National Academies Press: OpenBook

Relationship Between Erodibility and Properties of Soils (2019)

Chapter: Chapter 7. Correlation Equations Development

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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Suggested Citation:"Chapter 7. Correlation Equations Development." 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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Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

174 CHAPTER 7 7. CORRELATION EQUATIONS DEVELOPMENT The main goal of this research was to develop equations correlating soil erosion parameters defined in previous chapters (i.e. EC, , , , and ) to the common soil engineering properties. Since the majority of the erosion test data compiled in the TAMU-Erosion were comprised of the data obtained from the EFA, the JET, and the HET, and given the fact 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 (Figure 31) are populated with the hundreds of the EFA test data compiled in the TAMU-Erosion spreadsheet, and new equations are developed. Final phase of this project consists of performing two parallel statistical approaches: 1) Deterministic frequentists’ regression analysis, and 2) Probabilistic (Bayesian) analysis. The goal is to reach the best potential fits between erodibility parameters and geotechnical properties of soils. The experimental design, optimization, model selection, and final results of the deterministic frequentists’ regression approach is comprehensively presented in Section 7.3. The results of the probabilistic (Bayesian) approach is presented in Section 7.4. 7.1. Determining the Erosion Resistance Using the USCS As discussed in Chapter 1, the Erosion Function Charts are charts that show erosion categories in the Ż–v and Ż–τ space (Figure 3). These charts were conceptually designed to eliminate the need for site-specific erosion testing in the case of preliminary investigations and first order erosion analyses. It was first developed on the basis of EFA tests performed at Texas A&M University, after the 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 shows the values of τc and vc according to the erosion categories. This section discusses the intrdocution of zones that represent different soil types on the erosion charts. These zones are characterized by using the USCS categories. For coarse-grained soils, the erodibility 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 distribution, 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., 1986; Partheniades, 2009). It was also observed that the plasticity, clay content, and soil activity play the most dominant roles in erosion resistance of fine-grained soils. Plasticity parameters form the basis of the USCS classification for fine-grained soils and it is reasonable to think that, as such, the USCS category has good potential for distinguishing between erosion categories for fine-grained soils as well.

175 As discussed earlier in the previous chapters, about 330 EFA test results compiled in the TAMU-Erosion Spreadsheet were divided into different USCS classification groups. Table 32 below shows the USCS classification groups along with the number of EFA tests in each group. Table 32. List of the USCS categories associated with the 330 samples USCS Categories Number of samples Fat Clay (CH) 63 Lean Clay (CL) 131 Poorly Graded Sand (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 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 on each figure are proposed, after considering 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 the cases that the EFA data are not many to make an inclusive conclusion) so it is in a reasonable consistence with the previously proposed version of this chart by Briaud et al. (2008). It is observed that 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 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, that 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 TAMU-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

176 in Figure 145 apply solely to the erosion category and are not shown with respect to the erosion rate. Figure 131. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for fat clay (CH) soils

177 Figure 132. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for lean clay (CL) soils

178 Figure 133. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for poorly grqded gravel (GP) soils

179 Figure 134. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for clayey gravel (GC) soils

180 Figure 135. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for high plasticity silt (MH) soils

181 Figure 136. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for low plasticity silt (ML) soils

182 Figure 137. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for low plasticity silty clay (ML-CL) soils

183 Figure 138. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for clayey sand (SC) soils

184 Figure 139. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for clayey silty sand (SC- SM) soils

185 Figure 140. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for silty sand (SM) soils

186 Figure 141. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for poorly graded sand (SP) soils

187 Figure 142. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for poorly graded sand with clay (SP-SC) soils

188 Figure 143. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for poorly graded sand with silt (SP-SM) soils

189 Figure 144. Velocity-Erosion Rate and Shear Stress-Erosion Rate Plots for well graded sand with silt (SW-SM) soils

190 Figure 145. Erosion Category Charts with the USCS Symbols 0.1 1 10 100 1000 10000 100000 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 (30-150 mm Spacing) -Riprap -Jointed Rock (150-1500 mm Spacing) -Intact Rock -Jointed Rock (Spacing > 1500 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 1000 10000 100000 0.1 1 10 100 1000 10000 100000 SHEAR STRESS (Pa) Very High Erodibility I High Erodibility II MediumErodibility III Low Erodibility IV Very Low Erodibility V EROSION RATE (mm/hr) -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 (30-150 mm Spacing) -Riprap -Jointed Rock (150-1500 mm Spacing) -Intact Rock -Jointed Rock (Spacing > 1500 mm) MH SP-SM ML-CL Rock SW SW-SM SP-SC SP SM SC-SM SC ML GC CL GP CH

191 7.2. Plots of Critical Velocity and Shear Stress versus Mean Particle Size Briaud et al. (2001) and Briaud et al. (2017) 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 TAMU-Erosion Spreadsheet. It was observed that for soils with mean particle size larger than 0.3 mm, following relationships exist between the critical velocity/shear stress and mean particle size (Eqs. 61 and 62).  0.5500.315 ( )( / )c D mmv m s  (61) 50( ( ))c D mPa m  (62) It was also concluded that for fine-grained soils there is no direct relationship between critical 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 (Eqs. 63 to 66). Upper bound   1.45500.(m/ 7s) 0 ( )c Dv mm  (63) Lower bound   0.12500(m/ 1s) . ( )c Dv mm  (64) Upper bound   2.3500.(Pa 6) 0 ( )c D mm  (65) Lower bound   0.25500.(m/ 5s) 0 ( )c D mm  (66) One major difference of the updated plots with earlier versions (Figure 31) is that the boundary in which Eqs. 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. Eqs. 61 and 62 are strong equations for predicting the values of the critical shear stress and the critical velocity for coarse sand to very large gravels.

192 Figure 146. Mean particle size vs. critical velocity Figure 147. Mean particle size vs. critical shear stress

193 7.3. Deterministic (Frequentists) Regression Analysis As discussed in Chapter 1, the final phase of this project consisted of performing two parallel statistical approaches (i.e. “Frequentists’ Regression” and “Bayesian Inference”) with the goal of reaching the best potential fits between erodibility parameters and geotechnical properties of soils. This section is dedicated to the step by step process of Frequentists’ Regression. This approach was implemented in three major steps: 1. First Order Statistical Analysis 2. Second Order Statistical Analysis 3. Regression, Optimization, and Model Selection 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, range, quartiles, mean, median, mode, standard deviation, histograms, probability density functions, empirical cumulative density functions, etc.). As discussed earlier in Chapter 5, overall, there are 5 erodibility parameters (function variables). These parameters are the critical shear stress τc, the critical velocity vc, the initial slope Ev of the ̇ versus curve, the initial slope Eτ of the ̇ versus curve and the erosion function category (EC) in the Briaud erosion chart (2013). Up to 16 geotechnical engineering parameters (model variables) are also collected for each sample. These parameters include LL, PL, PI, Water Content (WC), Su, γ, D10, D30, D50, D60, Cc, Cu, Percent Fine (PF), Gs, Percent Clay (PC), and Soil Activity (A). Nearly 1000 erosion tests compiled in the TAMU-Erosion were studied in different groups. Chapter 5 showed the TAMU-Erosion incorporates more than ten different erosion tests; however, the three major erosion tests (i.e. EFA, HET, and JET) were chosen to further investigate the potential relationships. The study of the “first order statistical features” was started with the global dataset (including all different tests and all soil types all together). Next, the data was divided based on their erosion test type, and the “first order statistical features” for each sub-group was developed. Next, the data for each sub-group was again divided with regard to their soil type (coarse/fine) according to the USCS. The aforementioned groups are labeled and listed below: 1- TAMU/Global Dataset 2- TAMU/Fine Dataset 3- TAMU/Coarse Dataset 4- EFA/Global Dataset 5- EFA/Fine Dataset 6- EFA/Coarse Dataset

194 7- JET/Global Dataset 8- JET/Fine Dataset 9- JET/Coarse Dataset 10- HET/Global Dataset 11- HET/Fine Dataset 12- HET/Coarse Dataset A flowchart diagram of the grouping procedure is shown in Figure 148 below. Figure 148. Flowchart diagram of the grouping procedure As shown in Figure 148, twelve datasets were obtained, and the first order statistics analyses were performed on each dataset separately. The “first order statistical features” of all function and model variables are obtained for the aforementioned groups. Table 33 to Table 44 show these results. Table 33. First order statistics results for the TAMU Spreadsheet – TAMU/Global Dataset TAMU/Global EFA/Global EFA/Fine EFA/Coarse JET/Global JET/Fine JET/Coarse HET/Global HET/Fine HET/Coarse TAMU/Fine TAMU/Coarse EC Ev (mm- s/m-hr) Eτ (mm/hr- Pa) Vc (m/s) τc (Pa) LL (%) PL (%) PI (%) γ (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 std 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

195 Table 34. First order statistics results for the TAMU Spreadsheet – TAMU/Fine Dataset Table 35. First order statistics results for the TAMU Spreadsheet – TAMU/Coarse Dataset Table 36. First order statistics results for the TAMU Spreadsheet – EFA/Global Dataset EC Ev (mm-s/m-hr) Eτ (mm/hr- Pa) Vc (m/s) τc (Pa) LL (%) PL (%) PI (%) γ (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 std 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 EC Ev (mm- s/m-hr) Eτ (mm/hr- Pa) Vc (m/s) τc (Pa) LL (%) PL (%) PI (%) γ (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 std 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 EC Ev (mm- s/m-hr) Eτ (mm/hr- Pa) Vc (m/s) τc (Pa) LL (%) PL (%) PI (%) γ (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 std 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

196 Table 37. First order statistics results for the TAMU Spreadsheet – EFA/Fine Dataset EC Ev (mm-s/m-hr) Eτ (mm/hr- Pa) Vc (m/s) τc (Pa) LL (%) PL (%) PI (%) γ (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 std 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 38. First order statistics results for the TAMU Spreadsheet – EFA/Coarse Dataset EC Ev (mm-s/m-hr) Eτ (mm/hr- Pa) Vc (m/s) τc (Pa) LL (%) PL (%) PI (%) γ (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 std 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 39. First order statistics results for the TAMU Spreadsheet – HET/Global Dataset EC Ev (mm-s/m-hr) Eτ (mm/hr- Pa) Vc (m/s) τc (Pa) LL (%) PL (%) PI (%) γ (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 n/a 28.49 n/a 194.50 41.58 19.59 21.99 19.14 18.43 61.10 63.18 0.10 91.86 4.25 28.41 std 0.70 n/a 132.72 n/a 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 n/a 0 n/a 0.1 20 11 2.9 13.9 7.5 4.3 22 0.001 1.29 0.48 2 25% 3 n/a 0.13 n/a 15.305 30 15 15 18.3 12.56 43.1 42 0.0025 30 2.04082 13 50% 3.5 n/a 0.64 n/a 89.75 36 19 20 19.3 16.2 57.4 73.4 0.038 38 3.4087 24.84 75% 4 n/a 3.31 n/a 314.98 42 22 24 20.1 21.2 71.8 80.06 0.22 230 6.80272 44.75 max 4.75 n/a 1684.04 n/a 1158 148.1 33 125.8 22.9 44 143.5 100 0.29 230 7.74 77

197 Table 40. First order statistics results for the TAMU Spreadsheet – HET/Fine Dataset Table 41. First order statistics results for the TAMU Spreadsheet – HET/Coarse Dataset EC Ev (mm- s/m-hr) Eτ (mm/hr- Pa) Vc (m/s) τc (Pa) LL (%) PL (%) PI (%) γ (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 n/a 98.85 n/a 34.92 35.94 19.06 16.88 20.24 11.72 n/a 32.44 0.22 147.71 4.93 12.51 std 0.41 n/a 241.37 n/a 79.15 0.25 0.25 0.50 1.27 3.07 n/a 7.22 0.06 96.76 1.78 3.10 min 2 n/a 0.05 n/a 0.1 35 19 15 17.8 7.5 n/a 22 0.13 38 3.4087 7 25% 2.25 n/a 2.895 n/a 8.73 36 19 17 19.35 9.6 n/a 29 0.1525 38 3.4087 13 50% 2.75 n/a 12.82 n/a 13.35 36 19 17 19.9 10.8 n/a 34 0.23 230 3.4087 13 75% 3 n/a 85.355 n/a 23.815 36 19 17 21.3 14 n/a 42 0.29 230 6.94901 14 max 4.25 n/a 1684.04 n/a 513 36 20 17 22.9 18.9 n/a 46.5 0.29 230 6.94901 23.9 Table 42. First order statistics results for the TAMU Spreadsheet – JET/Global Dataset EC Ev (mm- s/m- hr) Eτ (mm/hr- Pa) Vc (m/s) τc (Pa) LL (%) PL (%) PI (%) γ (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 n/a 272.67 n/a 5.35 36.09 17.65 18.43 19.43 20.11 88.45 63.89 0.05 83.71 14.94 21.36 std 0.67 n/a 1082.02 n/a 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 n/a 0.03 n/a 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 n/a 1.29 n/a 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 n/a 4.92 n/a 1.12 36 16.55 19.1 19.45 14.4 79.6 68.8 0.0165 69.64 4.805 20 75% 3 n/a 29.53 n/a 7.2675 42 21.975 24 20.4 17.385 129.2 84 0.0625 85.83 16.995 29.7 max 3.75 n/a 6690.26 n/a 60.72 79.5 34.1 50.3 24.29 286.7 150.7 100 0.28 230 57.12 64.66 EC Ev (mm-s/m-hr) Eτ (mm/hr- Pa) Vc (m/s) τc (Pa) LL (%) PL (%) PI (%) γ (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 std 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

198 Table 43. First order statistics results for the TAMU Spreadsheet – JET/Fine Dataset EC Ev (mm- s/m-hr) Eτ (mm/hr- Pa) Vc (m/s) τc (Pa) LL (%) PL (%) PI (%) γ (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 n/a 58.39 n/a 6.38 39.38 17.14 22.18 19.43 23.18 91.77 77.29 0.02 70.55 3.83 26.38 std 0.63 n/a 238.47 n/a 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 n/a 0.03 n/a 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 n/a 0.86 n/a 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 n/a 2.92 n/a 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 n/a 11.59 n/a 8.32 47 21 31 20.35 18.68 129.4 87.3025 0.0461 85.83 4.805 30 max 3.75 n/a 1718.02 n/a 60.72 79.5 34.1 50.3 23.05 286.7 150.7 100 0.0733 85.83 7.74 64.66 Table 44. First order statistics results for the TAMU Spreadsheet – JET/Coarse Dataset EC Ev (mm-s/m-hr) Eτ (mm/hr- Pa) Vc (m/s) τc (Pa) LL (%) PL (%) PI (%) γ (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 n/a 705.68 n/a 3.29 28.52 18.84 9.69 19.43 11.33 68.55 26.38 0.21 96.86 26.06 10.87 std 0.63 n/a 1784.55 n/a 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 n/a 0.41 n/a 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 n/a 4.0325 n/a 0.115 22 16 5 18.475 10 39.85 25.1 0.15603 30.295 10.53 6.83 50% 2 n/a 20.875 n/a 0.45 30 17 6 19.4 11.2 57.45 25.1 0.204 53.45 20.08 8 75% 2.5625 n/a 208.705 n/a 2.8025 30 25 14 20.4 11.9804 86.15 25.1 0.25375 141.725 38.6 20 max 3.5 n/a 6690.26 n/a 28.92 39 25 22 24.29 17.8787 129.2 42.5 0.28 230 57.12 29 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 (PDF) and empirical cumulative density functions (ECDF) 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, PDF, and ECDF plots for the critical velocities of the TAMU/Global group in the TAMU-Erosion Spreadsheet, which are fitted with many statistical distribution models (i.e. Normal, Log-Normal, 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 ( ) can be well represented by Exponential distribution. Same approach was taken for all erodibility parameters (i.e. , τc, Ev, Eτ, and EC) as well as all 12 major geotechnical properties (i.e. LL, PL, PI, γ, WC, Su, PF, D50, Cu, Cc, PC, and Activity). The goal was to identify the best statistical distribution models that can best represent each parameter. The findings of this effort are not only very important in understanding of each erodibility parameter as well as geotechnical properties, but also a vital tool to perform Frequentist Regression and Bayesian Inference.

