**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

**Suggested Citation:**"Chapter 7 - Development of Correlation Equations." National Academies of Sciences, Engineering, and Medicine. 2019.

*Relationship Between Erodibility and Properties of Soils*. Washington, DC: The National Academies Press. doi: 10.17226/25470.

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168 The main goal of this research was to develop equations correlating soil erosion parameters defined in previous chapters (i.e., EC, Ev, Et, vc, and tc) with the common soil engineering properties. Since the majority of the erosion test data compiled in NCHRP-Erosion consists of data obtained from the erosion function apparatus (EFA), jet erosion test (JET), and hole erosion test (HET), and given that these tests are the main erosion tests, the regression analyses were focused on these three tests. Section 7.1 of this chapter presents the quickest method to estimate the erosion resistance of a soil using the Unified Soil Classification System (USCS). In section 7.2, the plots of critical shear stress/velocity versus mean particle size (Chapter 3, Figure 31) are populated with the hundreds of EFA test data compiled in the NCHRP-Erosion spreadsheet, and new equations are developed. The final phase of this project consisted of two parallel statistical approaches: deterministic frequentist regression analysis and probabilistic (Bayesian) analysis. The goal was to reach the best potential fit between erodibility parameters and geotechnical properties of soils. The experimental design, optimization, model selection, and final results of the deterministic frequentist regression approach are comprehensively presented in Section 7.3. The results of the probabilistic (Bayesian) approach are presented in Section 7.4. 7.1 Determining Erosion Resistance Using the USCS As discussed in Chapter 1, the erosion function charts are charts that show erosion categories in the z . versus v and the z . versus t space (see Chapter 1, Figure 3). These charts were conceptually designed to eliminate the need for site-specific erosion testing in the case of preliminary inves- tigations and first order erosion analyses. The erosion function charts were first developed on the basis of EFA tests performed at Texas A&M University after Hurricane Katrina in 2005. The categories in the charts are zoned according to boundaries that originate at the critical velocity of the corresponding erosion category. Table 1 (Chapter 1) shows the values of tc and vc according to the erosion categories. This section discusses the introduction of zones that represent different soil types on the erosion charts. These zones are characterized by using the USCS categories. The erodibility of coarse-grained soils is influenced mostly by gravity forces and, therefore, by the grain size. Since the USCS soil classification for coarse-grained soils is based primarily on grain size dis- tribution, it is thought to have good potential for distinguishing between erosion categories of coarse-grained soils. For fine-grained soils, parameters such as soil structure, orientation of clay particles, and aging may be important in characterizing erodibility (Lefebvre et al. 1985; Partheniades 2009). It was also observed that plasticity, clay content, and soil activity play the most dominant roles in the erosion resistance of fine-grained soils. Plasticity parameters form C H A P T E R 7 Development of Correlation Equations

Development of Correlation Equations 169 the basis of the USCS classification for fine-grained soils, and it is reasonable to think that the USCS categories therefore have good potential for distinguishing between erosion categories for fine-grained soils as well. As discussed in the previous chapters, nearly 330 EFA test results compiled in the NCHRP- Erosion spreadsheet were divided into different USCS classification groups. Table 32 lists the USCS classification groups along with the number of EFA tests in each group. Figure 131 through Figure 144 show the erosion functions of the samples plotted according to their USCS category in velocity space. The highlighted zones and the dashed red lines in each figure are proposed after consideration of the two following criteria: 1. The highlighted zone contains nearly 90% of the EFA test data for that specific USCS category. 2. The zone is adjusted (especially in cases in which there are not enough EFA data to make an inclusive conclusion) so that it is reasonably consistent with the previously proposed version of this chart by Briaud (2008). The erosion functions for soils with a given USCS category do not generally fall distinctly into a single erosion category but rather seem to plot approximately across two categories. Figure 145 summarizes all results into the two erosion category charts shown in Chapter 1, Figure 3. Figure 145 can be used as a preliminary step to estimate the erodibility of any sample, using the USCS. The width of each box, which is associated with a USCS category, represents the zone in which 90% of the EFA results performed on such samples would fall in the erosion category chart. For instance, if the soil type of a location in an arbitrary geotechnical site is classified as SM (silty sand) according to the USCS, it would most likely (with close to 90% confidence based on the EFA results compiled in NCHRP-Erosion) fall into the Category II (high erodibility) on Figure 145. Similarly, a soil classified as CH (fat clay) would most likely fall into the Category III (medium erodibility), and a SP (poorly graded sand) would fall within the Categories I and II (very high to high erodibility). Evidently, the wider the box is for a USCS category, the more the variability of the erodibility is for that particular soil type. It must be noted that the boxes shown in Figure 145 apply solely to the erosion category and are not shown with respect to the erosion rate. USCS Categories Number of Samples Fat clay (CH) 63 Lean clay (CL) 131 Poorly graded gravel (GP) 7 Clayey gravel (GC) 1 High-plasticity silt (MH) 14 Low-plasticity silt (ML) 24 Low-plasticity silty clay (ML-CL) 14 Clayey sand (SC) 28 Clayey silty sand (SC-SM) 8 Silty sand (SM) 17 Poorly graded sand (SP) 16 Poorly graded sand with clay (SP-SC) 3 Poorly graded sand with silt (SP-SM) 2 Well graded sand with silt (SW-SM) 1 Table 32. List of the USCS categories associated with the 329 samples.

170 Relationship Between Erodibility and Properties of Soils Figure 131. Velocityâerosion rate and shear stressâ erosion rate plots for fat clay (CH) soils. Figure 132. Velocityâerosion rate and shear stressâ erosion rate plots for lean clay (CL) soils.

Development of Correlation Equations 171 Figure 133. Velocityâerosion rate and shear stressâ erosion rate plots for poorly graded gravel (GP) soils. Figure 134. Velocityâerosion rate and shear stressâ erosion rate plots for clayey gravel (GC) soils.

172 Relationship Between Erodibility and Properties of Soils Figure 135. Velocityâerosion rate and shear stressâ erosion rate plots for high-plasticity silt (MH) soils. Figure 136. Velocityâerosion rate and shear stressâ erosion rate plots for low-plasticity silt (ML) soils.

Development of Correlation Equations 173 Figure 137. Velocityâerosion rate and shear stressâ erosion rate plots for low-plasticity silty clay (ML-CL) soils. Figure 138. Velocityâerosion rate and shear stressâ erosion rate plots for clayey sand (SC) soils.

174 Relationship Between Erodibility and Properties of Soils Figure 139. Velocityâerosion rate and shear stressâ erosion rate plots for clayey silty sand (SC-SM) soils. Figure 140. Velocityâerosion rate and shear stressâ erosion rate plots for silty sand (SM) soils.

Development of Correlation Equations 175 Figure 141. Velocityâerosion rate and shear stressâ erosion rate plots for poorly graded sand (SP) soils. Figure 142. Velocityâerosion rate and shear stressâ erosion rate plots for poorly graded sand with clay (SP-SC) soils.

Figure 143. Velocityâerosion rate and shear stressâ erosion rate plots for poorly graded sand with silt (SP-SM) soils. Figure 144. Velocityâerosion rate and shear stressâ erosion rate plots for well-graded sand with silt (SW-SM) soils.

Development of Correlation Equations 177 Figure 145. Erosion category charts with USCS symbols. 0.1 1 10 100 1,000 10,000 100,000 0.1 1.0 10 100 VELOCITY (m/s) EROSION RATE (mm/hr) Very High Erodibility I High Erodibility II Medium Erodibility III Low Erodibility IV Very Low Erodibility V -Fine Sand -Non-plastic Silt -Medium Sand -Low-Plasticity Silt - Increase in Compaction (well-graded soils) -Increase in Density -Increase in Water Salinity (clay) Non-Erosive VI -Fine Gravel -High-Plasticity Silt -Low-Plasticity Clay -All fissured clays -Jointed Rock (spacing < 30 mm) -Cobbles -Coarse Gravel -High-Plasticity Clay -Jointed Rock (30â150 mm spacing) -Riprap -Jointed Rock (150â1,500 mm spacing) -Intact Rock - Jointed Rock (spacing > 1,500 mm) MH SP-SM ML-CL Rock SW SW-SM SP-SC SP SM SC-SM SC ML GC CL GP CH 0.1 1 10 100 1,000 10,000 100,000 EROSION RATE (mm/hr) Very High Erodibility I High Erodibility II Medium Erodibility III Low Erodibility IV Very Low Erodibility V 0.1 1 10 100 1,000 10,000 100,000 SHEAR STRESS (Pa) -Fine Sand -Non-plastic Silt -Medium Sand -Low-Plasticity Silt - Increase in Compaction (well-graded soils) - Increase in Density - Increase in Water Salinity (clay) -Fine Gravel -High-Plasticity Silt -Low-Plasticity Clay -All fissured clays -Jointed Rock (spacing < 30 mm) -Cobbles -Coarse Gravel -High-Plasticity Clay -Jointed Rock (30â150 mm spacing) -Riprap -Jointed Rock (150â1,500 mm spacing) -Intact Rock -Jointed Rock (spacing > 1,500 mm) MH SP-SM ML-CL Rock SW SW-SM SP-SC SP SM SC-SM SC ML GC CL GP CH

178 Relationship Between Erodibility and Properties of Soils 7.2 Plots of Critical Velocity and Shear Stress Versus Mean Particle Size Briaud et al. (2001a) and Briaud et al. (2017b) proposed a set of equations to predict the critical velocity and critical shear stress of coarse-grained soils based on many EFA erosion tests performed at Texas A&M University. The number of data points used to generate the equations and corresponding plots were limited to few erosion test results. One of the goals of this NCHRP project was to update and possibly modify the older plots and equations using hundreds of new test results compiled in the NCHRP-Erosion spreadsheet. It was observed that for soils with mean particle size larger than 0.3 mm, following relation- ships exist between the critical velocity/shear stress and mean particle size (Equations 61 and 62). v Dc ( )( ) ( )= âm s 0.315 mm (61)50 0.5 Dc ( ) ( )t =Pa mm (62)50 It was also concluded that for fine-grained soils there is no direct relationship between criti- cal velocity/shear stress and the mean particle size. However, the data can be bracketed with an upper bound and a lower bound equation as follows (Equations 63 to 66). v Dc ( )( ) ( )= âUpper bound: m s 0.07 mm (63)50 1.45 v Dc ( )( ) ( )= âLower bound: m s 0.1 mm (64)50 0.12 Dc ( )( ) ( )t = âUpper bound: Pa 0.06 mm (65)50 2.3 Dc ( )( ) ( )t = âLower bound: Pa 0.05 mm (66)50 0.25 One major difference of the updated plots with earlier versions (Figure 31) is that the boundary in which Equations 61 and 62 are valid is shifted to D50 = 0.3 mm. In earlier versions of these plots, there was this wrong notion that for soils that are categorized as coarse-grained soils according to USCS classification system (D50 > 0.074 mm), direct relationships between critical velocity/shear stress and D50 exist. Figure 146 and Figure 147 show the scattered data for fine-grained soils with the defined upper and lower bound, as well as for the coarse-grained soils. These figures show clearly that mean particle size is not a sufficient parameter for soils that have a D50 smaller than 0.3 mm. Therefore, more parameters that specifically involve the plasticity behavior of fine-grained soils affect the critical velocity and critical shear stress of the soils. On the other hand, it is evident that once the soil has a relatively larger mean particle size (D50 > 0.3 mm), the mean particle size becomes the predominant parameter in showing the erosion resistance. Equations 61 and 62 are strong equations for predicting the values of the critical velocity and the critical shear stress, respectively, for coarse sand to very large gravels. 7.3 Deterministic (Frequentist) Regression Analysis As discussed in Chapter 1, the final phase of this project consisted of performing two parallel statistical approachesâfrequentist regression and Bayesian inferenceâwith the goal of reach- ing the best potential fits between erodibility parameters and geotechnical properties of soils.

Development of Correlation Equations 179 U.S. Army Corps of Engineers Figure 147. Mean particle size versus critical shear stress. U.S. Army Corps of Engineers Figure 146. Mean particle size versus critical velocity.

180 Relationship Between Erodibility and Properties of Soils This section is dedicated to the step-by-step process of frequentist regression. This approach was implemented in three major steps: 1. First order statistical analysis, 2. Second order statistical analysis, and 3. Regression, optimization, and model selection. 7.3.1 First Order Statistical Analysis The first step was to develop the first order statistical features. This step in statistics is known as a very crucial step to learn about all the details and complexities within the raw data themselves, before making any effort to generate relationships among them. The programming language of Python is used as the primary tool for this project and the software Stata and SPSS are used as alternative tools for overseeing the results. This step consisted of obtaining the primary statistical measures of our database (i.e., number of data points, range, quartiles, mean, median, mode, standard deviation, histograms, probability density functions, empirical cumu- lative density functions, etc.). As discussed earlier in Chapter 5, overall, there are 5 erodibility parameters (function variables). These parameters are the critical shear stress tc, the critical velocity vc, the initial slope Ev of the z . versus v curve, the initial slope Et of the z . versus t curve, and the erosion category (EC) in the Briaud erosion chart (2013). Up to 16 geotechnical engi- neering parameters (model variables) were also collected for each sample. These parameters include â¢ Liquid limit, LL; â¢ Plastic limit, PL; â¢ Plasticity index, PI; â¢ Water content, WC; â¢ Undrained shear strength, Su; â¢ Total unit weight, Î³ ; â¢ Particle size at which 10% of the particles are finer than this size, D10; â¢ Particle size at which 30% of the particles are finer than this size, D30; â¢ Mean particle size, D50; â¢ Particle size at which 10% of the particles are finer than this size, D60; â¢ Coefficient of curvature, Cc; â¢ Coefficient of uniformity, Cu; â¢ Percentage finer than #200 sieve, PF; â¢ Specific gravity, Gs; â¢ Percentage of clay (PC); and â¢ Soil activity (A). Nearly 1,000 erosion tests compiled in NCHRP-Erosion were studied in different groups. Chapter 5 showed that NCHRP-Erosion incorporates more than 10 different erosion tests; however, the three major erosion testsâEFA, HET, and JETâwere chosen to investigate the potential relationships further. The study of the first order statistical features was started with the global data set (including all different tests and all soil types all together). Next, the data were divided on the basis of their erosion test type, and the first order statistical features for each subgroup were developed. Next, the data for each subgroup were again divided with regard to their soil type (coarse or fine) according to the USCS. The aforementioned groups are labeled and listed below: 1. TAMU/Global data set, 2. TAMU/Fine data set,

Development of Correlation Equations 181 3. TAMU/Coarse data set, 4. EFA/Global data set, 5. EFA/Fine data set, 6. EFA/Coarse data set, 7. JET/Global data set, 8. JET/Fine data set, 9. JET/Coarse data set, 10. HET/Global data set, 11. HET/Fine data set, and 12. HET/Coarse data set. A flowchart diagram of the grouping procedure is shown in Figure 148 below. As shown in Figure 148, 12 data sets were obtained, and the first order statistics analyses were performed on each data set separately. The first order statistical features of all function and model variables were obtained for the aforementioned groups. Table 33 to Table 44 show these results. As part of the first order statistical analyses of the parameters, and in order to learn about the statistical traits of each parameter, the histograms, probability density functions (PDFs) and empirical cumulative density functions (ECDFs) were plotted for each parameter. After that, multiple statistical distribution models were fitted to the actual data with the goal of finding the best representative distribution for each parameter in each group. Figure 149 shows an example of a histogram and PDF and ECDF plots for the critical velocities of the TAMU/Global group in the NCHRP-Erosion spreadsheet, which are fitted with many statistical distribution models, namely, normal, lognormal, exponential, Rayleigh, alpha, gamma, and beta. Figure 150 and Figure 151 show the same set of plots for the TAMU/Coarse and TAMU/Fine, respectively. Results show that the critical velocity (vc) can be well represented by exponential distribution. The same approach was taken for all erodibility parameters (i.e., vc, tc, Ev, Et, and EC) as well as all 12 major geotechnical properties (i.e., LL, PL, PI, Î³, WC, Su, PF, D50, Cu, Cc, PC, and A). The goal was to identify the best statistical distribution models to best represent each parameter. The findings of this effort are not only very important to understanding each erodibility parameter and the geotechnical properties but are also a vital tool in performing frequentist regression and Bayesian inference. TAMU/Global EFA/Global EFA/Fine EFA/Coarse JET/Global JET/Fine JET/Coarse HET/Global HET/Fine HET/Coarse TAMU/Fine TAMU/Coarse Figure 148. Flowchart diagram of the grouping procedure.

