the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Recalculation of error growth models' parameters for the ECMWF forecast system

### Aleš Raidl

### Jiří Mikšovský

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 (*λ*=0.35 d^{−1}) and limit value of the predictability curve (*E*_{∞}=8.2) of the Lorenz system. Parameters are calculated from the quadratic
model with and without model error, as well as by the logarithmic,
general, and hyperbolic tangent models. The average value of the
largest Lyapunov exponent is estimated to be in the *<* 0.32;
0.41 *>* d^{−1} range for the ECMWF forecasting system; limit
values of the predictability curves are estimated with lower theoretically
derived values, and a new approach for the calculation of model error based on
comparison of models is presented.

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 *lower-bound predictability curve* (*L*). The
second type is calculated as the root mean square difference between pairs
of forecasts valid for the same time but with times differing by some fixed
time interval (the difference between two forecasts issued with 24 h lag but
valid at the same time is used in this article). This type, which is
historically referred to as the perfect model assumption, consists of initial
error, and we will call it the *upper-bound predictability curve* (*U*). Predictability curves of Lorenz's 05
system (L05; Lorenz, 2005) can be controlled by model parameters and by the
size of the initial error, and they are set to be as close to predictability
curves of ECMWF forecasting system as possible.

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_{β}) that is used today. An alternative, called the logarithmic
model (Lm), was introduced by Trevisan et al. (1992) and Trevisan (1993). The general model
(Gm) was introduced by Stroe and Royer (1993, 1994). All these models are
based on time derivatives of the error (error growth rate). Newer models
approximate the predictability curve directly by the hyperbolic tangent
(Tm and Tm_{β}) (Žagar et al., 2017).

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 *N* variables connected by
governing equations:

$n=\mathrm{1},\mathrm{\dots},N$.
*X*_{n−2}, *X*_{n−1}, *X*_{n}, and *X*_{n+1} are
unspecified (i.e., unrelated to actual physical variables) scalar
meteorological quantities, *F* is a constant representing external forcing, and
*t* is time. The index is cyclic so that ${X}_{n-N}={X}_{n+N}={X}_{n}$ and
variables can be viewed as existing around a circle. Nonlinear terms of Eq. (1) simulate advection. Linear terms represent mechanical and thermal
dissipation. The model quantitatively, to a certain extent, describes
weather systems, but, unlike the well-known Lorenz model of atmospheric
convection (Lorenz, 1963), it cannot be derived from any atmospheric dynamic
equations. The motivation was to formulate the simplest possible set of
dissipative chaotically behaving differential equations that share some
properties with the “real” atmosphere. One of the model's
properties is to have five to seven main highs and lows that correspond to
planetary waves (Rossby waves) and a number of smaller waves that correspond
to synoptic-scale waves. For Eq. (1) this is only valid for *N*=30 and that
is, as will be seen, not sufficient for the experimental setting.
Therefore, spatial continuity modification of the L05 system is used, whereby
Eq. (1) is rewritten to the form

where

If *L* is even, ${\sum}^{\prime}$ denotes a modified summation, in which the first and
last terms are to be divided by 2. If *L* is odd, ${\sum}^{\prime}$ denotes an ordinary
summation. Generally, *L* is much smaller than *N* and $J=L/\mathrm{2}$ if *K* is even and $J=(L-\mathrm{1})/\mathrm{2}$
if *L* is odd. For comparison with predictability curves of the ECMWF
forecasting system, we choose *N*=30, 60, 90, 120, 150, and 360. To keep a
desirable number of main pressure highs and lows, Lorenz (2005) suggested to
keep the ratio $N/L=\mathrm{30}$ and therefore *L*=1, 2, 3, 4, 5, and 12. For even values of
*L* we have *J*=1, 2, and 6, and for odd values of *L* we have *J*=0, 1, and 2. The parameter
*F*=15 is selected as a compromise between a Lyapunov exponent that is too high
(smaller *F*) and undesirable shorter waves (larger *F*). For this setting and by
the method of numerical calculation (Sprott, 2006), the largest global
Lyapunov exponents are calculated (Table 2). By the definition of Lorenz (1969): “A bounded dynamical system with a positive Lyapunov exponent is
chaotic”. Because the value of the largest Lyapunov exponent is positive
and the system under study is bounded, it is chaotic. Strictly speaking
(Aligood
et al., 1996), we also need to exclude the asymptotically periodic behavior,
but such a task is impossible to fulfill for the numerical simulation. The
choice of parameters *F* and time unit =5 d is made to obtain a similar
value of the largest Lyapunov exponent as the ECMWF forecasting system.