199 Figure 149. PDF, ECDF, and Histogram Plots for Critical Velocity in the TAMU/Global Dataset

200 Figure 150. PDF, ECDF, and Histogram Plots for Critical Velocity in the TAMU/Coarse Dataset

201 Figure 151. PDF, ECDF, and Histogram Plots for Critical Velocity in the TAMU/Fine Dataset Similar figures are included in the Appendix 3 for all five erodibility parameters as well as 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 ( , τc, Ev, Eτ, and EC) in each subgroup. Table 46 shows the best models to represent each major geotechnical properties (i.e. LL, PL, PI, γ, WC, Su, PF, D50, Cu, Cc, and PC) in each subgroup.

202 Table 45. Best simplest models to represent the erodibility parameters Best Simplest Statistical Model to Represent Each Parameter Number Subgroups EC Ev Eτ Vc τc 1 TAMU/Global Normal Exponential Log-Normal Exponential Exponential 2 TAMU/Fine Normal Exponential Log-Normal Exponential Exponential 3 TAMU/Coarse Normal Exponential Exponential Exponential Log-Normal 4 EFA/Global Normal Exponential Log-Normal Exponential Exponential 5 EFA/Fine Normal Exponential Log-Normal Exponential Exponential 6 EFA/Coarse Normal Exponential Exponential Exponential Log-Normal 7 JET/Global Normal - Log-Normal - Log-Normal 8 JET/Fine Normal - Log-Normal - Log-Normal 9 JET/Coarse Normal - Log-Normal - Alpha 10 HET/Global Normal - Log-Normal - Normal 11 HET/Fine Normal - Exponential - Normal 12 HET/Coarse Normal - Normal - Log-Normal Table 46. Best simplest models to represent the geotechnical properties Best Simplest Statistical Model to Represent Each Parameter Number Subgroups LL PL PI γ WC Su PF D50 Cu Cc PC 1 TAMU/Global Normal Normal Normal Normal Normal Exponential Normal Gamma Exponential Exponential Normal 2 TAMU/Fine Normal Normal Normal Normal Normal Exponential Normal Normal Exponential Exponential Normal 3 TAMU/Coarse Normal Normal Normal Normal Normal Exponential Normal Gamma Normal Exponential Normal 4 EFA/Global Normal Normal Normal Normal Normal Exponential Normal Gamma Exponential Log-Normal Exponential 5 EFA/Fine Normal Normal Normal Normal Normal Exponential Normal Exponential Exponential Log-Normal Normal 6 EFA/Coarse Normal Exponential Exponential Normal Normal Exponential Normal Log-Normal Exponential Log- Normal Normal 7 JET/Global Normal Normal Normal Normal Normal Normal Normal Gamma Normal Normal Normal 8 JET/Fine Normal Normal Normal Normal Exponential Normal Normal Normal Normal Normal Normal 9 JET/Coarse - - - Normal Normal - Normal - - - 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 - Normal Normal Alpha Normal Normal Second Order Statistical Analysis This step dealt with constructing the correlation matrices for the different parameters defined in each group and discussed in the previous step. Before generating the regression relationships, it was needed to learn about any potential inter-parameters relationships between the model variables (geotechnical parameters) and the function variables (erosion parameters). In order to do that, the correlation matrices were formed between the parameters and the Pearson’s Correlation Coefficient (PCC) was obtained for each matrix. PCC is defined as the covariance of the two variables divided by the product of their standard deviations. It ranges from -1 to +1 and reflects the linear dependency between two variables, with +1 showing a strong positive relationship, -1

203 indicating a solid negative relationship, and 0 referring to no relationship at all. Figure 152 shows an example of the results after the “second order statistical analysis” for EFA-Fine data. Such plots were created for all aforementioned groups explained in Figure 148 and compiled in the Appendix 3. The knowledge developed after the second order statistical analysis is very important for implementing the “Bayesian Inference” approach. Figure 152. Correlation Matrix for EFA-Fine Data

204 Experimental Design Model Expression After learning about each parameter, this step dealt with generating the relationship equations between the erosion parameters and the geotechnical engineering parameters. Two models were used in this research which are called the “Linear Model” and the “Power Model”. The Power Model for this research project is the multiple nonlinear model shown in Eq. 67. … (67) Where, Y refers to the dependent variable which in this case includes: , , , , and ; Pi s are the soil geotechnical properties which are used in the regressions; A and are the constant parameters defined in the model expression. The Linear Model on the other hand was the simple first degree linear model as described in Eq. 68. ⋯ (68) Where, Y refers to the dependent variable which in this case includes: , , , , and ; Pi s are the soil geotechnical properties which are used in the regressions; and are the constant parameters defined in the model expression. Several combinations between a function variable, say, , and model variables (i.e. LL, PL, PI, Water Content, Unit weight, …) can be selected to generate regression equations. In order to obtain the best fits, it was decided to study nearly all possible combinations and evaluate the goodness of each fit. Therefore, 135 regression combination groups are constructed for each function variable in each aforementioned subgroup (as shown in Figure 148). In addition to that, two model expressions (Eqs. 67 and 68) are used to develop the equations. In other words, the total number of regression equations is 2x5x135 (2 model expressions x 5 function parameters x 135 combination group) = 1350 regression equations for each subgroup (as shown in Figure 148). Measures of Statistical Significance Best models are selected after passing through a four-filter process: 1) R2 2) Mean Square Error (MSE) 3) 4) Cross-Validation Score Each generated model is checked with filters 1 to 4, one after another. Each model that does not qualify the requirements of one specific filter, is not going to be further studied in the following filters, and is thus eliminated from the best model sets. R-squared or R2 also called 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 R-squared is shown below:

205 1 (69) Where, are the data, is the mean value of the data set, and are the predicted values using the fitted regression model. In this study, R-squared values for each regression are reported. R- squared ranges from 1.0 being an absolute fit, to 0 being not any relationship. R squared can also sometimes be negative. Generally speaking, R-squared values more than 0.6 can be considered as potential correlations in many engineering applications. However, in many sources, R-squared values even more than 0.4 are accepted by researchers. The Mean Squared Error (MSE) is another statistical measure to evaluate the quality of a predicting model. MSE is also sometimes called the mean squared deviation. It always gets a non- negative 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: Σ (70) Where, are the data, refer to the predicted value using the estimator, and is number of the 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 R-squared is that it is highly dependent on the number of variables involved in the regression model. For example, when you have only one parameter to predict, and our data set includes only one data, any regression model can lead to an R-squared of 1. In order to reject the null hypothesis in a regression model, the F-value can be used. The F- value is obtained from Eq. 71: 1 (71) Where SSE(reg) is the sum of square errors for the regression values, SSE(res) is the sum of square errors for the residual values, D is the number of data points, and V is the number of variables. As can be seen from the equations, if the regression model has a lot of error (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 F statistic, a target value given in statistic tables. Note that the R-squared and MSE values represent the goodness of a fit evaluation, while the F test (comparing the F-value with the F-statistic) indicates whether there is enough data or not to propose the equation. The probability level for the F-test in this study is chosen as 5%. After the first three filters (R2, MSE, ) are passed successfully, cross-validation score of the selected models would be examined. Over-fitting is a common problem in many similar statistical analyses conducted in different fields of engineering. Over-fitting may happen when the equation is generated based on all the data points available in the dataset. 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 dataset, or removing some data from it can sometimes dramatically change the goodness of fit of the model. One approach to evaluate if this is a problem is to divide the dataset 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

206 testing subset. One issue with this approach is that our analysis is very dependent on the random “train” and “test” subsets that we chose in the beginning. Cross-validation is a technique used to assess the estimator performance by folding the dataset into ‘k’ random folds. The model is trained using the ‘k-1’ fold, and then validated using 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, and end up with the most robust equations. This four-filter model selection process was performed for each subgroup as shown in Figure 1. This process incorporates studying of thousands of generated models in the first step, narrowing down to smaller number of models after passing each filter. By the end of the fourth filter, the best models to correlate erodibility parameters to geotechnical properties were supposed to be achieved. For the “Coarse” datasets (subgroups of TAMU/Coarse, EFA/Coarse, HET/Coarse, and JET/Coarse) 105 soil parameter combination groups were chosen for each function variable; however, for “Global” and “Fine” datasets (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 regression analyses as well as list of these combination groups are presented in Appendix 4. After the requirements of all four filters are met successfully, the best fitting models are selected. If more than one fitting model meet the filter requirements, the authors’ engineering judgement helps narrow down the choices into one or at most two equations. More regarding the selected models is discussed in Chapter 8. Probability of Over/Under-Predicting (POO/POU) The best correlation equations that can estimate each erodibility parameter (i.e. , , , , and EC) are selected according to the four-filter process discussed in the previous section. These equations are 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 is to provide the engineer with a reliability-based approach to also assess whether the predicted values are conservative enough for engineering design purposes. This approach is called the “Probability of Over/Under-Predicting (POU/POO)” approach. As an instance, the engineer needs to know the probability that the predicted is smaller than the actual , in order to be on the safe side in the design problem. Similarly, critical velocity ( ) and erosion category (EC) are among those erodibility parameters that require the probability of under-predicting (POU). On the other hand, the engineer needs to know the probability that the predicted (or ) is larger than the actual (or ), in order to be on the safe side in the design problem. Therefore in such cases, the probability of over-predicting (POO) is presented. Eq. 72 shows the probabilistic model which consists of the selected deterministic predicted erodibility parameter and a correction factor, . (72)

207 Where refers to the new value for the erodibility parameter, refers to the correction factor, and is the deterministically predicted erodibility parameter. can also be inferred as the ratio of the “ ” over “ ”. Depending on the erodibility parameter, the “POU vs. ” or “POO vs. ” plots are developed for the best selected equations in the next section. These plots help the 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 ( ) are smaller than the measured values ( ). The black solid line in Figure 153 represents the case in which = . In this example, two different correction factors ( 0.6 and 1.0) are considered. The data points associated with ‘ 1.0’ are shown with black ‘o’ markers, and the data points associated with the ‘ 0.6’ are shown with red ‘+’ markers. The POU is then calculated by counting the number of data points above the black solid line (the data points in which ) and dividing that by the total number of data points. In this report, these plots are often accompanied with an offset value. As an example, the POO is calculated as the probability of “ ” instead of the probability of “ 0”. The “offset values” are small compared to the standard deviation of the parameter, and can often be neglected; however, due to the fact that 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 following section presents the process in which the best equations are selected for each erodibility parameter. The “POU vs. Correction Factor” or “POO vs. Correction Factor” plots are also presented for the selected equations. Figure 153. An example of how POU is obtained for two different correction factors

208 Regression, Optimization, and Model Selection The step-by-step procedure discussed in Section 7.3.3 to select the best fitting models is implemented for each erodibility parameter (i.e. , , , , EC) in the followings. The selected correlation equations in this section are repeated in the next Chapter, 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 the Appendix 4 of the appendices report. Table 47. Units and descriptions of the parameters used in regression analyses Parameter Description Unit A Soil Activity - Critical Shear Stress Pa Cc Coefficient of Curvature - Cu Coefficient of Uniformity - D50 Mean Particle Size mm EC Erosion Category - Ev Slope of Velocity-Erosion Rate mm-s/m-hr Eτ Slope of Shear-Erosion Rate Mm/hr-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 Velcoity m/s VST Vane Shear Strength kPa WC Water Content % Unit weight kN/m3 Critical Shear Stress ( ) First filter is R2. The study started from the very top subgroup in Figure 148, TAMU/Global. This database includes all existing data, regardless of test type or fine/coarse nature of the samples. Figure 154 shows the number of data in each of the 135 soil parameter combination groups. Figure 155 and Figure 156 show the results of R2 for these 135 soil parameter combination groups with the “Linear Model” (Eq. 68) and “Power Model” (Eq. 67), respectively. Figure 155 and Figure 156 show very poor R2 values (< 0.2) for the critical shear stress in the TAMU/Global dataset. Obviously, following through filters 2 to 4 are not reasonable for this dataset. The same poor results were observed when the dataset was changed to TAMU/Fine, TAMU/Coarse. These observations prove that the regression analyses needed to be narrowed down

209 to each test separately, and consecutively coarse or fine nature of the soils. Therefore, the subgroups of EFA/Global, EFA/Fine, EFA/Coarse, JET/Global, JET/Fine, JET/Coarse, HET/Global, HET/Fine, and HET/Coarse were used to implement the regression analyses, as described below. EFA Dataset It was observed that dividing the EFA/Global dataset into EFA/Fine and EFA/Coarse leads to better results. Figure 157 shows the number of data points in each of all the 135 combination groups in the EFA/Fine dataset. Also, Figure 158 and Figure 159 show the results of R2 for each combination group, for the “Linear Model” and “Power Model”, respectively. R2 values of groups 108 to 135 are generally higher than the rest for both “Linear” and “Power” models; however, a quick glance shows that the “Power” model is a better fit for the existing data in the case of critical shear stress in EFA/Fine dataset. Figure 154. Number of data in each 135 soil parameter combination groups for the TAMU/Global dataset – Critical Shear Stress Figure 155. R2 results for the “Linear Models” in TAMU/Global dataset – Critical Shear Stress

210 Figure 156. R2 results for the “Power Models” in TAMU/Global dataset – Critical Shear Stress Figure 157. Number of data in each 135 combination groups for the EFA/Fine dataset – Critical Shear Stress Figure 158. R2 results for the “Linear Models” in EFA/Fine dataset – Critical Shear Stress

211 Figure 159. R2 results for the “Power Models” in EFA/Fine dataset – Critical Shear Stress Next step was to select the best R2 values, and move forward with Filter 2 (Mean Square Error). Figure 160 and Figure 161 show the values of MSE for the “Linear” and “Power” models, respectively. As expected, the MSE values are generally lower for same group numbers (108 to 135) in both figures. After passing through filters 1 and 2 (R2 and MSE), “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 that F-value/F-stat is lower than 1, the model needs to be removed from the selection list. Otherwise, the model remains in the list. Table 48 shows the results of the selected models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). Figure 160. MSE results for “Linear Models” in EFA/Fine dataset – Critical Shear Stress

212 Figure 161. MSE results for “Power Models” in EFA/Fine dataset – Critical Shear Stress Table 48. Selected “Power” models for critical shear stress in the EFA/Fine dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 109 PC, γ, WC, Su, PF, D50 5.5 10 . . . . 50 . . 0.84 2.76 3.5 0.66 110 PC, WC, Su, PF, D50 1.5 10 . . . 50 . . 0.86 2.50 4.7 0.83 113 PC, γ, WC, Su, D50 7.9 10 . . . 50 . . 0.84 2.73 4.9 0.65 123 A, PL, γ, WC, Su 3.56 10 . . . . . 0.87 2.53 7.1 0.18 124 γ, A, WC, Su, PF, D50 158.06 . . . . 50 . 0.94 1.67 3.9 0.66 125 γ, WC, Su, PF, A 3.25 10 . . . . . 0.85 2.67 5.1 0.25 128 D50, γ, WC, Su, A 2.2 10 50 . . . . . 0.91 2.11 2.9 -0.08 Table 48 shows the selected models which meet the requirements of the first three filters; however, not all of them perform satisfactorily in the cross-validation test. As mentioned earlier, cross-validation is a technique used to assess the estimator performance by folding the dataset into ‘k’ random folds. The model is trained using the ‘k-1’ fold, and then validated using 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). Out of the 7 selected groups in Table 48, groups 110 and to some extent group 124 (highlighted in blue) are the ones that pass all four filters successfully. Section 7.3.3.3 and Eq. 72 discussed the reliability-based approach called the POU approach. The plots of “POU vs. ” are shown in Figure 162 and Figure 163 for groups 110 and 124,

