Developme of Kalma Filer ad Aalogs Schemes o Improve Numerical Weaher Predicios Luca Delle Moache *, Aimé Fourier, Yubao Liu, Gregory Roux, ad Thomas Warer (NCAR) Thomas Nipe, ad Rolad Sull (UBC) Wid Eergy Predicio - Research & Developme Workshop Boulder, CO -- May 00 * lucadm@ucar.edu
Oulie Esemble ad Kalma filerig (KF) o improve Numerical Weaher Predicios A ew mehod based o KF ad a aalog approach (ANKF, AN) Tess of KF, ANKF, ad AN o correc 0m wid speed Wavele filerig for hub-heigh wids Summary
PRED OBS KF day -7 day -6 day -5 day -4 day -3 day - day - = 0 Time KF-weigh 3
A Kalma filer bias correcio for deermiisic ad probabilisic ozoe predicios Summer of 004 (56 days) 8 phoochemical models 360 ozoe surface saios Sources: Delle Moache e al. (JGR, 006b) Delle Moache e al. (Tellus B, 008) Djalalova e al. (Amospheric Evirome, 00) 4
Esemble averagig ad Kalma Filerig effecs o sysemaic ad usysemaic RMSE compoes RMSE decomposiio (Willmo, Physical Geography 98): RMSE = RMSE RMSE, RMSE = (Pˆ i O i ), RMSE s u s = u i= i= Pˆ i = a boi, a ad b leas-squares regressio coefficies of P i ad O i (Pˆ P ) i i 5
Kalma Filerig effecs o probabilisic predicio reliabiliy 6
Kalma Filerig effecs o probabilisic predicio resoluio (6 %) (0. %) 7
PRED OBS KF KF-weigh day -7 day -6 day -5 day -4 day -3 day - day - = 0 Time NOTE This procedure is applied idepedely a each observaio locaio ad for a give forecas ime PRED OBS ANKF AN day -6 farhes aalog day -5 day -3 day - day - day -7 day -4 closes aalog Aalog Space 8
KF i aalog space () f is a forecas a ime ad a a give locaio, wih > 0 d = f A is a meric o measure he disace bewee f ad A i i i {A i } is a se of aalog forecass a a ime i, wih i < 0 {A i } are ordered wih respec o d i : d i - > d i, ad { i, N N : i N = { Ai } } We ca ow iroduce he Kalma filer bias correcio procedure as follows: The rue ukow forecas bias a ime ca me modeled by x = x η, η N( 0, η Ad he acual forecas error ca be expressed as ~ ) y i = A O = x i i i = x i η i i ε ε i, ε i ~ N( 0, ε i ) 9
0 The opimal recursive predicor of x ca be wrie as Where K, he Kalma gai is Ad p, he expeced mea square error is NOTE: The sysem of equaios is closed by: firs ruig he filer for (wih cosa) KF i aalog space () ) ˆ ( ˆ ˆ = x y K x x ) ( = p p K ε η η ) )( ( = K p p η = η ε r ε ε η ad
How o fid aalogs? () We ca defie a meric: Where, Nvar = = d f var ( ) i A w f i k Ai k var= var f k= var var N var is he umber of variable o compue he disace bewee w var is he weigh give o each variable while compuig he meric f var var is he sadard deviaio of he se { f } wih 0 0 ad is he half-widh of he ime widow over which differeces are compued f A i
How o fid aalogs? () We ca defie a meric: f var Nvar = = d f var ( ) i A w f i k Ai k var= var f k= var var i - i i 0 = 0 - Time
How o fid aalogs? (3) We ca defie a meric: Nvar = = d f var ( ) i A w f i k Ai k var= var f k= var var i i - 3
How o fid aalogs? (4) We ca defie a meric: Nvar = = d f var ( ) i A w f i k Ai k var= var f k= var var i i - 4
Modelig seigs for wid predicios Projec: Wid eergy predicios (Sposor: Xcel Eergy) D D D3 D: 30 km, 8x4 D: 0 km, 53x3 D3: 3.3 km, 54x57 NOTE: 37 verical levels, wih levels i he lowes -km WRF Model physics: Li e al. microphysics YSU for PBL Moi-Oboukov for SL Kai-Frisch CUP (Domai /) Noah Lad Surface Model 5
