Interpolation

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Jeff Jenness

jeffj@jennessent.com (928) 607-4638

Main Topics

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Main Topics

• What is Interpolation?

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Main Topics

• What is Interpolation?
• What is Not interpolation?

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Main Topics

• What is Interpolation?
• What is Not interpolation?
• Several Popular Methods

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What is Interpolation?

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

What is Interpolation?

• Estimate the value at a new location based on known values at other locations.

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

What is Interpolation?

• Estimate the value at a new location based on known values at other locations.
• A lot like Bilinear Interpolation, but with lots of variations.

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

Remember Bilinear Interpolation?

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Remember Bilinear Interpolation?

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What is Interpolation?

• Estimate the value at a new location based on known values at other locations.
• A lot like Bilinear Interpolation, but with lots of variations.
• We will mainly discuss tools available in Spatial Analyst.

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

What is Interpolation?

• Estimate the value at a new location based on known values at other locations.
• A lot like Bilinear Interpolation, but with lots of variations.
• We will mainly discuss tools available in Spatial Analyst.
• Most of the methods we will discuss have much more sophisticated versions in Geostatistical Analyst.

​

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

What is Interpolation?

• Estimate the value at a new location based on known values at other locations.
• A lot like Bilinear Interpolation, but with lots of variations.
• We will mainly discuss tools available in Spatial Analyst.
• Most of the methods we will discuss have much more sophisticated versions in Geostatistical Analyst.
• Generally these tools create new rasters, with values interpolated over the entire landscape.

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

What is not Interpolation?

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

What is not Interpolation?

• Not like most statistical models, where you predict a response variable from some combination of predictor variables.

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

What is not Interpolation?

• Not like most statistical models, where you predict a response variable from some combination of predictor variables.
• It’s got similarities, though…

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

What is not Interpolation?

• Not like most statistical models, where you predict a response variable from predictor variables.
• It’s got similarities, though…
• Also Not a method for predicting Rates of Occurrence, like the Point Density tools.

Popular Methods

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Popular Methods

• Nearest Sample Point (Voronoi, aka Thiessen, polygons)

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Popular Methods

• Nearest Sample Point (Voronoi, aka Thiessen, polygons)
• Produces Polygon Feature Class

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Popular Methods

• Nearest Sample Point (Voronoi, aka Thiessen, polygons)
• Produces Polygon Feature Class
• Weighted Values of Nearest Points

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Popular Methods

• Nearest Sample Point (Voronoi, aka Thiessen, polygons)
• Produces Polygon Feature Class
• Weighted Values of Nearest Points
• Natural Neighbor

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Popular Methods

• Nearest Sample Point (Voronoi, aka Thiessen, polygons)
• Produces Polygon Feature Class
• Weighted Values of Nearest Points
• Natural Neighbor
• IDW (Inverse Distance Weighted)

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Popular Methods

• Nearest Sample Point (Voronoi, aka Thiessen, polygons)
• Produces Polygon Feature Class
• Weighted Values of Nearest Points
• Natural Neighbor
• IDW (Inverse Distance Weighted)
• Kriging

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Popular Methods

• Nearest Sample Point (Voronoi, aka Thiessen, polygons)
• Produces Polygon Feature Class
• Weighted Values of Nearest Points
• Natural Neighbor
• IDW (Inverse Distance Weighted)
• Kriging
• Fitting Curves to Sample Points

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Popular Methods

• Nearest Sample Point (Voronoi, aka Thiessen, polygons)
• Produces Polygon Feature Class
• Weighted Values of Nearest Points
• Natural Neighbor
• IDW (Inverse Distance Weighted)
• Kriging
• Fitting Curves to Sample Points
• Spline: Local curves fitted to nearest points

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Popular Methods

• Nearest Sample Point (Voronoi, aka Thiessen, polygons)
• Produces Polygon Feature Class
• Weighted Values of Nearest Points
• Natural Neighbor
• IDW (Inverse Distance Weighted)
• Kriging
• Fitting Curves to Sample Points
• Spline: Local curves fitted to nearest points
• Trend: Global curve fit to all points

