Discussion - ETL-1110-1-1750003

Actions Required - ETL-1110-1-1750004

Chapter 1. Introduction - ETL-1110-1-1750007

Organization

Initial planning.

An Overview of Some Technical Aspects of Geostatistics

General considerations in spatial prediction

Variograms.

Kriging and kriging variance.

Indicator kriging. - ETL-1110-1-1750014

Chapter 2. Technical Aspects of Geostatistics

Figure 2-1. Diagrams showing A, hypothetical study area; B, stationary covariance function; and C, isotrophic covariance function

Example 1.

Example 1. (cont)

A covariance function is called stationary

Variograms

(Figure 2-1). Equation 2-20 is called an isotropic variogram and Equation 2-21 is a directional variogram at angle a.

Figure 2-2. Diagram showing variogram and features

Figure 2-3. Theoretical variograms showing A, exponential; B, spherical; C, Gaussian; and D, linear models

Ordinary kriging.

Ordinary kriging. (cont)

Effect of nugget.

Block kriging.

Co-kriging

Using Kriging to Assess Risk

Normal distributions and transformations

Indicator kriging. - ETL-1110-1-1750031

Chapter 3. Geostatistical Resources and Tools

Software

Software (cont)

Table 3-1. Geostatistical Software Characteristics

Chapter 4. Practical Aspects of Variogram Construction and Interpretation

General Computation of Empirical Variogram

Nonstationarity

Figure 4-2. Squared differencing of values for all possible pairs of points for Saratoga data

Figure 4-3. Initial sample variogram points for Saratoga data

Variogram Refinement

Figure 4-4. Sample variogram for ordinary-least squares trend residuals of Saratoga data

Figure 4-5. Sample variogram points for ordinary-least squares trend residuals of Saratoga data binned to 4 miles

Transformations and Anisotropy Considerations

Fitting a Theoretical Variogram to the Sample Variogram Points

Figure 4-6. Initial directional sample variogram points for raw Saratoga data--A, north-south and B, east-west

Exponential variogram.

Additional Trend Considerations

Outlier Detection

Calibration statistics.

Figure 4-7. Sample variogram points and theoretical spherical fit for iterated Saratoga residuals

Variogram-parameter adjustments.

Figure 4-8. Sample variogram points and theoretical Gaussian fit for iterated Saratoga residuals

Figure 4-9. Cross-validation probability plot for Saratoga data

Figure 4-10. Scatter plot of measured versus kriging estimates from cross-validation of Saratoga data

Chapter 5. Practical Aspects of Geostatistics in Hazardous, Toxic, and Radioactive Waste Site Investigations

Figure 5-1. Measured data for water-level examples--A, original data; B, original data without dropped sites; C, original data with added sites (added sites indicated with +) (Sheet 1 of 3)

Figure 5-1. Measured data for water-level examples--A, original data; B, original data without dropped sites; C, original data with added sites (added sites indicated with +) (cont) - ETL-1110-1-1750058

Figure 5-1. Measured data for water-level examples--A, original data; B, original data without dropped sites; C, original data with added sites (added sites indicated with +) (cont) - ETL-1110-1-1750059

Figure 5-2. Variogram and variogram cross-validation plots for residuals in water-level example--A, theoretical variogram; B, cross-validation scatterplot; C, cross-validation probability plot

Bedrock-Elevation Examples

Figure 5-2. Variogram and variogram cross-validation plots for residuals in water-level example--A, theoretical variogram; B, cross-validation scatterplot; C, cross-validation probability plot (cont)

Table 5-1. Variogram Characteristics and Cross-Validation Statistics

Figure 5-3. Kriging results for water-level examples--A, kriging estimates for original data

Figure 5-3. Kriging results for water-level examples--A, kriging estimates for original data (cont)

Figure 5-3. Kriging results for water-level examples--A, kriging estimates for original data (cont) - ETL-1110-1-1750066

Figure 5-3. Kriging results for water-level examples--A, kriging estimates for original data (cont) - ETL-1110-1-1750067

Water-Quality Examples

Figure 5-4. Measured data for bedrock-elevation examples-A, original data and B, restricted data

Figure 5-4. Measured data for bedrock-elevation examples-A, original data and B, restricted data (cont)

Figure 5-5. Variogram and variogram cross-validation plots for bedrock-elevation example-A,

Figure 5-5. Variogram and variogram cross-validation plots for bedrock-elevation example-A, (cont)

Figure 5-5. Variogram and variogram cross-validation plots for bedrock-elevation example-A, (cont)

Figure 5-6. Kriging results for bedrock-elevation examples-A,

Figure 5-6. Kriging results for bedrock-elevation examples-A, (cont)

Figure 5-6. Kriging results for bedrock-elevation examples-A, (cont) - ETL-1110-1-1750076

Figure 5-6. Kriging results for bedrock-elevation examples-A, (cont) - ETL-1110-1-1750077

Natural log transformations are routinely

Figure 5-7. Measured data for water-quality examples

Figure 5-8. Directional variograms and variogram cross-validation plots for water-quality example-A,

Figure 5-8. Directional variograms and variogram cross-validation plots for water-quality example-A, (cont)

Figure 5-8. Directional variograms and variogram cross-validation plots for water-quality example-A, (cont) - ETL-1110-1-1750082

Figure 5-8. Directional variograms and variogram cross-validation plots for water-quality example-A, (cont) - ETL-1110-1-1750083

Figure 5-9. Kriging results for water-quality example-A, kriging estimates back-transformed;

Figure 5-9. Kriging results for water-quality example-A, kriging estimates back-transformed; (cont)

Figure 5-9. Kriging results for water-quality example-A, kriging estimates back-transformed; (cont) - ETL-1110-1-1750086

Figure 5-9. Kriging results for water-quality example-A, kriging estimates back-transformed; (cont) - ETL-1110-1-1750087

The established variogram

Figure 5-10. Directional variogram plots for indicator kriging water-quality example

Figure 5-10. Directional variogram plots for indicator kriging water-quality example (cont)

Figure 5-11. Indicator kriging results for water-quality example

Chapter 6. Review of Kriging Applications

Important Elements of Kriging Applications

Errors in Measured Data

Chapter 7. Other Spatial Prediction Techniques

Simple Moving Average

Triangulation

Figure 7-1. Diagram showing Voronoi polygons

Trend-Surface Analysis

Simulation

Simulation (cont)

Appendix A: References - ETL-1110-1-1750102

Appendix A: References (cont)

Appendix B: Notation

Appendix B: Notation (cont)

ETL 1110-1-175