Can wavelets improve the representation of forecast error covariances in variational data assimilation?

Two wavelet-based control variable transform schemes are described and are used to model some important features of forecast error statistics for use in variational data assimilation. The first is a conventional wavelet scheme and the other is an approximation of it. Their ability to capture the position and scale-dependent aspects of covariance structures is tested in a two-dimensional latitude-height context. This is done by comparing the covariance structures implied by the wavelet schemes with those found from the explicit forecast error covariance matrix, and with a non-wavelet- based covariance scheme used currently in an operational assimilation scheme. Qualitatively, the wavelet-based schemes show potential at modeling forecast error statistics well without giving preference to either position or scale-dependent aspects. The degree of spectral representation can be controlled by changing the number of spectral bands in the schemes, and the least number of bands that achieves adequate results is found for the model domain used. Evidence is found of a trade-off between the localization of features in positional and spectral spaces when the number of bands is changed. By examining implied covariance diagnostics, the wavelet-based schemes are found, on the whole, to give results that are closer to diagnostics found from the explicit matrix than from the nonwavelet scheme. Even though the nature of the covariances has the right qualities in spectral space, variances are found to be too low at some wavenumbers and vertical correlation length scales are found to be too long at most scales. The wavelet schemes are found to be good at resolving variations in position and scale-dependent horizontal length scales, although the length scales reproduced are usually too short. The second of the wavelet-based schemes is often found to be better than the first in some important respects, but, unlike the first, it has no exact inverse transform.

A review of forecast error covariance statistics in atmospheric variational data assimilation. I: Characteristics and measurements of forecast error covariances

Ensemble prediction for nowcasting with a convection-permitting model - II: forecast error statistics

A 24-member ensemble of 1-h high-resolution forecasts over the Southern United Kingdom is used to study short-range forecast error statistics. The initial conditions are found from perturbations from an ensemble transform Kalman filter. Forecasts from this system are assumed to lie within the bounds of forecast error of an operational forecast system. Although noisy, this system is capable of producing physically reasonable statistics which are analysed and compared to statistics implied from a variational assimilation system. The variances for temperature errors for instance show structures that reflect convective activity. Some variables, notably potential temperature and specific humidity perturbations, have autocorrelation functions that deviate from 3-D isotropy at the convective-scale (horizontal scales less than 10 km). Other variables, notably the velocity potential for horizontal divergence perturbations, maintain 3-D isotropy at all scales. Geostrophic and hydrostatic balances are studied by examining correlations between terms in the divergence and vertical momentum equations respectively. Both balances are found to decay as the horizontal scale decreases. It is estimated that geostrophic balance becomes less important at scales smaller than 75 km, and hydrostatic balance becomes less important at scales smaller than 35 km, although more work is required to validate these findings. The implications of these results for high-resolution data assimilation are discussed.

A method for merging flow-dependent forecast error statistics from an ensemble with static statistics for use in high resolution variational data assimilation

The background error covariance matrix, B, is often used in variational data assimilation for numerical weather prediction as a static and hence poor approximation to the fully dynamic forecast error covariance matrix, Pf. In this paper the concept of an Ensemble Reduced Rank Kalman Filter (EnRRKF) is outlined. In the EnRRKF the forecast error statistics in a subspace defined by an ensemble of states forecast by the dynamic model are found. These statistics are merged in a formal way with the static statistics, which apply in the remainder of the space. The combined statistics may then be used in a variational data assimilation setting. It is hoped that the nonlinear error growth of small-scale weather systems will be accurately captured by the EnRRKF, to produce accurate analyses and ultimately improved forecasts of extreme events.