213 respectively. The vertical axes in these plots represent the probability that the predicted using the selected model is smaller than the actual , in percentage (with ± 0.5 Pa precision). The horizontal axes represent the correction factor ( ) that can multiply to the predicted value to reach a certain POU level (See Eq. 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 is smaller than the actual (with ± 0.5 Pa precision). However, if the engineer desires a higher confidence level, say, near 90%, then he/she needs to multiply the predicted value by 0.6. Figure 163, on the other hand, shows that by using the Group 124 correlation equation (see Table 48), there is almost 90% chance that the predicted value is smaller than the actual (with 0.5 Pa offset). These plots give the engineers flexibility in choosing the desired correction factors according to the design application. Figure 162. Plot of POU vs. correction factor for the Group 110 (Power) - in the EFA/Fine dataset POU = 0.73

214 Figure 163. Plot of POU vs. correction factor for the Group 124 (Power) - in the EFA/Fine dataset The same exact procedure was conducted in the EFA/Coarse dataset, and the best models are selected. Figure 164 shows the number of data points in each of all the 105 combination groups in the EFA/Coarse dataset. Also, Figure 165 and Figure 166 show the results of R2 for each combination group, for the “Linear Model” and “Power Model”, respectively. In contrary to the EFA/Fine dataset, both “Linear” and “Power” models show a few good groups in terms of R2 values; however, as shown in Figure 164, the number of data points in most groups are very lower compared to the EFA/Fine dataset. In other words, many regression groups might meet the requirements of the first two filters, R2 and MSE; however, they only might marginally pass through the F-test. Figure 167 and Figure 168 show the values of MSE for the “Linear” and “Power” models, respectively. Figure 164. Number of data in each 105 combination groups for the EFA/Coarse dataset – Critical Shear Stress

215 Figure 165. R2 results for the “Linear Models” in EFA/Coarse dataset – Critical Shear Stress Figure 166. R2 results for the “Power Models” in EFA/Coarse dataset – Critical Shear Stress Figure 167. MSE results for “Linear Models” in EFA/Coarse dataset – Critical Shear Stress

216 Figure 168. MSE results for “Power Models” in EFA/Coarse dataset – Critical Shear Stress 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 analyses. Also “Power” models associated with groups 34, 44, 46, 47, 51, 54, 56, 57, 58, 60, 64, 65, 66, 71, 77, 80, 83, 84, 86, 95, and 101 were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 49 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). As shown in Table 49, all selected models perform satisfactorily in the cross-validation test. Table 50 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The best models that also have a good cross-validation score are highlighted in blue in both “Linear” and “Power” forms. Out of all the highlighted 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 vs. ” for this model. The vertical axis in Figure 169 represent the probability that the predicted using the selected model is smaller than the actual , in percentage (with 0.3 Pa offset). In order to reach a 90% confidence that the predicted is smaller than the actual , the predicted value should be multiplied by 0.82. Table 49. Selected “Linear” models for critical shear stress in the EFA/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 51 Cc, γ, D50 0.016 0.09 0.86 50 1.12 0.92 1.05 5.1 0.89 54 Cc, WC, D50 0.009 0.03 0.82 50 0.83 0.92 1.06 5.1 0.89 57 PF, Cc, D50 0.015 0.01 0.88 50 0.05 0.93 1.01 5.4 0.88 60 Cc, γ, WC, PF 0.023 0.16 0.06 0.82 50 1.6 0.93 1.00 2.6 0.89

217 63 Cc, γ, PF, D50 0.015 0.06 0.01 0.88 50 0.89 0.93 1.03 2.9 0.87 66 Cc, WC, PF, D50 0.01349 0.06 0.03 0.85 50 0.62 0.93 0.98 2.9 0.88 71 Cc, γ, WC, PF, D50 0.024 0.13 0.08 0.03 0.84 50 1.38 0.94 0.93 2.1 0.88 77 Cu, γ, D50 0.001 0.06 0.86 50 0.71 0.92 1.06 5.1 0.9 80 Cu, WC, D50 0.001 0.03 0.83 50 0.72 0.92 1.05 5.1 0.88 83 PF, Cu, D50 0.015 0.0007 0.88 50 0.02 0.92 1.01 5.3 0.88 86 Cu, γ, WC, D50 0.001 0.11 0.05 0.83 50 1.03 0.93 1.02 3.3 0.89 89 Cu, γ, PF, D50 0.00025 0.04 0.01 0.88 50 0.55 0.93 1.05 3.3 0.86 92 Cu, WC, PF, D50 0.00037 0.06 0.03 0.85 50 0.56 0.93 0.99 3.3 0.87 97 Cu, γ, WC, PF, D50 0.001 0.09 0.07 0.03 0.85 50 0.88 0.94 0.96 2.4 0.86 Table 50. Selected “Power” models for critical shear stress in the EFA/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 34 PI, VST, PF, D50 32400.5 . . .50 . 0.99 0.191 140 -15.44 44 PI, , VST, PF, D50 7.8 10 . . . . 50 . 0.99 0.198 88 -1.4 46 PI, WC, VST, PF, D50 267267.2 . . . . 50 . 0.99 0.197 70.5 0.08 47 PI, , WC, VST, PF, D50 35.65 . . . . . 50 . 0.99 0.169 56.1 0.09 51 Cc, , D50 2.32 . . 50 . 0.93 1.045 36 0.98 52 Cc, WC, VST 1228385.72 . . . 0.98 0.178 38.6 0.1 54 Cc, WC, D50 1.67 . . 50 . 0.93 1.043 36.2 0.98 58 Cc, , WC, VST 0.22 . . . . 0.99 0.168 24.3 -0.5 60 Cc, , WC, D50 1.517 . . . 50 . 0.93 1.043 29.1 0.97 64 Cc, WC, VST, PF 1.09 10 . . . . 0.98 0.164 25.8 -1.5 77 Cu, , D50 1.58 . . 50 . 0.93 1.044 36.1 0.99 78 Cu, WC, VST 257.02 . . . 0.98 0.183 33.1 0.04 80 Cu, WC, D50 1.66 . . 50 . 0.93 1.044 36.1 0.98 86 Cu, , WC, D50 1.378 . . . 50 . 0.93 1.044 29.1 0.96 88 Cu, , VST, D50 5.845 10 . . . 50 . 0.97 0.236 21 0.03 95 Cu, , WC, VST, D50 2.24 10 . . . . 50 . 0.96 0.307 19 -11.3 101 Cc, Cu, PI 2.555 10 . . . 0.95 0.313 26.7 -0.44 104 Cc, Cu, VST 5.87 10 . . . 0.98 0.204 25.5 0.01

218 Figure 169. Plot of POU vs. correction factor for the Group 77 (Power) - in the EFA/Coarse dataset JET Dataset Similar approach was taken to select the best correlating 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 dataset are substantially low compared to the JET/Fine dataset. Figure 133 and Figure 134 show the number of data points in each of all the 135 combination groups in the JET/Global dataset and each of the 105 combination groups in the JET/Coarse dataset, respectively. 2) The R2 values for the JET/Global dataset (D50 < 0.3 mm), although low themselves, are still better than the R2 values for the JET/Fine dataset. Therefore, the regression results for the JET/Global dataset are presented as the best models.

219 Figure 170. Number of data in each 135 combination groups for the JET/Global dataset – Critical Shear Stress Figure 171. Number of data in each 105 combination groups for the JET/Coarse dataset – Critical Shear Stress Figure 172 and Figure 173 show the results of R2 for each combination group in the JET/Global dataset, for the “Linear Model” and “Power Model”, respectively. Both plots show relatively poor R2 for the JET/Global dataset. One of the major reasons behind the poor relationships for critical shear stress in the JET database is the variety of methods 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.

220 Figure 172. R2 results for the “Linear Models” in JET/Global dataset – Critical Shear Stress Figure 173. R2 results for the “Power Models” in EFA/Coarse dataset – Critical Shear Stress Figure 174. MSE results for “Linear Models” in JET/Global dataset – Critical Shear Stress

221 Figure 175. MSE results for “Power Models” in JET/Global dataset – Critical Shear Stress After passing through filters 1 and 2 (R2 and MSE), “Linear” models associated with groups 49, 76, 109, 112, and 113 in the JET/Global dataset were selected for further analyses. Also “Power” models associated with groups 53, 79, 109, and 124 in the JET/Global dataset were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 51 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). As shown in Table 51, almost none of the selected models perform well in the cross-validation test. Table 52 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The best models that also have a good cross- validation score are highlighted in blue in both “Linear” and “Power” forms. 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 vs. θ” for this model. The vertical axis in Figure 176 represent the probability that the predicted τc using the selected model is smaller than the actual τc, in percentage (with 1 Pa offset). In order to reach a 90% confidence that the predicted τc is smaller than the actual τc, the predicted value should be multiplied by 0.6.

222 Table 51. Selected “Linear” models for critical shear stress in the JET/Global dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 49 , WC, Su, PF, D50 0.769 0.08 0.04 0.07 46.05 50 28.36 50 0.3 0.44 3.36 1.318 0.05 76 LL, , Su, PF, D50 0.011 0.71 0.04 0.08 46.8 50 26.56 50 0.3 0.44 3.38 1.299 0.04 109 PC, , WC, Su, PF, D50 0.272 1.28 0.22 0.07 0.03 31.65 50 31.03 50 0.3 0.50 3.20 1.365 -0.02 112 PC, , Su, PF, D50 0.176 1.24 0.06 0.02 34.03 50 33.38 50 0.3 0.49 3.24 1.563 0.08 113 PC, , WC, Su, D50 0.248 1.23 0.21 0.07 36.89 50 31.82 50 0.3 0.50 3.20 1.647 0.1 Table 52. Selected “Power” models for critical shear stress in the JET/Global dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 53 PI, , WC, Su, PF, D50 2.697 . . . . . 50. 50 0.3 0.46 3.32 7.155 -1.5 79 LL, , WC, Su, PF, D50 7.412 . . . . . 50 . 50 0.3 0.45 3.35 7.005 -0.69 109 PC, , WC, Su, PF, D50 76.49 . . . . . 50 . 50 0.3 0.46 3.33 7.089 -1.62 124 , A, WC, Su, PF, D50 3.86 . . . . . 50 . 50 0.3 0.47 3.30 6.891 -0.77

223 Figure 176. Plot of POU vs. correction factor for the Group 113 (Linear) - in the JET/Global dataset HET Dataset Similar approach was taken to select the best correlating equation for critical shear stress in the HET database. On the HET database, there are also two important observations to notice: 1) The HET was primarily performed on the finer soils (D50 < 0.3 mm), and therefore, the number of data points in the HET/Coarse dataset are substantially low compared to the JET/Fine dataset. In fact, many combination groups on the HET/Coarse database have zero data points. Figure 177 and Figure 178 show the number of data points in each of all the 135 combination groups in the HET/Global dataset and each of the 105 combination groups in the HET/Coarse dataset, respectively. 2) The R2 values for the HET/Global dataset (D50 < 0.3 mm) are significantly better than the R2 values for the HET/Coarse and HET/Fine dataset. Therefore the regression results for the HET/Global dataset are presented as the best models.

224 Figure 177. Number of data in each 135 combination groups for the HET/Global dataset – Critical Shear Stress Figure 178. Number of data in each 105 combination groups for the HET/Coarse dataset – Critical Shear Stress Figure 179 and Figure 180 show the results of R2 for each combination group in the HET/Global dataset, for the “Linear Model” and “Power Model”, respectively. Both plots show that the best R2 for the HET/Global dataset are around 0.60 to 0.65. One of the major reasons behind the poor relationships for critical shear stress in the JET database is that the method used to calculate the erodibility parameters in the HET includes many crude judgements. More on the HET issues is discussed in Chapter 8. Figure 181 and Figure 182 show the values of MSE for the “Linear” and “Power” models, respectively.

225 Figure 179. R2 results for the “Linear Models” in HET/Global dataset – Critical Shear Stress Figure 180. R2 results for the “Power Models” in HET/Global dataset – Critical Shear Stress Figure 181. MSE results for “Linear Models” in HET/Global dataset – Critical Shear Stress

226 Figure 182. MSE results for “Power Models” in HET/Global dataset – Critical Shear Stress 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 dataset were selected for further analyses. Also “Power” models associated with groups 19, 35, 38, 40, 48, 50, and 52 in the HET/Global dataset were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 53 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). As shown in Table 53, almost none of the selected models perform well in the cross-validation test. Table 54 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The best models that also have a good cross- validation score are highlighted in blue in both “Linear” and “Power” forms. Out of all the correlation 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 vs. θ” for this model. The vertical axis in Figure 183 represent the probability that the predicted τc using the selected model is smaller than the actual τc, in percentage (with 0.4 Pa offset). In order to reach a 90% confidence that the predicted τc is smaller than the actual τc, the predicted value should be multiplied by 0.6.

227 Table 53. Selected “Linear” models for critical shear stress in the HET/Global dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 38 PI, WC, Su, D50 0.446 3.48 0.82 2153.5 50 98.12 0.63 23.0 1.945 0.01 48 PI, , WC, Su, D50 0.45 0.61 3.6 0.83 2161.4 50 89.22 0.63 23.0 1.741 -0.09 49 , WC, Su, PF, D50 3.66 6.55 0.98 0.61 3053.2 50 281.53 0.58 24.5 1.397 -0.05 52 PI, WC, Su, PF, D50 0.49 5.26 0.92 0.7 2958 50 201.85 0.66 22.0 1.977 0.1 53 PI, , WC, Su, PF, D50 0.479 1.43 5.1 0.91 0.73 2979.7 50 227.85 0.66 22.0 1.574 -0.13 71 LL, WC, Su, D50 0.416 3.86 0.85 2216.6 50 113 0.63 22.9 2.257 0.11 78 LL, WC, Su, PF, D50 0.423 5.4 0.93 0.55 2870.5 50 198 0.65 22.3 1.905 0.07 79 LL, , WC, Su, PF, D50 0.414 1.63 5.19 0.91 0.6 2897.1 50 227.7 0.65 22.3 1.519 0.01 124 , A, WC, Su, PF, D50 2.1 8.71 7.3 1.03 0.48 2808.5 50 265 0.63 23.1 1.353 -0.6 128 D50, , WC, Su, A 2257.2 50 0.57 6.34 0.99 9.34 171.52 0.61 23.5 1.616 0.06 Table 54. Selected “Power” models for critical shear stress in the HET/Global dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 19 PI, Su, D50 25.07 . . 50 . 0.64 22.7 16.049 0.43 35 PI, , Su, D50 495.9 . . . 50 . 0.65 22.4 12.907 -0.47 38 PI, WC, Su, D50 54.04 . . . 50 . 0.64 22.6 12.650 -0.1 40 PI, Su, PF, D50 38.24 . . . 50 . 0.64 22.7 12.580 -0.19 48 PI, , WC, Su, D50 562.6 . . . . 50 . 0.65 22.4 10.322 -0.14 50 PI, , Su, PF, D50 1430.36 . . . . 50 . 0.65 22.4 10.366 -0.48 52 PI, WC, Su, PF, D50 209.5 . . . . 50 . 0.64 22.6 10.176 -0.41 53 PI, , WC, Su, PF, D50 2491.8 . . . . . 50 . 0.65 22.3 8.410 -0.47

228 Figure 183. Plot of POU vs. correction factor for the Group 19 (Power) - in the HET/Global dataset Critical Velocity ( ) The same four-filter process discussed in Section 7.3.3 is implemented in this section to obtain the best models for critical velocity. First observation was that among the three erosion tests studied in this Chapter (EFA, JET, HET), only the EFA test has or critical velocity as one of its outputs. In other words, results of JET and HET consist of only three erodibility parameters (i.e. , , and EC) in contrary with the EFA results which include all five erodibility parameters (i.e. , , , , and EC). Therefore, the study of regression analysis for critical velocity is limited to only the EFA dataset. EFA Dataset Similar to the case of critical shear stress, it was observed that dividing the EFA/Global dataset into the EFA/Fine and the EFA/Coarse datasets would significantly improve the regression results. Figure 184 shows the number of data points in each of all the 135 combination groups in the EFA/Fine dataset. Also, Figure 185 and Figure 186 show the results of R2 for each combination group, for the “Linear Model” and “Power Model”, respectively. R2 values of groups 109 to 135 are generally higher than the rest for both “Linear” and “Power” models. Figure 187 and Figure 188 also show the results of the MSE for each of the 135 combination groups in the EFA/Fine dataset, for “Linear” and “Power” models, respectively.

