Ensemble-based diurnally varying background error covariances and their impact on short-term weather forecasting

Shiwei Zheng ,Yodeng Chen ,* ,Xing-Yu Hung ,Min Chen ,Xiny Chen ,Jing Hung

a Key Laboratory of Meteorological Disaster of Ministry of Education/Joint International Research Laboratory of Climate and Environment Change/Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science and Technology, Nanjing, China

b Institute of Urban Meteorology, China Meteorological Administration, Beijing, China

Keywords:Data assimilation Background error covariance Diurnal variation Ensemble method

ABsTRACT Background error covariance (BEC) plays an essential role in variational data assimilation.Most variational data assimilation systems still use static BEC.Actually,the characteristics of BEC vary with season,day,and even hour of the background.National Meteorological Center—based diurnally varying BECs had been proposed,but the diurnal variation characteristics were gained by climatic samples.Ensemble methods can obtain the background error characteristics that suit the samples in the current moment.Therefore,to gain more reasonable diurnally varying BECs,in this study,ensemble-based diurnally varying BECs are generated and the diurnal variation characteristics are discussed.Their impacts are then evaluated by cycling data assimilation and forecasting experiments for a week based on the operational China Meteorological Administration-Beijing system.Clear diurnal variation in the standard deviation of ensemble forecasts and ensemble-based BECs can be identified,consistent with the diurnal variation characteristics of the atmosphere.The results of one-week cycling data assimilation and forecasting show that the application of diurnally varying BECs reduces the RMSEs in the analysis and 6-h forecast.Detailed analysis of a convective rainfall case shows that the distribution of the accumulated precipitation forecast using the diurnally varying BECs is closer to the observation than using the static BEC.Besides,the cycle-averaged precipitation scores in all magnitudes are improved,especially for the heavy precipitation,indicating the potential of using diurnally varying BEC in operational applications.

1.Introduction

Data assimilation plays a key role in acquiring better initial conditions to improve numerical weather prediction (NWP).For most operational NWP systems,the variational assimilation method is adopted because of the high computational efficiency (Zhang et al.,2019).In variational assimilation,background error covariance (BEC) determines the spread of observation information and affects the weight of the background.Thus,the BEC has a great impact on variational data assimilation.

BEC must be estimated from background error samples because the true state of the atmosphere is unknown (Chen et al.,2021),making the appropriate estimation of background error samples a high priority.The background error samples are often estimated by the National Meteorological Center (NMC;Parrish and Derber,1992) method and the ensemble method (Houtekamer et al.,1996) through the difference of the forecasts.

With the development of computational technology,numerical models can compute more efficiently.To gain a more accurate prediction,rapid updated cycling (RUC) systems have been adopted in NWP centers(Benjamin et al.,2016).The cycling assimilation interval could be very short,such as three hours or even one hour.

Fig.1.Model domain configuration and the distribution of observations used in this study at 1200 UTC 16 August 2021: (a) outer domain and conventional observations;(b) inner domain and radar observations.

Fig.2.Vertical distribution of the ensemble spread and RMSEs of the ensemble mean in the 3-h forecast: (a) U (units: m s -1);(b) V (units: m s -1);(c) T (units: K).

For such a short assimilation interval,the BECs applied in these RUC systems are still the same.Actually,it has been proved that the characteristics of BEC vary with season,day,and even hour (Monteiro and Berre,2010 ;Wang et al.,2014 ;Lee and Huang,2020).Recently,Chen et al.(2021) estimated and researched the diurnally varying BECs that were generated via the NMC method based on three-month forecast error samples,and a better forecast of surface variables and accumulated precipitation was gained by cycling assimilation and forecasting,but the diurnal variation characteristics are gained by climatic samples.Ensemble methods may obtain the background error characteristics that suit the samples in the current moment (Pereira and Perre,2006).Therefore,to gain more reasonable diurnally varying BECs,in this study,ensemble-based diurnally varying BECs are generated,and the diurnal variation characteristics of ensemble forecast errors and BECs are discussed.Besides,the impact of the diurnally varying BECs on short-term weather forecasting is evaluated based on a one-week run and a convective rainfall case through cycling the operational China Meteorological Administration-Beijing (CMA-BJ) system (He et al.,2019).