EC Ev (mm-s/ m-h) Es (mm/ h-Pa) vc (m/s) sc (Pa) LL (%) PL (%) PI (%) f (kN/m3) WC (%) Su (kPa) PF (%) D50 (mm) Cu Cc PC (%) Count 831 314 810 319 807 675 674 676 729 729 244 683 483 172 172 584 Mean 2.86 129.07 88.18 0.90 64.33 43.14 19.93 23.17 19.17 22.15 51.97 66.24 0.21 64.85 4.07 26.58 SD 0.78 565.28 522.91 0.87 153.28 22.48 7.46 19.41 2.13 17.87 39.65 26.72 1.18 102.68 8.85 19.50 Min. 0.75 0.07 0 0.1 0.0001 14.5 6.3 1.5 11.4 1.02 2 0 0.0009 1.29 0.11 0 25% 2.5 3.2875 0.3425 0.29 0.56 30 15 11.8 18.1 14.2 20 42.95 0.0058 4.7949 1.0641 13 50% 3 8.885 1.425 0.6 5.7 37.6 19 21.9 19.2 18.5 38 75 0.0302 30 1.95541 20 75% 3.25 29.27 7.2275 1.065 32.255 48.2 23 30 20.3 26.88 76.05 87.165 0.13 49 3.4087 39.925 Max. 5.5 6300 6690.26 5.2 1158 264.1 77 238.8 25.13 286.7 150.7 100 19 850 82 96.39 Table 33. First order statistics results for NCHRP-Erosion spreadsheet: TAMU/Global data set. EC Ev (mm-s/ m-h) Es (mm/ h-Pa) vc (m/s) sc (Pa) LL (%) PL (%) PI (%) f (kN/m3) WC (%) Su (kPa) PF (%) D50 (mm) Cu Cc PC (%) Count 612 239 595 243 594 570 570 572 537 556 211 502 328 91 91 421 Mean 3.03 29.86 16.64 0.95 81.25 44.53 19.91 24.61 19.08 24.83 53.34 80.23 0.02 38.27 3.30 32.58 SD 0.74 87.38 105.43 0.87 173.24 23.28 7.48 20.19 2.15 19.06 40.12 13.25 0.02 90.46 8.64 19.66 Min. 0.75 0.07 0 0.1 0.01 15.2 6.3 1.5 11.4 7.51 3.3 48 0.0009 1.29 0.28 0 25% 2.5 2.775 0.22 0.34 0.9225 30 15 13 17.9 16.625 20 75 0.0025 5.41026 0.96 18.3 50% 3 6.56 0.79 0.7 6.125 39 18.8 22 19 20.75 39.2 81 0.016 30 1.57035 29.7 75% 3.5 16.915 3.005 1.155 64.8225 51 23 31 20.2 29.215 77.5 90.075 0.031 49 2.04082 48 Max. 5.5 761.8 1718.02 5.2 1158 264.1 77 238.8 25.13 286.7 150.7 100 0.075 850 82 96.39 Table 34. First order statistics results for NCHRP-Erosion spreadsheet: TAMU/Fine data set.

EC Ev (mm-s/ m-h) Es (mm/ h-Pa) vc (m/s) sc (Pa) LL (%) PL (%) PI (%) f (kN/m3) WC (%) Su (kPa) PF (%) D50 (mm) Cu Cc PC (%) Count 219 75 215 76 213 105 104 104 192 173 33 181 155 81 81 163 Mean 2.37 445.21 286.18 0.73 17.16 35.55 20.07 15.28 19.42 13.52 43.22 27.44 0.62 94.72 4.93 11.09 SD 0.67 1092.66 974.27 0.85 48.34 15.49 7.42 11.69 2.05 9.06 35.84 12.54 2.03 107.80 9.05 5.69 Min. 1 0.35 0.05 0.1 0.0001 14.5 7.7 2 12.3 1.02 2 0 0.074 1.42857 0.11 0 25% 2 8.13 2.46 0.23 0.23 24.5 16 5.7 18.63 9.7 22 22 0.13 3.88235 1.125 7 50% 2.5 30.79 9 0.325 3.46 36 19 15.05 19.6 11.6 32 29 0.22 38 3.4087 13 75% 2.75 296.67 109.05 0.9075 13.8 39 22.975 21.25 20.5 15.8 45.49 35.14 0.29 230 6.94901 15 Max. 4.25 6300 6690.26 4 513 90.6 43 56.8 24.29 82 132 65 19 500 57.12 29 Table 35. First order statistics results for NCHRP-Erosion spreadsheet: TAMU/Coarse data set. EC Ev (mm-s/ m-h) Es (mm/ h-Pa) vc (m/s) sc (Pa) LL (%) PL (%) PI (%) f (kN/m3) WC (%) Su (kPa) PF (%) D50 (mm) Cu Cc PC (%) Count 333 307 330 312 331 275 274 274 256 267 190 274 193 66 66 145 Mean 2.60 105.67 46.45 0.92 4.25 47.70 20.82 26.85 19.20 27.01 46.24 69.48 0.38 44.42 3.65 21.15 SD 0.70 476.80 285.49 0.87 7.92 27.67 9.34 24.62 2.77 14.41 37.20 31.72 1.85 125.83 12.20 19.74 Min. 0.75 0.07 0.01 0.1 0.05 14.5 6.3 1.5 12.3 1.02 2 0 0.001 1.32 0.11 0 25% 2.25 3.205 0.44 0.3 0.335 32.1 14.3 13 17.55 17.945 19 46.55 0.0082 3.2867 0.86595 5 50% 2.5 8.02 1.615 0.615 1.6 40.9 18.5 22.9 18.91 25.25 32.6 84.3 0.029 5.66536 1.11743 18.06 75% 3 26.725 5.74 1.0925 5.045 58.9 24.4 34.3 20.425 31.2 63.125 95.23 0.088 15 1.47322 28.88 Max. 5.5 6300 4470.97 5.2 88.35 264.1 77 238.8 25.13 88.1 143.5 100 19 850 82 96.39 Table 36. First order statistics results for NCHRP-Erosion spreadsheet: EFA/Global data set.

EC Ev (mm-s/ m-h) Es (mm/ h-Pa) vc (m/s) sc (Pa) LL (%) PL (%) PI (%) f (kN/m3) WC (%) Su (kPa) PF (%) D50 (mm) Cu Cc PC (%) Count 256 235 253 239 254 232 231 231 192 213 161 203 140 35 35 104 Mean 2.71 30.21 11.18 0.97 4.73 48.89 20.70 28.16 19.47 29.66 47.42 86.24 0.02 36.54 3.49 26.02 SD 0.65 88.10 46.26 0.87 8.68 28.56 9.14 25.80 2.81 13.36 37.73 13.78 0.02 143.54 13.67 20.92 Min. 0.75 0.07 0.01 0.1 0.06 15.2 6.3 1.5 13.87 9.8 3.3 48 0.001 1.32 0.44462 0 25% 2.5 2.665 0.36 0.34 0.435 34.225 14.4 15.35 17.6975 21.7 19 78.9 0.00567 3.97077 0.89663 7.5 50% 2.75 6.17 1.15 0.7 1.89 41 18.2 23.2 19.1 27.75 33.8 90.5 0.0172 5.71429 1.06786 21.98 75% 3 16.915 3.7 1.165 5.1075 59.25 24.2 35 20.625 32.9 67 97.3 0.03155 8.06818 1.35425 33.775 Max. 5.5 761.8 478 5.2 88.35 264.1 77 238.8 25.13 88.1 143.5 100 0.088 850 82 96.39 Table 37. First order statistics results for NCHRP-Erosion spreadsheet: EFA/Fine data set. EC Ev (mm-s/ m-h) Es (mm/ h-Pa) vc (m/s) sc (Pa) LL (%) PL (%) PI (%) f (kN/m3) WC (%) Su (kPa) PF (%) D50 (mm) Cu Cc PC (%) Count 77 72 77 73 77 43 43 43 64 54 29 71 53 31 31 41 Mean 2.22 351.95 162.34 0.75 2.69 41.29 21.49 19.81 18.39 16.54 39.72 21.58 1.32 53.32 3.83 8.77 SD 0.75 934.81 572.69 0.86 4.32 21.40 10.45 15.34 2.50 13.72 34.00 15.31 3.37 103.93 10.51 7.57 Min. 1 0.35 0.08 0.1 0.05 14.5 7.7 2 12.3 1.02 2 0 0.074 1.63 0.11 0 25% 1.75 7.155 1.81 0.23 0.21 24.5 13 6.1 16.67 9.6775 15.7 8.5 0.12 2.455 0.85932 1 50% 2 24.455 4.8 0.34 0.34 38 19.3 15.1 18.63 15.6 31.4 20 0.17 4.87805 1.125 8.5 75% 2.5 247.048 20.97 0.93 4.4 54.6 28.5 32.45 20.2 19.75 45.49 35.07 0.28 57.725 1.64528 15 Max. 4 6300 4470.97 4 20.03 90.6 43 56.8 23.3 82 132 50.36 19 500 57.12 25 Table 38. First order statistics results for NCHRP-Erosion spreadsheet: EFA/Coarse data set.

EC Ev (mm-s/ m-h) Es (mm/ h-Pa) vc (m/s) sc (Pa) LL (%) PL (%) PI (%) f (kN/m3) WC (%) Su (kPa) PF (%) D50 (mm) Cu Cc PC (%) Count 232 0 231 0 231 185 185 185 233 233 21 221 154 53 53 221 Mean 3.40 na 28.49 na 194.50 41.58 19.59 21.99 19.14 18.43 61.10 63.18 0.10 91.86 4.25 28.41 SD 0.70 na 132.72 na 237.10 18.41 6.00 15.17 1.56 8.32 38.76 21.99 0.11 92.71 2.33 18.91 Min. 1 na 0 na 0.1 20 11 2.9 13.9 7.5 4.3 22 0.001 1.29 0.48 2 25% 3 na 0.13 na 15.305 30 15 15 18.3 12.56 43.1 42 0.0025 30 2.04082 13 50% 3.5 na 0.64 na 89.75 36 19 20 19.3 16.2 57.4 73.4 0.038 38 3.4087 24.84 75% 4 na 3.31 na 314.98 42 22 24 20.1 21.2 71.8 80.06 0.22 230 6.80272 44.75 max 4.75 na 1684.04 na 1158 148.1 33 125.8 22.9 44 143.5 100 0.29 230 7.74 77 Note: na = not applicable. Table 39. First order statistics results for NCHRP-Erosion spreadsheet: HET/Global data set. EC Ev (mm-s/ m-h) Es (mm/ h-Pa) vc (m/s) sc (Pa) LL (%) PL (%) PI (%) f (kN/m3) WC (%) Su (kPa) PF (%) D50 (mm) Cu Cc PC (%) Count 169 0 168 0 168 169 169 169 170 170 21 158 92 25 25 158 Mean 3.68 n/a 2.10 n/a 254.35 42.12 19.64 22.48 18.73 20.91 61.10 75.44 0.02 29.30 3.49 34.76 SD 0.58 n/a 7.81 n/a 248.77 19.18 6.27 15.79 1.46 8.28 38.76 11.24 0.02 19.33 2.66 18.84 Min. 1 n/a 0 n/a 4.65 20 11 2.9 13.9 7.51 4.3 51 0.001 1.29 0.48 2 25% 3.25 n/a 0.0775 n/a 57.795 30 15 12 18 15.085 43.1 68.8 0.002 2.9 1.14 18.9 50% 3.75 n/a 0.29 n/a 192.93 36 19 22 18.87 18.5 57.4 79 0.0111 30 2.04082 30.9 75% 4 n/a 0.96 n/a 379.525 44 22 24 19.8 23.3235 71.8 84 0.0312 49 6.80272 49.5 Max. 4.75 n/a 70.81 n/a 1158 148.1 33 125.8 21.8 44 143.5 100 0.075 49 7.74 77 Table 40. First order statistics results for NCHRP-Erosion spreadsheet: HET/Fine data set.

EC Ev (mm-s/ m-h) Es (mm/ h-Pa) vc (m/s) sc (Pa) LL (%) PL (%) PI (%) f (kN/m3) WC (%) Su (kPa) PF (%) D50 (mm) Cu Cc PC (%) Count 63 0 63 0 63 16 16 16 63 63 0 63 62 28 28 63 Mean 2.67 na 98.85 na 34.92 35.94 19.06 16.88 20.24 11.72 na 32.44 0.22 147.71 4.93 12.51 SD 0.41 na 241.37 na 79.15 0.25 0.25 0.50 1.27 3.07 na 7.22 0.06 96.76 1.78 3.10 Min. 2 na 0.05 na 0.1 35 19 15 17.8 7.5 na 22 0.13 38 3.4087 7 25% 2.25 na 2.895 na 8.73 36 19 17 19.35 9.6 na 29 0.1525 38 3.4087 13 50% 2.75 na 12.82 na 13.35 36 19 17 19.9 10.8 na 34 0.23 230 3.4087 13 75% 3 na 85.355 na 23.815 36 19 17 21.3 14 na 42 0.29 230 6.94901 14 Max. 4.25 na 1684.04 na 513 36 20 17 22.9 18.9 na 46.5 0.29 230 6.94901 23.9 Table 41. First order statistics results for NCHRP-Erosion spreadsheet: HET/Coarse data set. EC Ev (mm-s/ m-h) Es (mm/ h-Pa) vc (m/s) sc (Pa) LL (%) PL (%) PI (%) f (kN/m3) WC (%) Su (kPa) PF (%) D50 (mm) Cu Cc PC (%) Count 145 0 145 0 144 122 122 123 118 108 28 76 29 6 6 108 Mean 2.47 na 272.67 na 5.35 36.09 17.65 18.43 19.43 20.11 88.45 63.89 0.05 83.71 14.94 21.36 SD 0.67 na 1082.02 na 9.44 12.63 4.90 12.17 1.86 36.46 39.37 25.31 0.07 77.59 21.88 12.54 Min. 1.25 na 0.03 na 0.0001 14.5 7.7 2.9 11.4 7.16526 22.2 17.12 0.0009 7.14 0.98 1.5 25% 2 na 1.29 na 0.18 27.725 14.3 6 18.7 11.8145 52.375 41.125 0.0053 43.3625 1.87 11.1575 50% 2.5 na 4.92 na 1.12 36 16.55 19.1 19.45 14.4 79.6 68.8 0.0165 69.64 4.805 20 75% 3 na 29.53 na 7.2675 42 21.975 24 20.4 17.385 129.2 84 0.0625 85.83 16.995 29.7 Max. 3.75 na 6690.26 na 60.72 79.5 34.1 50.3 24.29 286.7 150.7 100 0.28 230 57.12 64.66 Table 42. First order statistics results for NCHRP-Erosion spreadsheet: JET/Global data set.

