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Estimates of analysis and forecast error variances
derived from the adjoint of 4D-Var
Andrew Moore, Hernan Arango, Gregoire Broquet
To cite this version:
Andrew Moore, Hernan Arango, Gregoire Broquet. Estimates of analysis and forecast error variances
derived from the adjoint of 4D-Var. Monthly Weather Review, American Meteorological Society, 2012,
140 (10), pp.3183-3203. �10.1175/MWR-D-11-00141.1�. �hal-03206181�
Estimates of Analysis and Forecast Error Variances Derived
from the Adjoint of 4D-Var
ANDREWM. MOORE
Department of Ocean Sciences, University of California, Santa Cruz, Santa Cruz, California
HERNANG. ARANGO
Institute of Marine and Coastal Sciences, Rutgers University, The State University of New Jersey, New Brunswick, New Jersey
GREGOIREBROQUET
Laboratoire des Sciences du Climat et de l’Environnement, CEA, Gif-Sur-Yvette, France (Manuscript received 19 June 2011, in final form 2 March 2012)
ABSTRACT
A method is presented in which the adjoint of a four-dimensional variational data assimilation system (4D-Var) was used to compute the expected analysis and forecast error variances of linear functions of the ocean state vector. The power and utility of the approach are demonstrated using the Regional Ocean Modeling System configured for the California Current system. Linear functions of the ocean state vector were con-sidered in the form of indices that characterize various aspects of the coastal upwelling circulation. It was found that for configurations of 4D-Var typically used in ocean models, reliable estimates of the expected analysis error variances can be obtained both for variables that are observed and unobserved. In addition, the contribution of uncertainties in the model control variables to the forecast error variance was also quantified. One particularly powerful and illuminating aspect of the adjoint 4D-Var approach to the forecast problem is that the contribution of individual observations to the predictability of the circulation can be readily com-puted. An important finding of the work presented here is that despite the plethora of available satellite observations, the relatively modest fraction of in situ subsurface observations sometimes exerts a significant influence on the predictability of the coastal ocean. Independent checks of the analysis and forecast error variances are also presented, which provide a direct test of the hypotheses that underpin the prior error and observation error estimates used during 4D-Var.
1. Introduction
Data assimilation forms a critical component of all modern analysis and forecast systems for both the at-mosphere and ocean. As such, an important component of all data assimilation systems should be an estimate of the posterior or analysis error covariance associated with the best circulation estimate, as well as an estimate of the ensuing forecast error covariance. In addition, when data assimilation is applied sequentially, the re-sulting analysis typically becomes the prior for the next analysis cycle. Therefore, in addition to providing
information about the expected uncertainty in the anal-ysis, the analysis error covariance can be used to update the prior or background error covariance for the next data assimilation cycle.
While analysis error information can in principle be readily computed for statistical interpolation schemes and Kalman filters (Daley 1991), it presents a challenge for variational methods because the information needed is generally not available in a convenient form. In this paper, we will present a method by which formal pos-terior covariance information can be obtained using the adjoint of the entire variational data assimilation algo-rithm. To set the stage for our work, we will begin with a brief review of four-dimensional variational assimilation (4D-Var) of ocean data; however, the same ideas obvi-ously apply to the atmosphere as well, and to 3D-Var. Corresponding author address: Andrew M. Moore, Dept. of
Ocean Sciences, University of California, Santa Cruz, 1156 High St., Santa Cruz, CA 95064.
Following the notation introduced by Ide et al. (1997) and expanded by Daget et al. (2009) and Moore et al. (2011a, hereafter M11a), we denote by x the state vector of the ocean (or atmosphere). Data assimilation seeks to combine, in an optimal way, a background (or prior) estimate of the circulation, xb(t), with observations that are arranged in the vector yo. The background circula-tion is generally a solucircula-tion of a numerical model, and depends on background estimates of the initial condi-tions, xb(t0); surface forcing, fb(t); and open boundary
conditions, bb(t). The model solution will be denoted by xb(ti)5 M(ti, ti21)[xb(ti21), fb(ti), bb(ti)], where M(ti, ti21)
represents the model operators and fb(ti) and bb(ti)
represent the actions of the forcing and boundary con-ditions, respectively, during the interval [ti21, ti]. During
4D-Var the best circulation estimate is identified by adjusting a vector of control variables composed of the initial conditions, surface forcing, open boundary con-ditions, and corrections for model error. Sometimes, 4D-Var is implemented using the incremental formula-tion of Courtier et al. (1994), in which the control vector, denoted dz, is composed of increments to the priors, specifically increments dx(t0) to the state vector at initial
time t0, as well as increments to surface forcing, df(t);
open boundary conditions, db(t); and corrections for model error, h(t) at all times t spanning the data as-similation cycle. To first order, the state vector incre-ments dx(t) are governed by dx(ti)5 M(ti, ti21)du(ti21),
where du(ti21)5 [dxT(ti21), dfT(ti), dbT(ti),hT(ti)]T, and
M(ti, ti21) represents the perturbation tangent linear
model linearized about the time-evolving background xb(t).
Denoting as zb5 [xbT(t0),. . . , fbT(ti),. . . , . . . ,
bbT(ti),. . . , . . . , hbT(ti),. . . ]T the vector of background
fields (where the ellipses denote the sequence of all times ti in the assimilation interval), then the control
vector that yields the best circulation estimate can be conveniently expressed as ^z 5 zb1 dza, where dza 5
[dxT(t0), . . . , dfT(ti), . . . , . . . , dbT(ti), . . . , . . . ,
dhT(t
i),. . .]Tis the vector of analysis (or posterior)
in-crements that minimize the cost function:
J(dz)51 2dz
TD21dz11
2(Gdz 2 d)
TR21(Gdz 2 d), (1)
whereD and R are the background and observation error covariance matrices, respectively; d5 fyo2 H[xb(t)]g is the innovation vector; H[xb(t)] is the background eval-uated at the observation points via the observation
op-erator H; and G represents the tangent linear model
sampled at the observation points, and is essentially a convolution in time of M(ti, ti21) and the tangent
line-arization of H.
The analysis increments can be expressed as dza5 Kd, where
K 5 (D211 GTR21G)21GTR21 (2)
5 DGT(GDGT1 R)21 (3)
is the gain matrix. Equation (2) is sometimes referred to as the primal form ofK and arises from searching for dza in the full space spanned by the control vector dz. Al-ternatively, (3) is referred to as the dual form ofK and arises from a search for dzain the space of linear func-tions of dz spanned only by the observafunc-tions (i.e.,Gdz), usually referred to as observation space.
Using (2) or (3), the posterior or analysis error co-variance matrix,Ea, of the best estimate^z can be written in several different (although equivalent) forms; namely,
Ea5 (D211 GTR21G)21 (4)
5 [(I 2 KG)D(I 2 KG)T1 KRKT] (5)
5 (I 2 KG)D. (6)
It is important to realize that the analysis error co-variance matrix given by (4)–(6) will only be the true Eaif the prior error covariance matrix D and the
ob-servation error covariance matrixR are correctly spec-ified, and if the assumptions underlying the analysis system are correct, namely Gaussian, unbiased errors in the priors and the observations.
