For most statistical postprocessing schemes used to correct weather forecasts, changes to the forecast model induce a considerable reforecasting effort. We present a new approach based on response theory to cope with slight model changes. In this framework, the model change is seen as a perturbation of the original forecast model. The response theory allows us then to evaluate the variation induced on the parameters involved in the statistical postprocessing, provided that the magnitude of this perturbation is not too large. This approach is studied in the context of a simple Ornstein–Uhlenbeck model and then on a more realistic, yet simple, quasi-geostrophic model. The analytical results for the former case help to pose the problem, while the application to the latter provides a proof of concept and assesses the potential performance of response theory in a chaotic system. In both cases, the parameters of the statistical postprocessing used – the Error-in-Variables Model Output Statistics (EVMOS) method – are appropriately corrected when facing a model change. The potential application in an operational environment is also discussed.

A generic property of the atmospheric dynamics is its sensitivity to initial conditions. This implies that probabilistic forecasts is necessary to adequately describe this behaviour

Statistical postprocessing methods are used to correct the operational predictions of the atmospheric models. An important family of statistical techniques used to postprocess the forecasts are linear-regression techniques, possibly with multiple predictors

Despite their simplicity, most postprocessing schemes depend on the availability of a database of past forecasts, which allows one to train the regression algorithm by comparison with the observations database. Operational models are however subject to frequent evolution cycles, which are needed to improve their representation of the atmospheric processes. Therefore, there is a continuous need to recompute forecasts starting from past initial conditions with the latest model version to avoid a degradation of the postprocessing schemes due to model change. Such a recomputation of the past forecasts are called “reforecasts” and typically requires a huge data storage and management framework, as well as many computational resources

Recent research has investigated non-homogeneous regression with a time-adaptative training scheme, for which a trade-off between large training data sets for stable estimates and the benefit of a shorter training period for faster adaptation to data changes is considered

The present work investigates another research direction and considers a new technique to reduce the cost of adapting a postprocessing scheme to a model change. This method relies on the response theory for dynamical systems

In Sect.

In Sect.

In the last section, we discuss the implications that this new method could have on operational forecast postprocessing systems, as well as new research avenues.

The systems used to produce the weather forecasts are typically non-linear dynamical systems whose time evolution is governed by multi-dimensional ordinary differential equations:

We point the reader to recent articles dealing with the validity of the response theory for weakly hyperbolic systems and time series

We shall assume for simplicity that the system defined by Eq. (

When taking the gradient of a function

The linear perturbation

The tangent model provides thus the tool through which we will evaluate the impact of the model change on the average used by statistical postprocessing schemes. In other words, the tangent model will allow us to take into account the information on the model change (viewed as a perturbation of the initial model) to modify the previous postprocessing scheme and adapt it to the new model version. The solution to Eq. (

Equation (

We will thus also assume that the measures

Response theory is also valid for stochastic models with a well-defined stationary measure, as shown for instance in

In order to get a first impression of the impact of a model change on a postprocessing scheme, we consider two Ornstein–Uhlenbeck processes representing reality

These are therefore uncorrelated Ornstein–Uhlenbeck processes with noise amplitudes

We then consider a change

We have thus reality

We now consider a forecast situation where model

Since we are dealing with simple analytical models here, we can compute the theoretical values of the coefficient

For reality

Similarly, we get the same kind of results for model

For

Let us now assume that model change

After the model change, the forecasts are provided by model

The observables depend on the lead time

Here we consider that the observation are perfectly assimilated in the models and that there is no observation errors. However in operational setups, such errors are of course to be taken into account.

of reality defined by Eq. (Evaluating Eq. (

This expression is equivalent to the second term of Eq. (1) in

In order to investigate this research avenue on a case closer to those encountered in reality, we will now consider the application of postprocessing and response theory to a low-order atmospheric model displaying chaos.

A two-layer quasi-geostrophic atmospheric system on a

In the version proposed by

In the present work, the parameter

Dynamics of the reference system and model 0 of the postprocessing experiment with a modification of the friction coefficient (see Table

The main parameters used and modified in the experiments. Model 0 and model 1 are respectively the forecast model of reality before and after the model change.