To calculate predictability curves (Lorenz, 1996), arbitrary values of the
variables *X*_{n} are chosen, and, using a fourth-order Runge–Kutta method
with a time step Δ*t*=0.05 or 6 h, they are integrated forward for
14 400 steps or 10 years. Final values *X*_{0,n}, which should be free of
transient effect, are the initial values of “reality”. Initial values of
“prediction” are ${X}_{\mathrm{0},n}^{\prime}={X}_{\mathrm{0},n}+{e}_{\mathrm{0},n}$, where *e*_{0,n} is the
initial error and it is chosen randomly from a normal distribution *N**D*(*μ*;*σ*), where *μ*=0 is mean and *σ* is the
standard deviation, which is chosen from comparison with the ECMWF
forecasting system. From *X*_{0,n} and ${X}_{\mathrm{0},n}^{\prime}$ Eq. (2) is
integrated forward for 37.5 d (*K*=150 steps). For upper-bound
predictability curves, *X*_{n} and ${X}_{n}^{\prime}$ are chosen with the same
number of variables (*N*=30, 60, 90, 120, 150). For lower-bound
predictability curves, *X*_{n} is defined by *X*_{0,n} and by Eq. (2) with
*N*_{0}=360 and ${X}_{n}^{\prime}$ by ${X}_{\mathrm{0},n}^{\prime}$ and by Eq. (2) with *N*=30,
60, 90, 120, and 150. The size of the model error is corrected by the difference
of *N* for *X*_{n} and ${X}_{n}^{\prime}$. If, for example, *N*=120 then *X*_{n} is
compared with ${X}_{n}^{\prime}$ in each third point of *N*_{0}. This method was
presented by Lorenz (2005). Although not only resolution but also physical
parameterization affects the deficiencies of the ECMWF system, which make it
different from the real atmosphere, Buizza (2010) showed that a comparison
of predictability curves of the ECMWF system calculated from differences of
prediction and analysis as well as from two predictions of systems with different
horizontal resolutions leads to the same overall conclusions. Despite the
sub-differences mentioned by Buizza (2010), this method is sufficient for
comparing the L05 system and the ECMWF forecasting system.

In each time step Δ*t* of numerical integration *N* “real” and *N*
“predicted” values are obtained. The size of the error at a given time for
upper-bound predictability curves is ${e}_{n}\left(k\cdot \mathrm{\Delta}t\right)={X}_{k,n}^{\prime}-{X}_{k,n}$, where $k=\mathrm{1},\mathrm{\dots},K$ and
$n=\mathrm{1},\mathrm{\dots},N$ and for lower-bound predictability curves ${\mathit{\epsilon}}_{n}\left(k\cdot \mathrm{\Delta}t\right)={X}_{k,n}-{X}_{k,{n}^{\prime}}$,
where $k=\mathrm{1},\mathrm{\dots},K$, $n=\mathrm{1},\mathrm{\dots},N$ (except for *N*_{0}).
${n}^{\prime}=\mathrm{1},\mathrm{\dots},N$ (except for *N*_{0}) is the location of the value
${X}_{k,{n}^{\prime}}$ for *N*=360, where ${n}^{\prime}=n\cdot {N}_{\mathrm{0}}/N$ for *N*=30,
60, 90, 120, and 150. The predictability curves of the ECMWF forecasting system,
in this case, are obtained from annual averages of daily data. To simulate
that, the number of runs *M*=400 is made. In each new run, initial values
*X*_{0,n} are the last values *X*_{K,n} from the previous run. *M*⋅*N* values are obtained for each *k*. Final formulas of prediction errors that
constitute predictability curves by calculation with arithmetic mean (*A*)
are

Formulas to calculate prediction errors by geometric means (*G*) are

For an overview of the symbols see Table 1.