Equation (4) shows that Ea is equal to the inverse of the Hessian matrix,H, of the cost function J in (1). During applications of primal 4D-Var, the leading
ei-genvalues and eigenvectors ofH can be estimated and
used to compute a reduced-rank approximation ofEa. However, sinceEa5 H21, the leading eigenvectors of H are the least important eigenvectors of Ea and, in
practice, explain very little of the analysis error variance. Nonetheless, while such estimates may be expected to overestimate the analysis error, they are used routinely (e.g., Fisher and Courtier 1995) and provide informa-tion about geographic variainforma-tions of the error. Equa-tions (5) and (6) arise directly from the dual form of K in (3), and Daley and Barker (2001) [see also Gelaro et al. (2002) for more details] describe a routine
ap-proach for estimating Ea based on a Cholesky
fac-torization of the stabilized representer matrix P 5
(GDGT1 R).
In practice, the matrix inverse in (2) or (3) is evaluated iteratively by solving an equivalent system of linear equations using a conjugate gradient algorithm. In a num-ber of 4D-Var systems in current use (e.g., Fisher and
Courtier 1995; Tshimanga et al. 2008; Chua et al. 2009; M11a), the conjugate gradient algorithm is formulated in terms of a sequence of Lanczos vectors (Lanczos 1950), which are used to construct a reduced-rank approxima-tion ~K of K. By virtue of the orthonormal property of the Lanczos vectors, (6) leads to a convenient reduced-rank form forEa. Moore et al. (2011b, hereafter M11b) have used (6) in this capacity in an oceanographic ap-plication of 4D-Var to estimate analysis errors for the California Current circulation. However, since the Lanczos basis employed does not typically span the full space ofK, the resulting expected analysis error is still overestimated.
Monte Carlo methods (Bennett 2002) and ensemble techniques (Houtekamer et al. 1996; Zagar et al. 2005; Belo Pereira and Berre 2006; Berre et al. 2006) can also be used to provide estimates ofEain conjunction with 4D-Var, and the method we describe in this paper is inspired by this idea. In section 2 we describe how the adjoint of a 4D-Var system can be used to estimate the expected analysis error covariance without the need to explicitly compute an ensemble. The use of the adjoint of a data assimilation system is not new, and has tradi-tionally been used in numerical weather prediction to provide observation sensitivity and observation impact information pertaining to analyses (e.g., Zhu and Gelaro 2008) and forecasts (e.g., Langland and Baker 2004; Gelaro and Zhu 2009). In this study, however, the utility of the adjoint of a data assimilation system has been extended to provide formal error variance information about theoretical infinite-sized ensembles. Both our ap-proach and those of previous investigations, however, capitalize on the property of the adjoint of the data assimilation system to provide information about the sensitivity of scalar functions of the control vector to variations in the inputs to the system, in this case the observations and priors. The method has been tested in relation to the circulation of the California Current system (CCS), and the configuration of the model and 4D-Var system are presented in section 3. The adjoint of 4D-Var is used to estimate the expected errors as-sociated with various dynamical aspects of the coastal upwelling circulation that develops along the central California coast, and indices that quantify this circu-lation are introduced in section 4. The expected anal-ysis and forecast error variance of the circulation indices are explored in sections 5 and 6, respectively. Section 7 explores the predictability of the circulation attributable to the different observation platforms, and in section 8 we present a series of checks for consis-tency and statistical reliability for the derived error estimates. We end with a summary and conclusions in section 9.
2. Analysis error covariance from perturbed analyses
Consider an ensemble of analyses that are generated by perturbing the observations and background fields with perturbations drawn from normal distributions
with covariances R and D, respectively. It has been
proven (e.g., Zagar et al. 2005; Belo Pereira and Berre 2006; Daget et al. 2009) that the covariance of the perturbed analyses about the unperturbed analysis simulates the analysis error covariance. If the prior co-variancesR and D are equal to the true covariances, the covariance of the analysis ensemble will equal the true analysis error covariance. This is the principle behind ensemble 4D-Var currently used operationally at some numerical weather prediction (NWP) centers (e.g., Isaksen et al. 2010), and is the underlying premise behind the method presented here. In our case, however, instead of explicitly generating an ensemble of analyses, we use the adjoint of the analysis system in conjunction with the expectation operator and consider all possible analyses drawn from the appropriate distribution.
As shown by M11a (see also the appendix), the cir-culation analysis can be alternatively expressed as^z 5 zb1 K(d), where the analysis increment dza5 K(d) and K(d) represents the entire 4D-Var procedure, which, in general, is a nonlinear function of the innovation vector d by virtue of the algorithm employed to minimize J in (1). The ensemble 4D-Var approach outlined above involves perturbing the observations and background control vector, which yields perturbations dd in the innovations, and associated first-order changes in the analysis increments given by (›K/›d)dd, where ›K/›d represents the tangent linearization of the entire 4D-Var procedure. With this in mind, consider a 4D-Var anal-ysis spanning the interval [t0, t01 t]. The time evolution
of the resulting posterior state vector is given by xa(ti)5 M(ti, ti21)[xa(ti21), fa(ti), ba(ti),ha(ti)] . (7)
Consider now an ensemble of 4D-Var analyses ^zl,
where the lth ensemble member is generated by
add-ing perturbations dzb
l 5 [dx bT
l (t0),. . . , dfbTl (ti),. . . , . . . ,
dbbTl (ti),. . . , . . . , dhbTl (ti),. . .]T to the background
con-trol vector zband perturbations dyo
l to the observations,
where dzb
l and dyol are drawn from normal distributions
with covariances D and R, respectively. The
result-ing perturbation in the 4D-Var analysis control vector is given by d^zl5 dzbl 1 (›K/›d)ddl, where ddl5 (dyol2
Gdzb
l) is the perturbation to the innovation vector. The
tangent linear 4D-Var operator ›K/›d is linearized about the unperturbed 4D-Var sequence of iterations that yield the unperturbed analysis xa, fa, ba, andha. We can
also express the analysis perturbation control vector as d^zl5 [dxaTl (t0),. . . , dflaT(ti),. . . , . . . , dbaTl (ti),. . . , . . . ,
dhaT
l (ti),. . . ]T, so that from (7) the state vector of each
perturbed analysis evolves according to
xal(ti)5 M(ti, ti21)[xal(ti21), fal(ti), bal(ti),hal(ti)]
5 M(ti, ti21)[xa(ti21)1 dxal(ti21), fa(ti)1 dfla(ti), ba(ti)1 dbal(ti),ha(ti)1 dhal(ti)]
’ M(ti, ti21)[xa(ti21), fa(ti), ba(ti),ha(ti)]1 Ma(ti, ti21)dual(ti21) , (8)
whereMais the tangent linear model linearized about
the unperturbed time-evolving analysis (7). Since our primary focus in later sections is on the state vector x, we introduce an alternative form of the tangent linear model operator,Ma, that isolates the state vector
per-turbation dx arising from a control vector perper-turbation dz, so that dx(ti)5 Ma(ti, t0)dz. Recall that dual(ti) is the
vector of perturbations in the analysis at a single instant in time ti, while dzlare the perturbations in all the
con-trol vector elements (i.e., dx at t0and df, db, and dh at
all times during the assimilation interval). At the end of the analysis cycle, the perturbation in the analysis state vector will to first order be given by
dxal(t01 t) 5 Ma(t01 t, t0)d^zl 5 Ma(t01 t, t0)dzbl 1 Ma(t01 t, t0)›K ›d(dy o l 2 Gdzbl) . (9)
As noted earlier, if 4D-Var is run to complete conver-gence, the priorsR and D are correctly specified, and all of the underlying assumptions about 4D-Var are correct (e.g., an unbiased estimator), then the variance of the perturbed 4D-Var analyses (8) about the unperturbed analysis (7) equals the expected analysis error covariance. Therefore, using (9), the analysis error covariance of the state vectorEax(t01 t) 5 hdxla(t01 t)dxaTl (t01 t)i at time
t01 t is given by Ea x(t01 t) 5 Ma(t01 t, t0) " I 2›K ›dG D I 2›K ›dG T 1›K ›dR ›K ›d T# MT a(t01 t, t0) , (10)
where h i denotes the expectation operator and
(›K/›d)Trepresents the adjoint of the tangent linearized
4D-Var algorithm. Formally, the tangent linear model Mais linearized about the unperturbed 4D-Var analysis
(indicated by the subscript a). However, in the case where the linearized dynamics are always used to propagate information, as in the case of the representer method of
Bennett (1992) used in this study, thenMa[ Mband
the tangent linear model is linearized about the un-perturbed background (indicated by the subscript b).