These parameter changes induce slight modifications of the dynamics. In particular the system possesses two distinct weather regimes, depicted in Fig.

Attractors for the experiment with a modification of the friction coefficient:

The model described above with 10 modes (

Behaviour of the averages as a function of the lead time

Same as Fig.

In the framework of the EVMOS postprocessing scheme, the predictors and the predictands are the same nominal variable, and no other predictors are used. In both experiments considered, the postprocessing parameters

Let us consider again the response theory described in Sect.

Equation (

In what follows, we will numerically integrate Eq. (

For each of the two experiments detailed in Table

Corrections of the moments of

Corrections of the moments of

The response-theory approximations of the averages of model

Histograms of the solutions

Coefficients

Performance of the corrections on the variable

Performance of the corrections on the variable

The moments obtained by the response-theory approach are used to compute new EVMOS postprocessing

In conclusion, the correction of model 1 using the response-theory EVMOS matches almost perfectly the score of the “exact” EVMOS obtained with the forecasts of model 1 (dash-dotted green curve), up to a 4 d lead time. After that lead time, the errors due to the fat tails in the response of the first moments of the statistics induce errors in the variance needed to compute the

Comparison of the efficiency of the response-theory correction for different numbers

Statistical postprocessing techniques used to correct numerical weather predictions (NWP) require substantial past forecast and observation databases. In the case of a model change, which frequently occurs during the normal life cycle of an operational forecast model, one has to reforecast the entire database of past forecasts

Figure

Note however that in the context of this conceptual model, good estimates of the postprocessing coefficients

The response theory is efficient because the model changes are assumed to be small in comparison with the original parameterisation of the models. The method cannot improve a postprocessing scheme, but it can efficiently adapt it to a new model version. As such, the success of this method also depends on the quality of the past postprocessing scheme. There are situations where linear response theory is known to fail, but statistical tests which allow for the identification of its breakdown have been derived in

To test this approach, we have focused on the EVMOS statistical postprocessing method, but other methods could be considered as well. The only requirement is that the outcome of the minimisation of the cost function uses averages of the systems being considered. For instance, member-by-member methods that correct both the mean square errors and the spread of the ensemble while preserving the spatial correlation

The impact of initial-condition errors has not been addressed here, since the purpose was to demonstrate the applicability of the approach in a perfectly controlled environment. The main limiting issue of response theory in the present context is the presence of fat tails in the distribution of the perturbations

First, as suggested by

Secondly, another avenue would be to adapt the techniques based on the covariant Lyapunov vectors (CLVs) or on unstable periodic orbits (UPOs) to non-stationary dynamics. These techniques were recently introduced

In conclusion, the response-theory approach developed here is an effective method to deal with the problem of the impact of model change on the postprocessing scheme. Its main advantage is to be computed on the past model version and does not require reforecasts of the full model. Its operational implementation, however, is still an open question that should be addressed in the future.

The quasi-geostrophic model used is called QGS and was obtained by adapting the Python code of the MAOOAM ocean–atmosphere model

We consider a perturbed autonomous dynamical system

The ordinary differential equations of the model are given by

The supplement related to this article is available online at:

JD and SV developed the idea of using the response theory in the context of postprocessing, together with the overall experimental setup. JD made the analytical and numerical computations. Both authors contributed to the writing of the paper.

Stéphane Vannitsem is a member of the editorial board of the journal. Jonathan Demaeyer declares that he has no conflict of interest.

This article is part of the special issue “Advances in post-processing and blending of deterministic and ensemble forecasts”. It is not associated with a conference.

The authors warmly thank Lesley De Cruz for her suggestions throughout the paper. They also thank Michaël Zamo and the anonymous reviewer for their comments and suggested improvements.

This research has been supported by EUMETNET (Postprocessing module of the NWP Cooperation Programme).

This paper was edited by Maxime Taillardat and reviewed by Michaël Zamo and one anonymous referee.