Modelling the spatio-temporal repartition of right-truncated data: an application to avalanche runout altitudes in Hautes-Savoie

Abstract

In this paper, we propose a novel approach for generating avalanche hazard maps based on the spatial dependence of avalanche runout altitudes. The right-truncated data are described with a Bayesian hierarchical model in which the spatio-temporal process is assumed to be the sum of independent spatial and temporal terms. Topography is roughly taken into account according to valley altitude and path exposition, and the spatial dependence is modelled with a Matérn covariance function. An application is performed to the Haute-Savoie region, French Alps. A spatial dependence in runout altitudes is identified, and an effective range of about 10 km is inferred. The temporal trend extracted highlights the increase of avalanche runout altitudes from 1955, attributed to both anthropogenic factors and climate warming. In a cross validation scheme, spatial predictions are provided on undocumented paths using kriging equations. All in all, although our model is unable to take into account small topographic features, it is a first-ever approach that produces very encouraging results. It could be enhanced in future work by incorporating a numerical physically-based code into the modelling.

Appendices

Appendix 1: Empirical study of the spatio-temporal covariance structure

In order to investigate the presence of spatio-temporal dependence in runout altitudes, empirical covariance matrices are estimated in the non-separable, separable and additive cases (Fig. 9). Let $Y_{ict}$ be the runout altitude of avalanche i on path c the year t. The empirical spatio-temporal covariance at spatial lag $[h_{k-1},h_k]$ and time lag $[\tau _{l-1},\tau _l]$ is given by:

$$\begin{aligned} \hat{C}_Y(h_k,\tau _l)=\frac{1}{|N(h_k,\tau _l)|}\sum _{(ict,i'c't') \in N(h_k,\tau _l)}(Y_{ict}-\hat{\mu }_Y(c))(Y_{i'c't'}-\hat{\mu }_Y(c')) \end{aligned}$$

with $N(h_k,\tau _l)$ the set of avalanche pairs $(ict,i'c't')$ such that the distance between sites c and $c'$ is in the interval $[h_{k-1},h_k]$ and the lag time $t-t'$ belongs to the interval $[\tau _{l-1},\tau _l]$. $|N(h_k,\tau _l)|$ is the number of pairs. The reference mean $\hat{\mu }_Y(c)$ is calculated by linear regression on the valley altitude, the best covariate: $\hat{\mu }_Y(c)=6.3+ 7.4 10^{-4}h_c$. From the empirical non-separable covariance matrix, the additive and separable covariance matrices have been estimated by least squares. In details, for each spatial and time lag, estimates of $\gamma ^{T}_l$ and $\gamma ^S_k$ are found such that:

$$\begin{aligned} \hat{C}_Y(h_k,\tau _l) = \gamma ^{T}_l+\gamma ^S_k + \epsilon _{kl} \end{aligned}$$

for the additive covariance matrix, and:

$$\begin{aligned} \hat{C}_Y(h_k,\tau _l) = \gamma ^{T}_l\gamma ^S_k +\epsilon _{kl} \end{aligned}$$

for the separable covariance matrix, where $\epsilon _{kl}$ is the error.

Figure 9 shows that the non-separable covariance matrix is similar to the separable and additive ones, essentially because the spatial dependence is much higher than the temporal dependence. This pleads further in favor of our choice of a simple model with additive effects.

Fig. 9

Empirical non-separable (left), separable (center) and additive (right) spatio-temporal covariance matrices. The mean is only spatially indexed, and is computed from the valley altitude

Appendix 2: Decomposition of the predictive distribution

First we integrate the annual predictive distribution $[\mathbf {Y}_{c_0t}|\mathbf {Y}]$ for time t and paths $c_0$ over the time to make prediction for the entire study period:

$$\begin{aligned}{}[\mathbf {Y}_{c_0}|\mathbf {Y}]=\sum _{t=1}^{n_t} \frac{1}{n_{t}}[\mathbf {Y}_{c_0t}|\mathbf {Y}]\text {.} \end{aligned}$$

By using the hierarchical structure of the model we can express this distribution conditionally to the process level:

$$\begin{aligned}{}[\mathbf {Y}_{c_0t}|\mathbf {Y}]&=\int [\mathbf {Y}_{c_0t}|\mathbf {C}_{c_0},B_t,\alpha ,\sigma _{c_0}^2] [\mathbf {C}_{c_0},B_t,\alpha ,\sigma _{c_0}^2|\mathbf {Y}]\nonumber \\&\qquad \times d(\mathbf {C}_{c_0},B_t,\alpha , \sigma _{c_0}^2). \end{aligned}$$

(10)

