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NOTES and CORRESPONDENCE on the Deterministic Observation OCTOBER 2009 N O T E S A N D C O R R E S P O N D E N C E 3567 NOTES AND CORRESPONDENCE On the Deterministic Observation Impact Guidance: A Geometrical Perspective DACIAN N. DAESCU Portland State University, Portland, Oregon (Manuscript received 20 January 2009, in final form 26 April 2009) ABSTRACT An optimal use of the atmospheric data in numerical weather prediction requires an objective assessment of the value added by observations to improve the analyses and forecasts of a specific data assimilation system (DAS). This research brings forward the issue of uncertainties in the assessment of observation values based on deterministic observation impact (OBSI) estimations using observing system experiments (OSEs) and the adjoint-DAS framework. The state-to-observation space uncertainty propagation as a result of the errors in the verification state is investigated. For a quadratic forecast error measure, a geometrical perspective is used to provide insight and to convey some of the key aspects of this research. The study is specialized to a DAS implementing a linear analysis scheme and numerical experiments are presented using the Lorenz 40-variable model. 1. Introduction (Andersson et al. 1991; Bouttier and Kelly 2001; Kelly et al. 2007). Since an additional assimilation experiment is Development of efficient methodologies to quantify required for each new dataset being evaluated, in practice the observation impact (OBSI) on the analyses and only a few observing system components may be evalu- forecasts of a specific data assimilation system (DAS) is ated through OSEs. the focus of research at numerical weather prediction Observation sensitivity techniques were initially in- centers worldwide. An objective assessment of the ob- troduced in numerical weather prediction for applica- servation value is required to achieve an optimal use of tions to targeted observations (Baker and Daley 2000; the data from in situ platforms and satellites and to de- Doerenbecher and Bergot 2001) and adjoint-DAS tech- sign intelligent data-thinning procedures, cost-effective niques are currently implemented as an effective approach field experiments for targeted observations, and future (all at once) to estimate the impact of any data subset in observing system networks. the DAS on a specified forecast aspect and to monitor Traditionally (Atlas 1997), the OBSI estimation is per- the observation performance on short-range forecasts formed through observing system experiments (OSEs) (Fourrie´ et al. 2002; Cardinali et al. 2004; Langland and where selected datasets are systematically added to or Baker 2004; Langland 2005; Errico 2007; Gelaro et al. removed from the control DAS to obtain the experiment 2007; Zhu and Gelaro 2008; Tre´molet 2007, 2008; state. The OBSI on a forecast aspect of interest is then Daescu and Todling 2009). These studies revealed sev- quantified by comparison to the forecast issued from the eral practical issues in the adjoint-based OBSI estimates control DAS. The OSEs’ methodology requires a modest such as the difficulty to obtain an adjoint DAS fully software development, once the DAS is in place, and has consistent to the analysis scheme and the need for high- been used to assess the impact of various observing sys- order approximation measures. A particular source of tem components and the value of targeted observations uncertainty in the OBSI estimates is due to inconsis- tencies between the adjoint model implementation and the nonlinear model forecast (e.g., moist physical pro- Corresponding author address: Dr. Dacian N. Daescu, Depart- ment of Mathematics and Statistics, Portland State University, cesses are not properly incorporated in the adjoint P.O. Box 751, Portland, OR 97207. model; Ancell and Mass 2008). Nonlinearities in the E-mail: [email protected] DAS analysis scheme and/or in the forecast aspect DOI: 10.1175/2009MWR2954.1 Ó 2009 American Meteorological Society Unauthenticated | Downloaded 09/28/21 10:44 PM UTC 3568 MONTHLY WEATHER REVIEW VOLUME 137 measure and approximation errors may contribute to error covariance matrix R, and the linearized observa- further ambiguities in the OBSI interpretations. A sig- tion operator H. nificant software development effort is required to im- In the deterministic framework the OBSI calculations plement the adjoint DAS; however, this approach are performed for a specific forecast aspect and a typical provides detailed OBSI information that would be dif- scalar measure of the error in a forecast initiated from x ficult to be obtained by other means in a variational is defined as DAS. Comparative studies of observation impacts de- 2 5 f