Modelling the effects of fire and rainfall regimes on extreme erosion events in forested landscapes

Abstract

Existing models of post-fire erosion have focused primarily on using empirical or deterministic approaches to predict the magnitude of response from catchments given some initial rainfall and burn conditions. These models are concerned with reducing uncertainties associated with hydro-geomorphic transfer processes and typically operate at event timescales. There have been relatively few attempts at modelling the stochastic interplay between fire disturbance and rainfall as factors which determine the frequency and severity with which catchments are conditioned (or primed) for a hazardous event. This process is sensitive to non-stationarity in fire and rainfall regime parameters and therefore suitable for evaluating the effects of climate change and strategic fire management on hydro-geomorphic hazards from burnt areas. In this paper we ask the question, “What is the first-order effect of climate change on the interaction between fire disturbance and storms?” The aim is to isolate the effects of fire and rainfall regimes on the frequency of extreme erosion events. Fire disturbance and storms are represented as independent stochastic processes with properties of spatial extent, temporal duration, and frequency of occurrence, and used in a germ–grain model to quantify the annual area affected by extreme erosion events due to the intersection of fire disturbance and storms. The model indicates that the frequency of extreme erosion events will increase as a result of climate change, although regions with frequent storms were most sensitive.

Appendix: Germ and grain model

Suppose that we have two independent Boolean models in R^k. That is, let $\left\{ {\xi_{i} } \right\}$ and $\left\{ {\zeta_{i} } \right\}$ be independent stationary Poisson processes with intensities $\lambda_{\xi }$ and $\lambda_{\zeta }$, and let $\left\{ {X_{i} } \right\}$ and $\left\{ {Y_{i} } \right\}$ be mutually independent i.i.d. sequences of random sets, then our two models are ${\mathcal{X}} = \left\{ {\xi_{i} + X_{i} } \right\}$ and ${\mathcal{Y}} = \left\{ {\zeta_{i} + Y_{i} } \right\}$. Let Ω be a Borel subset of R^k then the intersection of Ω, $\mathcal{X}$ and $\mathcal{Y}$ is given by $A = \Omega \cap \left( { \cup_{i} \xi_{i} + X_{i} } \right) \cap \left( { \cup_{i} \xi_{i} + Y_{i} } \right)$. Let $\left\| A \right\|$ denote the content (Lebesgue measure) of A, then we have:

1.1 Proposition

If $\left\| \Omega \right\| < \infty^{{}}$ then ${\text{E}}\left\| A \right\| = \left\| \Omega \right\|\left( {1 - e^{{ - \lambda_{\xi} {\text{E}}\left\| {{X}} \right\|}} } \right)\left( {1 - e^{{ - \lambda_{\zeta} {\text{E}}\left\| {\text{Y}} \right\|}} } \right)$, where X and Y are random sets, distributed as the X _i and Y _i respectively. Moreover, let $\alpha_X\left( x \right) = {\text{E}}\left\| {\left( {x + X} \right) \cap X} \right\|$ and $\alpha_Y\left( x \right) = {\text{E}}\left\| {\left( {x + Y} \right) \cap Y} \right\|$, then

$${\text{Var}}\left\| A \right\| = \int\limits_{\Omega } {\int\limits_{\Omega }} \left( {\left( {1 - e^{ - \lambda_{\xi} \alpha_{X}(0)} } \right)^{2} e^{ - 2\lambda_{\zeta} \alpha_{Y}(0)} \left( {e^{{\lambda_{\zeta} \alpha_{Y} \left( {x - y} \right)}} - 1} \right)} + \left( {1 - e^{ - \lambda_{\zeta} \alpha_{Y} (0)} } \right)^{2} e^{ - 2\lambda_{\xi} \alpha_X(0)} \left( {e^{{\lambda_{\xi} \alpha_X\left( {x - y} \right)}} - 1} \right) + e^{{ - 2\left( {\lambda_{\xi} \alpha_X(0) + \lambda_{\zeta} \alpha_Y(0)} \right)}} \left( {e^{{\lambda_{\xi} \alpha_X\left( {x - y} \right)}} - 1} \right) \left( {e^{{\lambda_{\zeta} \alpha_{Y} \left( {x - y} \right)}} - 1} \right) \right) dx\;dy. $$

1.2 Proof

Let $1_{\mathcal {X}} \left( x \right) = \left\{ {\begin{array}{*{20}c} 1 & {x \in \cup_{i} \xi_{i} + X_{i} } \\ 0 & {\text{otherwise}} \\ \end{array} ,1_{\mathcal {Y}}\left( x \right)\left\{ {\begin{array}{*{20}c} 1 & {x \in \cup_{i} \zeta_{i} + Y_{i} } \\ 0 & {\text{otherwise}} \\ \end{array} } \right.} \right.$, then from Hall (1988) Equation (3.4) we have

