially blind yourself by getting elevation, population density, and
landcover information at only the usozonesites points and then interpolating them. To do this you
will need to do a ‘summarize zones’ command with ArcView using usozonesites and each of the
following three grids: nademlza, uspopden, and naigbplza. This will generate several tables which
you will have to read into excel, delete most columns, (keep ‘Monitor’ and ‘Median’), change name
of median to elevation, popden, and landcover as appropriate, and join these three tables back to
usozonesites.pat. Now you are ready for spatial interpolation.
The Elevation dataset
Get an IDW and a Kriged interpolated grid from the usozonesites pointcoverage from grid
Grid: idwelevation=IDW(usozonesites, elevation,. #, 2,
Sample, 11, 1000000, 10000)
Grid: krigelevation = kriging(usozonesites, elevation,. #, both, varelevation, linear, sample, 11,
1000000, 10000)
Questions
1) Look at these interpolated fields. How well did IDW or Kriging spatially interpolate elevation?
2) Does the varelevation grid seem honest with respect to characterizing the error of estimate?
3) Create two error grids with simple map algebra, from grid the commands would be:
Grid: idweleverror = nademlza – idwelevation
Grid: krigerror = nademlza – krigelevation
Display the grids idweleverror and krigeleverror in ArcView using standard deviation as a
classification and color scheme. How similar are the error patterns?
4) Now create the absolute error (so over and underestimates do not cancel each other out when you
calculate the mean). To do this use the absolute value function in grid:
Absidweleverror = abs(idweleverror)
Abskrigeleverror = abs(krigeleverror)
Which interpolator had the lowest Mean Absolute Deviation?
5) Comment on the usefulness of IDW and Kriging for interpolating Elevation data with point
coverages of the density of usozonesites.
The Population Density Dataset
Using the population density dataset for the conterminous United States that you used in the
nighttime satellite imagery lab repeat the same routine you did for elevation for population density.
Questions 1-B thru 5-B same as elevation dataset
The Landcover Dataset
Using the landcover dataset naigbplza dataset for the conterminous United States was in the
nighttime satellite imagery workspace repeat the same routine you did for elevation for landcover.
Note: smart students will not do this and explain to me that it is an inappropriate exercise.
6) Print the variograms for elevation and population density on a sheet of paper side by side. How do
they compare and how can they be interpreted in light of what you know about the spatial variability
of population and elevation in the conterminous United States.
Task #4: De-Trended Kriging of Ozone in Rocky Mountain National Park
Space is not the cause of variability in measurements of elevation, ozone concentration, etc. Kriging,
IDW, and other spatial interpolaters merely take advantage of the fact that spatial auto-correlation
exists. It is even better to build a model using spatially explicit measure of dependent variables,
apply them to extensive spatial coverages, look at the errors at known points and only krig the errors.
Approaches like this are sometimes called ‘de-trended’ kriging. For example, let’s consider ozone
concentration in Great Smoky Mountain National Park. Suppose you have a point coverage of ozone
measurements that you want an interpolated field of. Suppose you have a DEM, a land-cover
database, and slope. By intersecting the point coverage with these grids (summarize zones in
ArcView and a bunch of table joins) you will have a table from which you can build a single or
multiple regression to predict ozone from elevation and/or landcover and/or slope. Then apply this
regression to those grids. Compare this ‘model’ to the point values and get ‘errors’ at those point
values. Krig the ‘error’ values at those points and add the interpolated field of errors to your ‘model’.
Compare this to a simple ‘kriging’ and simple ‘IDWing’ of the original point data.
Procedures:
Produce a simple kriged estimate of ozone:
Grid: smpkriggsmnp = kriging(ozonelocs, adj_intsv_, # , both, varsmpkrig, linear, sample, 5, 500000,
200)
Produce a simple IDW estimate of ozone:
Grid: smpkriggsmnp = IDW(ozonelocs, adj_intsv_, #, 2,
Sample, 5, 500000, 10000)
Now you need to gather the data needed to build a model that uses more than simply the spatial
relationships of the ozone measurements at the ozone monitoring sites. To do this you need to read
in the ozonelocs point coverage, and the following grids: gsmnpelev, gsmnp_slope,
gsmnp_landcover. Do the GIS manipulations necessary to get a table with the following fields and
appropriate values in the records: Code4_site, Elevation, Adj_intsv_, gsmnpelev, gsmnp_slope,
gsmnp_landcover. Read that table into JMP. How does the Elevation variable correlate with the
gsmnpelev variable? This should be a strong relationship. It’s not perfect because the elevation
measurements were made with different devices. Now build a univariate or multivariate model that
predicts Adj_intsv_ using the other variables. Build a grid of that model estimate with ArcView or
Arc/INFO’s Grid module.
Questions:
1) How do these models compare visually?
2) How could you quantitatively compare the accuracy of these interpolations?
-for extra credit, brownie points, and my undying admiration…. Fire up the new Arc GIS 8.1 and see
if you can figure out how to get the Geo-Statistical analyst extension to do cross-validation.
LAST EXERCISE:
Use JMP to generate 3 random variables. 300 records with the following fields:
X coordinate 100*random Uniform Y Coord 100*random Uniform Z value Normal 20, 3
Create shapefile of points convert to point coverage. Krig these points and attach print-out of
variogram and interpret.