INTERPOLATION
Procedure to predict values of attributes at
unsampled points within the region sampled
Why?
Examples:
-Can not measure all locations:
- temperature
- acid rain deposition
- soil characteristics
- mining: gold deposits
•Time
•Money
•Impossible (physically, legally)
- Changing cell size
- Missing/unsuitable data
- Past date
(e.g. temperature)
Spatial Sampling:
- Gather observations representative of spatial
distribution of variable of interest.
Interpolation:
Use those sample points to predict values of
variable of interest at all other unsampled locations.
Sampling methods evaluated here:
- Systematic Sampling
- Random Sampling
- Cluster Sampling
- Adaptive Sampling
Systematic sampling
pattern
- Easy
- Samples spaced
uniformly at fixed X, Y
intervals
- Parallel lines
Advantages
- Easy to understand
Disadvantages
- All receive same
attention
- Difficult to stay on lines
- May be biased
Random Sampling
-Select random points
Advantages
- Less biased (unlikely to
match pattern in
landscape)
Disadvantages
- Does nothing to
distribute samples in
areas of high variation
- Difficult to explain,
location of points may be
a problem
Cluster Sampling
Cluster centers are
established (random or
systematic)
Samples arranged around
each center
Advantages
Reduced travel time
Less costly
Disadvantages
Less representative sampling
Adaptive sampling
- Higher density sampling where the
feature of interest is more variable.
- Requires some method of estimating
feature variation
Advantages
-Often efficient as large homogeneous
areas have few samples reserving
more for areas with higher spatial
variation.
Disadvantages
- If no method of identifying where
features are most variable then you
need to make several sampling visits;
Changes in sample density can not be
done on the spot
Spatial Sampling:
- Gather observations representative of spatial distribution of
variable of interest.
Interpolation:
- Use those sample points to predict values of variable of
interest at all other unsampled locations.
Interpolation methods evaluated here:
- Thiessen Polygons
- Fixed-radius – Local Averaging
- Inverse Distance Weighted
- Trend Surface
- Splines
- Kriging
INTERPOLATION
Many different methods
All methods use location and value at sampling locations to
estimate the variable of interest at unmeasured locations
Methods differ in weighting and number of observations used
Each method produces different results (even with same data)
No method best for every application
Accuracy is often judged by withheld sample points (difference
between the measured and interpolated values)
INTERPOLATION
Usually used for point-to-raster data
- Some methods produce contour lines (vector lines of uniform value)
Raster surface
•Values are measured at a set of sample points
•Raster layer boundaries and cell dimensions established
•Interpolation method estimate the value for the center of each
unmeasured grid cell
Contour Lines: Iterative process
•From the sample points estimate points of a value (e.g. 10° C)
•Connect these points to form a line
•Estimate the next value (e.g. 20 ° C), creating another line with
the restriction that lines of different temperatures do not cross.
Example Base
Elevation contours
Sampled locations and values
INTERPOLATION
Thiessen Polygon
Assigns interpolated value equal to the value found
at the nearest sample location
Conceptually simplest method
Only one point used (nearest)
Often called nearest sample or nearest neighbor
Thiessen Polygon
Start:
1)
3
1
1. Draw lines
connecting the
points to their
nearest neighbors.
2
5
4
2. Find the bisectors
of each line.
3. Connect the
bisectors of the
lines and assign
the resulting
polygon the value
of the center point
2)
3)
Sampled locations
and values
Thiessen polygons
INTERPOLATION
Thiessen Polygon
Advantage:
- Ease of application
- Appropriate for discrete (i.e., categorical) variables
Disadvantages:
- Accuracy depends largely on sampling density
- Boundaries often odd shaped at transitions
- Continuous variables often not well represented
INTERPOLATION
Fixed-Radius – Local Averaging
More complex than nearest sample
Cell values estimated based on the average of nearby samples
Samples used depend on search radius
(any sample found inside the circle is used in average, outside ignored)
•Specify output raster grid
•Fixed-radius circle is centered over a raster cell
Circle radius typically equals several raster cell widths
(causes neighboring cell values to be similar)
Several sample points used
Some circles many contain no points
Search radius important; too large may smooth the data too
much
INTERPOLATION
Fixed-Radius – Local Averaging
INTERPOLATION
Fixed-Radius – Local Averaging
INTERPOLATION
Fixed-Radius – Local Averaging
INTERPOLATION
Inverse Distance Weighted (IDW)
Estimates the values at unknown points using the
distance and values to nearby know points (IDW reduces
the contribution of a known point to the interpolated value)
Weight of each sample point is an inverse proportion to
the distance.