Computing the entire matrixEax(t01 t) given by (10)
presents a considerable challenge due to its large
di-mension. However, for linear scalar functions J 5
J [x(t)] (e.g., space–time integrals of x), computation of the expected analysis error variance ofJ presents much less of a challenge. Consider the generic linear scalar function of the discrete model state vector xk[ x(t01
kDt) given by J 5
å
k2k5k1h
T
kxk, where Dt is the model
time step and hkis an appropriate vector. The prior error
variance ofJ (i.e., before data assimilation) is given by
(sbJ)25
å
k2 k5k1 hTk(Mx)k Då
k2 l5k1 (Mx)Tlhl , (11) where (Mx)k [ Mx(t0 1 kDt, t0) is given by Ma orMbdepending on the 4D-Var algorithm used (see
be-low). Similarly, the expected posterior error variance in J after data assimilation is given by
(saJ)25
å
k2 k5k1 hTk(Mx)k (Eax)t 0å
k2 l5k1 (MTx)lhl , (12)where (Eax)t05 f[I 2 (›K/›d)G]D[I 2 (›K/›d)G]T1 (›K/›d)R(›K/›d)Tg. If 4D-Var is run to complete con-vergence, then for a linear data assimilation system (or the case of one outer loop; see section 4) ›K/›d 5 K, and (Ea
x)t0is identical to (5) for the true gain matrix.
The model used here is the Regional Ocean Modeling System (ROMS), for which an extensive suite of 4D-Var tools has recently been developed (M11a) based on both the primal and dual formulations. Two dual 4D-Var al-gorithms are currently available in ROMS: one that is based on the Physical-space Statistical Analysis System (PSAS; Cohn et al. 1998), and one that utilizes the in-direct representer method of Egbert et al. (1994), re-ferred to here as R4D-Var. In the case of primal 4D-Var or dual PSAS, the posterior circulation estimate is a
solution of the nonlinear model, in which caseMx [
Ma in (11) and (12). In the case of the dual indirect
representer method, the posterior circulation estimate is a solution of a finite-amplitude linearization of the
model (Bennett 2002) linearized about the prior circu-lation, in which caseMx[ Mbin (11) and (12). In the
present study we will focus our attention on R4D-Var.
3. Configuration of ROMS and 4D-Var
ROMS is a hydrostatic, primitive-equation, Boussinesq ocean general circulation model designed primarily for coastal applications by employing terrain-following ver-tical coordinates and horizontal orthogonal curvilinear coordinates (Shchepetkin and McWilliams 2005). The configuration of ROMS and R4D-Var used in the cal-culations reported below is described in detail by Broquet et al. (2009a,b, 2011) and M11b, so only a brief de-scription will be given here.
The ROMS CCS domain spans the region 318–488N to
1348–1168W with 30-km horizontal resolution and 30
terrain-following s levels in the vertical. While this res-olution is marginal for modeling the mesoscale circula-tion of the CCS, it provides a reasonable representacircula-tion of the large-scale circulation. The motivation for using a coarse resolution is that the sequential 4D-Var computa-tions presented here are computationally very demanding. Since the purpose of this paper is to demonstrate the ef-ficacy of using the adjoint of 4D-Var for estimating pos-terior errors of a sequential data assimilation system, the use of a coarse-resolution model is justified. The model domain and bathymetry are shown in Fig. 1.
The model forcing was derived from daily averaged output of atmospheric boundary layer fields from the Naval Research Laboratory’s (NRL) Coupled Ocean– Atmosphere Mesoscale Prediction System (COAMPS; Doyle et al. 2009) using the bulk formulations of Liu et al. (1979) and Fairall et al. (1996a,b). The resulting surface fluxes of momentum, heat, and freshwater rep-resent the background surface forcing, denoted fb(t) in section 1. The model domain has open boundaries at the northern, southern, and western edges where the tracer and velocity fields were prescribed, and the free surface and vertically integrated flow were subject to Chapman (1985) and Flather (1976) boundary conditions, respec-tively. 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 open boundary conditions denoted bb(t) in section 1. A sponge layer was also used adjacent to each open boundary in which viscosity increased linearly from 4 m2s21in the interior to 100 m2s21at the boundary
over a distance of;100 km.
Following M11b, ROMS R4D-Var was run sequen-tially for the period July 2002–December 2004 starting from a set of background initial conditions on 27 July
2002 derived from the 4D-Var sequence of analyses computed by Broquet et al. (2009a). The observations assimilated in the model were collected by various plat-forms, and include gridded sea surface height (SSH) analyses in the form of dynamic topography from Aviso at;1/38 resolution every 7 days; a blended SST product
with 10-km resolution, available daily, and consisting of 5-day means derived from the Goddard Earth Observ-ing System (GEOS), Advanced Very High Resolution Radiometer (AVHRR), and Moderate Resolution Im-aging Spectroradiometer (MODIS) satellite instruments [courtesy of D. Foley at the National Oceanic and At-mospheric Administration (NOAA) and available east of 1308W only at the time the experiments presented here were performed]; in situ hydrographic observations ex-tracted from the quality controlled EN3 (version v1d) data archive maintained by the Met Office as part of the European Union ENSEMBLES project (Ingleby and Huddleston 2007); and tagged elephant seal data from the Tagging of Pacific Pelagics (TOPP) program. To reduce data redundancy, all observations within each model grid cell, over a 6-h time window, were combined to form ‘‘superobservations,’’ and the standard deviation of the observations that contribute to the superobservation in each grid cell was used as an estimate of the error of representativeness.
Observation errors were assumed to be uncorrelated in space and time, resulting in a diagonal observation FIG. 1. The model domain and bathymetry. Also shown are the sections along which alongshore and cross-shore transports were computed, and the region in which upper-ocean heat content and SST were evaluated.
error covariance matrix R. The variances along the
main diagonal ofR 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: 0.02 m for SSH, 0.48C for SST, 0.18C for in situ T, and 0.01 for in situ S.
A background error standard deviation associated with xb(t0), fb(t), and bb(t) was estimated for each
cal-endar month. Following Weaver et al. (2005), the back-ground error for the initial conditions was decomposed into a balanced and unbalanced component, and the associated background error covariance matrix was fac-torized as the product of a univariate covariance matrix (describing the errors of the unbalanced flow) and a dy-namical balance operator. The dydy-namical balance op-erator yields cross-covariance information about the errors and is based on the T–S properties of the water column, hydrostatic balance, and geostrophic balance (Weaver et al. 2005; M11a). The background error standard deviations for the unbalanced components of the control vector were estimated each month based on the variance of the model run during the period 1999– 2004 subject only to surface forcing (i.e., no data as-similation). The temporal variability of the COAMPS surface forcing for the period 1999–2004 was used as the variance for the background surface forcing error, and the open boundary condition background error variances were chosen to be the variances of the ECCO fields at the boundaries. A balance operator was not applied to the surface forcing or open boundary condition incre-ments. In all of the experiments presented here, the cir-culation estimates were computed subject to the strong constraint, in which case the model is assumed to be free of error, and the correctionsh for model error were set to zero.
