# Scientific Background (Fires prior to 2016)

## Science Center Objects

The scientific background information for fires prior to 2016 for Emergency Assessment of Post-Fire Debris-Flow Hazards.

The preliminary hazard assessment relies upon empirical models to estimate the probability and volume of debris flows for selected basins in response to a design storm. The design storm is based on precipitation-frequency estimates for the burn area, which are used to estimate the storm recurrence interval (Bonnin and others, 2006; Miller and others, 1973). For example, a 10-year recurrence interval rainstorm is expected to have a 10% chance of happening in any given year.

The empirical models are based upon historical debris-flow occurrence and magnitude data, rainfall storm conditions, terrain and soils information, and burn-severity data from recently burned areas. Separate models have been developed for the regions of southern California (Rupert and others, 2008; Cannon, S.H., 2011, U.S. Geological Survey, unpublished data; Gartner, J.E., 2013, U.S. Geological Survey, unpublished data) and the intermountain western United States (Gartner and others, 2008; Cannon and others, 2010).

Post-fire debris-flow probability, volume, and combined hazards are estimated at both the drainage-basin scale and in a spatially distributed manner along the drainage network within each basin. The characteristics of basins affected by the fire were calculated using a geographic information system (GIS). Debris-flow probability and volume were estimated for each basin outlet as well as along the upstream drainage networks (pixels where the contributing area is greater than or equal to 0.02 km^{2}) using a method that has been applied in recently burned areas (for example, Verdin and others, 2012). Independent variable values were calculated for each pixel along the drainage network and summarized at the stream segment scale to obtain estimates of debris-flow probability and volume.

**Probability Model**

Probability estimates are based upon logistic regression models derived from region-specific databases. This model is designed to predict the probability of debris-flow occurrence at a point along the drainage network in response to a given storm by combining the following two equations:

(1) P = e^{x} / (1 + e^{x})

**Where**

- P is the probability of debris-flow occurrence in fractional form, and
- e
^{x}is the exponential function where e represents the mathematical constant 2.718.

For recently burned areas in southern California, equation 2 is used to calculate x:

(2) x = -5.22 + (0.003 × ElevRange) + (0.008 × HM50_{pct}) + (0.024 × bslp_{pct}) + (-0.007 × CC_{pct}) + (0.105 × i30)

**Where**

- ElevRange - is the range (maximum elevation–minimum elevation) of elevation values upstream of the point (in meters),
- HM50
_{pct}- is the percentage of the upstream watershed that was burned at high or moderate severity and has slope values in excess of 50 percent (in percent), - bslp
_{pct}- is the average gradient of the burned pixels upslope of the point (in percent), - CC
_{pct}- is the average clay content of the soils in the basin (in percent) (Schwartz and Alexander, 1995), and - i30 - is the spatially averaged upslope 30-min rainfall intensity for the design storm (in millimeters per hour [mm/h]).

Probabilities predicted by the equation potentially range from 0 (least likely) to 100 percent (most likely). The predicted probabilities are assigned to 1 of 5 equal (20 percent) interval classes for cartographic display.

For recently burned areas in the intermountain western United States, equation 3 is used to calculate x:

(3) x = -0.7 + (0.03 × Slp30_{pct}) - (1.6 × Rugged) + (0.06 × HM_{pct}) + (0.2 × CC_{pct}) – (0.4 × LL_{pct}) + (0.07 × i60)

**Where**

- Slp30
_{pct}– is the percentage area of the upstream area with slope gradients in excess of 30 percent (in percent), - Rugged – is the upslope ruggedness, which is equal to the total relief (in meters) of the upstream area divided by the square root of the total upslope area above the pixel (in square meters),
- HM
_{pct}- is the percentage of the upstream watershed that was burned at high or moderate severity (in percent), - CC
_{pct}- is the average clay content of the soils in the basin (in percent) (Schwartz and Alexander, 1995), - LL
_{pct}- is the average liquid limit of the soils in the basin (in percent) (Schwartz and Alexander, 1995), and - i60 - is the spatially averaged upslope 60-min rainfall intensity for the design storm (in millimeters per hour [mm/h]).

**Volume Model**

Debris-flow volumes both at the basin outlet and along the drainage network are predicted using multiple linear regression models for region-specific databases. The multiple linear regression models are used to estimate the volume (V, in m^{3}) of material that could issue from a point along the drainage network in response to a storm of a given rainfall intensity.

For recently burned areas in southern California, debris-flow volume is calculated with equation 4:

(4) ln(V) = 2.89 + (0.17 × sqrt(ElevRange)) + (0.3 × ln(HM_{km}) + (0.47 × sqrt(i15))

**Where**

- ElevRange is the range (maximum elevation–minimum elevation) of elevation values within the upstream watershed (in meters),
- HM
_{km}is the area upstream of the calculation point that was burned at high or moderate severity (in km^{2}), and - i15 is the spatially-average peak 15-min rainfall intensity for the design storm in the upstream watershed (in mm/h).

Volume estimates were classified in order of magnitude scale ranges 0–1,000 m^{3}; 1,000–10,000 m^{3}; 10,000–100,000 m^{3}; and greater than 100,000 m^{3} for cartographic display.

For recently burned areas in the intermountain western United States, debris-flow volume is calculated with equation 5:

(5) ln(V) = 7.5 + (0.6 × ln(Slp30_{km})) + (0.7 × sqrt(HM_{km})) + (0.2 × sqrt(r60))

**Where**

- Slp30
_{km}is the area upstream that has slope gradients in excess of 30 percent (in km^{2}), - HM
_{km}is the area upstream of the calculation point that was burned at high or moderate severity (in km^{2}), and - r60 is the spatially averaged 60-minute rainfall accumulation for the design storm in the upstream watershed (in mm).

**Combined Hazard**

Debris-flow hazards from a given basin can be considered as the combination of both probability and volume. For example, in a given setting, the most hazardous basins will show both a high probability of occurrence and a large estimated volume of material. Slightly less hazardous would be basins that show a combination of either relatively low probabilities and larger volume estimates or high probabilities and smaller volume estimates. The lowest relative hazard would be for basins that show both low probabilities and the smallest volumes.

We combined the results of the probability and the volume maps following the methods of Cannon and others (2010). A rank of 1 to 5 (with 5 being the highest) is assigned to each of the probability classes, and a rank of 1 to 4 is assigned to each of the volume classes. The ranks of the probability and volume classes are then added together to produce a map of the combined relative hazard ranking for each basin (with 9 being the highest combined hazard).

**References**

- Bonnin, G.M., Martin, D., Lin, B., Parzybok, T., Yekta, M., and Riley, D., 2006,Precipitation frequency atlas of the United States: Silver Spring, Md., National Weather Service, National Oceanic and Atmospheric Administration (NOAA) atlas 14, v. 1, version 5, accessed July 30, 2013, athttp://hdsc.nws.noaa.gov/hdsc/pfds/.
- Miller, J.F., Frederick, R.H., and Tracey, R.J., 1973, Precipitation frequency atlas of the western United States: United States: Silver Spring, Md., National Weather Service, National Oceanic and Atmospheric Administration (NOAA) atlas 2.
- Schwartz, G.E., and Alexander, R.B., 1995, Soils data for the conterminous United States derived from the NRCS State Soil Geographic (STATSGO) Database: U.S. Geological Survey Open-File Report 95–449, accessed July 2013, at https://water.usgs.gov/GIS/metadata/usgswrd/XML/ussoils.xml.