229 Figure 184. Number of data in each 135 combination groups for the EFA/Fine dataset – Critical Velocity Figure 185. R2 results for the “Linear Models” in EFA/Fine dataset – Critical Velocity Figure 186. R2 results for the “Power Models” in EFA/Fine dataset – Critical Velocity

230 Figure 187. MSE results for “Linear Models” in EFA/Fine dataset – Critical Velocity Figure 188. MSE results for “Power Models” in EFA/Fine dataset – Critical Velocity After passing through filters 1 and 2 (R2 and MSE), “Linear” models associated with groups 114, 124, 128, and 132 in the EFA/Fine dataset were selected for further analyses. Also “Power” models associated with groups 109, 110, 113, 114, 117, 118, 123, 124, 125, 128, 132, and 133 in the EFA/Fine dataset were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 55 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). As shown in Table 55, all four selected models perform acceptably in the cross-validation test. Table 56 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The best models that also have the best cross- validation scores are highlighted in blue in both “Linear” and “Power” forms. 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 vs. θ” for this model. The vertical axis in Figure 189 represent the probability that the predicted using the selected model is smaller than the actual , in percentage (with 0.2 m/s offset). In order to reach

231 a 90% confidence that the predicted is smaller than the actual , the predicted value should be multiplied by 0.8. Table 55. Selected “Linear” models for critical velocity in the EFA/Fine dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 114 PC, , WC, Su, PF 0.012 0.038 0.041 0.0042 0.008 0.328 0.77 0.42 21.5 0.58 124 , A, WC, Su, PF, D50 0.075 0.171 0.05 0.0042 0.02 11.33 50 2.41 0.72 0.49 16.6 0.64 128 D50, , WC, Su, A 2.561 50 0.022 0.051 0.005 0.146 0.384 0.69 0.51 14.9 0.62 132 A, WC, Su, D50 0.142 0.051 0.004 2.674 50 0.784 0.69 0.51 15.07 0.62 Table 56. Selected “Power” models for critical velocity in the EFA/Fine dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 109 PC, γ, WC, Su, PF, D50 40.218 . . . . . 50 . 0.81 0.39 21.16 0.87 110 PC, WC, Su, PF, D50 0.013 . . . . 50 . 0.81 0.4 23.74 0.87 113 PC, γ, WC, Su, D50 0.1085 . . . . 50 . 0.8 0.41 22.92 0.81 114 PC, , WC, Su, PF 2.699 . . . . . 0.77 0.42 23.23 0.83 117 PC, WC, Su, D50 2.518 10 . . . 50 . 0.80 0.41 26.05 0.80 118 PC, WC, Su, PF 1.8 10 . . . . 0.75 0.43 25.80 0.77 123 A, PL, , WC, Su 0.00875 . . . . . 0.81 0.40 21.56 0.78 124 , A, WC, Su, PF, D50 7.299 . . . . . 50 . 0.84 0.37 18.09 0.74 125 , WC, Su, PF, A 57.994 . . . . . 0.77 0.43 19.1 0.68 128 D50, , WC, Su, A 0.00728 50 . . . . . 0.82 0.39 22.9 0.50 132 A, WC, Su, D50 3.92 10 . . . 50 . 0.82 0.40 26.11 0.69 133 A, WC, Su, PF 7 10 . . . . 0.75 0.45 23.71 0.67

232 Figure 189. Plot of POU vs. correction factor for the Group 117 (Power) - in the EFA/Fine dataset The same procedure was conducted in the EFA/Coarse dataset, and the best models are selected for critical velocity. Figure 190 shows the number of data points in each of all the 105 combination groups in the EFA/Coarse dataset. Also, Figure 191 and Figure 192 show the results of R2 for each combination group, for the “Linear” and “Power” models, respectively. Both “Linear” and “Power” models show a few good groups in terms of R2 values; however, as shown in Figure 190, the number of data points in most groups are lower compared to the EFA/Fine dataset. Figure 193 and Figure 194 show the values of MSE for the “Linear” and “Power” models, respectively. Figure 190. Number of data in each 105 combination groups for the EFA/Coarse dataset – Critical Velocity

233 Figure 191. R2 results for the “Linear Models” in EFA/Coarse dataset – Critical Velocity Figure 192. R2 results for the “Power Models” in EFA/Coarse dataset – Critical Velocity Figure 193. MSE results for “Linear Models” in EFA/Coarse dataset – Critical Velocity

234 Figure 194. MSE results for “Power Models” in EFA/Coarse dataset – Critical Velocity After passing through filters 1 and 2 (R2 and MSE), “Linear” models associated with groups 44 and 47 were selected for further analyses. Also “Power” models associated with groups 8, 27, 30, 44, 45, 46, and 47 were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 57 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). As shown in Table 57, only one selected models perform satisfactorily in the cross-validation test. Table 58 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The best models that also have a good cross- validation score are highlighted in blue in both “Linear” and “Power” forms. Out of all the highlighted correlation equations in Table 57 and Table 58, the Group 27 correlation equations in “Power” form was selected as the most promising equation. Figure 195 shows the plot of “POU vs. ” for this model. The vertical axis in Figure 195 represent the probability that the predicted using the selected model is smaller than the actual , in percentage (with 0.1 m/s offset). In order to reach a 90% confidence that the predicted is smaller than the actual , the predicted value should be multiplied by 0.7. Table 57. Selected “Linear” models for critical velocity in the EFA/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 44 PI, , VST, PF, D50 0.002 0.1 0.01 0.09 13.6 50 7.21 0.074 50 0.3 0.93 0.16 2.43 0.67 47 PI, , WC, VST, PF, D50 0.002 0.11 0.007 0.009 0.092 13.846 50 7.46 0.93 0.16 0.75 -0.75

235 Table 58. Selected “Power” models for critical velocity in the EFA/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 8 PI, , D50 3 10 . . 50 . 0.074 50 0.3 0.83 0.26 9.56 0.74 27 PI, , WC, D50 3 10 . . . 50 . 0.074 50 0.3 0.88 0.22 10.73 0.72 30 PI, , PF, D50 4 10 . . . 50 . 0.074 50 0.3 0.84 0.25 7.09 0.62 44 PI, , VST, PF, D50 0.027 . . . . 50 . 0.074 50 0.3 0.97 0.09 7.10 0.91 45 PI, , WC, PF, D50 4 10 . . . . 50 . 0.074 50 0.3 0.88 0.21 7.97 0.28 46 PI, WC, VST, PF, D50 7162.39 . . . . 50 . 0.074 50 0.3 0.98 0.09 7.86 0.94 47 PI, , WC, VST, PF, D50 2.055 . . . . 50 . 0.074 50 0.3 0.98 0.08 4.97 0.90 Figure 195. Plot of POU vs. correction factor for the Group 27 (Power) - in the EFA/Coarse dataset Initial Slope of Erosion Rate-Shear 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 rate-shear stress curve ( ). The study of

236 regression analysis for is performed in three different sections for the EFA, HET, and JET separately. EFA Database Similar to the cases of critical shear stress and critical velocity, it was observed that dividing the EFA/Global dataset into the EFA/Fine and the EFA/Coarse datasets would significantly improve the regression results. Figure 196 shows the number of data points in each of all the 135 combination groups in the EFA/Fine dataset. Also, Figure 197 and Figure 198 show the results of R2 for each combination group, for the “Linear” and “Power” models, respectively. R2 values of “Power” models are generally higher than the “Linear” models. Also, it can be observed that the R2 values are considerably higher for groups 123 to 135 in Figure 198. Figure 199 shows the results of the MSE for each of the 135 combination groups in the EFA/Fine dataset, for “Power” models. Figure 196. Number of data in each 135 combination groups for the EFA/Fine dataset – Figure 197. R2 results for the “Linear Models” in EFA/Fine dataset –

237 Figure 198. R2 results for the “Power Models” in EFA/Fine dataset – Figure 199. MSE results for the “Power Models” in EFA/Fine dataset – After passing through filters 1 and 2 (R2 and MSE), “Power” models associated with groups 124, 126, 128, 130, 131, 132, and 134 were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 59 shows the results of the selected “Power” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). The best models that also have a good cross-validation score are highlighted in blue. The Group 134 correlation equations in “Power” form was selected as the most promising equation. Figure 200 shows the plot of “POO vs. ” for this model. The vertical axis in Figure 200 represents the probability that the predicted using the selected model is greater than the actual , in percentage (with 4 mm/hr-Pa offset). In order to reach a 87% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2.

238 Table 59. Selected “Power” models for Eτ in the EFA/Fine dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 124 , A, WC, Su, PF, D50 1.043059638 10 . . . . . 50 . 0.94 17.7 9.6 0.04 126 D50, , WC, PF, A 1.208024 10 50 . . . . . 0.94 14.26 21.3 0.51 128 D50, , WC, Su, A 2.3983097 10 50 . . . . 0.79 32.56 6.6 0.09 130 D50, Su, PF, A 789314.3 50 . . . . 0.93 17.2 11.7 -0.63 131 D50, WC, PF, A 86.707 50 . . 0.93 14.9 24.1 -0.21 132 A, WC, Su, D50 2.63681129 10 . . . 50 . 0.80 31.7 10.3 0.06 134 A, , PF, D50 1.429078 10 . . . 50 . 0.90 18.2 22.9 0.53 Figure 200. Plot of POO vs. correction factor for the Group 134 (Power) - in the EFA/Fine dataset The same procedure was conducted in the EFA/Coarse dataset, and the best models are selected for . Figure 201 shows the number of data points in each of all the 105 combination groups in the EFA/Coarse dataset. Also, Figure 202 and Figure 203 show the results of R2 for each combination group, for the “Linear” and “Power” models, respectively. Both “Linear” and “Power” models show a few good groups in terms of R2 values; however, as shown in Figure 201, the number of data points in most groups are lower compared to the EFA/Fine dataset. Figure 204 and Figure 205 show the values of MSE for the “Linear” and “Power” models, respectively.

239 Figure 201. Number of data in each 105 combination groups for the EFA/Coarse dataset – Figure 202. R2 results for the “Linear Models” in EFA/Coarse dataset – Figure 203. R2 results for the “Power Models” in EFA/Coarse dataset –

240 Figure 204. MSE results for the “Linear Models” in EFA/Coarse dataset – Figure 205. MSE results for the “Power Models” in EFA/Coarse dataset – After passing through filters 1 and 2 (R2 and MSE), “Linear” models associated with groups 38, 42, 43, and 47 were selected for further analyses. Also “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 selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 60 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). As shown in Table 60, none of the selected models perform satisfactorily in the cross-validation test. Table 61 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The Group 77 correlation equations in “Power” form was selected as the most promising equation. Figure 206 shows the plot of “POO vs. ” for this model. The vertical axis in Figure 206 represent the probability that the predicted using the selected model is greater than the actual , in percentage (with 15 mm/hr-Pa offset). In order to reach a 80% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2.5.

241 Table 60. Selected “Linear” models for Eτ in the EFA/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 38 WC, VST, PF, D50 79.308 7.52 38.22 20.35 50 374.67 0.88 359.3 7.71 -8.5 42 PI, , WC, VST, 50 0.669 5.06 1.89 0.31 112.8 50 149.23 0.73 10.3 0.64 -0.35 43 , WC, VST, PF, D50 31.207 78.76 7.86 36.82 39.92 50 180.81 0.88 356 5.98 -7.4 47 PI, , WC, VST, PF, D50 0.897 5.82 1.5 0.2 1.01 46.15 50 198.61 0.78 9.19 0.49 -1.5 Table 61. Selected “Power” models for Eτ in the EFA/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 12 PI, VST, PF 0.00009 . . . 0.95 17.39 26.4 -0.97 13 PI, VST, D50 1.096668 10 . .50 . 0.99 5.41 22.2 -0.35 16 , WC, D50 93090257.5 . . 50 . 0.79 349.1 19.6 -0.14 18 WC, VST, D50 4112.67 . . 50 . 0.94 243.5 23.9 -4.5 20 VST, , D50 604968.99 . . 50 . 0.92 281.8 18.1 -9.01 28 PI, , VST, PF 4630.2794 10 . . . . 0.95 17.58 19.9 -2.5 31 PI, WC, VST, PF 2.59644111 10 . . . . 0.87 5.9 18.1 -0.33 34 PI, VST, PF, D50 7.1386087 10 . . . 50 . 0.99 4.99 9.1 -1.53 39 PI, VST, PF, D50 7.13860971 10 . . . 50 . 0.99 4.99 8.8 -0.03 40 , WC, PF, D50 2.881 . . . 50 . 0.73 398.4 41.1 -0.53 41 PI, , WC, VST, PF 2.99273 10 . . . . . 0.85 6.23 13.3 -0.11rr 42 PI, , WC, VST, D50 1591199925 . . . . 50 . 0.96 4.05 8.3 -7.09 44 PI, , VST, PF, D50 127160023 . . . . 50 . 0.99 4.15 9.6 -0.37 74 Cu, , WC 3785948678096 . . . 0.84 167.8 16.7 0.58 77 Cu, , D50 3228.7 . . 50 . 0.91 126.3 28.02 0.64 80 Cu, WC, D50 34.62 . . 50 . 0.91 124.6 28.9 0.62 82 Cu, VST, D50 12.22 . . 50 . 0.99 22.6 10.04 -3.76 86 Cu, , WC, D50 0.148 . . . 50 . 0.91 124.3 23.4 -3.96 91 Cu, WC, VST, D50 45.82 . . . 50 . 0.99 22.6 9.5 -10.54

242 95 Cu, , WC, VST, D50 9.2 10 . . . . 50 . 0.99 37.13 11.1 -10.35 Figure 206. Plot of POO vs. correction factor for the Group 77 (Power) - in the EFA/Coarse dataset JET Database Similar to the case of the EFA database, it was observed that dividing the JET/Global dataset into the JET/Fine and the JET/Coarse datasets would significantly improve the regression results. Figure 207 shows the number of data points in each of all the 135 combination groups in the JET/Fine dataset. Figure 208 and Figure 209 show the results of R2 for each combination group, for the “Linear” and “Power” models, respectively. R2 values of “Power” models are generally higher than the “Linear” models. Figure 210 and Figure 211 show the results of the MSE for each of the 135 combination groups in the EFA/Fine dataset for “Power” and “Linear” models, respectively.