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Popular Methods

• Nearest Sample Point (Voronoi, aka Thiessen, polygons)
• Produces Polygon Feature Class
• Weighted Values of Nearest Points
• Natural Neighbor
• IDW (Inverse Distance Weighted)
• Kriging
• Fitting Curves to Sample Points
• Spline: Local curves fitted to nearest points
• Trend: Global curve fit to all points
• ANUDEM (aka Topo to Raster): Special Method for Topography

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Important Point

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Important Point

• Accuracy of any interpolation method depends on:

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Important Point

• Accuracy of any interpolation method depends on:
• Complexity and variation of the actual phenomenon you are analyzing

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Important Point

• Accuracy of any interpolation method depends on:
• Complexity and variation of the actual phenomenon you are analyzing
• How many sample points you have

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Important Point

• Accuracy of any interpolation method depends on:
• Complexity and variation of the actual phenomenon you are analyzing
• How many sample points you have
• How well distributed those sample points are

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Important Point

• Accuracy of any interpolation method depends on:
• Complexity and variation of the actual phenomenon you are analyzing
• How many sample points you have
• How well distributed those sample points are
• Make sure you are interpolating within the range of your sample points!

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Important Point

• Accuracy of any interpolation method depends on:
• Complexity and variation of the actual phenomenon you are analyzing
• How many sample points you have
• How well distributed those sample points are
• Make sure you are interpolating within the range of your sample points!
• Otherwise you are “Extrapolating”, not “Interpolating”!

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Database of Interpolation Samples

https://drive.google.com/file/d/1c6YIAhGjjzoF1VWzgDp0kk_5dD1-0TKK/view?usp=sharing

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Database of Interpolation Samples

https://drive.google.com/file/d/1c6YIAhGjjzoF1VWzgDp0kk_5dD1-0TKK/view?usp=sharing

• If you are interested, this database demonstrates interpolation methods using sample points from known phenomena:

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Database of Interpolation Samples

https://drive.google.com/file/d/1c6YIAhGjjzoF1VWzgDp0kk_5dD1-0TKK/view?usp=sharing

• If you are interested, this database demonstrates interpolation methods using sample points from known phenomena:
• Elevation

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Database of Interpolation Samples

https://drive.google.com/file/d/1c6YIAhGjjzoF1VWzgDp0kk_5dD1-0TKK/view?usp=sharing

• If you are interested, this database demonstrates interpolation methods using sample points from known phenomena:
• Elevation
• Human Population

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Database of Interpolation Samples

https://drive.google.com/file/d/1c6YIAhGjjzoF1VWzgDp0kk_5dD1-0TKK/view?usp=sharing

• If you are interested, this database demonstrates interpolation methods using sample points from known phenomena:
• Elevation
• Human Population
• Windspeed

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Database of Interpolation Samples

https://drive.google.com/file/d/1c6YIAhGjjzoF1VWzgDp0kk_5dD1-0TKK/view?usp=sharing

• If you are interested, this database demonstrates interpolation methods using sample points from known phenomena:
• Elevation
• Human Population
• Windspeed
• Vegetation Class

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Database of Interpolation Samples

https://drive.google.com/file/d/1c6YIAhGjjzoF1VWzgDp0kk_5dD1-0TKK/view?usp=sharing

• If you are interested, this database demonstrates interpolation methods using sample points from known phenomena:
• Elevation
• Human Population
• Windspeed
• Vegetation Class
• These allow you to directly compare the interpolated prediction with reality

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Database of Interpolation Samples

https://drive.google.com/file/d/1c6YIAhGjjzoF1VWzgDp0kk_5dD1-0TKK/view?usp=sharing

• If you are interested, this database demonstrates interpolation methods using sample points from known phenomena:
• Elevation
• Human Population
• Windspeed
• Vegetation Class
• These allow you to directly compare the interpolated prediction with reality
• Subtract original raster from interpolated version to get raster of interpolation error

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Voronoi (Thiessen) Polygons

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Voronoi (Thiessen) Polygons

• Simple method, does not require Spatial Analyst

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Voronoi (Thiessen) Polygons

• Simple method, does not require Spatial Analyst
• But it does require the Advanced license!

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Voronoi (Thiessen) Polygons

• Simple method, does not require Spatial Analyst
• But it does require the Advanced license!
• Just classify a point, or even a region, based on the nearest known point.

