Spatial Interpolation in GIS

Zhongwei Liu, Ph.D.

School of Environmental and Public AffairsUniversity of Nevada, Las Vegas

Zhongwei.Liu@unlv.edu2/18/2010

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OutlineSpatial interpolation

– Linear interpolation– Nonlinear interpolation

Case study

Tutorials

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Operations on surfaces Interpolation

– Linear interpolation

– Nonlinear interpolation

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Linear interpolation

Half way from A to B,Value is (A + B) / 2

A

BC

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Nonlinear interpolationBasic types

– Inverse Distance Weighted (IDW)

– Spline: fits a minimum-curvature surface through the input points

– Kriging: use virogram to determine the neighborhood for interpolation

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1. Inverse Distance

Weighted (IDW) Each input point has a local influence that diminishes

with distance an implementation of Tobler’s First Law of Geography Use inverse distance as weight for summation of values

in a neighborhood The new [Hmin, Hmax] is within the original [Hmin,

Hmax]

hx=???

h1 h2

h3

d1 d2

d3

w1=1/d1, w2=1/d2, w3=1/d3w=w1+w2+w3

hx=h1*w1/w+h2*w2/w+h3*w3/w =(h1*w1+h2*w2+h3*w3)/w

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A potentially undesirable characteristic of IDW

interpolation This set of six data

points clearly suggests a hill profile. But in areas where there is little or no data the interpolator will move towards the overall mean. Blue line shows the profile interpolated by IDW

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2. Spline Like bending a sheet of rubber to pass through

points while minimizing curvature of that sheet repeatedly applies a smoothing equation (polynomial) to the surface

Resulting surface passes through all points

Best for gently varying surfaces, not for rugged ones (can overshoot data values)

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Spline

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3. Kriging Use virogram to determine the neighborhood

for interpolation– Based on spatial auto-correlation– Use d* to define the neighborhood

Fits function to – Specified number of points OR– All points within a window of specified

radius Assumes distance or direction between sample

points shows a spatial correlation that help describe the surface.

Kriging differs from the methods discussed so far because kriging can assess the quality of prediction with estimated prediction errors.

d

variation

d*

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KrigingThe semi-variogram is based on modeling the (squared) differences in the z-values as a functionof the distances betweenall of the known points.

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Kriging

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Cross validationRemoving one of the n observation

points and using the remaining n-1 points to predict its value.

Error = observed - predicted

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IDW vs. Kriging

Kriging appears to give a more “natural” look to the data

Kriging avoids the “bulls eye” effect

Kriging gives us a standard error

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Which Method to Use? IDW - assumes variable decreases in

influence w/distance from sampled location– Interpolating a surface of consumer

purchasing power for a retail store

Spline - best for surfaces that are already smooth– Elevations, water table heights, etc.

Kriging - if you already know correlated distances or directional bias in data– Geology, soil science

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Interpolation SoftwareArcGIS 9.x with Geostatistical Analyst ArcView 3.xSurfer (Golden Software) Surface II package (Kansas Geological

Survey) GEOEAS (EPA) Spherekit (NCGIA, UCSB)Matlab

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The Everglades

10, 000 islands (tree islands)

6 Inches beneath sea level Average annual rainfall 130

cm Over 2,000 plant species

http://sofia.usgs.gov/eden

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Models based on spatial interpolation for Everglades

restorationEverglades

– Subtropical wetland

– Dry (Oct.- May) and wet (Jun.- Sept.) seasons

Everglades restoration– $7.8 billion Source: www.broward.edu.

sawgrass marsh

slough

alligator holes

tree islandswet

prairiewet prairie

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Alligator hole & water level and

depth American Alligator

– Top predator, keystone species, ecosystem engineer in Florida Everglades

Alligator Hole– Small but persistent ponds

excavated and maintained by alligators

– Dry-season refugia– Nest, colonization, and

foraging sites

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-260.0

-210.0

-160.0

-110.0

-60.0

-10.0

40.0

90.0

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

Sampling interval (m)

Elev

atio

n (c

m)

Water level Ground Bedrock

Cattails marshWillowhead Open water

Hole water depth

Sediment depth

Marsh water depth

Alligator hole profile

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Everglades Depth Estimation Network

(EDEN) Funded by Comprehensive

Everglades Restoration Plan (CERP) and USGS Priority Ecosystem Sciences (PES)

Integrated network of real-time water level monitoring, ground elevation modeling, and water-surface modeling

Daily water level/stage data from 253 gage stations

A marsh gage station

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EDEN Water-Surface Model

Developed by Pearlstine et al. (2007), validated by Liu et al. (2009)

Spatial interpolation of water levels at 240 gage stations in ArcGIS: radial basis function (RBF)

Basic model outputs– Water level/stage (direct output

)– Water depth (= water level –

DEM) 2000 – present

Cell resolution: 400 m

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EDEN DEM (Digital Elevation

Model) Developed by Jones and

Price (2007)

Spatial interpolation of High Accuracy Elevation Data (HAED) in ArcGIS: kriging

HAED elevation points collected via Airborne Height Finder and airboat

Cell resolution: 400 m

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EDEN water depth

= water level – DEM

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Revisions of EDEN Water-Surface Model

Modification to the canals files to better represent NE Shark River Slough in the area of Tamiami Trail and L67 Extension

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Revisions of EDEN Water-Surface Model

Reparameterization of the EDEN water-surface model – With new gage stations (including

coastal)

– With resurveyed gage information (locations, water levels) in NAVD88 datum

– RBF surface interpolation by EDEN sub-regions

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EDENInterpolation Method Radial Basis Functions (RBF)Kernel Function MultiquadricParameter 16.77Neighbors 1 Include at least 1Sector type 8Angle 350Major semiaxis 31000Minor semiaxis 30000Cross ValidationMean Prediction Error 0.25RMSE (m) 40.45

Revised model parameters

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EDEN DEM revision - WCA 1

Spatial trend

Kriging interpolation– Ordinary kriging– Universal kriging

(considering the trend)

– Cross-validation– Validation with

independent elevation data derived from measured depths (PI depth, n = 1,491)

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Kriging by 3 landscape units

Kriging by landscape unit (north, center, south)

Removed HAED elevation point based on SFWMD new vegetation/land use map

– HAED point falling on upland + others; and

– areal coverage of upland + others in the EDEN cell less than 33%

PI data

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North Center SouthKriging Method Universal Universal UniversalLag Size 400m 400m 400mNumber Lags 20 30 20Trend 1st 1st 1stAnisotropy Yes Yes YesSemivariogram Model Gaussian Spherical Gaussian#HAED Points (Used /Total) 526/526 1857/1857 935/936Cross Validation with HAED DataMean Prediction Error 0.0002 0.00009 -0.007

RMSE (m) 0.133 0.141 0.203Average Standard Error 0.137 0.142 0.212Validation with Elevation from PI Depth * 36 PI 602 PI 160 PIMean Prediction Error -0.0037 0.056 0.13RMSE (m) 0.0799 0.122 0.198Average Standard Error 0.129 0.138 0.194

Veg. map

FL GAP

Kriging Method

Ordinary

Trend NoRMSE (m)

0.162

RMSE - with PI depth (m)

0.36

Current released DEM

Revised