243 Figure 207. Number of data in each 135 combination groups for the JET/Fine dataset – Figure 208. R2 results for the “Linear Models” in JET/Fine dataset – Figure 209. R2 results for the “Power Models” in JET/Fine dataset –

244 Figure 210. MSE results for the “Linear Models” in JET/Fine dataset – Figure 211. MSE results for the “Power Models” in JET/Fine dataset – After passing through filters 1 and 2 (R2 and MSE), “Linear” models associated with groups 100 and 113 were selected for further analyses. Also “Power” models associated with groups 5, 12, 15, 20, 24, 38, 43, 71, 75, 78, 97, 123, 125, 128 were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 62 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). As shown in Table 62, none of the selected models perform satisfactorily in the cross-validation test. Table 63 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The Group 15 correlation equations in “Power” form was selected as the most promising equation. Figure 212 shows the plot of “POO vs. ” for this model. The vertical axis in Figure 212 represent the probability that the predicted using the selected model is greater than the actual , in percentage (with 6 mm/hr-Pa offset). In order to reach a 88% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2.

245 Table 62. Selected “Linear” models for Eτ in the JET/Fine dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 100 PL, , WC, Su, PF 2.468 9.40 3.30 0.38 0.39 287.11 0.93 6.29 2.31 -0.59 113 PC, , WC, Su, D50 0.707 7.10 2.89 0.39 162.63 50 218.4 0.94 6.14 2.54 -0.64 Table 63. Selected “Power” models for Eτ in the JET/Fine dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 5 LL, PL, , WC, Su 35913.13 . . . . . 0.94 5.94 6.98 -0.79 12 PI, , Su 0.085 . . . 0.93 6.29 8.63 -0.95 15 PI, WC, Su 396599.6 . . . 0.93 6.30 9.11 0.23 20 , WC, Su 4.363 10 . . . 0.93 6.27 8.74 -0.52 24 WC, Su, D50 1986216 . . 50 . 0.93 6.29 10.01 -2.08 38 PI, WC, Su, D50 13854336090024 . . . 50 . 0.94 5.89 8.21 -1.46 43 , Su, PF, D50 1.7 10 . . . 50 . 0.94 5.74 7.33 -8.18 71 LL, WC, Su, D50 21641385785 . . . 50 . 0.95 5.40 8.14 -1.43 75 LL, , WC, Su, D50 5.3 10 . . . . 50 . 0.97 4.35 7.99 -330 78 LL, WC, Su, PF, D50 5.22 10 . . . . 50 . 0.95 5.15 8.1 -2.95 97 PL, WC, Su, D50 11377689319 . . . 50 . 0.94 5.89 7.88 -76.1 123 A, PL, , WC, Su 5.8 10 . . . . . 0.98 3.27 8.90 -7.25 125 , WC, Su, PF, A 1.341 10 . . . . . 0.98 3.27 8.66 -2.5 128 D50, , WC, Su, A 1.898 10 50 . . . . . 0.98 3.23 8.59 -2.28

246 Figure 212. Plot of POO vs. correction factor for the Group 15 (Power) - in the JET/Fine dataset The same procedure was conducted in the JET/Coarse dataset, and the best models are selected for . Figure 213 shows the number of data points in each of all the 105 combination groups in the EFA/Coarse dataset. Also, Figure 214 and Figure 215 show the results of R2 for each combination group, for the “Linear” and “Power” models, respectively. Both “Linear” and “Power” models show many good groups in terms of R2 values; however, as shown in Figure 213, the number of data points in most groups are extremely low. Only a few combination groups consist of sufficient amount of data for regression analyses. Figure 213. Number of data in each 105 combination groups for the JET/Coarse dataset –

247 Figure 214. R2 results for the “Linear Models” in JET/Coarse dataset – Figure 215. R2 results for the “Power Models” in JET/Coarse dataset – After passing through filters 1 and 2 (R2 and MSE), “Linear” models associated with groups 5 and 26 were selected for further analyses. Also “Power” models associated with groups 5, 7, and 26 were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 64 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). As shown in Table 64, none of the selected models perform satisfactorily in the cross-validation test. Table 65 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The Group 5 correlation equations in “Power” form was selected as the most promising equation. Figure 216 shows the plot of “POO vs. ” for this model. The probability that the predicted using the selected model is greater than the actual , in percentage (with 5 mm/hr-Pa offset). In order to reach a 90% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 1.4.

248 Table 64. Selected “Linear” models for Eτ in the JET/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 5 PI, , WC 1.827 28.85 13.87 797.2 0.57 63.2 2.99 0.06 26 PI, , WC, PF 7 30.85 13.85 5.6722.2 0.57 70.5 1.22 -86.9 Table 65. Selected “Power” models for Eτ in the JET/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 5 PI, , WC 55637006351614 . .. 0.90 29.6 8.13 0.67 7 PI, , PF 14284687933437 . . . 0.77 51.42 6.56 -0.53 26 PI, , WC, PF 17383934656478 . .. . 0.92 31.3 4.35 0.26 Figure 216. Plot of POO vs. correction factor for the Group 5 (Power) - in the JET/Coarse dataset HET Database It was observed that dividing the HET/Global dataset into the HET/Fine and the HET/Coarse datasets 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 all the 105 combination groups in the HET/Coarse dataset. Many

249 combination groups as shown in Figure 217, do not include any data points. Figure 218 and Figure 219 show the results of 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 mm to 0.3 mm. Figure 217. Number of data in each 105 combination groups for the HET/Coarse dataset – Figure 218. R2 results for the “Linear Models” in HET/Coarse dataset –

250 Figure 219. R2 results for the “Power Models” in HET/Coarse dataset – After passing through filters 1 and 2 (R2 and MSE), “Power” models associated with groups 40 and 60 were selected for further analyses. “Linear” models did not lead to any better R2 values; 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 66 shows the results of the selected “Power” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). The Group 40 correlation equations in “Power” form was selected as the most promising equation. Figure 220 shows the plot of “POO vs. ” for this model. The probability that the predicted using the selected model is greater than the actual , in percentage (with 10 mm/hr-Pa offset). In order to reach a 80% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2. As mentioned earlier, it is important to note that the equation associated with the Group 40 should be used for coarse- grained soils with D50 ranging between 0.074 mm and 0.3 mm. Table 66. Selected “Power” models for Eτ in the HET/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 40 , WC, PF, D50 2.951 . . . 50 . 0.86 91.04 34.67 0.55 60 Cc, , WC, D50 4.4 10 . . . 50 . 0.87 121.8 15.56 0.20

251 Figure 220. Plot of POO vs. correction factor for the Group 40 (Power) - in the HET/Coarse dataset The same procedure was conducted in the HET/Global dataset (soils with D50 < 0.3 mm), and the best models are selected for . Figure 221 shows the number of data points in each of all the 135 combination groups in the HET/Global dataset. Also, Figure 222 and Figure 223 show the results of R2 for each combination group, for the “Linear” and “Power” models, respectively. Figure 224 and Figure 225 show the results of the MSE for each of the 135 combination groups in the EFA/Fine dataset for “Power” and “Linear” models, respectively. As expected, the combination groups with better R2 values possess a lower MSE value as well. Figure 221. Number of data in each 135 combination groups for the HET/Global dataset –

252 Figure 222. R2 results for the “Linear Models” in HET/Global dataset – Figure 223. R2 results for the “Power Models” in HET/Global dataset – Figure 224. MSE results for the “Linear Models” in HET/Global dataset –

253 Figure 225. MSE results for the “Power Models” in HET/Global dataset – 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 analyses. Also “Power” models associated with groups 5, 49, 53, 79, 105, 108, 109, 110, 112, 114, 115, and 118 were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 67 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). As shown in Table 67, many of the selected models perform satisfactorily in the cross-validation test. Table 68 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The best models that also have a good cross-validation score are highlighted in blue in both “Linear” and “Power” forms. 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 vs. ” for this model. The vertical axis in Figure 226 represent the probability that the predicted using the selected model is greater than the actual , in percentage (with 0 mm/hr-Pa offset). In order to reach a 90% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 1.45. It is noteworthy that the equation associated with the combination group 108 in “Power” form is best to be used for finer-grained soils with D50 smaller than 0.3 mm.

254 Table 67. Selected “Linear” models for Eτ in the HET/Global dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 5 LL, PL, , WC, Su 0.01 0.07 0.07 0.08 0.0045 0.16 0.71 0.26 2.581 0.37 49 , WC, Su, PF, D50 0.061 0.06 0.004 0.02 21.73 50 1.33 0.69 0.27 2.316 0.32 53 PI, , WC, Su, PF, D50 0.001 0.05 0.07 0.004 0.02 21.58 50 1.18 0.70 0.27 1.879 0.31 78 LL, WC, Su, PF, D50 0.002 0.06 0.0036 0.02 20.42 50 0.17 0.69 0.27 2.287 0.35 79 LL, , WC, Su, PF, D50 0.001 0.054 0.07 0.004 0.02 21.3 50 1.16 0.70 0.27 1.886 0.31 104 PL, WC, Su, PF, D50 0.024 0.06 0.0035 0.01 15.67 50 0.09 0.70 0.27 2.386 0.31 105 PL, , WC, Su, PF, D50 0.023 0.054 0.07 0.004 0.02 16.84 50 0.91 0.71 0.26 1.971 0.29 108 LL, PL, , PC, Su 0.005 0.06 0.1 0.02 0.0001 1.47 0.78 0.23 3.697 0.52 109 PC, , WC, Su, PF, D50 0.03 0.094 0.01 0.0003 0.02 8.86 50 2 0.77 0.23 2.791 0.46 110 PC, WC, Su, PF, D50 0.027 0.02 0.0008 0.01 8.95 50 0.32 0.75 0.25 3.068 0.45 112 PC, , Su, PF, D50 0.028 0.1 0.0003 0.02 8.31 50 1.86 0.77 0.23 3.134 0.51 113 PC, , WC, Su, D50 0.04 0.07 0.01 0.0004 10.12 50 0.26 0.74 0.25 2.689 0.48 114 PC, , WC, Su, PF 0.034 0.09 0.0045 0.00003 0.01 1.35 0.77 0.24 3.716 0.63 115 PC, Su, PF, D50 0.023 0.0002 0.01 8.08 50 0.03 0.75 0.25 3.679 0.51 118 PC, WC, Su, PF 0.031 0.01 0.0004 0.01 0.36 0.74 0.25 3.608 0.62 124 , A, WC, Su, PF, D50 0.045 0.09 0.05 0.003 0.02 19.41 50 1.16 0.72 0.26 3.009 0.27

255 Table 68. Selected “Power” models for Eτ in the HET/Global dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 5 LL, PL, , WC, Su 3.96 10 . . . . . 0.74 0.25 8.578 0.21 49 , WC, Su, PF, D50 4.1 10 . . . . 50 . 0.71 0.27 7.968 0.08 53 PI, , WC, Su, PF, D50 10 . . . . . 50 . 0.74 0.25 8.012 -1.53 79 LL, , WC, Su, PF, D50 1.7 10 . . . . . 50 . 0.72 0.26 5.568 -0.16 105 PL, , WC, Su, PF, D50 3.16 10 . . . . . 50 . 0.72 0.26 5.960 -2.92 108 LL, PL, , PC, Su 9 10 . . . . . 0.81 0.21 9.238 0.51 109 PC, , WC, Su, PF, D50 4 10 . . . . . 50 . 0.80 0.22 5.466 0.32 110 PC, WC, Su, PF, D50 0.006 . . . . 50 . 0.76 0.24 8.259 0.48 112 PC, , Su, PF, D50 8.5 10 . . . . 50 . 0.78 0.23 9.017 0.48 114 PC, , WC, Su, PF 2 10 . . . . . 0.76 0.24 8.881 -2.16 115 PC, Su, PF, D50 0.031 . . . 50 . 0.75 0.24 6.363 0.34 118 PC, WC, Su, PF 0.016 . . . . 0.72 0.26 7.113 0.48 124 , A, WC, Su, PF, D50 1.84 10 . . . . . 50 . 0.73 0.25 5.349 -0.24

256 Figure 226. Plot of POO vs. correction factor for the Group 108 (Power) - in the HET/Global dataset Initial Slope of Erosion Rate-Velocity ( ) 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 ( ), among the three erosion tests studied in this Chapter (EFA, JET, HET), only the EFA test can reflect Ev as one of its outputs. In other words, results of JET and HET consist of only three erodibility parameters (i.e. , , and EC) in contrary with the EFA results which include all five erodibility parameters (i.e. , , , , and EC). Therefore, the study of regression analysis for Ev is limited to only the EFA dataset. EFA Dataset Similar to the case of , it was observed that dividing the EFA/Global dataset into the EFA/Fine and the EFA/Coarse datasets would significantly improve the regression results for . Figure 227 shows the number of data points in each of all the 135 combination groups in the EFA/Fine dataset. Figure 228 and Figure 229 show the results of R2 for each combination group, for the “Linear Model” and “Power Model”, respectively. Results show that “Power” models in general perform better than the “Linear” models. Also, R2 values of groups 109 to 135 are generally higher than the rest for both “Power” models. Figure 230 and Figure 231 show the results of the MSE for each of the 135 combination groups in the EFA/Fine dataset, for “Linear” and “Power” models, respectively.

257 Figure 227. Number of data in each 135 combination groups for the EFA/Fine dataset – Figure 228. R2 results for the “Linear Models” in EFA/Fine dataset – Figure 229. R2 results for the “Power Models” in EFA/Fine dataset –

258 Figure 230. MSE results for the “Linear Models” in EFA/Fine dataset – Figure 231. MSE results for the “Power Models” in EFA/Fine dataset – After passing through filters 1 and 2 (R2 and MSE), “Power” models associated with groups 53, 79, 101, 105, 124, 126, 127, 131, and 134 were selected for further analyses. “Linear” models did not lead to any better R2 values; 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 meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). The Group 126 equations in “Power” form was selected as the most promising equation. Figure 232 shows the plot of “POO vs. ” for this model. The probability that the predicted using the selected model is greater than the actual , in percentage (with 10 mm- s/m-hr offset). In order to reach a 80% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 2.

259 Table 69. Selected “Power” models for Ev in the EFA/Fine dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 53 PI, , WC, Su, PF, D50 9785577.4 . . . . . 50 . 0.72 69.2 324.6 -5.9 79 LL, , WC, Su, PF, D50 142233608 . . . . . 50 . 0.71 70.6 320.1 -0.55 101 PL, , WC, Su, D50 662947356 . . . . 50 . 0.70 72.6 318.5 -0.43 105 PL, , WC, Su, PF, D50 3.12869 10 . . . . 50 . 0.70 71.8 316.3 -0.05 124 , A, WC, Su, PF, D50 1.4227 10 . . . . . 50 . 0.80 60.1 221.2 -2.33 126 D50, , WC, PF, A 1.682339 10 50 . . . . . 0.79 49.5 384.3 0.52 127 D50, , Su, PF, A 1.429 10 50 . . . . . 0.75 66.8 361.5 0.4 131 D50, WC, PF, A 4610.15 50 . . . . 0.73 56.4 355.6 -0.98 134 A, , PF, D50 2.0788 10 . . . 50 . 0.73 56.3 329.9 0.05 Figure 232. Plot of POO vs. correction factor for the Group 126 (Power) - in the EFA/Fine dataset The same procedure was conducted in the EFA/Coarse dataset, and the best models are selected for . Figure 233 shows the number of data points in each of all the 105 combination groups in the EFA/Coarse dataset. Figure 234 and Figure 235 show the results of R2 for each combination group, for the “Linear” and “Power” models, respectively.

