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  • View in gallery

    (a) Mean square error evolution for the Lorenz system, as obtained with an initial condition error sampled from a Gaussian of mean 0 and variance 10−3. The continuous and the short dashed lines refer to the MSE between the model and the reference system and between the MOS forecast and the reference system, respectively. The dashed–dotted line represents the difference. The inset displays the short-time evolution of the error in logarithmic scales. Temporal evolution of the variance of (b) x, (c) y, and (d) z for the model (continuous lines), the reference system (dotted lines), and the MOS forecast (dashed–dotted lines).

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    (a) Mean square error evolution for the x variable of the Lorenz model, as obtained with a perturbation of parameter a, a′ = a + 0.0001, for the model (continuous line) and MOS (short dashed line) forecasts. The full correction (drift + variability) is represented by the dashed–dotted line. (b) Short-term evolution of the drift (crosses) and variability (open dots) corrections for the x variable. The quadratic short-term evolution given in (24) and (28) is also plotted. (c) The variability correction with the 1-observable MOS scheme discussed in (b) and the one obtained with the 2-observable MOS scheme with model observables xm and ym. (d) The mean square error curves as in (a) (continuous and short dashed lines) but for the y component. (e) As in (b) but for the y component. (f)–(h) As in Figs. 1b–d but for the model perturbation of (a).

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    Mean square error evolution as obtained with a perturbation of one of the parameters of the Lorenz system, a′ = a + 0.01, and a perturbation of the initial conditions sampled from a Gaussian noise of mean 0 and variances ranging from 0.01 to 0.0001; σ2 = 0.01 (pluses), σ2 = 0.0025 (crosses), σ2 = 0.0009 (asterisks), and σ2 = 0.0001 (squares). The curves represent the analytical correction obtained for short times in section 3 up to the quadratic term.

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    Ratio of the mean square error of the MOS forecast and the model forecast for three different values of the parameter perturbation δa = a′ − a.

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    Mean square error along the y direction for a perturbation of parameter b (δb = 0.01). Three MOS schemes are used, 1-observable MOS (continuous line), 2-observable MOS with ym and xm (dashed line), and 2-observable MOS with ym and xmzm (dashed–dotted line).

  • View in gallery

    (a) Mean square error evolution as obtained with a perturbation of parameter η for the model (continuous line) and 1-observable MOS forecasts (dotted line). (b) The drift (crosses) and variability (open dots) corrections obtained with the 1-observable MOS scheme. The quadratic short-term evolution of the drift and variability corrections are also displayed; see (24) and (28). (c) As in (a) but for a perturbation of parameter α, δα = 0.01. (d) As in (b) but for a perturbation of parameter α, δα = 0.01. (e) Drift + variability corrections (lines with symbols) obtained with different MOS schemes for a perturbation of parameter α, δα = 0.01. The empty dots, crosses, and triangles correspond to the 1-observable MOS scheme, the 2-observable MOS scheme with two adjacent model observables, ψi and ψi−1, and the 2-observable MOS scheme with ψi and ∇2ψi, respectively. The lines without symbols represent the analytical short-term quadratic correction discussed in section 3.

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Dynamical Properties of Model Output Statistics Forecasts

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  • 1 Institut Royal Météorologique de Belgique, Brussels, Belgium
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Abstract

The dynamical properties of forecasts corrected using model output statistics (MOS) schemes are explored, with emphasis on the respective role of model and initial condition uncertainties. Analytical and numerical investigations of low-order systems displaying chaos indicate that MOS schemes are able to partly correct the impact of both initial and model errors on model forecasting. Nevertheless the amplitude of the correction is much more sensitive to the presence of (state dependent) model errors, and if initial condition errors are much larger than model uncertainties then MOS schemes become less effective. Furthermore, the amplitude of the MOS correction depends strongly on the statistical properties of the phase space velocity difference between the model and reference systems, such as its mean and its covariance with the model predictors in the MOS scheme. Large corrections are expected when the predictors are closely related to the sources of model errors. The practical implications of these results are briefly discussed.

Corresponding author address: S. Vannitsem, Institut Royal Météorologique de Belgique, Avenue Circulaire, 3, 1180 Brussels, Belgium. Email: svn@oma.be

Abstract

The dynamical properties of forecasts corrected using model output statistics (MOS) schemes are explored, with emphasis on the respective role of model and initial condition uncertainties. Analytical and numerical investigations of low-order systems displaying chaos indicate that MOS schemes are able to partly correct the impact of both initial and model errors on model forecasting. Nevertheless the amplitude of the correction is much more sensitive to the presence of (state dependent) model errors, and if initial condition errors are much larger than model uncertainties then MOS schemes become less effective. Furthermore, the amplitude of the MOS correction depends strongly on the statistical properties of the phase space velocity difference between the model and reference systems, such as its mean and its covariance with the model predictors in the MOS scheme. Large corrections are expected when the predictors are closely related to the sources of model errors. The practical implications of these results are briefly discussed.

Corresponding author address: S. Vannitsem, Institut Royal Météorologique de Belgique, Avenue Circulaire, 3, 1180 Brussels, Belgium. Email: svn@oma.be

1. Introduction

Operational numerical weather (or climate) predictions deteriorate as a function of lead time because of the presence of modeling and initial condition errors. To partly correct this decrease of skill postprocessors are commonly used based on (linear or nonlinear) statistical methods (see, e.g., Casaioli et al. 2003; Kalnay 2003; Marzban 2003; Wilks 2006). These are usually referred to as model output statistics (MOS) techniques. One of the most popular approaches consists in building a linear regression between a set of predictors provided by the NWP model and an observable at a certain lead time t and to use this statistical relation to perform corrected forecasts of this observable at the same lead time (Klein and Glahn 1974; Lemcke and Kruizinga 1988; Wilson and Vallée 2003; Dallavalle et al. 2004; Hart et al. 2004; Taylor and Leslie 2005).

