Banglin ZHANG,Vijay TALLAPRAGADA,Fuzhong WENG,Jason SIPPEL1，,and Zaizhong MA

1I.M.System Group,Inc.,College Park,MD 20740,USA

2NOAA NCEP Environmental Modeling Center,College Park,MD 20740,USA

3NOAA Center for Satellite Applications and Research,College Park,MD 20740,USA

4NOAA Joint Center for Satellite Data Assimilation,College Park,MD 20740,USA

Estimation and Correction of Model Bias in the NASA/GMAO GEOS5 Data Assimilation System:Sequential Implementation

Banglin ZHANG*1，2,Vijay TALLAPRAGADA2,Fuzhong WENG3,Jason SIPPEL1，2,and Zaizhong MA4

1I.M.System Group,Inc.,College Park,MD 20740,USA

2NOAA NCEP Environmental Modeling Center,College Park,MD 20740,USA

3NOAA Center for Satellite Applications and Research,College Park,MD 20740,USA

4NOAA Joint Center for Satellite Data Assimilation,College Park,MD 20740,USA

This study presents a simplified multivariate bias correction scheme that is sequentially implemented in the GEOS5 data assimilation system and compared against a control experiment without model bias correction.The results show considerable improvement in terms of the mean biases of rawinsonde observation-minus-background(OmB)residuals for observed water vapor,wind and temperature variables.The time series spectral analysis shows whitening of bias-corrected OmB residuals, and mean biases for rawinsonde observation-minus-analysis(OmA)are also improved.Some wind and temperature biases in the control experiment near the equatorial tropopause nearly vanish from the bias-corrected experiment.Despite the analysis improvement,the bias correction scheme has only a moderate impact on forecast skill.Significant interaction is also found among quality-control,satellite observation bias correction,and background bias correction,and the latter positively impacts satellite bias correction.

data assimilation,model bias,estimation and correction

1.Introduction

Data assimilation is typically a sequential step-by-step procedure in which a previous model forecast is compared with newlyreceivedobservationsto achievea“best”estimate ofthe modelstate orthe analysis.A newforecastis theninitiated from the analysis,and so on.Thoughthe goal of data assimilationis tocombinemodelbackgroundwithobservations in such a way that produces random,zero-mean errors,both models and observationsoften have systematic errors that are not represented by random noise.Model errors are caused by coarse model resolution,oversimplified physics schemes, inaccuratesurface forcing,and other imperfections.Observations also have systematic errors,especially for satellite observations where the approximations in the radiative transfer model depend on atmospheric profiles,and can cause complex state-dependent systematic errors in the assimilation,as well as complex interactions among observation quality control(QC),satellite radiance bias correction and model bias correction.

Despite efforts to remove biases from models and observations,their negative impacts on analyses and forecasts are still routinely seen.In the reanalysis datasets used for climate research it can be extremely difficult to separate real signals and trends form spurious ones caused by biases in models and observations.For numerical weather prediction, the presencesofresidualbiases means suboptimalinitial conditions are provided to forecast models.Bias-aware assimilation methods,which include assumptions about the sources and character of biases,are required to effectively remove biases during data assimilation.Similarly,the presence of persistent or repetitive patterns in the analysis increments produced during the assimilation indicates that there are systematic discrepanciesbetween model,observations,and possibly among different componentsof the observing system as well. These discrepancies can be effectively used for bias estimation and correction.

A model bias estimation and correction algorithm for sequential data assimilation system was first introducedby Dee and da Silva(1998).This algorithm was later implemented in GEOS2 for correctingspecific humidity(Dee and Todling, 2000),which resulted in significant improvement in moisture analyses.A full implementation scheme of the algo-rithm for all model variables was implemented in GEOS4 (Zhang et al.,2001)with promising preliminary results.Additionaltests oftheimplementedalgorithm,however,demonstrated mixed results.In particular,considerable reduction in the mean biases was often combined with deterioration in the residual standard deviations.Further studies indicated those errors were introduced primarily by a necessary postanalysis remapping operation from the analysis grid space to the model grid space.Since small differences between large quantities were involved,the magnitudes of the remapping errors were sometimes becoming larger than the magnitudes of the biases themselves.