Systematic model forecast error in Rossby wave structure

Diabatic processes can alter Rossby wave structure; consequently errors arising from model processes propagate downstream. However, the chaotic spread of forecasts from initial condition uncertainty renders it difficult to trace back from root mean square forecast errors to model errors. Here diagnostics unaffected by phase errors are used, enabling investigation of systematic errors in Rossby waves in winter-season forecasts from three operational centers. Tropopause sharpness adjacent to ridges decreases with forecast lead time. It depends strongly on model resolution, even though models are examined on a common grid. Rossby wave amplitude reduces with lead time up to about five days, consistent with under-representation of diabatic modification and transport of air from the lower troposphere into upper-tropospheric ridges, and with too weak humidity gradients across the tropopause. However, amplitude also decreases when resolution is decreased. Further work is necessary to isolate the contribution from errors in the representation of diabatic processes.

Analysing UK real estate market forecast disagreement

Given the significance of forecasting in real estate investment decisions, this paper investigates forecast uncertainty and disagreement in real estate market forecasts. Using the Investment Property Forum (IPF) quarterly survey amongst UK independent real estate forecasters, these real estate forecasts are compared with actual real estate performance to assess a number of real estate forecasting issues in the UK over 1999-2004, including real estate forecast error, bias and consensus. The results suggest that real estate forecasts are biased, less volatile compared to market returns and inefficient in that forecast errors tend to persist. The strongest finding is that real estate forecasters display the characteristics associated with a consensus indicating herding.

Breakdown of hydrostatic balance at convective scales in the forecast errors in the Met Office Unified Model

For data assimilation in numerical weather prediction, the initial forecast-error covariance matrix Pf is required. For variational assimilation it is particularly important to prescribe an accurate initial matrix Pf, since Pf is either static (in the 3D-Var case) or constant at the beginning of each assimilation window (in the 4D-Var case). At large scales the atmospheric flow is well approximated by hydrostatic balance and this balance is strongly enforced in the initial matrix Pf used in operational variational assimilation systems such as that of the Met Office. However, at convective scales this balance does not necessarily hold any more. Here we examine the extent to which hydrostatic balance is valid in the vertical forecast-error covariances for high-resolution models in order to determine whether there is a need to relax this balance constraint in convective-scale data assimilation. We use the Met Office Global and Regional Ensemble Prediction System (MOGREPS) and a 1.5 km resolution version of the Unified Model for a case study characterized by the presence of convective activity. An ensemble of high-resolution forecasts valid up to three hours after the onset of convection is produced. We show that at 1.5 km resolution hydrostatic balance does not hold for forecast errors in regions of convection. This indicates that in the presence of convection hydrostatic balance should not be enforced in the covariance matrix used for variational data assimilation at this scale. The results show the need to investigate covariance models that may be better suited for convective-scale data assimilation. Finally, we give a measure of the balance present in the forecast perturbations as a function of the horizontal scale (from 3–90 km) using a set of diagnostics. Copyright © 2012 Royal Meteorological Society and British Crown Copyright, the Met Office

Will Arctic sea ice thickness initialization improve seasonal forecast skill?

Arctic sea ice thickness is thought to be an important predictor of Arctic sea ice extent. However, coupled seasonal forecast systems do not generally use sea ice thickness observations in their initialization and are therefore missing a potentially important source of additional skill. To investigate how large this source is, a set of ensemble potential predictability experiments with a global climate model, initialized with and without knowledge of the sea ice thickness initial state, have been run. These experiments show that accurate knowledge of the sea ice thickness field is crucially important for sea ice concentration and extent forecasts up to 8 months ahead, especially in summer. Perturbing sea ice thickness also has a significant impact on the forecast error in Arctic 2 m temperature a few months ahead. These results suggest that advancing capabilities to observe and assimilate sea ice thickness into coupled forecast systems could significantly increase skill.

How is the balance of a forecast ensemble affected by adaptive and non-adaptive localization schemes?