Although the MOS approach is known to correct biases and some inaccuracies of forecasting systems, the nature of the errors that can be effectively corrected by the method (i.e., errors in the parameters in front of linear or nonlinear terms, truncation of a set of system’s variables, initial condition errors, etc.) needs to be further elucidated. In addition, although it has long been recognized that the (linear) MOS forecast converges to the climatological mean for long lead times, the connection between the correction and the underlying (deterministic) dynamics remains largely unexplored.

In this paper we investigate the impact of a (linear) MOS scheme on deterministic systems displaying chaos, sharing with the atmosphere the basic property of sensitivity to initial conditions. To this end, we adopt the probabilistic approach developed in a series of previous papers by the present authors (Nicolis et al. 1995; Vannitsem and Toth 2002; Nicolis 1992, 2003, 2004, 2005) in which the dynamics of the initial condition and of model errors were analyzed. The MOS schemes investigated will be as simple as possible to allow for a detailed analytical as well as numerical understanding of their impact on forecasting. To mimic the operational forecasting system models will be developed displaying small discrepancies with the original system’s equations and in which typical initial condition errors are present.

The MOS schemes used in the present paper are surveyed and some of their statistical properties are revisited in section 2. The short-time dynamical properties are then investigated analytically in section 3. The results of these investigations are confirmed and complemented in section 4 by a numerical analysis of the three-variable atmospheric Lorenz model (Lorenz 1984) and of a one-dimensional spatially distributed system known as the Kuramoto–Sivashinsky (KS) equation (Kuramoto and Tsuzuki 1976). The main conclusions are summarized in section 5.

2. The MOS technique revisited

The MOS technique aims at correcting current forecasts based on statistical information gathered from past forecasts (see, e.g., Wilks 2006). In its most popular form, it is based on a linear relation between the reference (the truth or the observation of the truth) variables that we want to predict, Xl, and a set of model observables (or predictors), {Vi(t)}, at a certain lead time t:
i1520-0493-136-2-405-e1
where Xl,C(t) is the corrected forecast. The parameters αl(t) and [βi,l(t)] are estimated using a set of past forecasts by minimizing a cost function J(t) = ΣKk=1[Xl,C,k(t) − Xl>,k(t)]2, where Xl,k(t) is the lth reference variable at time t at which the kth forecast is compared and K is the number of past predictions used to build the statistical relation (1). The minimization is performed by differentiating J(t) against the parameters. This leads to the following relations for the parameters:
i1520-0493-136-2-405-e2
i1520-0493-136-2-405-e3
where 〈〉 indicates a statistical average over the ensemble of forecasts K, or equivalently for an ergodic system (i.e., a system for which there exists a unique invariant set in phase space attracting the solution for almost all initial conditions) to the average over the ensemble of initial conditions on the attractor of the reference system, 〈f〉 = ∫ dX0 dY0ρ(X0, Y0) f, where f is any function of the variables Xl, {Vi} at time t and ρ(X0, Y0) is the invariant density of being at one particular point (X0, Y0) on the attractor of the reference solution.

In this context, two types of MOS equations can be developed depending on the problem of interest: (i) for predicting as close as possible one of the variables present in the reference system, Xi, and explicitly described by the model, Vi; and (ii) for evaluating an unresolved variable, Yj, by using the information provided by the model. The latter approach is not a correction of the forecast but is closer to a “downscaling” technique in which unresolved variables are evaluated based on the available model information. The first approach, which is closer to the idea of correcting, is the subject of the present paper.

The MOS equations can be used with as many variables as desired and in principle, the quality of the correction should improve when additional (well chosen) observables are used (e.g., Sokol 2003). In the present work we will focus on the properties of the two simple algorithms presented below, since they already capture the main properties of the technique.

a. 1-observable scheme

In the following the l index is dropped from (2) and (3) to simplify the notation. Using (2) and (3) for n = 1, one can compute the mean and variance of XC(t):
i1520-0493-136-2-405-e4
i1520-0493-136-2-405-e5
where σ2V(t) = 〈V(t)2〉 − 〈V(t)〉2 is the variance of the model observable at time t starting from the reference attractor, C[X(t), V(t)] is the covariance between X and V, and σ2X is the variance of the reference solution (independent of time for an autonomous ergodic system). Here, C[X(t), V(t)]2/σ2V(t)σ2X is the square of the correlation coefficient (the square of the anomaly correlation, ρ2), which ranges between 0 and 1. It is clear from these relations that although the nature of the correction to the mean cannot be anticipated, the variance of the corrected forecast is always smaller than or equal to the variance of the reference variable.

A central quantity in (5) is the covariance between the model and the reference system. If both systems are chaotic with sufficiently strong ergodic properties (i.e., any typical initial probability distribution will converge asymptotically toward the invariant probability distribution), the covariance between their respective solutions starting from close initial conditions will progressively decrease. In this case, one expects that the variance of the corrected forecast will decrease and eventually will be close to 0. Consequently the MOS forecast will be close to the mean for long lead time. This property of MOS schemes is precisely used to reduce the error of the forecasts of atmospheric variables for long lead times (see, e.g., Kalnay 2003; Wilks 2006), albeit at the expense of a good representation of variability.

