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 x ) C(x x ) x 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 section 4 using the Lorenz 40-variable model (Lorenz and xa,i 5 xb,i 1 Ki[yi 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 h(xb)], (1) ose def Iy (yi) 5 ey(xa) 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)1 (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:

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T T T 5 kkxy xt (11) dey 5 ey(xa) 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 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.

T f y a. OBSI uncertainty propagation in OSEs $xey(x) 5 2M (x)C(x x ), (7) In the deterministic OSEs’ framework, the uncer- T where M (x) is the adjoint of the tangent linear model tainty in the OBSI estimates is expressed as from t0 to t evaluated along the model trajectory ini- tialized from x. For example, the observation impact ose dIt,y (yi) 5 [et(xa) et(xa,i)] [ey(xa) ey(xa,i)]. (13) methodology introduced by Langland and Baker (2004) estimates the variation dey using sensitivity gradients From (3) and (10) it follows that along both background and analysis trajectories: f y 2 f y 2 1 1 ey(xa) ey(xa,i) 5kxa x kC kxa,i x kC g 5 $ e (x ) 1 $ e (x ), (8) y 2 x y a 2 x y b f f f f y 5hxa xa,i, xa 1 xa,i 2x iC (14) which provides a second-order accurate dey approximation (Errico 2007; Daescu and Todling 2009). In and practical applications the approximation in (6)–(8) was f t 2 f t 2 found to provide improved results as compared with et(xa) et(xa,i) 5kxa x kC kxa,i x kC first-order approximations (Gelaro et al. 2007; Gelaro f f 5hxf x , xf 1 x 2xti , (15) and Zhu 2009; Cardinali 2009). A measure of the con- a a,i a a,i C tribution of any data subset y in the DAS to the forecast i respectively, where h, i denotes the inner product in error reduction is obtained by taking the inner product C the state space, hu, vi 5 uTCv. After replacing (15) and between the innovation vector component dy and the C i (14) in (13): corresponding amplification factor in (6): ose f f T y t adj def T T dIt,y (yi) 5 2(xa xa,i) C(x x ) Iy (yi) 5 (dyi) (K gy)i. (9) 5 2kxf xf k kxy xtk cosu , (16) adj a a,i C C i Data components for which Iy (yi) , 0 contribute to the forecast error reduction (improve the forecast), f f where ui is the angle between the vectors xa xa,i and adj y t whereas data components with Iy (yi) . 0 will increase x 2 x . Equation (16) shows that in the OSEs’ frame- the forecast error (degrade the forecast). work the uncertainty in the OBSI estimation is deter- mined by the magnitude of the difference between the 3. Uncertainty analysis control forecast and the experiment forecast and by the y verification state error component along the direction In practice, the verification state x used to define the f f . The uncertainty propagation is thus specific forecast error measure in (3) is obtained after several xa xa,i Rn assimilation cycles to the verification time are per- to the dataset being evaluated. The hyperplane of passing through xt and normal to the vector xf xf : formed and represents only an estimation of the true a a,i atmospheric state xt at t. As such, the assessment of H 5 fxy : hxf xf , xy xti 5 0g (17) the impact of various data types in the DAS to reduce i a a,i C the actual forecast error must account for the state-to- identifies the set of verification states that are objec- observation space uncertainty propagation based on the tive in estimating the OBSI of the data component forecast error measure that involves the unknown true y , dIose(y ) 5 0. A geometrical illustration is provided atmospheric state xt: i t,y i in Fig. 1. It is also noticed that in a deterministic ap- f t T f t f t 2 proach verification states of equal precision may result et(x) 5 (x x ) C(x x ) 5 x x C. (10) in an ambiguous OBSI guidance; for example, Fig. 1 To account for the errors in the verification state, let n shows a configuration of equal-precision states xy, xw 2 t ose ose denote the dimension of the state vector and S(x , ) such that Iy (yi) , 0 and Iw (yi) . 0.

