PRACTICAL ASPECTS OF APPLYING GEOSTATISTICS AT HAZARDOUS, TOXIC, AND RADIOACTIVE WASTE SITES
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
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