Let us now turn to the mean square error (MSE) between the corrected forecast and the reference variable. Following the discussion above, one assumes that the predictor of the 1-observable MOS scheme is the same nominal variable as the forecasted one. The MSE can then be decomposed into three parts:
i1520-0493-136-2-405-e6
where the first term on the right-hand side is the mean square error between the model solution and the true system variable, and the two last terms are two negative corrections reducing the amplitude of the mean square error. Note that other decompositions can be used, such as in Marzban et al. (2006), but the advantage of the present one is to allow for the quantification of the correction as compared with the uncorrected forecasts.
Obviously the last term is associated with the correction of the mean of the forecast that can be referred to as a drift correction (DC). The nature of the second term, referred to as the variability correction (VC), is much less obvious and deserves further attention. VC is the central quantity of the present investigation. Let us rewrite it as −[β(t) − 1]2σ2V(t) = −[σC(t) − σV(t)]2. Note that if both systems are chaotic with sufficiently strong ergodic properties, σC(t) will decrease as already mentioned above, and consequently the correction to the mean square error will be mainly given by the variance of the model variable for long lead times. In addition the covariance can be expressed as
i1520-0493-136-2-405-e7
implying that in the long time limit (for which the model and the reference trajectories become linearly uncorrelated), the first and last term of the right-hand side of (6) reduce to σ2V(t) + σ2X. The mean square error between the corrected forecast and the reference solution becomes therefore equal to the variance of the reference system. Since the variance of the reference system, σ2X(t), is always smaller or equal to the mean square error between the uncorrected forecast and the reference trajectory for long lead times, an apparent improvement relative to the uncorrected forecast seems to occur.

b. 2-observable scheme

Usually, MOS schemes are not limited to a single model observable. The use of information from the model pertaining to (well chosen) additional variables that are different to the system variable under investigation may increase the MOS correction (see, e.g., Glahn and Lowry 1972; Sokol 2003). This aspect is also investigated in the context of the present analysis. To fix ideas and to simplify the analysis, we focus on the impact of two model observables. For two model observables, V1(t) and V2(t), the mean and variance of XC(t) can also be computed, 〈XC(t)〉 = 〈X(t)〉,
i1520-0493-136-2-405-eq1
where C[V1(t), V2(t)] is the covariance between the model observables. As for the 1-observable MOS scheme, the mean is well corrected, but the variance (for a chaotic system with sufficiently strong ergodic properties) decreases to 0 for long lead times.
The MSE between the corrected forecast and the reference solution can also be computed and we assume here without loss of generality that variable V1 is the same nominal variable as X. It can be decomposed as
i1520-0493-136-2-405-e8
where the first term is the MSE between the model and reference variables of interest, the second term is the DC, and the third term contains information on the variances and covariances between the different variables, here referred to as the VC as for the 1-observable MOS scheme. It can be checked that this variability correction is always negative.

3. The dynamics of the MOS forecasting: Short-time behavior

In the previous section, the statistical characteristics of the MOS technique have been revisited. This analysis did not provide information on the transient behavior of the correction (except in the long time limit) nor on the nature of the error that can be corrected. The present section is devoted to the analysis of the short-time behavior of the MOS forecast, which provides important information on these aspects, while being also amenable to a theoretical understanding.

Let the model equations be given by
i1520-0493-136-2-405-e9
where V are the variables included in the model, such as the coefficients of a truncated orthogonal expansion of the velocity, temperature, and density fields. Here, {λ′} is a set of parameters present in the equations like the eddy diffusivity coefficients, the orographic height at the chosen resolution, or parameters entering in the description of processes not resolved by the model like the impact of deep convection.
We next consider an enlarged set of equations supposed to describe a (typically unknown) reference system:
i1520-0493-136-2-405-e10
i1520-0493-136-2-405-e11
where X and V span the same phase space and Y are the variables that are unresolved by the model. These unresolved variables pertain to the small scales truncated from the original orthogonal expansion of the atmospheric fields or to processes that are not accounted at all at the level of the model, such as the balance of certain chemical species present in the atmosphere. Here, {μ} and {λ} denote the set of parameters present in the equations expected to represent completely the atmospheric dynamics, like molecular viscosity, gas law constants, or latent heat in phase changes.

We stress that the above formulation is quite general and may encompass most of the problems encountered in atmospheric dynamics since the unique hypothesis is that (coupled) dynamical equations can be written for both the set of resolved and unresolved processes.

The solutions of (9) and (10) can be formally written as
i1520-0493-136-2-405-e12
i1520-0493-136-2-405-e13
being understood that the model trajectories are starting close to or on the attractor of the reference system solution.

