This article provides a new estimate of error growth
models' parameters approximating predictability curves and their
differentials, calculated from data of the ECMWF forecast system over the
1986 to 2011 period. Estimates of the largest Lyapunov exponent are also
provided, along with model error and the limit value of the predictability
curve. The proposed correction is based on the ability of the Lorenz (2005) system to simulate the predictability curve of the ECMWF forecasting
system and on comparing the parameters estimated for both these systems, as
well as on comparison with the largest Lyapunov exponent (

Forecast errors in numerical weather prediction systems grow in time because of the inaccuracy of the initial state (initial error), chaotic nature of the system itself, and the model imperfections (model error). The growth of forecast error in weather prediction is exponential on average. As an error becomes larger, its growth slows down and then stops, with the magnitude saturating at about the average distance between two states chosen randomly from dynamically and statistically possible states (limit, or saturated, error). For very short lead times the error growth could be superexponential either due to small-scale processes (Zhang et al., 2019) or due to decorrelation between analysis and forecast errors. This average growth of forecast error as a function of time is called the predictability curve.

Predictability curves (Froude et al., 2013) of the European Centre for
Medium-Range Weather Forecasts (ECMWF) numerical weather prediction system
are calculated by the approach developed by Lorenz (1982), whereby two types
of error growth can be obtained (Lorenz, 1982). The first type is calculated
as the root mean square difference between forecast data of increasing lead
times and analysis data valid for the same time. This error growth estimate
consists of initial and model error that is often referred to as practical
predictability, but following Lorenz (1982) we will call it the

Over the years several error growth models approximating predictability
curves have been developed, aiming to quantify Lyapunov exponents, model
errors (for the imperfect model case in which the atmosphere is not perfectly
modeled), and limit (saturated) errors. The first, called quadratic (Km),
was designed by Lorenz (1969). Dalcher and Kalney (1987) added model error
to the quadratic model, and Savijarvi (1995) changed it to the form
(Km

Values of parameters calculated from error growth models are used to evaluate the improvement of the ECMWF forecasting system (Magnusson and Kallen, 2013), to estimate the predictability or the limit error (Bengtsson et al., 2008), to quantify impacts of different model resolutions (Buizza, 2010), and to study chaos and model error on different spatial–temporal scales (Žagar et al., 2015, 2017). They are also used by researchers when the need arises to estimate chaoticity, model error, or predictability, but their validity cannot be proven because standard methods (Sprott, 2006) to calculate the largest Lyapunov exponents for the ECMWF forecasting system cannot be used due to a large number of variables. An independent value estimated from forecast and analysis anomalies can be calculated for the limit error (Simmons et al., 1995), and its validity will be discussed. The need for correct values of error growth models' parameters is increasing because the quadratic model with model error is used to describe multiscale weather (Zhang et al., 2019); a parameter that usually measures model error represents the intrinsic upscale error growth and propagation from small scales here.

This article intends to provide a new estimate of parameters of error growth models in the ECMWF forecasting system calculated from data over the 1986 to 2011 period. The correction is based on comparing the parameters calculated from the error growth models for the L05 system and the ECMWF forecasting system as well as on comparison with the largest Lyapunov exponent and the limit value of the predictability curve of the L05 system that can be calculated independently and with sufficient accuracy. To make the correction valid, predictability curves of the ECMWF forecasting system and the L05 systems are compared for two different methods (arithmetic and geometric averages), and the number of variables of the L05 system pertaining to the best match of the predictability curves is identified. As a result, a new approach to the calculation of model error based on a comparison of models is presented.

This article is divided into seven sections. The second describes the experimental setting. The third describes calculation of the predictability curves. The fourth provides a comparison of predictability curves of the ECMWF forecasting system and the L05 system, and the fifth deals with the estimation of Lyapunov exponents, model, and limit errors of the ECMWF forecasting system based on the correction. Discussion and conclusions are then presented in the final two sections.

