Many applications require wind gust estimates at very different atmospheric height levels.
For example, the renewable energy sector is interested in wind and gust predictions at the hub height of a wind power plant.
However, numerical weather prediction models typically only derive estimates for wind gusts at the standard measurement height of 10

Severe wind events are one of the main weather hazards for humans and economies.
Extreme wind gusts cause damage to buildings, with effects from loose flying objects to uncovering complete roofs.
These hazards also affect whole forests, especially those with shallow-rooting trees such as spruce – the most used timber in Germany.
For the energy sector, wind prediction is becoming more relevant due to the growing demand in renewable energy, especially in wind power generation.
A steady strong wind is most efficient for the power production, as the power produced at wind plants is proportional to the cube of the horizontal wind speed.
The wind energy plant rotors react slowly to fluctuations in wind patterns; thus, they are not able to transform the higher energy of wind gusts into electricity.
On the contrary, if the shear forces due to gusts are too strong on the rotor, they can lead to the deactivation of the entire wind park.
For a stable electricity network, large wind variations are problematic; therefore, forecasts need to capture these variations.
The hubs of power plants reach heights above 150

Regional reanalyses provide a consistent retrospective data set of the three-dimensional (3-D) state of the atmosphere.
They are characterized by the fact that they incorporate observations via data assimilation into a numerical weather prediction (NWP) model.
The COSMO-REA6 regional reanalysis

Several approaches have been employed for the post-processing of wind and wind gusts. With the aim of applying this to risk assessment for off-shore wind farms,

In this study, we propose a post-processing method for the vertical structure of wind gusts at the location of the Hamburg Weather Mast

The remainder of this article is structured as follows:
in Sect.

Our target data are hourly gusts as measured at the Hamburg Weather Mast.
The Meteorological Institute at the University of Hamburg, partnered with the Max Planck Institute for Meteorology, operates the measuring site in Hamburg, Germany (tall mast: 53

The COSMO-REA6 regional reanalysis of the German Weather Service (DWD) was developed at the Hans Ertel Centre for Weather Research

We denote the hourly gust data as

Thus, we assume that

The cGEV parameters are then inferred using a maximum likelihood estimation (MLE) and the conditional independence assumption. In order to avoid overfitting and to assess sampling uncertainty, we apply a cross-validation procedure. For each year in the time sequence, the parameter estimation is performed on a reduced data set, where the respective year of data is left out. Thus, we obtain one set of parameter estimates for each of the 11 years that is independent of the data of the respective year. Further, the variability of the parameter estimates provides a measure of the sampling uncertainty.

The approximation using Legendre polynomials allows for an estimation using the data at all heights simultaneously. This post-processing model is denoted as “Legendre”. In order to assess the predictability in the vertical, an additional leave-one-out procedure is applied, where the layer to be predicted is withheld during the estimation procedure; this procedure is denoted as “leave-out”. We finally also estimate the parameter for each level independently, denoted as “layer-wise”, in order to quantify how well the approximation of the vertical variation of the parameter performs using Legendre polynomials.

As the number of covariates

The verification of the cross-validated predictive distribution is performed using proper scoring rules

For the calculations, we used the R statistical programming language

Residuals of the gust observations are derived using the cross-validated cGEV parameter estimates to transform the data to a standard GEV (e.g standard Gumbel with

Another assumption that is explicitly used in the MLE is the conditional independence of the gust observations at the different mast levels.
Although this assumption mainly concerns the uncertainty of the parameter estimates, conditional dependence will become relevant if one would like to draw realizations of the vertical gusts or derive aggregated measures (e.g. the probability of observing a gust at any level of the mast).
To assess the dependence of the gusts between different height levels, we use the bivariate Pickands dependence function

We consider the following variables as covariates:
the wind gust diagnostic at 10

List of preselected covariates from the COSMO-REA6 reanalysis.

The gust diagnostic in COSMO-REA6 is probably the most informative variable, as it aims as an estimate of the potential strength of a gust near the surface.
On the one hand, gusts are generated by turbulent deflection of upper air wind to the surface

Histogram of differences between observed gusts at 10

As gusts are naturally related to mean wind speed, we include the horizontal velocities at the station location. COSMO-REA6 has a staggered grid, so the wind velocity is given as the absolute velocity of the centred zonal and meridional velocities. To represent the state of the local vertical profile of the horizontal wind velocity in a height-independent variable, we use a principal component analysis. A principal component analysis of the wind velocity at the different heights reveals that most variability (about 92 %) is explained by a mode of variability where all wind anomalies have the same sign, with a slight increase in variability at higher levels. The second mode of variability, which explains about 6 % of the total variability, represents a dipole (i.e. baroclinic) structure with positive anomalies in the upper two levels and corresponding negative anomalies in the lowest three levels. The latter mode is called the baroclinic wind mode (Vh_EOF2), whereas the former – although not completely barotropic – is called the barotropic wind mode (Vh_EOF1).

