- [Instructor] Welcome back to module six.
In this part, we'll look at
Deterministic Interpolation Operations within ArcGIS.
In case you stepped away and forgot
what we were working with from the previous lectures,
I thought I'd show a quick slide with the data.
Once again, these are the recreation sites
for the state of Vermont.
And I display the point locations on the left
and then use that acreage attribute
that we've been working with and a graduated symbol size
to reveal where the difference in the size
of the individual recreation sites.
Now, the first of the deterministic methods
that we'll look at is known as Natural Neighbor.
And really this is an offshoot
of something we've already done,
which is to create Thiessen polygons.
So the simplest of all methods for interpolation
could be to create Thiessen polygons
and then assign the cell value
to that location that it's closest to on the landscape.
In this case, the Natural Neighbor approach
takes it one step further.
So first it constructs the Thiessen polygons
based on those sample point locations.
Let me see those polygons here drawn in black.
Then it draws a Thiessen polygon
for the interpolation point,
breaking down those individual polygons
from the first construct.
And the overlap
between those two different polygon datasets
define the weights that are used for estimation.
So the values that come out
of each one of those individual Thiessen polygons
is based on how much of that slice
comprises the total Thiessen polygon
from that round one calculation.
Now, this produces polygons of irregular size.
Where you have denser sampling, you'll get smaller polygons.
And this is an exact interpolator.
So, what's that look like
in terms of the Geoprocessing interface?
We enter our input point features here, recreation sites.
In this case, it's a Z value field
that we'll be working with.
It needs to be a numeric data type,
and we'll focus on the acreage attribute
of that recreation site's dataset.
If you switch over to the Environments tab,
you'll notice that the Cell Size
and Snap Raster Environments are honored, but not the Mask.
I set my output cell size to 500 meters in this case
since we're working on an interpolation
for the entire state.
And here's the output that we see.
Again, note that the mask parameter
from our environment settings has not been respected here
even though I have the state of Vermont boundaries
set as that mask.
If we zoom in, we get a closer look at the islands
that can form around the data from an approach like this.
We could also use
a fixed radius local averaging approach for interpolation.
It's a bit more complex than Natural Neighbor,
but certainly less complex
than really the rest of the others
that we have at our disposal.
The fixed radius approach smooths the sample data
when creating that surface.
It's not unlike a moving window analysis
that we've seen in previous exercises.
It computes the average of all the sample points
located within some search radius that you specify.
Too small of a search radius
and you'll have a lot of empty cells
resulting in no data values.
Too large and you might smooth out the dataset too much.
So, we see over on the right-hand side in the lower image
four cells for which we'll compute values.
The algorithm draws the search radius
around each one of those four cell locations,
identifies all of the neighbors within that search radius,
sums the value of those observations
and then computes the average
to assign to the output raster.
Note that if there are no observations
present within a search radius, no data value is assigned.
Because the fixed radius approach computes an average value,
it's an inexact interpolator,
meaning that the value at a known location,
the interpolated value at a sample location
might not match that sample value.
Next up is the
inverse distance weighted interpolation method.
This approach has an explicit assumption
that the things that are closer together are more alike
than those that are further apart.
Again, the basic tenant of spatial autocorrelation.
Now, the weight assignment here
is equal to the inverse distance
between the unknown location and the sample point location.
IDW also uses a power function.
The higher the P value,
the more rapid the decrease in the weight with distance,
meaning that you need to be closer to a location
in order to play a contributing factor
in the estimated value.
Now, this is an exact interpolator,
but you should use caution with both sampling or testing
when designating the number of neighbors
and the power function.
And we'll see an example here of the effect
that power function can have on the output surface.
The IDW interface is pretty straightforward.
Once again, I'm going to input my recreation sites
as my input point features
and specify acreage as my Z value field.
I've got a search distance of 2,500,
and then I'll put cell size of 500.
Now, my search radius is fixed and I'm using a power of two.
So that search radius is drawn
around that unknown value location,
the neighbors are identified and the value is assigned.
Now, I could also use a variable radius for my search.
In this case, I specify a minimum number of points
and, or a maximum distance.
ArcGIS then looks for the, in this case nearest 12 points
to that search location
and selects them for calculating the value
at that unknown point location.
We see here the output from an IDW operation
where we're looking for 12 neighbors
for each one of those interpolated points.