This paper investigates the effect on balance of a number of Schur product-type localization schemes which have been designed with the primary function of reducing spurious far-field correlations in forecast error statistics. The localization schemes studied comprise a non-adaptive scheme (where the moderation matrix is decomposed in a spectral basis), and two adaptive schemes, namely a simplified version of SENCORP (Smoothed ENsemble COrrelations Raised to a Power) and ECO-RAP (Ensemble COrrelations Raised to A Power). The paper shows, we believe for the first time, how the degree of balance (geostrophic and hydrostatic) implied by the error covariance matrices localized by these schemes can be diagnosed. Here it is considered that an effective localization scheme is one that reduces spurious correlations adequately but also minimizes disruption of balance (where the 'correct' degree of balance or imbalance is assumed to be possessed by the unlocalized ensemble). By varying free parameters that describe each scheme (e.g. the degree of truncation in the schemes that use the spectral basis, the 'order' of each scheme, and the degree of ensemble smoothing), it is found that a particular configuration of the ECO-RAP scheme is best suited to the convective-scale system studied. According to our diagnostics this ECO-RAP configuration still weakens geostrophic and hydrostatic balance, but overall this is less so than for other schemes.

A route to systematic error in forecasts of Rossby waves

Recent work has shown that both the amplitude of upper-level Rossby waves and the tropopause sharpness decrease with forecast lead time for several days in some operational weather forecast systems. In this contribution, the evolution of error growth in a case study of this forecast error type is diagnosed through analysis of operational forecasts and hindcast simulations. Potential vorticity (PV) on the 320-K isentropic surface is used to diagnose Rossby waves. The Rossby-wave forecast error in the operational ECMWF high-resolution forecast is shown to be associated with errors in the forecast of a warm conveyor belt (WCB) through trajectory analysis and an error metric for WCB outflows. The WCB forecast error is characterised by an overestimation of WCB amplitude, a location of the WCB outflow regions that is too far to the southeast, and a resulting underestimation of the magnitude of the negative PV anomaly in the outflow. Essentially the same forecast error development also occurred in all members of the ECMWF Ensemble Prediction System and the Met Office MOGREPS-15 suggesting that in this case model error made an important contribution to the development of forecast error in addition to initial condition error. Exploiting this forecast error robustness, a comparison was performed between the realised flow evolution, proxied by a sequence of short-range simulations, and a contemporaneous forecast. Both the proxy to the realised flow and the contemporaneous forecast a were produced with the Met Office Unified Model enhanced with tracers of diabatic processes modifying potential temperature and PV. Clear differences were found in the way potential temperature and PV are modified in the WCB between proxy and forecast. These results demonstrate that differences in potential temperature and PV modification in the WCB can be responsible for forecast errors in Rossby waves.

A note on atmospheric predictability

Using the method of Lorenz (1982), we have estimated the predictability of a recent version of the European Center for Medium-Range Weather Forecasting (ECMWF) model using two different estimates of the initial error corresponding to 6- and 24-hr forecast errors, respectively. For a 6-hr forecast error of the extratropical 500-hPa geopotential height field, a potential increase in forecast skill by more than 3 d is suggested, indicating a further increase in predictability by another 1.5 d compared to the use of a 24-hr forecast error. This is due to a smaller initial error and to an initial error reduction resulting in a smaller averaged growth rate for the whole 7-d forecast. A similar assessment for the tropics using the wind vector fields at 850 and 250 hPa suggests a huge potential improvement with a 7-d forecast providing the same skill as a 1-d forecast now. A contributing factor to the increase in the estimate of predictability is the apparent slow increase of error during the early part of the forecast.

