It has been documented several times that there are variables in the atmosphere that come from non-normal distributions (Biondini, 1976; López,1977; Mielke et al.,1977; Toth and Szentimrey, 1990; Sauvageot, 1994; Yang and Pierrehumbert, 1994; Miles et al., 2000; O'Neill et al., 2000; Harmel et al., 2002; Stephens et al., 2002; Foster and Bevis, 2003; Zhang et al., 2003; Cho et al., 2004; Sengupta et al., 2004; Foster et al., 2006; Perron and Sura, 2013). It is shown in Fletcher (2010), that atmospheric variables may come from different probability distributions depending on the season. If this is the case then the variables' inherent distributions could also be conditioned on large-scale climatic dynamics.

If a testing procedure can be established that determines the nature of a variable, then an appropriate analysis scheme may be chosen to suit the variable. Many numerical weather prediction centers include some form of variational data assimilation, or Kalman filter, for their analyses and forecasting scheme, that are dependent on a normal distribution assumption for the error description. These centers include the Met Office (Rawlins et al., 2000), the European Centre for Medium-Range Weather Forecasts (Rabier et al., 2000), Météo-France (Fischer et al., 2005), Meteorological Service of Canada (Gauthier et al., 2007), the Naval Research Laboratory (NRL) Atmospheric Variational Data Assimilation System-Accelerated Representer (Rosmond and Xu, 2006), and the National Centers for Environmental Predication's Gridpoint Statistical Interpolation (Kleist et al., 2009). For more thorough reviews of variational data assimilation see Fletcher (2010) and Fletcher and Jones (2014).

In addition to operational data assimilation systems, the normal distribution assumption for the modeling of errors is also made for satellite retrieval systems, for example in the National Oceanic and Atmospheric Administration's Microwave Integrated Retrieval System (MiRS) (Boukabara et al., 2011), but where a logarithmic transform is used to convert a lognormally distributed variable into a normally distributed variable. This transform approach is also used in the Canadian Middle Atmosphere Model to make the state more normally-distributed (Polavarapu et al., 2005). While moment statistics have been used to analyze atmospheric variables (Perron and Sura, 2013), as of this writing the authors are unaware of any testing procedure attempting to classify the statistical framework of mixing ratio, temperature, wind and surface pressure.

There have been previous studies that have shown that variables including precipitation (Biondini, 1976; Mielke et al., 1977; Sauvageot, 1994; Cho et al., 2004), total precipitable water (Foster and Bevis, 2003; Foster et al., 2006), extreme temperatures (Toth and Szentimrey, 1990; Harmel et al., 2002), cloud and radar echo populations (López, 1997) cloud droplet size (Miles et al., 2000), liquid water path (Sengupta et al., 2004; Stephens et al., 2002), aerosol optical depth (O'Neill et al., 2000), tropical water vapor (Zhang et al., 2003), and relative humidity (Yang and Pierrehumbert, 1994) do not conform to a normal distribution to describe their behavior. A climatology of nine variables' distributional characteristics is analyzed in Perron and Sura (2013). Some of the studies considered spatial data while others used time-series data. These studies used a variety of techniques to quantify the nature of these distributions including probability density-fitting via moment calculations, χ2 goodness-of-fit tests, moment statistics, the Shapiro–Wilk tests, and simply plotting histograms with prominent non-normal distribution features.

Across a variety of disciplines it is often convenient, and somewhat innocuous, to treat measured variables as normally distributed in nature. This can misrepresent the inherent summary statistics due to a loss of information (e.g., lack of higher statistical moment information), and can be harmful within certain applications of the data. If a model, or algorithm, incorrectly assumes that a random variable is normally distributed then the properties of this distribution may skew its output.

A variable's probability distribution dictates the probabilistic solution that is found when using data assimilation techniques. In 3-D variational assimilation the cost function is given by J(x)=12(x-xb)TB-1(x-xb)+12(y-h(x))TR-1(y-h(x)) where B is the background error covariance matrix, R is the observational error covariance matrix, y is the vector of observations, and h(x) is the non-linear observation operator. The background component of Eq. (1) includes the difference between the minimizing solution x and the background distribution xb. If both x and xb are assumed to be indepedent normally-distributed, then the difference of these variables is also a normally-distributed random variable. If these variables are not normally-distributed, or moreover lognormally-distributed, then the difference is not a normally-distributed random variable (Fletcher, 2010), however the ratio of the variables is a lognormally-distributed random variable (Casella and Berger, 2002). It is an implicit model assumption that the solution x and the background xb come from the same probability distribution. It should also be noted that if the “errors” are assumed to be unbiased then μtrue=μb, which means that the expected value of the minimizing solution x and background xb are centered at the same point but have different spread (variance) and therefore skewness. It has been previously observed that that mixing ratio errors are not normally distributed (Dee and da Silva, 2003) and in fact are lognormally distributed (Daley and Barker, 2001).