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^{∘} N) obtained daily from 1 January 1986 to 31 December 2011. Means are differences between operational
forecasts and analyses from ERA-Interim for a given day. Forecasts range
from 0.5 d ago relative to the given day to 10 d ago, with time step
0.5 d. The difference between operational analysis and analysis from
ERA-Interim is taken as the initial error. Upper-bound predictability curves
are calculated (Magnusson and Kallen, 2013) from 27 root mean
squares over the Northern Hemisphere (20–90^{∘}) obtained
daily from 1 January 1986 to 31 December 2011. Means are differences between
two operational forecasts issued with a 1 d lag but that are valid on the same
day. Specifically, the following differences are obtained for a given day
(hours): 0–24, 6–30, 12–36, 18–42, 24–48, 30–54, 36–60, 42–66,
48-72, 54–78, 60–84, 66–90, 72–96, 78–102, 84–108, 90–114, 96–120,
108–132, 120–144, 132–156, 144–168, 156–180, 168–192, 180–204,
192–216, 204–228, and 216–240. Prediction errors constituting the
predictability curves are calculated as annual averages of daily data.
Detailed information about calculating predictability curves of the ECMWF
forecasting system can be found in Lorenz (1982).

Comparisons of model predictability curves are done through values normalized by the limit (saturated) errors (${E}_{\mathrm{\infty},U}=\underset{t\to \mathrm{\infty}}{lim}{E}_{U}$, ${E}_{\mathrm{\infty},L}=\underset{t\to \mathrm{\infty}}{lim}{E}_{L})$. Because the maximum forecast time for the ECMWF forecasting system is 10 d, presented predictability curves do not reach their limit value. An independent measure of limit error can be calculated as

where $\stackrel{\mathrm{\u203e}}{\left(f-c\right)}$ is the time-averaged anomaly with
respect to climate and $\stackrel{\mathrm{\u203e}}{\left(a-c\right)}$ is the
time-averaged analysis anomaly with respect to climate. The climate is
defined from ERA-Interim daily climatology. *E*_{∞,U} and *E*_{∞,L} differ if the ECMWF forecasting system does not sufficiently describe
the variability of the atmosphere (model error). More information can be
found in Simmons et al. (1995). Because it will be shown that values of
limit error calculated by this method are not correct,
predictability curves of the ECMWF forecasting system are normalized by
values calculated by Eq. (15).

Predictability curves of the ECMWF (26 annual averages) and L05 systems are
compared to find a setting of the L05 system (number of variables *N*, the
size of the initial errors, preference of arithmetic or geometric mean) that
gives the most similar progress of systems' predictability curves.

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 *E* in the
text is provided in Table 1. Initial values ${E}_{U}^{\mathrm{L}\mathrm{05}}\left(\mathrm{0}\right)$
and ${E}_{L}^{\mathrm{L}\mathrm{05}}\left(\mathrm{0}\right)$ or equivalently standard deviations
*σ* from a normal distribution *N**D*(*μ*;*σ*) of
the L05 system are calculated from a comparison of initial values of the
ECMWF system (26 annual averages) that are normalized (*E*_{Norm}) by limit
(saturated) errors ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (15). Upper-bound
predictability curves start for the ECMWF forecasting system on day 1 (the
difference between 1 d prediction and the analysis), and therefore
${E}_{U}^{\mathrm{L}\mathrm{05}}\left(\mathrm{0}\right)$ is calculated from predictability curves
that are close on the first day: $\left({E}_{\mathrm{Norm}}^{\mathrm{L}\mathrm{05}}\left(\mathrm{1}\right)={E}_{\mathrm{Norm}}^{\mathrm{EFS}}\left(\mathrm{1}\right)\right)$. Initial values for the
L05 system are computed for *N*=60, 90, 120, and 150. Normalized predictability
curves with *N*=30 exhibit different evolution compared to predictability
curves of the ECMWF forecasting system, and they are not displayed. Initial
prediction errors ${E}_{U}^{\mathrm{L}\mathrm{05}}\left(\mathrm{0}\right)$ calculated by arithmetic
and geometric mean as well as *N*=60, 90, 120, and 150 have the same values, and these
values are in the interval ${E}_{U}^{\mathrm{L}\mathrm{05}}\left(\mathrm{0}\right)\in \u2329\mathrm{0.3};\mathrm{0.8}\u232a$, where lower values correspond to initial
prediction errors of the ECMWF system from later years and higher values
pertain to early years. For lower-bound predictability curves of the ECMWF
forecasting system, the initial error ${E}_{L}^{\mathrm{EFS}}\left(\mathrm{0}\right)$ is
computed as a difference between analysis from the operational forecasting
system and analysis from ERA-Interim. Initial errors of the L05 system
${E}_{L}^{\mathrm{L}\mathrm{05}}\left(\mathrm{0}\right)$ are calculated as ${E}_{L}^{\mathrm{L}\mathrm{05}}\left(\mathrm{0}\right)={E}_{\mathrm{\infty},L}^{\mathrm{L}\mathrm{05}}\cdot {E}_{L}^{\mathrm{EFS}}\left(\mathrm{0}\right)/{E}_{\mathrm{\infty},L}^{\mathrm{EFS}}$ and ${E}_{L}^{\mathrm{L}\mathrm{05}}\left(\mathrm{0}\right)\in \u2329\mathrm{0.2};\mathrm{0.7}\u232a$, where lower values correspond to initial
prediction errors of the ECMWF system from later years and higher values
pertain to early years. Initial values are the same for all *N* as well as arithmetic
and geometric mean.

