Accurate estimates of analysis and short-range forecast error variances are critical to numerical weather prediction (NWP) for assessing the performance of analysis and forecast systems, tuning initial perturbations in ensemble forecasting, and specifying background forecast error variance in data assimilation (DA). Pena and Toth (2014) developed a Statistical Analysis and Forecast Error (SAFE) estimation algorithm to assess actual error variances reflecting the differences between analysis or forecast and true states of the atmosphere. One of the assumptions of SAFE is that all errors in the analysis initially expand exponentially in ensuing NWP forecasts. While decaying or neutral errors, if present in the analysis fields originating e.g. from random observational errors, by definition will not have a significant impact on forecast errors, potentially they can still accumulate in and influence NWP DA – forecast cycles. Recognizing the presence of decaying and neutral errors in NWP analyses, in this study we relax the restrictive error evolution assumption of SAFE and beyond growing errors, introduce decaying errors in the statistical error estimation. With the generalization, the error evolution model (SAFE-4) has four independent parameters: the growing and decaying components of the analysis error variance and the associated exponential growth and decay factors. Like SAFE, the new method uses an additional simple assumption about error behavior in DA - forecast cycles, along with measurements of perceived error variances (differences between forecasts and corresponding verifying analyses) to arrive at estimates of the four unknown parameters. SAFE-4 has been applied to estimate and decompose analysis and forecast error variances in the Global Forecast System (GFS) of the National Centers for Environmental Prediction. Estimates of total 500hPa height analysis error variance averaged over the Northern Hemisphere extratropics for the September – November 2015 period is about 31 m2, 13 m2 of which is exponentially growing at a 6-h amplification rate of 1.18 The other, 18 m2 part of the analysis error drops below 1 % of its initial variance within 24 h lead time. The growing errors are assumed to amplify with a rate close to the leading nonlinear Lyapunov Vectors (LVs), while the contraction of the other error component reflects its projection on the corresponding spectrum of neutral and trailing LVs.
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