At the present time, ROMS 4D-Var supports only homogeneous error correlations that are separable in the horizontal and vertical. The horizontal and vertical correlation functions are modeled using the diffusion operator approach described by Weaver and Courtier (2001) (see also M11a for details of the implementation in ROMS). The decorrelation length scales used to model the background errors of all initial condition control variable components ofD were 50 km in the horizontal and 30 m in the vertical. Horizontal correlation scales chosen for the background surface forcing error
com-ponents ofD were 300 km for wind stress and 100 km
for heat and freshwater fluxes. The correlation lengths for the background open boundary condition error com-ponents ofD were 100 km in the horizontal and 30 m in the vertical. Explicit temporal correlations of the back-ground errors are not included in the current version of
ROMS 4D-Var. However, the surface forcing and bound-ary condition increments, df(t) and db(t), were only computed daily and interpolated to each intervening model time step, which effectively introduces some tem-poral correlation of the errors in fb(t) and bb(t). A dis-cussion of the choice of the aforementioned background error covariance parameters can be found in Broquet et al. (2009a,b, 2011) and M11b.
The R4D-Var model was run sequentially for the period July 2002–December 2004 with 1 outer loop and 60 inner loops1spanning 7-day assimilation cycles, which is sufficient to guarantee a significant level of convergence of J toward its minimum value as demon-strated by M11b. Following El Akkraoui and Gauthier (2010), the minimum residual method of Paige and Saunders (1975) was used to minimize the cost function J during each data assimilation cycle, and the analysis at the end of each assimilation cycle, xa(t01 7), becomes
the background initial condition for the next assimila-tion cycle. Figure 2 shows the ratio of the final and initial values of J for the R4D-Var sequence, and indicates that an order of magnitude reduction in J during each cycle is typical.
4. Coastal transport, heat content, and SST
The central California coastal circulation is charac-terized by a pronounced seasonal cycle of coastal up-welling (Checkley and Barth 2009). Equatorward of Cape Mendocino (about 408N), the winds are along-shore and upwelling favorable year round. However, FIG. 2. A time series of the ratio of the final value of the cost function, Jf, to the initial value of the cost function, Ji, for the
se-quence of R4D-Var data assimilation cycles spanning the period July 2002–December 2004.
1The 4D-Var method proceeds by iteratively solving a sequence
of linear, least squares minimization problems. The first member of the sequence is created via a series of inner loops linearized about the prior circulation. Subsequent members of the sequence are generated by updating the circulation about which the inner loops are linearized. This is done by rerunning the nonlinear circulation model using the newly computed increments from the last series of inner loops. The updates of the nonlinear model circulation in this way are referred to as outer loops.
during late spring and early summer as the atmospheric subtropical high pressure system builds over the ocean, the alongshore winds intensify and the coastal upwelling reaches a peak. With these characteristics of the cir-culation in mind, we will consider four linear functions J (x). These functions also serve to demonstrate how (10) can provide error information not only about ocean-state variables that are frequently and directly observed (such as ocean temperature) but also error information
about variables that are not directly observed. The fol-lowing four functions are considered.
a. Alongshore transport crossing 378N, J37N
The pronounced seasonal changes in the alongshore winds near the central California coast modulate the strength of the California Current system, as well as the cross-shore pressure gradients that develop in response to Ekman divergence at the coast and Ekman pumping FIG. 3. Time series of (a)J37N(solid line) andDJ37N(line with circles), (b)J500m(solid line)
andDJ500m(line with circles), (c)JHC(solid line) andDJHC(line with circles), and (d)JSST
associated with wind stress curl. As a representative measure of these changes, we consider the 7-day-average
transport crossing 378N between the coast and 1278W
over the upper 500 m of the water column during each R4D-Var assimilation cycle. This section is indicated in Fig. 1. Specifically, following the notation introduced in section 2, we will denote this transport over the interval [t0, t01 7] as J37N5
å
k2
k5k1(h37N)
T
kxk, where t0 [ k1Dt
and t01 7 [ k2Dt. The elements of the vector (h37N)k
are zero except for those that correspond to the velocity grid points that contribute to the transport normal to the 378N section for the prescribed longitude and depth range.
Figure 3a shows a time series ofJ37N(xb) computed
from the background circulation xb(t) during each R4D-Var cycle. A pronounced seasonal cycle is apparent with poleward flow during winter and spring, and equator-ward flow during summer and fall, and a peak-to-peak transport of about 65 Sv (1 Sv [ 106m3s21). Also
shown in Fig. 3a is the increment in 378N transport
DJ37N5 J37N(xa)2 J37N(xb), the transport difference
between the analysis, xa(t) and the background xb(t) due to assimilating the observations during each
R4D-Var cycle. The mean transport increment is 20.35 SV
with a range of;62 SV.
b. Cross-shore transport at the continental shelf break,J500m
The wind stress along the coast and wind stress curl farther offshore drive Ekman transport and Ekman pumping in the upper ocean, resulting in a cross-shore transport. As a representative measure of these pro-cesses during each R4D-Var assimilation cycle, we con-sidered the 7-day-average transport in the upper 15 m of the water column, between 358 and 40.58N, crossing over the continental shelf break, taken here to be the 500-m isobath. This section is indicated in Fig. 1. Specif-ically we will denote this transport over the interval [t0, t01 7] as J500m5
å
k2
k5k1(h500m)
T
kxk, where the
ele-ments of the vector (h500m)kare zero except for those
corresponding to the velocity grid points that contribute to the transport normal to the 500-m isobath for the prescribed longitude, latitude, and depth range. The vector (h500m)k is defined so that offshore transport
corresponds toJ500m. 0. Figure 3b shows time series
of J500m(xb) computed from the background
circula-tion xb(t) of each R4D-Var cycle, and shows that the transport is generally offshore during most of the year,
with peak-to-peak variations;61 Sv. The cross-shore
transport incrementsDJ500m5 J500m(xa)2 J500m(xb)
are also shown in Fig. 3b, and have a mean close to zero, as well as peak-to-peak variations of;60.5 Sv.
c. SST and upper-ocean heat content,JSSTandJHC
The seasonal cycle of coastal upwelling leads to pro-nounced changes in the thermal structure of the upper ocean. To quantify these changes, we consider the 7-day mean SST averaged over the target area shown in Fig. 1,
and given by JSST5
å
k2
k5k1(hSST)
T
kxk, where the
ele-ments of (hSST)kare zero except for those surface
tem-perature grid points that fall within the target area. The target area encompasses the region where the largest seasonal variations in SST occur associated with the seasonal variations in the wind. In addition, we also consider the 7-day mean temperature in the same tar-get area averaged over the upper 50 m of the water
column; namely,JHC5
å
k2
k5k1(hHC)
T
kxk, where (hHC)k
is appropriately defined. The linear function JHC is
also proportional to the heat content in the same vol-ume of water. Time series ofJHC(xb) andJSST(xb) for
the R4D-Var background circulation are shown in Figs. 3c and 3d, respectively. The two time series are similar, and show that the lowest temperatures associated with the peak in upwelling occur during the spring and early summer. Following the weakening of the subtropical high, the alongshore upwelling-favorable winds decrease in strength around midsummer and the temperature of the surface waters increases. The incrementsDJHCand
DJSSTare also shown in Figs. 3c and 3d, and the mean
increment in both cases is;0.38C, indicating a cold bias in the background. While JHC(xb) and JSST(xb) provide
similar information, important differences in their error variance properties will become apparent later.