260 Both “Linear” and “Power” models show some good groups in terms of R2 values; however, as shown in Figure 233, the number of data points in most groups are very low. Figure 233. Number of data in each 105 combination groups for the EFA/Coarse dataset – Figure 234. R2 results for the “Linear Models” in EFA/Coarse dataset – Figure 235. R2 results for the “Power Models” in EFA/Coarse dataset –

261 After passing through filters 1 and 2 (R2 and MSE), “Linear” models associated with groups 38,42, 43, and 45 were selected for further analyses. Also “Power” models associated with groups 18, 20, 28, 34, 36, 44, 46, 56, 65, 74, 77, 80, 82, 86, and 88 were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 70 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). As shown in Table 70, none of the selected models perform satisfactorily in the cross-validation test. Table 71 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The Group 86 correlation equations in “Power” form was selected as the most promising equation. Figure 236 shows the plot of “POO vs. ” for this model. The probability that the predicted using the selected model is greater than the actual , in percentage (with 10 mm-s/m-hr offset). In order to reach a 80% confidence that the predicted is greater than the actual , the predicted value should be multiplied by 5. Table 70. Selected “Linear” models for Ev in the EFA/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 38 WC, VST, PF, D50 101.665 4.65 55.46 85.9 50 355.9 0.85 643.1 5.46 -2.3 42 PI, , WC, VST, 50 4.543 35.21 9.16 2.03 1277.9 50 901.4 0.83 49.23 5.11 -4.6 43 , WC, VST, PF, D50 59.184 101.11 5.77 54.22 45 50 1391 0.85 635.7 2.09 -22.4 45 PI, , WC, PF, D50 6.62 48.82 2.92 10.46 216.18 50 1497 0.65 88.3 0.96 -0.03 Table 71. Selected “Power” models for Ev in the EFA/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 18 WC, VST, D50 9280.99 . . 50 . 0.87 570 8.62 -2.8 20 VST, , D50 1481.13 . . 50 . 0.87 577 8.40 -0.58 28 PI, , VST, PF 2.991 10 . . . 0.79 58.0 4.9 -0.7 34 PI, VST, PF, D50 805281.14 10 . . . 50 . 0.99 4.85 0.93 -0.57 36 , WC, VST, D50 2 10 . . . 50 . 0.91 491 8.8 -5.02 44 PI, , VST, PF, D50 820969.5 10 . . . . 50 . 0.99 2.02 0.89 0.82 46 PI, WC, VST, PF, D50 84388620 . . . . 50 . 0.99 3.21 0.91 -0.51 56 Cc, VST, D50 5401.92 . . 50 . 0.81 399 2.98 -0.25

262 65 Cc, WC, VST, D50 10199544.5 . . . 50 . 0.97 155.6 13.4 -1.98 74 Cu, , WC 23300967450 . . . 0.79 389.2 9.9 0.29 77 Cu, , D50 4489.56 . . 50 . 0.86 323.5 19.2 0.37 80 Cu, WC, D50 152.16 . . 50 . 0.85 324 17.9 0.01 82 Cu, VST, D50 95.18 . . 50 . 0.97 155.9 21.7 0.03 86 Cu, , WC, D50 88969.4 . . . 50 . 0.86 319.4 15.9 0.64 88 Cu, , VST, D50 7.6 10 . . . 50 . 0.97 155.6 13.43 -1.28 Figure 236. Plot of POO vs. correction factor for the Group 86 (Power) - in the EFA/Coarse dataset Erosion Category (EC) The same four-filter process discussed in Section 7.3.3 is implemented in this section to obtain the best models for erosion category (EC). The study of regression analysis for EC is performed in three different sections for the EFA, HET, and JET separately. EFA Database It was observed that dividing the EFA/Global dataset into the EFA/Fine and the EFA/Coarse datasets would improve the regression results. Figure 237 shows the number of data points in each of all the 135 combination groups in the EFA/Fine dataset. Figure 238 and Figure 239 show the results of R2 for each combination group, for the “Linear” and “Power” models, respectively. R2 values for both “Linear” and “Power” models are not very high (up to 0.6). Also, it can be observed that the better R2 values are observed in the combination groups 109 to 135. Figure 240

263 and Figure 241 show the results of the MSE for each of the 135 combination groups in the EFA/Fine dataset, for “Linear” and “Power” models, respectively. It is observed that in general, “Power” and “Linear” models are not noticeably different. Figure 237. Number of data in each 135 combination groups for the EFA/Fine dataset – EC Figure 238. R2 results for the “Linear Models” in EFA/Fine dataset – EC

264 Figure 239. R2 results for the “Power Models” in EFA/Fine dataset – EC Figure 240. MSE results for the “Linear Models” in EFA/Fine dataset – EC Figure 241. MSE results for the “Power Models” in EFA/Fine dataset – EC

265 After passing through filters 1 and 2 (R2 and MSE), “Linear” models associated with groups 109, 110, 113, 117, and 124 were selected for further analyses. Also “Power” models associated with groups 124, 128, and 132 were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 72 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). As shown in Table 72, most of the selected models perform satisfactorily in the cross-validation test. Table 73 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The Group 132 correlation equations in “Power” form was selected as the most promising equation. Figure 242 shows the plot of “POU vs. ” for this model. The probability that the predicted EC using the selected model is smaller than the actual EC, in percentage. In order to reach a 90% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by 0.75. Table 72. Selected “Linear” models for EC in the EFA/Fine dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 109 PC, , WC, Su, PF, D50 0.02 0.112 0.04 0.0005 0.0012 3.09 50 1.27 0.59 0.47 3.96 0.33 110 PC, WC, Su, PF, D50 0.022 0.03 0.0016 0.0029 3.85 50 1.2 0.56 0.48 4.13 0.35 113 PC, , WC, Su, D50 0.02 0.11 0.04 0.0005 3.853 50 1.088 0.59 0.46 4.64 0.43 117 PC, WC, Su, D50 0.023 0.03 0.0017 1.845 50 0.8566 0.56 0.48 4.97 0.43 124 , A, WC, Su, PF, D50 0.125 0.072 0.04 0.0001 0.0086 16.12 50 0.259 0.56 0.47 3.34 0.33 Table 73. Selected “Power” models for EC in the EFA/Fine dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 124 , A, WC, Su, PF, D50 0.193 . . . . . 50 . 0.59 0.46 6.09 0.58 128 D50, , WC, Su, A 0.0237 50 . . . . . 0.56 0.48 8.23 0.53 132 A, WC, Su, D50 0.1933 . . . 50 . 0.55 0.49 7.99 0.53

266 Figure 242. Plot of POU vs. correction factor for the Group 132 (Power) - EC in the EFA/Fine dataset The same procedure was conducted in the EFA/Coarse dataset, and the best models are selected for EC. Figure 243 shows the number of data points in each of all the 105 combination groups in the EFA/Coarse dataset. Figure 244 and Figure 245 show the results of R2 for each combination group, for the “Linear” and “Power” models, respectively. Both “Linear” and “Power” models show some good groups in terms of R2 values; however, as shown in Figure 243, the number of data points are very low in many combination groups. Figure 243. Number of data in each 105 combination groups for the EFA/Coarse dataset – EC

267 Figure 244. R2 results for the “Linear Models” in EFA/Coarse dataset – EC Figure 245. R2 results for the “Power Models” in EFA/Coarse dataset – EC After passing through filters 1 and 2 (R2 and MSE), “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 analyses. Also “Power” models associated with groups 32, 42, 46, 47, 65, 69, 91, and 95 were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 74 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). Table 75 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The Group 91 correlation equations in “Power” form was selected as the most promising equation. Figure 246 shows the plot of “POU vs. ” for this model. The probability that the predicted EC using the selected model is smaller than the actual EC, in percentage. In order to reach a 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 the Group 91 combination group is best to be used for soils with D50 ranging between 0.074 and 0.3 mm.

268 Table 74. Selected “Linear” models for EC in the EFA/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 42 PI, , WC, VST, D50 0.02 0.03 0.02 0.0027 5.586 50 3.917 0.80 0.27 0.93 0.08 46 PI, WC, VST, PF, D50 0.017 0.01 0.0049 0.023 7.27 50 4.01 0.82 0.25 1.11 -0.5 47 PI, , WC, VST, PF, D50 0.014 0.05 0.01 0.006 0.027 7.36 50 5.231 0.84 0.24 0.70 -2.12 55 Cc, VST, PF 0.01 0.0094 0.0137 1.105 0.85 0.24 2.78 0.33 61 Cc, , VST, PF 0.009 0.02 0.0095 0.0145 1.458 0.85 0.24 1.68 0.01 64 Cc, , PF, D50 0.0047 0.03 0.0031 0.028 1.524 0.92 0.18 1.59 0.55 67 Cc, VST, PF, D50 0.015 0.0092 0.02 2.69 50 0.534 0.87 0.23 1.94 -0.16 68 Cc, , WC, VST, PF 0.003 0.03 0.03 0.0032 0.03 1.995 0.93 0.17 2.00 0.61 70 Cc, , VST, PF, D50 0.014 0.01 0.0092 0.021 2.578 50 0.75 0.87 0.22 1.07 -0.5 72 Cc, WC, VST, PF, D50 0.0041 0.04 0.003 0.028 0.23 50 1.58 0.92 0.18 1.85 -0.13 73 Cc, , WC, VST, PF, D50 0.0009 0.03 0.04 0.0027 0.029 0.74 50 2.24 0.93 0.17 0.94 0.14 87 Cu, , VST, PF 0.001 0.07 0.01 0.02 2.2 0.84 0.25 1.56 -0.76 90 Cu, WC, VST, PF 0.0001 0.04 0.0015 0.032 1.55 0.91 0.18 3.04 0.27 94 Cu, , WC, VST, PF 0.0005 0.047 0.036 0.004 0.032 2.338 0.93 0.17 2.10 0.35 98 Cu, WC, VST, PF, D50 0.00006 0.04 0.002 0.029 1.2 50 1.85 0.92 0.18 1.79 0.14 99 Cu, , WC, VST, PF, D50 0.0004 0.044 0.036 0.003 0.031 0.32 50 2.37 0.93 0.17 0.98 -1.55 Table 75. Selected “Power” models for EC in the EFA/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score

269 32 PI, WC, VST, D50 0.722 . . . 50 . 0.074 50 0.3 0.87 0.22 40.8 0.49 42 PI, , WC, VST, D50 0.1685 . . . . 50 . 0.074 50 0.3 0.87 0.21 27.5 0.54 46 PI, WC, VST, PF, D50 1.055 . . . . 50 . 0.074 50 0.3 0.90 0.19 36.2 0.22 47 PI, , WC, VST, PF, D50 0.44 . . . . . 50 . 0.074 50 0.3 0.91 0.18 21.8 -0.16 65 Cc, WC, VST, D50 0.7423 . . . 50 . 0.074 50 0.3 0.87 0.23 18.1 0.65 69 Cc, , WC, VST, D50 2.22 . . . . 50 . 0.88 0.22 11.3 -0.08 91 Cu, WC, VST, D50 1.12 . . . 50 . 0.074 50 0.3 0.92 0.18 29.9 0.80 95 Cu, , WC, VST, D50 0.222 . . . . 50 . 0.074 50 0.3 0.94 0.16 21.2 0.79 Figure 246. Plot of POU vs. correction factor for the Group 91 (Power) - EC in the EFA/Coarse dataset JET Dataset Similar approach was taken to select the best correlating equation for EC in the JET database. On the JET database, there are two important observations to notice:

270 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 dataset are substantially low compared to the JET/Fine dataset. In fact, many combination groups on the JET/Coarse database have zero data points. 2) The R2 values for the JET/Global dataset (D50 < 0.3 mm) are significantly better than the R2 values for the JET/Coarse and JET/Fine dataset. Therefore, the regression results for the JET/Global dataset are presented as the best models. Figure 247 shows the number of data points in each of all the 135 combination groups in the JET/Global dataset. Figure 248 and Figure 249 show the results of R2 for each combination group, for the “Linear” and “Power” models, respectively. Both “Linear” and “Power” models show some reasonable groups in terms of R2 values. Figure 247. Number of data in each 135 combination groups for the JET/Global dataset – EC Figure 248. R2 results for the “Linear Models” in JET/Global dataset – EC

271 Figure 249. R2 results for the “Power Models” in JET/Global dataset – EC 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 analyses. Also “Power” models associated with groups 87, 93, 100, 102, 109, 112, and 129 were selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 76 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). As shown in Table 76, almost all of the selected models perform satisfactorily in the cross-validation test. Table 77 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The best models that also have a good cross-validation score are highlighted in blue in both “Linear” and “Power” forms. Out of all of the highlighted correlation equations in Table 76 and Table 77, the Group 88 correlation equation in “Linear” form was selected as the most promising equation. Figure 250 shows the plot of “POU vs. ” for this model. The vertical axis in Figure 250 represent the probability that the predicted EC using the selected model is smaller than the actual EC, in percentage. In order to reach a 90% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by 0.85. It should be noted that the proposed equations are best for soils with D50 smaller than 0.3 mm. Table 76. Selected “Linear” models for EC in the JET/Global dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 30 PF, Su, D50 0.007 0.003 6.48 50 3.5 0.68 0.26 5.58 0.58 40 PI, Su, PF, D50 0.003 0.0029 0.009 6.73 50 3.6 0.68 0.26 4.39 0.58 49 , WC, Su, PF, D50 0.019 0.01 0.003 0.006 6.37 50 3.95 0.68 0.26 3.52 0.28 50 PI, , Su, PF, D50 0.003 0.0004 0.003 0.009 6.73 50 3.61 0.68 0.26 3.54 0.47

272 52 PI, WC, Su, PF, D50 0.006 0.013 0.002 0.007 6.65 50 3.72 0.69 0.25 3.55 0.52 53 PI, , WC, Su, PF, D50 0.006 0.01 0.013 0.002 0.007 6.37 50 3.83 0.69 0.25 3.00 0.42 88 PL, Su, D50 0.022 0.0031 5.5 50 3.34 0.70 0.25 6.23 0.58 94 PL, , Su, D50 0.023 0.01 0.003 5.4 50 3.64 0.70 0.25 4.96 0.33 97 PL, WC, Su, D50 0.029 0.009 0.0037 5.25 50 3.23 0.71 0.25 5.11 0.58 104 PL, WC, Su, PF, D50 0.027 0.011 0.0038 0.002 5.74 50 3.35 0.71 0.24 5.13 0.24 124 , A, WC, Su, PF, D50 0.017 0.2 0.008 0.003 0.005 6.54 50 3.75 0.72 0.24 3.96 0.47 129 LL, , A, Su, PF 0.013 0.05 0.31 0.0054 0.018 2.6 0.73 0.23 5.03 0.20 Table 77. Selected “Power” models for EC in the JET/Global dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 87 PL, Su, PF 0.8 . . . 0.67 0.26 286.6 0.40 93 PL, , Su, PF 1.639 . . . . 0.68 0.26 235.2 0.05 100 PL, , WC, Su, PF 2.33 . . . . . 0.68 0.26 195.7 0.10 102 PL, , Su, PF, D50 1.644 . . . . 50 . 0.68 0.26 196.3 0.05 109 PC, , WC, Su, PF, D50 4.345 . . . . . 50 . 0.70 0.25 163.8 0.14 112 PC, , Su, PF, D50 3.282 . . . . 50 . 0.70 0.25 165.7 0.17 129 LL, , A, Su, PF 4.35 . . . . . 0.73 0.23 190.4 0.25

273 Figure 250. Plot of POU vs. correction factor for the Group 88 (Linear) - EC in the JET/Global dataset HET Database Although the number of data points in the HET/Coarse combination groups are very low in most cases, it was observed that dividing the HET/Global dataset into the HET/Fine and the HET/Coarse datasets would improve the regression results. Figure 251 shows the number of data points in each of all the 135 combination groups in the HET/Fine dataset. Figure 252 and Figure 253 show the results of R2 for each combination group, for the “Linear” and “Power” models, respectively. R2 values for both “Linear” and “Power” models are not very high (up to 0.7). Figure 254 and Figure 255 show the results of the MSE for each of the 135 combination groups in the HET/Fine dataset, for “Linear” and “Power” models, respectively. It is observed that in general, “Power” and “Linear” models are not noticeably different.