EC Ev (mm-s/ m-h) Es (mm/ h-Pa) vc (m/s) sc (Pa) LL (%) PL (%) PI (%) f (kN/m3) WC (%) Su (kPa) PF (%) D50 (mm) Cu Cc PC (%) Count 97 0 97 0 96 85 85 86 80 80 24 56 25 3 3 73 Mean 2.63 na 58.39 na 6.38 39.38 17.14 22.18 19.43 23.18 91.77 77.29 0.02 70.55 3.83 26.38 SD 0.63 na 238.47 na 10.40 13.10 4.83 11.99 1.89 41.98 38.55 13.04 0.03 26.46 3.39 11.42 Min. 1.25 na 0.03 na 0.01 15.2 9.9 2.9 11.4 7.55 22.2 51.5 0.0009 40 1.87 5.3 25% 2.25 na 0.86 na 0.31 29 13 11.325 18.8225 13.6925 55.375 68.8 0.0049 62.915 1.87 18.9 50% 2.75 na 2.92 na 1.78 36.7 16 22.2 19.64 15.935 83.15 79.8 0.0082 85.83 1.87 28 75% 3.25 na 11.59 na 8.32 47 21 31 20.35 18.68 129.4 87.3025 0.0461 85.83 4.805 30 Max. 3.75 na 1718.02 na 60.72 79.5 34.1 50.3 23.05 286.7 150.7 100 0.0733 85.83 7.74 64.66 Table 43. First order statistics results for NCHRP-Erosion spreadsheet: JET/Fine data set. EC Ev (mm-s/ m-h) Es (mm/ h-Pa) vc (m/s) sc (Pa) LL (%) PL (%) PI (%) f (kN/m3) WC (%) Su (kPa) PF (%) D50 (mm) Cu Cc PC (%) Count 48 0 48 0 48 37 37 37 38 28 4 20 4 3 3 35 Mean 2.13 na 705.68 na 3.29 28.52 18.84 9.69 19.43 11.33 68.55 26.38 0.21 96.86 26.06 10.87 SD 0.63 na 1784.55 na 6.78 7.20 4.94 7.08 1.81 2.37 44.03 5.14 0.07 117.60 28.54 7.15 Min. 1.25 na 0.41 na 0.0001 14.5 7.7 3.9 15.8 7.16526 30.1 17.12 0.1351 7.14 0.98 1.5 25% 1.6875 na 4.0325 na 0.115 22 16 5 18.475 10 39.85 25.1 0.15603 30.295 10.53 6.83 50% 2 na 20.875 na 0.45 30 17 6 19.4 11.2 57.45 25.1 0.204 53.45 20.08 8 75% 2.5625 na 208.705 na 2.8025 30 25 14 20.4 11.9804 86.15 25.1 0.25375 141.725 38.6 20 Max. 3.5 na 6690.26 na 28.92 39 25 22 24.29 17.8787 129.2 42.5 0.28 230 57.12 29 Table 44. First order statistics results for NCHRP-Erosion spreadsheet: JET/Coarse data set.

188 Relationship Between Erodibility and Properties of Soils Figure 149. PDF, ECDF, and histogram plots for critical velocity in TAMU/Global data set.

Development of Correlation Equations 189 Figure 150. PDF, ECDF, and histogram plots for critical velocity in the TAMU/Coarse data set.

190 Relationship Between Erodibility and Properties of Soils Figure 151. PDF, ECDF, and histogram plots for critical velocity in the TAMU/Fine data set.

Development of Correlation Equations 191 Similar figures are included in Appendix 3 for all five erodibility parameters as well as the 12 major geotechnical properties for each subgroup (see Figure 148). The results of the best simplest statistical model for each parameter and in each subgroup are obtained and shown in Table 45 and Table 46. Table 45 shows the best models to represent each erodibility parameter (vc, tc, Ev, Et, and EC) in each subgroup. Table 46 shows the best models to represent each major geotechnical property (i.e., LL, PL, PI, Î³, WC, Su, PF, D50, Cu, Cc, and PC) in each subgroup. Best Simplest Statistical Model to Represent Each Parameter No. Subgroup EC Ev Es vc sc 1 TAMU/Global Normal Exp. ln Exp. Exp. 2 TAMU/Fine Normal Exp. ln Exp. Exp. 3 TAMU/Coarse Normal Exp. Exp. Exp. Exp. 4 EFA/Global Normal Exp. ln Exp. Exp. 5 EFA/Fine Normal Exp. ln Exp. Exp. 6 EFA/Coarse Normal Exp. Exp. Exp. ln 7 JET/Global Normal na ln na ln 8 JET/Fine Normal ns ln na ln 9 JET/Coarse Normal na ln na Alpha 10 HET/Global Normal na ln na Normal 11 HET/Fine Normal na Exp. na Normal 12 HET/Coarse Normal na Normal na ln Note: Exp. = exponential; ln = lognormal; na = not applicable. Table 45. Best simplest statistical model to represent erodibility parameters. Best Simplest Statistical Model to Represent Each Parameter No. Subgroup LL PL PI f WC Su PF D50 Cu Cc PC 1 TAMU/Global Normal Normal Normal Normal Normal Exp. Normal Gamma Exp. Exp. Normal 2 TAMU/Fine Normal Normal Normal Normal Normal Exp. Normal Normal Exp. Exp. Normal 3 TAMU/Coarse Normal Normal Normal Normal Normal Exp. Normal Gamma Normal Exp. Normal 4 EFA/Global Normal Normal Normal Normal Normal Exp. Normal Gamma Exp. ln Exp. 5 EFA/Fine Normal Normal Normal Normal Normal Exp. Normal Exp. Exp. ln Normal 6 EFA/Coarse Normal Exp. Exp. Normal Normal Exp. Normal ln Exp. ln Normal 7 JET/Global Normal Normal Normal Normal Normal Normal Normal Gamma Normal Normal Normal 8 JET/Fine Normal Normal Normal Normal Exp. Normal Normal Normal Normal Normal Normal 9 JET/Coarse na na na Normal Normal na Normal na na na Normal 10 HET/Global Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal 11 HET/Fine Normal Normal Normal Normal Normal Normal Normal Normal Alpha Normal Normal 12 HET/Coarse Normal Normal Normal Normal Normal na Normal Normal Alpha Normal Normal Note: Exp. = exponential; ln = lognormal; na = not applicable. Table 46. Best simplest statistical model to represent geotechnical properties.

192 Relationship Between Erodibility and Properties of Soils 7.3.2 Second Order Statistical Analysis The second order statistical analysis dealt with constructing correlation matrices for the different parameters defined in each group and discussed in the previous step. Before the regression relationships could be generated, it was necessary to learn about any potential inter- parameter relationships between the model variables (geotechnical parameters) and the function variables (erosion parameters). To do that, correlation matrices were formed between the parameters, and Pearsonâs correlation coefficient was obtained for each matrix. Pearsonâs correlation coefficient is defined as the covariance of the two variables divided by the product of their standard deviations. It ranges from â1 to +1 and reflects the linear dependency between two variables, with +1 indicating a strong positive relationship, â1 indicating a solid negative relationship, and 0 indicating no relationship at all. Figure 152 shows an example of the results of the second order statistical analysis for EFA/Fine data. Such plots were created for all aforementioned groups explained in Figure 148 and compiled in Appendix 3. The knowledge developed after the second order statistical analysis is very important for implementing the Bayesian inference approach. 7.3.3 Experimental Design 7.3.3.1 Model Expression The next step in the process dealt with generating the relationship equations between the erosion parameters and the geotechnical engineering parameters. Two models were used in this research: the linear model and the power model. The power model was the multiple nonlinear model shown in Equation 67. . . . (67)1 2 3 4 1 2 3 4Y A P P P P( ) ( ) ( ) ( )= Ã Ã Ã Ã ÃÎ± Î± Î± Î± where Y = dependent variable, which in this case includes tc, vc, Ev, Et, and EC; Pi = soil geotechnical properties used in the regressions; and A and Î±i = constant parameters defined in the model expression. The linear model, on the other hand, was the simple first-degree linear model, as described in Equation 68. . . . (68)1 1 2 2 3 3Y P P P[ ]( ) ( ) ( )= Î± Ã + Î± Ã + Î± Ã + + Î² where Y = dependent variable, which in this case includes tc, vc, Ev, Et, and EC; Pi = soil geotechnical properties used in the regressions; and Î±i and Î² = constant parameters defined in the model expression. Several combinations between a function variable, say, tc, and model variables (e.g., LL, PL, PI, water content, unit weight, etc.) can be selected to generate regression equations. To obtain the best fits, the investigators decided to study nearly all possible combinations and evaluate the goodness of each fit. Therefore, 135 regression combination groups were constructed for each function variable in each subgroup shown in Figure 148. In addition, the two model expressions (Equations 67 and 68) were used to develop the equations. That is, the total number of regres- sion equations was 2 Ã 5 Ã 135 (2 model expressions Ã 5 function parameters Ã 135 combination groups) = 1,350 regression equations for each subgroup shown in Figure 148.

Figure 152. Correlation matrix for EFA/Fine data. (continued on next page)

Figure 152. (Continued).

Development of Correlation Equations 195 7.3.3.2 Measures of Statistical Significance The best models were selected after passing through a four-filter process: Filter 1. R2, Filter 2. Mean square error (MSE), Filter 3. F-value/F-statistic, and Filter 4. Cross-validation score. Each model generated was checked with Filters 1 to 4, one after another. Each model that did not meet the requirements of a given specific filter was not further studied with the subsequent filters and thus was eliminated from the best model sets. R2, also called the coefficient of determination, is one of the most well-known statistical measures of fit. It represents how far the actual data stands from the fitted regression model. The mathematical definition of R2 is as follows: 1 (69)2 2 2R y f y y i i i â â ( ) ( ) = â â â where yi = the data, yâ = mean value of the data set, and fi = predicted values using the fitted regression model. This study reports R2 values for each regression. R2 ranges from 1.0 (absolute fit), to 0 (no relationship). R2 can also sometimes be negative. Generally speaking, R2 values greater than 0.6 can be considered as potential correlations in many engineering applications. However, in many sources, even R2 values greater than 0.4 are accepted by researchers. The MSE is another statistical measure for evaluating the quality of a predictive model. MSE is also sometimes called the mean square deviation. It always gets a nonnegative value and reflects the average of the squared difference between the predicted value and the actual value. The mathematical definition of MSE is shown below: MSE 1 Ë (70) 2 n Y Yi iâ( )= â where Yi = the data, YÌi = predicted value using the estimator, and n = number of predicted data points. The main concern with R2 and MSE is that they cannot completely represent the statistical significance of an estimator; therefore, an additional statistical measure is used in this study. The main problem with R2 is that it is highly dependent on the number of variables involved in the regression model. For example, if there is only one parameter to predict, and the data set includes only one data point, any regression model can lead to an R2 of 1. The F-value can be used to reject the null hypothesis in a regression model. The F-value is obtained from Equation 71: SSE reg SSE res 1 (71)valueF D V ( ) ( ) = Ã âï£« ï£ï£¬ ï£¶ ï£¸ï£·

196 Relationship Between Erodibility and Properties of Soils where SSE(reg) = sum of square errors for the regression values, SSE(res) = sum of square errors for the residual values, D = number of data points, and V = number of variables. As can be seen from the equations, if the regression model has a lot of error [i.e., SSE(res) is large] then F will be small. Also, if the number of data points is close to the number of variables, then F will be small again. The goal is to maximize the value of F and in any case to make it higher than the F-statistic (F-stat), a target value given in statistical tables. The R2 and MSE values represent the goodness of fit evaluation, while the F-test, which compares the F-value with the F-stat, indicates whether there are enough data to propose the equation. The probability level for the F-test in this study was chosen as 5%. After the first three filters (R2, MSE, F-value/F-stat) were successfully met, the cross-validation score of the selected models was examined. Overfitting is a common problem in many similar statistical analyses conducted in different fields of engineering. Overfitting may occur when the equation is generated on the basis of all the data points available in the data set. In this case, the proposed model is not necessarily designed for yet unseen data. As a result, adding a few data points to the existing data set or removing some data from it can sometimes dramatically change the goodness of fit of the model. One approach to evaluate whether this is a problem is to divide the data set into two random subsets and call them training and testing subsets (i.e., 85% and 15% of the data, respectively). After that, the model is trained using the training subset, and validated through the testing subset. One issue with this approach is that the analysis is very dependent on the random training and testing subsets chosen in the beginning. Cross-validation is a technique used to assess the estimator performance by folding the data set into k random folds. The model is trained using the k-1 fold and then validated with the remaining folds. The final score of the cross-validation is the average of all scores for each time that the model is trained and validated (the average of k scores). Cross-validation was used to further sieve the best models to obtain the most robust equations. This four-filter model selection process was performed for each subgroup as shown in Figure 148. This process incorporates studying thousands of generated models in the first step and narrowing down to a smaller number of models after each filter is passed. By the end of the fourth filter, the best models for correlating erodibility parameters with geotechnical properties were supposedly achieved. For the coarse data sets (subgroups of TAMU/Coarse, EFA/Coarse, HET/Coarse, and JET/ Coarse), 105 soil parameter combination groups were chosen for each function variable; how- ever, for the Global and Fine data sets (subgroups of TAMU/Global, TAMU/Fine, EFA/Global, EFA/Fine, HET/Global, HET/Fine, JET/Global, JET/Fine), 135 soil parameter combination groups were selected for each function variable. Units of the parameters used in the regression analyses as well as list of these combination groups are presented in Appendix 4. After the requirements of all four filters were met successfully, the best-fitting models were selected. If more than one fitting model met the filter requirements, the authorsâ engineering judgement helped narrow down the choices to one or at most two equations. More regarding the selected models is discussed in Chapter 8. 7.3.3.3 Probability of Over- or Underpredicting The best correlation equations that can estimate each erodibility parameter (i.e., tc, vc, Et, Ev, and EC) were selected according to the four-filter process discussed in the previous section.