Predictability curves calculated by arithmetic and geometric mean show a
significant difference for the L05 system (all *N*), which agrees with Ruiqiang
and Jianping (2011), and a minor difference for the ECMWF forecasting system.
For the L05 system and upper- and lower-bound predictability curves, the
maximal difference is between 6.5 % and 10.5 % of ${E}_{\mathrm{\infty},U}^{\mathrm{L}\mathrm{05}}$ or ${E}_{\mathrm{\infty},L}^{\mathrm{L}\mathrm{05}}$, and these maximal values occur between 5 and 9 d of forecast length. For the ECMWF forecasting system and upper- and lower-bound predictability curves, the maximal difference is 2 % of ${E}_{\mathrm{\infty},U}^{\mathrm{EFS}}$ or ${E}_{\mathrm{\infty},L}^{\mathrm{EFS}}$, and these maximal values occur at the
end of the forecast length (10 d). The choice of the averaging method does
not significantly change the evolution of the ECMWF forecasting system's
predictability curves, and it does not change values of parameters of the
approximations. For the L05 system, the choice of averaging method is
significant, and it changes values of the parameters. The reason for this
sensitivity can be found in the spread of values that are used for
averaging. For the ECMWF forecasting system, the values are closer to each
other than for the L05 system, and from the definition of means, it leads to
the aforementioned difference. Calculating predictability curves by
arithmetic and geometric mean, although it does not affect predictability
curves of the ECMWF forecasting system, is mentioned because it affects the
calculation of predictability curves of the L05 system, and this then affects
the comparison of predictability curves, which is important for
recalculation of error growth models' parameters for the ECMWF forecast
system.

The comparison of predictability curves is done with given initial values.
Predictability curves of the ECMWF forecasting system are normalized by
${E}_{\mathrm{\infty},U}^{\mathrm{EFS}}$ or ${E}_{\mathrm{\infty},L}^{\mathrm{EFS}}$ (Fig. 7, black full curves)
and for the L05 system by ${E}_{\mathrm{\infty},U}^{\mathrm{L}\mathrm{05}}$ and ${E}_{\mathrm{\infty},L}^{\mathrm{L}\mathrm{05}}$
displayed in Table 2 (for a description of the symbols see Table 1). For the
L05 system predictability curves are calculated with *N*=60, 90, 120, and 150
variables and by arithmetic and geometric mean (for lower-bound
predictability curves this sets different values of the model error). For
the ECMWF forecasting system only arithmetic mean is used.

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 *N*=90 (the
fact that this would mean unrealistic values of the model error for the
ECMWF forecasting system is further discussed). For upper-bound
predictability curves (Fig. 1), predictability curves for the L05 system
with *N*=90 are the most similar to the year 1999 for predictability
curves of the L05 system calculated by geometric mean and after 1999 by the
arithmetic mean.

