1 Department of Ocean Sciences, 1156 High Street. University of California, Santa Cruz CA

Transcription

1 The Regional Ocean Modeling System (ROMS) 4-Dimensional Variational Data Assimilation Systems: II - Performance and Application to the California Current System Andrew M. Moore 1, Hernan G. Arango 2, Gregoire Broquet 3, Chris Edwards 1, Milena Veneziani 1, Brian Powell 4, Dave Foley 5, James D. Doyle 6, Dan Costa 7 and Patrick Robinson 7 1 Department of Ocean Sciences, 1156 High Street University of California, Santa Cruz CA Institute of Marine and Coastal Sciences, Rutgers University, 71 Dudley Road, New Brunswick NJ Laboratoire des Sciences du Climat et de l Environnement, CEA-Orme des Merisiers, F GIF-SUR-YVETTE CEDEX, France. 4 Department of Oceanography, University of Hawai i at Manoa, Honolulu HI Environmental Research Division, NOAA Southwest Fisheries Science Center, Pacific Grove, California. 6 Naval Research Laboratory, Monterey, California. 7 Department of Ecology and Evolutionary Biology, Long Marine Laboratory University of California, Santa Cruz CA November 14, 2010

2 Abstract The Regional Ocean Modeling System (ROMS) 4-dimensional variational (4D-Var) data assimilation systems have been systematically applied to the mesoscale circulation environment of the California Current to demonstrate the performance and practical utility of the various components of ROMS 4D-Var. In particular, we present a comparison of three approaches to 4D-Var, namely: the primal formulation of the incremental strong constraint approach; the dual formulation physical-space statistical analysis system ; and the dual formulation indirect representer approach. In agreement with theoretical considerations all three approaches converge to the same ocean circulation estimate when using the same observations and prior information. However, the rate of convergence of the dual formulation was found to be inferior to that of the primal formulation. Other aspects of the 4D-Var performance that relate to the use of multiple outer-loops, preconditioning, and the weak constraint are also explored. A systematic evaluation of the impact of the various components of the 4D-Var control vector (i.e. the initial conditions, surface forcing and open boundary conditions) is also presented. It is shown that correcting for uncertainties in the model initial conditions, surface wind stress forcing, and for model errors exerts the largest influence on the ability of the model to fit the available observations. Various important diagnostics of 4D-Var are also examined, including estimates of the posterior error, the information content of the observation array, and innovation-based consistency checks on the prior error assumptions. Using these diagnostic tools, we find that more than 90% of the observations assimilated into the model provide redundant information, a symptom of the large percentage of satellite data used and data processing. i

3 1 Introduction Ocean data assimilation is an essential component of ocean modeling, and is recognized as a valuable and indispensable tool for obtaining seamless estimates of the ocean circulation. However, data assimilation is a technically and computationally challenging problem, and the development of state-of-the-art data assimilation systems requires substantial effort and commitment of resources. The development of ocean data assimilation systems for community ocean models is therefore an important and critical undertaking if such methods are to be widely accessible. In Moore et al., (2010a), hereafter referred to as Part I, we presented a review of current state-of-the-art 4-dimensional variational (4D-Var) data assimilation systems and related diagnostic tools that are being increasingly used for oceanographic applications, and that have been developed for the Regional Ocean Modeling System (ROMS). ROMS is a widely used community ocean model (Haidvogel et al., 2009), and all of the 4D-Var capabilities described in Part I are freely available to the ocean modeling community ( ROMS is unique in that it is the only community regional ocean model for which such a comprehensive suite of 4D-Var based data assimilation tools are available. As described in Part I, ROMS users can choose between three incremental 4D-Var systems: a strong constraint, primal formulation, (see Courtier et al., 1994) referred to as I4D-Var; a dual formulation based on physical-space statistical analysis, (see Da Silva et al., 1995) referred to as 4D-PSAS; and a dual formulation based on the indirect representer algorithm (see Egbert et al., 1994), referred to as R4D-Var. Both 4D-PSAS and R4D-Var also support the weak constraint, and the choice of 4D-Var scheme that should be used for a particular 1

4 problem is dictated by several practical considerations (e.g. number of observations, the linear stability of the circulation, etc). In this paper, we demonstrate the performance and utility of ROMS 4D-Var as applied to the California Current System (CCS). The CCS is characterized by a pronounced seasonal cycle of upwelling and by energetic mesoscale circulations that are governed by various physical processes (Hickey, 1998; Checkley and Barth, 2009), and therefore provides a challenging test bed for demonstrating the capabilities of ROMS 4D-Var. The configurations of ROMS CCS and 4D-Var are described in section 2. A systematic comparison of the performance and utility of ROMS I4D-Var, 4D-PSAS, and R4D-Var is presented in section 3 under the assumption of the strong constraint, while in section 4 the strong constraint is relaxed and the influence of model errors are considered. Section 5 deals with the important issue of posterior error estimation, while in section 6 we consider various important diagnostics of 4D-Var, including: innovation-based consistency checks of the analysis system; quantification of the effective number of degrees of freedom of the observation array; and the degree of reachability of observed ocean states, and array modes. A summary of important findings and conclusions is presented in section 7. 2 Model Configuration and Experimental Set-up The ROMS CCS domain is described by Veneziani et al. (2009a,b) and Broquet et al. (2009a,b) and spans the region 134 W to 116 W and 31 N to 48 N. Two different configurations were used in the experiments presented here: one with 30 km horizontal resolution and 30 terrain-following σ-levels, and one with 10 km resolution and 42 σ-levels. The 30 km 2

5 model has been used extensively for development and testing of the 4D-Var algorithms, while the 10 km model provides significantly better resolution of the mesoscale circulation features and presents a more significant challenge for 4D-Var because of the much larger computational cost. When necesary, the 30 km model will be used to illustrate the overall performance and characteristics of the ROMS 4D-Var algorithms, while select experiments will be presented from the 10 km model to illustrate the generality of the conclusions derived from the 30 km model. The 30 km and 10 km models will be referred to as WC30 and WC10 respectively, and the model domain and bathymetry for each configuration is shown in Fig. 1. The model forcing was derived from daily averaged output of boundary layer fields from the Naval Research Laboratory s (NRL) Coupled Ocean-Atmosphere Mesoscale Prediction System (COAMPS) (Doyle et al., 2009). The ocean surface fluxes were derived using the bulk formulations of Liu et al. (1979) and Fairall et al. (1996a,b), and represent the background (or prior) surface forcing f b (t) in the incremental formulation of 4D-Var introduced in Part I (section 3). The model domain has open boundaries at the northern, southern, and western edges, and at these boundaries the tracer and velocity fields were prescribed, while the free surface and vertically integrated flow are subject to Chapman (1985) and Flather (1976) boundary conditions respectively. The prescribed open boundary solution was taken from the Estimating the Circulation and Climate of the Ocean (ECCO) global data assimilation product (Wunsch and Heimbach, 2007), and represents the background (prior) boundary conditions b b (t) in the incremental formulation of 4D-Var. A sponge layer was also used adjacent to each open boundary where viscosity increased linearly from 4 m 2 s 2 in the interior to 400 m 2 s 2 at the boundary over a distance of 100 km. 3

6 ROMS 4D-Var is generally run sequentially, and the starting point for the experiments presented here was taken from the 4D-Var sequence performed by Broquet et al. (2009a). Before commencing the first cycle of the sequence in the CCS, Broquet et al. (2009a) first spun-up the model for 7 years using climatological forcing from the Comprehensive Ocean-Atmosphere Data Set (COADS) and using Levitus climatology at the open boundaries. Strong constraint I4D-Var was performed sequentially starting on 1 Jan 1999, using 14 day assimilation windows adjusting only the initial conditions as described by Broquet et al. (2009a). The background initial conditions for the first assimilated cycle for the sequential experiments presented here was taken on 27 July 2002 from the Broquet et al. (2009a) experiments. The model background initial conditions x b (t 0 ) for each subsequent data assimilation cycle was the best circulation estimate derived from the previous 4D-Var assimilation cycle. 2.1 Observations Observations from various different instruments and satellite platforms were assimilated into ROMS CCS. These include: Aviso Dynamic Topography (DT): This product combines sea level anomaly data (a merged product comprised of Ssalto-Duacs data from TOPEX/Poseidon, Jason-1, Envisat and GFO measurements) and the mean dynamic topographic data estimated by Rio et al. (2005), provided by Aviso 188 and the Centre National d Etudes Spatiales. Data are available every 7 days with horizontal resolution 1/3. The steric signal was removed using the Willis et al. (2004) database, since ROMS is a Boussinesq model. DT was compared to the model SSH and calibrated to ensure that the spatio-temporal mean of DT and the free model are equal. 4

7 Blended SST data: The SST product used is a blend of data from the GEOS, AVHRR and MODIS satellite instruments and was developed at the CoastWatch/NOAA Fisheries in Pacific Grove, CA. Blended SST estimates were available from July 2002 every day as a 5 day mean product with horizontal resolution 10 km, and data were available only east of 130 W at the time of our experiments. In situ T and S data: Quality controlled data from the European Union ENSEMBLES project (EN3) archived at the UK Met Office (Ingleby and Huddleston, 2007) were also used. Within the domain spanned by ROMS CCS, EN3 contains in situ observations from a variety of instruments, platforms and sources that include mechanical and expendable bathythermographs (denoted XBT), Argo floats, and Conductivity, Temperature, Depth (CTD) profiles from the World Ocean Data (WOD) 2005, the Global Temperature-204 Salinity Profile Program (GTSPP), and the USGODAE Argo Global Data Assembly Centers (GDAC). Many of the CTD profiles were collected during the California Cooperative Ocean Fisheries Investigation (CalCOFI), GLOBal ocean ECosystems dynamics experiment (GLOBEC), and the NE Pacific Long Term Observation Program (LTOP) survey cruises, which offer the most regular sampling within the region (Bograd and Lynn, 2003). Other miscellaneous hydrographic data from volunteer observing ships are also included in EN3. In addition, temperature observations from tagged elephant seals as part of the Tagging of Pacific Pelagics program (TOPP, were also assimilated. Before assimilation, all observations within each model grid cell, over a 6h time window, were combined to form super observations to reduce data redundancy. The standard deviation of the observations that contribute to each super observation was used as a measure of the error of representativeness. 5

8 2.2 4D-Var configuration Before running ROMS 4D-Var, it is necessary to specify the background error covariances for the observations, initial conditions, surface forcing, open boundary conditions, and model errors in the case of weak constraint 4D-Var. In addition, choices must be made regarding the number of inner- and outer-loops that will be used, and the form of second level preconditioning that will be employed, if any. Decisions regarding the number of iterations and preconditioning will be discussed in section 3 in relation to specific experiments. In this section we will summarize the choice of parameters for modeling the background error covariance matrix, D, and the observation error covariance matrix, R. Further details regarding the choice of prior error covariances and decorrelation lengths can be found in Broquet et al. (2009a, 2010). Observation errors were assumed to be uncorrelated in space and time, and the variances along the main diagonal of R were assigned as a combination of measurement error and the error of representativeness. Measurement errors were chosen independent of the data source, with the following standard deviations: 2 cm for DT; 0.4 C for SST; 0.1 C for in situ T ; and 0.01 for in situ S. At the present time, ROMS 4D-Var supports only homogeneous error correlations that are separable in the horizontal and vertical. The decorrelation length scales used to model the background errors of all initial condition control variable components of D were 50 km in the horizontal and 30 m in the vertical. Horizontal correlation scales chosen for the background surface forcing error components of D were 300 km for wind stress and 100 km for heat and freshwater fluxes. The correlation lengths for the background open boundary 6