Wid speed: saisics as fucio of ime 6 mohs period 500 surface saios ~ = wvar =, var Variables o search aalogs: spd, dir, T, P, Q @ surface 6
Wid speed: RMSE (m/s) as fucio of space Raw KF ANKF AN 7
Sesiiviy o daa se legh Couresy of Josh Hacker (of NPS) ad Dara Rife (of NCAR) NWP model: MM5 year period surface saios i NM ~ = w var =, var Variables: u, v, T @ surface KF vs ANKF Skill Score (%) 8
Wavele filerig Time Series Raw Pred OBS Correced Pred Sum wavele Compoes Smoohes Compoe 5 mi. 30 mi. h h 4 h 9
Wavele filerig: RMSE ad Correlaio a XXXX Wid Farm 30 days 4 urbies ~ = wvar =, var Variables o search aalogs: spd, dir, T, P, Q @ surface NOTE: more o ANKF/AN applied o wid farms i Gregory Roux s alk (ex)
Summary Esemble ad Kalma filerig (KF) o improve Numerical Weaher Predicios New mehods based o KF ad a aalog approaches (ANKF, AN) Tes KF, ANKF, ad AN o correc 0m wid speed ANKF ad AN beas KF over a rage of merics ANKF gai vs KF grows wih legh of daa se The combiaio of ANKF ad AN wih a wavele decomposiio furher improve predicio wih oisy daa (@ a wid farm)
Esemble filers mai seps: Possible sources of error model space obs space (3) observaio gross error k y 0 H H H () forward operaor k () backgroud forecass ( prior ) (4) updaes y 0 (5) regressio io sae vecor k Sources of error: Model error: () Forward operaor (H) errors (e.g., ierpolaio, flow-depede behavior of represeaiveess errors, ec.): () Isrumeal errors, rerieval ad rasmissio of obs: (3) Gaussia ad oher (depedig o he mehod) assumpios: (4) Samplig errors (aalogs as a way o populae a esemble): (4), (5) Errors from liear regressio bewee obs ad sae variables icremes: (5) (ex slide) Source: Adaped from Aderso (Fig. Physica D, 007)
KF i aalog space () f is a forecas a ime ad a a give locaio, wih > 0 d = f A is a meric o measure he disace bewee f ad A i i i {A i } is a se of aalog forecass a a ime i, wih i < 0 {A i } are ordered wih respec o d i : d i - > d i, ad { i, N N : i N = { Ai } } We ca ow iroduce he Kalma filer bias correcio procedure as follows: The rue ukow forecas bias a ime ca me modeled by x = x η, η N( 0, η Ad he acual forecas error ca be expressed as ~ ) y i = A O = x i i i = x i η i i ε ε i, ε i ~ N( 0, ε i ) 3
4 The opimal recursive predicor of x ca be wrie as Where K, he Kalma gai is Ad p, he expeced mea square error is NOTE: The sysem of equaios is closed by: firs ruig he filer for (wih cosa) KF i aalog space () ) ˆ ( ˆ ˆ = x y K x x ) ( = p p K ε η η ) )( ( = K p p η = η ε r ε ε η ad
Kalma Filer predicor bias correcio XXXX Sie, 9-5 Augus 004 FRI = RawFcss KEK RawFcss Obs Fracioal Relaive Improveme Delle Moache e al., Joural of Geophysical Research (006b) 5
KF resuls: XXXX Sie ENSEMBLE MEMBERS ENSEMBLE MEAN CRMSE OBS 6
A Kalma filer bias correcio for deermiisic ad probabilisic PM.5 predicios Source: Djalalova e al. (Amospheric Evirome, 00) 7
Error-raio sesiiviy ess 8
Global Saisics 9
Wid speed: Correlaio as fucio of space Raw KF ANKF AN 30
Saisics i space: MAE (m/s) Raw KF ANKF AN 3
Saisics i space: BIAS (m/s) Raw KF ANKF AN 3
Wid speed: PDFs (observaios vs. predicios) Raw KF ANKF AN 33