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 $\left(\left(E\left(t+\mathrm{\Delta}t\right)+E\left(t\right)\right)/\mathrm{2};\left(E\left(t+\mathrm{\Delta}t\right)-E\left(t\right)\right)/\mathrm{\Delta}t\right)$, where *t* is time and
Δ*t*=0.25 d (Figs. 3 and 4). Error growth models considered here are

where parameters of Tm are *α*=2*a*, ${E}_{lim}=\mathrm{2}A$, and

where parameters of Tm_{β} are $\mathit{\alpha}=a\left(A+B\right)/A$, $\mathit{\beta}=a\left({A}^{\mathrm{2}}-{B}^{\mathrm{2}}\right)/A$, and ${E}_{lim}=A+B$. *E* is an average forecast
error. *t* represents time, *α* is the estimate of the Lyapunov exponent
*λ*, *β* is the parameter of model error ($\mathrm{d}E/\mathrm{d}t$ when
*E*=0), *E*_{lim} is the limit (saturated) value of *E* (value of *E* when $\mathrm{d}E/\mathrm{d}t=\mathrm{0}$, theoretically *E*_{∞}; Figs. 3 and 4), and *p*, *A*, *B*, *a*, and *b* are
parameters.

The calculation is done for the ECMWF forecasting system (26 annual
averages) and the L05 system (*N*=90) for arithmetic (*A*) and geometric (*G*)
means, upper-bound predictability curves (*U*), and lower-bound
predictability curves (*L*). See Tables 3 and 4 for root mean square (rms)
values of parameters $\stackrel{\mathrm{\u203e}}{\mathit{\alpha}}$, ${\stackrel{\mathrm{\u203e}}{E}}_{lim}$, $\stackrel{\mathrm{\u203e}}{\mathit{\beta}}$, and $\stackrel{\mathrm{\u203e}}{p}$ that are calculated over all initial errors used for
the L05 system and all calculated years for the ECMWF forecasting system.

The average values of parameters $\stackrel{\mathrm{\u203e}}{\mathit{\alpha}}$ and ${\stackrel{\mathrm{\u203e}}{E}}_{lim}$ are
higher for the lower-bound predictability curves than for the upper-bound
predictability curves. Upper-bound predictability curves should not include
model error (theoretically *β*=0), but from Table 4 it can be seen that
for the L05 system (arithmetic mean) the values are even higher than for the
lower-bound predictability curves. For the ECMWF forecasting system the
values of $\stackrel{\mathrm{\u203e}}{\mathit{\beta}}$ are higher for lower-bound predictability curves,
which is theoretically more acceptable, but $\stackrel{\mathrm{\u203e}}{\mathit{\beta}}$ is not zero for
the upper-bound predictability curves. A possible explanation can be the
sensitivity to correct approximation (cases with higher *β* have lower
*α*), but this cannot fully explain the discrepancy. For $\stackrel{\mathrm{\u203e}}{p}$
the values of upper- and lower-bound predictability curves are similar to
each other (L05 system and ECMWF forecasting system).

There are significant differences of parameters $\stackrel{\mathrm{\u203e}}{\mathit{\alpha}}$, ${\stackrel{\mathrm{\u203e}}{E}}_{lim}$, $\stackrel{\mathrm{\u203e}}{\mathit{\beta}}$, and $\stackrel{\mathrm{\u203e}}{p}$ between predictability curves calculated by arithmetic and geometric mean for the L05 system (for the ECMWF forecasting system only arithmetic mean is presented). The most significant differences are detected for $\stackrel{\mathrm{\u203e}}{\mathit{\beta}}$ and $\stackrel{\mathrm{\u203e}}{p}$; for $\stackrel{\mathrm{\u203e}}{\mathit{\beta}}$ values are closer to zero for geometric mean and values of predictability curves calculated by arithmetic mean are 2 or 3 times higher. Values of parameter $\stackrel{\mathrm{\u203e}}{p}$ are closer to $\stackrel{\mathrm{\u203e}}{p}=\mathrm{1}$ for geometric mean. This means that differences of predictability curves calculated by geometric mean have a shape that is close to a symmetric parabola (for example, Fig. 3a), but for the arithmetic mean the parabolic shape is skewed to the left (for example, Fig. 3c).