9 condition error components of D were chosen to be 100 km in the horizontal and 30 m in the vertical. The surface forcing and boundary condition increments, δf(t) and δb(t), were computed daily, and interpolated to each intervening model time step. In all calculations presented here, the multivariate balance operator, K b, (or the T S sub-component of K b ) described in Part I (section 5.2) was used. The background error standard deviations for the initial condition components of the control vector were estimated each month based on the variance of the model run during the period subject only to surface forcing (i.e. no data assimilation). The temporal variability of the COAMPS surface forcing for the period was used as the variance for background surface forcing error, and the open boundary condition background error variances were chosen to be the variances of the ECCO fields. A discussion of the prior model error covariances will be postponed until section 4. The sensitivity of 4D-Var in the CCS to the choice of error covariance parameters was explored by Broquet et al. (2009a,b, 2010), and the parameters used here yield the best results. The influence and ramifications of the choices of error covariances are discussed throughout in relation to the data assimilation results. 3 Strong Constraint 4D-Var We will present first the results of experiments performed using I4D-Var, 4D-PSAS and R4D-Var subject to the strong constraint (i.e. assuming, a priori, an error-free model). This allows us to illustrate some important properties relating to the equivalence of the resulting circulation estimates from the inner-loops, and the convergence of each assimilation 7

10 algorithm. The strong constraint will be relaxed in section Observation-space versus control vector-space searches As discussed in Part I (section 3), the best ocean circulation estimate is identified by searching for solutions of the model that minimize the cost function J given by: J(δz) = 1 2 δzt D 1 δz (Gδz d)t R 1 (Gδz d). (1) where, in the case of strong constraint 4D-Var, δz is the control vector comprised of increments to the background initial conditions, δx(t 0 ), background surface forcing, δf(t), and background boundary conditions, δb(t). The matrix G is the operator that samples the tangent linear model at observation points; and d = (y H(x b (t))) is the innovation vector where y is the vector of observations, and H(x b ) is the background x b (t) evaluated at the observation points via the observation operator H. One approach is to perform a search for the minimum of J in the full space spanned by the control vector. This is the option exercised in ROMS I4D-Var. However, in the incremental form of J, the observations that are central to the second term on the righthand side (rhs) of (1) provide information only about linear combinations (via Gδz) of some of the elements of the state-vector increment, δx(t). Therefore, an alternative approach is to perform a search for the minimum of J in the subspace spanned by the linear combinations of the observed model variables. The associated space is referred to as the dual or observation-space, and this is the method employed in 4D-PSAS and R4D-Var. As shown in Part I, minimization of J in the space of the control vector leads to the 8

11 following expression for the analysis increments δz a : δz a = (D 1 + G T R 1 G) 1 G T R 1 d (2) also referred to as the primal formulation. Alternatively, minimization of J in the dual space leads to the equivalent expression: δz a = DG T (GDG T + R) 1 d. (3) In I4D-Var, (2) is identified by direct minimization of J (see Part I, section 3.2.1), while in 4D-PSAS and R4D-Var (3) is identified by minimizing an auxilliary function (see Part I, sections and 3.2.3). In either case, the minimization procedure is equivalent to inverting a large matrix. In the case of I4D-Var the matrix inverse (D 1 + G T R 1 G) 1 is required (i.e. the inverse of the Hessian of J) and has the dimension of the control vector, which for the WC30 and WC10 is O(10 5 ) and O(10 7 ) respectively. For 4D-PSAS and R4D-Var, however, the matrix inverse (GDG T + R) 1 is required (i.e. the inverse of the stabilized representer matrix) which has the dimension of the number of observations, which for WC30 and WC10 is O(10 4 ) and O(10 5 ) respectively. In practice, these inversions are performed iteratively by identifying the solutions of equivalent systems of linear equations. As described in Part I, the solution procedure involves a sequence of inner-loop iterations which perform a minimization of J. The circulation, forcing, and boundary conditions about which the inner-loops are linearized can be updated via a series of outer-loops. Equations (2) and (3) suggest that if the iterations in control vector-space and observation-space are continued to convergence, the solutions obtained for the analysis increments in both cases will be the 9

12 same A single outer-loop The performance of the three ROMS 4D-Var algorithms will be demonstrated here by considering first a representative case study, and then showing the generality of the result when the system was run sequentially. We consider first a representative period during March 2003 and show a comparison of I4D-Var, 4D-PSAS and R4D-Var using a 4-day, 7-day and 14-day assimilation window, all starting on 3 March In each case the control vector was comprised of increments to the model initial conditions, δx(t 0 ), surface forcing, δf(t), and open boundary conditions, δb(t). As described in Part I (section 3), the nonlinear ROMS is used during the outer-loops of I4D- Var and 4D-PSAS to update the x(t) about which each sequence of inner-loops is minimized. However, during R4D-Var the finite-amplitude tangent linear model is used instead which means that I4D-Var, 4D-PSAS and R4D-Var will only yield identical inner-loop solutions when using a single outer-loop. When more than one outer-loop is used, identical solutions for R4D-Var and I4D-Var/4D-PSAS cannot be guaranteed, so we will restrict our attention to the case of a single outer-loop for the time being. Cases using multiple outer-loops will be considered in section Figures 2 and 3 show plots of log 10 (J) vs the number of inner-loops m = 1,..., 100 from I4D-Var for the three different assimilation windows using WC30 and WC10 respectively. It is evident that in all cases J decreases monotonically and asymptotes to a near constant value, indicating that effective convergence of J to the minimum value has been achieved. Figures 2 and 3 indicate that the number of iterations required to reach the asymptote increases with the length of the assimilation window, but is 10

13 relatively unaffected by resolution. Also shown in Figs. 2 and 3 is log 10 (J) for dual 4D-Var calculations using 4D-PSAS and R4D-VAR. The inner-loop cost functions J for both 4D-PSAS and R4D-Var are identical and Figs. 2 and 3 indicate that both converge to the minimum of J more slowly than I4D- Var. However, in accordance with equations (2) and (3) all three assimilation algorithms yield the same solution for a single outer-loop, irrespective of the length of the assimilation window. This was also confirmed by inspection of the circulation estimates from each case (not shown). It is important to note that the circulation estimates obtained using the primal formulation I4D-Var and dual formulation of either 4D-PSAS or R4D-Var are only equivalent for the inner-loop increments at the minimum of J. If the inner-loop iterations are terminated before the asymptote in J has been reached, the solutions obtained using the primal and dual formulations are, in general, very different. The control space searches of I4D-Var typically visit physically realistic parts of the solution space as the minimum of J is approached monotonically, so terminating the inner-loop before complete convergence to the asymptotic minimum of J yields acceptable circulation estimates (e.g. Broquet et al., 2009a). Dual space searches, on the other hand, often visit physically unrealistic parts of solution space before arriving at the minimum of J, meaning that termination of the inner-loops in 4D- PSAS and R4D-Var before convergence has been reached is not advisable. El Akkraoui and Gauthier (2010) have recently explored this issue and the slow convergence rate of dual 4D- Var compared to primal 4D-Var in numerical weather prediction, and offer some practical solutions to these issues which will be implemented in ROMS at a future date. As noted, 4D-PSAS and R4D-Var are identical at the end of each inner-loop when using 11

14 a single outer-loop, and they differ only during the final computation of the ocean circulation estimate x a (t). In the case of 4D-PSAS, x a (t) is computed using the nonlinear model, while for R4D-Var the finite-amplitude tangent linear model (RPROMS) is used 1. The generality of the results in Figs. 2 and 3 is illustrated in Fig. 4 which shows time series of the initial and final values of log 10 (J) when I4D-Var, 4D-PSAS and R4D-Var were run sequentially using WC30 for the period July 2002 to Dec 2004 using a 7 day assimilation window, 1 outer-loop, and 50 inner-loops. The best estimate circulation on day 7 of each window was used as the background initial condition x b (t) for the next assimilation cycle. The time series of J from each 4D-Var algorithm are very similar, because the circulation estimates are very close to one another. Differences between solutions are associated with (i) the inevitable differences in the nonlinear model trajectories about which the inner-loops were linearized due to rounding errors, (ii) the tangent linear assumption employed in the outer-loop of RPROMS in the case of R4D-Var (as evidenced by the blue curves in Fig. 4 which generally differ most from I4D-Var and 4D-PSAS, the latter two being always very similar to each other), and (iii) the fact that 50 inner-loops may not always provide adequate convergence of J to the minimum value in the case of the dual formulations. Figures 2 and 3 also show two other important diagnostics for the chosen assimilation cycle. First, for later reference in section 6.2, the dashed line in each panel represents the theoretical minimum value of J that should be reached by each assimilation cycle in the event that the prior hypotheses about the various sources of error embodied in D and R 1 For a single outer-loop, the finite-amplitude tangent linear model solution is simply x b (t) + δx(t) where δx(t) is the solution of the perturbation tangent linear model initialized with initial condition δx a (0), forced by δf(t), and subject to the boundary conditions δb(t). 12

15 are correct. The expected minimum value of J is N obs /2, half the number of observations (Bennett et al., 1993). Second, Figs. 2 and 3 also show the value of the nonlinear cost function J NL given by: J NL = 1 2 (z zb ) T D 1 (z z b ) (y H(x))T R 1 (y H(x)) (4) where z and z b are respectively the vectors of full control variables and full background control variables (as opposed to the increments). J NL was computed after the last innerloop using the best circulation estimate z = z b + δz a derived from the increments δz a. In the case of I4D-Var and 4D-PSAS the circulation component (x) of z is a solution of NLROMS. However, for R4D-Var the x component of z is a solution of RPROMS meaning that in this case J NL will be identical to the value of J during the final inner-loop given by (1). Figures 2 and 3 indicate that for I4D-Var and 4D-PSAS, J NL is significantly larger than J indicating that the tangent linear assumption employed in the inner-loops yields an over optimistic measure of departures of the analysis increments from the background and observations Multiple outer-loops and second-level preconditioning In this section we illustrate the impact of using multiple outer-loops on 4D-Var performance. Multiple outer-loops are generally considered advantageous in 4D-Var for several reasons. First, updating the nonlinear model solution about which the inner-loops are linearized should yield a more reliable best estimate circulation. Second, for a given total number of iterations (i.e. number of outer-loops number of inner-loops) the potential exists to accelerate convergence to the cost function minimum. Finally, when more than one outerloop is used, 2nd level preconditioning predicated on the previous outer-loops may be used 13