y T f y 5 f y rived from OSEs and adjoint-DAS techniques are pro- ey(x) (x À x ) C(x À x ) x À x C, (3) vided in the work of Gelaro and Zhu (2009) and Cardinali (2009). The consensus is that adjoint-based techniques where xf 5 M(x) is the nonlinear model forecast at time y are effective in measuring the response in a functional t initiated at t0 , t from x, x is the verifying analysis at aspect of a short-range forecast due to any perturbation time t, the superscript T denotes the transpose operator, in the observing system, whereas OSEs are effective in and C is a symmetric and positive definite matrix that measuring the impact of a single data component on any defines the metric on the state space (e.g., an appropri- forecast aspect. Forecasts sensitivity to observations and ate energy norm). The subscript ey is used to emphasize OBSI estimation in an ensemble Kalman filter is discussed the dependence of the measure in (3) on the verification by Liu and Kalnay (2008) and Torn and Hakim (2008). state xy. It is noticed that the right side of (3) is quadratic f This research note brings forward an issue that has not in terms of x , however, the nonlinearity of ey(x)asa been properly addressed in previous studies; namely, the function of the initial conditions is of a more general uncertainty in the OBSI guidance derived from de- type because of the nonlinearities in the model forecast. terministic forecast error measures using OSEs and a. OSE’s framework adjoint-DAS techniques. The OSEs’ framework and the adjoint-DAS approach to observation impact estima- Let y denote the set of all observations assimilated in tion in a DAS implementing a linear analysis scheme are the control DAS in (1)–(2), yi y denote the data subset briefly reviewed in section 2. The state-to-observation whose impact is being evaluated, and yi 5 ynyi denote space uncertainty propagation in the OBSI assessment the complement set of yi with respect to y (i.e., the set of as a result of the errors in the verification state is dis- observations in the DAS after removing the data com- cussed in section 3. For a quadratic forecast error mea- ponent yi). In the OSEs’ data denial framework an ad- sure, a geometrical perspective is used to provide insight ditional assimilation is performed using only data yi to and to convey some of the key aspects of this research. obtain the experiment analysis state xa,i: Illustrative numerical experiments are presented in sec- tion 4 using the Lorenz 40-variable model (Lorenz and xa,i 5 xb,i 1 Ki[yi À hi(xb,i)]. (4) Emanuel 1998). Concluding remarks are in section 5. Equation (4) accounts for the change in the observation- related input from (y, R, h) in the control DAS to 2. Deterministic OBSI estimation (yi, Ri, hi)intheOSEs,aswellasforthemodified The deterministic approach to observation impact background-related input from (xb, B) in the control estimation using OSEs and the adjoint-DAS metho- DAS to (xb,i, Bi) in the OSEs. The latter must be con- dology is briefly reviewed in this section. For simplicity, sidered when several data assimilation cycles are per- consider a linear analysis scheme such as a Kalman filter formed to assess the impact of data removal. Once xa,i is and variational methods implementing a single outer available, the impact of the selected data subset yi within loop iteration (Daley 1991; Kalnay 2002): the control DAS on any forecast aspect of interest ey may be easily evaluated by xa 5 xb 1 K[y À h(xb)], (1) ose def Iy (yi) 5 ey(xa) À ey(xa,i), (5) where xb is a background estimate to the initial condi- ose tions, y is the vector of observational data, h is the ob- such that Iy (yi) measures the value added by the data yi servation operator, and to the observing system yi. b. Adjoint-DAS framework K 5 BHT(HBHT 1 R)À1 (2) Current adjoint-based OBSI techniques are based on is the optimal gain matrix expressed in terms of the an observation-space estimation of the forecast impact background error covariance matrix B, the observation due to assimilation of all data in the DAS: Unauthenticated | Downloaded 09/28/21 10:44 PM UTC OCTOBER 2009 N O T E S A N D C O R R E S P O N D E N C E 3569 T T T 5 kkxy À xt (11) dey 5 ey(xa) À ey(xb) ’ (dxa) gy 5 (dy) K gy, (6) denote a measure of the precision (accuracy) of xy. The where dy 5 y 2 h(xb) is the innovation vector, dxa 5 Kdy (n 2 1) sphere of center xt and radius is the analysis increment in (1), and gy is a properly de- fined vector measuring the forecast sensitivity to initial S(xt, ) 5 fxy 2 Rn : kxy À xtk 5 g (12) conditions (Gelaro et al. 2007). Associated to the measure in (3), the gradient to initial conditions is expressed as is the set of verification states of precision.
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