Our aim now is to obtain information from the dynamics on the nature of the correction to the forecast. A first answer would be to consider the evolution of the moments of the model variables (Nicolis 2005). However, the dynamics of the first two moments, mean and variance, is coupled to the higher-order moments leading to a problem of closure of an infinite system of (coupled) equations. In view of the difficulty of this problem, we adopt an alternative approach by focusing on the short-time evolution of the mean and variance using a Taylor expansion for short times and assessing its impact on the initial stages of the MOS correction.

a. 1-observable MOS scheme

For the 1-observable MOS scheme (section 2a), we start from the definition of the covariance entering in (5) and relations (12) and (13) for the specific variables X(t) and V(t). The covariance can be written as
i1520-0493-136-2-405-e14
where for compactness, the dynamical equations corresponding to the variables X(t) and V(t), Fi[X(t), Y(t), {λ}] and Gi[V(t), {λ′}], are written as F[X(t)] and G[V(t)], respectively. Note that the (possible) presence of unresolved variables Y and parameters are always taken into account in the subsequent analyses.
Similarly, one can write the variance of the model variable, σ2V(t), as
i1520-0493-136-2-405-e15
Let us develop the VC term of (6) in a Taylor series around t = 0:
i1520-0493-136-2-405-e16
Decomposing V(0) = X(0) + ϵ(0), where ϵ(0) is the initial condition error, and using relations (14) and (15), the role of the early stages of the dynamics of both systems on the MOS correction can be determined. It is found that the first coefficient of the Taylor expansion, S0, is given by
i1520-0493-136-2-405-e17
where σ2ϵ(0) = 〈ϵ(0)2〉 − 〈ϵ(0)〉2, σ2X(0) = 〈X(0)2〉 − 〈X(0)〉2, and C(X(0), ϵ(0)) = 〈ϵ(0)X(0)〉 − 〈ϵ(0)〉〈X(0)〉. If there is no error in the initial conditions, ϵ(0) = 0 and S0 = 0. If the error in the initial conditions is an uncorrelated random noise {i.e., C[X(0), ϵ(0)] = 0}, then
i1520-0493-136-2-405-e18
indicating that the correction at initial time is very small, proportional to the square of the mean square error.
Turning to the second coefficient in (16), one gets
i1520-0493-136-2-405-e19
where
i1520-0493-136-2-405-e20
in which R is either V(0) or X(0) and L, G[V(0)] or F[X(0)]. When there is no error in the initial conditions,
i1520-0493-136-2-405-eq2
and hence S1 = 0. When one assumes that small initial condition errors are present, S1 is given by
i1520-0493-136-2-405-e21
where (∂G/∂V)[X(0)] are the rows of the Jacobian matrix at X(0) and 〈δR{(∂G/∂V)[X(0)]ϵ(0)}〉 are defined as in (20) in which (∂G/∂V)[X(0)]ϵ(0) corresponds to L. Assuming further that the initial condition errors are coming from a small-amplitude unbiased white noise,
i1520-0493-136-2-405-e22

In (22), the impact of the two error sources is displayed explicitly. When no model error is present, the correction is of the order of O[ϵ(0)4], a very small correction when the error in the initial conditions is small. On the other hand, the correction will be highly sensitive to the presence of model errors since S1 is related linearly to the model error amplitude.

Note, however, that when this model error is a process uncorrelated with the state of the system, the first (dominant) term in the parentheses disappears. In particular for a constant or a white noise model error, this immediately follows by considering their impact at the level of (14).

The third coefficient S2 in (16) is given by
i1520-0493-136-2-405-e23
Based on the definitions of the covariance and variance given in (14) and (15), it is possible to evaluate the second derivatives appearing in (23). In addition, assuming that the initial condition error is a small-amplitude unbiased white noise process, one gets
i1520-0493-136-2-405-e24
where the terms have been reordered as a function of ϵ(0)2. If a model error is present and ϵ(0) is sufficiently small, S2 is well approximated by neglecting terms of O[ϵ(0)4], provided the model error source, GF, is not a process uncorrelated with the state of the system. When no model errors are present, the only term left is O[ϵ(0)4]. Note that the second line of (24) involving the time derivative of GF can be written as
i1520-0493-136-2-405-eq3
When S2 vanishes, that is, the functions F and G are identical and no initial conditions are present, higher-order terms should be computed. In the absence of initial condition errors, the fourth coefficient, S3, will also vanish and one must go to the fifth coefficient multiplying the quartic term given by
i1520-0493-136-2-405-e25
where {∂F/∂X} are the elements of the row of the Jacobian matrix corresponding to the variable X of interest. The correction will now depend on the velocity difference in the full phase space, or in other words on the model errors affecting the other variables.
A similar expansion for short times can be performed for the DC term of (6) yielding
i1520-0493-136-2-405-e26
i1520-0493-136-2-405-e27
i1520-0493-136-2-405-e28
where 〈X(0)〉 − 〈V(0)〉 is usually referred as the systematic error. If this systematic error is 0, the first nontrivial term is S2 = 2{〈F[X(0)] − G[V(0)]〉}2, which is the only one to survive when there are no initial condition errors. Here also the correction will depend on the difference between the velocities between the model and the reference system. Furthermore, if the model equation G is equivalent to the system equation F and in the absence of systematic errors, the first nontrivial term is the fourth-order term given by
i1520-0493-136-2-405-e29

b. 2-observable MOS scheme

For the 2-observable scheme, a similar analytical computation has been performed. For short times and in the presence of model errors only, the Taylor series expansion gives
i1520-0493-136-2-405-e30
It can be easily checked that in the absence of initial condition errors, correction S2 is always larger than the correction obtained with one variable (24), indicating the benefit in introducing an additional variable. Relation (30) reveals further that the correction depends on the covariances between variables V1, V2, and the model error GF. If the model error is proportional to one of the two variables, the gain reached with the correction can be very large. This aspect is discussed further in the numerical investigation of section 4.

c. Summary of the main results and discussion

In this section, the short-time behavior of two MOS schemes has been investigated with emphasis on their ability to correct the impact of model and/or initial condition errors. Some generic trends have emerged, as summarized below.

