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

• Published in: Quarterly journal of the Royal Meteorological Society 2020-04, Vol.146 (728), p.1302-1321
• Main Authors: Feng, Jie, Toth, Zoltan, Peña, Malaquias, Zhang, Jing
• Format: Article
• Language: English
• Subjects: Components; Data assimilation; Data collection; Decay; ensemble forecasts; error estimation; Errors; Forecast errors; forecast verification; uncertainty of analysis; Variance analysis; Vectors; Weather forecasting
• ISSN: 0035-9009; 1477-870X
• DOI: 10.1002/qj.3738

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. Modified statistical analysis and forecast error estimation algorithm recognizes the approximately exponential growing and decaying components in analysis errors, and improves the accuracy of estimated analysis and forecast error variances. Perceived error variance gives a rather poor estimate of the true error variance (e.g. 3–4‐fold underestimation at 6 hr lead time) and a related overestimation of the error growth rate within the first 2 days.