Spatial Interpolation in GIS

transcript

Zhongwei Liu, Ph.D.

School of Environmental and Public AffairsUniversity of Nevada, Las Vegas

Zhongwei.Liu@unlv.edu2/18/2010

OutlineSpatial interpolation

– Linear interpolation– Nonlinear interpolation

Case study

Tutorials

Operations on surfaces Interpolation

– Linear interpolation

– Nonlinear interpolation

Linear interpolation

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

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

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,

hx=???

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

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

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100

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)

Spline

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.

variation

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

Kriging

Cross validationRemoving one of the n observation

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

Error = observed - predicted

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

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

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

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

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

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

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

-260.0

-210.0

-160.0

-110.0

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

Sampling interval (m)

Water level Ground Bedrock

Cattails marshWillowhead Open water

Hole water depth

Sediment depth

Marsh water depth

Alligator hole profile

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

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

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

EDEN water depth

= water level – DEM

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

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

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

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)

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

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)

RMSE - with PI depth (m)

Current released DEM

Revised

Spatial Interpolation in GIS

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