Geostatistics in Practice: Interpolation
Through Examples Prahlad Jat
Eric Krause
Sessions of note…Tuesday
• Interpolating Surfaces in ArcGIS (1:00-2:00 SDCC Rm33C)
• Kriging: An Introduction to Concepts and Applications (2:30-3:30 SDCC Rm33C)
• Geostatistical Analyst: An Introduction (4:00-5:00 SDCC Rm30C)
Wednesday
Thursday
• Surface Interpolation in ArcGIS (11:15-12:00 SDCC Demo Theater 10)
• Empirical Bayesian Kriging and EBK Regression Prediction in ArcGIS (2:30-3:15 SDCC Demo Theater 10)
• Geostatistics in Practice: Learning Interpolation Through Examples (8:30-9:30 SDCC Rm30A)
• Polygon-to-Polygon Predictions Using Areal Interpolation (11:15-12:00 SDCC Demo Theater 10)
• Geostatistical Analyst: An Introduction (1:00-2:00 SDCC Rm30A)
• Using Living Atlas Data and ArcGIS Pro for 3D Interpolation (2:30-3:30 SDCC Rm 30C)
• Interpolating Surfaces in ArcGIS (4:00-5:00 SDCC Rm15A)
• Kriging: An Introduction to Concepts and Applications (4:00-5:00 SDCC Rm15B)
2
Geostatistical Analyst Resourceshttp://esriurl.com/GeostatGetStarted
• GeoNet – community.esri.com
- Blogs
- Free textbook and datasets
- Spatial Statistical Analysis For GIS Users
- Lots of discussions and Q&A
• Learn GIS – learn.arcgis.com
- Model Water Quality Using Interpolation
- Analyze Urban Heat Using Kriging
- Interpolate 3D Oxygen Measurements in Monterey Bay
Outline
• Interpolation
• Demo: interpolation with barriers
• Geostatistical interpolation
• Steps in geostatistical interpolation
• Demo: geostatistical interpolation (impact of mean trend)
• Advanced geostatistical Interpolation (EBK, Regression EBK)
• Demo: EBK, Regression EBK
• From 2D to 3D
• Demo: 3D
• Questions
What is interpolation?
• Predict values at unknown locations using values at measured locations
• Why: Cost prohibitive & impractical to measure values everywhere
Interpolation methods
Deterministic method:
- Solely based on mathematical functions (not based on statistical theory)
- IDW (Inverse Distance Weighted), Spline interpolation
- Not able to estimate prediction error
Geostatistical method:
- Based on both mathematical functions and statistical models (spatial autocorrelation)
- Can account for direction dependent weighting
- Kriging
Demo
Interpolation with
Barriers
Why Geostatistical Methods for Interpolation?
✓ Theory based advanced interpolation methods
✓ Quantify the spatial autocorrelation
✓ Account for the spatial configuration of measured sample values (directionality in data)
✓ Unlike deterministic methods, they also provide the uncertainty of predictions
Geostatistical Interpolation Assumptions
✓ Data is normally distributed
✓ Data has spatial autocorrelation
✓ Data has no local trend
✓ Data exhibits stationarity
✓ Mean stationarity: mean is constant between samples & is independent of location
✓ Intrinsic stationarity: the variance of the difference is the same between any two points
that are at the same distance and direction apart no matter which two points you choose.
Steps in Geostatistical Interpolation
❑ Exploratory spatial data analysis (ESDA)
❑ Mean trend analysis
❑ Modeling autocorrelation (semivariogram)
❑ Search neighborhood and performing interpolation
❑ Cross validation
Steps in Geostatistical Interpolation
❑ Exploratory spatial data analysis (ESDA)
Purpose: to maximize insight into the data
- To detect outliers
- To explore the distribution of data (determine: data transformation)
Techniques:
- Data visualization
- Charting/plotting (histogram, QQ plot)
Steps in Geostatistical Interpolation
❑ Mean trend analysis
Mean trend: Systematic and gradual changes across study domain
Z(s) = µ(s) + ε(s)
trend random errorWhy: Identifying and removing mean trend may improve interpolation accuracy
Challenge: No magical way to decompose data uniquely into trend & random error
Risk: Overfitting
Steps in Geostatistical Interpolation
❑ Modeling semivariogram (autocorrelation)
Steps in Geostatistical Interpolation
❑ Search neighborhood and perform interpolation
Neighborhood Prediction Prediction error
Steps in Geostatistical Interpolation
❑ Cross validation
Cross validation: Technique to evaluate the reliability of the model
Why: Predictions of every interpolation method are different
Method: leave-one-out cross validation (LOOC)
- Iteratively discard each sample
- Use remaining points to estimate value at measured location
- Compare predicted versus measured value
Steps in Geostatistical Interpolation
❑ Cross validation
Cross validation statistics:
Error = (predicted value - true value )
- Root-Mean-Square (RMS): root of average squared errors
- Root-mean-square standardized (RMSS): standardized RMSE
- Mean error: average of the errors
- Mean standardized error: standardized mean errors
Demo
Geostatistical
Interpolation
Impact of mean trend on
- variogram modeling
- cross validation
Advanced geostatistical Interpolation
Limitations with traditional geostatistical methods:
✓ Modeled semivariogram perfectly captures spatial autocorrelation
✓ A single semivariogram can truly represent autocorrelation
in the entire study area
(spatial dependency is equally distributed over the whole study area)
Solution: EBK (Empirical Bayesian Kriging) , Regression EBK
EBK (Empirical Bayesian Kriging), Regression EBK
EBK
✓ Easier to run: requires minimal interaction
✓ Better handles small and nonstationary datasets
✓ Doesn’t assume one semivariogram model fits the entire data
Regression EBK
✓ Use explanatory variable to improve predictions
✓ Handles multicollinearity using PCA (principle component analysis)
Demo
Geostatistical
InterpolationEBK/Regression EBK
- stationarity
- regression
From 2D to 3D
More and more 3D data are collected
Datasets in geosciences have samples in 3D space
Example: Oceanographic data
EBK in 3D (new in Pro2.3): Empirical Bayesian Kriging in 3D space)
Demo
Geostatistical
Interpolation
EBK3D