Projecting Flood-Inducing Precipitation with a Bayesian Analogue Model

Abstract

The hazard of pluvial flooding is largely influenced by the spatial and temporal dependence characteristics of precipitation. When extreme precipitation possesses strong spatial dependence, the risk of flooding is amplified due to catchment factors such as topography that cause runoff accumulation. Temporal dependence can also increase flood risk as storm water drainage systems operating at capacity can be overwhelmed by heavy precipitation occurring over multiple days. While transformed Gaussian processes are common choices for modeling precipitation, their weak tail dependence may lead to underestimation of flood risk. Extreme value models such as the generalized Pareto processes for threshold exceedances and max-stable models are attractive alternatives, but are difficult to fit when the number of observation sites is large, and are of little use for modeling the bulk of the distribution, which may also be of interest to water management planners. While the atmospheric dynamics governing precipitation are complex and difficult to fully incorporate into a parsimonious statistical model, non-mechanistic analogue methods that approximate those dynamics have proven to be promising approaches to capturing the temporal dependence of precipitation. In this paper, we present a Bayesian analogue method that leverages large, synoptic-scale atmospheric patterns to make precipitation forecasts. Changing spatial dependence across varying intensities is modeled as a mixture of spatial Student-t processes that can accommodate both strong and weak tail dependence. The proposed model demonstrates improved performance at capturing the distribution of extreme precipitation over Community Atmosphere Model (CAM) 5.2 forecasts.

Supplementary materials accompanying this paper appear online.

MCMC Details

Metropolis–Hastings MCMC algorithms were implemented for making posterior draws of the parameters in both the occurrence and intensity models using the R programming language (http://ww.r-project.org).

1.1 Occurrence Model Metropolis–Hastings Algorithm

The parameters $\rho$, $\nu$, and $\theta$ were updated using variable-at-a-time random walks. Truncated normal Gibbs updates are available for the latent Gaussian process at observation locations ${\mathbf {Z}}_t = (Z_t({\varvec{s}}_1), \ldots , Z_t({\varvec{s}}_n))$. Denoting occurrence indicators ${\mathbf {O}}_t = (O_t({\varvec{s}}_1), \ldots , O_t({\varvec{s}}_n))$ and mean function $\varvec{\mu }_t = (\mu _t({\varvec{s}}_1), \ldots , \mu _t({\varvec{s}}_n))$, the Gibbs updates for the latent Gaussian process are

$$\begin{aligned} {\mathbf {Z}}_t|{\mathbf {O}}_t, \varvec{\mu }_t, \rho , \nu \sim \mathrm {TN}_n(\varvec{\mu }_t, \Sigma _{\nu , \rho }, {\mathbf {l}}_t, {\mathbf {u}}_t) \end{aligned}$$

where $\Sigma _{\nu , \rho }$ is the spatial covariance matrix for the n observation locations, lower bounds ${\mathbf {l}}_t = (l({\varvec{s}}_1), \ldots , l({\varvec{s}}_n))$ have elements

$$\begin{aligned} l({\varvec{s}}_i) = {\left\{ \begin{array}{ll} 0, &{}\text { if } O_t({\varvec{s}}_i) = 1\\ -\infty &{}\text { if } O_t({\varvec{s}}_i) = 0 \end{array}\right. } \end{aligned}$$

and upper bounds ${\mathbf {u}}_t = (u({\varvec{s}}_1), \ldots , u({\varvec{s}}_n))$ have elements

$$\begin{aligned} u({\varvec{s}}_i) = {\left\{ \begin{array}{ll} \infty , &{}\text { if } O_t({\varvec{s}}_i) = 1\\ 0 &{}\text { if } O_t({\varvec{s}}_i) = 0 \end{array}\right. }\qquad . \end{aligned}$$

Gibbs updates are also available for both the location parameters $\gamma _t^{(O)}, t = 1, \ldots , T$ and $\varvec{\beta }^{(O)}_t$. For the location parameter, the Gibbs update is

$$\begin{aligned}&\gamma _t^{(O)}|\nu , \rho , \theta , \varvec{\gamma }_{-t}^{(O)}, \sigma ^2_\gamma , \varvec{\beta }^{(O)}_t, {\mathbf {Z}}_t \sim \mathrm {N}\biggl \{\biggl ({\mathbf {1}}'\Sigma _{\nu , \rho }^{-1}[{\mathbf {Z}}_t - \varvec{\psi }'\varvec{\beta }^{(O)}_t] + \frac{{\mathbf {w}}_t'\varvec{\gamma }_{-t}^{(O)}}{\sigma ^2_{\gamma ^{(O)}}}\biggr ) \\&\quad \times \,\biggl ({\mathbf {1}}'\Sigma _{\nu , \rho }^{-1}{\mathbf {1}}+ \frac{1}{\sigma ^2_{\gamma ^{(O)}}}\biggr )^{-1},\biggl ({\mathbf {1}}'\Sigma _{\nu , \rho }^{-1}{\mathbf {1}} + \frac{1}{\sigma ^2_{\gamma ^{(O)}}}\biggr )^{-1}\biggr \}. \end{aligned}$$