5. Posterior error estimates The prior (sb
J)2and posterior (saJ)2error variance of
each linear functionJ can be computed using (11) and (12), and Fig. 4 shows time series of the expected error variances of the various circulation indices introduced in section 4. Figure 4 shows that for all J , (sa
J)2 is
consistently lower than (sb
J)2, indicating the expected
positive impact of data assimilation. Recall that (sa J)2
will only be the true error variance ifD and R are cor-rectly specified. Therefore, compared to the prior esti-mate, it is possible that the posterior estimate might actually be degraded by assimilating data if theD and R are not correct, yet the expected (sa
J)2may indicate
otherwise. Nonetheless, in the case of the transports J37NandJ500m(Figs. 4a and 4b), the posterior error is
;40%–80% lower than the prior error, and both (sa
J)2
and (sb
J)2tend to be larger during winter, a reflection
of the seasonal dependence imposed onD (verified by
comparison with cases where there were no seasonal
JSST, the (saJ)2 are two orders of magnitude smaller
than the (sbJ)2.
Figure 4 also shows an estimate of the posterior error
computed using the reduced-rank approximation ~K of
the gain matrix according to (~saJ)25 f
å
k2k5k1h T k(Mx)kg ~Eaf
å
k2 l5k1(M Tx)lhlg, where ~Ea5 [(I 2 ~KG)D(I 2 ~KG)T1
~KR~KT]. As discussed in the appendix, ~K can be
com-puted using the Lanczos vectors derived from the m inner loops employed in R4D-Var. However, Fig. 4 in-dicates that (~saJ)2 is only marginally less than (sb
J)2
for the transports, and (~saJ)2is significantly larger than (sa
J)2in the case ofJHCandJSST. This is because ~K is
based on m Nobs, the number of observations, and
represents a single realization of ~K that typically will span only a small subspace of observation space. As such, ~Eatends to overestimate the true expected poste-rior error variance as discussed in M11b. However, even though the tangent linearization of 4D-Var, (›K/›d), is
computed using the same number of inner loops m
Nobs, each realization of the perturbation in the analysis
increment (›K/›d)ddlwill span a different set of m
di-rections in observation space. In the case of (12), all possible realizations of ddl, drawn from a normal
dis-tribution with covariance (GDGT1 R), are considered FIG. 4. Time series of the background (prior) error variance (sb
J)2(solid line) and the
ex-pected analysis (posterior) error (sa
J)2computed from (12) (line with closed circles), and (~saJ)2
computed using the practical gain matrix ~K (line with open circles) for (a) J37N, (b)J500m,
by virtue of the expectation operator used to deriveEax, and (sa
J)2represents a more reliable estimate of the true
expected posterior error variance than (~sa
J)2, as is
dem-onstrated in the appendix.
6. Forecast errors
Consider now the adjacent analysis and forecast cycles illustrated schematically in Fig. 5a, where forecast cycle j is initialized with the circulation estimate at the end of analysis cycle j. Also, consider different forecast re-alizations resulting from perturbations in the forecast initial conditions of forecast cycle j that are consistent with the expected 4D-Var analysis error at the end of
analysis cycle j. In addition, assume that the forecast surface forcing and open boundary conditions are also perturbed. The expected error covariance of the state vector relative to the unperturbed forecast is given by
Ef k5 hdx f kdx fT ki 5 (Mf)kDf(MTf)k, (13)
where (Mf)k is the tangent linear model at forecast
time kDt, linearized about the unperturbed forecast that is initialized using the unperturbed 4D-Var analysis, dxkf[ dxf(kDt) ’ (M
f)kdxa(k2Dt) is the forecast
per-turbation derived from the analysis increment at the end of analysis cycle j, andDf5 diag(Eax,Bf,Bb), whereEaxis
given by (10) and represents the state-vector forecast initial condition error covariance at the forecast start time, t0j1 7 5 k2Dt. Here, it is assumed that the forcing
and boundary condition perturbations are drawn from distributions with the prior error covariances,BfandBb.
In practice, however, these would be the error co-variances of the forecast products used to drive the ocean forecasts. In general,Df would also include a component due to model errors, but in keeping with the strong constraint 4D-Var calculations of section 3, the model error covariance term is not included for sim-plicity. It is important to remember that the prior error covariance of the background for each data assimilation cycle is alwaysD, as prescribed in section 3, which in the present experiments is never updated using Eax from (10). Therefore, the error estimates for each pair of consecutive analysis and forecast cycles can be con-sidered to be independent.
Equation (13) is based on forecast realizations cen-tered about a forecast initialized by the unperturbed analysis and is equivalent to the usual forecast ensemble employed at operational weather centers (e.g., Leith 1974; Molteni et al. 2006). Therefore, (13) also describes the expected forecast error covariance of a perfect model (i.e., no model error).
The covariance of any linear function J (x) of the
forecast state vector xf(t) derived from all possible fore-cast realizations is given by (sfJ)25 f
å
k4k5k3h T k(Mf)kg Dff
å
k4 l5k3(M Tf)lhlg, where k4Dt denotes the final forecast
verification time (Fig. 5a). Figure 6 shows time series of (sfJ)2 for each index J , which now are defined as the averages of the circulation over the last 48 h [tj01 12, tj01 14] [ [k3Dt, k4Dt] of 7-day forecasts
initial-ized every week.
Also shown in Fig. 6 are time series of the contribution to (sfJ)2 of uncertainties in the initial conditions, sur-face forcing, and boundary conditions. With regard to the expected errors in the forecast transports, Fig. 6a reveals that the alongshore transportJ37Nis controlled
FIG. 5. (a) A schematic of adjacent analysis and forecast cycles. Shown are a 14-day forecast and a 7-day forecast initialized at the end of consecutive 4D-Var analysis cycles ending on days tj0and tj01 7, respectively, with both forecasts verifying on the same day, tj01 14. The times tj0, tj01 7, tj01 12, and t0j1 14 are also referred to in the main text as k1Dt, k2Dt, k3Dt, and k4Dt, respectively. (b) As
in (a), but illustrating different 14- and 7-day forecast realizations initialized from the analysis cycles ending on days t0jand tj01 7. The standard deviations (‘‘spread’’) of the 7- and 14-day forecast dis-tributions are denoted as sf7and sf14, respectively.
primarily by errors in the initial conditions, with errors in the surface forcing and open boundary conditions playing a minor role. The expected forecast errors for J37Nare generally largest during fall and winter. On the
other hand, forecast error variance in the cross-shore
transportJ500mis largely a result of uncertainties in the
surface forcing (Fig. 6b). The biggest contributor by far in this case is errors in the alongshore wind stress (not shown), confirming the important role of Ekman trans-port. In this case uncertainties in the initial conditions FIG. 6. Time series (solid line) of (sfJ)2for (a)J37N, (b)J500m, (c)JHC, and (d)JSST. In each
case the contributions to (sfJ)2of uncertainties in the initial conditions (line with open circles), surface forcing (line with closed circles), and boundary conditions (dashed line) are indicated, although in (b)–(d) the boundary condition contribution is negligible and is not shown. (e) Time series showing the contributions of uncertainties in the meridional wind stress (line with closed circles) and surface heat flux component (solid line) to (sfJ)2in the case ofJSST.
account for about 25% of (sfJ)2, and the forecast error variance is typically largest during the spring.