Zooming in, we see some more definition around the edges
where we saw a smoother pattern previously
with the Natural Neighbors approach.
Let's look at the impact
of the power parameter on the output.
All of these examples use the same choropleth color scheme
to render the values.
The one difference is that I changed the power
between each iteration of the IDW approach.
So here we are with a power factor of 0.5,
one, two, 10 and 25.
Remember, as the power increases,
the weight decreases quite rapidly,
meaning you need to be closer to that unknown location
to make a contribution to the estimated value.
That's why we see islands around some of these points
that have larger acreages assigned to them
and a lot of green area out there
which is the low end value on the color scale in this case
Now, you can also use the Geostatistical Wizard
to compute an inverse distance weighting surface.
And this is slightly different and a bit more complex
than what we just saw.
In the previous version,
ArcGIS makes quite a few assumptions for you.
When you use the Geostatistical Wizard,
you have a lot more control
over the way that your interpolation approach proceeds.
So, I've selected my inverse distance weighting
and input my recreation sites.
The next thing I see is a plot of the dataset
where the hash mark
is centered over an unknown point location
and the radius is drawn around it.
From that we can also see which of the other features
have been selected to be included in the estimation
of that value at the unknown location.
I can change my sector type to four with a 45 degree offset,
which now means that ArcGIS
will look for a minimum number of 10 neighbors
in each one of the quadrants that I've specified.
Next, we can look at the cross validation pane,
and in this case see that the model that's been built
does not necessarily fit all that well
and it might be due to the wild distribution
of the dataset itself.
From here, I can click Finish
and I can review my method report.
This gives me all the details
of the parameters that I set via the Geostatistical Wizard,
and I can also see the output over on the right-hand side.
Now, one thing to note,
when you're using the Geostatistical Wizard,
the output is a Geostatistical analyst layer file.
They're still easy to work with,
but if you want to share them
or do some further advanced calculations,
you might need to export those to a raster dataset.
If you right click,
I think you can find the right path to achieving that.
The next approach we'll look at is known as the Spline.
Now, this is a mathematical function
that attempts to minimize the overall surface curvature.
So think of it as sort of bending a piece of rubber
or a piece of paper
to fit through all of the observation points.
It evaluates a specified number of neighbors,
again, determined by the user.
Now, there's two different kinds of Spline approaches.
One is the regularized.
This is a smooth, gradually changing surface
whose values may exceed or fall below
the maximum and minimum values in the dataset respectively.
It also uses 1st, 2nd and 3rd order derivatives
in the calculation,
but don't worry, that happens behind the scenes.
It's not something I'm expecting you to learn.
When you specify the weight parameter here,
the higher the weight,
the smoother the surface that will result.
The range is usually between zero and five,
although it can go higher,
but typically you'll see the weight value
ranging between zero and 0.5.
The tension approach only uses 1st
and 2nd order derivatives.
It considers more points and produces a smoother surface,
but it increases the processing time.
And in this case, the higher the weight,
the coarser the surface.
The tighter it's going to be snapped
to those sample value locations.
The weight must be greater than zero.
Now, the Spline operations are both exact interpolators,
but they can be relaxed the exactness.
This is particularly helpful with the tension approach
because of that increased processing time that is required
if you want the surface to snap
directly to the sample values.
Here's a look at the regularized Spline acreage attribute
from the recreation sites dataset
where I'm using a weight of 0.1
and looking for 12 points within my neighborhood.
Contrast that with the Tension Spline.
Same set up here with a weight of 0.1
and 12 neighbor points.
And lastly, if we look at all four of these approaches
that we evaluated here,
I'm using the same choropleth shading scheme
for all four of these, and then I know the value ranges
for each of the output datasets.
So with our Natural Neighbor,
that data set range estimates from five to 152,000,
IDW, five to 154,000.
My Regularized Spline, -91,000 acres.
Not exactly sure what that means, all the way up to 512,000.
And lastly, the Tension Spline,
- 72,000, all the way up to 229,000.
So, you can see the difference in the output
from the various approaches.
Of course, if you were doing this in a real-world setting,
you'd want to get to know the data a lot more
before making decisions
about your neighborhood sizes and shapes.
Well, that's it for deterministic methods.
There's certainly a lot more options
within the ArcGIS toolkit that you could explore,
but I think this is a good place to start and stop for now.
Next up, we'll look at some Geostatistical approaches
to round out this week's lectures.