A probabilistic framework for assessing vulnerability to climate variability and change: the case of the US water supply system

Abstract

We introduce a probabilistic framework for vulnerability analysis and use it to quantify current and future vulnerability of the US water supply system. We also determine the contributions of hydro-climatic and socio-economic drivers to the changes in projected vulnerability. For all scenarios and global climate models examined, the US Southwest including California and the southern Great Plains was consistently found to be the most vulnerable. For most of the US, the largest contributions to changes in vulnerability come from changes in supply. However, for some areas of the West changes in vulnerability are caused mainly by changes in demand. These changes in supply and demand result mainly from changes in evapotranspiration rather than from changes in precipitation. Importantly, changes in vulnerability from projected changes in the standard deviations of precipitation and evapotranspiration are of about the same magnitude or larger than those from changes in the corresponding means over most of the US, except in large areas of the Great Plains, in central California and southern and central Texas.

Appendix A - vulnerability analysis

In general, vulnerability of water supply to shortage may be defined as the probability that the supply is less than the demand; that is,

$$ V= Pr\left[S<D\right]= Pr\left[S-D<0\right] $$

(A.1)

where S is water supply, and D is water demand. Also, in general,

$$ S=P-E+I+{Q}_{div}+ SV $$

(A.2)

where P is precipitation, E is actual evapotranspiration, I is the net input from upstream, Q _div is the net diversions (difference between diversions into and diversions out of the system), and SV is reservoir and other storage.

Defining the surplus, Z, as the difference between supply and demand, Eq. (A.1) can be rewritten as,

$$ V= Pr\left[Z<0\right]= Pr\left[\frac{Z-{\mu}_Z}{\sigma_Z}<-\frac{\mu_Z}{\sigma_Z}=-\beta \right] $$

(A.3)

where μ _Z  = μ _S  − μ _D , σ ² _Z  = σ ² _S  + σ ² _D  − 2cov(S, D), β = μ _Z /σ _Z , and μ _S , μ _D, σ _S , and σ _D, cov(S, D), are the mean, standard deviation and covariance of water supply and water demand.

Therefore, as is clear from Eq. (A.3), the vulnerability of water supply as defined in Eq. (A.1) is a function of the mean, standard deviation and covariance of supply and water demand, that is, μ _S , μ _D, σ _S , and σ _D, cov(S, D).

We may then express the total change in vulnerability, dV, as a function of the individual contributions of changes in μ _S , μ _D, σ _S , and σ _D, cov(S, D), as follows,

$$ dV=\frac{\partial V}{\partial {\mu}_S}d{\mu}_S+\frac{\partial V}{\partial {\mu}_D}d{\mu}_D+\frac{\partial V}{\partial {\sigma}_S}d{\sigma}_S+\frac{\partial V}{\partial {\sigma}_D}d{\sigma}_D+\frac{\partial V}{\partial cov\left(S,D\right)} dcov\left(S,D\right) $$

(A.4)

where each of the partial derivatives represents the sensitivity of the vulnerability to unit changes in each of the independent variables μ _S , μ _D, σ _S , and σ _D, cov(S, D). In addition, each of the five terms of Eq. (A.4) represents the contribution to the total change in vulnerability resulting from the changes in μ _S , μ _D, σ _S , and σ _D, cov(S, D).

For the case of normally distributed surplus, Eq. (A.1) can be rewritten as,

$$ V= Pr\left[Z<0\right]= Pr\left[\frac{Z-{\mu}_Z}{\sigma_Z}<-\frac{\mu_Z}{\sigma_Z}=-\beta \right]={\left[2\pi {\sigma}_Z^2\right]}^{-0.5}{\displaystyle \underset{-\infty }{\overset{0}{\int }}{e}^{-\frac{{\left(z-{\mu}_Z\right)}^2}{2{\sigma}_Z^2}}} dz $$

(A.5)

In the case of non-Gaussian Z, Eq. (A.5) corresponds to a First Order Second Moment approximation.

Carrying out the integral of Eq. (A.5) yields,

$$ V\left({\mu}_S,{\mu}_D,{\sigma}_Z\right)=\frac{1}{2}+\frac{1}{2} erf\left[\frac{\left(-{\mu}_S+{\mu}_D\right)}{\sqrt{2{\sigma}_Z^2}}\right] $$

(A.6)

where erf() is the Gauss error function (also known as the probability integral).

The partial derivatives appearing in Eq. (A.4) are obtained by differentiating Eq. (A.6) with respect to μ _S , μ _D, σ _S , and σ _D, cov(S, D). The resulting derivatives are shown in Eqs. (A.7–A.11).

$$ \frac{\partial V}{\partial {\mu}_S}=-{\left[2\pi \left({\sigma}_Z^2\right)\right]}^{-0.5}{e}^{-\frac{{\left(-{\mu}_S+{\mu}_D\right)}^2}{2{\sigma}_Z^2}} $$

(A.7)

$$ \frac{\partial V}{\partial {\mu}_D}={\left[2\pi \left({\sigma}_Z^2\right)\right]}^{-0.5}{e}^{-\frac{{\left(-{\mu}_S+{\mu}_D\right)}^2}{2{\sigma}_Z^2}} $$

(A.8)

$$ \frac{\partial V}{\partial {\sigma}_S}=-{\sigma}_S\left(-{\mu}_S+{\mu}_D\right){\left[2\pi {\left({\sigma}_Z^2\right)}^3\right]}^{-0.5}{e}^{-\frac{{\left(-{\mu}_S+{\mu}_D\right)}^2}{2{\sigma}_Z^2}} $$

(A.9)

$$ \frac{\partial V}{\partial {\sigma}_D}=-{\sigma}_D\left(-{\mu}_S+{\mu}_D\right){\left[2\pi {\left({\sigma}_Z^2\right)}^3\right]}^{-0.5}{e}^{-\frac{{\left(-{\mu}_S+{\mu}_D\right)}^2}{2{\sigma}_Z^2}} $$

(A.10)

$$ \frac{\partial V}{\partial cov\left(S,D\right)}=\left(-{\mu}_S+{\mu}_D\right){\left[2\pi {\left({\sigma}_Z^2\right)}^3\right]}^{-0.5}{e}^{-\frac{{\left(-{\mu}_S+{\mu}_D\right)}^2}{2{\sigma}_Z^2}} $$

(A.11)

Although not included here for brevity, the above analysis can be easily extended to define changes in vulnerability as a function of changes in the probabilistic characteristics of P, and E, explicitly.