16 to accelerate the convergence of the cost function to its minimum value. Figure 5 shows log 10 (J) versus the total number of iterations for a series of WC30 calculations using I4D-Var and different combinations of the number of outer-loops, n, and inner-loops, m, where the total number of iterations, nm, is approximately the same and 100 in each case. In each case, a 4 day assimilation window was used corresponding to the period 3-7 March, No second level preconditioning was used. Also shown in Fig. 5 are the values of J NL evaluated at the end of each outer-loop. While the final value of inner-loop cost J is similar for each combination of n and m, Fig. 5 shows that the values of J NL at the end of the final outer-loop are quite different. In general, more frequent updates of the nonlinear ROMS solution yield lower final values of J NL, although there are exceptions. For example, the cases n = 4 and n = 5 actually lead to an increase in J NL. This is particularly evident in the case n = 4 where J NL at the end of the 4 th outer-loop is significantly larger than all other cases even though J NL during the preceding outer-loops is much smaller. While there is no guarantee that increasing n will improve the final value of J NL, the problem encountered here for n = 4, m = 25 can be alleviated using second level preconditioning. As described in Part I (section 6), ROMS employs two levels of preconditioning. The first level of preconditioning is always applied, while second level preconditioning is controlled by the user. The influence of second level preconditioning for the 3-7 March period is illustrated in Table 1 which shows the final value of J NL for n = 4, m = 25 when second level Ritz preconditioning was employed (as described in Part I, section 6.2) using different numbers of Ritz vectors. Table 1 shows that J NL improves dramatically when the number of Ritz vectors used increases from 1 to 2. Adding a third and fourth Ritz vector to the preconditioner reduces J by a further 2%. However, for a larger number of 14

17 Ritz vectors, J NL begins to increase again due to inaccuracies in the additional members of the Ritz spectrum. 2 Qualitatively similar findings have been reported by Tshimanga et al. (2008). 3.2 Control vector impacts on J In all of the strong constraint experiments discussed so far, the control vector δz was comprised of increments to the initial conditions, δx(t 0 ), surface forcing, δf(t), and open boundary conditions, δb(t). However, it is of considerable interest to quantify the relative impact that the adjustments to each of the control variables has on the resulting circulation estimates as measured by J. This is illustrated in Fig. 6a which shows the final value of J o = 1/2(Gδz d) T R 1 (Gδz d), the contribution of the observations to J in (1), from a sequence of experiments using R4D-Var, 1 outer-loop and 50 inner-loops in WC30 and the same 7 day assimilation window 3-10 March, In each experiment the increment control vector δz is comprised of a different combination of δx(t 0 ), δf(t) = (δτ T, δq T, δf T ) T and δb(t), where δτ, δq, and δf are the increments in surface wind stress, heat flux, and net freshwater flux respectively. The value of J o when no data were assimilated is also shown for reference. We consider J o in Fig. 6a, rather than J, because the the values of J for each experiment cannot be directly compared since the size of the first-term on the rhs of (1) depends on the number of control elements of δz. However, since N obs is the same in all cases, the values of J o from each experiment can be directly compared. Nonetheless, all of the comments below about J o also apply to J. 2 As described in Part I, the Ritz vectors were identified using an iterative Lanczos method meaning that not all vectors identified are reliable. The accuracy of the Ritz vectors is discussed further in section

18 Figure 6a indicates that by far the largest decrease in J o (and J) is always associated with adjustments in the initial conditions (Experiments B, C, D, E, F and O) indicating that a good fit of the model to the observations is always possible when the initial conditions are included as part of the control vector. Adjustments of the surface wind forcing can also lead to significant reductions in J o (and J) (Experiments G, H, I, J, M and N), while adjustments to the surface heat flux, fresh water flux and open boundary conditions typically yield very small reductions in J o (and J) (e.g. Experiments K and L). Also, in all cases where the initial conditions are adjusted J is closer to J min (not shown). The generality of the results from the single assimilation cycle in Fig. 6a are confirmed in Fig. 6b which shows time series of the ratio of the final and initial values of J o from the sequence of WC30 R4D-Var cycles when the control vector was comprised of either δx(t 0 ) only, different components of δf(t) only, or only δb(t). Figure 6b shows that during all cycles, the largest fractional change in J o is associated with adjusting the initial conditions. During most cycles, adjusting the wind stress component of δz is the next most effective control variable, although adjustments in surface fluxes of heat and freshwater are some times as equally effective as adjustments in the wind stress for fitting the model to the observations. 4 Weak Constraint 4D-Var So far we have side-stepped the issue of model error by invoking the strong constraint, thereby requiring that the best circulation estimates are exact solutions of the ROMS equations. However, model errors are undoubtedly present and it behooves us to account for their effects 16

19 during 4D-Var, in which case the resulting best estimate circulations are only approximate solutions of the ROMS equations. It should be stated at the outset that the character, structure, and properties of model errors are poorly understood, and prescription of the prior model error covariance Q is a considerable challenge (Bennett, 2002; Trémolet, 2006). As in the case of the prior errors for x b (t 0 ), f b (t), and b b (t), we cannot hope to capture completely the correct statistical properties of model errors in the presence of complex geophysical fields. However, any reasonable and well founded choice of prior model error should yield a more reliable ocean circulation estimate. In this section we will present results from one attempt to account for model error in ROMS CCS. Illustrative examples only will be presented here, with a more detailed treatment and analysis forthcoming in a separate manuscript. Veneziani et al. (2009a) have demonstrated that WC10 tends to overestimate SST along the coast, particularly during upwelling seasons, which is indicative of a source of error of unknown origin. Broquet et al. (2009b, 2010) found that strong constraint 4D-Var tends to compensate for this error via corrections in both δx(t 0 ) and δf(t), to produce a closer fit of the model to the SST observations, and a lower J. Independent analyses of the COAMPS surface winds used to drive ROMS by Doyle et al. (2009) reveal that they agree well with independent observations not assimilated into COAMPS, and with scatterometer measurements. Therefore, it seems likely that ROMS CCS possesses a source of error near the coast which should be explicitly accounted for during 4D-Var rather than attempting to partially compensate for it via corrections to the surface forcing. Therefore, as a first attempt, a prior model error covariance Q = K b WΣ x CΣ T x W T K T b was assumed where, as described in Part I (section 5), C is a univariate correlation matrix 17

20 for the unbalanced component of the model error; Σ x is the diagonal matrix of prior model error standard deviations; K b is a multivariate balance operator; and W is a diagonal weight matrix with non-zero elements that reflect errors in temperature only, so that the model error standard deviations WΣ x are a maximum at the coast, and decrease linearly to zero 300 km offshore. The model temperature error variance, Σ 2 x, was assumed to be 5% of the initial condition background temperature error variance standard deviation, Σ 2, which recall is based on the model climatological variance. While there are most certainly sources of model error other than those acting on T near the coast and in other variables, the approach used here at least allows us to target, somewhat directly, the known nearshore model deficiencies in temperature. Weak constraint 4D-Var is only possible in ROMS using the dual formulation. The results of a weak constraint assimilation cycle beginning 3 March, 2003 in WC30 and WC10 using the aforementioned prior specification for Q in R4D-Var are shown in Figs. 2 and 3 (black curves) where the convergence rates of J towards the minimum can be directly compared with the strong constraint experiments discussed in Section 3. Figures 2 and 3 shows that in the weak constraint cases, J asymptotes to a constant value which is typically lower than that reached during the strong constraint experiments (e.g. Fig. 3c which shows the WC10 case for 1 outer-loop and 200 inner-loops). However, the use of the weak constraint reduces the rate of convergence of J to the minimum value. Experimentation with the level assumed for Σ 2 x revealed that increasing Σ 2 x pushes J to lower asymptotic values, although for Σ 2 x 15 20% of Σ 2, some aspects of the circulation estimates appear to be unphysical. For the Σ 2 x used in Figs. 2 and 3, the circulation estimates appear reasonable, and are significantly different to those obtained under the assumption of the strong constraint (not 18

21 shown). The influence of the weak constraint on J o in the context of the control vector impact is also shown in Fig. 6a as Expt O, and comparing Expts O and F shows that correcting for model error allows for an even better fit to the observations. 5 Analysis Error Estimates An important diagnostic that should accompany all 4D-Var circulation estimates is the expected posterior or analysis error. However, the analysis error can be very challenging to compute, and is seldom reported. As shown in Part I (section 7.1), it is possible to compute a reduced rank estimate Ẽa of the expected analysis error covariance matrix E a of the resulting 4D-Var analysis by making direct use of the Lanczos vectors derived from each inner-loop. In practice, however, the dimension of Ẽa is very large and manipulation and storage of the entire matrix is unwieldly and in most cases prohibitively expensive. However, the diagonal of Ẽa, which represents the expected analysis error variance, can be readily computed during dual 4D-Var for little additional computational effort. In addition, the leading EOFs of Ẽa can also be readily computed and used to construct further reduced-rank approximations of Ẽa that are more easily manipulated than Ẽa, and from which cross-covariance error information can be derived. We will present in this section example calculations of the diagonal of Ẽa and the EOFs of Ẽa. 5.1 Analysis error variance With regard to the analysis error variance, Fig. 7 shows (σ b σ a )/σ b at time t 0 for ocean temperature from the representative strong constraint R4D-Var cycle spanning 3-10 March, 19

22 2003 using WC10, where σ b and σ a are respectively the background and analysis error standard deviations for temperature. Therefore (σ b σ a )/σ b represents the fractional reduction in standard deviation of the analysis error relative to the background error. Recall that SST is an assimilated model variable east of 130 W, and Fig. 7a shows that the expected analysis error in SST is significantly less than the background error over much of the model domain. In some areas, the reduction in σ a relative to σ b extends well below the surface, most often in the vicinity of in situ subsurface observations (locations are shown in Fig. 7), such as along the Oregon coast at 45 N associated with observations from the GLOBEC and LTOP cruises, and in deep water when Argo floats are present. Analysis errors in sea surface salinity exhibit some similar features to the error in SST that are clearly associated with some in situ observation platforms (Fig. 8a). Significant reductions in σ a relative to σ b are also evident in variables for which there is no assimilated data. For example, Fig. 8b shows (σ b σ a )/σ b for the meridional component of velocity, and reveals significant reductions in error just upstream of the GLOBEC and LTOP lines. Significant reductions in σ a relative σ b can also be attributed to increments in other control variables, such as surface wind stress and net surface heat flux as shown in Fig. 9. The impact of the in situ observations on the wind stress and heat flux errors is clearly evident in Fig. 9 also. It is important to note that the estimates obtained for Ẽa are approximations of the true E a, and will improve as the number of inner-loops, m, increases. In the limit that m N obs, the expected analysis error covariance Ẽa will converge to that given by the true Kalman gain matrix for the assumed prior hypotheses. To illustrate, Fig. 10 shows the difference between σa 2 for T at various depths obtained using 800 inner-loops and 100 inner-loops, in this case using WC30, strong constraint R4D-Var, and a 7 day assimilation window. Clearly 20

23 the estimates of σ 2 a obtained using 100 inner-loops are a significant over estimate of the expected posterior error in T in some areas, especially at the surface, and the influence of subsurface in situ observations is again evident. Over estimation of σa 2 is also an issue with attempts to compute analysis error variance based on the smallest eigenvalues of the cost function Hessian, although the approach used here seems to fair better (based on Fig. 7) than previously reported results (Fisher and Courtier, 1995; Powell and Moore, 2009). Nonetheless the spatial variations of σ a derived from a practical number of inner-loops are a useful qualitative and quantitative guide of the geographic variations of the expected error in the resulting circulation estimates. The offset between the location of some of the observations and the local maxima apparent in Figs. 7, 8 and 9 can be attributed to the influence of advection and wave dynamics since the analysis errors shown are for time t 0, while the observations are distributed throughout the assimilation interval. The estimated analysis error covariance is given by Ẽa = (I KG)D where K is expanded in terms of the inner-loop Lanczos vectors. Therefore integrations of the tangent linear model, represented by G, are required to evaluate Ẽa. If the ocean circulation is linearly unstable, G may possess expressions of unstable eigenmodes, and as such (I KG) may become negative. Care must therefore be exercised when using information derived from Ẽa in these non-physical cases where the tangent linear assumption is violated. 5.2 Analysis error EOFs While the manipulation and storage of the entire matrix Ẽa is generally prohibited, the product of Ẽa with any arbitrary vector is a straightforward calculation in ROMS 4D-PSAS 21