?Institute of Atmospheric Physics/Chinese Academy of Sciences，and Science Press and Springer-Verlag Berlin Heidelberg 2016

The experiments reported in the present paper were carried out with the GEOS5 data assimilation system(GEOS5 DAS)(Rienecker et al.,2008).This system consists of a global atmospheric model developed at NASA GSFC and an analysis system developed jointly by the NCEP and NASA GMAO.Remapping in GEOS5 is not necessary,which removes one possible negative impact on the bias correction scheme.We will explore several interesting topics,including the removal of the diurnal components of bias in GEOS5 and the assessment of complex interactions among observation QC,satellite radiance bias correction and model bias correction.The second topic is particularly interesting given that the GSI uses the Community Radiative Transfer Model developed by the Joint Center for Satellite Data Assimilation, wherein the atmospheric profile inputs depend on the model background.

The reportingof this studyproceedsas follows:Evidence of model bias is shown in section 2.The bias estimation and correction procedures performed online in GEOS5 assimilation cycles are described in section 3.Results are discussed and compared against a control experiment without bias correction in section 4.The interaction among QC,satellite radiance bias and model bias correction is discussed in section 5,and conclusions are drawn in section 6.

2.Model bias in GEOS5 GCM

Systematic model errors can be detected by comparing forecasts against observations and calculating regional averages of the non-zero mean residuals with a large set of stations in a reasonably long time period(Dee and Todling, 2000).The solid curves in Fig.1 are the monthly mean and standard deviation of temperature,water vapor,zonal wind,and meridional wind residuals of observation-minusbackground(OmB)over four different regions in February 2006 for the control experiment(CTL),where no bias correction is applied to the GEOS5 GSI analysis and model integration.Figure 1 shows that the systematic component of the OmB residuals is substantial,which could lead to analysis bias if the bias in the model background was not removed(Dee and da Silva,1998,Dee andTodling,2000).The statistics from February 2006 of observation-minus-analysis (OmA)for CTL indicate the amplitude of the mean bias is smaller(not shown);however,it is still quite significant,especially for tropical upper-tropospheric zonal wind and temperature.This is whyGEOS5 DASanalyseswithoutbias correction have strong upper-tropospheric easterlies in the tropics andare typicallytoo coldnearthe tropicaltropopause(not shown).

Another way to detect systematic errors is to assume the time average of analyses as the true state of the atmosphere andto calculate the time averageof forecasterrorsagainstthe analyses for a given lead time(Kamga et al.,2000).These systematic errors are called the“climate drift”in the GCM community.Figure 2 shows the structures and magnitudes of the model climate drift,with vertical–latitude cross sections of zonally averaged five-day forecast errors of geopotential height,temperature,zonal wind,and specific humidity,based on 1800 UTC forecasts from February 2006 initial conditions,which are analyses without bias correction.It is important to remember that the analyses have biases too,but they are much smaller than model systematic biases.

There are many areas of significant climate drift in Fig. 2.For example,negative geopotential height drift(Fig.2a)is present almost everywhereand has the biggest bias in the upper troposphere and throughout the stratosphere in the tropics.Furthermore,high latitudes in the NH(70°–90°N)have large bias through the depth of the atmosphere.The zonally averaged temperature forecast error in Fig.2b reveals three areas of strong positive bias.Two are equatorial at 400 and 200 hPa,with respective peak values of+3°C and+2°C.Another area,near 40°N and at the 500 hPa level,has a peak value of+2°C.Meanwhile,a cool bias of-2°C is observed close to 75°N at 500 hPa.In addition,there are a few areas of weaker cool bias in both hemispheres.For zonal wind,the most organized bias(Fig.2c)is near 200 hPa in the tropics, where easterly wind in the model is too strong by over 1 m s-1.Finally,moisture in the tropics exhibits a dry bias from the surface to about 700 hPa and a wet bias from 700 to 300 hPa(Fig.2d).

Figure 3 shows the vertical–time cross section of global averaged forecast errors of geopotential height,temperature, zonal wind,and specific humidity for lead forecast times fromsix hours upto five days.Forall fourvariables,the forecast errors quasi-linearly increase up to a certain lead-time and then gradually become saturated.However,the length of time before saturation is quite different depending on the altitude and variable.For example,850-hPa specific humidity error takes about two days to saturate,whereas the forecast error of 10-hPa zonal wind is still increasing at day 5.