Let $\varvec{\theta }_1=(B_t,\alpha ,\sigma _{c_0}^2)'$, we decompose the distribution $[\mathbf {C}_{c_0},\varvec{\theta }_1|\mathbf {Y}]$ in the same way to make the spatial term $\mathbf {A}_{c_0}$ appear,

$$\begin{aligned} [\mathbf {C}_{c_0},\varvec{\theta }_1|\mathbf {Y}] &= \int [\mathbf {C}_{c_0},\varvec{\theta }_1|\mathbf {A}_{c_0},a,b,o_S,\rho ^2] [\mathbf {A}_{c_0},a,b,o_S,\rho ^2|Y]\\ &\quad \times d(\mathbf {A}_{c_0},a,b,o_S,\rho ^2). \end{aligned}$$

(11)

Finally, we express $[\mathbf {A}_{c_0},\varvec{\theta }_2|\mathbf {Y}]$, with $\varvec{\theta }_2=(a,b,o_S,\rho ^2)'$, conditionally to $\mathbf {A}$ rather than $\mathbf {Y}$:

$$\begin{aligned}{}[\mathbf {A}_{c_0},\varvec{\theta }_2|\mathbf {Y}]=\int [\mathbf {A}_{c_0},\varvec{\theta }_2|\mathbf {A},\phi ,\tau ^2] [\mathbf {A},\phi ,\tau ^2|\mathbf {Y}]d(\mathbf {A},\phi ,\tau ^2)\text {.} \end{aligned}$$

(12)

The joint Gaussian distribution for the vector $[\mathbf {A},\mathbf {A}_{c_0}|\phi ,\tau ^2]$ is:

$$\begin{aligned} \left[ \begin{array}{c} \mathbf {A}\\ \mathbf {A}_{c_0} \end{array} \right] \sim N \left( \left[ \begin{array}{c} 0\\ 0 \end{array} \right] , \tau ^2 \left[ \begin{array}{c c} \varvec{\Sigma }_{\phi }^{..} &{} \varvec{\Sigma }_{\phi }^{.0}\\ \varvec{\Sigma }_{\phi }^{0.} &{} \varvec{\Sigma }_{\phi }^{00} \end{array} \right] \right) \text {,} \end{aligned}$$

where $\varvec{\Sigma }_{\phi }^{..}$ and $\varvec{\Sigma }_{\phi }^{00}$ are the variance-covariance matrices of $\mathbf {A}$ and $\mathbf {A}_{c_0}$, respectively, and where $\varvec{\Sigma }_{\phi }^{.0}$ and $\varvec{\Sigma }_{\phi }^{0.}$ are covariance matrices between elements of $\mathbf {A}$ and $\mathbf {A}_{c_0}$. Then the formulas for conditional Gaussian distributions give:

$$\begin{aligned}E(\mathbf {A}_{c_0}|\mathbf {A},\phi ,\tau ^2)&=\varvec{\Sigma }_{\phi }^{0.}\varvec{\Sigma }_{\phi }^{..-1}\mathbf {A}\\ V(\mathbf {A}_{c_0}|\mathbf {A},\phi , \tau ^2)&= \varvec{\Sigma }_{\phi }^{00}-\varvec{\Sigma }_{\phi }^{0.}\varvec{\Sigma }_{\phi }^{..-1}\varvec{\Sigma }_{\phi }^{.0} .\end{aligned}$$

By combining Eqs. (11) and (12) within Eq. (10), and using the Bayes’ rule we obtain

$$\begin{aligned} [\mathbf {Y}_{c_0,t}|\mathbf {Y}] &=\int [\mathbf {Y}_{c_0t}|\mathbf {C}_{c_0},\varvec{\theta }_1] [\mathbf {C}_{c_0}|\mathbf {A}_{c_0},\varvec{\theta }_2][\mathbf {A}_{c_0}|\varvec{\theta }_3]\nonumber \\ & \quad \times [\varvec{\theta }_1,\varvec{\theta }_2,\varvec{\theta }_3|\mathbf {Y}]d(\varvec{\theta }_1,\varvec{\theta }_2,\varvec{\theta }_3),\end{aligned}$$

(13)

with $\varvec{\theta }_3$ the vector $(\mathbf {A},\phi ,\tau ^2)'$.

Appendix 3: Algorithm details

We detail the algorithm to draw $\mathbf {C}$ under the constraint $\sum _{c=1}^{n_c}C_c=0$, the same method is used to draw $\mathbf {B}$ under the constraint $\sum _{t=1}^{n_t}B_t=0$.