  • For short times, the impact of both initial condition and model errors are partly corrected by the 1-observable MOS scheme. The correction of uncorrelated random initial condition errors is small as already revealed by (18), which indicates that the percentage of correction (S0/〈ϵ(0)2〉) is proportional to the initial mean square error itself. For model errors, the (drift and variability) corrections depend on the mean of the velocity difference, GF, and its covariance with the model observable, respectively. This last result reveals that for sufficiently strong covariance between the velocity difference and model observable, the dominant contribution to the variability correction is coming from the model error. In other words, the corrections are potentially more sensitive to the model error than to the initial condition uncertainties, except when model error sources are uncorrelated with the state of the system.For realistic forecasting models it is quite difficult to know what is the proportion of the model error, which is uncorrelated with the model solution itself. However, several investigations tend to suggest that substantial state-dependent model errors are present in atmospheric prediction models (Leith 1978; D’Andrea and Vautard 2000; Danforth et al. 2007 for a recent discussion).
  • When only model errors are present, a quadratic or a quartic short-term behavior is found depending on whether the model error affects the equation of the variable of interest or another variable of the reference system.
  • As expected from the statistical theory, the use of two observables instead of one in the MOS scheme does improve the correction. More importantly the use of an observable, which is proportional to the model error source, appears to have a strong positive impact on the correction. This result provides a useful guideline when dealing with a real forecasting problem in which one is usually confronted with a large number of potential predictors, namely, to use predictors closely related with the model error source. As an example, one can assume that the surface fluxes (known to be a large source of model errors) can be successfully used as predictors for MOS corrections (see, e.g., Termonia and Deckmyn 2007). Analyses in this spirit could be very promising.

4. Numerical investigations

In this section, we investigate the impact of MOS on two typical systems displaying chaotic dynamics: the atmospheric Lorenz model (Lorenz 1984) and a one-dimensional spatially distributed system describing the dynamics of the flow in the vicinity of a convective instability, known as the Kuramoto–Sivashinsky equation (see, e.g., Manneville 1990). For both systems model errors and/or initial condition errors are introduced and the MOS equations are built based on a set of previous forecasts starting from the reference attractor. The number of forecasts for both the development of the linear relations and the subsequent forecast verification are fixed to 100 000 for the Lorenz system and 20 000 for the Kuramoto–Sivashinsky equation. We limit ourselves to build-in parametric errors, although the analysis of the previous section applies when truncation errors are present as well.

a. The Lorenz system

The system’s equations read
i1520-0493-136-2-405-e31
i1520-0493-136-2-405-e32
i1520-0493-136-2-405-e33
where x refers to the amplitude of the westerly wind, and y and z refer to the phases of the large-scale atmospheric waves affecting x. Here, F and G are thermal forcings, a is the inverse of a damping time scale, and b characterizes the strength of the displacement of the eddies by the westerly current. The variables are nondimensionalized and the damping time scale of the wave variables y and z is set to 1 time unit [third term in (32) and (33)]. Since these waves correspond to the large-scale baroclinic waves whose typical lifetime is known to be of the order of 5 days, we consider from now on that the physical time unit is of the same order. The equations are integrated with a second-order Runge–Kutta scheme with a time step of 0.01 time units.

This system can display a wide range of solutions from stable fixed points to chaotic solutions. Since in the present study we are interested in having a solution that displays a dynamics showing strong similarities with the one of the atmosphere, we fix the parameter values in such a way to get a chaotic solution. The current reference trajectory is obtained using the parameter values α = 0.25, F = 16, G = 3, and b = 6.

To investigate the impact of the prediction errors on the correction provided by MOS, we have built a model in which the parameters have been slightly perturbed. The MOS forecasts for variables x, y, and z are computed separately using the 1- and 2-observable MOS schemes in which the model observables are the variables xm, ym, and zm of the model or their products. Consequently for each system’s variable, two or three MOS parameters are estimated. The mean square error between the reference variables and the model (or corrected) variables is obtained from 100 000 realizations starting from different initial conditions on the attractor of the reference system.

Figure 1a displays the mean square error evolution obtained when the model is perfect and the initial conditions are slightly perturbed by a Gaussian white noise with zero mean. The continuous line represents the error between the model solution and the reference trajectory, the short dashed line represents the error of the 1-observable MOS forecast, and the dashed–dotted line represents the correction (the difference between the two mean error curves). Clearly the correction is very small initially. This result is in full agreement with the theoretical considerations presented in section 3a, (18), and is almost negligible up to a time scale at which the error becomes large (after about 1 time unit).

Beyond this time scale the 1-observable MOS forecast provides, in the mean, large improvements, but its variance is rapidly decreasing (Figs. 1b–d). Although this variance depletion is effectively exploited to get the best information available for the forecast, it reduces the possibility to predict extreme events. This behavior was expected for long lead times for chaotic systems with sufficiently strong ergodic properties (see section 2), but it is interesting to note that this decrease is concomitant with the rapid error amplification seen in Fig. 1a.

To clarify this evolution, let us first recall the fundamental stages present in the evolution of initial condition errors (Nicolis 1992): (i) an initial regime during which the error is small and whose dynamics can be described by linearized equations (that can be referred to as the “exponential” regime and which is valid up to a time scale associated with the inverse of the dominant Lyapunov exponent); (ii) a second regime during which the error amplifies in a linear way and reaches large values; and (iii) a saturation phase of the error associated with the finite size of the system’s attractor. For the two last regimes the nonlinearities are playing a central role on the error dynamics. In Fig. 1a, the exponential phase is indeed valid up to a time scale of the order of the time scale associated with the dominant Lyapunov exponent (1/2λ+ ≈ 0.9 time units). Beyond this time scale the error amplifies rapidly before the slow convergence toward the asymptotic plateau.