The expected forecast error variances for upper-ocean heat content,JHC, and SST,JSST(Figs. 6c and
6d), are also largest in the summer, particularly during the wind relaxation season in the case ofJSST(Fig. 6d,
note the log scale), and are almost exclusively due to uncertainties in the surface forcing. The alongshore wind stress is generally the largest contributor to (sfJ)2 forJSSTduring the summer, although uncertainties in
the surface heat flux play a significant role also (Fig. 6e), especially during the winter. In the case ofJHC, errors in
alongshore wind stress are the largest contributor to (sfJ)2during summer, while at other times of the year initial condition errors are equally important. The large summertime contributions of surface forcing to forecast error are partly a reflection of the seasonal dependence imposed on the surface forcing prior error variances.
7. Predictability
To quantify the influence of data assimilation on the predictability of the circulation, consider again the overlapping analysis and forecast cycles j depicted in Fig. 5b. Consider now all of the possible realizations of 14- and 7-day forecasts initialized on days tj0and tj01 7, respectively, that verify on day t0j1 14. Each forecast realization is associated with uncertainties in the fore-cast initial conditions, surface forcing, and open bound-ary conditions drawn from a normal distribution with error covarianceDf. The state vectors for a few of the 7- and 14-day forecast realizations are illustrated in Fig. 5b, and are denoted as xf7and x
f
14, respectively. The
difference in the variance of the xf7and x f
14realizations
will be due solely to the assimilation of observations during the interval [tj0, tj01 7], assuming there are no model errors (or the effect of the model error is the same in both ensembles). If we denote the 7- and 14-day forecast variances ofJ as (sf7)2and (sf14)2, respectively, then it can be shown that
(sf14)22 (sf7)25 gT[2(›K/›d)GD 2 (›K/›d) 3 (GDGT1 R)(›K/›d)T ]g , (14) where g5 MTa
å
k4 k5k2(M14) T khk and [k2Dt, k4Dt] [[tj01 7, tj01 14] is the overlapping forecast period. The propagatorM14is linearized about the 14-day forecast
xf14, andMais the propagator for the jth analysis cycle
spanning the interval [tj0, tj01 7]. In the case of R4D-Var used here,Ma[ Mb[ M14, as discussed in section 2.
Given that (sf7)2 and (sf14)2 are measures of the
predictability ofJ for xf7and xf14, then (14) quantifies the change in predictability due to assimilating observa-tions. As noted in section 3, if 4D-Var is run to complete convergence, then (›K/›d) 5 K, the true gain matrix, so using (3) then (14) reduces to (sf14)22 (sf7)25 gTKGDg
as required according to (6).
Figure 7 shows a time series for each pair of consec-utive analysis–forecast cycles of r5 100[(sf14)22 (sf7)2]/ (sf14)2, which represents the percentage change in the forecast uncertainty due to assimilating observations.
The case r5 100% represents the situation where the
uncertainty in the 7-day forecast is reduced to zero by
4D-Var, while r 5 0 corresponds to the case where
4D-Var yields no change in the uncertainty. When r, 0, this indicates that (sf7)2. (sf14)2 and that 4D-Var de-grades the predictability ofJ . Therefore, r . 0 indicates a positive impact of 4D-Var on the predictability ofJ .
Figure 7 shows that r . 0 for all J indicating that 4D-Var increases the predictability of J . For J37N,
JHC, and JSST, the reduction in uncertainty is
sub-stantial, while forJ500mit is more modest. In the case of
JSST, the fraction r indicates a pronounced seasonal
cycle in predictability, and data assimilation has the least impact on the uncertainty during wind relaxation pe-riods, the same periods when forecast errors are typi-cally largest (Fig. 6d). Since the vector (›K/›d)g in (14) resides in observation space, the fraction that each ob-servation platform contributes to r can also be com-puted, and is indicated in Fig. 7 by the colored bars. For the alongshore transport, Fig. 7a shows that satellite SSH observations contribute most to the predictability, while for the cross-shore transport both SST and SSH contribute about equally on average. Not surprisingly, SST exerts the largest influence on the predictability of upper-ocean heat content and SST. However, for allJ , the in situ observations often exert considerable influence on r, despite being only about;10% of the available data (M11b). We note here though that the impact of the observations on r will be influenced by the hypotheses
embodied in D, R, and G. Therefore, for example,
overfitting of the observations or the presence of ob-servations that, for any reason, are incompatible with the model may exert an overly large or undue influence on r.
As noted in the introduction, the use of the adjoint of a data assimilation system has recently found favor in NWP to quantify the impact of the observing system components on individual forecasts (e.g., Langland and Baker 2004; Gelaro and Zhu 2009). The focus of these studies has typically been on assessing the impact of each observation on a measure of forecast error, usually defined as the squared difference between the forecast and the verifying analysis at the same forecast lead time.
In the approach used here, however, we consider the changes that arise in the spread of two theoretical, infinite-sized ensembles by using the adjoint of the 4D-Var algorithm to provide information about the
expected errors that arise from perturbations in d drawn from the appropriate distribution. Thus, the information about the contribution of each observation to the pre-dictability provided by our approach is quite different FIG. 7. Time series of r5 100[(sf14)22 (sf7)2]/(sf14)2for (a)J37N, (b)J500m, (c)JHC, and
(d)JSST. The colored bars show the contribution of each observation platform to r during each
adjacent pair of analysis–forecast cycles. Key: SSH, Aviso SSH (red); SST, blended satellite SST (light green); Txbt, T from XBT casts (dark blue); Tctd, Sctd, T or S from CTD casts (brown and light blue); Targo, Sargo, T or S from Argo drifting floats (orange and yellow); and TOPP, T from tagged elephant seals (purple).
from observation impact information that has been con-sidered in previous studies.
8. Consistency checks
In this section we present a series of consistency and reliability checks for the posterior and forecast errors derived in sections 5–7. It is important to reiterate that Ea andEf given by (10) and (13) will be correct only
if the prior error covariance D and observation
er-ror covarianceR are correctly specified. The consistency checks presented here therefore provide an independent evaluation of the validity of the prior hypotheses about the errors in the background and the observations. a. Prior minus posterior error
The expected covariance of the increments for any analysis cycle is given by h(^z 2 zb)(^z 2 zb)T
i 5 hdzadzaTi 5 D 2 Ea. By the same token, for unbiased
analyses h(Ja2 Jb)2i 5 (sb
J)22 (saJ)2, which shows
that for any analysis cycle (Ja2 Jb) is drawn from
a distribution with a variance (sb
J)22 (saJ)2. Therefore,
if (Ja2 Jb) and (sb
J)22 (saJ)2are consistent with each
other, the single realizations of DJ 5 (Ja2 Jb) for each R4D-Var cycle shown in Fig. 3 should lie within a distribution with a variance (sb
J)22 (saJ)2 given by
(11) and (12).
Figure 8 shows (Ja2 Jb) from each R4D-Var
anal-ysis cycle for the 7-day-average transport, heat content, and SST indices. Also shown in Fig. 8 are the standard
deviation (std) ranges of 61 std and 62 std based on
[(sb
J)22 (saJ)2]1/2in Fig. 4. As noted in section 4, some
of theJ s are biased, so Fig. 8 shows the bias-corrected values of (Ja2 Jb) for each index. For all indices, 35%–
45% of the values of (Ja2 Jb) fall within61 std, and
69%–82% are within 62 std, suggesting that for the
most part, (Ja2 Jb) and (sb
J)22 (saJ)2are fairly
con-sistent. For a large sample size, we might statistically expect;95% of the values of (Ja2 Jb) to fall within 62 std, so the 69%–82% range reported above indicates that some of the hypotheses that underpinD and R are incorrect and must be refined. Figure 8, however, does not offer any guidance on how to do this, only that there is some level of inconsistency.
b. Forecast error
The consistency and statistical reliability of the fore-cast error estimates can be assessed using ideas bor-rowed from probabilistic NWP (e.g., see Buizza et al. 2005). With this in mind, consider again the hypothetical infinite set of all possible forecast realizations in sec-tion 6 drawn from a normal distribusec-tion with error co-varianceDf. If the forecasts are statistically reliable, then
the true ocean state should belong to this same set of forecast realizations. In NWP the verifying analysis for the forecast time interval is usually taken as a surrogate for the truth and should, therefore, be a member of the same distribution as the forecasts that are initialized from the analysis at the end of the previous assimilation window. Therefore, on average, the distance between the mean forecast and the verifying analysis (e.g., the root-mean-square difference) should equal the average distance between the forecast mean and the forecast realizations (i.e., the forecast standard deviation). This situation is illustrated schematically in Fig. 9.