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FIG. 1. Illustration of the uncertainty in the observation impact FIG. 2. Illustration of the uncertainty in the observation impact estimation using deterministic OSEs. The verification state error estimation in the deterministic adjoint-DAS framework. Verifica- f f component along the direction xa xa,i contributes to the uncer- tion states of equal-precision and equal-overall data impact may tainty propagation. Verification states of equal-precision xy and xw provide an ambiguous observation impact guidance through the f y may provide an ambiguous observation impact guidance. vector xm 2 x .

In a probabilistic framework, by expressing xf xf 5 f f T y t a a,i de de 5 2(xa x ) C(x x ) (20) f t f t t y b (xa x ) (xa,i x ) in (17), it is noticed that an objective OBSI assessment should be based on verification such that the hyperplane: states whose errors are unbiased and are statistically independent from the errors in the forecasts produced by y f f y t Hde 5 fx : hxa xb, x x iC 5 0g (21) the DAS. t b. OBSI uncertainty propagation in adjoint DAS identifies the set of verification states of equal-overall data impact: In the deterministic adjoint-DAS framework the uncertainty in the verification state is propagated into the dey 5 det. (22) OBSI estimates of all the data subsets being evaluated. The uncertainty in the OBSI estimation (9) of a data In the deterministic adjoint-DAS approach verification subset yi is expressed in a general form as states of equal-precision () and equal-overall data impact (de ) located on the (n 2 2) sphere of Rn: adj adj adj T T t dIt,y (yi) 5 It (yi) Iy (yi) 5 (dyi) [K (gt gy)]i. (18) t S,de 5 S(x, ) \Hde For example, associated with the measure in (7)–(8): t t y Rn y t f f y t 5fx 2 :kx x k5 , hxa xb, x x iC 5 0g adj T T T T y t dIt,y (yi) 5 (dyi) [K (Ma 1 Mb )C(x x )]i, (19) (23) which shows that, accounting for the errors in xy only, will lead to OBSI estimates that will vary in magnitude the uncertainty in the OBSI guidance is a result of the and that for certain data subsets may also vary in sign. In backward propagation of the errors in the verification general, as illustrated in Fig. 2, S consists of two points ,det state through the adjoint model, the adjoint-DAS oper- for n 5 2, is a circle for n 5 3, and so on. For simplicity, ator, and amplified by the innovation vector components. consider the case of linear dynamics xf 5 Mx where M To provide a geometrical perspective, it is noticed that is a state-independent matrix operator. In this frame- by analogy to the derivation in (13)–(17) in the adjoint- work, the measure in (3) is a quadratic function of the DAS approach: initial conditions and the approximation in (6)–(8) is thus

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 3571 exact. Theoretically, for quadratic measures all second- system provides four data types y(1), y(2), y(3),andy(4), order adjoint-based OBSI estimates are equivalent each being a 10-dimensional vector, and that data type y(i) (Daescu and Todling 2009). The vector gy defined by is taken at locations 4k 1 i, k 5 0, 1, ... , 9. Compo- (7)–(8) is expressed as nentwise, the structure of the observation vector is (1) (2) (3) (4) (1) (2) (3) (4) (1) (2) (3) (4) T thus y 5 [y1 y1 y1 y1 y2 y2 y2 y2 ...y10 y10 y10 y10 ] . T y T f y (i) gy 5 M C(Mxa 1 Mxb 2x ) 5 2M C(xm x ), (24) The observation errors are normally distributed N [0, s ] with the standard deviation prescribed as s(1) 5 0.1, s(2) where 5 0.2, s(3) 5 0.4, and s(4) 5 0.8. The DAS implements an extended Kalman filter algorithm that provides the f 1 1 f f background estimate at time ti1 and the associated xm 5 (Mxa 1 Mxb) 5 (xa 1 xb) (25) 1 2 2 background error covariance matrix according to f f denotes the midpoint of the segment from xb to xa. For t 5 t y f y xb( i11) Mt !t [xa( i)] and (27) each verification state x 2S the vector xm 2 x in i i11 ,det (24) will vary in magnitude and orientation for example, B(t ) 5 M(t )A(t )MT(t ) 1 Q(t ), (28) Fig. 2 illustrates a configuration where verification states of i11 i i i i equal-precision and equal-overall data impact may pro- where x (t ) is the analysis (1) at time t , vide an ambiguous OBSI guidance in (9) since in this case a i i

f y f w A(ti) 5 [I K(ti)H(ti)]B(ti) (29) (xm,2 x2)(xm,2 x2 ) , 0.