Note that the description of symbols that indicate the type of parameters of
error growth models *α*, *β*, *p*, and *E*_{lim} in the text is
provided in Table 1. The Lyapunov exponent of the ECMWF forecasting system
is recalculated by the formula

where *α*^{EFS} and *α*^{L05} are parameters of error growth
models and *λ*^{L05}=0.35 d^{−1}. The formula (Eq. 14) is based on
the assumption that if normalized predictability curves of the L05 system
and the ECMWF forecasting system are similar, then the differences between
true values of the largest global Lyapunov exponents (*λ*^{EFS},
*λ*^{L05}) and values determined from error growth models (*α*^{EFS}, *α*^{L05}) are similar (${\mathit{\lambda}}^{\mathrm{EFS}}-{\mathit{\alpha}}^{\mathrm{EFS}}\approx {\mathit{\lambda}}^{\mathrm{L}\mathrm{05}}-{\mathit{\alpha}}^{\mathrm{L}\mathrm{05}}$). Similarity of differences *λ*−*α*
allows us to estimate the largest global Lyapunov exponents of the ECMWF
forecasting system. For upper-bound predictability curves (the L05 system
with *N*=90 to the year 1999 calculated by geometric mean and after 1999
by arithmetic mean), the average value ${\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{U}^{\mathrm{EFS}}$ over all
error growth models is in the range 〈0.33;0.41〉 d^{−1} (Fig. 5a). Lm is not used because this error
growth model is not sufficient to approximate predictability curves. Root mean square errors (RMSEs)
of ${\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{U}^{\mathrm{EFS}}$ are mostly about 0.01 d^{−1} only in the
years 1991, 1995, and 1997; a 1999 RMSE is about 0.02 d^{−1}. For comparison,
RMSEs of ${\stackrel{\mathrm{\u203e}}{\mathit{\alpha}}}_{U}^{\mathrm{EFS}}$ are in the range 〈0.02;0.07〉 d^{−1} (Fig. 5a). For lower-bound
predictability curves (the L05 system with *N*=90 calculated by arithmetic
mean), the average value ${\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{U}^{\mathrm{EFS}}$ over all error growth
models is in the range 〈0.32;0.41〉 d^{−1} (Fig. 5b). RMSEs of ${\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}_{L}^{\mathrm{EFS}}$ are in the range
〈0.01;0.02〉 d^{−1}. For comparison,
RMSEs of ${\stackrel{\mathrm{\u203e}}{\mathit{\alpha}}}_{L}^{\mathrm{EFS}}$ are in the range 〈0.03;0.07〉 d^{−1} (Fig. 5b). The average value
${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}}^{\mathrm{EFS}}$ over upper- and lower-bound predictability
curves is shown in Fig. 6, and RMSEs of ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}}^{\mathrm{EFS}}$ are
mostly about 0.01 d^{−1}. Low values of RMSEs of ${\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}^{\mathrm{EFS}}$
compared to RMSEs of ${\stackrel{\mathrm{\u203e}}{\mathit{\alpha}}}^{\mathrm{EFS}}$ and similar values of
${\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}^{\mathrm{EFS}}$ for upper- and lower-bound predictability curves (low
values of RMSEs of ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}}^{\mathrm{EFS}}$) prove the validity of
${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}}^{\mathrm{EFS}}$. Values of ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}}^{\mathrm{EFS}}$ and ${\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}^{\mathrm{EFS}}$ are generally closer to parameters
*α*^{EFS} of Km_{β}, Tm_{β}, and Gm than to *α*^{EFS} of Km, Tm, and Lm, but none of the error growth models
approximate ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}}^{\mathrm{EFS}}$ (Fig. 6).

New limit values ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ are calculated from the error growth models by the formula

where ${E}_{lim}^{\mathrm{EFS}}$ and ${E}_{lim}^{\mathrm{L}\mathrm{05}}$ are values from error growth
models and ${E}_{\mathrm{\infty}}^{\mathrm{L}\mathrm{05}}=\mathrm{8.2}$. As in calculating *λ*^{EFS}, Eq. (15) is based on the assumption that if normalized predictability curves of
the L05 system and the ECMWF forecasting system are similar, then the
differences between true limit values (${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$, ${E}_{\mathrm{\infty}}^{\mathrm{L}\mathrm{05}}$) and values determined from error growth models (${E}_{lim}^{\mathrm{EFS}}$, ${E}_{lim}^{\mathrm{L}\mathrm{05}}$) are similar. In this case, however, only normalized
values can be compared:

Similarity of normalized differences ($\left({E}_{\mathrm{\infty}}-{E}_{lim}\right)/{E}_{\mathrm{\infty}}$) allows us to estimate new limit values of the ECMWF forecasting system.
For upper-bound predictability curves (the L05 system with *N*=90), the average
value over all error growth models ${\stackrel{\mathrm{\u203e}}{E}}_{\mathrm{\infty},U}^{\mathrm{EFS}}$ is in the
range 〈96;133〉 m (Fig. 7a). Lm is not
used because this error growth model is not sufficient to approximate
predictability curves. RMSEs of ${\stackrel{\mathrm{\u203e}}{E}}_{\mathrm{\infty},U}^{\mathrm{EFS}}$ are mostly
about 1 m only in the years 1987, 1988, 1995, 1997, and 2003, and in 2011 it is
about 2 m. For comparison, RMSEs of ${\stackrel{\mathrm{\u203e}}{E}}_{lim,U}^{\mathrm{EFS}}$ are in the
range 〈2;6〉 m (Fig. 7a). For lower-bound
predictability curves (the L05 system with *N*=90 calculated by arithmetic
mean), the average value over all error growth models ${\stackrel{\mathrm{\u203e}}{E}}_{\mathrm{\infty},L}^{\mathrm{EFS}}$ is in the range 〈114;134〉 m
(Fig. 7b). Lm is not used because this error growth model is not
sufficient to approximate predictability curves. RMSEs of ${\stackrel{\mathrm{\u203e}}{E}}_{\mathrm{\infty},L}^{\mathrm{EFS}}$ are mostly 3 m, and after the year 2004, they are 4 m. RMSEs of
${\stackrel{\mathrm{\u203e}}{E}}_{lim,L}^{\mathrm{EFS}}$ are in the range 〈3;6〉 m (Fig. 7b). Lower values of RMSEs of ${\stackrel{\mathrm{\u203e}}{E}}_{\mathrm{\infty},U}^{\mathrm{EFS}}$ and ${\stackrel{\mathrm{\u203e}}{E}}_{\mathrm{\infty},L}^{\mathrm{EFS}}$ calculated by Eq. (15)
compared to RMSEs of ${\stackrel{\mathrm{\u203e}}{E}}_{lim,U}^{\mathrm{EFS}}$ and ${\stackrel{\mathrm{\u203e}}{E}}_{lim,L}^{\mathrm{EFS}}$ prove the validity of ${\stackrel{\mathrm{\u203e}}{E}}_{\mathrm{\infty},U}^{\mathrm{EFS}}$ and
${\stackrel{\mathrm{\u203e}}{E}}_{\mathrm{\infty},L}^{\mathrm{EFS}}$.

The argument that favors ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (15) (Fig. 7, black full curves) instead of ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (7)
(Fig. 7, black dashed curves) is based on the parameter of model error
*β*. The most similar predictability curves of the L05 system and the
ECMWF forecasting system with ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (15)
are found for the L05 system with *N*=90 (for lower-bound predictability
curves calculated by arithmetic mean and for upper-bound predictability
curves calculated by geometric mean to 1999 and after by arithmetic mean).
The most similar predictability curves of the L05 system and the ECMWF
forecasting system with ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (7) are found
for the L05 system with *N*=90 by the arithmetic mean for upper- and lower-bound predictability curves. It means that if the comparison is valid and
model error is constant for the L05 system (same number of variables over
years in the L05 system means constant model error over years), it must also be
constant for the ECMWF forecasting system, but the calculation of
parameters ${\mathit{\beta}}_{L}^{\mathrm{EFS}}$ shows a decreasing trend with increasing time
(Fig. 8b). But parameters ${\mathit{\beta}}_{U}^{\mathrm{EFS}}$ have nonzero values (Fig. 8a)
that are close to ${\mathit{\beta}}_{L}^{\mathrm{EFS}}$ for some years, and that is
inconsistent with the theoretical expectation that upper-bound
predictability curves should be without model error; therefore, *β*
should be 0 m d^{−1}. This inconsistency can be solved by the new definition of
the model error. From Fig. 6 it can be seen that a closer value of *α*^{EFS} to ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}}^{\mathrm{EFS}}$ for *α*^{EFS} is approximated
from error growth models Km_{β}, Tm_{β}, and Gm than for
*α*^{EFS} approximated from error growth models Km, Tm, and Lm.
Gm has parameter *p* that defines skewness of the originally parabolic
shape of the difference of predictability curves. *p*=1 pertains to
symmetrical parabolic shape (Gm becomes Km) and *p*=0 means the greatest
skewness to the left (Gm becomes Lm). Parameters *β* also skew the
originally parabolic shape (Figs. 3 and 4). The model error can be seen as a
difference between skewness of upper- and lower-bound predictability curves,
and the new definition of model error would be