For the basis coefficients $\varvec{\beta }^{(O)}_t$, the Gibbs update is

$$\begin{aligned}&\varvec{\beta }^{(O)}_t|{\mathbf {Z}}_t, \gamma _t^{(O)}, \nu , \rho , \sigma ^2_{\beta ^{(O)}} \sim N_p\biggl \{\biggl (\varvec{\psi }'\Sigma _{\nu , \rho }^{-1}\varvec{\psi } + \frac{1}{\sigma _{\beta ^{(O)}}^2}I_p\biggr )^{-1}\\&\quad \times \, \biggl (\varvec{\psi }'\Sigma _{\nu , \rho }^{-1}[{\mathbf {Z}}_t - \gamma _t^{(O)}{\mathbf {1}}]\biggr ), \biggl (\varvec{\psi }'\Sigma _{\nu , \rho }^{-1}\varvec{\psi } + \frac{1}{\sigma _{\beta ^{(O)}}^2}I_p\biggr )^{-1}\biggr \}. \end{aligned}$$

Inverse-gamma Gibbs updates are used for $\sigma ^2_{\gamma ^{(O)}}$ and $\sigma ^2_{\beta ^{(O)}}$. Using inverse gamma parameterized with shape a and scale b, having density $f(x; a, b) = \frac{b^a}{\Gamma (a)} x^{-(a + 1)} \exp (- \frac{b}{x}), \, x > 0$. With prior $\sigma _{\gamma ^{(O)}}^2 \sim \mathrm {IG}(a_\gamma ,b_\gamma )$, the Gibbs update for $\sigma _{\gamma ^{(O)}}^2$, is

$$\begin{aligned} \sigma _{\gamma ^{(O)}}^2|\varvec{\gamma }^{(O)}_t \sim \mathrm {IG}\left\{ a_\gamma + n/2, b_\gamma + \frac{1}{2}\sum _{t =1}^T(\gamma _t - {\mathbf {w}}_t'\varvec{\gamma _{-t}}^{(O)})^2\right\} , \end{aligned}$$

and the Gibbs update for $\sigma ^2_{\beta ^{(O)}}$, assuming prior $\sigma _{\beta ^{(O)}}^2 \sim \mathrm {IG}(a_\beta ,b_\beta )$ is

$$\begin{aligned} \sigma _{\beta ^{(O)}}^2|\varvec{\beta }^{(O)}_{1:T} \sim \mathrm {IG}\left\{ a_\beta + np/2, b_\beta + \frac{1}{2}\sum _{t = 1}^{T}\varvec{\beta }^{(O)'}_t\varvec{\beta }^{(O)}_t\right\} , \end{aligned}$$

1.2 Intensity Model Metropolis–Hastings Algorithm

The parameters $\rho , \nu , a_k, b_k, \varvec{\alpha _k}$ for $k = 1, \ldots , K$, and $\theta$ were updated using variable-at-a-time random walks. The cluster labels $\xi _t$ were also updated variable-at-a-time, but with discrete uniform independence proposals, each on 1:K. The location offset and basis functions Gibbs updates are similar to those in the occurrence model but are also dependent on the mixture labels.

$$\begin{aligned}&\gamma _t^{(I)}|\nu _k, \rho _k, \theta , \varvec{\gamma }_{-t}^{(I)}, \xi _t = k, \sigma ^2_\gamma , \varvec{\beta }^{(I)}_t, {\mathbf {Z}}_t \sim \mathrm {N}\biggl \{\biggl ({\mathbf {1}}'\Sigma _{k}^{-1}[{\mathbf {Y}}_t - \varvec{\psi }'\varvec{\beta }^{(I)}_t] + \frac{{\mathbf {w}}_t'\varvec{\gamma }_{-t}^{(I)}}{\sigma ^2_{\gamma ^{(I)}}}\biggr )\\&\quad \times \,\biggl ({\mathbf {1}}'\Sigma _{k}^{-1}{\mathbf {1}}+ \frac{1}{\sigma ^2_{\gamma ^{(I)}}}\biggr )^{-1},\biggl ({\mathbf {1}}'\Sigma _{k}^{-1}{\mathbf {1}} + \frac{1}{\sigma ^2_{\gamma ^{(I)}}}\biggr )^{-1}\biggr \}. \end{aligned}$$

where the covariance matrix $\Sigma _{k}$ is calculated using dependence parameters $\rho _k$ and $\nu _k$ corresponding to mixture class k.

Similarly, the basis coefficients $\varvec{\beta }^{(I)}_t$ have Gibbs update

$$\begin{aligned}&\varvec{\beta }^{(I)}_t|{\mathbf {Y}}_t, \gamma _t^{(I)},\nu _k, \rho _k, \xi _t = k, \sigma ^2_{\beta ^{(I)}} \sim N_p\biggl \{\biggl (\varvec{\psi }'\Sigma _{k}^{-1}\varvec{\psi } + \frac{1}{\sigma _{\beta ^{(I)}}^2}I_p\biggr )^{-1}\\&\quad \times \,\biggl (\varvec{\psi }'\Sigma _{k}^{-1}[{\mathbf {Y}}_t- \gamma _t^{(I)}{\mathbf {1}}]\biggr ), \biggl (\varvec{\psi }'\Sigma _{k}^{-1}\varvec{\psi } + \frac{1}{\sigma _{\beta ^{(I)}}^2}I_p\biggr )^{-1}\biggr \}. \end{aligned}$$

The Gibbs updates for the prior variances $\sigma ^2_{\gamma ^{(I)}}$ and $\sigma ^2_{\beta ^{(I)}}$ are completely analogous to those in the occurrence model.