1、Backgro und error for NCEPs GSI an alysis in regi onal modeWan-Shu WuNational Centers for Environmental Prediction, Environmental Modeling Center, Washington, D. C.NOAA/NWS/NCEP, W/NP, Room 207, 5200 Auth Rd., Camp Springs, MD 20746,SummaryTo apply NCEP s Gridpoint Statistical Inter

2、polation (GSI) to the WRF -NMM regional system, the regional background error was needed for the unified 3D Var. The error statistics from a global system was setup as the default values for applications to various location and resolutions. The NMC method was also used to estimate the statistics fro

3、m the WRF-NMM forecasts of the cen tral US doma in. Spectral calculati ons were used to find the stream function and velocity potential from the forecast differences of the wind fields. The horizontal scales derived from the derivatives of the non-hydrostatic model fields were very un stable and cha

4、 nge sharply from layer to layer. Auto correlation was found to be the most robust and stable in estimatingthe horizontalstructure. The characteristics of the error statistics were differe nt from those of the default. It was show n that the multivariate projecti ons from the two systems produced si

5、milar results and the varia nces and the structure fun cti ons from a differe nt forecasts system could be tuned to produce comparable results.1. IntroductionA 3D Var formulated in model grid space was proposed in Wu et. al.(2002). The formulation allowed greater flexibility (in homoge neity and ani

6、 sotropy) for backgrou nd error statistics. The in itial tests, a simpler formulation:inhomogeneous only inlatitude directi on, in NCEPs global data assimilatio n showed promising results. It was straight forward to apply this physical-spaced 3D Var to a regional doma in. The un ified grid-po int st

7、atistical in terpolati on (GSI) analysis in NCEP was constructed to include both global and regi onal applicati ons.The objective of this work was to estimate the background error statistics for the data assimilation system of NCEPs regional forecast model NMM. Since the regional model could be appl

8、ied to various domains and resolutions, the error statistics of the global forecast system were setup as default for the movableassimilatio nsystem. The statistics werein terpolated to the latitude and height of the regi onal doma in in GSI and the differe nces in the resoluti ons were take n in to

9、accou nt.In comparis onthe NMCmethod was also used to estimate the regional backgrounderror statistics directly from the WRF-NMM forecasts. Spectral calculations were used to find the stream function and velocity potential from the wind fields. Derived with the method proposed in the paper, the hori

10、z on talscales of the nonhydrostatic model fields were unstable and change sharply from layer to layer. Auto correlationwasfound to be the most robust and stable in estimati ng the horiz on tal structure.Si nee the global errorstatistics were in terpolatedto the regionaldomain, the differencesin the

11、backgrounderrors from these two very different models could be compared directly and the impact on the forecasts from the in itial con diti ons gen erated with the two sets of error covaria nee was exam in ed.2. Estimatingbackground error in regionaldomai nThe differe nces of the WRF-NMM forecasts o

12、n the cen tral US domai n with 8 km resoluti on from differe nt lead time were used to estimate the background error covarianee.The error statistics included themultivariate projectio n betwee n the temperature, surface pressure, velocity potential and the stream function, the variances and the scal

13、es of the analysis variables. Thean alysisvariables were streamfun cti on, un bala need velocity pote ntial,un bala needtemperature,normalizedrelative humidity (Holm etal. 2002) and un bala need surface pressure.The stream functionand the velocity potentialfields were calculated in spectral space fr

14、om the forecasts differencesof the wind components. Thewi nd comp onents were in terpolated from E to A grid and filled to Fast Fourier Transform acceptable grid numbers by tapering to zero near the boundaries. The fields were transformed to the spectral space and vorticity and diverge nee fields we

15、re calculated. Applied 2 to find stream function and velocitypotential and transformed the two fields back to physical space. The practice could be verified by taking derivatives of the stream fun cti on and velocity pote ntialin physical space to find the windcomp onen ts. The winds can the n becom

16、pared withthe original wind fields.The horizontal scales of the structure function derived from the horizontal gradients of the forecastdifferences, as used in the original paper, varied sharply from layer to layer. The problem of the statistics might relate to the facts that the regional model was

17、non-hydrostaticand the NJ啊33Hl15IQFig 1. The error variances of the stream function derived from the global system (above) and the regional NMM (below) as function of latitude and vertical level.resolution washigh.Among the newmethods tested,autocorrelation was found to be the most stable and robust

18、 in estimati ng the horiz on tal structure. Appl ying auto correlationto the entire horizontaldomain,however, meant that the scale statistics were only height depe ndent for the regi onal doma in.The backgrou nd error statistics of the global GFS were used as the default for regional application of

19、GSI. The variances and scales were defined as function of latitude and height in physicalunits. Thestatistics were in terpolated to the user defi ned domain and resolution. Separate tuning coefficients for variances and scales were definedfor eachvariable. The balanee projectionmatrices were alsoint