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Voronoi (Thiessen) Polygons

• Simple method, does not require Spatial Analyst
• But it does require the Advanced license!
• Just classify a point, or even a region, based on the nearest known point.
• Good for Categorical data.

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Voronoi (Thiessen) Polygons

• Simple method, does not require Spatial Analyst
• But it does require the Advanced license!
• Just classify a point, or even a region, based on the nearest known point.
• Good for Categorical data.
• Creates a Polygon feature class, where each polygon receives the attribute values from the sample points.

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Voronoi (Thiessen) Polygons

• Simple method, does not require Spatial Analyst
• But it does require the Advanced license!
• Just classify a point, or even a region, based on the nearest known point.
• Good for Categorical data.
• Creates a Polygon feature class, where each polygon receives the attribute values from the sample points.
• Use the Create Thiessen Polygons tool.

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Fun with Voronoi Polygons

• Nothing to do with Interpolation, but Voronoi polygons can also be used to separate the landscape into roughly equal-sized and equal-shaped units.

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• Iterative process, called Adaptive Voronoi Polygons.

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1. Create Voronoi Polygons around points
2. Create centroid points from Voronoi Polygons
3. Repeat until you are satisfied with results…

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• See https://youtu.be/PZ1dz7IGDm0

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Initial set of points can be randomly generated

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Initial set of points can be randomly generated

Generate Voronoi Polygons around points

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Initial set of points can be randomly generated

Generate Voronoi Polygons around points

Find centroid points of Voronoi Polygons

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Initial set of points can be randomly generated

Generate Voronoi Polygons around points

Find centroid points of Voronoi Polygons

Generate Voronoi Polygons around centroids

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Initial set of points can be randomly generated

Generate Voronoi Polygons around points

Find centroid points of Voronoi Polygons

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Initial set of points can be randomly generated

Generate Voronoi Polygons around points

Find centroid points of Voronoi Polygons

Rinse and repeat…

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Initial set of points can be randomly generated

Generate Voronoi Polygons around points

Find centroid points of Voronoi Polygons

Rinse and repeat…

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Initial set of points can be randomly generated

Generate Voronoi Polygons around points

Find centroid points of Voronoi Polygons

Rinse and repeat…

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Initial set of points can be randomly generated

Generate Voronoi Polygons around points

Find centroid points of Voronoi Polygons

Rinse and repeat…

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Initial set of points can be randomly generated

Generate Voronoi Polygons around points

Find centroid points of Voronoi Polygons

Rinse and repeat…

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Initial set of points can be randomly generated

Generate Voronoi Polygons around points

Find centroid points of Voronoi Polygons

Rinse and repeat…

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Initial set of points can be randomly generated

Generate Voronoi Polygons around points

Find centroid points of Voronoi Polygons

Rinse and repeat…

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Continue iterations until you are satisfied with arrangement…

Initial set of points can be randomly generated

Generate Voronoi Polygons around points

Find centroid points of Voronoi Polygons

Rinse and repeat…

Interpolating by Proximity-Weighted Values

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Interpolating by Proximity-Weighted Values

• Natural Neighbor: Voronoi-Based approach

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Interpolating by Proximity-Weighted Values

• Natural Neighbor: Voronoi-Based approach
• IDW (Inverse-Distance Weighted): Weighted purely on distance to point.

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Interpolating by Proximity-Weighted Values

• Natural Neighbor: Voronoi-Based approach
• IDW (Inverse-Distance Weighted): Weighted purely on distance to point.
• Kriging: Weighted by Distance and autocorrelation among nearby points.