An examinationof the time-averagedanalysis increments reveals the systematic nature of the analysis correction being made in GEOS5 DAS,which can also be attributed to model bias(Schubert and Chang,1996;Takacs,1996).For example,Fig.4 shows the mean analysis increment in February 2006 for 200-hPa zonal wind,850-hPa specific humidity and 150-hPa temperature in CTL.For 200-hPa zonal wind, thereare two maximanearthe equatorat 20°–80°W and 80°–120°E,where analyses systematically add a westerly zonal wind component to correct the easterly bias in GEOS5.This is consistent with the OmF statistics of rawinsondewind data in Fig.1c,which show substantial mean OmF biases fromCTL.In another example,850-hPaspecific humidity near the equator systematically increases over a large area to correct thecorrespondingdrymodelbias.Thisis alsoconsistentwith the OmF moisture biases in Fig.1b.For the temperature field in Fig.4,the maximum and minimum mean analysis increments at 150 hPa also correct the model bias.

Fig.1.Mean bias(light lines)and standard deviation(heavy lines)of(a)temperature,(b)water vapor,(c)zonal wind,and (d)meridional wind residuals of OmB over four different regions in February 2006.Observations are from quality-controlled rawinsonde reports.The solid curves are for the experiment without bias correction(CTL),and the dashed lines are for the experiment with sequential bias correction(SBC).Dashed light lines are also for zero values.

Fig.2.Vertical–latitude cross section of zonally averaged 120-h forecast errors against analysis of(a)geopotential height,(b)temperature,(c)zonal wind,and(d)specific humidity in greyscale.The forecasts were run from 1800 UTC February 2006 initial fields of CTL.The units are 10 gpms in(a),K in(b),m s-1in(c),and g kg-1in(d).

Fig.3.Vertical–timecross sections of global-averaged forecast errors against analysis of(a)geopotential height, (b)temperature,(c)zonal wind,and(d)specific humidity in greyscale.The forecasts were run from 1800 UTC February 2006 initial fields of CTL.The units are 10 gpms in(a),K in(b),m s-1in(c),and g kg-1in(d).

The model bias has both a persistent componentand periodically varying components,such as a diurnal cycle.As an example,Fig.5 shows the February2006monthlymean 850-hpPa temperature analysis increment at(a)0000 UTC,(b) 0600 UTC,(c)1200 UTC,and(d)1800 UTC.The monthly mean increment reveals a very strong diurnal cycle that is largest over Siberia.At that location there is a positive analysis increment at 0000 UTC and 1800 UTC,and a negative analysis increment at 0600 UTC and 1200 UTC.

Fig.4.February 2006 monthly mean analysis increment of 200-hPa zonal wind (top,m s-1),850-hPa specific humidity(middle,g kg-1)and 150-hPa temperature(bottom,K)for CTL.

The diurnal signal is also evident in the time series of OmB residuals.The light solid curves in Fig.6 display the averagenormalizedpower spectra of OmB residuals fromthe temperature and water vapor mixing ratio time series.The statistics were calculated from February 2006 using global rawinsonde data at different pressure levels in CTL.Ideally, these curves should be close to the unit line as an indication of whiteness of the OmB residuals.In reality,however,this is rarely the case,and much of the work here and elsewhere in the literature on data assimilation aims to bring these curves to their desired levels.In this example there is an excess of power in periods longerthan 10 days and a large spike for the one-day period at all levels for temperature and at levels below 850 hPa for moisture.The long-period power is related to the persistence component of bias,while that at one day is a result of diurnal bias.Since there are both persistent and diurnal components of OmB residuals,it is better to correct both components together.