is the analysis error covariance matrix, and M(ti)is the state-dependent Jacobian matrix of the numerical 4. Numerical experiments model from ti to ti11. The model error covariance matrix The Lorenz 40-variable model (Lorenz and Emanuel Q is taken to be time invariant, diagonal and with con- 1998) stant entries, and by trial and error the specification of the diagonal entries Qjj 5 0.01 was found to provide dx improved analyses as compared to other selections. j 5 (x x )x x 1 F, where j 5 1, 2, ..., n, dt j11 j2 j1 j Numerical experiments are set with two data assimila- (26) tion systems that differ only in the specification of the observation error covariance R: in a first set of experi- n 5 40, x21 5 xn21, x0 5 xn, and xn11 5 x1, is used in ments, hereinafter referred to as DAS-I, R is taken to be OSEs and the adjoint-DAS approach to investigate the diagonal and with entries Rjj that are statistically con- uncertainty in the OBSI estimation as a result of the sistent with the observation errors in each data type, as errors in the verification state. For this model, Liu and described above; in a second set of experiments, here- Kalnay (2008) provided a comparative analysis of the after referred to as DAS-II, all the diagonal entries are OBSI derived from the adjoint-DAS method in (6)–(8) taken Rjj 5 0.04, which corresponds to an observation and an ensemble sensitivity method. The system in (26) error standard deviation of 0.2. The DAS-I attempts to is integrated with a fourth-order Runge–Kutta method simulate an optimal DAS, whereas in the DAS-II the and a constant time step Dt 5 0.05 that in the data as- errors in data type y(1) are overestimated and the errors similation experiments is identified to a 6-h time period. in data types y(3) and y(4) are underestimated. Each DAS The time evolution of the true state xt is obtained by configuration is run for 7560 analysis cycles and time- taking the external forcing to be F 5 8 and a model error averaged estimates of the OBSI on the 24-h forecast is considered in the forecast model by taking F 5 7.6. An error are collected using OSEs and the adjoint-DAS t initial state x0 is obtained by a 90-day (360 time step) approach over the last N 5 7200 analysis cycles (a 5-yr integration started from xj 5 8 for j 6¼ n/2 and xn/2 5 time period). The time-averaged relative error in the t 8.008; a background estimate xb to x0 is prescribed by analyses over this period was found to be of ;2.6% in t introducing random perturbations in x0 taken from the the DAS-I and of ;5.1% in the DAS-II. Therefore, the standard normal distribution N (0, 1). Observational relative error in the verification states xy produced by data is generated for each state component (at each grid the DAS is about 2 times larger in the DAS-II as com- point) and at each time step, such that the observation pared to the DAS-I. OSEs are set by removing a single operator is the identity. To investigate the uncertainty in data point from the observing system, for each data lo- the OBSI estimation when various data types are in- cation, for a total of n additional data assimilation exper- cluded in the DAS, it is assumed that the observing iments. The adjoint-DAS approach to OBSI estimation

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FIG. 3. Time-averaged adjoint-DAS and OSEs’ observation impact estimates on the 24-h forecast error in the DAS-I setup. Results with a verification state provided by the DAS (ey measure, open circles) and with a verification state taken to be the true state (et measure, crosses). For each approach, the figure on the right displays the relative error in the OBSI estimation as a result of the errors in the verification state.

