Identifying non-normal and lognormal characteristics of temperature, mixing ratio, surface pressure, and wind for data assimilation systems

Abstract. Data assimilation systems and retrieval systems that are based upon a maximum likelihood estimation, many of which are in operational use, rely on the assumption that all of the errors and variables involved follow a normal distribution. This work develops a series of statistical tests to show that mixing ratio, temperature, wind and surface pressure follow non-normal, or in fact, lognormal distributions thus impacting the design-basis of many operational data assimilation and retrieval systems. For this study one year of Global Forecast System 00:00 UTC 6 h forecast were analyzed using statistical hypothesis tests. The motivation of this work is to identify the need to resolve whether or not the assumption of normality is valid and to give guidance for where and when a data assimilation system or a retrieval system needs to adapt its cost function to the mixed normal-lognormal distribution-based Bayesian model. The statistical methods of detection are based upon Shapiro–Wilk, Jarque–Bera and a χ² test, and a new composite indicator using all three measures. Another method of detection fits distributions to the temporal-based histograms of temperature, mixing ratio, and wind. The conclusion of this work is that there are persistent areas, times, and vertical levels where the normal assumption is not valid, and that the lognormal distribution-based Bayesian model is observationally justified to minimize the error for these conditions. The results herein suggest that comprehensive statistical climatologies may need to be developed to capture the non-normal traits of the 6 h forecast.