 # Regression Equations

The volume-based approach of the model involves the following controls. Initiation occurs for a user-defined initial failure volume in the first reach of the event. Thereafter, the morphology of each subsequent reach determines the flow behaviour (UF, CF or TF), and slope angle of the reach determines the mode of flow (entrainment +dV, or deposition -dV). The volume of entrainment and (or) deposition is calculated using regression equations. Accordingly, the cumulative flow volume ΣVi at each reach i can be determined from the reach morphology (unconfined, confined, or transition flow), and a set of measured and derived predictor variables for the reach. The measured predictor variables are :

Length of the reach, Li
Width of the entrainment or deposition, Wi
Slope angle of the reach, THi

The derived predictor variables are :

Incoming flow volume, ΣVi-1
Bend-angle function, BAFi

The BAFi is determined by :

   BAFi = cos(dTHi) cos(dAZi) ln(ΣVi-1)

where,

   dTHi = | THi - THi-1 |

   dAZi = | AZi - AZi-1 | The change in flow volume is determined by one of five equations, depending on the slope angle and flow type. Each equation determines the change in volume for a reach i. This change is negative in the case of deposition (flow volume decreases) and positive in the case of entrainment (flow volume increases). ## Unconfined flow (UF)

For unconfined flow on a slope angle between 29.5° and 55°, the flow will entrain a volume dVi, where dVi can be determined from :

   ln(dVi) = 0.728 + 1.31ln(Wi) + 0.742ln(Li) - 0.0464(THi)

For unconfined flow on a slope angle between 18.5° and 29.5°, the flow will entrain a volume dVi, where dVi can be determined from :

   ln(dVi) = 1.13ln(Wi) + 0.787ln(Li) - 0.0636ln(ΣVi-1)

For unconfined flow on a slope angle less than 18.5°, the flow will deposit a volume dVi, where dVi can be determined from :

   ln(dVi) = 0.514 + 0.988ln(Wi) + 0.101(BAFi) + 0.731ln(Li) - 0.0155(THi) The Blueberry Creek flow began with an unconfined flow on an open slope, and then became a confined flow once it entered the forested area.

## Confined flow (CF)

For confined flow on a slope angle between 10.5° and 55°, the flow will entrain a volume dVi, where dVi can be determined from :

   ln(dVi) = 0.344 + 0.851ln(Wi) + 0.898ln(Li) - 0.0162(THi)

For confined flow on a slope angle less than 10.5°, there is no change in flow volume (neither deposition nor entrainment).

## Transition flow (TF)

For transition flow on a slope angle greater than 20.5°, there is no change in flow volume (neither deposition nor entrainment).

For transition flow on a slope angle less than 20.5°, the flow will deposit a volume dVi, where dVi can be determined from :

   ln(dVi) = 1.54ln(Wi) + 0.90ln(Li) - 0.123(BAFi)