24 and R4D-Var. The Lanczos algorithm (see Part I, section 4) can therefore be used to compute any number of the leading eigenvectors (i.e. EOFs) of Ẽa. An example for the representative strong constraint R4D-Var data assimilation cycle 3-10 March, 2003 using WC30 is shown in Fig. 11 which depicts the cumulative analysis error variance explained by different numbers of the leading EOFs. While the dimension of Ẽa is O(10 5 ) in this case, Fig. 11 shows that 30% of the posterior variance is explained by 0.2% of the total EOF spectrum. Figure 11 also gives the impression of a relatively flat EOF spectrum, meaning that no single EOF (or group of EOFs) accounts for a large fraction of the error variance. This is qualitatively the type of EOF spectrum expected for an error field that is random in space and time. 6 4D-Var Diagnostics It is important to remember that the ocean circulation estimates computed using the 4D-Var algorithms will only be minimum variance or maximum likelihood estimates if the hypotheses embodied in the prior (background) error covariance matrices are correct. Clearly it is important to test the validity of the prior hypotheses. 6.1 Consistency Checks Desroziers et al. (2005) (hereafter D05) present a useful set of diagnostics based on the innovation vector that are easy to compute routinely, and which provide information about the consistency of the 4D-Var analysis scheme with the underlying prior hypotheses. Specifically D05 consider the projection of the background and analysis into observation space, and derive a series of diagnostics based on the observation minus background, observation minus anal- 22

25 ysis, and analysis minus background differences. Based on D05 we can define the following difference vectors: d = (y H(x b (t))), d o a = (y H(x a (t))), and d a b = (H(xa (t)) H(x b (t))), where y is the observation vector, H is the observation operator that maps x into observation space, and d is the innovation vector. D05 show that the expected value E(d a b dt ) GDG T and E(d o ad T ) R T if the prior choices of D and R are the true covariances. Thus d a b dt and d o ad T provide a check of the consistency of the background and observation error covariances. Specifically, following D05 we consider the sub-traces of the matrices d a b dt and d o ad T given by: (σ i b)2 = (d a b) T i (d) i /p i = 1 p i (yj a y p j)(y b j o yj) b (5) i j=1 (σ i o)2 = (d o a) T i (d) i /p i = 1 p i (yj o yj a )(yj o y p j) b (6) i where i refers to the observation type (i.e. temperature, salinity, SSH); p i is the number j=1 of observations of type i; y o j is the value of the j th observation of type i, and y a j and y b j are the analysis and background counterparts in observation space. Thus (σ b i )2 and (σ o i )2 are the expected background and observations variances for the p i observations j = 1,..., p i if D and R are correctly specified. The level of agreement between (σ i b)2 and (σ i o)2 and the variances specified a priori in D and R, provides a measure of the consistency of the assimilation system, and (σ i b)2 and (σ i o)2 can be readily computed. Figure 12 shows time series of σ i b and σ i o for all observations of SSH, temperature, and salinity from WC30 when strong constraint R4D-Var is run sequentially for the period July 2002-Dec Also shown in Fig. 12 are time series of the specified standard deviations 23

26 σ b i and σo i averaged over the same observation points 3. The extent to which σ b i and σ o i, and σi b and σi o agree is a measure of the validity of the prior assumptions embodied in D and R. Figure 12a shows that for SSH σ o and σ o agree quite well, while σ b is generally always significantly larger than σ b. For temperature on the other hand, Fig. 12b shows that σ and σ are within a factor of 2 in the case of the background and the observations. With regard to salinity, σ o and σ o generally agree quite well on average (Fig. 12c), although in general σ b > σ b during most cycles (Fig. 12d). The level of agreement between σ and σ in Fig. 12 is fairly typical of that reported in both meteorological (e.g. D05; Houtekamer and Mitchell, 2005) and oceanographic (e.g. Daget et al., 2009) applications of data assimilation. However, care must be exercised when interpreting Fig. 12 since σ b and σ o are not the true or optimal values, and both depend on the a priori choices of σ b and σ o also. Nonetheless D05 and Desroziers et al. (2009) have shown how the prescribed variances used to compute D and R can be optimized to yield better agreement with the diagnosed values σ b and σ o. While the optimized variances do not necessarily represent the true background and observation variances, Desroziers et al. (2009) find that they generally have a positive impact on forecast skill. 6.2 Degrees of Freedom and Information Content Bennett et al., (1993) showed that if all of the hypotheses about the nature of the various sources of error described by the background error covariance matrix D and observation error covariance R are correct, the cost function in (1) has an expected mean minimum value of 3 To be consistent with the definitions of (σ b i ) and (σ o i ) in (5) and (6), σb i and σo i are defined as the square roots of the mean variances. 24

27 J min = N obs /2. As discussed by Talagrand (1999), J min can be a useful diagnostic in 4D-Var as a test of the prior hypotheses and the optimality of the resulting circulation estimates. However, while significant departures of J from J min indicate that the prior hypotheses described by D and R are incorrect, J min alone does not provide sufficient information about how to refine the prior hypotheses. The cost function (1) is comprised of two terms. The first term on the rhs of (1) relates to the departures of the control vector δz from the background, and is usually denoted J b. The second term on the rhs of (1) is a measure of the departure of the state-vector x from the observations, and is denoted J o as in section 3.2. Following Talagrand (1999), Chapnick et al. (2006) and Desroziers et al. (2009) showed that the expected mean minimum values of J b and J o are given by, (J b ) min = Tr(GK)/2 and (J o ) min = (N obs Tr(GK))/2 respectively, so that J min = (J b ) min +(J o ) min = N obs /2 as required 4. Furthermore, Cardinali et al (2004) showed that (J b ) min is in fact a direct measure of the number of degrees of freedom (dof) of the observations, and is a useful measure of the information content of the observation array. Similarly, (J o ) min is a measure of the dof or information content of the background in observation space. These distinctions become clear if one considers the extreme cases (J b ) min = 0 or (J o ) min = 0. In the former case, (J b ) min = 0 corresponds to the situation where δz = 0 meaning that z = z b, and hence the observations contain no useful information. This case is also equivalent to the situation where the observed ocean state is unreachable by the ocean model (Antoulas, 2005, section 4.2.1). Conversely, the case (J o ) min = 0 corresponds to the situation when x evaluated at the observation points agrees 4 The expressions reported by Chapnick et al. (2006) and Desroziers et al. (2009) do not contain the factor of 1/2 because this factor is absent from their definition of the cost function. 25

28 exactly with the observations, y, in which case there is no information in the background in observation space. Following Part I (section 7.1), the dual form of the practical gain matrix K for a single outer-loop can be written as K = DG T R 1 2 V m T 1 m V T mr 1 2, where V m is the N obs m matrix of Lanczos vectors of the preconditioned stabilized representer matrix ˆP = (R 1 2 GDG T R I) resulting from the m inner-loops, and T m is a diagonal matrix of known coefficients. In the limit m N obs then K K = DG T R 1 2 VT 1 V T R 1 2 where K is the true gain matrix, and V is the N obs N obs matrix of the complete set of orthonormal Lanczos vectors. In this limit, it can be shown (see the appendix, section A.1) that: (J b ) min = 1 2 (N obs Tr(T 1 )) = 1 ( N obs ) N obs λ 1 i 2 (J o ) min = 1 2 Tr(T 1 ) = 1 2 N obs i=1 i=1 (7) λ 1 i (8) where λ i are the eigenvalues of T and ˆP. While it is not practical, or indeed desirable, to perform N obs inner-loops, Tr(T 1 ) in (7) and (8) can be estimated from the leading members of the eigenspectrum of T as described in the appendix (section A.2). Each panel in Fig. 13 shows time series of J min = N obs /2 and estimates of the upper and lower bounds of (J b ) min (Fig. 13a) and (J o ) min (Fig. 13b) when strong constraint R4D-Var was applied sequentially during the period using 1 outer-loop and 200 innerloops in WC30. During each cycle, (J b ) min and (J o ) min were estimated as described in the appendix (section A.2) using the extrapolated eigenspectrum λ i obtained by fitting a curve of the form log 10 (λ i ) = ae bi to the directly computed eigenvalues λ i. In each case a and b were determined using the λ i for the range i = 50 I where I is cycle dependent and based on the 26

29 error criteria ɛ i 10 8 (see appendix section A.2). The upper and lower bounds in (J b ) min and (J o ) min shown in Fig. 13 are based on an error of 25% in the estimated eigenvalues that are significantly greater than 1 as described in the appendix (section A.2). Figure 13a shows that for the given prior hypotheses the number of dof of the observations, (J b ) min, is 4 6% of the total number of observations, indicating a large degree of redundancy in the observation array. This is perhaps to be expected given that the majority of the data available during each assimilation cycle takes the form of satellite SST. In addition, the assimilated SST is 5 day composites from different satellite platforms, and compositing effectively averages the data leading to a loss of information. However, the patterns of SST associated with surface circulation features such as eddies and fronts are persistent over periods of several days, which of course is the philosophy underlying the blending of data from different platforms. Therefore, the amount of new information introduced every day by each set of SST observations after the first complete realization during a given 4D-Var cycle will be small, implying a large amount of redundancy. The level of redundancy of the satellite observations increases with model resolution as shown in appendix A.2 which suggests that in WC10, as much as 98% of the observational data may be redundant. Also shown in Fig. 13a is the value of J b at the end of each R4D-Var cycle. Figure 13a reveals that J b > (J b ) min for all cycles, indicating that, for the given priors D and R, R4D-Var is over estimating the number of dof of the observations, and is generally placing too much confidence in the observations. Similarly, Fig. 13b shows time series of (J o ) min and J o at the end of each cycle. Figure 13b, reveals that J o < (J o ) min for many cycles, indicating that R4D-Var is significantly underestimating the number of dof of the background (prior) at the observation points. Taken together, Figs. 13a and 13b suggest that R4D-Var is over- 27

30 fitting x to the observations when J min > J b > (J b ) min and, at the same time, J o < (J o ) min and J o < J min. It is important to realize that the number of dof of the observations depends on the priors D and R. If the decorrelation lengths assumed for D are large, the influence of each observation will be spread over a large distance, and if the regions of influence of each datum significantly overlap with each other this will reduce the dof (i.e. the independence) of all the observations. Conversely, if the decorrelation lengths of D are short, the number of dof of the observations will increase. In the present case, R is a diagonal matrix which for satellite observations is probably a poor choice since along track instrument errors are likely to be correlated. In the case of correlated observations, the non-diagonal structure of R will further impact the number of dof of the observations. A time series of J = J o + J b is shown in Fig. 13c. While the averages J and J min over all assimilation cycles are similar, the disagreement between J and J min in in Fig. 13c indicates that one or more of the prior hypotheses about x(t 0 ), f(t), b(t) or η(t) are incorrect and must be refined in future applications. Recall though that we have assumed: (i) that the model is error free, (which we know to be incorrect); (ii) that D is assumed to be time invariant; and (iii) that observation errors are uncorrelated. The flow dependence of D in 4D-Var is a non-trivial problem and is an area of active research (Belo Pereira and Berre, 2006; Daget et al., 2009). Substantial improvements in the fit of J to the theoretical minimum can also be afforded by updating D as the assimilation cycles proceed to account for a reduction in error of x b, which is derived from the previous assimilation cycle (e.g. Powell et al., 2008; Moore et al., 2010c). It should be noted, however, that it is a tall order to expect to accurately prescribe the background error covariances D, so while the expected minimum 28