Results (Fig. 9a) show good agreement for ${\mathit{\beta}}_{L-U}^{\mathrm{EFS}}$ (Eq. 16)
calculated from Km_{β} and Tm_{β}, a decreasing trend of
${\mathit{\beta}}_{L-U}^{\mathrm{EFS}}$ with increasing time for predictability curves with
${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (15), and almost constant values of
${\mathit{\beta}}_{L-U}^{\mathrm{EFS}}$ with increasing years (slight decrease can be due to
the error of approximations) for predictability curves with ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (7). There is also good agreement with trends of
|*p*_{L}−*p*_{U}| (Fig. 9b). Because constant values of
*β*_{L−U} for predictability curves with ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$
calculated by Eq. (7) are not theoretically possible, predictability curves
with ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (15) are favored. The reason for
the decreasing trend of ${\mathit{\beta}}_{L-U}^{\mathrm{L}\mathrm{05}}$, which is found for predictability
curves of the L05 system with *N*=90 that are the most similar to
predictability curves of the ECMWF forecasting system normalized by
${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (15), is that they are partly
calculated by geometric and partly by the arithmetic mean.

These arguments are taken as proof of the validity of ${\stackrel{\mathrm{\u203e}}{\stackrel{\mathrm{\u203e}}{\mathit{\lambda}}}}^{\mathrm{EFS}}$ and ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (15). The reason for the overestimation of ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (7) (Fig. 7) can be found in the multiscale behavior of weather. If some events are predictable on a timescale longer than 10 d (for example, long-lived anomalies in sea surface temperature or soil moisture), then they would not be captured by medium-range weather forecast (Simmons et al., 1995; Brisch and Kantz, 2019). It is also possible that the overestimation is due to the different source of data used for calculation of ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ by Eqs. (7) and (15). For ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (7) only data from ERA-Interim (Janoušek, 2011) are used, but for ${E}_{\mathrm{\infty}}^{\mathrm{EFS}}$ calculated by Eq. (15) data from the operational forecast are employed.

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_{β} in the ECMWF forecasting system to estimate the influence
of different spatiotemporal scales with the parameter *β* newly representing
the intrinsic upscale error growth and propagation from small scales and
*α* representing synoptic-scale error growth. The results of our analysis
support this approach by the new definition of model error (Eq. 16)
and by showing the errors of approximations for individual error growth
models.

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
(*N*=90, arithmetic mean for lower-bound predictability curves; geometric
mean up to 1999 and arithmetic mean after 1999 for upper-bound
predictability curves). It is also based on knowledge of the largest
Lyapunov exponent (*λ*=0.35 d^{−1}) and the limit value of the
predictability curve (*E*_{∞}=8.2) of 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 〈0.33;0.41〉 d^{−1} (Fig. 5a) and RMSEs are mostly about 0.01 d^{−1}. For lower-bound predictability curves the average value over all error growth models is in
the range 〈0.32;0.41〉 d^{−1} (Fig. 5b). RMSEs are in the range 〈0.01;0.02〉 d^{−1}. The average value over upper- and lower-bound predictability
curves is shown in Fig. 6 and RMSEs are mostly about 0.01 d^{−1}. Values
of the Lyapunov exponent are generally closer to parameters *α*^{EFS} of
Km_{β}, Tm_{β}, and Gm than to *α*^{EFS} of Km,
Tm, and Lm (Fig. 6).

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 〈96;133〉 m (Fig. 7a) and RMSEs are mostly about 1 m. For lower-bound predictability curves the average value over all error growth models is in the range 〈114;134〉 m (Fig. 7b) and RMSEs are mostly 3 m.

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 https://doi.org/10.17605/OSF.IO/CEK32 (Bednář, 2020).

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.

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- Abstract
- Introduction
- Experimental setting
- Calculation of predictability curves
- Comparison of predictability curves
- Estimation of parameters
- Discussion
- Conclusions
- Code and data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References

- Abstract
- Introduction
- Experimental setting
- Calculation of predictability curves
- Comparison of predictability curves
- Estimation of parameters
- Discussion
- Conclusions
- Code and data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References