Development of Correlation Equations 197 These equations were obtained through a deterministic statistical analysis, meaning that they reflect the predicted value as a single number, with no quantification of the possible error associated with predicting the future event. One of the major goals of this study was to provide the engineer with a reliability-based approach to also assess whether the predicted values are conservative enough for the purpose of engineering design. This approach is called the probability of underpredicting (POU)/probability of overpredicting (POO) approach. As an instance, the engineer needs to know the probability that the predicted tc is smaller than the actual tc to be on the safe side in the design problem. Similarly, critical velocity, vc, and erosion category, EC, are among those erodibility parameters that require the POU. On the other hand, the engineer needs to know the probability that the predicted Et or Ev is larger than the actual Et or Ev, in order to be on the safe side in the design problem. Therefore in such cases, the POO is presented. Equation 72 shows the probabilistic model, which consists of the selected deterministic predicted erodibility parameter and a correction factor, q. (72)new detZ Z= q Ã where Znew = new value for erodibility parameter, q = correction factor, and Zdet = deterministically predicted erodibility parameter. q can also be inferred as the ratio of Znew over Zdet. Depending on the erodibility parameter, the POU versus q or POO versus q plots were developed for the best selected equations, as discussed in the next section. These plots help engineers find the best correction factors associated with a confidence level in design problems. Figure 153 gives an example of how POU is calculated for a function parameter such as EC. POU refers to the probability that the predicted values (Zp) are smaller than the measured values (Zm). The black solid line in Figure 153 represents the case in which Zm = Zp. In this example, two different correction factors (q = 0.6 and q = 1.0) are considered. The data points Figure 153. Example of how POU is obtained for two different correction factors.

198 Relationship Between Erodibility and Properties of Soils associated with q = 1.0 are shown as black open circles, and the data points associated with q = 0.6 are shown as red plus signs. The POU is then calculated by counting the number of data points above the black solid line (the data points in which Zm > Zp) and dividing that by the total number of data points. In this report, these plots are often accompanied by an offset value. For example, the POO is calculated as the probability that Zm â Zp < offset value instead of the probability that Zm â Zp < 0. The offset values are small compared with the standard deviation of the parameter and can often be neglected; however, because the R2 for selected equations were very high and the parameter values were very small in many cases, the lack of an offset value could lead to unrealistic POU or POO values. The next section presents the process in which the best equations were selected for each erodibility parameter. The POU versus correction factor or POO versus correction factor plots are also presented for the selected equations. 7.3.4 Regression, Optimization, and Model Selection The step-by-step procedure discussed in Section 7.3.3 for selecting the best-fitting models was implemented for each erodibility parameter (i.e., tc, vc, Et, Ev, EC). The selected correlation equations in this section are repeated in the Chapter 8, and recommendations on how they should be used are provided. The units and description of all the parameters used in the following equations in this section are listed in Table 47, as well as in Appendix 4 of the appendices report. Parameter Description Unit A Soil activity na c Critical shear stress Pa Cc Coefficient of curvature na Cu Coefficient of uniformity na D50 Mean particle size mm EC Erosion category na Ev Slope of velocityâerosion rate mm-s/m-h E Slope of shearâerosion rate Mm/h-Pa LL Liquid limit % PC Clay percentage (<0.002 mm) % PF Percent finer than sieve #200 % PI Plasticity index % PL Plastic limit % PP Pocket penetrometer strength kPa Su Undrained shear strength kPa vc Critical velocity m/s VST Vane shear strength kPa WC Water content % Unit weight kN/m3 Table 47. Units and descriptions of parameters used in regression analyses.

Development of Correlation Equations 199 7.3.4.1 Critical Shear Stress The first filter was R2. The study began with the first subgroup, TAMU/Global (see Figure 148). This database includes all existing data, regardless of test type or the fine/coarse nature of the samples. Figure 154 shows the number of data points in each of the 135 soil parameter combination groups. Figure 155 and Figure 156 show the results for R2 for these 135 soil param- eter combination groups with the linear model (Equation 68) and power model (Equation 67), respectively. Figure 155 and Figure 156 show very poor R2 values (<0.2) for critical shear stress, tc, in the TAMU/Global data set. Therefore, continuing with Filters 2 through 4 was not reasonable for this data set. The same poor results were observed when the data set was changed to TAMU/ Fine and TAMU/Coarse. These observations proved that the regression analyses needed to be narrowed down to each test separately and, consecutively, to the coarse or fine nature of the soil. Therefore, the subgroups EFA/Global, EFA/Fine, and EFA/Coarse; JET/Global, JET/Fine, and JET/Coarse; and HET/Global, HET/Fine, and HET/Coarse were used to implement the regression analyses, as described below. EFA Data Set. It was observed that dividing the EFA/Global data set into EFA/Fine and EFA/Coarse led to better results. Figure 157 shows the number of data points in each of the Figure 154. Number of data points in each of 135 soil parameter combination groups for TAMU/Global data set: critical shear stress. Figure 155. R2 results for linear models in TAMU/Global data set: critical shear stress.

200 Relationship Between Erodibility and Properties of Soils 135 combination groups in the EFA/Fine data set. Figure 158 and Figure 159 show the results for R2 for each combination group for the linear and power models, respectively. The R2 values of Groups 108 to 135 were generally higher than those of the other groups for both the linear and the power models; however, a quick glance shows that the power model was a better fit for the existing data in the case of critical shear stress in the EFA/Fine data set. The next step was to select the best R2 values, and move forward with Filter 2 (MSE). Figure 160 and Figure 161 show the values of MSE for the linear and power models, respectively. As expected, the MSE values were generally lower for same group numbers (Groups 108 to 135) in both figures. After passing through Filters 1 and 2 (R2 and MSE), the power models associated with Groups 109, 110, 113, 123, 124, 125, and 128 were selected for further analysis. Filter 3, F-value/ F-stat, was determined for each group. In cases in which F-value/F-stat was lower than 1, the model was removed from the selection list. Otherwise, the model remained in the list. Table 48 shows the results of the selected models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. Table 48 shows the selected models that met the requirements of the first three filters; however, not all of them performed satisfactorily in the cross-validation test. As mentioned Figure 156. R2 results for power models in TAMU/Global data set: critical shear stress. Figure 157. Number of data points in each of 135 combination groups for the EFA/Fine data set: critical shear stress.

Development of Correlation Equations 201 Figure 158. R2 results for the linear models in EFA/Fine data set: critical shear stress. Figure 159. R2 results for the power models in EFA/Fine data set: critical shear stress. Figure 160. MSE results for linear models in EFA/Fine data set: critical shear stress.

202 Relationship Between Erodibility and Properties of Soils earlier, cross-validation is a technique used to assess the estimator performance by folding the data set into k random folds. The model is trained by using the k-1 fold and then validated by using the remaining folds. The final score of the cross-validation test is the average of all scores for each time that the model is trained and validated (the average of k scores). Out of the seven selected groups in Table 48, Group 110 and, to some extent, Group 124 (both shaded and highlighted in blue) were the ones that passed all four filters successfully. Section 7.3.3.3 and Equation 72 discussed the reliability-based approach called the POU approach. The plots of POU versus q are shown in Figure 162 and Figure 163 for Groups 110 and 124, respectively. The vertical axes in these plots represent the probability that, when the selected model is used, the predicted tc will be smaller than the actual tc, in percentage (with Â± 0.5 Pa precision). The horizontal axes represent the correction factor (q) that can be multiplied by the predicted value to reach a certain POU level (See Equation 72). For instance, Figure 162 shows that by using the Group 110 correlation equation (see Table 48), there is near 73% chance that the predicted tc will be smaller than the actual tc (with Â± 0.5 Pa precision). However, Figure 161. MSE results for power models in EFA/Fine data set: critical shear stress. Table 48. Selected power models for critical shear stress in EFA/Fine data set.

Development of Correlation Equations 203 if the engineer desires a higher confidence level, say, near 90%, then he or she needs to multiply the predicted value by 0.6. Figure 163, however, shows that by using the Group 124 correlation equation (see Table 48), there is almost a 90% chance that the predicted value will be smaller than the actual tc (with 0.5 Pa offset). These plots give the engineers flexibility in choosing the desired correction factors according to the design application. The same procedure was conducted in the EFA/Coarse data set, and the best models were selected. Figure 164 shows the number of data points in each of the 105 combination groups in the EFA/Coarse data set. Figure 165 and Figure 166 show the results for R2 for each combination group for the linear and power models, respectively. Unlike the EFA/Fine data set, both the linear and power models showed a few good groups in terms of R2 values; however, as shown in Figure 164, the number of data points in most groups POU = 0.73 Figure 162. Plot of POU versus correction factor for Group 110 (power model): sc in EFA/Fine data set. Figure 163. Plot of POU versus correction factor for Group 124 (power model): sc in EFA/Fine data set.

204 Relationship Between Erodibility and Properties of Soils Figure 164. Number of data points in each of 105 combination groups for the EFA/Coarse data set: critical shear stress. Figure 165. R2 results for linear models in EFA/Coarse data set: critical shear stress. Figure 166. R2 results for the power models in EFA/Coarse data set: critical shear stress.

Development of Correlation Equations 205 were very low as compared with the EFA/Fine data set. That is, many regression groups might meet the requirements of the first two filters, R2 and MSE, but only marginally pass through the F-test. Figure 167 and Figure 168 show the values of MSE for the linear and power models, respectively. After passing through Filters 1 and 2 (R2 and MSE), linear models associated with Groups 51, 54, 57, 60, 63, 66, 71, 77, 80, 83, 86, 89, 92, and 97 were selected for further analysis. Power models associated with Groups 34, 44, 46, 47, 51, 52, 54, 58, 60, 64, 77, 78, 80, 86, 88, 95, 101, and 104 were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 49 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. As shown in Table 49, all selected models performed satisfactorily in the cross-validation test. Table 50 shows the results of the selected power models after the requirements of the first three filters were met. The best models that also had a good cross- validation score are shaded and highlighted in blue in Tables 49 and 50. Out of all the high- lighted correlation equations in Table 49 and Table 50, the Group 77 correlation equation in power form was selected as the most promising equation. Figure 169 shows the plot of POU versus q for this model. The vertical axis in Figure 169 represents the probability that, when the selected model is used, the predicted tc will be smaller than the actual tc, in percentage Figure 168. MSE results for power models in EFA/Coarse data set: critical shear stress. Figure 167. MSE results for linear models in EFA/Coarse data set: critical shear stress.

206 Relationship Between Erodibility and Properties of Soils (with 0.3 Pa offset). To reach 90% confidence that the predicted tc is smaller than the actual tc, the predicted value should be multiplied by 0.82. JET Data Set. A similar approach was taken to select the best correlation equation for critical shear stress in the JET database. However, there are two important notes to notice about the JET database: 1. The JET was primarily performed on the finer soils (D50 < 0.3 mm), and, therefore, the number of data points in the JET/Coarse data set was substantially low compared with the JET/Fine data set. Figure 170 and Figure 171 show the number of data points in each of the 135 combination groups in the JET/Global data set and each of the 105 combination groups in the JET/Coarse data set, respectively. 2. The R2 values for the JET/Global data set (D50 < 0.3 mm), although low themselves, were still better than the R2 values for the JET/Fine data set. Therefore, the regression results for the JET/Global data set are presented as the best models. Figure 172 and Figure 173 show the results for R2 for each combination group in the JET/Global data set, for the linear and power models, respectively. Both plots show relatively poor R2 for the JET/Global data set. One of the major reasons behind the poor relationships for critical shear stress in the JET database is the variety of methods used to interpret the raw data and obtain the critical shear stress. Section 4.3.1.4 of this report discussed these different methods. More on the JET issues is discussed in Chapter 8. Figure 174 and Figure 175 show the values of MSE for the linear and power models, respectively. Group No. Independent Variables Model Expression R2 MSE F-value/ F-stat Cross- Validation Score 51 Cc, Î³ , D50 0.92 1.05 5.1 0.89 54 Cc, WC, D50 0.92 1.06 5.1 0.89 57 PF, Cc, D50 0.93 1.01 5.4 0.88 60 Cc, Î³ , WC, PF 0.93 1.00 2.6 0.89 63 Cc, Î³ , PF, D50 0.93 1.03 2.9 0.87 66 Cc, WC, PF, D50 0.93 0.98 2.9 0.88 71 Cc, Î³ , WC, PF, D50 0.94 0.93 2.1 0.88 77 Cu , Î³ , D50 0.92 1.06 5.1 0.9 80 Cu, WC, D50 0.92 1.05 5.1 0.88 83 PF, Cu, D50 0.92 1.01 5.3 0.88 86 Cu, Î³ , WC, D50 0.93 1.02 3.3 0.89 89 Cu, Î³, PF, D50 0.93 1.05 3.3 0.86 92 Cu, WC, PF, D50 0.93 0.99 3.3 0.87 97 Cu, Î³, WC, PF, D50 0.94 0.96 2.4 0.86 Table 49. Selected linear models for critical shear stress in the EFA/Coarse data set.

Development of Correlation Equations 207 Group No. Independent Variables Model Expression R2 MSE F-value/ F-stat Cross- Validation Score 34 PI, VST, PF, 0.99 0.191 140 â15.44 44 PI, , VST, PF, 0.99 0.198 88 â1.4 46 PI, WC, VST, PF, 0.99 0.197 70.5 0.08 47 PI, , WC, VST, PF, 0.99 0.169 56.1 0.09 51 Cc, , 0.93 1.045 36 0.98 52 Cc, WC, VST 0.98 0.178 38.6 0.1 54 Cc, WC, 0.93 1.043 36.2 0.98 58 Cc, , WC, VST 0.99 0.168 24.3 â0.5 60 Cc, , WC, 0.93 1.043 29.1 0.97 64 Cc, WC, VST, PF 0.98 0.164 25.8 â1.5 77 Cc, , 0.93 1.044 36.1 0.99 78 Cc, WC, VST 0.98 0.183 33.1 0.04 80 Cc, WC, 0.93 1.044 36.1 0.98 86 Cc, , WC, 0.93 1.044 29.1 0.96 88 Cc, , VST, 0.97 0.236 21 0.03 95 Cc, , WC, VST, 0.96 0.307 19 â11.3 101 Cc, Cu, PI 0.95 0.313 26.7 â0.44 104 Cc, Cu, VST 0.98 0.204 25.5 0.01 Table 50. Selected power models for critical shear stress in the EFA/Coarse data set. Figure 169. Plot of POU versus correction factor for Group 77 (power): sc in the EFA/Coarse data set.

208 Relationship Between Erodibility and Properties of Soils Figure 170. Number of data points in each of 135 combination groups for the JET/Global data set: critical shear stress. Figure 171. Number of data points in each of 105 combination groups for the JET/Coarse data set: critical shear stress. Figure 172. R2 results for the linear model in JET/Global data set: critical shear stress.

Development of Correlation Equations 209 Figure 173. R2 results for the power models in JET/Global data set: critical shear stress. Figure 174. MSE results for linear models in JET/Global data set: critical shear stress. Figure 175. MSE results for power models in JET/Global data set: critical shear stress.