274 Figure 251. Number of data in each 135 combination groups for the HET/Fine dataset – EC Figure 252. R2 results for the “Linear Models” in HET/Fine dataset – EC Figure 253. R2 results for the “Power Models” in HET/Fine dataset – EC

275 Figure 254. MSE results for the “Linear Models” in HET/Fine dataset – EC Figure 255. MSE results for the “Power Models” in HET/Fine dataset – EC After passing through filters 1 and 2 (R2 and MSE), “Linear” models associated with groups 48, 53, 74, 79, 100, 109, 112, 114, and 129 were selected for further analyses. Also “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 selected for further analyses. Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 78 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2, MSE, and F-value/F-stat). Table 79 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The best models that also have a good cross- validation score are highlighted in blue in both “Linear” and “Power” forms. The Group 12 correlation equation in “Power” form was selected as the most promising equation. Figure 256 shows the plot of “POU vs. ” for this model. The vertical axis in Figure 256 represent the probability that the predicted EC using the selected model is smaller than the actual EC, in percentage. In order to reach a 100% confidence that the predicted EC is smaller than the actual EC, the predicted value should be multiplied by 0.95.

276 Table 78. Selected “Linear” models for critical shear stress in the HET/Fine dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 48 PI, , WC, Su, D50 0.002 0.05 0.02 0.004 5.43 50 1.59 0.60 0.13 1.548 -0.18 53 PI, , WC, Su, PF, D50 0.002 0.05 0.03 0.0037 0.001 6.82 50 1.36 0.60 0.13 1.242 -0.39 74 LL, , WC, Su, PF 0.002 0.05 0.01 0.0029 0.002 2.05 0.64 0.12 2.097 -0.50 79 LL, , WC, Su, PF, D50 0.002 0.051 0.03 0.0037 0.0007 6.48 50 1.36 0.60 0.13 1.220 -0.52 100 PL, , WC, Su, PF 0.012 0.05 0.02 0.0033 0.0045 1.82 0.61 0.13 1.929 0.12 109 PC, , WC, Su, PF, D50 0.014 0.044 0.0015 0.002 0.0038 13.1 50 1.46 0.66 0.12 2.416 0.25 112 PC, , Su, PF, D50 0.014 0.04 0.0021 0.0038 13.17 50 1.44 0.66 0.12 3.018 0.32 114 PC, , WC, Su, PF 0.008 0.04 0.01 0.0015 0.0016 2.44 0.67 0.12 2.052 0.09 129 LL, , A, Su, PF 0.003 0.05 0.04 0.0021 0.0014 2.17 0.69 0.11 1.623 -0.42 Table 79. Selected “Power” models for critical shear stress in the HET/Fine dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 5 LL, PL, , WC, Su 1.87 . . . . . 0.69 0.12 796.3 0.26 12 PI, , Su 1.67 . . . 0.70 0.11 1172.2 0.54 18 PI, Su, PF 3.23 . . . 0.70 0.11 1172.3 0.57 19 PI, Su, D50 2.57 . . 50 . 0.68 0.12 917.4 0.54 31 PI, , WC, Su 1.74 . . . . 0.69 0.11 911.5 0.38 34 PI, , Su, PF 2.221 . . . . 0.71 0.11 968.3 0.40 35 PI, , Su, D50 1.68 . . . 50 . 0.69 0.11 968.6 0.24 37 PI, WC, Su, PF 3.221 . . . . 0.70 0.11 963.5 0.39 40 PI, Su, PF, D50 3.47 . . . 50 . 0.70 0.11 963.5 0.33 47 PI, , WC, Su, PF 2.211 . . . . . 0.71 0.11 781.6 0.24 50 PI, , Su, PF, D50 2.4 . . . . 50 . 0.71 0.11 780.2 0.15 52 PI, WC, Su, PF, D50 3.708 . . . . 50 . 0.70 0.11 771.9 -0.26 53 PI, , WC, Su, PF, D50 2.547 . . . . . 50 . 0.71 0.11 638.6 -0.11

277 108 LL, PL, , PC, Su 2.064 . . . . . 0.70 0.11 622.9 0.32 109 PC, , WC, Su, PF, D50 1.861 . . . . . 50 . 0.68 0.12 608.8 0.39 112 PC, , Su, PF, D50 1.919 . . . . 50 . 0.68 0.12 778.3 0.51 113 PC, , WC, Su, D50 1.854 . . . . 50 . 0.68 0.12 778.1 0.44 129 LL, , A, Su, PF 2.123 . . . . . 0.69 0.11 779.6 0.19 Figure 256. Plot of POU vs. correction factor for the Group 12 (Power) - EC in the HET/Fine dataset The same procedure was conducted in the HET/Coarse dataset, and the best models are selected for EC. Figure 257 shows the number of data points in each of all the 105 combination groups in the EFA/Coarse dataset. Figure 258 and Figure 259 show the results of R2 for each combination group, for the “Linear” and “Power” models, respectively. Both “Linear” and “Power” models show some good groups in terms of R2 values; however, as shown in Figure 257, the number of data points are very low in most combination groups. In fact, close to half of the combination groups do not have any data points. After passing through filters 1 and 2 (R2 and MSE), “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 analyses. Also “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 selected for further analyses.

278 Figure 257. Number of data in each 105 combination groups for the HET/Coarse dataset – EC Figure 258. R2 results for the “Linear Models” in HET/Coarse dataset – EC Figure 259. R2 results for the “Power Models” in HET/Coarse dataset – EC Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 80 shows the results of the selected “Linear” models after meeting the requirements of the first three filters (R2,

279 MSE, and F-value/F-stat). Table 81 also shows the results of the selected “Power” models after meeting the requirements of the first three filters. The best models that also have a good cross- validation score are highlighted in blue in both “Linear” and “Power” forms. The Group 48 correlation equation in “Power” form was selected as the most promising equation. Figure 260 shows the plot of “POU vs. ” for this model. The vertical axis in Figure 260 represent the probability that the predicted EC using the selected model is smaller than the actual EC, in percentage. To reach a 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 the Group 48 should be used for soils with D50 ranging from 0.074 to 0.3 mm. Table 80. Selected “Linear” models for critical shear stress in the HET/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 4 PI, WC 0.569 0.07 11.94 0.79 0.17 6.29 0.04 5 PI, , WC 0.557 0.12 0.05 9.41 0.80 0.16 4.58 0.11 7 PI, , PF 0.014 0.21 0.09 3.6 0.78 0.17 6.29 0.13 10 PI, WC, PF 0.015 0.07 0.09 1.1 0.79 0.17 4.6 0.04 15 , WC, PF 0.128 0.02 0.05 3.38 0.62 0.25 4.59 0.55 26 PI, , WC, PF 0.014 0.12 0.05 0.092.78 0.80 0.16 3.11 0.10 48 Cc, , WC 0.138 0.05 0.0037 2.35 0.77 0.17 4.58 0.70 50 Cc, , PF 0.004 0.04 0.02 0.92 0.77 0.17 4.59 0.73 51 Cc, , D50 0.145 0.04 0.0045 50 2.5 0.77 0.17 4.59 0.73 53 Cc, WC, PF 0.005 0.004 0.04 1.55 0.77 0.17 4.56 0.71 54 Cc, WC, D50 0.177 0.004 0.0055 50 3.41 0.77 0.17 4.57 0.71 57 PF, Cc, D50 0.03 0.01 0.0002 50 1.61 0.77 0.17 4.59 0.74 59 Cc, , WC, PF 0.004 0.05 0.0037 0.02 0.89 0.77 0.17 3.13 0.69 60 Cc, , WC, D50 0.138 0.05 0.0037 0.0043 50 2.35 0.78 0.16 3.08 0.69 63 Cc, , PF, D50 0.0044 0.04 0.02 0.0001 50 0.92 0.77 0.16 3.11 0.73 66 Cc, WC, PF, D50 0.0054 0.004 0.03 0.0002 50 1.55 0.76 0.17 3.06 0.71 71 Cc, , WC, PF, D50 0.004 0.05 0.0037 0.024 0.0001 50 0.9 0.77 0.17 2.84 0.69 74 Cu, , WC 0.003 0.05 0.0037 1.3 0.77 0.17 4.58 0.64 76 Cu, , PF 0.003 0.04 0.0003 1.33 0.77 0.17 4.59 0.73 79 Cu, WC, PF 0.003 0.004 0.0003 2.1 0.77 0.17 4.56 0.71 85 Cu, , WC, PF 0.003 0.05 0.0037 0.0003 1.28 0.77 0.16 3.10 0.64 100 Cc, Cu 0.00006 0.003 2.11 0.76 0.16 6.3 0.68 102 Cc, Cu, 0.00005 0.003 0.04 1.34 0.77 0.16 4.45 0.67 103 Cc, Cu, WC 0.0006 0.003 0.004 2.1 0.76 0.17 4.40 0.66

280 Table 81. Selected “Power” models for critical shear stress in the HET/Coarse dataset Group No. Independent Variables Model Expression R 2 MSE F- value/F- stat Cross- Validation Score 4 PI, WC 1675.1 . . 0.79 0.17 419.5 0.51 5 PI, , WC 151.5 . . . 0.80 0.16 329.3 -0.10 7 PI, , PF 7.31 . . . 0.78 0.17 330.1 -0.20 15 , WC, PF 3.47 . . . 0.61 0.26 321.8 0.51 26 PI, , WC, PF 1.848 . . . . 0.80 0.16 294.5 -1.19 48 Cc, , WC 1.045 . . . 0.77 0.17 328.6 0.78 50 Cc, , PF 27.15 . . . 0.77 0.17 328.6 0.75 51 Cc, , D50 1.23 . . 50 . 0.77 0.17 328.8 -22.7 57 PF, Cc, D50 0.272 . . 50 . 0.76 0.17 323.5 0.65 59 Cc, , WC, PF 5.834 . . . . 0.77 0.16 291.5 0.67 60 Cc, , WC, D50 1.16 . . . 50 . 0.77 0.17 292.6 -18.62 71 Cc, , WC, PF, D50 0.06 . . . . 50 . 0.77 0.17 266.4 0.12 74 Cu, , WC 0.45 . . . 0.77 0.17 329.5 0.66 76 Cu, , PF 0.98 . . . 0.77 0.17 328.3 0.55 77 Cu, , D50 0.02 . . 50 . 0.77 0.17 330.1 0.62 79 Cu, WC, PF 34.54 . . . 0.76 0.17 330.1 0.54 83 PF, Cu, D50 0.21 . . 50 . 0.76 0.17 326.6 -8.3 85 Cu, , WC, PF 9.902 . . . . 0.77 0.17 329.5 0.65 86 Cu, , WC, D50 3.97 . . . 50 . 0.77 0.16 295.6 0.55 97 Cu, , WC, PF, D50 0.19 . . . . 50 . 0.77 0.17 258.1 0.55 102 Cc, Cu, 0.006 . . . 0.77 0.17 328.9 0.53 103 Cc, Cu, WC 3.42 . . . 0.76 0.17 329.5 0.57 105 Cc, Cu, PF 0.0032 . . . 0.76 0.17 329.5 -9.5

281 Figure 260. Plot of POU vs. correction factor for the Group 48 (Power) - EC in the HET/Coarse dataset 7.4. Probabilistic (Bayesian) Analysis The preceding section focused on selecting the ‘optimal’ vector of model parameters that maximize the likelihood of fitting experimental observations. This section introduces a series of Bayesian probabilistic calibrations carried out on a set of regression (empirical) models proposed as ‘optimal’ to capture only the model’s 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) when presented to physical evidence generated by the same erosion tests presented in the deterministic regression analysis discussed in the previous section. That is, Bayesian regression allows to quantify varying uncertainty scenarios resulting from different sources of evidence (i.e. varying experimental observations, varying model complexity and varying expert’s judgment), providing further inferences to better understand the performance of a given regression model. This is achieved by providing a full characterization of all possible solutions of the model parameters and their relative probabilities while simultaneously providing a systematic and transparent approach to assess the performance of the proposed regression model. The major difference between the Bayesian analysis and deterministic frequentists’ regression approach is in how each one of them interprets the observed data and model parameters. Frequentists’ regression assumes that the model parameters are in fact fixed unknown parameters, and the observed data are random repeatable samples. On the other hand, the Bayesian inference approach assumes that the observed data are fixed values, while the model parameters are random parameters. Therefore, a frequentists’ regression approach results in fixed values attributed to the model parameters, whereas the Bayesian approach would result in posterior probability distributions for each model parameter rather than a fixed value.

282 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 to assess 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 predicting model. The Bayesian approach departs from standard deterministic calibrations (i.e. least squares) by populating “all” the likely combinations of parameters of a predicting model suitable to represent the mean of the process of interest (i.e. fit of a given set of experimental observations), as opposed to proposing a single parameters combination (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 pdf (i.e. each model parameter is represented by its own pdf), and a full probabilistic description of the parameters’ correlation structure (when taken two at the time). It should also be noted that the concepts of “confidence interval” and “hypothesis testing” in frequentists’ regression and Bayesian inference are different. As an illustration, when in interpreting the results of a frequentists’ regression approach, the confidence interval for a parameter is reported as 90%, it 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 intervals 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 typically called: the “credible interval” in the Bayesian inference) for a parameter, that means there is 90% chance that the credible interval contains the parameter. Motivation One main objective of the TAMU-Erosion spreadsheet (Chapter 5) is to assimilate information collected from a broad range of erodibility tests and correlating those results with primary soil geotechnical properties through statistical modeling. However, regardless of the level of model sophistication, broadness of experimental observations, or accuracy of expert’s judgement, there is a finite amount of uncertainty associated with every mathematical representation – none of the modeling results can predict the property of interest with complete confidence. This is due to several practical limitations:  The effect of ‘missing data’ from the database may introduce bias on the proper model calibration;  The proposed empirical models do not capture the ‘mean of the process of interest’, or does not show a proper correlation between erodibility and geotechnical parameters;  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); Understanding this varying evidence condition motivates a specialized calibration that points to systematic and transparent assessment of prediction confidence based on all available evidence.