An important index to capture vertical instability is the lifted index

We further include information on the atmospheric circulation above the boundary layer at 700 hPa surrounding the Hamburg Weather Mast.
The wind velocities at the closest 25 grid cells are used to calculate an averaged horizontal (Mean

The annual cycle is represented by a linear combination of a sine and cosine function with a period of 1 year (AC_COS and AC_SIN).

Several decisions are needed before setting up the post-processing approach.
The first concerns the threshold for censoring.
We choose the 50 % quantile of the observations at each level respectively, which corresponds to 5.79

Estimates of the regression coefficients using the Legendre model with

Diagnostics for Legendre model without VAR

The next step is the selection of the most important predictors.
The variable selection is performed using the LASSO approach including cross-validation, providing 11 sets of penalized regression coefficients.
The value of

Table

The influence of the covariates on

The interpretation of the role of the covariates is not straightforward, as the selected covariates are correlated.
This is particularly the case for the 10

Post-processing of gusts on 26 August 2011 at 10

The covariate Var

Verification skill scores for the Legendre model against climatology

Same as in Fig.

The post-processing method is assessed using proper verification skill scores.
We first assess the effect of the Legendre approximation.
Figure

Decomposition of the QSS of the predictive 99 % quantile at 10 m (black) and 110 m (grey) into scaled resolution (RES/UNC) and scaled reliability (REL/UNC) for the layer-wise, Legendre, and leave-out models. The crosses show the range of the 100-member bootstrap samples. The grey dashed lines indicate the QSS. The QSS amount is given on the upper and the right axes (grey numbers).

Histogram of differences between observed gusts at 10

The advantage of the Legendre model is the possibility to provide predictions at levels where no observations are available.
Figure

Post-processing of gusts on 29 February and 1 March 2008 at 10

The post-processing method aims at an improved 10

Vertical post-processing of gusts using the Legendre model for times highlighted in Fig.

Pickands dependence function of 10 and 110

Pickands dependence function at

To illustrate the post-processing using the Legendre model, we have a closer look at storm Emma between 29 February and 1 March 2008.
During Emma, we observe the largest gusts at 10

Figure

The dependence between residual gusts at 10 m and higher levels decreased with distance in the vertical, as indicated by the value of the Pickands dependency function at

This study presents a post-processing approach for hourly wind gusts at different vertical heights from observations at the Hamburg Weather Mast. The post-processing model is based on a conditional censored Gumbel-type GEV distribution. The censoring threshold is defined as the 50 % quantile of the observations at each mast level respectively. The censoring approach performs well and leads to a good representation of the larger gusts.

A LASSO approach is used to select the most informative covariates.
The selected variables are the COSMO-REA6 wind gust diagnostic at 10

Vertical variations of the cGEV parameters are approximated using the three lowest-order Legendre polynomials.
Although the best scores are obtained if the post-processing is performed for each level independently, the unified description only results in a slight degradation of skill at the intermediate layers.
The unified description induces a small bias at 10

Our post-processing strategy is applicable to NWP forecasts without relevant changes, except for the selection of the covariates.
Particularly, if applied to ensemble forecasts, additional predictors such as the predictive uncertainty, quantiles, or probabilities for threshold exceedances as derived from the ensemble may be considered.
For an example of how to include ensemble statistics into the post-processing approach, see

The strength of the spatial dependency of gusts is assessed using the Pickands dependence function. The gusts at the different heights are highly dependent. Conditioning the gusts on the COSMO-REA6 covariates reduces the dependency of the residuals between heights. This reduction in dependence is significantly modulated by the stability of the atmosphere as given by the lifted index in the sense that an unstable atmosphere increases mixing and, therefore, dependency. Dependency is not simply a function of distance. For a full spatial model description of the gusts, dependency needs to be modelled as a function of atmospheric condition as well as height.

The post-processing model as estimated for the Hamburg Weather Mast should, in principle, be transferable to other locations.
This may be tested using observations from other weather masts in the model region. However, difficulties may arise because observations from different masts might be processed differently or made with different instruments. Furthermore, different topography or other local parameters could introduce systematic biases. Moreover, at other locations, only measurements of the 10

The wind gust observations from the Hamburg Weather Mast were provided by Ingo Lange from the Meteorological Institute of the University of Hamburg (further information and contact:

JS and PF jointly developed the concept and methodology for this work. JS carried out the post-processing and the visualization of the results, and PF supervised the process. JS was the lead author on the paper with input from PF.

The authors declare that they have 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.

We are grateful to Ingo Lange and the Meteorological Institute at the University of Hamburg for the provision of the wind data from the Hamburg Weather Mast. Special thanks go to Sebastian Buschow for helpful discussions and valuable ideas. We wish to thank Stéphane Vannitsem and the two anonymous reviewers for their constructive comments.

This work has been conducted in the framework of the Hans Ertel Centre for Weather Research funded by the German Federal Ministry for Transportation and Digital Infrastructure (grant no. BMVI/DWD 4818DWDP5A).

This paper was edited by Stéphane Vannitsem and reviewed by two anonymous referees.