Let $\mathbf {X}_C$ the $n\times n_c$ matrix such that ${\mathbf {X}_C}_{ij}=1$ if the avalanche i occurs on path j, 0 either, $\mathbf {D}$ the diagonal variance-covariance matrix of $\mathbf {Z}$, $\mathbf {D}=\text {diag}(\sigma _{c_i}^2)_{i=1,n}$, with $c_i$ the path label for avalanche i, and $\mathbf {1}_c$ the $n_c$ length column vector of ones.

The complete posterior distribution of $\mathbf {C}$ is given by,

$$\begin{aligned} \begin{array}{l} \mathbf {C}|\mathbf {Z},\mathbf {B},\alpha ,\mathbf {D},E[\mathbf {C}],\rho ^2 \sim \mathcal {N}(\mathbf {m},\varvec{\Sigma })\\ \text {with}\quad \left\{ \begin{array}{l} \mathbf {m}=\left(\mathbf {X}_C'\mathbf {D}^{-1}\mathbf {X}_C+\frac{1}{\rho ^2}\mathbf {I}_c\right)^{-1}(\mathbf {X}_C'D^{-1}(\mathbf {Z}-\alpha -\mathbf {X}_B\mathbf {B}) +\frac{1}{\rho ^2}E(\mathbf {C}))\\ \Sigma =\left(\mathbf {X}_C'\mathbf {D}^{-1}\mathbf {X}_C+\frac{1}{\rho ^2}\mathbf {I}_{c}\right)^{-1} \end{array} \right. \end{array} \end{aligned}$$

At a second step we write the joint distribution of $\mathbf {C}|\mathbf {Z},\mathbf {B},\alpha ,\mathbf {D},E[\mathbf {C}],\rho ^2$, noted $\mathbf {C}|\varvec{\gamma }$ and its constraint $\mathbf {1}_{c}\mathbf {C}=0$,

$$\begin{aligned} \left[ \begin{array}{c} \mathbf {C}|\varvec{\gamma }\\ \mathbf {1}_{c}'\mathbf {C}\end{array} \right] \sim N \left( \left[ \begin{array}{c} \mathbf {m}\\ \mathbf {1}_{c}'\mathbf {m}\end{array} \right] , \left[ \begin{array}{c c} \varvec{\Sigma }&{} \varvec{\Sigma }\mathbf {1}_{c}'\\ \mathbf {1}_{c}'\varvec{\Sigma }&{} \mathbf {1}_{c}'\varvec{\Sigma }\mathbf {1}_{c}' \end{array} \right] \right) \text {.} \end{aligned}$$

The conditional distribution is then:

$$\begin{aligned} \begin{array}{l} \mathbf {C}|\mathbf {Z},\mathbf {B},\alpha ,\mathbf {D},E[\mathbf {C}],\rho ^2,\mathbf {1}_{c}'\mathbf {C}=0 \sim \mathcal {N}(\mathbf {m}_0,\varvec{\Sigma }_0)\\ \text { with } \left\{ \begin{array}{l} \mathbf {m}_0=\mathbf {m}-\frac{\varvec{\Sigma }\mathbf {1}_{c}\mathbf {1}_{c}'\mathbf {m}}{\mathbf {1}_{c}'\varvec{\Sigma }\mathbf {1}_{c}}\\ \varvec{\Sigma }_0=\varvec{\Sigma }- \frac{\varvec{\Sigma }\mathbf {1}_{c}\mathbf {1}_{c}'\varvec{\Sigma }}{\mathbf {1}_{c}'\varvec{\Sigma }\mathbf {1}_{c}} \end{array} \right. \end{array} . \end{aligned}$$

(14)

Only the $n_c-1$ first components of $\mathbf {C}$ are sampled, $C_{n_c}$ is given by $C_{n_c}=-\sum _{c=1}^{n_c-1}C_c$.

Then we constrain the mean runout altitude by path, $\alpha +C_c$. We have $h_c<\alpha +C_c<s_c$ for all $c\in \{1,\cdots n_c\}$. In a Gibbs sampler, $\mathbf {C}$ and $\alpha$ are drawn successively in their complete conditional distributions. We describe the method to take into account the constraint for $\mathbf {C}$, $\alpha$ is sampled in a similar way. The $n_c-1$ first components of $\mathbf {C}$ must fullfill the constraints:

$$\begin{aligned} \left\{ \begin{array}{l} h_1 - \alpha< C_1< s_1 - \alpha \\ \vdots \\ h_{n_c-1} - \alpha< C_{n_c-1}< s_{n_c-1} - \alpha \\ h_{n_c}- \alpha< -\sum _{c=1}^{n_c-1}C_c < s_{n_c}-\alpha \text {,} \end{array}\right. \end{aligned}$$

