3D Estimates of Analysis and Short-Range Forecast Error Variances

Transcription

1 3D Estmates of Analyss and Short-Range Forecast Error Varances Je Feng, Zoltan Toth Global Systems Dvson, ESRL/OAR/NOAA, Boulder, CO, USA Malaquas Peña Envronmental Modelng Center, NCEP/NWS/NOAA, College Park, MD, USA Acknowledgment to Hongl Wang, Yuanfu Xe, Scott Gregory and Isdora Jankov 7 th EnKF Data Assmlaton Workshop PSU,

2 Outlne Motvaton Method and expermental setup Error estmaton n QG model OSSE Prelmnary results from GFS model Dscussons and future work

3 Motvaton Accurate estmates of error varances n numercal analyses and forecasts are crtcal: Evaluaton of forecast system Tunng of data assmlaton (DA) system Proper ntalzaton of ensemble forecasts Tradtonal methods: Observatons as proxy Sparse observatons no grdded nformaton Fraught wth observatonal error (ncludng representatveness error) DA schemes themselves Computatonally expensve Affected by same assumptons used n DA scheme, potentally based/naccurate estmates Short-range forecasts (forecast mnus analyss) Ignore model forecast related uncertantes 3

4 Statstcal Analyss and Forecast Error (SAFE) Estmaton Peña and Toth (014) True Error Forecast Truth Perceved Error Analyss A F1 F F3 β 3 β β 1 T ρ 1 =cosβ 1; ρ =cosβ ; Forecast State Analyss State Analyss Error d ( F A) (( F T ) ( A T )) ( x x ) 0 Perceved Error True State Forecast Error Measurements d x x x x 0 0 Estmated quanttes Can we estmate unknown parameters wth observed quanttes?

5 Cost Functon and Relevant Assumptons d x x x x 0 0 Measurements Estmated quanttes Cost Functon J max( d dˆ ) w 1 max : L norm Samplng standard error of the mean (SEM) Mnmzaton: Lmted-memory BFGS SEM sd N f (1 r )(1 r ) w SEM SEM 1 1 f Peña and Toth (014) Connect measurements to estmates: (1) How true error grows n tme; () How true forecast errors get decorrelated from true analyss errors wth ncreasng lead tme. = 1 x x Exponental x e 0 Logstc S t e α : Growt h Rate t c c c x / ( S x ) 0 0 S :Saturato n Value 5

6 Analyss / Forecast Error Correlaton Wth no DA step, analyss & forecast errors correlate at 1.0 Wth one DA step, errors become de-correlated, 1 > ρ1 >0; Wth multple () DA steps, -Assumng effectveness of observng & DA systems statonary n tme Note same analyss system used for both Intalzaton & verfcaton A F 1 F F 3 β 3 β β 1 T = 1 ρ 1 =cosβ 1; ρ =cosβ ;

7 Expermental Setup Perfect model OSSE envronment - Truth s known; Develop and test SAFE method that can be used n real world envronment (w/o knowng truth). Forecasts Analyss Truth Model: Quas-geostrophc model (T1L3; Marshall and Molten, 1993) DA: Ensemble Kalman Flter (EnKF) 00-member ensemble; 1.69 nflaton of background covarance, no localzaton; Setup: 30-day forecast every 1hrs over 90-day perod (180 cases). 7

8 Exponental Error Growth 3D spatal and temporal mean error varance of GHT *SEM~95% confdence nterval (uncertanty bar) mn, 1 1 ( ) ( ) m n t 1, j1 k 1 t x 0 α ρ 1 Actl 53.0, 0.38, 0.85 Est 48.4, 0.39, 0.84 x 0 4% dfference Assumptons consstent wth data Dfferences between measured and modeled values may because: (1) Intal decay of analyss error not presented n SAFE; () Lnear exponental growth s an approxmaton; (3) Samplng errors of fnte samples