This paper is organized as follows: Section 2 gives a description of the CMA-BJ system and the method for generating the diurnally varying BECs.Section 3 demonstrates the diurnal variation in the standard deviation (STD) of the ensemble forecasts and the statistics of the diurnally varying BECs,analyzing the diurnal variation characteristics.An evaluation of the performance of the diurnally varying BECs in a one-week run is presented in section 4.Their impact on precipitation forecasting is further evaluated in section 5 via a convective rainfall case,and a summary and conclusions are given in section 6.

2.Model configuration and methods

2.1.CMA-BJ system

This study is based on a 3-h cycled CMA-BJ system.The model domains are shown in Fig.1.Double one-way nested domains are configured,and the horizontal grid points in the outer domain (d01) and the inner domain (d02) are 649 ×500 and 550 ×424 with resolutions of 9 km and 3 km,respectively.Both domains are configured with 59 vertical levels with a model top of 10 hPa.Version 4.1.2 of WRF-ARW is used as the NWP model,configured with the Thompson microphysics parameterization scheme,RRTMG shortwave and longwave radiation schemes,and YSU boundary layer scheme.The New Tiedtke cumulus parameterization scheme is only applied in d01.

2.2.Methods

The ensemble-based diurnally varying background error is considered by the following expression:

Fig.3.Vertical distribution of the averaged STD of (a) U (units: m s -1),(b) V (units: m s -1),and (c) T (units: K) at each time i.

Fig.4.Diurnal variation in the eigenvalues and horizontal length scales of (a,d) U (units: m 2 s -2),(b,e) V (units: m 2 s -2),and (c,f) T (units: K 2) of the first mode.

whereb(i),andare the background error,ensemble forecasts,and the ensemble mean at timei,wherei=0000,0300,0600,…,2100 UTC.The initial and boundary conditions are ECMWF analysis and forecasts,the initial condition perturbations are generated using the RANDOMCV method (Chen et al.,2014),and the physics tendencies perturbation is generated via stochastically perturbed physics tendencies(SPPT;Berner et al.,2015),and the SPPT perturbs the net physical tendency of temperature,humidity,and the wind components at each time step in the forecast.Due to the high computational and storage costs of the BEC matrix,the BEC is generated via the control variable transform method (Barker et al.,2004).

In this study,b(i) means the background error at 0000,0300,0600,…,and 2100 UTC.The ensemble forecasts are within a 7-day period valid from 0000 UTC 15 August 2021 to 2100 UTC 21 August 2021,and the ensemble number is 30.Thus,the background error sample number is 30 ×7=210 in every momenti.The vertical distribution of the ensemble spread and the root-mean-square errors (RMSEs) of the ensemble mean of zonal and meridional winds (UandV) and temperature (T) are shown in Fig.2.The RMSEs are calculated against the fifth major global reanalysis produced by ECMWF (ERA5).It can be seen that the ensemble spread is reasonably close to the RMSEs of the ensemble mean forUandVin the lower and middle troposphere,where the weather systems are active,and forTat most levels,indicating that the ensemble forecast spread can be used to estimate BECs.

Fig.5.One-week-averaged RMSE of (a) U (units: m s -1),(b) V (units: m s -1),(c) T (units: K),and (d) Q (units: g kg -1) in the analysis against ERA5.The error bars are the 95% confidence interval.

3.Diurnally varying BECs

3.1.STD of ensemble forecast error samples

It is crucial to investigate the diurnal variation features in the ensemble forecast error samples because the background error is approximately estimated by forecast error (Chen et al.,2021).Therefore,the averaged STD ofU,V,andTat each moment is shown in Fig.3.It can be seen that there is clear diurnal variation in the three variables.The STD of the three variables varies more at low levels (below 700 hPa)than high levels (above 700 hPa),indicating that there is more obvious diurnal variation at low levels.This illustrates that there is diurnal variation in forecast errors,and the variation is clearer at low levels,while the forecast errors are more stable at high levels.