20、erpolated.Note that the projection matrix fromstreamfunctionto temperature required doubleinterpolation in the vertical.Fig 2. Horizontal scales of the structure function derived from the global system (above) and from NMM forecasts (below) as function of vertical level of the model. Black curves re

21、present stream function, green: velocity potential, yellow: temperature and red: normalized relative humidity.The horiz on tal scales were very differe nt betwee nthe two systems. Besides the characteristics of the forecastmodels, there were issues of differentmethods used to derive the statistics a

22、nd the statistical robustness of the methods,i.e. samplingerrors. The multi-Gaussian setup in GSI made the backgrou nd error spectrum wider tha n whe n a sin gle Gaussian was used so that the analysis response was less sen sitive to the tuning of horiz on tal scales.The variances of the stream funct

23、ion derived from the two systems in Fig. 1 showed differences in amplitude but with similar patter ns. The varia nee from both set of statistics in creased with height and has maximums around tropopause. The amplitudes also in creasedslightly with latitude. The verticalscales in Fig. 2 showed maximu

24、ms betwee n z=10 to 15 but the decrease in the lowest 5 levels were much sharper in the regional statistics than those in the global. Interpolation from the lower resolution global system to the higher resolution NMM introduced errors in the rapidly cha nging layers n ear the surface.The vertical gr

25、id structures of the 42-layer global was much coarser than that of the 60-layer regional model. There were 8 layers below sigma=0.9 in the global system and 19 layers in the WRF-NMM.To find out the differences of the balanee projectionsfrom a global spectral model and aregional 8km non-hydrostatic m

26、odel, the analysis results of one single U observation at 250 mb were exam in ed.The vertical cross secti ons of the Ucomp onent in creme nts andthe projectedTin creme nts from a 1m/s westerly wi nd inno vati on with regi onal backgrou nd statistics and global backgrou nd error statistics interpolat

27、edto the regional domainwere shown in Fig. 3. The tuning of the statistics resulted in similar an alysis resp onse in wind fields to the innovation. The responses in temperature field, no separate tuning in volved, through bala nee projection in the upper layers were also similar through the project

28、ion matrices from the two very different systems. The vertical cross sections of the T in creme nts and the corresp ondingU in creme nts(not shown) from a 1 temperatureobservationalresidual at 1000 mb showed that the temperature in creme nts exte ndedmuch higher in the doma inwhe n the global statis

29、tics were used which is con siste nt with the estimated vertical scales showed in Fig. 2. The projected wi nd in creme ntswereopposite in sign between the analysis results using the two backgrounderror covarianee. However, theprojected wi nd in creme ntsn ear the surface weremuch weaker than those i

30、n the higher levels. This differenee was shown to have minor impact on the forecasts.3. Analysis results and forecast impactNorth-south cross secti ons of thean alysisin creme nts of U at the cen ter of the regi onal domai n were show n in Fig. 4. There are differe nces betwee nthe an alysis results

31、 from the two sets of backgrou nd error statistics. However the gen eral patter ns of the an alysis in creme nts are similar. The differe nces are more significantnear the surface in temperaturein creme nts (n ot show n). The vertical scales in the lower atmosphere are smaller with the NMM statistic

32、s.Fig. 3. The vertical cross sections of the analysis increments of U component wind (color contours) and the temperature (black contours) with global (above) and with regional background statistics (below) from a 1 m/s westerly wind innovation at 250mb.Data assimilation over the centralUS domainusi

33、ng the global and the NMM background error statistics were cycled for 12 hours (four analysis cycles with 3-hour in terval) before 48-hour free forecasts. The no rmalized total pen alties of the 2-day forecasts fit to conventional data were shown in Fig.5. With all the differencesin the backgrounder

34、rors, theexperime ntsproduced comparableforecasts and the one with NMM statistics produced small positive impact on the forecasts.differe nt. The differe nces in the vertical grid structures of the two systems created very different an alysisin creme ntsn ear the surface. However,using the backgrou

35、nd error covaria nee derived from a very differentsystem had only smallnegativeimpact on the 2-day forecasts.AcknowledgmentsThe author would like to thank David Parrish, Russ Treadon and Dayle Kleist for many helpful discussions.Refere ncesWu, W.-S., R. J. Purser, and D. F. Parrish, 2002: Three dime

36、nsional variational analysis with spatially inhomogeneous covariance.Mon. Wea. Rev. 130.2905-2916.Holm, E., E. Andersson, A. Beljaars, P. Lopez, J-F. Mahfouf, A.J. Simmons and J-N. Thepaut, 2002: Assimilation and modeling of the hydrological cycle: ECMWF s status and plans.ECMWF Tech Memo 3834-3JOEOSC mikl 210 2M 279300 JXJW 4 2D 4M 4S0Fig. 4.vertical cross sec