Biases could also be introduced in data assimilation and retrieval systems that assume variables, and hence their errors, are normally distributed when they actually follow a non-normal distribution in nature. A clear example of where this can be problematic is if a computed value is physically impossible, such as relative humidity taking a negative value. This dubious value may be incorrectly incorporated into the analyses, or reset to a lower bound near zero. In either case this is certainly less desirable than solving for the correct value using an appropriate scheme that incorporates its correct underlying probability distribution. Recently, mixed normal-lognormal variational data assimilation methods have been developed in 3-D (Fletchera and Zupanski, 2006a, b, 2007), and 4-D in Fletcher (2010). These initial full field formulations were not consistent with the current operational incremental configurations. However, a derivation and testing of a mixed multiplicative and additive incremental 3-D- and 4-D-VAR for a control vector that contains both normal and lognormally distributed variables is presented in Fletcher and Jones (2014).

Evidence of how an assimilation scheme improves based on the distribution of the observational errors is shown in Fletcher and Jones (2014). Using the Lorenz '63 chaotic model the authors show that a lognormal-based cost function performs better than the current normal formulation given lognormal errors. Those conclusions result from testing observations of varying accuracy, sparseness in time, and over different window lengths.

Given that there is now a mathematical framework for assimilating mixed normal-lognormally distributed variables/errors, techniques are needed that can inform the user of a mixed system when to switch between a full normal distribution-based version or a mixed normal-lognormal-based version to optimize the performance of the system and to make it consistent with the “current” observed probabilistic behavior.

Therefore, the motivation of this work is to design a set of tests that can be performed offline between cycles or windows such that the configuration for the approximation for the background error covariance matrix, cost function, Jacobian and approximations to the Hessian, if used, can be ready for the next minimization step.

Given the motivation to detect a non-normal, specifically a lognormal signal, we use 1 ∘ resolution data from the National Oceanic and Atmospheric Administration (NOAA) Global Forecast System (GFS) 00:00 UTC 6 h forecast between 1 January 2005 and 31 December 2005 defined on a 181×360 grid. The forecasts, which are the GFS outputs, at each grid point form the time series. The data is analyzed over the entire year as well as on a “seasonal” basis by considering 3 months at a time (January–March, April–June, July–September, and October–December). The sample sizes are consistent with the suggested size from Croarkin and Tobias (1999) of 2n25, where n is the total number of observations available. The chosen variables include mixing ratio, temperature, surface pressure and wind at levels 100, 200, 300, 500, 700, 850, and 1000 hPa.

While there are transformation techniques employed by operational centers for moisture (Bocquet et al., 2010), the Navy Operational Global Atmospheric Prediction System (NOGAPS) previously used the logarithm of specific humidity (Eckermann et al., 2004), which is equivalent to mixing ratio (Dee and da Silva, 2003) analyzed in this study. In Fletcher and Zupanski (2007) it is shown that a logarithmic transformation finds the median in multivariate lognormal space, which is positively biased relative to the mode, or the most likely state.

In this work we propose using easily calculable statistics and hypothesis testing to show that these variables described above show strong evidence of a non-normal nature, or more specifically, a lognormal behavior. The hypothesis tests considered in this paper include the Jarque–Bera, Shapiro–Wilk, and χ2 goodness-of-fit. In addition, a composite hypothesis test is proposed that includes all of the decisions made by the aforementioned tests. Such tests are a requirement of advanced methods (Fletcher and Zupanski, 2007; Fletcher, 2010; Song et al., 2012; Fletcher and Jones, 2014) that are able to use multiple probability models.

The format of the remainder of this paper proceeds as follows: Sect. 2 describes the formulation of the hypothesis tests as well as the test statistics. In Sect. 3 results of these tests are presented. In Sect. 4 conclusions and a discussion of the results of Sect. 3 are presented.

