Bringing statistics to storylines: rare event sampling for sudden, transient extreme events

Abstract

A leading goal for climate science and weather risk management is to accurately model both the physics and statistics of extreme events. These two goals are fundamentally at odds: the higher a computational model's resolution, the more expensive are the ensembles needed to capture accurate statistics in the tail of the distribution. Here, we focus on events that are localized in space and time, such as heavy precipitation events, which can start suddenly and decay rapidly. We advance a method for sampling such events more efficiently than straightforward climate model simulation. Our method combines elements of two recent approaches: adaptive multilevel splitting (AMS), a rare event algorithm that generates rigorous statistics at reduced cost, but that does not work well for sudden, transient extreme events; and "ensemble boosting" which generates physically plausible storylines of these events but not their statistics. We modify AMS by splitting trajectories well in advance of the event's onset following the approach of ensemble boosting, and this is shown to be critical for amplifying and diversifying simulated events in tests with the Lorenz-96 model. Early splitting requires a rejection step that reduces efficiency, but nevertheless we demonstrate improved sampling of extreme local events by a factor of order 10 relative to direct sampling in Lorenz-96. Our work makes progress on the challenge posed by fast dynamical timescales for rare event sampling, and it draws connections with existing methods in reliability engineering which, we believe, can be further exploited for weather risk assessment.

Figures (23)

• Figure 1: Schematic of the splitting step in (a) AMS and (b) TEAMS. Black curves represent an initial ensemble member, or ancestor, which exceeds the first level $\ell_1$ and has been selected for cloning in the first round. In AMS, the perturbation is applied at the instant $t_0(\ell_1)$ when the ancestor first exceeds $\ell_1$, resulting in a descendant trajectory (blue) which essentially replicates the extreme event because the separation timescale is longer than the event itself. On the other hand, in TEAMS (right) we apply the perturbation in advance, by some margin $\delta>0$. This can sometimes result in rejection (blue descendant), i.e., failure to cross $\ell_1$. However, when a descendant is accepted (red) it will be more distinct from the ancestor than the corresponding descendant in AMS and have the potential to reach a substantially higher peak value.
• Figure 2: Time evolution of the L96 model expressed as timeseries of $x_0(t)$ (left column) and Hovmöller diagrams (right column) with three different levels of stochastic forcing. (a,b) have $F_4=0$ (the deterministic system); (c,d) have $F_4=1$ (moderate forcing); (e,f) have $F_4=3$ (strong forcing).
• Figure 3: Steady-state statistics of the L96 model as a function of noise strength, calculated from a long simulation of length $1.28\times10^6$. (a) Histograms of the model variable at one site ($x_0$) and (b) return level vs. return period for (twice) the local energy $x_0^2$. Shading in (b) represents 95% bootstrapped confidence intervals from the modified block maximum method. See text for details.
• Figure 4: Scores for single ancestors and their descendents within the AMS algorithm (special case of TEAMS with $\delta=0$). For each stochastic forcing amplitude, 56 independent runs of AMS were carried out (indexed 0-55) with $N=128$ ensemble members (0-127). (a) Time-dependent score function $R(X(t))$ for the 7th initial ensemble member (ancestor) of run 14 for $F_4=3$. A black circle indicates the scalar score $S(X)=\max_tR(X(t))$. $R(X(t)$ and $S(X)$ are also shown for a single lineage (path down the family tree) in a sequence of brightening colors, ending with the highest scoring descendant's score in red. (b) Scores in gray dots, with the horizontal axis numbering all descendants from ancestor 7 of run 14 for $F_4=3$. Colored circles indicate those descendants in the lineage from (a). The dashed gray curve indicates the levels $\ell$ from which each descendant was split. (c,e,g) are the same as (a), and (d,f,h) are the same as (b), but with stochastic forcing strength decreasing to $F_4=1,0.5$, and 0.25 respectively. In each case, the run and ancestor were hand-selected among the ancestors with the maximum boosting.
• Figure 5: Performance of the AMS algorithm (special case of TEAMS with $\delta=0$). (a) Return level vs. return period plots for $F_4=3$. Blue lines show estimates from the initial 128 members of each AMS run; red lines show estimates from the completed AMS runs; black line shows DNS. (b) Return level vs. return period for a pooled AMS ensemble containing all $56\times1024$ members. Blue and red envelopes indicate 95% confidence intervals (see text for details). Gray envelope is a 95% confidence interval based on subsets of DNS equal in total cost to the 56 AMS runs. Thus, the dashed red line and shading from AMS is of equal cost to the gray shading from DNS. (c) Unweighted histogram of scores for AMS initialization (blue), completed AMS (red), and DNS (black). Following rows are same as first row, but with noise decreasing to $F_4=1,0.5$, and 0.25, respectively. The slight variability in TEAMS costs listed to the left are due to the early halting criterion of one single ancestor remaining (see section \ref{['sec:subset']}).