31 value of the cost function is a useful gauge of how well the assimilation is performing, it will in general be a difficult target to reach. With regard to observation errors, errors in satellite measurements will undoubtedly be correlated, so the choice of a diagonal form for R is probably inappropriate. The prescription of correlated observation errors for satellite platforms is also an active area of research (Brankart et al., 2009). Clearly these are all issues which must be addressed and are being actively pursued in ROMS. 6.3 The Degree of Reachability and Array Modes Bennett (1985) also explored the level of redundancy of observation arrays by examining the eigenspectrum of the stabilized representer matrix, P. He notes that the number of eigenvalues λ i of P for which λ i αλ 1 is a measure of the number of independent observations where λ 1 denotes the largest eigenvalue. Based on previous experience, Bennett and McIntosh (1982) adopted an ad hoc choice of α = 0.01 which Bennett (1985) refers to as the 1% rule. This measure of observation redundancy is closely related to the concept of reachability used in control theory. As discussed in Part I (section 3), if the model is forced with random noise with a covariance described by D, then GDG T is the covariance of the model response in observation space. Following Antoulas (2005, p80) the positive definite quantity wr T (GDG T ) 1 w r can be thought of as a measure of the energy required to steer the tangent linear model from a state of rest to the observation space increment w r (i.e. the state of the system evaluated at the observation points). The state with minimum energy wr T (GDG T ) 1 w r is a measure of the degree of reachability and, according to the Rayleigh quotient, corresponds to the largest eigenvalue of GDG T. Stated another way, the model circulation increments w r that require least energy to reach correspond to the eigenvectors 29

32 of GDG T with largest eigenvalue. Conversely, those states with smallest eigenvalues require the most energy to reach, and for sufficiently small eigenvalues become, to all intents and purposes, unreachable without a great deal of effort. The number of significant eigenvalues of GDG T is therefore a measure of all the observed states that are reachable by the model during the assimilation process. The eigenvectors of GDG T are the same as those of ˆP, the preconditioned stabilized representer matrix, and the eigenvalues differ by 1 and can be readily computed as described in section 6.2. Figure 13a also shows an estimate of the number of independent observations during each assimilation cycle based on the 1% rule, and agrees qualitatively with the measure of dof using (J b ) min : namely the number of independent observations determined via this measure is typically small compared to N obs, reaffirming the conclusion that there is a large degree of redundancy in the observation array. Upper and lower bounds are also shown for the 1% rule curves in Fig. 13a but are almost indistinguishable. Following Bennett (1985), the posterior state-vector increment δx a (t) can be expressed in terms of the eigenpairs (λ i, ŵ i ) of the preconditioned stabilized representer matrix, ˆP, according to: δx a (t) = U(t)R 1 2 WΛ 1 W T R 1 2 d (9) = N obs i=1 λ 1 i (ŵ T i R 1 2 d)ψi (t) (10) where U(t) = (u j (t)) is the matrix of the representers u j (t), j = 1,..., N obs ; W = (ŵ j ) is the matrix of eigenvectors ŵ j of ˆP; Λ is the diagonal matrix of eigenvalues λj ; and Ψ i (t) = N obs j=1 w jiu j (t) are called the array modes where w i = (w ji) = R 1 2 ŵ i. Each of 30

33 the N obs array modes are comprised of a linear combination of the representers weighted by the elements w ji of the vectors w i. If the eigenpairs (λ i, ŵ i ) are arranged in order of descending eigenvalues, equation (10) shows that the first array mode, Ψ 1, is weighted in δx a by λ 1 1 and nominally contributes least to the analysis circulation increment. Since the array modes Ψ j (t) are independent of the observation values and depend only on the observation locations, the prior covariances, and the prior circulation, then Ψ 1 represents the most stable component of the circulation with respect to variations in the innovations d, and is of interest for this reason. As noted by Bennett (1990), the array modes are the natural frame in which to consider the contribution of the observations to the analysis increments because the Ψ j (t) of an observation array can be evaluated and analysed before the observations are collected and assimilated. Even though the representers are not explicitly computed during R4D-Var, the array modes can be computed a posteriori using the indirect representer method according to Ψ j (t) = M(t 0, t)dg T R 1 2 ŵ j where M(t 0, t) denotes an integration of the tangent linear model over the interval [t 0, t]. Figures 14 and 15 show the initial and final structure of the array mode Ψ 1 (t) for the representative R4D-Var assimilation cycle spanning the period 3-7 March, 2003, using WC10. At initial time (Fig. 14), Ψ 1 (t) is characterized by coherent structures in all components of the state-vector with scales km. Coastally confined features are also evident, which extend well below the surface, particularly in T (Figs. 14e and 14f), and are reminiscent of circulation changes induced by offshore wind stress curl. Over time, the background circulation x b (t), surface forcing f b (t), and boundary conditions b b (t) lead to a more complex array mode pattern and a predominance of smaller scales (Fig. 15). Imperfections in the open boundary conditions are also evident in Ψ 1 (t) at 31

34 the final time, as revealed in Fig. 15 which clearly shows evidence of a spurious boundary wave along the northern open boundary. The array mode Ψ(t) 1 is one manifestation in observation space of the increment w r that requires the least energy to reach. Each array mode, Ψ j (t), can be decomposed into the contribution arising from information pertaining to the background initial conditions, background surface forcing, and background boundary conditions and their associated error covariances. This is illustrated in Fig. 16 for Ψ 1 (t) which shows the respective contributions of the priors to SST and SSH at the final time. Figure 16 reveals that in this example the surface forcing prior contributes most to Ψ 1 (t) (Figs. 16b and 16e). In particular, the spurious wave signature along the northern open boundary is associated entirely with the surface forcing priors (Fig. 16e). The initial condition priors also contribute significantly (Figs. 16a and 16d) to Ψ 1 (t), while the boundary condition prior contribution is negligible (Figs. 18c and 18f). It is possible to decompose Ψ 1 (t) further into contributions from each individual component of the state vector, surface forcing, and boundary condition prior to diagnose the relative importance of specific elements of the background control vector. Since there is no formal requirement for the array modes to be orthogonal, there is often a considerable degree of overlap between the structures of different modes. This is illustrated in Fig. 17 which shows the SST structure of array modes 2-5 at initial time. While each array mode has a very distinct structure, there are obvious areas of large amplitude that are common to each array mode. These are robust features in the most stable interpolation patterns of the array that, in general, do not appear to be obviously related to the location of the in situ observations (which are indicated in Figs. 14a and Fig. 17). The structures of the leading array modes are therefore governed primarily by a combination of the satellite 32

35 data locations and the priors. According to the definition Ψ j (t) = M(t 0, t)dg T R 1 2 ŵ j, the array mode structure at initial time t 0 is strongly influenced by the smoothing imposed by prior error covariance D. At later times during the assimilation interval, the array modes become considerably contorted by the background (prior) circulation and surface forcing via M(t 0, t) (e.g. Figs. 15) due, for example, to the straining and shearing components of the flow. Therefore, the small scales introduced into the circulation estimates during the latter stages of the analysis cycle should be treated with caution due to their strong dependence on z b. 6.4 Clipped Analyses Equation (10) indicates that the analysis state vector increment δx a (t) can be represented as a linear combination of array modes Ψ j (t), each weighted by λ 1 j. Since the eigenvalues are arranged in descending order, the weights λ 1 j will increase with increasing index j and the corresponding array modes Ψ j (t) are characterized by finer and finer scales. Therefore the higher array modes may contribute significant small scale structure to the analysis increment which may not be physically realistic. In light of this, Bennett (1990) suggests that the array mode summation in (10) should be truncated or clipped so that: δx a (t) = M i=1 λ 1 i (ŵ T i R 1 2 d)ψi (t) (11) where M < N obs is chosen so that unphysical array modes do not contribute to the analysis. Bennett (1990, 2002) discuss various criteria that can be used for selecting M based on size of λ M relative to the observation error variances. In ROMS, the indirect representer algorithm 33

36 is exploited to evaluate (11) according to δx a = M(t 0, t)dg T M i=1 λ 1 i (ŵ T i R 1 2 d)ŵ i. An example of a clipped analysis using (11) is shown in Fig. 18, in this case for a 4 day strong constraint R4D-Var cycle spanning the period 3-7 March, 2003, using WC30 with 1 outer-loop and 800 inner-loops (the case analysed in the appendix, section A.2). The analysis increments for SST computed directly from R4D-Var are shown in Figs. 18a and 18b on 3 March and 7 March respectively, and a significant amount of small scale structure is evident on both days. The eigenvalues corresponding to the array modes are the same as those shown in Fig. A1a for the case of 800 inner-loops. Inspection revealed that the array modes with λ j < 0.01λ 1 (i.e. those violating the 1% rule of section 6.3) have spatial scales that are typically smaller than the expected spatial resolution of the WC30 model grid. Therefore, we would consider any contribution of these array modes to the analysis increment δx a as suspicious and nonphysical. Figures 18c and 18d show the SST increments of clipped analyses on 3 March and 7 March resulting from (11) using the 1% rule as the criteria for selecting M, and in the example here M = 255. The resulting clipped analysis increments are noticably smoother than the original increments, and locally the differences exceed 1 C as shown in Figs. 18e and 18f. 7 Summary and Conclusions This is the companion paper to Moore et al. (2010a) and both papers serve as a useful and comprehensive review of state-of-the-art 4D-Var methods and diagnostic analyses, applied here in the context of ROMS in a meso-scale ocean circulation environment. In Part I we presented a comprehensive description and overview of the ROMS 4D-Var system and the 34

37 various diagnostic tools that are available. In this companion paper, we discuss some additional diagnostic tools and present an extensive series of calculations from two configurations of ROMS for the CCS, one with relatively low horizontal resolution, 30 km (WC30), and another with moderate resolution, 10 km (WC10). Calculations illustrating the performance of each 4D-Var platform were presented for a randomly chosen representative data assimilation cycle using WC10 and WC30. The generality of the performance was demonstrated by presenting results from calculations in which each 4D-Var algorithm was applied sequentially using WC30 over a 2.5 year time period spanning While the strong constraint was imposed in most cases on the resulting circulation estimates, an example of weak constraint 4D-Var was also considered. ROMS is a widely used and respected community ocean model, and all of the utilities and tools described here are freely available. However, while the performance characteristics and example calculations presented here therefore serve as an indispensable guide for all users of the ROMS 4D-Var system, we also expect our results to be generally applicable to oceanographic applications of 4D-Var in general, so this paper is of broader interest beyond the ROMS community. A number of important conclusions follow from the calculations presented here. First, this paper represents the first comparison (of which the authors are aware) of the primal and dual formulations of 4D-Var algorithms as applied to a meso-scale ocean circulation environment. The primal formulation is referred to here as I4D-Var, while the dual formulation is presented in two forms, 4D-PSAS and R4D-Var. In keeping with theory, for a single outerloop all three algorithms yield the same ocean circulation estimates when using the same observations and prior information, and they also track each other very closely when applied sequentially. However, the performance of the dual algorithm as currently formulated in 35