In section 6, the tangent linear assumption was em-ployed to derive the forecast error covariance (13), in which case the mean forecast is simply the unperturbed forecast, xf(see Fig. 9). For each of the circulation in-dices,J , we therefore expect the difference between the unperturbed forecast and the verifying analysis, (Jf2
Ja), to be drawn from a normal distribution with
a standard deviation of sfJ based on (13). SinceJfalso represents the mean forecast, it lies at the center of the forecast distribution, so significant departures of (Jf2 Ja) from the expected distribution indicates that
the verifying analysis,Ja, is not a member of the fore-cast distribution, for which events the system is consid-ered to be statistically unreliable.
Figure 10 shows the bias-corrected (Jf2 Ja) from each
forecast cycle for the 2-day-average transport, heat con-tent, and SST indices. Also shown in Fig. 10 are the ranges 6sf
J and62sJf of Fig. 6 for each cycle. The schematic in
Fig. 9 indicates that for a statistically reliable forecast en-semble, we require (DxTDx)1/2 ’ sf, where Dx is the
difference between the unperturbed forecast state vector and the state vector of the verifying analysis. For the linear scalar functions considered here, statistical reliability re-quires [(Jf2 Ja)2 ]1/25 (
å
k2 k¼k1å
k2 l¼k1Dx T khkhTlDxl)1/2[ j(Jf2 Ja)j ’ sfJfor each forecast cycle. The average
ofj(Jf2 Ja)j over all cycles is referred to as the mean absolute difference (MAD), while the average of sfJ over all cycles represents the mean standard deviation (MSTD), and both are indicated in Fig. 10 for each circulation index. ForJ37NandJ500m, Fig. 10 indicates
that 91% and 97% of the occurrences of (Jf2 Ja),
respectively, fall within6sfJand62sfJ, indicating that (Jf2 Ja) and sf
J are consistent most of the time. On
the whole the system is statistically reliable for these indices since the verifying analysis Ja is statistically indistinguishable from a typical forecast realization,
and MAD; MSTD. On the other hand, Fig. 10 reveals
that only 40%–60% of the (Jf2 Ja) for heat content
and SST fall within the 62sfJ range, indicating that (Jf2 Ja) and sf
Jare inconsistent in these cases. ForJHC
because for most cycles the verifying analysisJa is an outlier. Here, the unreliable nature of the forecast system is due to inappropriate choices ofD and R. Given the large contribution of uncertainties in wind stress and heat flux to the forecast error inJHCandJSST(cf. Fig.
6), it seems likely that these components ofD are good candidates for improvement. However, model error has not been accounted for and may be the cause of much of the bias inJHCandJSST.
c. Posterior error
If sufficient observations are available, the analog ^J ofJ can be evaluated in observation space. If D and R
are correctly prescribed, then following Desroziers et al. (2005) h( ^Ja2 ^Jb)( ^Jo2 ^Ja)i 5 hT
oGE aGT
ho5 (bsaJ)2,
where hois the equivalent of h in observation space. The
scalar functions ^Ja and ^Jb are evaluated using the
posterior estimate (xa) and the prior estimate (xb),
re-spectively, evaluated at the observation points, and ^Jo
is computed from the observations (yo). The variance (bsaJ)2 is the equivalent of (12) only now evaluated in observation space. Thus, h( ^Ja2 ^Jb)( ^Jo2 ^Ja)i
pro-vides a consistency check for (sa
J)2 computed from
(12). Only a single realization of ( ^Ja2 ^Jb)( ^Jo2 ^Ja) is
available during each analysis cycle, but if it is consis-tent with (12), we would expect ( ^Ja2 ^Jb)( ^Jo2 ^Ja) to
FIG. 8. Time series of (Ja2 Jb) (squares) from each R4D-Var analysis cycle for (a)J37N,
(b)J500m, (c)JHC, and (d)JSST. Also shown in each case are time series of6[(sbJ)22 (saJ)2]1/2
(solid lines) and62[(sb
be drawn from a distribution with the variance (sa J)2
shown in Fig. 4.
In the present case, there are only sufficient obser-vations available to estimate ^JSST, the observation
space equivalent ofJSST. Figure 11 showsj( ^Ja2 ^Jb)
( ^Jo2 ^Ja)j for ^J
SSTcomputed from each analysis cycle
for the 7-day-average SST index. Due to the large range in ( ^Ja2 ^Jb)( ^Jo2 ^Ja), the absolute values are plotted
so that a log scale can be used. Also shown in Fig. 11 are the variances (sJa)2and 4(sJa)2computed from (12) for each analysis cycle. Figure 11 reveals that the majority of the occurrences of ( ^Ja2 ^Jb)( ^Jo2 ^Ja) lie outside the
range of 4(sa
J)2, indicating that the hypotheses embodied
inD and R are in need of refinement. d. Caveats
While the diagnostics presented here provide a useful indication of the consistency between the resulting anal-yses and forecast, and the assumptions made about the prior error covariance matrixD and the observation error covariance matrix R, they are probably not infallible.
For example, the covariancesD and R can be tuned to
yield analysis errors and a cost function pattern of be-havior that is closer to optimal, without the necessity of D and R being correct as demonstrated, for example, by Chapnik et al. (2004, 2006) and Desroziers et al. (2009).
9. Summary and conclusions
We have demonstrated how the adjoint of a 4D-Var data assimilation system can be used to compute esti-mates of the expected analysis and forecast error vari-ances of linear functions of the ocean circulation. The use of adjoint 4D-Var in this way is a significant de-parture from methods currently used in meteorology and oceanography for estimating expected errors. Be-cause the number of inner loops used during 4D-Var is typically much smaller than the number of observations, the posterior error covariance estimates based on a sin-gle, low-rank approximation ~K of the true gain matrix do not effectively span the entire observation space, lead-ing to an overestimate of the expected error. Alterna-tively, the adjoint of 4D-Var can be used to compute the sensitivity of the analysis increments to changes in the innovation vector resulting from changes in the control vector. By considering the expectation of all possible realizations of the analysis increments drawn from the appropriate normal distribution, the adjoint of 4D-Var can also be used to compute the posterior error co-variance. Since the combination of the adjoint of 4D-Var and the expectation operator effectively spans the entire observation space, the resulting posterior error covariance estimates are more reliable than those based
on ~K. Due to the computational challenge involved,
our approach is restricted at the present time to com-puting analysis (and forecast) error variance estimates for scalar functions of the state vector. However, the potential for using the adjoint of 4D-Var to estimate the full posterior and forecast error covariance matrices deserves further investigation. The expected error var-iance in quadratic scalar functions can also be estimated using the same approach if some additional assumptions are made. Quadratic functions will be discussed in a fu-ture publication.