9 Key Ponts Grd-Pont Error Estmaton (1) Much smaller sample sze, noser nput data, more dffcult estmaton; () ρ 1 vares n space wth the observng network and the DA scheme, present large-scale characterstcs. Practcal approach Step1. Estmate ρ 1 usng spatally smoothed data; Step. Estmate other parameters wth ρ 1 specfed from spatally smoothed estmates. GB NH SH TRO X 0 α ρ 1 Actl Etm Actl Etm Actl Etm ρ 1 vares only moderately: Estmated spatal mean ρ 1 of GHT500 over GB, NH, SH and TRO are all wthn 95% confdence nterval (1.96*SEM of ρ 1 ). 9

10 Practcal Estmaton Black dots: estmates are out of 95% level defned by 1.96*SEM of x 0 3% Spatal corr wth the truth: x Estmated x 0 Actual x 0 Estmated α Actual α Black dots: estmates are out of 95% level defned by 1.96*SEM of α 16% Spatal corr wth the truth: α 0.85 Global mean of grd-pont value: x / 4. Prescrbed ρ Rato of grd-pont wthn 95% level Global mean of grd-pont value: α 0.37 / 0.38

11 Error estmaton n GFS operatonal forecasts Perod: 1Sep-30Nov, 015; Varable: GHT500; Spatal Resoluton: 1 o X1 o NH (30 o -90 o ) True Error Perceved Error SH (30 o -90 o ) True Error Perceved Error NH (30 o -90 o ) Correlaton ρ SH (30 o -90 o ) Correlaton ρ NH SH x α ρ Analyss error varance NH<SH; Error growth rate NH>SH, NH stronger baroclnc nstablty ρ SH>NH, sparser observatons 11

12 Grd-pont Error estmaton of 500hPa GH Ana Error Varance Error Growth Rate Prescrbed ρ NH (30 o -90 o ) Klest and Ide (015) Estmaton from weak Gaussan smoothed data (prescrbe ρ) SH (30 o -90 o ) stream. Estmated perceved errors at each grd pont for all.5dy lead tme are wthn 95% confdence nterval Drect estmated ρ from very strong Gaussan smoothed data x 0 and ρ are closely related to the observatonal network. NH: evdent land & ocean dfference SH: bascally zonal dstrbuton large α s related to polar & subtropcal jet 1

13 Ongong and Future Work Assessment of statstcal devaton from unknown truth may be possble wth some accuracy. The SAFE s cheap and ndependent of each DA scheme. Descrbe ntal decay of random analyss error varance n error growth model to mprove accuracy of estmates; Spatal mean and 3D grd-pont estmaton of GFS total energy, wnd, temperature, etc. other varables; Applcaton areas: (1) Specfy frst guess error varance n any DA scheme. () Set ntal ensemble varance n any ensemble generaton scheme. 13

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15 Comparson wth EnKF & NMC error estmates EnKF (ensemble spread) Estmates of analyss and forecast error varance NMC Estmates of background forecast error varance Actl Analyss Error Varance(m ): 4. Actl Background Error Varance1h (m ): 50.3 EnKF NMC SAFE Spatal Corr 0.9 NA 0.90 Error Varance(m )/ Devaton of Est Before nflaton: 19.0/55% After nflaton: 3.1/4% Spatal Corr 0.90 Error Varance(m )/ Devaton of Est Before nflaton:.8/55% After nflaton: 38.6/3% NA 39.8/6% 48hr4hr : hr1hr : hr4hr : 48.9/ 3% 4hr1hr : 18.9/ 6% /6% EnKF: spatal dstrbuton (good), magntude (severely underestmated); Correlaton may be lower when used wth other DA schemes (e.g., hybrd GSI) NMC: spatal dstrbuton (bad), magntude (good, tuned n operatonal forecast systems); Both magntude and spatal dstrbuton reasonably estmated by SAFE At very low CPU cost compared to EnKF n operatonal settng Estmates ndependent of DA scheme used 15