38 ROMS is inferior to the primal formulation in that it requires considerably more iterations to achieve convergence of the cost function to its minimum value. Further work is needed to accelerate the convergence rate of the dual-space search algorithm, and some promising solutions are available capitalizing on very recent experience gained in meteorological applications (e.g. El Akkraoui and Gauthier, 2010). A second important finding of this work in relation to the practical application of 4D- Var is that while multiple outer-loops often yield a closer fit of the nonlinear model to the observations, this is by no means guaranteed (and has never been formally proved) and not always the case, and examples can often be found in which multiple outer-loops actually degrade the fit of the model to the data. However, it was found here that second level preconditioning using the Ritz vectors of the Hessian matrix as basis functions can be beneficial and partly alleviate this problem. Care must be exercised, however, and only those Ritz vectors that are sufficiently accurate should be used for preconditioning, a finding in general agreement with that of Tshimanga et al. (2008). An open question in ocean data assimilation for some time has been the relative importance of uncertainties in the initial conditions, surface forcing and open boundary conditions on ocean circulation estimates. Here we have explored this question in relation to the ocean mesoscale circulation in a coastal upwelling environment by examining the relative impact of different elements of the 4D-Var control vector on the cost function, J. It was found that corrections for uncertainties in the initial conditions have by far the largest influence on J, although corrections to the surface wind stress are also significant, as are corrections for model error. In Moore et al. (2010b), we shall find that these conclusions are fairly general where we explore the impact of the different components of the control vector on specific 36

39 physical aspects of the resulting circulation estimates. Important information that should always accompany any estimate of the ocean circulation is some indication of the reliability of the estimate, in particular analysis error covariance information. In most current ocean applications of 4D-Var, this is usually not available, since computation of analysis error estimates is challenging and computationally demanding. The Lanczos method employed in ROMS 4D-Var, however, renders tractable the computation of analysis error covariance information. We have demonstrated here that analysis error information can be obtained routinely as part of ROMS 4D-PSAS and R4D-Var at relatively little additional computational cost. However, the posterior error information obtained should be interpreted with care for various reasons. First, and foremost, the expected analysis error estimates will only be reliable if the prior errors are correctly specified. Second, the expected analysis error variances are likely to be overestimates of the true variances due to the limited number of inner-loops that can be practically employed in 4D-Var. Finally, because the computation of the expected analysis error in ROMS is predicated on the tangent linear assumption, violations of this assumption and the growth of linearly unstable modes of the circulation can lead to spurious analysis error estimates. Nonetheless, despite these short comings, geographical variations in the analysis error estimates will likely prove very useful. An alternative and more reliable method for computing the expected analysis error variance of linear functions of the circulation (e.g. transport, heat content) using the adjoint of 4D-Var is described by Moore et al. (2010c). We also presented the results of a number of very important diagnostic calculations that shed light on the consistency of the 4D-Var circulation estimates and the number of dof of the observing array. To our knowledge, these are the first and most comprehensive 37

40 calculations of their kind for the CCS observing systems currently in place. An important finding of this work is that there is a great deal of redundancy in the existing observational network. This arises because of the plethora of satellite observations, particularly SST, compared to subsurface observations. Over the typical weekly time scales of each data assimilation cycle the surface SST will not change very much, so after the first set of satellite SST observations have been assimilated, there is little new information to be gleaned from subsequent satellite passes. This will compounded when using blended SST products where data from multiple platforms are combined to maximize the benefits of the various available satellite sensors. The blending procedure necessarily involves combining information over a time window spanning several days which is tantamount to time averaging. In the current study, we also used gridded satellite altimetry products. The gridding process involves objective mapping techniques which will introduce further redundancy in the resulting fields because the objective mapping will not (and should not) introduce additional information about the circulation between satellite swaths. In future applications of ROMS 4D-Var to the CCS we plan to use raw along-track satellite SSH and SST data from individual satellite platforms instead of blended and gridded products. However, the effective use of along-track SSH presents new issues and challenges for 4D-Var and requires the introduction of temporal background error correlations. While the high level of redundancy of the current observing array may not be suprising, we have presented here robust methods for quantifying this aspect of observing systems. In WC30 we found that less then 10% of the currently available observations provide independent information. Observation redundancy increases with model resolution, and preliminary calculations with WC10 suggest that only 1-2% of observations are independent. While 38

41 these conclusions depend on the prior hypotheses about the nature of the various sources of error described by D and R, they do at least provide a useful order of magnitude estimate of observation redundancy since rather radical changes in D and R would be required to significantly influence the degrees of freedom of the observing system. The level of observation redundancy and efficiency of the observing arrays was also explored in terms of the degree of reachability of ocean states, an idea borrowed from control theory, and in terms of the array modes. The array modes are the most natural representation of the analysis increments, and those that are most stable to variations in the innovations are most preferred for interpolating the observations. By jettisoning the array modes with nonphysical scales it is possible to refine the resulting circulation analyses. The diagnostic calculations of section 6 highlight an important but often overlooked point: ocean data assimilation is a process of learning and refinement, and repeated efforts are required to obtain the best possible analyses and predictions. This can only come from an honest and unbiased assessment of the system performance such as from informed diagnostic calculations like those presented here. Other tools that are available for ROMS in support of 4D-Var include the adjoint of each 4D-Var system. These tools allow us to explore the impact that different observation platforms and individual observations have on particular aspects of the ocean circulation. The 4D-Var adjoint can also be used to assess the sensitivity of the resulting circulation estimates to changes or uncertainties in the observations. The ROMS observation impact and observation sensitivity analysis tools are described in Part I, and systematic studies in relation to the CCS are presented in Moore et al. (2010b) which forms the third companion paper. 39

42 Acknowledgements We are grateful for the continued and unwavering support of the Office of Naval Research (N , N , N , N ). Development of R4D-Var and 4D-PSAS was also supported by the National Science Foundation (OCE , OCE , OCE ). Any opinions, findings, and conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the National Science Foundation. Part of this work was also supported by the National Ocean Partnership Program (NA05NOS ). J. Doyle acknowledges support through the Office of Naval Research s Program Element N. We are also indebted to Anthony Weaver at CERFACS for countless discussions and much advice on various 4D-Var issues. COAMPS R is a registered trademark of the Naval Research Laboratory. 40

43 Appendix A.1 The Trace of GK For the cost function given by (1), Chapnick et al. (2006) and Desroziers et al. (2009) showed that the expected mean minimum values of J b and J o are given by, (J b ) min = Tr(GK)/2 and (J o ) min = (N obs Tr(GK))/2. However, as discussed in Part I (section 4), the gain matrix K can be expressed as: K = DG T (GDG T + R) 1 = DG T R 1 2 (R 1 2 GDG T R I) 1 R 1 2 = DG T R 1 2 VT 1 V T R 1 2 (A1) where V is the matrix of the complete set of Lanczos vectors of the preconditioned stabilized representer matrix, there being one column for each vector, and T is a tridiagonal matrix of the coefficients of the Lanczos recursion relation given by equation (13) of Part I. Using (A1) then Tr(GK) can be written as: Tr(GK) = Tr(GDG T R 1 2 VT 1 V T R 1 2 ). (A2) However, according to the Lanczos recursion relation, (R 1 2 GDG T R 1 2 +I)V = VT in which case GDG T R 1 2 V in (A2) can be expressed as GDG T R 1 2 V = R 1 2 (VT V). Making this substitution in (A2) yields: Tr(GK) = Tr(R 1 2 (VT V)T 1 V T R 1 2 ) 41

44 = Tr((VT V)T 1 V T R 1 2 R 1 2 ) = Tr((VT V)T 1 V T ) = Tr(VV T VT 1 V T ) = Tr(V T V) Tr(T 1 V T V) = N obs Tr(T 1 ). (A3) Recalling that the trace of any matrix is the sum of the eigenvalues, and that the eigenvalues of an inverse matrix are the inverse eigenvalues of the matrix, it follows that Tr(T 1 ) = Nobs i=1 λ 1 i where λ i are the eigenvalues of T. Therefore we can write (J b ) min = 1 2 (N obs Nobs i=1 λ 1 i ) and (J o ) min = 1 Nobs 2 i=1 λ 1 i. It is not practical, or indeed desirable, to perform N obs inner-loops, however Tr(T 1 ) can be estimated from the leading members of the eigenspectrum of T as described in section A.2. A.2 Estimating Tr(T 1 ) Figure A1a shows estimates of the leading eigenvalues λ i of T, arranged in descending order, from a representative 7 day strong constraint R4D-Var data assimilation cycle using WC30 with 1 outer-loop and different numbers, m, of inner-loops ranging from 100 to 800. Figure A1b shows the accuracy of each estimated eigenpair (λ i, ŵ i ) in the four cases measured according to ɛ i = λ 1 1 ˆP m ŵ i λ i ŵ i / ŵ i where ŵ i are the eigenvector estimates of ˆP, the preconditioned stabilized representer matrix. Figure A1b indicates that the error of each estimated eigenpair increases with i. Choosing ɛ i 10 8 as an acceptable accuracy for λ i, then Fig. A1b indicates that 50% of the leading computed eigenvalues are acceptable for a given choice of m. Since the eigenvalues of ˆP have a minimum value of 1, closer 42

45 inspection of the leading portion of the eigenspectrum for each case in Fig. A1a for which ɛ i 10 8 reveals that for i 50 the eigenspectrum is well approximated by a relationship of the form log 10 (λ i ) = ae bi. This is illustrated in Fig. A2a for the case of 1 outerloop and 800 inner-loops. The coefficients a = and b = were determined from a least-squares best fit to the directly computed eigenvalues λ i, i = for which ɛ i Using the resulting values of a and b in this case and extrapolating the eigenspectrum to obtain the remaining (N obs 500) eigenvalues yields (J b ) min = 387 and (J o ) min = However, it is not practical to perform many hundreds of inner-loops for each assimilation cycle, especially in the case of WC10. Therefore it is of interest to ascertain whether reliable estimates of the entire eigenspectrum λ i of T (and ˆP) can be estimated from a smaller number of inner-loops. Figure A2b shows the case where the coefficients a and b were estimated from λ i, i = 50 90, for which ɛ i 10 8, computed directly using 200 inner-loops. The resulting coefficients a and b were used to estimate the portion of the eigenspectrum λ i for i = that was computed directly using 800 inner-loops, and the estimates λ i are compared in Fig. A2b with the directly computed λ i. Figure A2b shows that on the whole the extrapolated eigenspectrum λ i is rather good as quantified in Fig. A2c which shows the percentage error 100(λ i λ i )/λ i for each extrapolated eigenvalue λ i. Figure A2c indicates that the error in the extrapolated λ i asymptotes to 22% as i 500. Despite this asymptotic level of error, using this case to extrapolate all remaining (N obs 90) members of the eigenspectrum yields estimates of (J b ) min and (J o ) min that differ from those reported above by only 2%. Nonetheless, Fig. A2c indicates that the extrapolated eigenvalue estimates λ i based on log 10 (λ i ) = ae bi are typically underestimates of the directly computed eigenvalues. Underestimates of λ i translate to overestimates of λ 1 i, 43