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Natural Neighbor

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Natural Neighbor

• Based on Voronoi polygons again

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Natural Neighbor

• Based on Voronoi polygons again
• New values calculated from weighted values of neighbors

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Natural Neighbor

• Based on Voronoi polygons again
• New values calculated from weighted values of neighbors
• Weighting done by proportion of overlapping Voronoi regions…

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Natural Neighbors

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Natural Neighbors

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Natural Neighbors

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Natural Neighbors

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Natural Neighbors

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IDW (Inverse Distance Weighted)

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IDW (Inverse Distance Weighted)

• As with Natural Neighbors, new values calculated from weighted values of neighbors

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IDW (Inverse Distance Weighted)

• As with Natural Neighbors, new values calculated from weighted values of neighbors
• Weighting done by inverse distance to neighbors

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IDW (Inverse Distance Weighted)

• As with Natural Neighbors, new values calculated from weighted values of neighbors
• Weighting done by inverse distance to neighbors
• Close neighbors weighted higher than distant neighbors

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IDW (Inverse Distance Weighted)

• As with Natural Neighbors, new values calculated from weighted values of neighbors
• Weighting done by inverse distance to neighbors
• Close neighbors weighted higher than distant neighbors
• You specify number of neighbors, either by count or threshold distance

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IDW (Inverse Distance Weighted)

• As with Natural Neighbors, new values calculated from weighted values of neighbors
• Weighting done by inverse distance to neighbors
• Close neighbors weighted higher than distant neighbors
• You specify number of neighbors, either by count or threshold distance
• In my experience, makes a kind of spiky landscape.

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IDW (Inverse Distance Weighted)

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IDW (Inverse Distance Weighted)

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IDW (Inverse Distance Weighted)

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IDW (Inverse Distance Weighted)

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IDW (Inverse Distance Weighted)

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Kriging

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Kriging

• As with Natural Neighbor and IDW, new values calculated from weighted values of neighbors

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Kriging

• As with Natural Neighbor and IDW, new values calculated from weighted values of neighbors
• Weighting done by building semivariogram

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Kriging

• As with Natural Neighbor and IDW, new values calculated from weighted values of neighbors
• Weighting done by building semivariogram
• A fitted model showing how variance among sample points changes by distance

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Kriging

• As with Natural Neighbor and IDW, new values calculated from weighted values of neighbors
• Weighting done by building semivariogram
• A fitted model showing how variance among sample points changes by distance
• Weights are based on distance and model value

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Kriging

• As with Natural Neighbor and IDW, new values calculated from weighted values of neighbors
• Weighting done by building semivariogram
• A fitted model showing how variance among sample points changes by distance
• Weights are based on distance and model value
• Because weights are based on Model values rather than Sample Point values, Kriging interpolations can produce values outside the range of the original data.

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Kriging

• As with Natural Neighbor and IDW, new values calculated from weighted values of neighbors
• Weighting done by building semivariogram
• A fitted model showing how variance among sample points changes by distance
• Weights are based on distance and model value
• Because weights are based on Model values rather than Sample Point values, Kriging interpolations can produce values outside the range of the original data.
• You specify number of neighbors, either by count or threshold distance

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Kriging

• Quantifies Tobler’s observation:

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

Kriging

• Quantifies Tobler’s observation:

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• Calculates variance between every point, and all points in distance bands from every point.

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

Kriging

• Quantifies Tobler’s observation:

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• Calculates variance between every point, and all points in distance bands from every point.

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• Assumes that the general overall variance over entire region is homogeneous, but nearby features have lower variance.

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Everything is related to everything else, but near things are more related than distant things.

Waldo Tobler, The First Law of Geography

Kriging

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Kriging

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Kriging

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Kriging

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Kriging

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Kriging

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https://gis.stackexchange.com/questions/270274/how-to-calculate-kriging-weights/270935#270935

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Interpolating by Fitting Mathematical Curves

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Interpolating by Fitting Mathematical Curves

• These methods find the best-fitting curves through the sample points.

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Interpolating by Fitting Mathematical Curves

• These methods find the best-fitting curves through the sample points.
• The Spline method is similar to the neighbor-weighting approaches in that the interpolated landscape goes exactly through the sample points.

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Interpolating by Fitting Mathematical Curves

• These methods find the best-fitting curves through the sample points.
• The Spline method is similar to the neighbor-weighting approaches in that the interpolated landscape goes exactly through the sample points.
• The Trend method is similar to a linear regression model, which simply finds the best-fitting line through a scatterplot.