210 Relationship Between Erodibility and Properties of Soils After passing through Filters 1 and 2 (R2 and MSE), the linear models associated with Groups 49, 76, 109, 112, and 113 in the JET/Global data set were selected for further analysis. Power models associated with Groups 53, 79, 109, and 124 in the JET/Global data set were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 51 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. Only one of the selected models in Table 51 performed well in the cross-validation test and is shaded and highlighted in blue. Table 52 shows the results of the selected power models after the requirements of the first three filters were met. Out of all the correlation equations in Table 51 and Table 52, the Group 113 correlation equation in linear form was selected as the most promising equation. Figure 176 shows the plot of POU versus q for this model. The vertical axis in Figure 176 represents the probability that, when the selected model is used, the predicted tc will be smaller than the actual tc, in percentage (with 1 Pa offset). To reach 90% confidence that the predicted tc is smaller than the actual tc, the predicted value should be multiplied by 0.6. Table 51. Selected linear models for critical shear stress in the JET/Global data set. Table 52. Selected power models for critical shear stress in the JET/Global data set.

Development of Correlation Equations 211 HET Data Set. A similar approach was taken to select the best correlation equation for critical shear stress in the HET database. In the HET database, there are also two important observations to notice: 1. The HET was primarily performed on finer soils (D50 < 0.3 mm), and, therefore, the number of data points in the HET/Coarse data set was substantially low compared with the HET/Fine data set. In fact, many combination groups in the HET/Coarse database had zero data points. Figure 177 and Figure 178 show the number of data points in each of the 135 combination groups in the HET/Global data set and each of the 105 combination groups in the HET/Coarse data set, respectively. 2. The R2 values for the HET/Global data set (D50 < 0.3 mm) were significantly better than the R2 values for the HET/Coarse and HET/Fine data sets. Therefore the regression results for the HET/Global data set are presented as the best models. Figure 179 and Figure 180 show the results for R2 for each combination group in the HET/Global data set, for the linear and power models, respectively. Both plots show that the best R2 values for the HET/Global data set were around 0.60 to 0.65. One of the major reasons Figure 176. Plot of POU versus correction factor for Group 113 (linear): sc in the JET/Global data set. Figure 177. Number of data points in each of 135 combination groups for the HET/Global data set: critical shear stress.

212 Relationship Between Erodibility and Properties of Soils Figure 178. Number of data points in each of 105 combination groups for the HET/Coarse data set: critical shear stress. Figure 179. R2 results for the linear models in HET/Global data set: critical shear stress. Figure 180. R2 results for the power models in HET/Global data set: critical shear stress.

Development of Correlation Equations 213 behind the poor relationships for critical shear stress in the HET database was that the method used to calculate the erodibility parameters in the HET includes many crude judgements. More on the issues affecting the HET is discussed in Chapter 8. Figure 181 and Figure 182 show the values of MSE for the linear and power models, respectively. After passing through Filters 1 and 2 (R2 and MSE), linear models associated with Groups 38, 48, 49, 52, 53, 71, 78, 79, 124, and 128 in the HET/Global data set were selected for further analysis. Power models associated with Groups 19, 35, 38, 40, 48, 50, 52, and 53 in the HET/Global data set were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 53 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. Table 54 shows the results of the selected power models after the requirements of the first three filters were met. Only one of the selected models in Table 54 performed well in the cross-validation test and is shaded and highlighted in blue. Of all the cor- relation equations in Table 53 and Table 54, the Group 19 correlation equation in power form was selected as the most promising equation. Figure 183 shows the plot of POU versus q for this model. The vertical axis in Figure 183 represents the probability that, when the selected model is used, the predicted tc will be smaller than the actual tc, in percentage (with 0.4 Pa offset). To reach 90% confidence that the predicted tc is smaller than the actual tc, the predicted value should be multiplied by 0.6. Figure 182. MSE results for power models in HET/Global data set: critical shear stress. Figure 181. MSE results for linear models in HET/Global data set: critical shear stress.

214 Relationship Between Erodibility and Properties of Soils Table 53. Selected linear models for critical shear stress in the HET/Global data set. Table 54. Selected power models for critical shear stress in the HET/Global data set.

Development of Correlation Equations 215 7.3.4.2 Critical Velocity The same four-filter process discussed in Section 7.3.3 was implemented to obtain the best models for critical velocity. The first observation was that of the three erosion tests studied in this chapter (EFA, JET, HET), only the EFA test has critical velocity, vc, as one of its outputs. That is, the results of the JET and HET consist of only three erodibility parameters (i.e.,tc, Et, and EC), whereas the EFA results include all five erodibility parameters (i.e.,tc, vc, Et, Ev, and EC). Therefore, regression analysis for critical velocity was limited to the EFA data set. As in the case of critical shear stress, it was observed that dividing the EFA/Global data set into the EFA/Fine and EFA/Coarse data sets would significantly improve the regression results. Figure 184 shows the number of data points in each of the 135 combination groups in the EFA/Fine data set. Figure 185 and Figure 186 show the results for R2 for each combination group for the linear and power models, respectively. The R2 values of Groups 109 to 135 were generally higher than the rest for both the linear and power models. Figure 187 and Figure 188 show the MSE results for each of the 135 combination groups in the EFA/Fine data set for the linear and power models, respectively. Figure 183. Plot of POU versus correction factor for Group 19 (power): sc in the HET/Global data set. Figure 184. Number of data points in each of 135 combination groups for the EFA/Fine data set: critical velocity.

216 Relationship Between Erodibility and Properties of Soils Figure 185. R2 results for the linear models in EFA/Fine data set: critical velocity. Figure 186. R2 results for the power models in EFA/Fine data set: critical velocity. Figure 187. MSE results for linear models in EFA/Fine data set: critical velocity.

Development of Correlation Equations 217 After passing through Filters 1 and 2 (R2 and MSE), the linear models associated with Groups 114, 124, 128, and 132 in the EFA/Fine data set were selected for further analysis. Power models associated with Groups 109, 110, 113, 114, 117, 118, 123, 124, 125, 128, 132, and 133 in the EFA/Fine data set were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 55 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. As shown in Table 55, all four selected models performed accept- ably in the cross-validation test. Table 56 shows the results of the selected power models after the requirements of the first three filters were met. The best models that also had the best cross-validation scores are shaded and highlighted in blue in Tables 55 and 56. Out of all the correlation equations in Table 55 and Table 56, the Group 117 correlation equation in power form was selected as the most promising equation. Figure 189 shows the plot of POU versus q for this model. The vertical axis in Figure 189 represent the probability that, when the selected model is used the predicted vc will be smaller than the actual vc, in percentage (with 0.2 m/s offset). To reach 90% confidence that the predicted vc is smaller than the actual vc, the predicted value should be multiplied by 0.8. The same procedure was conducted in the EFA/Coarse data set, and the best models were selected for critical velocity. Figure 190 shows the number of data points in each of the 105 com- bination groups in the EFA/Coarse data set. Figure 191 and Figure 192 show the results for R2 for each combination group for the linear and power models, respectively. Figure 188. MSE results for power models in EFA/Fine data set: critical velocity. Table 55. Selected linear models for critical velocity in the EFA/Fine data set.

218 Relationship Between Erodibility and Properties of Soils Table 56. Selected power models for critical velocity in the EFA/Fine data set. Figure 189. Plot of POU versus correction factor for Group 117 (power): vc in the EFA/Fine data set.

Development of Correlation Equations 219 Figure 190. Number of data points in each of 105 combination groups for the EFA/Coarse data set: critical velocity. Figure 191. R2 results for the linear models in EFA/Coarse data set: critical velocity. Figure 192. R2 results for the power models in EFA/Coarse data set: critical velocity.

220 Relationship Between Erodibility and Properties of Soils Both the linear and power models showed a few good groups in terms of R2 values; however, as shown in Figure 190, the number of data points in most groups was lower as compared with the EFA/Fine data set. Figure 193 and Figure 194 show the values of MSE for the linear and power models, respectively. After passing through Filters 1 and 2 (R2 and MSE), the linear models associated with Groups 44 and 47 were selected for further analysis. Power models associated with Groups 8, 27, 30, 44, 45, 46, and 47 were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 57 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. As shown in Table 57, only one selected model performed satisfactorily in the cross-validation test. Table 58 shows the results of the selected power models after the requirements of the first three filters were met. The best models that also had a good cross-validation score are shaded and highlighted in blue in Tables 57 and 58. Out of all the highlighted correlation equations in Table 57 and Table 58, the Group 27 correlation equation in power form was selected as the most promising. Figure 195 shows the plot of POU versus q for this model. The vertical axis in Figure 195 represents the probability that, when the selected model is used, the predicted vc will be smaller than the actual vc, in percentage (with 0.1 m/s offset). To reach 90% confidence that the predicted vc is smaller than the actual vc, the predicted value should be multiplied by 0.7. Figure 193. MSE results for linear models in EFA/Coarse data set: critical velocity. Figure 194. MSE results for power models in EFA/Coarse data set: critical velocity.

Development of Correlation Equations 221 Table 57. Selected linear models for critical velocity in the EFA/Coarse data set. Table 58. Selected power models for critical velocity in the EFA/Coarse data set. Figure 195. Plot of POU versus correction factor for Group 27 (power): vc in the EFA/Coarse data set.

222 Relationship Between Erodibility and Properties of Soils 7.3.4.3 Initial Slope of Erosion RateâShear Stress The same four-filter process discussed in Section 7.3.3 is implemented in this section to obtain the best models for the initial slope of the erosion rateâshear stress curve, Et. Regression analysis for Et was performed for the EFA, HET, and JET separately. EFA Database. As with the cases of critical shear stress and critical velocity, it was observed that dividing the EFA/Global data set into the EFA/Fine and the EFA/Coarse data sets would significantly improve the regression results. Figure 196 shows the number of data points in each of the 135 combination groups in the EFA/Fine data set. Figure 197 and Figure 198 show the results for R2 for each combination group for the linear and power models, respectively. The R2 values of the power models were generally higher than those of the linear models. It can be observed that the R2 values were considerably higher for Groups 123 to 135 in Figure 198. Figure 199 shows the MSE results for the power models for each of the 135 combination groups in the EFA/Fine data set. After passing through Filters 1 and 2 (R2 and MSE), the power models associated with Groups 124, 126, 128, 130, 131, 132, and 134 were selected for further analysis. Filter 3, F-value/ F-stat, was determined for each group mentioned above. Table 59 shows the results of the Figure 196. Number of data points in each of 135 combination groups for the EFA/Fine data set: Es. Figure 197. R2 results for the linear models in EFA/Fine data set: Es.

Development of Correlation Equations 223 Figure 198. R2 results for the power models in EFA/Fine data set: Es. Figure 199. MSE results for the power models in EFA/Fine data set: Es Table 59. Selected power models for shear stress slope in the EFA/Fine data set.

224 Relationship Between Erodibility and Properties of Soils selected power models after the requirements of the first three filters (R2, MSE, and F-value/ F-stat) were met. The best models that also had a good cross-validation score are shaded and highlighted in blue. The Group 134 correlation equation in power form was selected as the most promising equation. Figure 200 shows the plot of POO versus q for this model. The vertical axis in Figure 200 represents the probability that, when the selected model is used, the predicted Et will be greater than the actual Et, in percentage (with 4 mm/h-Pa offset). To reach an 87% confidence that the predicted Et is greater than the actual Et, the predicted value should be multiplied by 2. The same procedure was conducted in the EFA/Coarse data set, and the best models were selected for Et. Figure 201 shows the number of data points in each of the 105 combination groups in the EFA/Coarse data set. Figure 202 and Figure 203 show the results for R2 for each combination group for the linear and power models, respectively. Both the linear and power models showed a few good groups in terms of R2 values; however, as seen in Figure 201, the number of data points in most groups was lower as compared with the EFA/Fine data set. Figure 204 and Figure 205 show the values of MSE for the linear and power models, respectively. Figure 200. Plot of POO versus correction factor for Group 134 (power): Es in the EFA/Fine data set. Figure 201. Number of data points in each of 105 combination groups for the EFA/Coarse data set: Es.

Development of Correlation Equations 225 Figure 202. R2 results for the linear models in EFA/Coarse data set: Es. Figure 203. R2 results for the power models in EFA/Coarse data set: Es. Figure 204. MSE results for the linear models in EFA/Coarse data set: Es.

226 Relationship Between Erodibility and Properties of Soils After passing through Filters 1 and 2 (R2 and MSE), linear models associated with Groups 38, 42, 43, and 47 were selected for further analysis. Power models associated with Groups 12, 13, 16, 18, 20, 28, 31, 34, 39, 40, 41, 42, 44, 74, 77, 80, 82, 86, 91, and 95 were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 60 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. As shown in Table 60, none of the selected models performed satisfactorily in the cross-validation test. Table 61 shows the results of the selected power models after the requirements of the first three filters were met. The Group 77 correlation equation in power form was selected as the most promising equation. Figure 206 shows the plot of POO versus q for this model. The vertical axis in Figure 206 represents the probability that, when the selected model is used, the predicted Et will be greater than the actual Et, in percentage (with 15 mm/h-Pa offset). To reach 80% confidence that the predicted Et is greater than the actual Et, the predicted value should be multiplied by 2.5. JET Database. As with the case of the EFA database, it was observed that dividing the JET/Global data set into the JET/Fine and the JET/Coarse data sets would significantly improve the regression results. Figure 207 shows the number of data points in each of the 135 combina- tion groups in the JET/Fine data set. Figure 208 and Figure 209 show the results for R2 for each combination group for the linear and power models, respectively. Figure 205. MSE results for the power models in EFA/Coarse data set: Es. Table 60. Selected linear models for shear stress slope in the EFA/Coarse data set.

Development of Correlation Equations 227 EÏ Table 61. Selected power models for shear stress slope in the EFA/Coarse data set. Figure 206. Plot of POO versus correction factor for Group 77 (power): Es in the EFA/Coarse data set.

228 Relationship Between Erodibility and Properties of Soils Figure 207. Number of data points in each of 135 combination groups for the JET/Fine data set: Es. Figure 208. R2 results for the linear models in JET/Fine data set: Es. Figure 209. R2 results for the power models in JET/Fine data set: Es.

Development of Correlation Equations 229 The R2 values of the power models were generally higher than those of the linear models. Figure 210 and Figure 211 show the MSE results for each of the 135 combination groups in the EFA/Fine data set for the linear and power models, respectively. After passing through Filters 1 and 2 (R2 and MSE), the linear models associated with Groups 100 and 113 were selected for further analysis. Power models associated with Groups 5, 12, 15, 20, 24, 38, 43, 71, 75, 78, 97, 123, 125, 128 were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 62 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. As shown in Table 62, none of the selected models performed satisfactorily in the cross-validation test. Table 63 shows the results of the selected power models after the requirements of the first three filters were met. The Group 15 correlation equation in power form was selected as the most promising equation. Figure 212 shows the plot of POO versus q for this model. The vertical axis in Figure 212 represents the probability that, when the selected model is used, the predicted Et will be greater than the actual Et, in percentage (with 6 mm/h-Pa offset). To reach 88% confidence that the predicted Et is greater than the actual Et, the predicted value should be multiplied by 2. The same procedure was conducted with the JET/Coarse data set, and the best models were selected for Et. Figure 213 shows the number of data points in each of the 105 combination Figure 210. MSE results for the linear models in JET/Fine data set: Es. Figure 211. MSE results for the power models in JET/Fine data set: Es.