Regional Differences in the Prediction of Extratropical Cyclones by the ECMWF Ensemble Prediction System

A regional study of the prediction of extratropical cyclones by the European Centre for Medium-Range Weather Forecasts (ECMWF) Ensemble Prediction System (EPS) has been performed. An objective feature-tracking method has been used to identify and track the cyclones along the forecast trajectories. Forecast error statistics have then been produced for the position, intensity, and propagation speed of the storms. In previous work, data limitations meant it was only possible to present the diagnostics for the entire Northern Hemisphere (NH) or Southern Hemisphere. A larger data sample has allowed the diagnostics to be computed separately for smaller regions around the globe and has made it possible to explore the regional differences in the prediction of storms by the EPS. Results show that in the NH there is a larger ensemble mean error in the position of storms over the Atlantic Ocean. Further analysis revealed that this is mainly due to errors in the prediction of storm propagation speed rather than in direction. Forecast storms propagate too slowly in all regions, but the bias is about 2 times as large in the NH Atlantic region. The results show that storm intensity is generally overpredicted over the ocean and underpredicted over the land and that the absolute error in intensity is larger over the ocean than over the land. In the NH, large errors occur in the prediction of the intensity of storms that originate as tropical cyclones but then move into the extratropics. The ensemble is underdispersive for the intensity of cyclones (i.e., the spread is smaller than the mean error) in all regions. The spatial patterns of the ensemble mean error and ensemble spread are very different for the intensity of cyclones. Spatial distributions of the ensemble mean error suggest that large errors occur during the growth phase of storm development, but this is not indicated by the spatial distributions of the ensemble spread. In the NH there are further differences. First, the large errors in the prediction of the intensity of cyclones that originate in the tropics are not indicated by the spread. Second, the ensemble mean error is larger over the Pacific Ocean than over the Atlantic, whereas the opposite is true for the spread. The use of a storm-tracking approach, to both weather forecasters and developers of forecast systems, is also discussed.

The impact of targeted observations made during the Greenland Flow Distortion Experiment

The impact of targeted sonde observations on the 1-3 day forecasts for northern Europe is evaluated using the Met Office four-dimensional variational data assimilation scheme and a 24 km gridlength limited-area version of the Unified Model (MetUM). The targeted observations were carried out during February and March 2007 as part of the Greenland Flow Distortion Experiment, using a research aircraft based in Iceland. Sensitive area predictions using either total energy singular vectors or an ensemble transform Kalman filter were used to predict where additional observations should be made to reduce errors in the initial conditions of forecasts for northern Europe. Targeted sonde data was assimilated operationally into the MetUM. Hindcasts show that the impact of the sondes was mixed. Only two out of the five cases showed clear forecast improvement; the maximum forecast improvement seen over the verifying region was approximately 5% of the forecast error 24 hours into the forecast. These two cases are presented in more detail: in the first the improvement propagates into the verification region with a developing polar low; and in the second the improvement is associated with an upper-level trough. The impact of cycling targeted data in the background of the forecast (including the memory of previous targeted observations) is investigated. This is shown to cause a greater forecast impact, but does not necessarily lead to a greater forecast improvement. Finally, the robustness of the results is assessed using a small ensemble of forecasts.

Moist singular vectors and the predictability of some high impact European cyclones

The ECMWF full-physics and dry singular vector (SV) packages, using a dry energy norm and a 1-day optimization time, are applied to four high impact European cyclones of recent years that were almost universally badly forecast in the short range. It is shown that these full-physics SVs are much more relevant to severe cyclonic development than those based on dry dynamics plus boundary layer alone. The crucial extra ingredient is the representation of large-scale latent heat release. The severe winter storms all have a long, nearly straight region of high baroclinicity stretching across the Atlantic towards Europe, with a tongue of very high moisture content on its equatorward flank. In each case some of the final-time top SV structures pick out the region of the actual storm. The initial structures were generally located in the mid- to low troposphere. Forecasts based on initial conditions perturbed by moist SVs with opposite signs and various amplitudes show the range of possible 1-day outcomes for reasonable magnitudes of forecast error. In each case one of the perturbation structures gave a forecast very much closer to the actual storm than the control forecast. Deductions are made about the predictability of high-impact extratropical cyclone events. Implications are drawn for the short-range forecast problem and suggestions made for one practicable way to approach short-range ensemble forecasting. Copyright © 2005 Royal Meteorological Society.