In addition, the variance of the corrected forecasts is directly related to the covariance between the two trajectories [see (7)]. If 〈(XV)2〉 is rapidly increasing as is the case in the linear phase of growth (stage ii) identified in Nicolis (1992), the covariance should substantially decrease. This will in turn induce a considerable decrease of the MOS forecast variance.

Figure 2 summarizes the results for a perturbation of the parameter a in the absence of initial condition errors. Figure 2a displays the mean square error evolution (for variable x) between the solutions generated by the reference system and the model for which the perturbed parameter a′ = a + 0.0001 (continuous line) and the mean square error evolution between the 1-observable MOS forecast and the reference solution (the short dashed line). The correction is quite substantial for short times during which a large gain of several orders of magnitude can be reached. Beyond this period, both curves are closer up to a time scale of the order of 3–4 time units where the correction starts again to increase rapidly (see the dashed–dotted curve). As in Fig. 1a, the correction increase is very fast when the nonlinearities start to play an important role in the error dynamics and coincides with the time at which the decrease of variance of the MOS forecast begins (Figs. 2f–h), confirming the link between these dynamical characteristics.

The initial evolution of the two corrections (DC and VC) of the 1-observable MOS forecast is split in Fig. 2b. Clearly, DC (crosses) is substantial and is increasing up to a time scale of the order of 0.3 time units, after which it almost saturates (not shown). The initial amplification of this term is quadratic as revealed by the continuous line for which the coefficient has been evaluated using relation (28). The quadratic evolution stops approximately beyond a time scale associated with the most negative Lyapunov exponent of the system [λ = −1.4 (time units)−1, see Nicolis et al. 1995], 1/(|2λ_|) ≈ 0.3 time units, a time scale known to play a central role in the short-time evolution of model errors (Nicolis 2003). For VC (open dots in Fig. 2b), the quadratic evolution deduced from the Taylor expansion (24) is also shown. Again the short-term relation deduced analytically is in agreement with the experiment.

For times between 1/(|2λ_|) and the time at which nonlinear terms will start to play a predominant role, the correction progressively saturates (or oscillates) around an intermediate error level (see Figs. 2b,c). The description of this evolution is more involved since it will depend on higher-order terms of the Taylor expansion. This saturation of both corrections while the error continues to increase indicates that the MOS approach becomes progressively less effective at this stage of the error growth.

In Fig. 2c, the VC obtained with the 2-observable MOS scheme with xm and ym is compared with the 1-observable MOS result already displayed in Fig. 2b. As expected from the discussion of section 3b the VC is larger than the one obtained using one variable only. However, the improvement using a second observable is not very substantial. The reason lies in the fact that a large portion of the model error has already been corrected by the 1-observable MOS scheme because the model observable used is the one appearing in the model error term. This point will be further addressed at the end of the section when perturbing parameter b.

The second component of the mean square error (along direction y) is displayed in Fig. 2d. The gain is much less substantial than in Fig. 2a. The corrections shown in Fig. 2e display an initial quartic evolution as expected from section 3, see (25)(29), which are also plotted on the figure.

Figure 3 shows the evolution of the correction when both model and initial condition errors are present. Different curves are shown with fixed model error amplitude but different initial condition perturbation variances. The symbols represent the experiments and the lines represent the theoretical curves deduced in section 3. The agreement is very good for the early stages of the growth. Clearly, when both errors have a similar amplitude, the correction can hardly be distinguished with the quadratic curve obtained without initial condition errors. The impact of these errors on the correction only shows up for larger amplitude of initial condition perturbations. This behavior indicates the larger sensitivity of the correction to the presence of model errors.

One additional question is to know how the behavior shown in Fig. 2a is modified when the amplitude of the model error is increased. Figure 4 displays the ratio between the mean square error of the corrected forecast and the error of the model forecast. When the parameter difference increases, the ratio becomes smaller for intermediate and long lead times, suggesting that the convergence toward the mean of the corrected forecast (and the variance decrease) starts sooner. This confirms further the central role played by the nonlinearities whose impact on the error dynamics is also felt sooner when the difference increases.

As discussed in section 3, the impact of the use of a second well-chosen observable in the MOS scheme can improve substantially the correction. To illustrate this point, parameter b is perturbed and the y component of the mean square error is displayed in Fig. 5 for three different MOS schemes, 1-observable MOS, 2-observable MOS with xm and ym, and 2-observable MOS with ym and xmzm. The latter quantity was chosen since it is the term on the right-hand side of (32) affected by the parametric error. The improvement obtained with the third scheme is spectacular, reflecting the necessity to choose additional MOS observables linked to the sources of the model errors.

b. The Kuramoto–Sivashinsky equation

The extended KS equation (Kuramoto and Tsuzuki 1976) pertains to a class of partial differential equations that describe the flow dynamics in the vicinity of a convective instability threshold of two-dimensional stationary rolls transverse to the x axis. This prototype system displays a complex spatiotemporal dynamics with strong similarities with the convective structures that are formed in mesoscale atmospheric flows (e.g., Agee 1984). It is written as
i1520-0493-136-2-405-e34
where ψ(x, t) is the convective velocity and η is a damping parameter. An additional parameter α has been introduced in front of the second spatial derivative to evaluate the impact of a parametric perturbation in front of spatial derivatives common in numerical weather prediction equations.