230 Relationship Between Erodibility and Properties of Soils Table 62. Selected linear models for shear stress slope in the JET/Fine data set. EÏ Table 63. Selected power models for shear stress slope in the JET/Fine data set.

Development of Correlation Equations 231 groups in the EFA/Coarse data set. Figure 214 and Figure 215 show the results for R2 for each combination group for the linear and power models, respectively. Both the linear and power models showed many good groups in terms of R2 values; however, as shown in Figure 213, the number of data points in most groups was extremely low. Only a few combination groups contained sufficient data for regression analysis. After passing through Filters 1 and 2 (R2 and MSE), the linear models associated with Groups 5 and 26 were selected for further analysis. Power models associated with Groups 5, 7, and 26 were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 64 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. As shown in Table 64, none of the selected models performed satisfac- torily in the cross-validation test. Table 65 shows the results of the selected power models after the requirements of the first three filters were met. The Group 5 correlation equation in power Figure 212. Plot of POO versus correction factor for Group 15 (power): Es in the JET/Fine data set. Figure 213. Number of data points in each of 105 combination groups for the JET/Coarse data set: Es.

232 Relationship Between Erodibility and Properties of Soils Figure 214. R2 results for the linear models in JET/Coarse data set: Es. Figure 215. R2 results for the power models in JET/Coarse data set: Es. Table 64. Selected linear models for shear stress slope in the JET/Coarse data set. EÏ Table 65. Selected power models for shear stress slope in the JET/Coarse data set.

Development of Correlation Equations 233 form was selected as the most promising equation. Figure 216 shows the plot of POO versus q for this model. The vertical axis in Figure 216 shows the probability that, when the selected model is used, the predicted Et will be greater than the actual Et, in percentage (with 5 mm/h-Pa offset). To reach 90% confidence that the predicted Et is greater than the actual Et, the predicted value should be multiplied by 1.4. HET Database. It was observed that dividing the HET/Global data set into the HET/Fine and the HET/Coarse data sets would not significantly improve the regression results; however, the HET/Coarse database included a few combination groups with acceptable R2 values. Figure 217 shows the number of data points in each of the 105 combination groups in the HET/Coarse data set. As shown in Figure 217, many combination groups had no data points. Figure 218 and Figure 219 show the results for R2 for each combination group for the linear and power models, respectively. It is noteworthy that as the HET is limited to finer-grained soils, the HET/Coarse database corresponds to soils with D50 ranging from 0.074 to 0.3 mm. After passing through Filters 1 and 2 (R2 and MSE), the power models associated with Groups 40 and 60 were selected for further analysis. Linear models did not lead to R2 values that were any better; therefore, the study of best-fitting models was limited to power models. Figure 216. Plot of POO versus correction factor for Group 5 (power): Es in the JET/Coarse data set. Figure 217. Number of data points in each of 105 combination groups for the HET/Coarse data set: Es.

234 Relationship Between Erodibility and Properties of Soils Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 66 shows the results of the selected power models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. The Group 40 correlation equation in power form was selected as the most promising equation. Figure 220 shows the plot of POO versus q for this model. The vertical axis in Figure 220 shows the probability that, when the selected model is used, the predicted Et will be greater than the actual Et, in percentage (with 10 mm/h-Pa offset). To reach 80% confidence that the predicted Et is greater than the actual Et, the predicted value should be multiplied by 2. It is important to note that the equation associated with Group 40 should be used for coarse-grained soils with D50 ranging between 0.074 mm and 0.3 mm. The same procedure was conducted with the HET/Global data set (soils with D50 < 0.3 mm), and best models were selected for Et. Figure 221 shows the number of data points in each of the 135 combination groups in the HET/Global data set. Figure 222 and Figure 223 show the results for R2 for each combination group for the linear and power models, respectively. Figure 224 and Figure 225 show the MSE results for each of the 135 combination groups in the HET/Global data set for the linear and power models, respectively. As expected, the combination groups with better R2 values possess a lower MSE value as well. After passing through Filters 1 and 2 (R2 and MSE), linear models associated with Groups 5, 49, 53, 78, 79, 104, 105, 108, 109, 110, 112, 113, 114, 115, 118, and 124 were selected for further analysis. Power models associated with Groups 5, 49, 53, 79, 105, 108, 109, 110, 112, 114, 115, 118, and 124 were also selected for further analysis. Figure 218. R2 results for the linear models in HET/Coarse data set: Es. Figure 219. R2 results for the power models in HET/Coarse data set: Es.

Development of Correlation Equations 235 Table 66. Selected power models for shear stress slope in the HET/Coarse data set. Figure 220. Plot of POO versus correction factor for Group 40 (power): Es in the HET/Coarse data set. Figure 221. Number of data points in each of 135 combination groups for the HET/Global data set: Es.

236 Relationship Between Erodibility and Properties of Soils Figure 222. R2 results for the linear models in HET/Global data set: Es. Figure 223. R2 results for the power models in HET/Global data set: Es. Figure 224. MSE results for the linear models in HET/Global data set: Es.

Development of Correlation Equations 237 Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 67 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. As shown in Table 67, many of the selected models perform satisfactorily in the cross-validation test. Table 68 shows the results of the selected power models after the requirements of the first three filters were met. The best models that also had a good cross-validation score are shaded and highlighted in blue in Tables 67 and 68. Out of all the highlighted correlation equations, the Group 108 equation in power form was selected as the most promising equation. Figure 226 shows the plot of POO versus q for this model. The vertical axis in Figure 226 represent the probability that, when the selected model is used, the predicted Et will be greater than the actual Et, in percentage (with 0 mm/h-Pa offset). To reach 90% confidence that the predicted Et is greater than the actual Et, the pre- dicted value should be multiplied by 1.45. It is noteworthy that the equation associated with the com bination Group 108 in power form is best to be used for finer-grained soils with D50 smaller than 0.3 mm. Figure 225. MSE results for the power models in HET/Global data set: Es. Table 67. Selected linear models for shear stress slope in the HET/Global data set. (continued on next page)

238 Relationship Between Erodibility and Properties of Soils Table 67. (Continued). Table 68. Selected power models for shear stress slope in the HET/Global data set.

Development of Correlation Equations 239 7.3.4.4 Initial Slope of Erosion RateâVelocity The same four-filter process discussed in Section 7.3.3 is implemented in this section to obtain the best models for Ev. Similar to the case of critical velocity, vc, among the three erosion tests studied in this chapter (EFA, JET, HET), only the EFA test can reflect the initial slope of the erosion rate, Ev, as one of its outputs. In other words, results of JET and HET consist of only three erodibility parameters (i.e., tc, Et, and EC) in contrary with the EFA results which include all five erodibility parameters (i.e., tc, vc, Et, Ev, and EC). Therefore, regression analysis for Ev is limited to only the EFA data set. EFA Data Set. Similar to the case of Et, it was observed that dividing the EFA/Global data set into the EFA/Fine and the EFA/Coarse data sets would significantly improve the regression results for Ev. Figure 227 shows the number of data points in each of the 135 combination groups in the EFA/Fine data set. Figure 228 and Figure 229 show the results for R2 for each combination group for the linear and power models, respectively. Figure 226. Plot of POO versus correction factor for Group 108 (power): Es in the HET/Global data set. Figure 227. Number of data points in each of 135 combination groups for the EFA/Fine data set: Ev.

240 Relationship Between Erodibility and Properties of Soils Results show that power models in general perform better than the linear models. R2 values of Groups 109 to 135 are generally higher than the rest for both power models. Figure 230 and Figure 231 show the MSE results for each of the 135 combination groups in the EFA/Fine data set for linear and power models, respectively. After passing through Filters 1 and 2 (R2 and MSE), the power models associated with Groups 53, 79, 101, 105, 124, 126, 127, 131, and 134 were selected for further analysis. The linear models did not lead to R2 values that were any better; therefore, the study of best-fitting models was limited to power models. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 69 shows the results of the selected power models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. The Group 126 equation in power form was selected as the most promising equation. Figure 232 shows the plot of POO versus q for this model. The vertical axis in Figure 232 shows the probability that, when the selected model is used, the predicted Ev will be greater than the actual Ev, in percentage (with 10 mm-s/m-h offset). To reach 80% confidence that the predicted Ev is greater than the actual Ev, the predicted value should be multiplied by 2. The same procedure was conducted with the EFA/Coarse data set, and the best models were selected for Ev. Figure 233 shows the number of data points in each of the 105 combination groups in the EFA/Coarse data set. Figure 234 and Figure 235 show the results for R2 for each combination group for the linear and power models, respectively. Figure 228. R2 results for the linear models in EFA/Fine data set: Ev. Figure 229. R2 results for the power models in EFA/Fine data set: Ev.

Development of Correlation Equations 241 Figure 230. MSE results for the linear models in EFA/Fine data set: Ev. Figure 231. MSE results for the power models in EFA/Fine data set: Ev. Table 69. Selected power models for velocity slope in the EFA/Fine data set.

242 Relationship Between Erodibility and Properties of Soils Figure 232. Plot of POO versus correction factor for Group 126 (power): Ev in the EFA/Fine data set. Figure 233. Number of data points in each of 105 combination groups for the EFA/Coarse data set: Ev. Figure 234. R2 results for the linear models in EFA/Coarse data set: Ev.

Development of Correlation Equations 243 Both the linear and power models showed some good groups in terms of R2 values; however, as shown in Figure 233, the number of data points in most groups was very low. After passing through Filters 1 and 2 (R2 and MSE), the linear models associated with Groups 38, 42, 43, and 45 were selected for further analysis. Power models associated with Groups 18, 20, 28, 34, 36, 44, 46, 56, 65, 74, 77, 80, 82, 86, and 88 were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 70 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. As shown in Table 70, none of the selected models performed satisfactorily in the cross-validation test. Table 71 shows the results of the selected power models after the requirements of the first three filters were met. The Group 86 correlation equation in power form was selected as the most promising equation. Figure 236 shows the plot of POO versus q for this model. The vertical axis in Figure 236 shows the probability that, when the selected model is used, the predicted Ev will be greater than the actual Ev, in percentage (with 10 mm-s/m-h offset). To reach 80% confidence that the predicted Ev is greater than the actual Ev, the predicted value should be multiplied by 5. 7.3.4.5 Erosion Category The four-filter process discussed in Section 7.3.3 was implemented to obtain the best models for erosion category, EC. Regression analysis for EC was performed for the EFA, HET, and JET separately. Figure 235. R2 results for the power models in EFA/Coarse data set: Ev. Table 70. Selected linear models for velocity slope in the EFA/Coarse data set.

244 Relationship Between Erodibility and Properties of Soils Table 71. Selected power models for velocity slope in the EFA/Coarse data set. Figure 236. Plot of POO versus correction factor for Group 86 (power): Ev in the EFA/Coarse data set.

Development of Correlation Equations 245 EFA Database. It was observed that dividing the EFA/Global data set into the EFA/Fine and the EFA/Coarse data sets would improve the regression results. Figure 237 shows the number of data points in each of the 135 combination groups in the EFA/Fine data set. Figure 238 and Figure 239 show the results for R2 for each combination group for the linear and power models, respectively. The R2 values for both the linear and power models were not very high (up to 0.6). It can be observed that the better R2 values were observed in Combination Groups 109 to 135. Figure 240 and Figure 241 show the MSE results for each of the 135 combination groups in the EFA/Fine data set, for linear and power models, respectively. In general, the power and linear models were not noticeably different. After passing through Filters 1 and 2 (R2 and MSE), the linear models associated with Groups 109, 110, 113, 117, and 124 were selected for further analysis. Power models associated with Groups 124, 128, and 132 were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 72 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. As shown in Table 72, most of the selected models performed satisfactorily in the cross-validation test. Table 73 shows the results of the selected power models after the requirements of the first three filters were met. The Group 132 correlation equation in power form was selected as the most promising equation. Figure 242 shows the plot of POU Figure 237. Number of data points in each of 135 combination groups for the EFA/Fine data set: EC. Figure 238. R2 results for the linear models in EFA/Fine data set: EC.

246 Relationship Between Erodibility and Properties of Soils Figure 239. R2 results for the power models in EFA/Fine data set: EC. Figure 240. MSE results for the linear models in EFA/Fine data set: EC. Figure 241. MSE results for the power models in EFA/Fine data set: EC.

Development of Correlation Equations 247 Table 72. Selected linear models for erosion category in the EFA/Fine data set. Table 73. Selected power models for erosion category in the EFA/Fine data set. Figure 242. Plot of POU versus correction factor for Group 132 (power): EC in the EFA/Fine data set.

248 Relationship Between Erodibility and Properties of Soils versus q for this model. The vertical axis in Figure 242 shows the probability that, when the selected model is used, the predicted EC will be smaller than the actual EC, in percentage. To reach 90% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by 0.75. The same procedure was conducted with the EFA/Coarse data set, and the best models were selected for the EC. Figure 243 shows the number of data points in each of the 105 combination groups in the EFA/Coarse data set. Figure 244 and Figure 245 show the results for R2 for each combination group for the linear and power models, respectively. Both the linear and power models showed some good groups in terms of R2 values; however, as shown in Figure 243, the number of data points was very low in many combination groups. After passing through Filters 1 and 2 (R2 and MSE), the linear models associated with Groups 42, 46, 47, 55, 61, 64, 67, 68, 70, 72, 73, 87, 90, 94, 98, and 99 were selected for further analysis. Power models associated with Groups 32, 42, 46, 47, 65, 69, 91, and 95 were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 74 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. Table 75 shows the results of the selected power models after the Figure 243. Number of data points in each of 105 combination groups for the EFA/Coarse data set: EC. Figure 244. R2 results for the linear models in EFA/Coarse data set: EC.

Development of Correlation Equations 249 Figure 245. R2 results for the power models in EFA/Coarse data set: EC. Table 74. Selected linear models for erosion category in the EFA/Coarse data set.

250 Relationship Between Erodibility and Properties of Soils requirements of the first three filters were met. The Group 91 correlation equation in power form was selected as the most promising equation. Figure 246 shows the plot of POU versus q for this model. The vertical axis in Figure 246 shows the probability that, when the selected model is used, the predicted EC will be smaller than the actual EC, in percentage. To reach 90% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by 0.84. It is also noteworthy that the equation associated with Group 91 is the best one to use for soils with D50 ranging between 0.074 and 0.3 mm. Table 75. Selected power models for erosion category in the EFA/Coarse data set. Figure 246. Plot of POU versus correction factor for Group 91 (power): EC in the EFA/Coarse data set.

