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