In this study we have used the adjoint of ROMS 4D-Var to compute the analysis error variances of four indices that describe different aspects of the central California coast upwelling circulation pattern during a 2.5-yr sequence of 7-day 4D-Var data assimilation cy-cles. It was found that the expected error variances of the analyses from data assimilation are significantly lower than the prior error estimates. This is particularly true in the case of indices for which direct observations are available, as in the case of temperature, where the FIG. 9. A schematic that illustrates an ensemble of 7-day
fore-casts xfof the state vector for the interval [tj0, t0j1 7]. Each ensemble member results from perturbing the forecast initial conditions, surface forcing, and boundary conditions with perturbations drawn from a distribution with covarianceDf. The unperturbed forecast is initialized at time tj0 using the posterior state vector estimate obtained at the end of analysis cycle j2 1 (see Fig. 5a). A few representative ensemble members are indicated, along with the unperturbed forecast, which is also equivalent to the ensemble mean in the case of the tangent linear assumption. Also indicated is the verifying analysis, which is the posterior state vector estimate ob-tained using 4D-Var during the interval [tj0, t0j1 7], namely analysis cycle j (Fig. 5a). The difference between the unperturbed forecast (ensemble mean) and the verifying analysis is indicated asDx, while the standard deviation of the ensemble about the ensemble mean (unperturbed forecast) is indicated as sf. For a statistically reliable forecast ensemble, (DxTDx)1/2 ’ sf. For each circulation index
J referred to in the main text, Jf
is computed from the last 2 days of the unperturbed forecast (i.e., the interval [tj01 5, tj01 7]), while Ja
ratio of the posterior and the prior error variance reflects at least two orders of magnitude reduction.
The expected error variances of forecasts initialized from the 4D-Var analyses were also computed for the four circulation indices. A powerful aspect of the adjoint 4D-Var approach is that the contribution to the forecast error variance of uncertainties in the different com-ponents of the state-vector, surface forcing, and open boundary conditions can be readily computed. While some aspects of the forecast error variance behavior were expected, such as the large contribution of alongshore
wind stress errors on the forecast skill of coastal SST, other findings were not anticipated, such as the apparent lack of impact of surface forcing uncertainties on the alongshore transport, or the negligible role played by errors in the open boundary conditions in all cases.
Another aspect of the adjoint 4D-Var approach with considerable utility is the ability to partition the pre-dictability of the circulation indices (based on the stan-dard deviation of all possible forecast realizations) into the contributions associated with each observation. This is in contrast to previous applications of data FIG. 10. Time series of (Jf2 Ja) (squares) from each R4D-Var analysis cycle for (a)J37N,
(b)J500m, (c)JHC, and (d)JSST. Also shown in each case are time series of6sJf (solid lines)
and62sJf (dashed lines). In addition, the MAD given byjJf2 Jaj over all cycles, and the MSTD given by sJf over all cycles, are indicated. When MAD’ MSTD, the forecast system is considered to be statistically reliable.
assimilation adjoints for computing the impact of ob-servations on estimates of forecast error based on departures from a verifying analysis. In the system con-sidered here, satellite data represent, by far, the largest fraction of available observations (M11b), and as such exert considerable influence on the forecast errors and predictability of the circulation. However, despite the relatively small fraction of subsurface observations, these data also sometimes exert a significant control on the forecast errors and predictability. The partitioning of the change in forecast uncertainty associated with each ob-servation also allows the obob-servations that impact the forecast skill the most to be identified, including those observations that are detrimental to the forecast.
A number of consistency checks were also presented, which provide an independent evaluation of the re-liability of the analysis and forecast error estimates de-rived from the 4D-Var adjoint. On the whole, these checks indicate that the prior estimates of the back-ground error covariance matrixD and observation error
covariance R are in need of refinement. Nonetheless,
despite the obvious and inevitable shortcomings of D
andR, and the coarse resolution of the model used here, the utility of the adjoint 4D-Var approach is obvious, and we believe that the aforementioned conclusions are robust.
If a modular 4D-Var system is already in place, con-struction of the adjoint of 4D-Var is generally not dif-ficult since all of the components required will already exist. In the case of ROMS, the inner-loop structure is composed of separate calls to the adjoint model, prior error covariance model, tangent linear model, and ob-servation error covariance [see Fig. 3 in Moore et al. (2011a)], which facilitates calculation of the action of the stabilized representer matrix (GDGT1 R) on an input vector during R4D-Var. The minimization algorithm is also a separate module in ROMS, and the user may choose between the Lanczos formulation of the con-jugate gradient algorithm or the minimum residual method (minres). Therefore, since the stabilized rep-resenter matrix is symmetric, the only additional model component required in ROMS for the calculations
presented here was the adjoint of the minimization algorithm employed for identifying the minimum of the cost function, which is available for both the Lanczos algorithm and minres. The approach that we are ad-vocating here is therefore practical for implementation in other models.
Acknowledgments. We are grateful for the continued support of the Office of Naval Research (N000140110209, N000140810556), and for support from the National Sci-ence Foundation (OCE-0628690) and the National Ocean Partnership Program (NA05NOS4731242). Any opin-ions, findings, and conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors wish to thank Jim Doyle for the COAMPS surface forcing data, Dave Foley for the blended SST data, and Dan Costa and Patrick Robinson for the TOPP data that were all used in this study. Finally, we wish to thank the four anonymous reviewers, who provided valuable comments and feedback on earlier versions of this manuscript.
APPENDIX
Expected Analysis Error Covariance Estimates In this appendix we will compare the expected analysis error estimates based on the reduced-rank gain matrix ~K
and the adjoint dual 4D-Var operator (›K/›d)T. The
reduced-rank gain matrix ~K is given by ~K 5 DGTV
mWmV T
m, (A1)
whereVmis the matrix of m orthonormal Lanczos vectors,
qi, arising from the m inner loops of dual 4D-Var.
Fol-lowing Paige and Saunders (1975) and El Akkraoui and Gauthier (2010), the weight matrixWm5 (LTm)21~DmQm,
whereLmandQmrepresent the LQ factorization of the
m3 m symmetric tridiagonal matrix Tmwith diagonal
el-ements di5 qTiPqiand off-diagonal elements gi5 (aTiai)1/2,
where ai5 Pqi2 diqi2 gi21qi21andP is the stabilized
FIG. 11. Time series ofj( ^Ja2 ^Jb)( ^Jo2 ^Ja)j (squares) from each R4D-Var analysis cycle for
^
representer matrix introduced in section 1. The matrix ~Dmis
an m3 m diagonal matrix with leading diagonal elements equal to 1 except for the mth element, which is given by Lm,m/(L2m,m1g2m11)1/2, where Lm,m is the mth diagonal
element ofLm. Each Lanczos vector qiobeys the recurrence
relationPqi 5 giqi11 1 diqi1 gi21qi21, where the first
member of the sequence q15 d/jd j. Therefore, since Vm
depends explicitly on d, the analysis increments ~Kd can also be expressed as a functionK(d) as noted in section 2. Using
the Lanczos vector expansion (A1), the tangent lineariza-tion ofK(d) can be expressed as
›K ›d dd[ DGT[VmWmVmTdd1 VmWmdVTmd 1 VmdWmVT md1 dVmWmV T md] , (A2)
where dVmis the perturbation Lanczos vector matrix
arising from the perturbations dd to the innovation FIG. A1. The ratio s2
a/s2bvs the number of inner loops derived using ›K/›d (circles) and using
~K (squares) for (a) J37N, (b)J500m, (c)JHC, and (d)JSST. The horizontal dashed line denotes
the value of s2