Development of Correlation Equations 251 JET Data Set. A similar approach was taken to select the best correlation equation for EC in the JET database. In the JET database, there are two important observations: 1. The JET was performed primarily on the finer soils (D50 < 0.3 mm), and, therefore, the number of data points in the JET/Coarse data set is substantially low compared with the JET/Fine data set. In fact, many combination groups in the JET/Coarse database have zero data points. 2. The R2 values for the JET/Global data set (D50 < 0.3 mm) are significantly better than those for the JET/Coarse and JET/Fine data sets. Therefore, the regression results for the JET/Global data set are presented as the best models. Figure 247 shows the number of data points in each of the 135 combination groups in the JET/Global data set. Figure 248 and Figure 249 show the results for R2 for each combination group for the linear and power models, respectively. Both the linear and power models show some reasonable groups in terms of R2 values. After passing through Filters 1 and 2 (R2 and MSE), linear models associated with Groups 30, 40, 49, 50, 52, 53, 88, 94, 97, 104, 124, and 129 were selected for further analysis. Power models associated with Groups 87, 93, 100, 102, 109, 112, and 129 were also selected for further analysis. Figure 247. Number of data points in each of 135 combination groups for the JET/Global data set: EC. Figure 248. R2 results for the linear models in JET/Global data set: EC.

252 Relationship Between Erodibility and Properties of Soils Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 76 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. As shown in Table 76, almost all of the selected models performed satisfactorily in the cross-validation test. Table 77 shows the results of the selected power models after the requirements of the first three filters were met. The best models that also had a good cross-validation score are shaded and highlighted in blue in Tables 76 and 77. Out of all of the highlighted correlation equations in Table 76 and Table 77, the Group 88 correlation equation in Figure 249. R2 results for the power models in JET/Global data set: EC. Table 76. Selected linear models for erosion category in the JET/Global data set.

Development of Correlation Equations 253 linear form was selected as the most promising equation. Figure 250 shows the plot of POU versus q for this model. The vertical axis in Figure 250 represents the probability that, when the selected model is used, the predicted EC will be smaller than the actual EC, in percentage. To reach 90% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by 0.85. The proposed equations are best for soils with D50 smaller than 0.3 mm. HET Database. Although the number of data points in the HET/Coarse combination groups was very low in most cases, it was observed that dividing the HET/Global data set into the HET/Fine and the HET/Coarse data sets would improve the regression results. Figure 251 shows the number of data points in each of the 135 combination groups in the HET/Fine data set. Figure 252 and Figure 253 show the results for R2 for each combination group for the linear and power models, respectively. The R2 values for both the linear and power models were not very high (up to 0.7). Figure 254 and Figure 255 show the MSE results for each of the 135 combination groups in the HET/Fine data set for the linear and power models, respectively. In general, the power and linear models were not noticeably different. Table 77. Selected power models for erosion category in the JET/Global data set. Figure 250. Plot of POU versus correction factor for Group 88 (linear): EC in the JET/Global data set.

254 Relationship Between Erodibility and Properties of Soils Figure 251. Number of data points in each of 135 combination groups for the HET/Fine data set: EC. Figure 252. R2 results for the linear models in HET/Fine data set: EC. Figure 253. R2 results for the power models in HET/Fine data set: EC.

Development of Correlation Equations 255 After passing through Filters 1 and 2 (R2 and MSE), the linear models associated with Groups 48, 53, 74, 79, 100, 109, 112, 114, and 129 were selected for further analysis. Power models associated with Groups 5, 12, 18, 19, 31, 34, 35, 37, 40, 47, 50, 52, 53, 108, 109, 112, 113, and 129 were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 78 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. Table 79 shows the results of the selected power models after the requirements of the first three filters were met. The best models that also had a good cross-validation score are shaded and highlighted in blue in Tables 78 and 79. The Group 12 correlation equation in power form was selected as the most promising equation. Figure 256 shows the plot of POU versus q for this model. The vertical axis in Figure 256 represents the probability that, when the selected model is used, the predicted EC will be smaller than the actual EC, in percentage. To reach 100% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by 0.95. The same procedure was conducted with the HET/Coarse data set, and the best models were selected for EC. Figure 257 shows the number of data points in each of the 105 combination groups in the EFA/Coarse data set. Figure 258 and Figure 259 show the results for R2 for each combination group for the linear and power models, respectively. Figure 254. MSE results for the linear models in HET/Fine data set: EC. Figure 255. MSE results for the power models in HET/Fine data set: EC.

256 Relationship Between Erodibility and Properties of Soils Table 78. Selected linear models for erosion category in the HET/Fine data set. Table 79. Selected power models for erosion category in the HET/Fine data set.

Development of Correlation Equations 257 Figure 256. Plot of POU versus correction factor for Group 12 (power): EC in the HET/Fine data set. Figure 257. Number of data points in each of 105 combination groups for the HET/Coarse data set: EC. Figure 258. R2 results for the linear models in HET/Coarse data set: EC.

258 Relationship Between Erodibility and Properties of Soils Both the linear and power models showed some good groups in terms of R2 values; however, as shown in Figure 257, the number of data points was very low in most combination groups. In fact, close to half of the combination groups did not have any data points. After passing through Filters 1 and 2 (R2 and MSE), the linear models associated with Groups 4, 5, 7, 10, 15, 26, 48, 50, 51, 53, 54, 57, 59, 60, 63, 66, 71, 74, 76, 79, 85, 100, 102, and 103 were selected for further analysis. Power models associated with Groups 4, 5, 7, 15, 26, 48, 50, 51, 57, 59, 60, 71, 74, 76, 77, 79, 83, 85, 86, 97, 102, 103, and 105 were also selected for further analysis. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 80 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) were met. Table 81 shows the results of the selected power models after the requirements of the first three filters were met. The best models that also had a good cross-validation score are shaded and highlighted in blue in Tables 80 and 81. The Group 48 correlation equation in power form was selected as the most promising equation. Figure 260 shows the plot of POU versus q for this model. The vertical axis in Figure 260 represents the probability that, when the selected model is used, the predicted EC will be smaller than the actual EC, in percentage. To reach 90% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by 0.85. It is very important to note that the proposed equation associated with Group 48 should be used for soils with D50 ranging from 0.074 to 0.3 mm. 7.4 Probabilistic (Bayesian) Analysis The preceding section focused on selecting the optimal vector of model parameters that maximizes the likelihood of fitting experimental observations. This section introduces a series of Bayesian probabilistic calibrations carried out on a set of regression (empirical) models pro- posed as optimal to capture only the modelâs first order statistics (expected or mean behavior). The Bayesian regression analysis introduces a methodology to fully assess both first and second order statistics (expected or mean and variance and covariance, respectively) generated by the same erosion tests presented in the deterministic regression analysis discussed in the previous section. That is, Bayesian regression allows quantification of varying uncertainty scenarios resulting from different sources of evidence (i.e., varying experimental observations, varying model complexity, and varying expert judgment), providing further inferences to better understand the performance of a given regression model. This is achieved by providing a full Figure 259. R2 results for the power models in HET/Coarse data set: EC.

Development of Correlation Equations 259 Table 80. Selected linear models for erosion category in the HET/Coarse data set.

260 Relationship Between Erodibility and Properties of Soils Table 81. Selected power models for erosion category in the HET/Coarse data set. Figure 260. Plot of POU versus correction factor for Group 48 (power): EC in the HET/Coarse data set.

Development of Correlation Equations 261 characterization of all possible solutions of the model parameters and their relative probabilities while simultaneously providing a systematic and transparent approach to assess the perfor- mance of the proposed regression model. The major difference between Bayesian analysis and deterministic frequentist regression is in how each approach interprets the observed data and model parameters. Frequentist regression assumes that the model parameters are, in fact, fixed unknown parameters and the observed data are random repeatable samples. However, the Bayesian inference approach assumes that the observed data are fixed values, while the model parameters are random parameters. There- fore, a frequentist regression approach results in fixed values attributed to the model parameters, whereas the Bayesian approach results in posterior probability distributions for each model parameter rather than a fixed value. The main benefit of the use of the Bayesian inference approach is the definition of a metric of confidence on the model predictions. This permits assessment of both the performance of competing models given a set of experimental observations based on a given experimental method and the performance of competing experimental methods given a predictive model. The Bayesian approach departs from standard deterministic calibrations (i.e., least squares) by populating âallâ the likely combinations of parameters of a predictive model suitable to represent the mean of the process of interest (i.e., the fit of a given set of experimental observations), as opposed to proposing a combination of single parameters (i.e., the optimal, which may not be unique). As a result, it is then possible to generate a full probabilistic description of the model parameters in the form of marginal probability density functions (PDFs)âthat is, each model parameter is represented by its own PDFâand a full probabilistic description of the parametersâ correlation structure (when taken two at a time). It should also be noted that the concepts of the confidence interval and hypothesis testing in frequentist regression and Bayesian inference are different. As an illustration, when, in the interpretation of the results of a frequentist regression approach, the confidence interval for a parameter is reported as 90%, this means that if the exact same experiment is repeated several times, and for each time the confidence interval is obtained, then 90% of the obtained inter- vals include the parameter. It does not mean that there is 90% chance that the parameter is within the confidence interval. On the other hand, when a Bayesian approach reports a 90% confidence interval (or, as it is typically called in Bayesian inference, the âcredible intervalâ) for a parameter, this means that there is 90% chance that the credible interval contains the parameter. 7.4.1 Motivation One main objective of the NCHRP-Erosion spreadsheet (see Chapter 5) is to assimilate information collected from a broad range of erodibility tests and correlate those results with the geotechnical properties of primary soil by means of statistical modeling. However, regard- less of the level of model sophistication, broadness of experimental observations, or accuracy of expert judgement, there is a finite amount of uncertainty associated with every mathematical representationânone of the modeling results can predict the property of interest with complete confidence. This is due to several practical limitations: â¢ Missing data from the database may introduce bias into the proper model calibration. â¢ The proposed empirical models do not capture the mean of the process of interest or do not show a proper correlation between erodibility and geotechnical parameters. â¢ One single model selection, the optimal, may represent only one single fit of possibly millions of likely combinations that may produce the same degree of curve fitting (i.e., the regression analysis is an ill-posed problem).

262 Relationship Between Erodibility and Properties of Soils Understanding this varying evidence condition motivates a specialized calibration that points to systematic and transparent assessment of prediction confidence based on all available evidence. 7.4.2 Hypotheses The foregoing regression analysis selected several groups of experimental observations and corresponding statistical models that showed satisfying model/observation comparisons. The proposed probabilistic (Bayesian) calibration method (Medina-Cetina 2006) is introduced here to complement the previous deterministic regression approach. Several hypotheses are proposed and are thoroughly discussed in Chapter 8: â¢ The probabilistic approach allows a full characterization of sources and propagation of model uncertainty and relative probabilities through systematic assimilation of available evidence (i.e., experimental observations, model prediction, and expert judgment). â¢ Varying estimation confidence levels of parameters can exhibit in the parametric space. â¢ Varying correlation structures among parameters can be shown in the calibration results, which will help reveal the model nature. â¢ The probabilistic characteristics of the proposed models will be depicted in multidimensional âphysical domainsâ (composed of model independent variables); the goodness of the modeling depends not only on capturing each available observation (erodibility test results), but also on the confidence level of the estimation. 7.4.3 Methodology 7.4.3.1 Uncertainty Quantification Framework The uncertainty quantification of an inverse problem aims to identify, characterize, and simulate the various sources of uncertainty inherently participating in the physical process of interest (Medina-Cetina 2006). The expected output of the true process is represented by a set of random vectors, d, which is based on the definition of the physical process at prescribed control points. In practice, experimental observations, dobs, can be retrieved from lab or field measurements and compared with predicting outcomes of the same process, dpred. These are quantifiable vectors formed as a priori information used to approximate the true process, d. Accordingly, the uncertainty quantification framework in terms of d, dobs and dpred can be summarized as follows: d d d d d d d d d d = + Î = + Î â = Î â Î (73) obs obs pred pred obs pred obs pred Equation 73 illustrates the tradeoff between the scientific evidence dobs and dpred through the gradients, Îd. The involved uncertainty incorporates random vectors both in experimental observations and theoretical predictions. Herein, dpred comprises mathematical predictions stem- ming from a forward model, g(q), which is governed or characterized by a set of parameters, q. Notice that q can represent geometric and statistical properties, such as shape parameters or, in this case, linear/power parameters. Similarly, the uncertainty involved in the calibration of vectors, q, can be defined as Ë (74)q = q + Îq

Development of Correlation Equations 263 where qÌ denotes mean of parameters and Îq represents the uncertainty component. Other than the deterministic methodology, the probabilistic calibration allows for an exhaustive exploration of all potential combinations of the model parameters that best resemble the experimental observations. As a result, the correlation structures of model parameters are popu- lated, which ultimately can be translated into a better understanding of uncertainty inherent to model nature (Îdpred) with respect to the experimental observations. Notice that in this work, the model prediction is assumed to be unbiased with respect to the process of interest, which implies E(Îdobs â Îdpred) = 0. 7.4.3.2 Bayesian Probabilistic Calibration The proposed uncertainty quantification framework requires assessment of plausible solu- tions of model parameters q conditioned on the data, dobs, which generates the need of math- ematical mapping regarding this inverse problem. From a deterministic standpoint, this can be accomplished by selecting an optimal set of model parameters that maximize the likelihood of fitting dobs. However, the proposed calibration can be an ill-posed inverse problem, since many combinations of the model parameters can lead to the same experimental response. To tackle this problem, the proposed probabilistic calibration method follows a Bayesian approach, which accounts for the full probabilistic description of the model parameters through probability maps. This starts from an expertâs belief setting up the prior [p(q)] about model parameters q before the experimental evidence is presented to the mechanical model (forward model). This prior knowledge ideally facilitates calibration of model parameters by limiting and defining plausible values in the form of probability distribution, which is later updated systematically via quantifying the likelihood (p(dobs|q)) between available observations dobs and model parameters q. From the basic definition of Bayes theorem: , , , (75)obs obs obs obsd p d g p d g d p d g â« ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )p q = q q p q q q p q q â q q p q Q The posterior p(q|dobs) is the probability proportional to the prior p(q) and the likelihood, p(dobs|q). This is because the integral of the denominator is a normalizing constant over the parametric space Q so that the integral of the posterior p(q|dobs) can be 1. One important note about implementing Bayesian inference is that the computations often involve very complex integrations that cannot be handled analytically. Also, the posterior distribution hardly is in explicit form and requires simulations in order to be achieved. Markov Chain Monte Carlo (MCMC) sampling methods are typically used to solve for complex posterior models. The most typical MCMC sampling method is the MetropolisâHastings (Metropolis et al. 1953; Hastings 1970). This sampling method follows an algorithm that generates values from a posterior distribution and converges to a predetermined target dis- tribution. Further information on MCMC and its algorithm can be found in Hastings (1970). When any MCMC sampling method is used to approximate the posterior distributions, verifying its convergence becomes a vital step in the Bayesian inference approach. In this study, the sampling of the posterior is based on an MCMC approach coupled with Metropolisâ Hastings criteria, which makes it possible to draw samples from a proposing distribution to infer the target posterior distributions. 7.4.4 Probabilistic Calibration for Varying Data Scenarios The probabilistic calibrations discussed in this section were conducted on the proposed model/data scenarios produced by the deterministic regression results populated in the pre- vious section. Table 82 to Table 86 list deterministic regression calibrations with the highest

264 Relationship Between Erodibility and Properties of Soils an = number of data points. bParameter values given by deterministic regression. Table 82. Selected models for critical shear stress, sc. an = number of data points. bParameter values given by deterministic regression. Table 83. Selected models for critical velocity, vc. an = number of data points. bParameter values given by deterministic regression. Table 84. Selected models for erosion category, EC.

