Adjoint Estimation of the Forecast Impact of Observation Error - PowerPoint PPT Presentation

Adjoint Estimation of the Forecast Impact of Observation Error Correlations Derived from A Posteriori Covariance Diagnosis DACIAN N. DAESCU Portland State University, Portland, OR ROLF H. LANGLAND NRL Marine Meteorology Division, Monterey, CA 6 th WMO Symposium on Data Assimilation – 2013

Outline A posteriori NRL NAVDAS-AR/NAVGEM error covariance diagnosis T319L50 resolution (Desroziers et al. 2001+) 4D-VAR, 6-hour window Radiance σ 2 o estimates Time period: spatial obs error 2012 10/15-11/01, all UTC correlations interchannel Merging information is Adjoint-DAS forecast error necessary sensitivity (Daescu&Langland 2013) feasible error covariance sensitivity a priori guidance & forecast SYNERGISTIC APPROACH TO impact estimates ERROR COVARIANCE TUNING

Error Covariance Diagnosis: x a = x b + K [ y − h ( x b )] � b ) T � � � − 1 � � HBH T + R HB t H T + R t � d o a ( d o R = E = R � b ) T � = HBH T � � − 1 � � HBH T + R HB t H T + R t H � BH T d a b ( d o = E Innovation error covariance consistency valid for any ( R , B ) BH T + � R = HB t H T + R t H � Error covariance estimates valid if and only if BH T = HB t H T ⇔ � R = R t ⇔ HK = HK t H � What is the analysis impact/benefit of the ( � R , � B ) iteration? BH T � � − 1  BH T + � Hx a = H � H � H � H �  = x a K R  HBH T � � − 1 HBH T + R =  H = I ⇒ x a = �  x a = HK

� � x a ( x b , B , y , R ) Adjoint-DAS Forecast Error Sensitivity e Which DAS components contribute most to forecast uncertainties? o Identify high-impact Σ R y error covariance o parameters C e(x a ) K x a Provide the steepest b Σ descent directions M T B x b −∇ R e, −∇ B e b C A priori guidance to the gain in the forecast skill of ( ˜ R , ˜ B ) � ∂e � � ∂e � e [ x a ( ˜ R , ˜ B )] − e [ x a ( R , B )] ≈ ∂ R , δ R + ∂ B , δ B � �� � p × p n × n gain The optimal corrections δ R ⋆ , δ B ⋆ not provided

Forecast ( R , B ) -sensitivity: x a = x b + K [ y − h ( x b )] Daescu and Langland (2013) � � HBH T + R y − h ( x b ) = ( solver ) z x b + BH T z x a = ( post − multiplication ) Background sensitivity † Observation sensitivity ∂ y = K T ∂e ∂e ∂ x b = ∂e ∂e ∂ x a − H T ∂e ∂ x a ∂ y Forecast R -sensitivity Forecast B -sensitivity � � T ∂ R = − ∂e ∂e ∂ B = ∂e ∂e ∂ yz T H T z ∂ x b δ R -impact estimation δ B -impact estimation � � T ∂e δe ≈ − ( δ R z ) T ∂e δ B H T z δe ≈ ∂ x b ∂ y

1 i ) 2 = A posteriori σ o -estimates: (˜ σ o [ y − h ( x a )] T i [ y − h ( x b )] i nobs i AMSU-A IASI - 51 channels The estimates ˜ σ o are much lower than the assigned σ o In consensus with estimates at other NWP centers Will forecasts benefit from a ˜ σ o -specification?

Forecast error σ 2 o -weight sensitivity guidance: ( s o i ) σ 2 o 24-hr moist total energy error norm, e ( x a ) = � x a f − x a v � 2 E AMSU-A IASI Guidance to maximize the forecast impact of σ o -tuning Reduce assigned σ o for AMSU-A channels 5-8 Inflate assigned σ o for most IASI channels (error correlations !?)

OSE validation of σ o -tuning impact: AMSUA, IASI, AIRS self-validation: e(EXP) - e(CTL) radiosonde T: EXP/CTL mean J o ( x b EXP ) − J o ( x b CT L ) σ o = 1 EXP1: AMSUA Ch 5-8 ˜ 2 σ o EXP2: IASI ˜ σ o = 2 σ o EXP3: σ o = 1 AMSUA Ch 5-8 ˜ 2 σ o IASI, AIRS ˜ σ o = 2 σ o

� � � � IASI spatial error correlations � [ y − h ( x a )] i y − h ( x b ) R ( r ) = E j For each channel assume C ( ǫ o i , ǫ o j ) = φ ( dist { y i , y j } ) d 1 < dist ( y i , y j ) < d 2 25Km separation bin

A priori forecast error � R -impact estimation: IASI

o - and � σ 2 A priori ˜ R -impact estimation: IASI

IASI inter-channel error correlations diagnosis/impact � � � R ( i, j ) = E [ y − h ( x a )] ch # i [ y − h ( x b )] ch # j forecast error impact error correlation estimates A priori forecast impact assessment of the error correlations � ∂e � R covariance model: e ( ˜ ˜ R ) − e ( R ) ≈ ∂ R , δ R

Summary & Research Prospects Design of improved error covariance models A posteriori error covariance consistency diagnosis/estimates Adjoint-DAS sensitivity & a priori guidance to covariance tuning impact Merging information is both necessary and feasible Future research & adjoint-DAS applications Adaptive B -optimization, hybrid ensemble/4D-Var Model error covariance Q -diagnosis/sensitivity/impact There is a need for improved validation procedures Forecast error metric e , verification state x v selection

Summary & Research Prospects Design of improved error covariance models A posteriori error covariance consistency diagnosis/estimates Adjoint-DAS sensitivity & a priori guidance to covariance tuning impact Merging information is both necessary and feasible Future research & adjoint-DAS applications Adaptive B -optimization, hybrid ensemble/4D-Var Model error covariance Q -diagnosis/sensitivity/impact There is a need for improved validation procedures Forecast error metric e , verification state x v selection

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