3.Sequential correction of persistent and diurnal biases

Based on a bias correction scheme first developedby Dee anddaSilva(1998),Radakovichetal.(2001)proposedasim-plified sequential bias correction scheme:

Fig.5.February 2006 monthly mean analysis increment of 850-hPa(temperature)(K)for four different analysis times:(a) 0000 UTC;(b)0600 UTC;(c)1200 UTC;(d)1800 UTC.

whereδwa，kis the analysis incrementat time tk,Kkis analysis weight,wo，kcontains observations,Hkis the observation operator,which maps model variables to observables,wf，kand wa，kare the forecast and analysis,bf，kis the updated bias estimate,bf，k-1is the bias estimated from the previous analysis increment δwa，k,and γ is a weight parameter that needs to be tuned.The sequential(i.e.,step-by-step)implementation scheme first estimates the analysis increment δwa，kat time tkby assuming the bias bf，k-1is known from the previous time. It then estimates the bias bf，kwith the derived analysis increment δwa，kand the previous bias bf，k-1.The sequential implementation is much simpler than the simultaneous bias estimation/correction scheme where the bias variables need be augmented into the analysis model.One problem for this scheme is that the bias estimates remain constant if no observations are available.A small damping to relax the bias mode,Eq.(3),can be added to obtain

where the constant parameter β specifies the rate of decay of the bias estimate in the absence of observations.

The above relaxing scheme is good at removing the persistent part of bias,but it cannot remove the diurnal component shown in Fig.5 and Fig.6.The long-term integration of Eq.(4)is actually an averaging operator,will smooth out diurnal cycles and any other temporal cycles in the bias,and yields only the persistent component.To remove the diurnal component,a highly truncated Fourier series expansion developed by Radakovich et al.(2001)was used as a bias model.Only the persistent bias and bias from the one-day cycle in the Fourier series expansion was applied:

where a0,acand asare the coefficients for the Fourier series, t is time in hours,and ω=2π/24 is the frequency.The estimation of the three amplitude coefficients is similar to in Eq. (4):

4.Evaluation of the sequential bias correction scheme

Fig.6.Average normalized power spectra of(a)temperature and(b)water vapor mixing ratio OmB residual times series from February 2006 global rawinsonde data for CTL (light solid)and SBC(heavy dashed).

The NASA GEOS5 DAS was used to evaluate the sequential bias correction scheme.In GEOS5 DAS,the conventional data and a large amount of satellite radiance data fromRTOVS(Revised TIROS Operational Vertical Sounder) (HIRS-2,MSU),ATOVS[HIRS-3,HIRS-4,AMSU-A, AMSU-B,MHS(microwave humidity sounder)],and EOS/ Aqua[AIRS(atmospheric infrared sounder),AMSU-A],as well as solar backscatter UV data,are processed into BUFR files and passed directly into the GSI,which undertook QC for the satellite data.The experiments with(SBC)and without bias correction(CTL)used the same GSI data assimilation system,and the only difference was the bias correction.Both experiments were run on half-degree resolution from restart files from 2100 UTC 31 December 2005 through 2100 UTC 5 March 2006.For SBC,the initial bias was set to zero,and January 2006 was used as a spin-up period.The model output in February 2006 was used to evaluate the SBC scheme against CTL.

4.1.OmB and OmA

The impact of bias correction is easy to see from the rawinsonde OmB and OmA statistics.The dashed lines in Fig. 1 show the monthly mean and standard deviation of temperature,water vapor,zonal wind,and meridional wind OmB residuals over four different regions in February 2006 for SBC.Comparing with the solid curves from CTL,there is an overall reduction of mean biases in SBC,although the reduction varies for different levels,regions,and variables.For example,the reduction of temperature bias is greatest in the tropics for all levels.However,there is a small bias increase in 200-hPa temperature OmB residuals in the NH.For zonal wind,the biases are cut by about half at almost all levels in the tropics.Despite these differences,there is no noticeable difference in the OmB standard deviation between SBC and CTL.As for the statistics of the OmA residuals over four different regions in February 2006 for SBC,there is an overall reduction of the mean OmA bias but no noticeable changes in standard deviation(not shown).

Fig.7.Average five-day forecast errors against analysis in greyscale of(a)500-hPa geopotential height,(m2s-2)(b)50-hPa geopotential height,(c)200-hPa zonal wind,and(d)200-hPa temperature for SBC(top at each pair)and CTL(bottom at each pair).