Development of Correlation Equations 265 goodness of fit for each erodibility variable (i.e., critical shear stress, tc; critical velocity, vc; erosion category, EC; velocity slope, Ev; and shear stress slope, Et, respectively). It must be noted that not all of the selected equations in the previous section could be converged in the probabilistic calibration. Tables 82 to 86 show only the equations that were successfully calibrated. The results of the probabilistic calibrations for all of the equations presented in Tables 82 to 86 are presented in Appendix 5 of the appendices report. However, in this section, one case is discussed to illustrate the applicability of the method, labeled with Group 132 (See Table 84). In Appendix 5, the same type of results as the ones described in this section are generated for all of the selected equations. 7.4.4.1 Power Model for Erosion Category, EC, EFA/Fine Data Set The selected case is the power model created to predict the erodibility parameter, EC, for the EFA/Fine data set (Group 132). Four independent variables are considered: soil activity, A; water content, WC; undrained shear strength, Su; and mean particle size, D50. A total of 44 data observations were available, each consisting of four geotechnical properties as input and EC as the erodibility property or output. Table 84 provides the optimization result of model parameters through nonlinear regression. Taking these parameter values, the generated model predictions, along with the observed data plotted along each variable domain, are as shown in Figure 261. It is shown that the vector of experimental observations is scattered along each variable domain and that the proposed model overall captures the mean of the process. Step 1. Optimization. In the proposed probabilistic calibration framework, the optimization results are able not only to provide the initial guess of random parameter values (as presented in Figure 261), but also to retrieve the shape of the error between model predictions and observa- tions, which leads to the selection of the probability function to be considered for the likelihood. Figure 262a shows a histogram and kernel density estimate of residuals of model prediction, a nominal Gaussian distribution of the mean around zero, seemingly a reasonable fit of empirical Table 85. Selected model for velocity slope, Ev. an = number of data points. bParameter values given by deterministic regression. Table 86. Selected models for shear stress slope, Es.

266 Relationship Between Erodibility and Properties of Soils (a) (b) (c) (d) Figure 261. Experimental observations and model predictions along variable domains: (a) soil activity, (b) water content, (c) undrained shear strength, and (d) mean particle size (dobs = experimental observations; dopt = d optimized). (a) (b) Figure 262. (a) Histogram and kernel density estimate of error and (b) ECDF and Gaussian fit of error.

Development of Correlation Equations 267 distribution, which indicates also the unbiased character of the proposed model. However, the change of bin size and starting position of the histogram may result in a variation to its shape. Even though the kernel density estimate can eliminate the effect of the starting position of the plot, one still needs to determine the kernel width, which is generally difficult in practice. Figure 262b indicates a better solution to the problem. Plotting the ECDF of error allows a mono- tonic increase of probability to provide a unique description of error distribution. A Gaussian distribution presents a suitable approximation of error distribution; thus, it is adopted as the likelihood function and used in the Bayesian formulation. A noninformative distribution is considered as a prior for the Bayesian formulation. Step 2. Probabilistic Calibration and Convergence Diagnosis. For the sake of a better presentation of the calibration results for Group 132 (Table 84), each model parameter was named following the variable it serves, that is, Î²0, Î²A, Î²WC, and Î²Su, where Î²0 refers to the scaling factor at the beginning of the equation. The selected prior for all parameters is a vague Gaussian prior with the mean equal to zero and a standard deviation of 106. Figure 263 presents MCMC sample sequences of each parameter for 100,000 iterations. A stationary state was achieved for each parameter as illustrated by Figure 264 and Figure 265 are cumulative mean and cumulative standard deviations of sample sequences for each parameter. These plots are used to validate the convergence of MCMC achieved at a stationary condition and to define the burn-in point after which statistics about the model performance are com- puted. The principle of stationary should be achieved for both cumulative mean and standard deviation plots. In this case, the burn-in point is set as 40,000. Step 3. Posterior Statistics. Once the MCMC posterior sampling reaches a stationary state, statistical inferences can be generated. Figure 266 shows a depiction of the joint relative frequency histograms across all model parameters as well as the marginal PDFs of each param- eter presented along the figuresâ matrix diagonal. Most probability distributions indicate an asymptotic normality in its shape. The validity of this rests on the central limit theorem. Table 87 gives posterior statistics in terms of mean, standard deviation (SD), coefficient of variation (CV), mode, and 95% highest posterior density (HPD) region. The mean values are close to what has been obtained from regression analysis, but one should adopt these with caution. For instance, the distribution of Î²0 is not symmetric and skews to the right (i.e., positive skewness), which yields its mean value greater than the mode. A more rational choice would be the latter, as it represents higher probability by the probabilistic calibration. Lower triangular plots in Figure 266 are the parametersâ cross-correlation investigations, with two at a time. Since the PDF of Î²0 is asymmetric, nonlinear correlation structure is observed between Î²0 and other parameters. A significant negative correlation is shown between Î²A and Î²WC, indicating the strong association of soilâs water content and activity in predicting erosion category. Scatter plots of posterior samples and Pearson correlation coefficients are given in the upper triangular matrix, which provides direct delineations of correlation type and degree among various parameter combinations. It is worth noting that in Table 87, the CVs are given as complementary information of parameter variability. Additionally, a 95% HPD region, which denotes the credible range where a given parameter exhibits higher probability of occurrence, is given for each parameter. Step 4. Probabilistic Realizations and Assessment of Model Performance. Finally, the probabilistic calibration approach allows production of a metric of the model performance. Once the posterior distribution is populated, likely realizations of the model predictions can be computed by sampling random parameter combinations from it. These model responses provide numerical evidence to estimate first and second order statistics in regard to model performance.

268 Relationship Between Erodibility and Properties of Soils (a) (b) (c) (e) (d) Figure 263. Random samples of model parameters: (a) a0 , (b) aA, (c) aWC, (d) aSu, and (e) aD50.

Development of Correlation Equations 269 (a) (b) (c) (e) (d) Figure 264. Cumulative mean of sample sequences for each parameter: (a) a0, (b) aA, (c) aWC, (d) aSu, and (e) aD50.

270 Relationship Between Erodibility and Properties of Soils (a) (b) (c) (e) (d) Figure 265. Cumulative standard deviation of sample sequences for each parameter: (a) a0, (b) aA, (c) aWC, (d) aSu, and (e) aD50.

Figure 266. Joint relative frequency histogram of model parameters, two at a time.

272 Relationship Between Erodibility and Properties of Soils To achieve this goal, composition of a multidimensional mesh grid retrieved across all independent variables is required, enabling the assessment of the model prediction along each domain of all independent variables versus the dependent variable or erodibility parameter. Each domain range is decided by minimum and maximum values retrieved from experimental observations; additionally, 10 uniform discretization steps were chosen for each variable to ensure computational efficiency and sufficiency of the inferences at the same time. For example, in the case of Group 132, there are four independent variables, and the mesh grid is discretized with 10 steps, meaning 104 = 10,000 points where the hypersurface produced by the model will be repeatedly evaluated for different parametersâ combinations. That is, from the posterior, 1,000 random parameter samples are taken after the burn-in point; these produce the ensemble of model predictions (107 model predictions along the mesh grid). Probabilistic and deterministic model realizations along each independent variable domain are shown in Figure 267. Once the ensemble of model predictions is plotted along each variable domain, a one-to-one mapping is then possible where multiple model outputs can correspond to one prescribed variable value, as it shown in Figure 267. This shows 1,000 probabilistic realiza- tions as opposed to 1 deterministic realization. The progress of model realizations along the A and D50 domains shows a distinct trend compared with WC and Su. This is due to the mean of the parameters retrieved from the posterior being positive for Î²WC and Î²Su but negative for Î²A and Î²D50. First order statistics for this model can be computed along the domain of each independent variable (physical domain) as presented in Figure 268, which shows that the mean computed from the ensemble of simulations produced by the posterior converges to the optimal estimate produced by the deterministic calibration. However, a heteroscedastic condition is shown, or variance variability along each domain of all independent variables. This is a reflection of the non-Gaussian nature of the variation of the mean of the EC process. Figure 268 shows larger HPD interval areas above the mean of the model predictions, which indicates that the EC predictions are skewed to the upper side for all independent variables. It is very important to notice that these first and second order statistics correspond to the given model realizations (Group 132), which represent the credibility of the mean, not of the whole population. The yellow and red curves represent the mean of the probabilistic calibration and optimal model predictions at each independent variable. The mean produced from the probabilistic calibra- tion stemmed from an exhaustive sampling process across all the parametric space (MCMC) as opposed to a limited sampling to assess the deterministic calibration to produce the optimal vector of the model parameters. Figure 268b presents the least overall uncertainty among the four independent variables, which shows WC as the best predictor for this group number. As with the POU/POO analyses performed on the basis of the deterministic regression results, it is also possible to associate the correction factors with confidence levels with respect to the probabilistic results. Figure 269 presents the measured EC versus predicted EC in an Parameter Mean SD CV Mode 95% HPD region Lower Bound Upper Bound Î²0 0.27 0.13 0.50 0.19 0.07 0.53 Î²A â0.04 0.05 â1.02 â0.04 â0.13 0.05 Î²WC 0.46 0.10 0.22 0.49 0.27 0.66 Î²Su 0.07 0.06 0.77 0.07 â0.03 0.19 Î²D50 â0.12 0.04 â0.34 â0.12 â0.19 â0.04 Table 87. Statistics of probabilistic calibrated model parameters.

Development of Correlation Equations 273 equal aspect ratio context, where results lying along the 45Â° line indicate a perfect fit. Similar to the 1,000 posterior ensemble of realizations, this figure shows 1,000 model predictions at the same location of the available experimental EC observations. Box plots at each of these points are presented to indicate the model variation or variation of the mean of the EC process. From the same 1,000 realizations, the correction factors (q) are computed to produce POU levels, as shown in Figure 270a. Figure 270b presents the mean and HPD intervals for the same real- izations, showing a growing uncertainty inherent in POU with the increase of the correction factor. This is a significant improvement over the previous deterministic plots, since these add a confidence metric on the assessment of the correction factors associated with every POU value. 7.4.4.2 Linear Model for Erosion Category, EC, EFA/Fine Data Set A second probabilistic calibration is presented for the linear model for erodibility parameter EC, EFA/Fine data set (Group 117). The main difference between the two cases discussed here is the proposed regression model used to predict EC. Both first and second order statistics for each model parameter are evaluated. Only results after Step 2 of the probabilistic calibration (c) (d) (a) (b) Figure 267. Model realizations coupled with observed data set along each variable domain: (a) soil activity, (b) water content, (c) undrained shear strength, and (d) mean particle size (dpro-pred = d probabilistically predicted; ddet-pred = d deterministically predicted).

274 Relationship Between Erodibility and Properties of Soils (c) (d) (a) (b) Figure 268. Mean and standard deviations of model predictions versus observed data: (a) soil activity, (b) water content, (c) undrained shear strength, and (d) mean particle size. 0.0 Figure 269. Measured EC versus predicted EC based on optimization and probabilistic calibration results.

Development of Correlation Equations 275 are presented for this model, as convergence of the MCMC posterior sample was also achieved. The matrix of plots presenting the posteriorâs joint relative frequency histograms of this modelâs parameters is introduced in Figure 271. A uniform negative correlation between intercept Î²0 and other parameters is presented in the bottom row, with correlation coefficients ranging from â0.491 to â0.841. In contrast with the power model, all marginal PDFs along the diagonal follow a Gaussian shape without skewness, which results from the free boundary of possible EC parameter values in the linear model (negative values of Î²0 in the power model are by default not sampled during MCMC). Higher CV values of Î²D50 and Î²Su are expected to cause bigger varia- tions along the D50 and Su domains, which will be examined through realizations and statistical analysis in each variable domain. Table 88 shows the model characteristics and optimization result on the basis of the determin- istic approach. Table 89 gives posterior statistics for the same group of parameters in terms of the mean, standard deviation, coefficient of variation, mode, and 95% highest posterior density (HPD) region. Figure 272 and Figure 273 display 1,000 model realizations from the sampling of the posterior and its corresponding first and second order statistics along each domain of the independent variables. The mean of the EC process produced by the probabilistic calibration and the optimal realization produced by the deterministic regression show significant agreement. In addition, comparison of HPD intervals for the linear and power models indicates uncertainty is more uniformly distributed in both domains and codomains for the former and also quantitatively smaller. However, it is worth noticing that using the linear model may yield negative values of EC that might not make sense in reality. A rational choice in such a case would be a power model instead or to further restrict the variation of EC during the MCMC sampling process of the posterior. Finally, a reliability-based analysis is conducted to assess the model performance. Even though both linear and power models show a good capacity to capture the mean of the EC process, a more uniform prediction variation is presented for the linear model (Figure 274), which is consistent with the findings in Figure 273. In regard to the plots of POU versus q (see Figure 275), the results are similar for both models. For instance, seeking 90% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by a correction factor equal to 0.80 for both scenarios. (a) (b) Figure 270. Probability of underprediction: (a) 1,000 realizations of POU versus correction factor and (b) mean and HPD intervals of POU.

Figure 271. Joint relative frequency histogram of model parameters, two at a time.

Group No. Independent Variables Data Seta Model Expression (parameter values given by deterministic regression) R2 Cross- Validation Score 117 PC, WC, Su, D50 EFA/Fine (n = 44) 0.56 0.43 a n = number of data points. Table 88. Model characteristics and optimization result. Parameter Mean SD CV Mode 95% HPD region Lower Bound Upper Bound Î²PC 0.021 0.008 0.378 0.021 0.006 0.037 Î²WC 0.031 0.005 0.160 0.030 0.021 0.040 Î²Su 0.002 0.003 1.313 0.002 â0.003 0.007 Î²D50 â2.396 5.349 â2.232 â1.137 â12.858 8.083 Î²0 0.922 0.440 0.477 1.004 0.053 1.770 Table 89. Statistics of probabilistic calibrated model parameters. (a) (b) (c) (d) Figure 272. Model realizations coupled with observed data set along each variable domain: (a) percentage of clay, (b) water content, (c) undrained shear strength, and (d) mean particle size.

278 Relationship Between Erodibility and Properties of Soils (c) (d) (a) (b) Figure 273. Mean and standard deviations of model predictions versus observed data: (a) percentage of clay, (b) water content, (c) undrained shear strength, and (d) mean particle size. Figure 274. Measured EC versus predicted EC, based on optimization and probabilistic calibration results.

Development of Correlation Equations 279 (a) (b) Figure 275. Probability of underprediction: (a) 1,000 realizations of POU versus correction factor and (b) mean and HPD intervals of POU.