46 meaning that the trace estimate Tr(T 1 ) = N obs 1 i=1 λ i obtained from the curve fit can be regarded as an approximate upper bound. To account for errors resulting from the curve fit errors (and remembering that the minimum value of λ i is 1) a conservative estimate of the lower bound on Tr(T 1 ) = N obs i=1 λ 1 i estimate λ i that is is underestimated by 25%. was computed by assuming that each eigenvalue For the representative WC30 example depicted in Figs. A2a-A2c, the estimated ranges based on these upper and lower bounds for (J o ) min are , and for (J b ) min are During this cycle, N obs /2 = 3366 so (J b ) min indicates that for the chosen prior assumptions embodied in D and R, only 11 18% of the observations yield independent information about the linear combinations of the model state-vector increments in observation space. Similar calculations using WC10 and based on Fig. A2d reveal that (J o ) min = and (J b ) min = Since N obs /2 = in this case then (J b ) min implies that only 1 2% of the observations provide independent information in the case of WC10. These calculations show that much of the observational information is redundant. The level of redundancy increases with horizontal resolution because the SST observations have 10 km resolution so there are almost an order of magnitude more SST observations assimilated in WC10 than in WC30. 44

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53 Table Caption Table 1: The final value of the non-linear cost function J NL using 4 outer-loops and 25 inner-loops for the representative 4 day strong constraint I4D-Var assimilation cycle starting on 3 March, 2003 using WC30 versus the number of Ritz vectors used in the second level Ritz preconditioner. The case with no second level preconditioning is represented by the column labeled 0. 51

54 Figure Captions Figure 1: The ROMS CCS model domain and bathymetry used in (a) WC30 and (b) WC10. Figure 2: log 10 (J) versus number of inner-loop iterations for data assimilation windows of (a) 4 days, (b) 7 days, and (c) 14 days duration for a representative assimilation cycle starting 3 March 2003 using I4D-Var (red curve) and R4D-Var (blue), assuming a strong constraint and using WC30. The value of the non-linear cost function J NL at the end of inner-loop 100 is also indicated for I4D-Var (red cross) and for 4D-PSAS (green cross). Also shown is a case using R4D-Var and the weak constraint (black curve). The horizontal dashed line in each case shows the theoretical minimum value, J min = N obs /2, of J for each assimilation interval (for later reference in section 6.2). In each case a single outer-loop was used. Figure 3: Same as Fig. 2 but using WC10. In this case, only the results from R4D-Var are shown for the dual formulation. Figure 4: Time series of the initial value of log 10 (J) (thin lines) and final values of log 10 (J) (bold lines) for 7 day assimilation cycles when 4D-Var was applied sequentially using I4D- Var (black curves), R4D-Var (blue curves), and 4D-PSAS (red curves) using WC30. In each case 1 outer-loop and 50 inner-loops were used, and the strong constraint was employed. Figure 5: The inner-loop cost function, log 10 (J), (bold curves) at the end of each innerloop for different combinations of inner- and outer-loops using I4D-Var and WC30 for a representative assimilation cycle. The non-linear cost function, log 10 (J NL ), (circles) is also shown at the end of each outer-loop. The abscissa refers the total combined number of inner- 52

55 and outer-loops in each case. Figure 6: (a) The final value of J o for different R4D-Var experiments for the representative 7 day assimilation cycle starting on 3 March, 2003, using WC30. In each experiment, the increment control vector was comprised of different combinations of x(t 0 ), f(t) = (δτ T, δq T, δf) T, and b(t), where δτ, δq, and δf are the increments of surface wind stress, net surface heat flux, and net surface freshwater flux respectively. The increment control vector was comprised of the following components: Expt A- No assimilation; Expt B- x(t 0 ) only; Expt C- x(t 0 ) and δτ ; Expt D- x(t 0 ), δτ and δq; Expt E- δx(t 0 ), δτ, δq, and δf; Expt F- δx(t 0 ), δτ, δq, δf and b(t); Expt G- δτ only; Expt H- δτ and δq; Expt I- δτ, δq, and δf; Expt J- δτ, δq, δf and b(t); Expt K- δq only; Expt L- b(t) only; Expt M- b(t) and δτ ; Expt N- b(t), δτ and δq; and Expt O- same as F but subject to weak constraint. The numerical values of J o are also indicated for some experiments to aid comparison where the bar heights are very similar. (b) The ratio (J o ) f /(J o ) i of the final value of J o and the initial value of J o for a sequence of R4D-Var cycles using WC30 for the case where only δx(t 0 ) is adjusted (black curve), only δτ is adjusted (blue curve), only δq and δf are adjusted (red curve), and only δb(t) is adjusted (green curve). Figure 7: The ratio (σ b σ a )/σ b for temperature at various depths, where σ b is the background error standard deviation in T and σ a is the analysis error standard deviation in T. The results shown are for the representative 7 day cycle starting on 3 March 2003 using R4D-Var and WC10 with 1 outer-loop and 100 inner-loops. Figure 8: Same as Fig. 7 but for (a) sea surface salinity, and (b) meridional velocity v in the vicinity of the Oregon and Washington coast. 53

56 Figure 9: Same as Fig. 7 except for (a) zonal wind stress, (b) meridional wind stress, and (c) net surface heat flux. Figure 10: Maps of the ratio (σ σ 2 800)/σ where σ and σ are the posterior error variances when using 800 inner-loops or 100 inner-loops respectively, and 1 outer-loop with R4D-Var and WC30. Figure 11: The percentage of cumulative analysis error variance explained by the sum of different numbers of the leading eigenvectors (EOFs) of the analysis error covariance matrix (blue curve). The case shown is for the representative 7 day R4D-Var cycle starting on 3 March 2003 using WC30. The trace of Ẽa was estimated using the randomization method of Bai et al. (1995) in which vectors with randomly chosen elements ±1 are used to estimate Tr(Ẽa ). The red curves show the cumulative variance associated with Tr(Ẽa ) ± σ Tr where σ Tr is the standard deviation of the resulting trace estimate based on a sample size of 400 randomized trace estimates. The error ɛ i = ν 1 1 Ẽaˆx i ν iˆx i / ˆx i of each eigenpair (ν i, ˆx i ) can be used to quantify the reliability of each EOF ˆx i that is computed. Choosing ɛ i = 10 8 as an acceptable lower bound of the error, then only the leading 200 EOFs are acceptable which together account for 30% of the analysis error variance. Figure 12: Time series of σ b (solid black curves) and σ o (solid blue curves) computed using (5) and (6) for all observations of (a) SSH, (b) temperature, and (c,d) salinity. Also shown are time series of the average standard deviations σ b (dashed black curves) and σ o (dashed blue curves) (at the same observation points) that were prescribed a priori in D and R. Computations are based on strong constraint R4D-Var run sequentially using WC30 for the period July 2002-Dec

57 Figure 13: Time series of log 10 of various statistics of the cost function J from WC30 run sequentially during the period July 2003-July 2004 using a 7 day assimilation cycle, strong constraint R4D-Var, 1 outer-loop and 200 inner-loops. The solid black curve shows the theoretical minimum value of the cost function J min = N obs /2. The solid red curves show the anticipated upper and lower bounds for the theoretical minimum value (J o ) min = 1 2 Nobs i=1 of J o, where λ i are the eigenvalues of the preconditioned stabilized representer matrix ˆP derived from curve fits to the directly computed eigenvalues of the form log 10 ( λ i ) = ae bi. The coefficients a and b are cycle dependent and were derived from the fit to the computed eigenvalues λ i for the range i = 50 I where i = I +1 is the first computed eigenvalue with an error > 10 8 (see appendix, section A.2). The solid blue curves show the anticipated upper and lower bounds for the theoretical minimum value (J b ) min = (N obs /2 (J o ) min ) of J b. The dashed black, red and blue curves show the actual value of J, J o and J b respectively for each assimilation cycle. Also shown is the anticipated range in the number of eigenvalues of ˆP for which λ i /λ 1 ɛ, for ɛ = (the 1% rule of Bennett and McIntosh (1982)) (solid green curves), where λ 1 is the largest eigenvalue. Following Bennett (1989) this represents an estimate of the number of independent linear combinations of the model state variables provided by the observations. λ 1 i Figure 14: The structure of array mode Ψ 1 at initial time from a representative strong constraint R4D-Var cycle for the period 3-7 March, 2003, using WC10 and a 4 day assimilation interval: (a) SST, (b) SSH, (c) surface zonal velocity, (d) surface meridional velocity, (e) temperature at 100m depth, and (f) temperature at 200m depth. Each field has the physical units of the variable, and the corresponding values of the physical fields are determined by the total weight λ 1 i (ŵi T R 1 2 d) for Ψ 1 in equation (10). The diamonds in (a) indicate the 55

58 location of the in situ hydrographic observations during this 4 day period. Figure 15: Same as Fig. 14 except showing the structure of Ψ 1 at final time. Figure 16: The contribution of the initial condition (IC), surface forcing (FC), and boundary condition (BC) information to array mode Ψ 1 at final time for SST (a-c) and SSH (d-f). Figure 17: The SST structure of array modes 2-5 at initial time. The diamonds indicate the location of the in situ hydrographic observations during this 4 day period. Figure 18: The analysis increment for SST from a typical 4 day cycle of strong constraint R4D-Var at (a) initial time, and (b) final time. The clipped analysis increments using only those array modes for which λ i > 0.01λ 1 are shown in (c) at initial time, and (d) at final time. The difference between the R4D-Var analysis increment and the clipped analysis increment at initial and final time is shown in (e) and (f) respectively. Figure A1: (a) log 10 (λ i ) versus eigenvalue number i of the preconditioned stabilized representer matrix P resulting from 1 outer-loop and 100, 200, 400 and 800 inner-loops. (b) log 10 (ɛ i ) where ɛ i = λ 1 1 Pŵ i λ i ŵ i / ŵ i is the error associated with each eigenpair (λ i, ŵ i ) in each case, and only values of ɛ i > are shown. The horizontal dashed line represents the ɛ i = 10 8 error threshold used to select reliable eigenvalues. Figure A2: (a) The black curve shows log 10 (λ i ) for the leading 500 eigenvalues λ i of the preconditioned stabilized representer matrix ˆP computed using 1 outer-loop and 800 innerloops in WC30 which Fig. A1b shows have an error The red curve shows the eigenvalues λ i for the range i = that were extrapolated from a curve of the form log 10 ( λ i ) = ae bi, (a = and b = ) where a and b were determined from a 56

59 best fit of the curve to the directly computed eigenvalues in the range i = (b) The black curve shows log 10 (λ i ) for the leading 500 eigenvalues λ i of the preconditioned stabilized representer matrix ˆP computed using 1 outer-loop and 800 inner-loops in WC30 which Fig. A1b shows have and error The red curve shows the eigenvalues λ i for the range i = extrapolated from a curve of the form log 10 ( λ i ) = ae bi, (a = and b = ) where a and b were determined from a best fit of the curve to the directly computed eigenvalues in the range i = from the case using 1 outer-loop and 200 inner-loops. In this case i = 90 represents the number of the eigenvalues for which the error (c) The percentage error 100(λ i λ i )/λ i where λ i are the directly computed eigenvalues (i.e. the true eigenvalues) and λ i denotes the eigenvalues extrapolated from the curve fit in (b) using WC30, for the range i = (d) Same as (a) but for WC10 using 1 outer-loop and 200 inner-loops, with a curve fit based on the directly computed eigenvalues i = The curve fit coefficients in this case are a = 3.08 and b =