A detailed description of the way this type of equations is obtained as well as their dynamical properties is provided in Manneville (1990). The dynamics of this model is explored in Vannitsem and Nicolis (1994). This equation is integrated with rigid boundary conditions ψ(0, t) = ψ(L, t) = 0 and ∂xψ(0, t) = ∂xψ(L, t) = 0, where L is the size of the system. For the numerical integration one uses a semi-implicit Adams–Bashford Cranck–Nicholson scheme with a time step of 0.1 time units and a space grid dimension of 0.5 space units. For the present analysis, the length is fixed to L = 100, giving rise to a system possessing 201 grid points (or equivalently to a system living in a 201-dimensional phase space). In this model, we adopt the same experimental approach as in the Lorenz system. Two or three MOS parameters are calculated at each grid point (199 grid points in the domain), except at the boundaries where values are fixed to 0. The values of η and α for the reference system are 0.01 and 1, respectively.

Figure 6 illustrates the mean square error evolution for the model and the 1-observable MOS forecasts for the whole domain, averaged over 20 000 realizations, when parametric errors are introduced in parameters η and α. No initial condition errors are present. The mean square error and the corrections displayed in the different panels of Fig. 6 are summed over all the grid points. In Figs. 6a and 6c similar phases as the ones found for the Lorenz system can be isolated: (i) an initial phase during which the correction is substantial and (ii) a long lead time phase during which the nonlinear terms play a prominent role and during which the VC increases rapidly (Figs. 6b,d).

Contrary to the Lorenz model, the amplitude of VC is much larger than DC. The divergence of the mean of the model is thus not necessarily the dominant effect when dealing with an imperfect representation of the reference system.

The short-term quadratic evolutions given by (24)(28), summed over all the grid points, are also shown in Figs. 6b and 6d, in good agreement with the short-term evolution of the corrections. It is also worth mentioning here that since the model error is affecting all the grid points, the first nontrivial term of the Taylor expansion around t = 0 is quadratic at all the grid points.

Finally, the impact of adding a second variable has been investigated. Figure 6e shows the correction using three different MOS schemes when parameter α is perturbed: a 1-observable MOS scheme, a 2-observable MOS scheme with model observables ψi and ψi−1, and a 2-observable MOS scheme with model observables ψi and ∇2ψi, the latter quantity corresponding to the term affected by the model error. The correction of the third scheme is about 30% larger than for the two other schemes, leading to a very small value of the mean square error after 0.1 time units (2.4 × 10−7). The large improvement obtained with the third scheme is fully in line with the guideline proposed in section 3c, in which the use of observables related to the model error source is recommended.

5. Conclusions

Using the probabilistic approach recently developed by one of the present authors (Nicolis 2004) the respective roles of initial condition and model errors on the corrections of the MOS forecasts have been disentangled. Analytical results were first obtained for short and long lead times, which were confirmed and complemented by the numerical investigations of the 84-Lorenz model and the Kuramoto–Sivashinsky equation. Several generic properties of MOS forecasts have been brought out:

  • MOS schemes are able to partly correct the effect of both initial condition and model errors. But the correction of uncorrelated random initial condition errors is small. For model errors, the corrections depend for short times on the mean of the velocity difference between the model and the reference system and on its covariance with the model observable(s). These analytical results, corroborated by the numerical analysis performed in section 4, indicate that the corrections are potentially more sensitive to the model error than to the initial condition uncertainties. This conclusion should be slightly tempered for model errors uncorrelated with the state of the system; the mean of the velocity difference and its covariance with the model observable cancel out, implying, in turn, that the dominant corrections associated with these quantities will disappear. However, several investigations indicate that large model errors are state dependent (e.g., Danforth et al. 2007). One can therefore expect that these can be substantially corrected by the MOS technique.
  • The MOS correction can be decomposed into two parts: (i) a drift term correcting the mean of the model forecast and (ii) a term containing information on the variances and covariances of the model observables and system variables. When only model error is present, the short-term evolution of the correction of a specific forecast variable is quadratic or quartic depending on the fact that model error affects the dynamical equation of this variable or other variables of the system.
  • When more than one observable is used, the correction is highly sensitive to the specific choice of the additional observables. Statistical predictor selection algorithms constitute primary tools to perform this choice. However, our work indicates that if one of them is proportional to the phase space velocity difference between the model and reference system, the correction is large (when the initial condition errors do not dominate). This result provides a very useful guideline for real applications when choosing model observables in a bunch of potential predictors. The choice is not at all obvious in the context of atmospheric forecasting models, but the intuition can sometimes help in choosing suitable observables, such as surface fluxes for correcting the 2-m temperature (e.g., Termonia and Deckmyn 2007).
  • The mean square error between the corrected forecast and the reference trajectory shows improvements for long lead times. These are related to the (well known) decrease of the variance of the corrected forecasts (convergence of MOS prediction toward the mean). The numerical investigations suggest that this variance depletion is concomitant with the time at which the nonlinear terms start to play a central role in the error dynamics.

From these results, it is clear that MOS schemes are mainly useful when the forecasting problem is strongly affected by model errors. This is, for instance, the case for the atmospheric boundary layer. But if the errors are too large, the correction is made at the expense of the variability of the corrected forecasts since the variance is substantially decreasing in this case. An analysis of the correction obtained with 1-observable MOS for the atmospheric temperature at midlatitudes forecasted by the European Centre for Medium-Range Weather Forecasts (ECMWF) forecasting system confirms these results: (i) the MOS scheme provides a substantial correction for 2-m temperature but not for the temperature at 500 and 850 hPa and (ii) the variance of the corrected forecast rapidly decreases (after about 24 h), reflecting the fact that the error enters in the nonlinear growth regime. A similar difficulty in correcting the moments of model variables was recently reported for ensemble prediction calibrations (Hamill and Whitaker 2007).