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Interpolating by Fitting Mathematical Curves

• Spline: Fits local curves around specified numbers of points

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Interpolating by Fitting Mathematical Curves

• Spline: Fits local curves around specified numbers of points
• Trend: Fits global polynomial curve for entire region

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Spline

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Spline

• New values calculated by fitting curves to neighbors

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Spline

• New values calculated by fitting curves to neighbors
• You specify number of neighbors

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Spline

• New values calculated by fitting curves to neighbors
• You specify number of neighbors
• Optionally you can set barriers that mark distinct changes in the landscape

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Spline

• Esri describes it as “bending a sheet of rubber to pass through the points, while minimizing the total curvature of the surface.”

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Spline

• Esri describes it as “bending a sheet of rubber to pass through the points, while minimizing the total curvature of the surface.”
• Appropriate for phenomena that change gradually over the landscape.

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Spline

• Not appropriate if there are large changes in the surface within a short horizontal distance, because it can overshoot estimated values.

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Trend

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East Kaibab Monocline, from Mike Collier (http://www.knau.org/post/land-lines-east-kaibab-monocline)

Trend

• New values calculated from Global Polynomial Model created from all sample points

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East Kaibab Monocline, from Mike Collier (http://www.knau.org/post/land-lines-east-kaibab-monocline)

Trend

• New values calculated from Global Polynomial Model created from all sample points

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• Good for estimating phenomena that cover large portions of your analysis area, like geologic features

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East Kaibab Monocline, from Mike Collier (http://www.knau.org/post/land-lines-east-kaibab-monocline)

Trend

• New values calculated from Global Polynomial Model created from all sample points

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• Good for estimating phenomena that cover large portions of your analysis area, like geologic features
• 1^st Order (Flat)

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Trend

• New values calculated from Global Polynomial Model created from all sample points

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• Good for estimating phenomena that cover large portions of your analysis area, like geologic features
• 1^st Order (Flat)
• 2^nd Order (1 bend)

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Trend

• New values calculated from Global Polynomial Model created from all sample points

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• Good for estimating phenomena that cover large portions of your analysis area, like geologic features
• 1^st Order (Flat)
• 2^nd Order (1 bend)
• 3^rd Order (2 bends)

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Trend

• New values calculated from Global Polynomial Model created from all sample points

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• Good for estimating phenomena that cover large portions of your analysis area, like geologic features
• You can specify higher orders if you want (up to 12).

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Trend

• New values calculated from Global Polynomial Model created from all sample points

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• Good for estimating phenomena that cover large portions of your analysis area, like geologic features
• You can also calculate a Logistic Regression trend

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Trend

• New values calculated from Global Polynomial Model created from all sample points

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• Good for estimating phenomena that cover large portions of your analysis area, like geologic features
• You can also calculate a Logistic Regression trend

• Good for predicting Presence/Absence

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Topo to Raster

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Topo to Raster

• Uses the ANUDEM algorithm

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Topo to Raster

• Uses the ANUDEM algorithm
• Specialized for Topography

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Topo to Raster

• Uses the ANUDEM algorithm
• Specialized for Topography
• ANUDEM stands for Australian National University Digital Elevation Model

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Topo to Raster

• Uses the ANUDEM algorithm
• Specialized for Topography
• ANUDEM stands for Australian National University Digital Elevation Model
• Can accept multiple elevation source datasets, including elevation points, contour lines and streams.

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Topo to Raster

• Uses the ANUDEM algorithm
• Specialized for Topography
• ANUDEM stands for Australian National University Digital Elevation Model
• Can accept multiple elevation source datasets, including elevation points, contour lines and streams.
• Tries to be Hydrologically Correct, but still creates sinks when creating a flow path “contradicts the input elevation data”.

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Topo to Raster

• Uses the ANUDEM algorithm
• Specialized for Topography
• ANUDEM stands for Australian National University Digital Elevation Model
• Can accept multiple elevation source datasets, including elevation points, contour lines and streams.
• Tries to be Hydrologically Correct, but still creates sinks when creating a flow path “contradicts the input elevation data”.
• Optional outputs include stream polylines

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Which methods are most accurate?

• Depends on the data, of course…
• Based on my 3 sample rasters of Elevation, Wind Speed and Arizona Population,
• And using randomly-distributed sample points at n=20, 200 and 4000
• And using default tool parameters
• Mean errors over entire raster (calculated as the mean of the absolute values) are as follows:

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