The L05 model is based on the low-dimensional atmospheric system presented by
Lorenz (1996). It is a nonlinear model, with

To calculate predictability curves (Lorenz, 1996), arbitrary values of the
variables

In each time step

To calculate predictability curves for the ECMWF forecasting system (EFS)
values of 500 hPa geopotential height are used. Data were obtained from
ECMWF (Magnusson, 2018). Lower-bound predictability curves are calculated
(Magnusson and Kallen, 2013) from 21 root mean squares over the
Northern Hemisphere (20–90

Description of symbols that indicate types of predictability curve,
types of mean and systems for prediction error

Values of the largest global Lyapunov exponents

Comparisons of model predictability curves are done through values
normalized by the limit (saturated) errors (

Predictability curves of the ECMWF (26 annual averages) and L05 systems are
compared to find a setting of the L05 system (number of variables

Predictability curves of the L05 system show negative growth for the first time step (6 h) but turn into an increase thereafter. At the second time step (12 h) values of predictability curves reach approximately the same values as they had initially. A possible explanation could be that initial errors set the initial state off the attractor and a decrease occurs because the first tendency is to get on the attractor (Brisch and Kantz, 2019). With an increase in average errors, chaotic behavior becomes dominant. Predictability curves of the ECMWF forecasting system do not exhibit this type of behavior. This may be because of larger time steps or methods of objective analysis. We aim to get the most similar predictability curves of both models, and therefore the first two time steps (up to 12 h) of the L05 model's predictability curves are filtered out.

A description of symbols that indicate the type of prediction error

Predictability curves calculated by arithmetic and geometric mean show a
significant difference for the L05 system (all

The comparison of predictability curves is done with given initial values.
Predictability curves of the ECMWF forecasting system are normalized by

A comparison of lower-bound predictability curves (Fig. 2) shows the most
similar predictability curves of the ECMWF forecasting system and the L05
system for the L05 system calculated by arithmetic mean with

Comparison of upper-bound predictability curves

Comparison of lower-bound predictability curves

Parameters of error growth models are the Lyapunov exponent, model error,
and limit error. They are estimated from approximations of predictability
curves or differences of predictability curves

The calculation is done for the ECMWF forecasting system (26 annual
averages) and the L05 system (

The rms values calculated over all initial errors used for the L05
system (

The rms values calculated over all initial errors used for the L05
system (

The average values of parameters

There are significant differences of parameters

Approximations of differences of upper-bound predictability curves
(representative samples).

Approximations of differences of lower-bound predictability curves
(representative samples).

Note that the description of symbols that indicate the type of parameters of
error growth models

Lyapunov exponents

Average values over upper- and lower-bound predictability curves of
Lyapunov exponents

New limit values

Limit values

The argument that favors

Parameters

Absolute values of differences of parameters

These arguments are taken as proof of the validity of

At the end of this section, it is important to remind the readers about the
importance of the correct values of the parameters. Zhang et al. (2019)
used Km

The values of error growth models' (Eqs. 8–13) parameters that
approximate predictability curves and their differences (Figs. 3 and 4) in
the ECMWF forecast system (Tables 3 and 4) were recalculated. This is based on
similarities of normalized upper- and lower-bound predictability curves
(Figs. 1 and 2) of the ECMWF forecasting system (annual arithmetic mean of
geopotential heights of 500 hPa from years 1986–2011) and the L05 system
(

Lyapunov exponents of the ECMWF forecasting system were recalculated by Eq. (14). The average value over all error growth models for upper-bound
predictability is in the range

New limit values were calculated from the error growth models by Eq. (15).
For upper-bound predictability curves, the average value over all error
growth models is in the range

The argument that favors limit values calculated by Eq. (15) (Fig. 7, black full curves) instead of limit values calculated by Eq. (7) (Fig. 7, black dashed curves) is based on the new definition of model error (Eq. 16) that shows a decreasing trend with increasing years for predictability curves with limit values calculated by Eq. (15) and an almost constant trend with increasing time (slight decrease can be due to the error of approximations) for predictability curves with limit values calculated by Eq. (7), which is theoretically impossible (Fig. 9a). This new model error calculated as a difference of model error parameters between the upper- (Fig. 8a) and lower-bound (Fig. 8b) predictability curves supports model error parameters calculated for upper-bound predictability curves that are used to represent the intrinsic upscale error growth and propagation from small scales (Zhang et al., 2019).

The ECMWF forecasting system dataset was obtained from the personal
repository of Linus Magnusson (Magnusson, 2013). The L05 system dataset,
products from the ECMWF forecasting system dataset, codes, and figures were
conducted in Wolfram Mathematica, and they are permanently stored at

HB proposed the idea, carried out the experiments, and wrote the paper. AR and JM supervised the study and co-authored the paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are grateful to Linus Magnusson for offering the dataset.

This research has been supported by the Czech Science Foundation (grant no. 19-16066S).

This paper was edited by Richard Neale and reviewed by two anonymous referees.