3.2.Characteristics of diurnally varying BECs

Eigenvalues,eigenvectors,and horizontal length scales represent the major characteristics of the BEC (Chen et al.,2016).In this study,no apparent diurnal variation is found in the eigenvectors (not shown).Therefore,to analyze the diurnal variation characteristics of the BECs,eigenvalues and length scales are discussed.

The diurnal variation in the eigenvalues at each moment are shown in Fig.4,and only the first mode is presented because it can represent the main characteristics of the BEC.It can be seen that the eigenvalues ofUandVerrors are high in the afternoon,while they are low in the morning.This may be because the convection,which is diffi-cult to predict,is more active during the period.The eigenvalues ofTerrors are high in the daytime,but low and stable at nighttime.This may be because of the diurnal variation in solar radiation.The horizontal length scales of all three variables in the afternoon are relatively smaller than at other times,indicating more short-scale activities during this period.No obvious diurnal variation in the eigenvalues and horizontal length scales of relative humidity (RH) errors is found(not shown).

Fig.6.Flow chart of the assimilation and forecast of the convective rainfall case in Beijing on 16 August 2021.

Fig.7.Distribution of 6-h accumulated precipitation from observations and forecasts from 0900 UTC to 1500 UTC and from 1200 UTC to 1800 UTC: (a,d)observation;(b,e) CONTROL experiment;(c,f) DIURNAL experiment.The black box covers Beijing and its surrounding area,used for computing the verification scores of the precipitation forecasts.

4.Results of one-week partial cycling data assimilation and forecasting

Two parallel 3-h partial cycling data assimilation and forecasting experiments,CONTROL and DIURNAL,are carried out from 12 August to 18 August 2021 based on the CMA-BJ system,to evaluate the influence of the diurnally varying BEC on short-term weather forecasting.In CONTROL,the static operational BEC is applied,and the 3DVar method is used for assimilation;whereas in DIURNAL,under the 3DVar framework,the diurnally varying BECs are implemented in d02 by replacing the eigenvalues and horizontal length scales within the operational BEC by the diurnally varying counterparts calculated from the ensemble.A cold-start forecast run is generated on 1800 UTC from 11 August to 17 August.After the 6-h spin-up,conventional observations (examples are shown in Fig.1 (a)) are assimilated in both domains every three hours from 0000 UTC to 2100 UTC,and a 6-h forecast is carried out after each assimilation.

The averaged RMSEs of the analysis against ERA5 in the one-week assimilation and forecasting are shown in Fig.5.It can be seen that the RMSEs ofU,V,andTbelow 700 hPa are all reduced,indicating that the diurnally varying BECs can improve the analysis.Little difference is found in the RMSEs of specific humidity (Q) at all levels andU,V,andTabove 700 hPa,which may be because there is no obvious diurnal variation in RH and other variables above the planetary boundary layer.The RMSEs in the 6-h forecast (not shown) ofUandTbelow 700 hPa in DIURNAL,though less obvious than in the analysis,are still less than in CONTROL,while those in other variables and at other levels show little difference.This means that the positive impact of diurnally varying BECs can last to the 6-h forecast.

Fig.8.Five-cycle-averaged (a) TS and (b) FAR for 6-h accumulated precipitation in Beijing and its surrounding area.The horizontal axis is the threshold of accumulated precipitation (units: mm),and the error bars represent the 95% confidence interval.

5.Results of a convective rainfall case

To further evaluate the performance of the diurnally varying BECs on precipitation forecasting,a convective rainfall case that occurred in Haidian,Beijing,is selected for cycling data assimilation and forecasting.The rainfall mainly occurred from 1230 UTC to 1900 UTC on 16 August 2021,with maximum precipitation over 80 mm from 1300 UTC to 1400 UTC,causing two deaths.