The further away the point, the less the weight in
helping define the unsampled location
INTERPOLATION
Inverse Distance Weighted (IDW)
Zi is value of known point
Dij is distance to known point
Zj is the unknown point
n is a user selected exponent
(often 1,2 or 3)
Any number of points may be
used up to all points in the
sample; typically 3 or more
INTERPOLATION
Inverse Distance
Weighted (IDW)
INTERPOLATION
Inverse Distance Weighted (IDW)
Factors affecting interpolated surface:
•Size of exponent, n affects the shape of the surface
(larger n means the closer points are more influential)
•A larger number of sample points results in a smoother
surface
INTERPOLATION
Inverse Distance Weighted (IDW)
INTERPOLATION
Inverse Distance Weighted (IDW)
INTERPOLATION
Trend Surface Interpolation
Fitting a statistical model, a trend surface, through the
measured points. (typically polynomial)
Where Z is the value at any point x
Where ais are coefficients estimated in a regression
model
INTERPOLATION
Trend Surface Interpolation
INTERPOLATION
Splines
Name derived from the drafting tool, a flexible ruler, that
helps create smooth curves through several points
Spline functions (also called splines) are use to interpolate
along a smooth curve. (similar to the flexible ruler)
Force a smooth line to pass through a desired set of
points
Constructed from a set of joined polynomial functions
INTERPOLATION : Splines
INTERPOLATION
Kriging
A statistically based estimator of spatial variables
Components:
•Spatial trend
(an increase/decrease in a variable that depends
on direction, e.g. temperature may decrease
toward the northwest)
•Autocorrelation
(the tendency for points near each other to have
similar values)
•Random
(statistically defined by probability function)
Creates a mathematical model which is used to estimate
values across the surface
Kriging
Concept of
Lag distance
Where:
Zi is a variable at a sample point
hi is the distance between sample
points
Every possible set of pairs Zi,Zj
defines a distance hij, and is different
by the amount
Zi – Zj.
The distance hij is know as the lag
distance between point i and j. Also
there is a subset of points in a sample
set that are a given lag distance apart
h=4
3
2
6
3
2
Kriging
Concept of
Spatial Autocorrelation
Higher autocorrelations
indicates points near
each other are alike.
This provides
substantial information
about nearby locations
h = width of 1 cell
INTERPOLATION
Kriging
Concept of
Semi-variance
Where Zi is the measured variable at one point
Zj is another at h distance away
n is the number of pairs that are approximately h distance apart
Semi-variance may be calculated for any h
(When nearby points are similar (Zi-Zj) is small so the semi-variance is small. High
spatial autocorrelation means points near each other have similar Z values)
INTERPOLATION (cont.)
Kriging
When calculating the semi-variance of a particular h often a
tolerance is used (as few h values will be identical).
Plot the semi-variance of a range of lag distances in a variogram
Variogram
Semi-variance is usually small at small lag distances and
increases to a constant value as the lag distance h
increases
Variogram
•A nugget is the initial semi-variance when the autocorrelation typically is highest
•The sill is the point where the variogram levels off; background noise; where there
is little autocorrelation
•The range is the lag distance at which the sill is reached
INTERPOLATION (cont.)
Kriging
•A set of sample points are used to estimate the shape of the
variogram
•Variogram model is made
(A line is fit through the set of semi-variance points)
•The Variogram model is then used to interpolate the entire
surface
INTERPOLATION (cont.)
Kriging
Similar to Inverse Distance Weighting (IDW)
Kriging uses the minimum variance method to calculate the
weights rather than applying an arbitrary or less precise weighting
scheme
INTERPOLATION
Kriging
Interpolation in ArcGIS: Spatial Analyst
Interpolation in ArcGIS: Geostatistical Analyst
Interpolation in ArcGIS: arcscripts.esri.com
INTERPOLATION (cont.)
Exact/Non Exact methods
(Is there a difference at the sample locations?)
Exact
Thiessen
IDW
Non Exact
Fixed-Radius (averages several points near the sample location)
Trend surface (surface typically does not pass through the measured points)
Spline
Kriging
Class Vote: Which method works best for this example?
Systematic
Random
Original Surface:
Cluster
Adaptive
Class Vote: Which method works best for this example?
Thiessen
Polygons
Fixed-radius –
Local Averaging
IDW: squared,
12 nearest points
Original Surface:
Trend Surface
Spline
Kriging
Core Area Identification
• Commonly used when we have
observations on a set of objects, want to
identify regions of high density
• Crime, wildlife, pollutant detection
• Derive regions (territories) or density
fields (rasters) from set of sampling
points.