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Spatial Interpolation in GIS Zhongwei Liu, Ph.D. School of Environmental and Public Affairs University of Nevada, Las Vegas 2/18/2010

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2 Outline Spatial interpolation –Linear interpolation –Nonlinear interpolation Case study Tutorials

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Operations on surfaces Interpolation –Linear interpolation –Nonlinear interpolation 3

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Linear interpolation 4 Half way from A to B, Value is (A + B) / 2 A B C

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Nonlinear interpolation Basic types –Inverse Distance Weighted (IDW) –Spline: fits a minimum-curvature surface through the input points –Kriging: use virogram to determine the neighborhood for interpolation 5

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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] 6 hx=??? h1h2 h3 d1d2 d3 w1=1/d1, w2=1/d2, w3=1/d3 w=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 7

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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) 9

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10 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. 12 d variation d*

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Kriging 13 The semi-variogram is based on modeling the (squared) differences in the z-values as a function of the distances between all of the known points.

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14 Kriging

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Cross validation Removing one of the n observation points and using the remaining n-1 points to predict its value. Error = observed - predicted 16

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17 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 18

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Interpolation Software ArcGIS 9.x with Geostatistical Analyst ArcView 3.x Surfer (Golden Software) Surface II package (Kansas Geological Survey) GEOEAS (EPA) Spherekit (NCGIA, UCSB) Matlab 19

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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 20

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21 Models based on spatial interpolation for Everglades restoration Everglades –Subtropical wetland –Dry (Oct.- May) and wet (Jun.- Sept.) seasons Everglades restoration –$7.8 billion Source: sawgrass marsh slough alligator holes tree islands wet 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 22

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23 Alligator hole profile

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24 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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25 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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26 EDEN DEM () 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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27 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 28

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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 29

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30 EDEN Interpolation MethodRadial Basis Functions (RBF) Kernel FunctionMultiquadric Parameter16.77 Neighbors1 Include at least1 Sector type8 Angle350 Major semiaxis31000 Minor semiaxis30000 Cross Validation Mean Prediction Error0.25 RMSE (m)40.45 Revised model parameters

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31 EDEN DEM revision - WCA 1 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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33 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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34 NorthCenterSouth Kriging MethodUniversal Lag Size400m Number Lags Trend1st AnisotropyYes Semivariogram ModelGaussianSphericalGaussian #HAED Points (Used /Total)526/ / /936 Cross Validation with HAED Data Mean Prediction Error RMSE (m) Average Standard Error Validation with Elevation from PI Depth *36 PI602 PI160 PI Mean Prediction Error RMSE (m) Average Standard Error Veg. mapFL GAP Kriging Method Ordinary TrendNo RMSE (m)0.162 RMSE - with PI depth (m) 0.36 Current released DEM Revised

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