The flow chart of the case is shown in Fig.6.The cold-start forecast run,in this case,is carried out at 1800 UTC 15 August 2021,using the ECMWF global analysis and forecast fields.After the 6-h spinup,conventional (surface synoptic,sounding,ship,buoy,aircraft reports,wind profiles,and satellite atmospheric motion vector observations;Fig.1 (a)) and radar observations (examples of radar observations are shown in Fig.1 (b)) are assimilated from 0000 UTC to 1200 UTC 16 August (total of five cycles),and a 6-h forecast is carried out after each assimilation.

5.1.Distribution of accumulated precipitation

The observed and forecasted 6-h accumulated precipitation from 0900 UTC to 1500 UTC and from 1200 UTC to 1800 UTC are shown in Fig.7,covering the period of the heaviest rainfall in this case.It can be seen from the accumulated precipitation from 0900 UTC to 1500 UTC that in the observed precipitation center in Beijing,the precipitation forecast in CONTROL is weak,and in the south of Shandong Province,the precipitation is overestimated.In DIURNAL,more precipitation is forecasted in Beijing,and the forecast rainfall center hit the observation.The overestimation in Shandong is also weakened.In the accumulated precipitation from 1200 UTC to 1800 UTC,in addition to the similar forecast improvements found in Beijing and Shandong,the overestimation in south of Beijing is also weakened in DIURNAL.

5.2.Verification of the precipitation forecast

Fig.8 shows the five-cycle-averaged threat score (TS) and false alarm ratio (FAR) of the 6-h accumulated precipitation forecast in Beijing and its surrounding area (38°—42°N,114°—119°E;black box in Fig.7) verified against the accumulated precipitation at ground observation stations.It can be seen that both the TS and FAR of DIURNAL are better than those of CONTROL in all categories of accumulated precipitation,especially for heavy precipitation.This is because the weight of the background is fixed at different moments when using the static BEC,and the weight varies at different moments and generates different analysis increments when using the diurnally varying BECs.Thus,the observation information is absorbed more reasonably when applying the diurnally varying BECs,making the atmospheric state forecast more accurate,and a better precipitation forecast is gained.

6.Summary and discussion

BEC plays an essential role in variational data assimilation.To provide RUC systems with more reasonable diurnally varying BECs,this study introduces ensemble-based diurnally varying BECs.The diurnal variation characteristics are discussed,and cycling assimilation and forecasting experiments for a convective rainfall case are carried out based on the operational CMA-BJ system.The conclusions are as follows:

The analysis of the ensemble forecast and diurnally varying BECs indicates that there is clear diurnal variation in the ensemble forecast STD and the eigenvalues and horizontal length scales ofU,V,andT,and the diurnal variation of the eigenvalues and horizontal length scales can be related to the diurnal variation characteristics of the atmosphere.

The results of one-week cycling data assimilation and forecasting show that the use of diurnally varying BECs can reduce the RMSEs of variables with relatively clearer diurnal variation in the analysis and 6-h forecast,indicating that the diurnally varying BECs can enhance the analysis and forecast.

The results of the rainfall case demonstrate that the application of diurnally varying BECs can improve the forecast of the precipitation distribution and the precipitation forecast scores,especially for heavy precipitation.

In the future,longer parallel experiments will be carried out for different seasons in an operational environment to assess the applicability of the diurnally varying BECs for operational implementations.Moreover,it is an initial application of the diurnally varying BECs;thus,the diurnally varying BECs are only applied in d02,and it may be valuable to apply the diurnally varying BECs in all model domains.

Funding

This work was jointly sponsored by the National Natural Science Foundation of China [grant number 42075148],the Outreach Projects of the State Key Laboratory of Severe Weather [grant number 2021LASWA08],and the Outreach Projects of the Key Laboratory of Meteorological Disaster [grant number KLME202209].The numerical calculations for this study were supported by the High-Performance Computing Center of Nanjing University of Information Science and Technology (NUIST).