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Forecast error growth: A dynamic-stochastic model

Eviatar Bach, Dan Crisan, Michael Ghil

TL;DR

The paper addresses forecast-error growth in chaotic systems by introducing a nonlinear stochastic differential equation that adds multiplicative noise to the Dalcher–Kalnay framework. It nondimensionalizes the model, derives a stationary distribution via the Fokker–Planck equation, and analyzes first-passage times to quantify forecast-skill horizons. Through calibration to ECMWF TIGGE ensemble data using ensemble Kalman inversion, the authors demonstrate that the model captures both the mean error growth and its probabilistic structure, including error-distribution evolution and first-passage-time behavior. This provides a probabilistic, interpretable framework for predictability in NWP and offers a template for applying similar stochastic error-growth analyses to other complex prediction problems.

Abstract

There is a history of simple forecast error growth models designed to capture the key properties of error growth in operational numerical weather prediction (NWP) models. We propose here such a scalar model that relies on the previous ones and incorporates multiplicative noise in a nonlinear stochastic differential equation (SDE). We analyze the properties of this SDE, including the shape of the error growth curve for small times and its stationary distribution, and prove well-posedness and positivity of solutions. Next, we fit this model to operational NWP error growth curves, and show good agreement with both the mean and probabilistic features of the error growth. These results suggest that the dynamic-stochastic error growth model proposed herein and similar ones could play a role in many other areas of the sciences that involve prediction.

Forecast error growth: A dynamic-stochastic model

TL;DR

The paper addresses forecast-error growth in chaotic systems by introducing a nonlinear stochastic differential equation that adds multiplicative noise to the Dalcher–Kalnay framework. It nondimensionalizes the model, derives a stationary distribution via the Fokker–Planck equation, and analyzes first-passage times to quantify forecast-skill horizons. Through calibration to ECMWF TIGGE ensemble data using ensemble Kalman inversion, the authors demonstrate that the model captures both the mean error growth and its probabilistic structure, including error-distribution evolution and first-passage-time behavior. This provides a probabilistic, interpretable framework for predictability in NWP and offers a template for applying similar stochastic error-growth analyses to other complex prediction problems.

Abstract

There is a history of simple forecast error growth models designed to capture the key properties of error growth in operational numerical weather prediction (NWP) models. We propose here such a scalar model that relies on the previous ones and incorporates multiplicative noise in a nonlinear stochastic differential equation (SDE). We analyze the properties of this SDE, including the shape of the error growth curve for small times and its stationary distribution, and prove well-posedness and positivity of solutions. Next, we fit this model to operational NWP error growth curves, and show good agreement with both the mean and probabilistic features of the error growth. These results suggest that the dynamic-stochastic error growth model proposed herein and similar ones could play a role in many other areas of the sciences that involve prediction.

Paper Structure

This paper contains 19 sections, 1 theorem, 46 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Stochastic treatment of error growth
  3. Previous error growth models in NWP
  4. A stochastic error growth model
  5. Stochastic model formulation
  6. Nondimensionalization
  7. Fokker--Planck equation and stationary distribution
  8. First passage times and forecast skill horizons
  9. Initial curvature
  10. Error growth in the SDE model and in NWP models
  11. Forecasts and reanalysis data
  12. Fitting procedure
  13. Identical twins
  14. Scale analysis
  15. Error with respect to reanalysis
  16. ...and 4 more sections

Key Result

Theorem 1

Equation eq:nondim has a unique global solution $\{v^{\ast }(t^*): t^*\ge 0\}$, $P$-almost surely for any initial value $v^{\ast } (0) >0$. Moreover the solution remains positive $P$-almost surely.

Figures (9)

  • Figure 1: A typical pattern of error growth for a chaotic dynamical system. The system used here is the chaotic convection model of Lorenz (1963)lorenz_deterministic_1963. The error growth is averaged over many initial conditions and a moving average filter is applied.
  • Figure 2: Sample realizations of the SDE \ref{['eq:SDE_add']}. Here $v(0) = 30$ and the parameters are picked to be $v_\infty = 9000$, $\alpha = 0.6$, $s = 80$, and $\sigma = 0.2$.
  • Figure 3: Empirical PDFs of first passage times for solutions of the SDE, for increasing threshold values, from $v_c = 0.5 v_{\infty}$ to $v_c = 0.95 v_{\infty}$. Time in nondimensional units. The PDFs are estimated from data using kernel density estimation hastie_elements_2009; this is a likely reason for the small apparent deviations from unimodality.
  • Figure 4: Example of a geopotential height forecast ensemble member.
  • Figure 5: Mean error growth curves. (a) For identical twins in an operational NWP forecast (blue curve) and the SDE model (red curve); (b) error growth of the individual NWP forecast pairs; and (c) realizations of the fitted SDE model.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Remark 1
  • Theorem 1
  • proof