Partition of analysis and forecast error variance into growing and decaying components

Jie Feng, Zoltan Toth, Malaquias Peña, Jing Zhang

Research output: Contribution to journalArticle

1 Citation (Scopus)


Due to the scarcity of and errors in observations, direct measurements of errors in numerical weather prediction (NWP) analyses and forecasts with respect to nature (i.e. “true” error) are lacking. Peña and Toth (2014) introduced an inverse method called SAFE-I where true errors are (a) theoretically assumed to follow exponential error growth, and (b) estimated from the perceived errors (i.e. forecast minus verifying analysis) that they affect. While decaying or neutral errors, by definition will not have a significant impact on longer-range forecast errors, they can still accumulate in, and negatively influence NWP data assimilation–forecast cycles. In a new, generalized version of the inverse method (SAFE-II), analysis and forecast error variance is decomposed into exponentially growing and decaying components, assuming they are independent as they comprise vectors from the leading and trailing ends of the Lyapunov spectrum, respectively. SAFE-II uses the initial variance and decay rate associated with non-growing perturbations to describe and estimate their behaviour. The assumptions behind SAFE-II are first validated in a simulated environment. SAFE-II is then applied to estimate the error variance in both simulated and operational analyses/forecast environments. Perceived error measurements are found to be statistically consistent (at the 95% significance level) with the SAFE-II error behaviour model, which offers a more accurate description of error variance than SAFE-I that neglects decaying errors. At various levels and for different variables, decaying errors are found to constitute up to 60% of the total analysis error variance, much of which decays during the first 12–18 hr of forecast integrations.

Original languageEnglish
Pages (from-to)1302-1321
Number of pages20
JournalQuarterly Journal of the Royal Meteorological Society
Issue number728
Publication statusPublished - Apr 1 2020


  • data assimilation
  • ensemble forecasts
  • error estimation
  • forecast verification
  • uncertainty of analysis

ASJC Scopus subject areas

  • Atmospheric Science

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