U.S. DEPARTMENT OF THE INTERIOR

U.S. GEOLOGICAL SURVEY

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Open-File Report 03-500

Logistic regression was used to predict the probability of debris flows occurring in areas recently burned by wildland fires. Multiple logistic regression is conceptually similar to multiple linear regression because statistical relations between one dependent variable and several independent variables are evaluated. In logistic regression, however, the dependent variable is transformed to a binary variable (debris flow did or did not occur), and the actual probability of the debris flow occurring is statistically modeled. Data from 399 basins located within 15 wildland fires that burned during 2000-2002 in Colorado, Idaho, Montana, and New Mexico were evaluated. More than 35 independent variables describing the burn severity, geology, land surface gradient, rainfall, and soil properties were evaluated. The models were developed as follows: (1) Basins that did and did not produce debris flows were delineated from National Elevation Data using a Geographic Information System (GIS). (2) Data describing the burn severity, geology, land surface gradient, rainfall, and soil properties were determined for each basin. These data were then downloaded to a statistics software package for analysis using logistic regression. (3) Relations between the occurrence/non-occurrence of debris flows and burn severity, geology, land surface gradient, rainfall, and soil properties were evaluated and several preliminary multivariate logistic regression models were constructed. All possible combinations of independent variables were evaluated to determine which combination produced the most effective model. The multivariate model that best predicted the occurrence of debris flows was selected. (4) The multivariate logistic regression model was entered into a GIS, and a map showing the probability of debris flows was constructed. The most effective model incorporates the percentage of each basin with slope greater than 30 percent, percentage of land burned at medium and high burn severity in each basin, particle size sorting, average storm intensity (millimeters per hour), soil organic matter content, soil permeability, and soil drainage. The results of this study demonstrate that logistic regression is a valuable tool for predicting the probability of debris flows occurring in recently-burned landscapes.

Univariate correlations can be a good initial indicator of which independent variables might be significant in the multivariate models. Univariate correlations are usually observed first to explore the data (table 1).

Multivariate models were constructed by evaluating every possible combination of variables shown in table 1. Several logistic regression models were constructed using different combinations of variables, and then the most effective model was selected.

Table 1. Summary of univariate and multivariate correlations between
debris flow occurrence and independent variables describing burn severity,

geology, land surface gradient, rainfall, soil properties, and variable
interactions using data from 15 wildland fires in Colorado, Idaho, Montana,
and

New Mexico, 2000-2002.

[GE, greater than or equal to; mm, millimeters; hr, hour; >, greater
than; < less than; X, denotes statistically significant variable in
multivariate correlations]

(a) Spearman's rho is a nonparametric rank-order test, similar to the
student's t-test. Variables and models with no correlation with debris
flows have spearman's rho numbers near zero, and variables with perfect
correlations have numbers near plus or minus one.

(b) The Wilcoxon test is a nonparametric test to determine if there is
a significant difference between areas with and without debris flows.
As the effectiveness of the models improves, the Wilcoxon p-values approach
zero.

(c) McFadden's rho is calculated by logistic regression, and conceptually
is similar to an r-squared in linear regression. A McFadden's rho value
of zero means there is no correlation; values between 0.2 and 0.4 denote
significant correlation.

Probability of Debris Flow= e (-29.693 + 2.864*SL + 10.697*BS - 9.875*SO
+ 0.208*SI + 5.729*OM - 0.957*P + 9.351*D - 8.335*BS*OM + 4.669*SO*D -
0.174*SL*SI)

(-29.693 + 2.864*SL + 10.697*BS - 9.875*SO + 0.208*SI + 5.729*OM - 0.957*P
+ 9.351*D - 8.335*BS*OM + 4.669*SO*D - 0.174*SL*SI)1 + e

Where:

SL = Percent of basin with slope greater than or equal to 30 percent

BS = Burn severity category

SO = Partical size sorting, in Phi units

SI = Average storm intensity, in millimeters per hour

OM = Soil organic matter, in percent by weight

P = Soil permeability, in inches per hour

D = Soil drainage category

Hosmer, D.W., and Lemeshow, S., 1989, Applied logistic regression: New
York, John Wiley & Sons, Inc., 307 p.

Schwarz, G.E. and Alexander, R.B. 1995, State Soil Geographic (STATSGO)
Data base for the conterminous United States: U.S. Geological Survey Open-File
Report 95-449, http://water.usgs.gov/lookup/getspatial?ussoils

Montana Bureau of Mines and Geology, U.S.D.A. Forest Service, U.S. Geological Survey

Figure 1. Basins with and without debris flows, Missionary Ridge Fire, 2002, near Durange, Colorado.

Figure 2. Burn severity at the Missionary Ridge Fire, 2002.

Figure 3. Probability of a debris flow occurring in response to a 25-year rain event (33 mm in one hour), Missionary Ridge Fire, 2002, using model M1.

Table 1. Summary of univariate and multivariate correlations between debris flow occurrence and independent variables describing burn severity, geology, land surface gradient, rainfall, soil properties, and variable interactions using data from 15 wildland fires in Colorado, Idaho, Montana, and New Mexico, 2000-2002.

Water
Resources of Colorado

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