The powerspectra of the OmB time series is another metric used to examine the impact of bias correction.The heavy dashed curvesin Fig.6 displaythe averagenormalizedpower spectra of temperature and water vapor mixing ratio from the OmB residualtime series fromtheFebruary2006globalrawinsonde data for SBC.Comparison with the thin solid curves from CTL shows that big spikes related to the diurnal cycle are greatly reduced in SBC.The excess of power related to the persistentbias is also effectivelyremovedsincethe curves are muchflatter than thosefor CTL.It is worthnotingthat the diurnal bias is not completely removed at some levels,and at other levels it is over-corrected.This occurs because the amplitude of the diurnal cycle changes between levels,regions, and variables,and in this study only one set of global parameters was used to test the bias correction scheme.These parameters need to be fine-turned to account for the spatial and temporal variability of OmBs in the future.

4.2.Reduction of systematic error

Fig.8.As in Fig.2 but for SBC.

To reduce the kind of systematic model errors demonstrated in Figs.2 and 3,Saha(1992)used a constant forcing term estimated from one-day mean forecast errors to correct model forecasts and improve forecast skill.The biases estimated by the online sequential bias correction scheme for data assimilation could also be used to correct model forecasts.The advantage in this study is that the estimated biases are promptly available after each analysis time and can be easily used to make corrections to the forecasts as follows:

Fig.9.As in Fig.3 but for SBC.

Although the formula is quite simple,it is not trivial to write the code and implement the bias correction scheme for forecasts.For testing purposes,the forecasts can be obtained by running data assimilation without observations.This permits us to use the same bias model,bf，k=βbf，k-1-γδwa，k,with analysis increment δwa，kas zero,and sequentially correct the forecast biases every six hours.

Fig.10.RMSEs versus lead time(days)for 500-hPa geopotential height averaged over four regions:NH extratropics from 20°N to 80°N;SH extratropics from 80°S to 20°S;tropics from 20°S to 20°N;and global.The solid curve is for CTL,and the dashed line is for SBC.The forecasts were run from 1800 UTC February 2006 initial fields.

The forecasts were run from all 1800 UTC February 2006 initial fields for both SBC and CTL.The five-day forecast systematic errors with respect to the analysis of 500-hPa geopotentialheight,50-hPageopotentialheight,200-hPa zonal wind,and 200-hPa temperature for SBC(top pair)and CTL(bottom pair)are shown in Fig.7.From the figure,it is obvious that the forecasts corrected by model biases have a significant reduction in systematic errors.The maximum value of 500-hPa height errors is about 80 m in CTL and about 40 m in SBC.The regions with errors less than 10 m cover about 50%of the globe in the bias-corrected case, whereas they only cover about 15%of total area in CTL. In addition,the maximum error of 50 hPa near(60°N,0°) is reduced by about 20 m,and the regions of larger errors are much smaller in the bias-corrected forecasts.Finally,the 200-hPa zonal wind and temperature errors are reduced by about 30%on average.

The vertical–latitude cross section of zonally averaged five-day forecast errors from SBC is shown in Fig.8 for geopotential height,temperature,zonal wind,and specific humidity.The contour and shading intervals are the same as in Fig.2 for CTL.A comparison between Fig.2 and Fig. 8 shows that the bias-corrected forecasts have dramatically reduced the five-day forecast errors for height,temperature, and zonal wind.The reduction in specific humidity error is also visible but much smaller.

Figure 9 displays the vertical–time cross section of global-averaged forecast errors for geopotential height,temperature,zonal wind,and specific humidity from SBC.Once again,compared to CTL in Fig.3 there is a dramatic reduction in forecast errors for all variables except specific humidity,especially at 850 hPa.

4.3.Forecast skill

One of the most convincingargumentswe can present for motivating the use of bias correction is by showing the positive impact on some well-known forecast skill scores(von Storch and Zwiers,1999).Here,we show the RMSEs between forecasts and analyses for 500-hPageopotential height in Fig.10.For 500-hPa height,we see immediately that the RMSEs are smaller in the NH extratropics from 20°N to 80°N,in the tropics from 20°S to 20°N,and in the global mean.Meanwhile,there is a slight degradationin RMSEs for the SH extratropics from 80°S to 20°S.