$${\text{E}}\left\| A \right\| ={\text{E}}\int\limits_{\Omega}{1_{\mathcal{X}}\left(x \right)1_{\mathcal{Y}}\left(x \right)dx\;\; =\;\;\int\limits_{\Omega} {{\text{E1}_{{\mathcal{X}}}}\left(x\right){\text{E1}_{{\mathcal{Y}}}}\left(x \right)}} dx \hfill =\int\limits_{\Omega}{{\text{P}}\left({x\;{\text{covered}}\;{\text{by}}\;{\mathcal{X}}}\right){\text{P}}\left({x\;{\text{covered}}\;{\text{by}}\;{\mathcal{Y}}}\right)\;\; = \;\;\int\limits_{\Omega} {\left({1 - e^{{-\lambda_{\xi} {\text{E}}\left\| {X} \right\|}}} \right)\left({1 - e^{{- \lambda_{\zeta} {\text{E}}\left\| {Y}\right\|}}} \right)}} = \left\| \Omega \right\|\left({1 -e^{{- \lambda_{\xi} {\text{E}}\left\| {X} \right\|}}}\right)\left({1 - e^{{- \lambda_{\zeta} {\text{E}}\left\| {Y}\right\|}}} \right). $$

Note that the result still holds when ${\text{E}}\left\| X \right\| = \infty$ or ${\text{E}}\left\| Y \right\| = \infty$.

For the variance we note first that

$$ {\text{E}}\left\| A \right\|^{2} \;\; = \;\;{\text{E}}\int\limits_{\Omega} {1_{\mathcal{X}} \left(x \right)1_{\mathcal{Y}} \left(x \right)dx\int\limits_{\Omega} {1_{\mathcal{X}} \left(y \right)1_{\mathcal{Y}} \left(y \right)dy\;\; = \;\;\int\limits_{\Omega} {\int\limits_{\Omega} {{\text{E1}}_{\mathcal{X}} \left(x \right)1_{\mathcal{X}} \left(y \right){\text{E}}1_{\mathcal{Y}} \left(x \right)1_{\mathcal{Y}} \left(y \right)dx\;dy}}}} $$

From Hall (1988) Equation (3.6) and preceding calculations

$$ {\text{E1}}_{\mathcal{X}} \left(x \right)1_{\mathcal{X}} \left(y \right) ={\text{P}}\left({x\;{\text{and}}\;y\;{\text{covered}}\;{\text{by}}\;{\mathcal{X}}} \right) = 1 - {\text{P}}\left({x\;{\text{not}}\;{\text{covered}}\;{\text{by}}\;\mathcal{X}} \right) - {\text{P}}\left({y\;{\text{not}}\;{\text{covered}}\;{\text{by}}\;\mathcal{X}} \right) + {\text{P}}\left({{\text{neither}}\;x\;{\text{nor}}\;y\;{\text{covered}}\;{\text{by}}\;\mathcal{X}} \right) \hfill = \;1 - 2e^{-\lambda_{\xi} \alpha_X\left(0 \right)} + e^{{- 2\lambda_{\xi} \alpha_X\left({0} \right) + \lambda_{\xi} \alpha_X\left({x - y} \right)}} $$

Thus

$${\rm{Var}}\left\| A \right\| = E{\left\| A \right\|^2} -{(E\left\| A \right\|)^2} = {\int\limits_{\Omega}} {\int\limits_{\Omega}} \left( {{\left( {1 - 2{e^{ - {\lambda _{\xi} }{\alpha _X}(0)}} + {e^{- 2{\lambda _{\xi} }{\alpha _X}(0) + {\lambda _{\xi} }{\alpha _X}(x- y)}}} \right)}}\times{\left( {1 - 2{e^{ - {\lambda _{\zeta}}{\alpha _Y}(0)}} + e^{ - 2\lambda_{\zeta} \alpha_{Y}(0) + \lambda_{\zeta} \alpha _Y\left( {x - y} \right)}} \right)} - \left( {1 - e^{ - \lambda _{\xi}\alpha_{X}(0)}}\right)^{2} \left( {1- e^{ - \lambda_{\zeta} \alpha_{Y}(0)}} \right)^{2} \right) dx\;dy = {\int\limits_{\Omega}} {\int\limits_{\Omega}} \left( {\left( {1 - e^{ - \lambda_{\xi} \alpha _{X}(0)}} \right)^{2} e^{- 2\lambda_{\zeta}\alpha _{Y}(0)} {\left( {e^{{\lambda_{\zeta}\alpha _{Y}\left( {x - y} \right)}} - 1} \right)} + {\left( {1 - e^{ - \lambda_{\zeta}\alpha_{Y}(0)}} \right)^{2}} e^{ - 2\lambda_{\xi}\alpha_{X}(0)} {\left( {e^{{\lambda _{\xi}\alpha_{X}\left( {x - y} \right)}} - 1} \right)} + e^{{ - 2\left( {\lambda_{\xi} \alpha _X(0) + \lambda_{\zeta} \alpha _Y(0)} \right)}} {\left( {e^{{\lambda_{\xi} \alpha _X\left( {x - y} \right)}} - 1} \right)} \left( {e^{\lambda_{\zeta} \alpha_Y \left( {x - y} \right)} - 1} \right)} \right) dx\;dy.$$