(15)

that we can write matrix-wise $\mathbf {V}\mathbf {C}<\mathbf {v}$ with $\mathbf {V}$ the $k\times n_{c}-1$ matrix of constraints. Here $k=2n_c$, since there are two constraints for each line of the system Eq. (15), one bears on $s_c$, the other one on $h_c$. The posterior distribution of $\mathbf {C}$ is thus a multivariate truncated distribution whose mean and variance for the complete version are given by Eq. (14) and whose support $S=\{\mathbf {x}\in \mathbb {R}^{n_c-1} {:}\, \mathbf {V}\mathbf {x}<\mathbf {v}\}$. We follow the two-step algorithm of Rodriguez-Yam et al. (2004). In the first step, we remove the dependence between the components of $\mathbf {C}$ by considering the vector $\mathbf {w}=\mathbf {L}\mathbf {C}$, with $\mathbf {L}$ such that $\mathbf {L}\varvec{\Sigma }_0 \mathbf {L}'=\mathbf {I}_{n_c-1}$ and $\mathbf {I}_{n_c-1}$ the identity matrix. It has the following truncated distribution:

$$\begin{aligned} \begin{array}{l} \mathbf {w}|\mathbf {Z},\mathbf {B},\alpha ,\mathbf {D},E[\mathbf {C}],\rho ^2,\mathbf {1}_{c}'\mathbf {C}=0 \sim \mathcal {N}_{S}(\mathbf {L}\mathbf {m}_0, \mathbf {I}_{n_c-1})\\ \text {with\,support}{:} \,S=\{ \mathbf {w}\in \mathbb {R}^{n_c-1} {:}\, \mathbf {V}\mathbf {L}^{-1}\mathbf {w}<\mathbf {v}\} \text {.}\end{array} \end{aligned}$$

In the second step, each component of $\mathbf {w}$ is drawn successively conditionally to the others in a Gibbs sampling. In order to avoid the computation of the entire vector of constraints $\mathbf {V}\mathbf {L}^{-1} \mathbf {w}$ at each draw, we may notice that for each constraint $i \in \{1, \ldots k\}$, and each component of $\mathbf {w}$ $j_0\in \{1, \ldots n_c-1\}$

$$\begin{aligned} (\mathbf {V}\mathbf {L}^{-1} \mathbf {w})_{i}= & {} \sum _{j=1}^{n_c-1} \left( \sum _{l=1}^{n_c-1} \mathbf {V}_{il}\mathbf {L}_{lj}^{-1}\right) w_j\\= & {} \underbrace{\sum _{j\ne j_0}^{n_c-1} \left( \sum _{l=1}^{n_c-1} \mathbf {V}_{il}\mathbf {L}_{lj}^{-1}\right) w_j}_{u_{i}^{-j_0}} + \underbrace{\left( \sum _{l=1}^{n_c-1} \mathbf {V}_{il}\mathbf {L}_{l{j_0}}^{-1}\right) }_{u_{i}^{j_0}} w_{j_0}\text {.} \end{aligned}$$

The constraint i for the $j_0$ component of $\mathbf {w}$ is written $u_{i}^{j_0}w_{j_0} < v_i - u_{i}^{-j_0}$. To update $w_{j_0}$, one only has to compute $u_{i}^{j_0}$ for each constraint. This step demands $k(n_c-1)$ elementary operations, instead of $k(n_c-1)^2$ operations when $\mathbf {V}\mathbf {L}^{-1}$ is computed naively.

Finally we implement the following algorithm for sampling the $n_c-1$ components of $\mathbf {C}$ from their truncated Gaussian distributions:

• Compute the vector $\mathbf {u}= \mathbf {V}\mathbf {C}$;

• Initiate the matrix $\mathbf {L}$, its inverse $\mathbf {L}^{-1}$, and the vector $\mathbf {w}=\mathbf {L}\mathbf {C}$;

• For each j in $\{1,\ldots n_c-1\}$

  • Compute the k-dimension vectors $\mathbf {u}^j=(u_1^j,\ldots u_k^j)$, and $\mathbf {u}^{-j}= \mathbf {u}- u^jw_j$;

  • Find the interval $[ a_j, b_j ]$ in which $w_j$ is drawn to satisfy the k constraints;

  • Sample $w_j$ in the normal truncated distribution $\mathcal {N}((\mathbf {L}\mathbf {m}_{0})_j,1)$ with support $[ a_j, b_j ]$. Here, we use the method and the code proposed by Chopin (2011);

  • Update $\mathbf {u}$ as $\mathbf {u}=\mathbf {u}^{-j}+\mathbf {u}^jw_j$;

• Set $\mathbf {C}$ as $\mathbf {C}=\mathbf {L}^{-1}\mathbf {w}$.