283 Hypotheses The foregoing regression analysis has selected several groups of experimental observations and corresponding statistical models that showed satisfying model/observation comparisons. Herein, the proposed Probabilistic (Bayesian) Calibration Method (Medina-Cetina, 2006) is introduced to complement the previous deterministic regression approach. Several hypotheses are proposed and intent to be thoroughly discussed in the following chapter:  The probabilistic approach allows a full characterization of sources and propagation of model uncertainty, and their relative probabilities, through systematic assimilation of available evidences (i.e. experimental observations, model prediction and expert’s judgment).  Varying estimation confidence levels of parameters can exhibit in the parametric space.  Varying correlation structures among parameters can be shown in the calibration results, which will help reveals the model nature.  The probabilistic characteristics of the proposed models will be depicted in multi- dimensional ‘physical domains’ (comprised of model independent variables), the goodness of modeling does not solely depend on capturing each available observation (erodibility test results), but also on the confidence levels of estimations. Methodology Uncertainty quantification framework The uncertainty quantification (UQ) 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 which based on the definition of the physical process at prescribed control points. In practice, experimental observations, , can be retrieved from lab or field measurements, and compared to predicting outcomes of the same process, . These are quantifiable vectors formed as a-priori information used to approximate the true process, . Accordingly, the uncertainty quantification framework in terms of , and can be summarized by: ∆ ∆ ∆ ∆ (72) Eq. (72) illustrates the tradeoff between the scientific evidence and through the gradients, ∆ . The involved uncertainty incorporates random vectors both in experimental observations and theoretical predictions. Herein, is comprised of mathematical predictions stemming from a forward model , which is governed or characterized by a set of parameters, . Noticed that can represent geometric and statistical properties, such as shape parameters or

284 linear/power parameters in this case. Similarly, the uncertainty involved in the calibration of vectors can be defined as: ∆ (73) Where denotes mean of parameters and ∆ represent 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 populated, which ultimately be translated in a better understanding of uncertainty inherent to model nature (∆ ) with respect to the experimental observations. Noticed that in this work, the model prediction assumed to be unbiased with respect to process of interest, which implies ∆ ∆ 0. Bayesian probabilistic calibration The proposed UQ framework requires assessment of plausible solutions of model parameters conditioned on the data, , which generates the need of a mathematical mapping regarding this inverse problem. From a deterministic standpoint, this can be accomplished by selecting an ‘optimal’ set of model parameters, which maximize the likelihood of fitting . 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 expert’s belief setting up the prior ( ) about model parameters 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 later updated systematically via quantifying likelihood ( | ) between available observations and model parameters . From the basic definition of Bayes theorem: | ,, ∝ | , (74) The posterior | is the probability proportional to the prior and the likelihood, | . This is because of the integral of denominator is a normalizing constant over the parametric space Θ so that the integral of the posterior | 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 Metropolis-Hasting (MH) (Metropolis et al., 1953; Hasting, 1970). This sampling method follows an algorithm that generates values from a posterior distribution and converge to a pre-determined target distribution. 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

285 in the Bayesian inference approach. In this study, the sampling of the posterior is based on a Markov Chain Monte Carlo approach coupled with Metropolis Hastings criteria (MCMC- MH), which makes it possible to draw samples from a proposing distribution to infer the target posterior distributions. Probabilistic calibration for varying data scenarios The probabilistic calibrations discussed in this section are conducted on the proposed model/data scenarios produced by the deterministic regression results populated in the previous section. Table 82 to Table 86 list ‘deterministic regression calibrations’ with the highest goodness of fit for each erodibility variable (i.e., critical shear stress τ , critical velocity , erosion category EC, velocity slope , and shear stress slope 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 only show 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 the Appendix 5 of the appendices report. However, in this section, one case is discussed to illustrate the applicability of the method, labeled with Group No. 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. Table 82. Selected models for critical shear stress Group No. Independent variables Dataset/ No. of data Model expression (parameter values given by deterministic regression) Cross- validation score 124 γ, A, WC, Su, PF, D50 EFA/Fine 44 158.06 . . . . 50 . 0.94 0.66 77 Cu, , D50 EFA/Coarse 28 1.58 . . 50 . 0.93 0.99 113 PC, , WC, Su, D50 JET/Global 28 0.248 1.23 0.21 0.07 36.89 50 31.82 50 0.3 0.50 0.10 19 PI, Su, D50 HET/Global 21 25.07 . . 50 . 50 0.3 0.64 0.43 Table 83. Selected models for critical velocity Group No. Independent variables Dataset/ No. of data Model expression (parameter values given by deterministic regression) Cross- validation score 117 PC, WC, Su, D50 EFA/Fine 46 2.518 10 . . . 50 . 0.80 0.80 44 PI, , VST, PF, D50 EFA/Coarse 10 0.002 0.1 0.01 0.09 13.6 50 7.21 0.074 50 0.3 0.93 0.67

286 Table 84. Selected models for erosion category EC Group No. Independent variables Dataset/ No. of data Model expression (parameter values given by deterministic regression) Cross- validation score 132 A, WC, Su, D50 EFA/Fine 44 0.1933 . . . 50 . 0.55 0.53 117 PC, WC, Su, D50 EFA/Fine 44 0.023 0.03 0.0017 1.845 50 0.8566 0.56 0.43 91 Cu, WC, VST, D50 EFA/Coarse 11 1.12 . . . 50 . 0.074 50 0.3 0.92 0.80 13 PI, , PF JET/Fine 56 0.00375 . . . 0.51 0.47 88 PL, Su, D50 JET/Global 28 0.022 0.0031 5.5 50 3.34 50 0.3 0.70 0.58 12 PI, , Su HET/Fine 21 1.67 . . . 0.70 0.54 Table 85. Selected models for velocity slope Group No. Independent variables Dataset/ No. of data Model expression (parameter values given by deterministic regression) Cross- validation score 86 Cu, , WC, D50 EFA/Coarse 28 88969.4 . . . 50 . 0.86 0.64 Table 86. Selected models for shear stress slope Group No. Independent variables Dataset/ No. of data Model expression (parameter values given by deterministic regression) Cross- validation score 77 Cu, , D50 EFA/Coarse 28 3228.7 . . 50 . 0.91 0.64 40 , WC, PF, D50 HET/Coarse 62 2.951 . . . 50 . 0.86 0.55 Power model for erosion category EC, EFA/Fine dataset The selected case is the power model created to predict erodibility parameter EC, for EFA/Fine dataset (group number 132). Four independent variables are considered: A, WC, Su, and D50 which denote soil activity, water content, undrained shear strength, and mean particle size, respectively. A total of 44 data observations are available where each consist of 4 geotechnical properties as input, and EC as the erodibility property or output. Table 84 provides optimization result of model parameters through non-linear regression. Taking these parameter values, the generated model predictions, along with observed data plotted along each variable domain as

287 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. (a) (b) (c) (d) Figure 261. Experimental observations and model predictions along variable domains, (a) soil activity, (b) water content, (c) undrained shear strength, (d) mean particle size. Step1. Optimization In the proposed probabilistic calibration framework, the optimization result are not only able to provide the initial guess of random parameter values (as presented in Figure 261), but it also can retrieve the shape of the error between model predictions and observations, which leads to the selection of the probability function to be considered for the likelihood. Figure 262 (a) shows histogram and kernel density estimate of residuals of model prediction, a nominal Gaussian distribution of mean around zero seemingly a reasonable fit of empirical distribution, which indicates also that unbiased character of the proposed model. However, the change of bin size and starting position of histogram may result in a variation to its shape. Even though the kernel density estimate can eliminate the effect of starting position of plot, one still need to determine the kernel width which is difficult to follow a systematic approach in practice. Figure 262 (b) indicates a

288 better solution to the problem. By plotting empirical cumulative density function (eCDF) of error, a monotonic increase of probability is able 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 into the Bayesian formulation. A non-informative distribution is considered as a prior for the Bayesian formulation. (a) (b) Figure 262. (a) Histogram and kernel density estimate of error (b) eCDF and Gaussian fit of error Step 2. Probabilistic calibration and convergence diagnose For the sake of a better presentation of the calibration results for the group number 132 (Table 84), each model parameter is named following the variable it serves to, that is β , β , β , and β . Where, refers to the scaling factor at the beginning of the equation. The selected prior for all parameters are vague Gaussian prior, with mean equals zero and standard deviation of 10 . Figure 263 present 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 computed. By the principle of stationary should be achieved for both cumulative mean and standard deviation plots, the burn-in point set as 40,000.

289 (a) (b) (b) (d) (e) Figure 263. Random samples of model parameters, (a) , (b) , (c) , (d) , (e)

290 (a) (b) (c) (d) (e) Figure 264. Cumulative mean of sample sequences for each parameter, (a) , (b) , (c) , (d) , (e)

291 (a) (b) (b) (d) (e) Figure 265. Cumulative standard deviation of sample sequences for each parameter, (a) , (b) , (c) , (d) , (e)

292 Step3. Posterior statistics Once the MCMC posterior sampling reached 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 parameter presented along the figures’ 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, coefficient of variation, 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 is not symmetric and skew to the right (i.e., positive skewness), which yields its mean value bigger 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 parameters cross-correlation investigations, with two at a time. Since ’s pdf is asymmetric, non-linear correlation structure observed between and other parameters. A significant negative correlation is shown between and , indicating the strong association of soil’s water content and activity in predicting erosion category. The scatter plots of posterior samples and Pearson correlation coefficients are given in the upper triangular matrix, which provide direct delineations of correlation type and degree among various parameter combinations. It is worth noting that in Table 87, the coefficients of variation are given as complementary information of parameter variability. Additionally, a 95% HPD region is given for each parameter which denotes the ‘credible range where a given parameter exhibits higher probability of occurrence. Figure 266. Joint relative frequency histogram of model parameters, two at a time.

293 Table 87. Statistics of probabilistic calibrated model parameters Parameters Mean SD CoV mode 95% HPD region (lower and upper bound) 0.27 0.13 0.50 0.19 0.07 0.53 -0.04 0.05 -1.02 -0.04 -0.13 0.05 0.46 0.10 0.22 0.49 0.27 0.66 0.07 0.06 0.77 0.07 -0.03 0.19 -0.12 0.04 -0.34 -0.12 -0.19 -0.04 Step4. Probabilistic realizations and assessment of model performance Finally, the probabilistic calibration approach allows to produce 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. To achieve this goal, it is required to compose a multi-dimensional mesh grid retrieved across all independent variables, enabling the assessment if the model prediction along each domain of all independent variables vs. the dependent variable or erodibility parameter. Each domain range is decided by minimum and maximum values retrieved from experimental observations, besides 10 uniform discretization steps was chosen for each variable to ensure computational efficiency and sufficiency on the inferences at the same time. For example, for the case of group No. 132 there are 4 independent variables and the mesh grid is discretized with 10 steps, meaning 10 10,000 ‘points’ where the hyper-surface produced by the model will be repeatedly evaluated for different parameters’ combinations. That is, from the posterior, 1000 random parameter samples are taken after the burn-in point which produce ensemble of model predictions (10 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-n 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 realizations as opposed to 1 deterministic realization. The progress of model realizations along A and D50 domains shows a distinct trend compared to WC and Su. This is due to the mean of parameters retrieved from posterior are positive for and , whereas negative for and .

294 (a) (b) (b) (d) Figure 267. Model realizations coupling with observed dataset along each variable domain, (a) soil activity, (b) water content, (c) undrained shear strength, (d) mean particle size. First order statistics for this model can be computed along the domain of each independent variable (physical domain) as presented in Figure 268, which show 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 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 No. 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 calibration stemmed from an exhaustive sampling process across all the parametric space (MCMC) as opposed to a limited

295 sampling to assess the deterministic calibration to produce the ‘optimal’ vector of the model parameters. Figure 268 (b) presents the least overall uncertainty among the four independent variables, which shows WC as the best predictor for this group number. (a) (b) (c) (d) Figure 268. Mean and standard deviations of model predictions vs. observed data, (a) soil activity, (b) water content, (c) undrained shear strength, (d) mean particle size. Similar to the POU/POO analyses performed based on 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 equal aspect ratio context, where results laying along the 45-degree line means 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. Boxplots 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 ( ) are computed to produce POU levels as shown in Figure 270 (a). Figure 270 (b) presents the mean and HPD intervals for the same realizations, showing a growing uncertainty inherent in POU with the increase of the correction factor. This is a significant

296 improvement over the previous deterministic plots, since these add a confidence metric on the assessment of the correction factors associated with every POU values. Figure 269. Measured EC vs. Predicted EC, based on optimization and probabilistic calibration results (a) (b) Figure 270. (a) 1,000 realizations of POU vs. correction factor, (b) Mean and HPD intervals of POU Linear model for erosion category EC, EFA/Fine dataset A second probabilistic calibration is presented for the linear model for erodibility parameter EC, EFA/Fine dataset (Group No. 117). The main difference between the two cases discussed here

297 is the proposed regression model used to predict the erosion category EC. Both first and second order statistics for each model parameter are evaluated. Only results after ‘step 2’ of the probabilistic calibration will be presented for this model since convergence of the MCMC posterior sample was also achieved. The matrix of plots presenting the posterior’s joint relative frequency histograms of this model parameters is introduced in Figure 271. A uniform negative correlation between intercept and other parameters is presented at the bottom row, with correlation coefficients range from -0.491 to -0.841. In contrast with power model, all marginal pdfs along the diagonal are following Gaussian shape without skewness, which results from free boundary of possible EC parameter values in linear model (negative values of in the power model are by default not sampled during MCMC). Higher CoV values of and are expected to cause bigger variations along D50 and Su domains which will be examined through realizations and statistical analysis in each variable domain. Figure 271. Joint relative frequency histogram of model parameters, two at a time Table 88. Model characteristics and optimization result Group No. Independent variables Dataset/ No. of data Model expression (parameter values given by deterministic regression) Cross- validation score 117 PC, WC, Su, D50 EFA/Fine 44 0.023 0.03 0.0017 1.845 50 0.8566 0.56 0.43

298 Table 89. Statistics of probabilistic calibrated model parameters Parameters Mean SD CoV mode 95% HPD region (lower and upper bound) 0.021 0.008 0.378 0.021 0.006 0.037 0.031 0.005 0.160 0.030 0.021 0.040 0.002 0.003 1.313 0.002 -0.003 0.007 -2.396 5.349 -2.232 -1.137 -12.858 8.083 0.922 0.440 0.477 1.004 0.053 1.770 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 a significant agreement. In addition, comparison of HPD intervals for linear and power models indicates uncertainty is more uniformly distributed in both domains and co-domains for the former, and also quantitatively smaller. However, it is worth noticing that by using the linear model, it may yield negative values of EC that might not make sense in reality. A rational choice in such case would be power model instead or to further restrict the variation of EC during the MCMC sampling process of the posterior.

299 Figure 272. Model realizations coupling with observed dataset along each variable domain, (a) percent clay, (b) water content, (c) undrained shear strength, (d) mean particle size Figure 273. Mean and standard deviations of model predictions vs. observed data, (a) percent clay, (b) water content, (c) undrained shear strength, (d) mean particle size 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 regards to the plots of “POU vs. ”, 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 equals to 0.80 for both scenarios.

300 Figure 274. Measured EC vs. Predicted EC, based on optimization and probabilistic calibration results (a) (b) Figure 275. (a) 1,000 realizations of POU vs. correction factor, (b) Mean and HPD intervals of POU

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Relationship Between Erodibility and Properties of Soils Get This Book
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TRB’s National Cooperative Highway Research Program (NCHRP) has released the pre-publication version of NCHRP Research Report 915: Relationship Between Erodibility and Properties of Soils, which provides reliable and simple equations quantifying the erodibility of soils based on soil properties.

The report a detailed analysis of the issue. In addition, the project that developed the report also produced a searchable spreadsheet that uses statistical techniques to relate geotechnical properties to soil erodibility. The spreadsheet, NCHRP Erosion, includes a searchable database that includes compiled erosion data from the literature review and a plethora of erosion tests. It contains equations which may be used to estimate the erosion resistance of soil and determine whether erosion tests are needed.

The following appendices to NCHRP Report 915 were published online in a single Appendices Report.

Appendix 1 – Erosion Test Results Spreadsheets

Appendix 2 – Geotechnical Properties Spreadsheets

Appendix 3 – First and Second Order Statically Analysis Results

Appendix 4 – Deterministic Frequentists’ Regression Analysis

Appendix 5 – Probabilistic Calibration Results

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