The specific time scale of 1 day reported above corresponds to the time scale during which the linearized regime is valid for detailed global numerical weather prediction models (Gilmour et al. 2001). So the short-term model error correction reported in the present work can only be obtained for times smaller than this limit, or in other words for short-time weather predictions. In particular this could be very useful in defining more accurate boundary conditions of high-resolution regional forecasting models.

The present work was mainly limited to simple MOS schemes, still tractable for analytical investigations. Additional investigations with more observables, or with other linear or nonlinear techniques, are worth performing as in Casaioli et al. (2003) or Marzban et al. (2006). In addition, in the numerical investigations the emphasis was placed on the correction of parametric errors. However, many other types of model errors can arise like the truncation of subgrid-scale variables or the use of a numerical integration scheme. Although our analytical findings are general enough to encompass these errors, it would be worth investigating quantitatively the ability of these MOS schemes to correct such types of errors in more complex systems like the ones used in Teixeira et al. (2007).

Another way to pursue this work is to investigate the properties of the corrections of the ensemble model output statistics (EMOS) technique designed recently (Gneiting et al. 2005) to cope with the underdispersive characteristic of the ensemble prediction systems currently operational in different meteorological centers (Buizza et al. 2005).

Acknowledgments

We thank Tom Hamill and an anonymous reviewer for their constructive comments. This work is partly supported by the European Commission under Contract 12975 (NEST) and the Belgian Federal Science Policy Office under Contract MO/34/017.

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Fig. 1.
Fig. 1.

(a) Mean square error evolution for the Lorenz system, as obtained with an initial condition error sampled from a Gaussian of mean 0 and variance 10−3. The continuous and the short dashed lines refer to the MSE between the model and the reference system and between the MOS forecast and the reference system, respectively. The dashed–dotted line represents the difference. The inset displays the short-time evolution of the error in logarithmic scales. Temporal evolution of the variance of (b) x, (c) y, and (d) z for the model (continuous lines), the reference system (dotted lines), and the MOS forecast (dashed–dotted lines).

Citation: Monthly Weather Review 136, 2; 10.1175/2007MWR2104.1

Fig. 2.
Fig. 2.

(a) Mean square error evolution for the x variable of the Lorenz model, as obtained with a perturbation of parameter a, a′ = a + 0.0001, for the model (continuous line) and MOS (short dashed line) forecasts. The full correction (drift + variability) is represented by the dashed–dotted line. (b) Short-term evolution of the drift (crosses) and variability (open dots) corrections for the x variable. The quadratic short-term evolution given in (24) and (28) is also plotted. (c) The variability correction with the 1-observable MOS scheme discussed in (b) and the one obtained with the 2-observable MOS scheme with model observables xm and ym. (d) The mean square error curves as in (a) (continuous and short dashed lines) but for the y component. (e) As in (b) but for the y component. (f)–(h) As in Figs. 1b–d but for the model perturbation of (a).

Citation: Monthly Weather Review 136, 2; 10.1175/2007MWR2104.1

Fig. 3.
Fig. 3.

Mean square error evolution as obtained with a perturbation of one of the parameters of the Lorenz system, a′ = a + 0.01, and a perturbation of the initial conditions sampled from a Gaussian noise of mean 0 and variances ranging from 0.01 to 0.0001; σ2 = 0.01 (pluses), σ2 = 0.0025 (crosses), σ2 = 0.0009 (asterisks), and σ2 = 0.0001 (squares). The curves represent the analytical correction obtained for short times in section 3 up to the quadratic term.

Citation: Monthly Weather Review 136, 2; 10.1175/2007MWR2104.1

Fig. 4.
Fig. 4.

Ratio of the mean square error of the MOS forecast and the model forecast for three different values of the parameter perturbation δa = a′ − a.

Citation: Monthly Weather Review 136, 2; 10.1175/2007MWR2104.1

Fig. 5.
Fig. 5.

Mean square error along the y direction for a perturbation of parameter b (δb = 0.01). Three MOS schemes are used, 1-observable MOS (continuous line), 2-observable MOS with ym and xm (dashed line), and 2-observable MOS with ym and xmzm (dashed–dotted line).

Citation: Monthly Weather Review 136, 2; 10.1175/2007MWR2104.1

Fig. 6.
Fig. 6.

(a) Mean square error evolution as obtained with a perturbation of parameter η for the model (continuous line) and 1-observable MOS forecasts (dotted line). (b) The drift (crosses) and variability (open dots) corrections obtained with the 1-observable MOS scheme. The quadratic short-term evolution of the drift and variability corrections are also displayed; see (24) and (28). (c) As in (a) but for a perturbation of parameter α, δα = 0.01. (d) As in (b) but for a perturbation of parameter α, δα = 0.01. (e) Drift + variability corrections (lines with symbols) obtained with different MOS schemes for a perturbation of parameter α, δα = 0.01. The empty dots, crosses, and triangles correspond to the 1-observable MOS scheme, the 2-observable MOS scheme with two adjacent model observables, ψi and ψi−1, and the 2-observable MOS scheme with ψi and ∇2ψi, respectively. The lines without symbols represent the analytical short-term quadratic correction discussed in section 3.

Citation: Monthly Weather Review 136, 2; 10.1175/2007MWR2104.1

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