In this section the statistical methods that are used to detect a non-normal distribution signal are presented along with tests to see if the distribution is a lognormal distribution. The random sample x1,…,xn∈X of independent and identically distributed (iid) observations is taken from the GFS data for each of the hypothesis tests that all rely on a significance level of α=0.01. This value of α indicates a 99 % confidence-level in the results of the testing procedures.

The samples' autocorrelation has been checked in order to verify the iid assumption for the hypothesis tests. While there is some autorcorrelation in the samples, we attempt to minimize its effect by choosing such a small α. Histograms of the data are also presented in order to verify the validity of the results of the hypothesis tests as well. The iid assumption on any data set found in nature is difficult to assert and it is also noted that many methods, including the National Meteorological Center (NMC) method (Parrish and Derber, 1992), make no correction for autocorrelation.

The results of the time-series hypothesis tests for mixing ratio and temperature resulted in numerous figures and data plots displaying the non-normal and lognormal nature of the GFS data. An overview of these results is presented along with a more detailed analysis of specific points of interest. Instead of presenting the results of the Shapiro–Wilk, Jarque–Bera, and Chi-squared tests only the results of the Composite Test are shown which incorporate all of the results simultaneously.

For each point of the GFS data, an forecast from each day between 1 January 2005 through 31 December 2005 makes up the random variable X for one year. This data is also broken down into four “seasons,” i.e. 1 January 2005 through 31 March 2005 (denoted as JFM in all figures), 1 April 2005 through 30 June 2005 (denoted AMJ), 1 July 2005 through 30 September 2005 (denoted JAS), and 1 October 2005 through 31 December 2005 (denoted OND).

Since mixing ratio and temperature have been shown to be non-normally distributed and in many cases appear to be lognormally distributed, 3-D- and 4-D-VAR data assimilation schemes that include lognormal cost functions for both the observations and the apriori background may be required for more accurate results. This would have implication on the forecast skill of a DA system, or for a retrieval system, as the analysis state from the minimization of the mixed distribution cost function should be consistent with the probabilistic behavior of the true state. The normal assumption, while convenient and easily adaptable, may need to be more carefully considered in light of these results.

While it is true that a lognormal distribution with a small variance looks very similar to a normal distribution, the detection methods used in this paper attempt to operationally handle large amounts of data similar to the resolution of an inner loop in incremental data assimilation schemes. It is in this end that these statistical procedures have been demonstrated in order to understand the true nature of atmospheric variables.

The time-series data clearly indicates data for mixing ratio and temperature will follow a lognormal distribution in certain areas. These results give light to the fact that the normal distribution assumption is not a valid assumption for the basis of the data assimilation and variational based retrieval systems and suggests that more research is needed to study the impact of assuming a normal distribution fit on forecast skills, variational observational quality control as well as the gross error check (Lorenc and Hammon, 1988).

Therefore this work suggests that statistical climatology tests need to be developed on a seasonal, or possibly a monthly basis, as the distributions that are found for specific variables indicate which distribution's cost function should be used in the assimilation schemes as a function of space and time. Ideally a real-time decision of how the data is statistically structured would be determined, ensuring that the correct scheme is chosen. In either case, it is the goal that an objective decision methodology be available for an appropriate scheme based on the nature of the data. The choice under what observational conditions to apply alternative Baysian models is now made as an objective decision through the procedure used and demonstrated in this work.

Future work can consider longer time-series, more vertical levels, other atmospheric variables such as column water vapor when a boundary layer cloud is present as seen in Fletcher (2010), and other statistical methods including the Akaike information criterion (Akaike, 1974). The possible future benefit of the Akaike information criterion (AIC) is that it detects the best distribution for a random variable based on information theory which could then give guidance for what other distributions need to be included in the variational cost function. AIC balances the goodness-of-fit of a distribution while minimizing the number of model parameters.

It has been shown in Fletcher and Jones (2014) that there is a negative impact on the performance of a normal distribution only incremental 4-D-VAR when lognormal forecasts are assimilated. However, when the same observations were assimilated in a lognormal-based incremental 4-D-VAR, then there was no negative impact on the analysis error. Therefore, the need to determine which distribution the observations and their errors come from is important to minimize the impact of these errors on the analysis of a DA system and the subsequent forecast. In this paper methodologies have been developed and tested with the 2005 GFS 00:00 UTC 6 h forecast and it has been shown that there are lognormal signals in the forecasts. This therefore suggests a need for statistical climatologies to be developed and for these climatologies to also be linked in near real-time with the data assimilation and retrieval systems.