60 Table 1: The final value of the non-linear cost function J NL using 4 outer-loops and 25 inner-loops for the representative 4 day strong constraint I4D-Var assimilation cycle starting on 3 March, 2003 using WC30 versus the number of Ritz vectors used in the second level Ritz preconditioner. The case with no second level preconditioning is represented by the column labeled 0. Ritz # J

61 a b Figure 1: The ROMS CCS model domain and bathymetry used in (a) WC30 and (b) WC10. 59

62 a b c Figure 2: log 10 (J) versus number of inner-loop iterations for data assimilation windows of (a) 4 days, (b) 7 days, and (c) 14 days duration for a representative assimilation cycle starting 3 March 2003 using I4D-Var (red curve) and R4D-Var (blue), assuming a strong constraint and using WC30. The value of the non-linear cost function J NL at the end of inner-loop 100 is also indicated for I4D-Var (red cross) and for 4D-PSAS (green cross). Also shown is a case using R4D-Var and the weak constraint (black curve). The horizontal dashed line in each case shows the theoretical minimum value, J min =N obs /2, of J for each assimilation interval (for later reference in section 6.2). In each case a single outer-loop was used. 60

63 a b c Figure 3: Same as Fig. 2 but using WC10. In this case only results from R4D-Var are shown for the dual formulation. In (c), the black curve shows the result of a weak constraint R4D-Var calculation using 1 outer-loop and 200 inner-loops, so the iteration number on the abscissa should be multiplied by a factor of 2 in this case. 61

64 Figure 4: Time series of the initial value of log 10 (J) (thin lines) and final values of log 10 (J) (bold lines) for 7 day assimilation cycles when 4D-Var was applied sequentially using I4D-Var (black curves), R4D-Var (blue curves), and 4D-PSAS (red curves) using WC30. In each case 1 outer-loop and 50 inner-loops were used, and the strong constraint was employed. 62

65 Figure 5: The inner-loop cost function, log 10 (J), (bold curves) at the end of each inner-loop for different combinations of inner- and outer-loops using I4D-Var and WC30 for a representative assimilation cycle. The nonlinear cost function, log 10 (J NL ), (circles) is also shown at the end of each outer-loop. The abscissa refers the total combined number of inner- and outer-loops in each case. 63

66 a b Figure 6: (a) The final value of J o for different R4D-Var experiments for the representative 7 day assimilation cycle starting on 3 March, 2003, using WC30. In each experiment, the increment control vector was comprised of different combinations of δx(t 0 ), δf(t)=(δτ T,δQ T,δF T ) T, and δb(t), where δτ, δq, and δf are the increments of surface wind stress, net surface heat flux, and net surface freshwater flux respectively. The increment control vector was comprised of the following components: Expt A- No assimilation; Expt B- δx(t 0 ) only; Expt C- δx(t 0 ) and δτ; Expt D- δx(t 0 ), δτ and δq; Expt E- δx(t 0 ), δτ, δq, and δf; Expt F- δx(t 0 ), δτ, δq, δf and δb(t); Expt G- δτ only; Expt H- δτ, and δq; Expt I- δτ, δq, and δf; Expt J- δτ, δq, δf and δb(t); Expt K- δq only; Expt L- δb(t) only; Expt M- δb(t) and δτ; Expt N- δb(t), δτ and δq; and Expt O- same as F but subject to weak constraint. The numerical values of J o are also indicated for some experiments to aid comparison where the bar heights are very similar. (b) The ratio (J o ) f /(J o ) i of the final value of J o and the initial value of J o for a sequence of R4D-Var cycles using WC30 for the case where only δx(t 0 ) is adjusted (black curve), only δτ is adjusted (blue curve), only δq and δf are adjusted (red curve), and only δb(t) is adjusted (green curve). 64

67 a b c d Figure 7: The ratio (σ b - σ a )/ σ b for temperature at various depths, where σ b is the background error standard deviation in T and σ a is the analysis error standard deviation in T. The results shown are for the representative 7 day cycle starting on 3 March 2003 using R4D-Var and WC10 with 1 outer-loop and 100 inner-loops. The white diamonds indicate the locations of the in situ observations. 65

68 a b Figure 8: Same as Fig. 7 but for (a) sea surface salinity, and (b) meridional velocity v in the vicinity of the Oregon and Washington coast. 66

69 a b c Figure 9: Same as Fig. 7 except for (a) zonal wind stress, (b) meridional wind stress, and (c) net surface heat flux. 67

70 a b c d Figure 10: Maps of the ratio ( σ σ )/ σ where σ 800 and σ are the posterior error variances when using 800 inner-loops or 100 inner-loops respectively, and 1 outer-loop with R4D-Var and WC30. The white diamonds show the positions of the in situ observations. 68

71 Figure 11: The percentage of cumulative analysis error variance explained by the sum of different numbers of the leading eigenvectors (EOFs) of the analysis error covariance matrix (blue curve). The case shown is for the a representative 7 day strong constraint R4D-Var cycle starting on 3 March 2003 using WC30. The trace of E was estimated using the randomization method of Bai et al. (1995) in which vectors with randomly chosen elements +1 a or -1 are used to estimate Tr( E a ). The red curves show the cumulative variance associated with Tr( E ) ± σ Tr where σ Tr is the standard deviation of the resulting trace estimate based on a sample size of 400 randomized trace 1 a estimates. The error ε ˆ ˆ ˆ i = ν1 Ex i νixi / xi of each eigenpair ( ν, ˆ i x i) can be used to quantify the reliability of each EOF x ˆ i that is computed. Choosing ε =10 8 as an acceptable lower bound of the error, then only the leading 200 EOFs are acceptable which together account for ~30% of the analysis error variance. 69

72 a b c d b o Figure 12: Time series of ( σ ) (solid black curves) and ( ) σ (solid blue curves) computed using (5) and (6) for all observations of (a) SSH, (b) temperature, and (c,d) salinity. Also shown are time series (dashed curves) of the b o average standard deviations σ (dashed black curves) and σ (dashed blue curves) (at the same observation points) that were prescribed a priori in D and R. Computations are based on strong constraint R4D-Var run sequentially using WC30 for the period July 2002-Dec

73 a b c Figure 13: Time series of log 10 of various statistics of the cost function J from WC30 run sequentially during the period July 2003-July 2004 using a 7 day assimilation cycle, strong constraint R4D-Var, 1 outer-loop and 200 innerloops. In each panel, the solid black curve shows the theoretical minimum value of the cost function J min =N obs /2. In (a) the solid blue curves show the anticipated upper and lower bounds for the theoretical minimum value N 1 obs 1 ( Jb ) min = Nobs λ i of J b, where λ i are the eigenvalues of the preconditioned stabilized representer matrix 2 i= 1 ˆP derived from curve fits to the directly computed eigenvalues of the form log 10 (λ i ) = ae bi. The coefficients a and b are cycle dependent and were derived from the fit to the computed eigenvalues λ i for the range i=50-i where i=i+1 is the first computed eigenvalue with an error > 10-8 (see appendix, section A.2). The dashed blue curve shows the actual value of J b at the end of each assimilation cycle. Also shown is the anticipated range (upper and lower bounds) in the number of eigenvalues of ˆP for which λ i / λ 1 ε, for ε= (the 1% rule of Bennett and McIntosh (1982)) (solid green curves), where λ 1 is the largest eigenvalue. Following Bennett (1989) this represents an estimate of the number of independent linear combinations of the model state variables provided by the observations. In (b) the solid red curves show the anticipated upper and lower bounds for the theoretical minimum value ( Jo ) min = ( Nobs /2 ( Jb ) min ) of J o, while the dashed red curve shows the actual value of J o at the end of each assimilation cycle. In (c) the dashed black curve shows the actual value of J at the end of each assimilation cycle. 71

74 a b c d e f Figure 14: The structure of array mode Ψ 1 at initial time from a representative strong constraint R4D-Var cycle for the period 3-7 March, 2003, using WC10 and a 4 day assimilation interval: (a) SST, (b) SSH, (c) surface zonal velocity, (d) surface meridional velocity, (e) temperature at 100m depth, and (f) temperature at 200m depth. Each field has the physical units of the variable, and the corresponding values of the physical fields are determined by the 1 T / total weight λ ( wr ˆ dfor ) Ψ 1 in equation (10). The diamonds in (a) indicate the location of the in situ hydrographic observations during this 4 day period. 72

75 a b c d e f Figure 15: Same as Fig. 14 except showing the structure of Ψ 1 at final time. 73

76 a d b e c f Figure 16: The contribution of the initial condition (IC), surface forcing (FC), and boundary condition (BC) information to array mode Ψ 1 at final time for SST (a-c) and SSH (d-f). 74

77 a b c d Figure 17: The SST structure of array modes 2-5 at initial time. The diamonds indicate the location of the in situ hydrographic observations during this 4 day period. 75

78 a b c d e f Figure 18: The analysis increment for SST from a typical 4 day cycle of strong constraint R4D-Var at (a) initial time, and (b) final time. The clipped analysis increments using only those array modes for which λ ι >0.01λ 1 are shown in (c) at initial time, and (d) at final time. The difference between the R4D-Var analysis increment and the clipped analysis increment at initial and final time is shown in (e) and (f) respectively. 76

79 a b Figure A1: (a) log 10 ( λ i ) versus eigenvalue number i of the preconditioned stabilized representer matrix ˆP resulting 1 from 1 outer-loop and 100, 200, 400 and 800 inner-loops. (b) log 10 (ε i ) where ε = λ P ˆwˆ λwˆ / wˆ is the i 1 i i i i error associated with each eigenpair ( λ, w ˆ ) in each case, and only values of ε i > are shown. The horizontal i dashed line represents the ε i =10-8 error threshold used to select reliable eigenvalues. i 77

80 a b c d Figure A2: (a) The black curve shows log 10 (λ i ) for the leading 500 eigenvalues λ i of the preconditioned stabilized representer matrix ˆP computed using 1 outer-loop and 800 inner-loops in WC30 which Fig. A1b shows have an error The red curve shows the eigenvalues λ i for the range i= that were extrapolated from a curve of the form log 10 (λ i ) = ae bi, (a=1.684 and b= ) where a and b were determined from a best fit of the curve to the directly computed eigenvalues in the range i= (b) The black curve shows log 10 (λ i ) for the leading 500 eigenvalues λ i of the preconditioned stabilized representer matrix ˆP computed using 1 outer-loop and 800 innerloops in WC30 which Fig. A1b shows have and error The red curve shows the eigenvalues λ i for the range i= extrapolated from a curve of the form log 10 (λ i )=ae -bi, (a=1.752 and b= ) where a and b were determined from a best fit of the curve to the directly computed eigenvalues in the range i=50-90 from the case using 1 outer-loop and 200 inner-loops. In this case i=90 represents the number of the eigenvalues for which the error (c) The percentage error 100(λi λ i)/λi, where λ i are the directly computed eigenvalues (i.e. the true eigenvalues) and λ i denotes the eigenvalues extrapolated from the curve fit in (b) using WC30, for the range i= (d) Same as (a) but for WC10 using 1 outer-loop and 200 inner-loops, with a curve fit based on the directly computed eigenvalues i= The curve fit coefficients in this case are a=3.08 and b=