Trajectory learning for ensemble forecasts via the continuous ranked probability score: a Lorenz '96 case study

TL;DR

The paper tackles forecast uncertainty from unresolved dynamics by learning stochastic sub-grid parametrizations using the continuous ranked probability score (CRPS) within a trajectory-learning framework. It develops both additive and multiplicative stochastic models based on coupled Ornstein–Uhlenbeck processes and evaluates them on the two-scale Lorenz '96 system, showing improved short-term accuracy and ensemble sharpness over derivative-fitting baselines. Key findings include that CRPS-based trajectory learning yields models that are accurate and sharp at short lead times, with the self-spread term in CRPS providing a stabilizing influence on training and forecasts, and that the approach holds promise for data assimilation in geophysical flows. The work also documents trade-offs in long-term stability, the impact of trajectory length during training, and potential benefits of extending the framework with neural-state–dependent parametrizations for future, more complex systems.

Abstract

This paper demonstrates the feasibility of trajectory learning for ensemble forecasts by employing the continuous ranked probability score (CRPS) as a loss function. Using the two-scale Lorenz '96 system as a case study, we develop and train both additive and multiplicative stochastic parametrizations to generate ensemble predictions. Results indicate that CRPS-based trajectory learning produces parametrizations that are both accurate and sharp. The resulting parametrizations are straightforward to calibrate and outperform derivative-fitting-based parametrizations in short-term forecasts. This approach is particularly promising for data assimilation applications due to its accuracy over short lead times.

Figures (11)

• Figure 1: Example of CRPS-based trajectory learning for ensemble predictions. The true trajectory of a single variable in the L96 system serves as a reference. The trajectory length equals 25 coarse-grid time steps (left) and 100 coarse-grid time steps (right), with each ensemble consisting of 20 members. After training, the ensemble exhibits reduced mean absolute error and spread.
• Figure 2: POD modes for $c=4$ (left) and $c=10$ (right) that serve as the basis for the global stochastic parametrization. The pairs of POD modes are shown with a vertical offset for visibility, all modes have zero mean. The mean profile $\xi_0$ does not have zero mean and is not offset.
• Figure 3: Evolution of the loss per batch over the training duration, for $c=4$ (left column) and $c=10$ (right column), for the additive noise model (top row) and the multiplicative noise model (bottom row). The displayed values are normalised by the initial value for visual comparison, and a moving average is applied for clarity.
• Figure 4: Comparison of the average MSE for short lead times, for $c=4$ (left) and $c=10$ (right). The MSE is measured per $X$-variable in ensembles consisting of 50 members, simulated from 100 different initial conditions.
• Figure 5: Comparison of the average CRPS for short lead times, for $c=4$ (left) and $c=10$ (right). The CRPS is measured per $X$-variable in ensembles consisting of 50 members, simulated from 100 different initial conditions.