USGS Debris Flow Likelihood Model

Inputs

• S: Slope (in degrees).
• R: Rainfall intensity (mm/hr).
• B: Burn severity index (unitless, typically normalized).
• A: Area of the watershed (km²).
• T: Terrain ruggedness (unitless).

Probability of Debris Flow

The likelihood of debris flow Pdf for a given cell or watershed can be computed as:

Pdf(i,j) = σ(α S(i,j) + β R(i,j) + γ B(i,j) + δ A(i,j) + ε T(i,j) + η)
    

Where:

• (i,j) are the spatial indices of the cell.
• σ(x) = 1 / (1 + e-x): Sigmoid function for mapping to probability.
• α, β, γ, δ, ε, η: Coefficients calibrated from historical data.

Flow Accumulation

To model the cumulative debris flow likelihood downstream, use a recursive accumulation:

Ldf(i,j) = Pdf(i,j) + Σ(m,n) ∈ upstream(i,j) wm,n ⋅ Ldf(m,n)
    

Where:

• upstream(i,j): Set of upstream neighbors of cell (i,j).
• wm,n: Weight based on contributing area or flow fraction.

Implementation Notes

• Input grids S, R, B, A, T are raster datasets.
• The model is typically implemented using a hydrological flow direction map to guide accumulation.
• Parameters α, β, γ, δ, ε, η are derived through regression on observed debris flow events.