implements the second-order measure in (6)–(9) and the approach the relative error in the ey-based OBSI as time-averaged relative error in the dey approximation in compared to the et-based OBSI remains in general (6) was found to be of ;3%. This additional source of within a small factor from the relative error in the ver- uncertainty is characteristic to the current formulation ification state; however, it is noticed that data type y(4) of the adjoint-DAS approach and in operational systems (grid point locations 4, 8, ... , 40) may be subject to a was found to range from 15% to 25% as a result of the larger OBSI uncertainty. The complementary nature of difficulties of the practical implementation (Langland the OBSI information extracted from OSEs and the 2005; Gelaro et al. 2007). For both OSEs and the adjoint- adjoint-DAS approach is discussed in detail in the work DAS approach the uncertainty in the OBSI estimation of Gelaro and Zhu (2009) and Cardinali (2009) and it is due to the errors in the verification state is monitored by not further addressed here. Since the adjoint-DAS OBSI comparing the results from a forecast error measure ey operator in (9) is linear in the observation space, the based on a verification state xy produced by the corre- OBSI information from the adjoint DAS in Fig. 3 allows a sponding DAS to the OBSI estimates from a forecast quantification of the impact of any data subset whereas t error measure et based on the true state x . additional experiments are required in the OSEs to an- The results obtained in the DAS-I setup are displayed alyze the OBSI of other data subsets. in Fig. 3. It is noticed that in this idealized DAS both the The results obtained in the DAS-II setup are dis- OSEs and the adjoint-DAS properly identify each data- played in Fig. 4. Both OSEs and the adjoint-DAS OBSI type impact on the forecast error reduction, at all data estimates reveal an increased uncertainty at all locations locations. Negative OBSI values indicate that each data of data type y(4) whose observation error is largely un- is of benefit to the forecasts, with data type y(1) having the derestimated in the DAS-II. It is also noticed that in largest contribution (grid point locations 1, 5, ... ,37), these experiments the adjoint-DAS OBSI based on the followed in decreasing order of magnitude by data types ey measure attributes a benefic forecast impact from the y(2), y(3), and y(4). For both OSEs and the adjoint-DAS assimilation of any subset of data type y(4), whereas both

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FIG. 4. As in Fig. 3, but for the OBSI estimation in the DAS-II setup. in the adjoint-DAS approach and OSEs the observation data assimilation systems (Tan et al. 2007; Ancell and impacts derived from the measure et identify the as- Hakim 2007). An objective OBSI assessment relies on similation of this data type as detrimental to the fore- the ability to provide verification states whose errors are casts. The compounded propagation of errors in the unbiased and are statistically independent from the er- verification state and in the dey adjoint-based approxi- rors in the forecasts produced by the DAS. mation measure in the OBSI estimation needs to be further investigated. Acknowledgments. This work was supported by the NASA Modeling, Analysis, and Prediction Program 5. Conclusions under Award NNG06GC67G. This research brings forward the issue of objective assessment of observations value based on the obser- REFERENCES vation impact calculations from deterministic OSEs and Ancell, B. C., and G. J. Hakim, 2007: Comparing ensemble and the adjoint-DAS approach. A geometrical perspective is adjoint sensitivity analysis with applications to observation provided to the uncertainty in the OBSI estimation in targeting. Mon. Wea. Rev., 135, 4117–4134. the presence of errors in the verification state. Numeri- ——, and C. F. Mass, 2008: The variability of adjoint sensitivity cal experiments with a simple model are used to illus- with respect to model physics and basic-state trajectory. Mon. trate that the practical difficulty of providing an accurate Wea. Rev., 136, 4612–4628. Andersson, E., A. Hollingsworth, G. Kelly, P. Lo¨ nnberg, representation of the true atmospheric state in the J. Pailleux, and Z. Zhang, 1991: Global observing system ex- forecast error measure may lead to an increased un- periments on operational statistical retrievals of satellite certainty in the OBSI estimation derived from deter- sounding data. Mon. Wea. Rev., 119, 1851–1865. ministic forecast error measures, in particular when Atlas, R., 1997: Atmospheric observations and experiments to as- suboptimal input error statistics are specified in the sess their usefulness in data assimilation. J. Meteor. Soc. Japan, 75, 111–130. DAS. To increase the confidence in the OBSI guidance, Baker, N. L., and R. Daley, 2000: Observation and background a probabilistic framework may be considered using an adjoint sensitivity in the adaptive observation-targeting ensemble of verification states and/or an ensemble of problem. Quart. J. Roy. Meteor. Soc., 126, 1431–1454.

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