When considering the RMSEs for all variables at all levels(not shown here),the RMSEs are generally smaller in SBC,throughout the entire atmosphere,in both the tropics and NH,and they are also smaller above 100 hPa in the SH. However,the RMSEs increase below 100 hPa in the SH.

5.Interaction among QC,satellite radiance bias and model bias correction

Fig.11.Time series of global-averaged NOAA-17 AMSU-B brightness temperature(K)(a)observations,(b)estimated bias, (c)bias-corrected observations,and(d)the residuals of bias-corrected-OmB.The solid curve is for CTL,and the dashed line for SBC.

GEOS5 DAS utilizes a global angle and an air mass correction scheme to correct the biases between simulated and observed satellite observations,which are directly caused by biased observations,inadequacies in the characterization of the instruments,and deficiencies in the forward models.Thesatellite biases are also indirectly impacted by model biases in the background.Since the satellite bias correction is recalibrated prior to each analysis,a feedback process between model bias correctionand satellite bias correctionis possible. GEOS5 DAS applies QC to eliminate the observations that are considered to be out of the expected error range,which could also interact with satellite bias correction(Aulign′e and McNally,2007).As a result,interaction is possible among QC,satellite radiance bias correction,and model bias correction.

Figure 11 shows the time series of globally averaged brightness temperature from the NOAA-17 AMSU-B channels 1,2,3,4 and 5,the estimated bias of these observations,bias-corrected observations,and the residuals of biascorrected-OmB for CTL(solid)and SBC(dashed).There is a visible difference in the time series of the original observations for the five channels in Fig.11a.It is worth noting that QC is applied to bias-corrected observations,and Fig. 11a shows the time series of observations without bias correction to illustrate the impacts of data selection from QC. The estimated satellite biases in Fig.14b are quite different for CTL and SBC,which shows that model bias corrections have a significant effect on satellite bias estimates.The differencebetweenbias-correctedobservationsinCTLandSBC shown in Fig.11c includes impacts from both QC and satellite bias correction.Non-negligible values indicate that although OmAs are small,the analysis fields themselves could have big differences.The impact of model bias correction is to bring OmBs much closer to zero in SBC than in CTL for most channels(Fig.11d).

The results from AIRS and AMSU-A data also illustrate that the interaction among QC,satellite bias and background bias correction is quite significant(not shown).The OmBs of NOAA-18 AMSU-A from SBC are closer to zero than in CTL for most channels.The model bias correction has the largest impact on channel 14,whose weight peaks at 2.3 hPa. At this altitude,model bias is large,satellite bias estimates have their biggest differences,and model bias correction has its biggest impact.

6.Conclusions

Inthisstudyweimplementedasimplifiedfullbiascorrection scheme in GEOS5 DAS(SBC)and compared the results to a control experiment where no backgroundbias correction was applied(CTL).Considerable improvement was found in both rawinsonde OmB and OmA residuals for all observed variables in SBC.A spectral analysis was applied to the time series of the OmB residuals,and the results showed whitening of the time series in SBC as compared with CTL.Forecasts fromCTL werealso comparedwith forecasts fromSBC that had been sequentially corrected by bias estimates.The results showed a positive impact on forecast skill and a dramaticreductioninsystematicerrors.We alsofoundthatinteraction exists among QC,satellite bias correction and model bias correction,and the model bias correction has a highly positive effect on satellite bias correction.

The bias correction scheme implemented here is sequential and makes bias corrections only at the analysis time in a data assimilation system.The correspondingforecasts experiment was also corrected sequentially at six-hour intervals. We thus have an incomplete bias correction,especially for diagnostics not directly assimilated.In the future,we need to implement an incremental full bias correctionscheme similar to Takacs(1996)and Dee(2005).

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10.1007/s00376-015-5155-y.

26 June 2015;revised 16 November 2015;accepted 10 December 2015)

Banglin ZHANG

Email:banglin.zhang@noaa.gov

Acknowledgements.The wishes to thank Dick DEE of the ECMWF for introducing him to this work and giving him all the necessary help for a very long period of eight years.The authors would like to thank Ronald GELARO,Arlindo da SILVA of the GMAO,and John DERBER of the NCEP,for their many helpful discussions.