Using Green's Functions to Calibrate an Ocean General Circulation Model

7/27/2019 Using Green's Functions to Calibrate an Ocean General Circulation Model

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Using Greens Functions to Calibrate an Ocean General Circulation Model

DIMITRISMENEMENLIS, ICHIROFUKUMORI, ANDTONGLEE

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California

(Manuscript received 29 June 2004, in final form 27 September 2004)

ABSTRACT

Greens functions provide a simple yet effective method to test and to calibrate general circulation model(GCM) parameterizations, to study and to quantify model and data errors, to correct model biases andtrends, and to blend estimates from different solutions and data products. The method is applied to an oceanGCM, resulting in substantial improvements of the solution relative to observations when compared to priorestimates: overall model bias and drift are reduced and there is a 10%30% increase in explained variance.Within the context of this optimization, the following new estimates for commonly used ocean GCMparameters are obtained. Background vertical diffusivity is (15.1 0.1) 106 m2 s2. Background vertical

viscosity is (18 3) 106

m2

s2

. The critical bulk Richardson number, which sets boundary layer depth,is Ric 0.354 0.004. The threshold gradient Richardson number for shear instability vertical mixing is Ri00.699 0.008. The estimated isopycnal diffusivity coefficient ranges from 550 to 1350 m2 s2, with thelargest values occurring at depth in regions of increased mesoscale eddy activity. Surprisingly, the estimatedisopycnal diffusivity exhibits a 5%35% decrease near the surface. Improved estimates of initial andboundary conditions are also obtained. The above estimates are the backbone of a quasi-operational,global-ocean circulation analysis system.

1. Introduction

General circulation models (GCMs) resolve only aminute fraction of the climate-system degrees of free-dom (e.g., Holloway 1999). Subgrid-scale processes,which are not resolved by these models, must therefore

be represented using statistical or empirical parameter-izations. The discussion herein concerns a method,based on the computation of model Greens functions,for calibrating these parameterizations. For illustrationpurposes, the method is applied to an ocean GCMwithin the context of a global-ocean data assimilationproject.

Example subgrid-scale parameterizations in oceanGCMs are those used to represent the role of eddies,internal waves, small-scale turbulence, etc. For the spe-cific application example discussed here, these pro-cesses have been represented using the isopycnal mix-ing schemes of Redi (1982) and Gent and McWilliams(1990) and the vertical mixing scheme of Large et al.

(1994). These schemes contain empirical diffusioncoefficients, critical Richardson numbers, etc., whosecareful calibration is key to obtaining a realistic repre-sentation of the physical processes that have been pa-rameterized.

The conventional approach for calibrating empiricalparameterizations is to adjust one parameter at a timeusing GCM sensitivity studies and comparisons withdata. But this approach is suboptimal because estimatesof these empirical parameters depend on each other

and on model configuration, initial conditions, surfaceboundary conditions, etc. Therefore an optimal set ofparameters can only be obtained through the simulta-neous adjustment of all of these conditions, a dauntingtask.

A recent study by Stammer et al. (2003) demon-strates that, using the adjoint method, it is possible tosimultaneously adjust the initial and surface boundaryconditions of an ocean GCM in order to fit a widevariety of data products. The above study is being ex-tended to include the estimation of the GCMs mixingcoefficients (D. Stammer 2003, personal communica-tion). Powerful though it is, the adjoint method doeshave some drawbacks: it is computationally expensive,

its implementation is technically demanding, and itdoes not easily accommodate error analysis and chaoticsystems.

The Greens function approach discussed here pro-vides a different set of trade-offs between optimality,computational cost, error description, and ease ofimplementation. Key advantages relative to the adjointmethod are 1) simplicity of implementation, 2) the pos-sibility of obtaining complete a posteriori error statis-tics for the parameters being estimated, and 3) im-

Corresponding author address: D. Menemenlis, Jet PropulsionLaboratory, California Institute of Technology, Mail Stop 300323, 4800 Oak Grove Drive, Pasadena, CA 91109.E-mail: [email protected]

1224 M O N T H L Y W E A T H E R R E V I E W VOLUME133

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proved robustness in the presence of nonlinearities.The major drawback of the Greens function approachis that computational cost increases linearly with thenumber of control parameters. By comparison, the costof the adjoint method, while substantial, is largely in-dependent from the number of control parameters.

More will be said later about these various trade-offs.Greens functions were first used to solve partial dif-

ferential equations describing electrical, magnetic, me-chanical, and thermal phenomena (Challis and Sheard2003). Greens functions have also been used to linear-ize and to solve a wide variety of geophysical inverseproblems (e.g., Fan et al. 1999; Gloor et al. 2001; Grayand Haine 2001; Wunsch 1996; and references therein).Application examples that are closely related to thepresent discussion are those of Stammer and Wunsch(1996) and Menemenlis and Wunsch (1997), in whichmodel Greens functions were used to estimate thelarge-scale Pacific Ocean circulation. What sets apartthe present discussion from the work of Stammer and

Wunsch (1996) and Menemenlis and Wunsch (1997) isthe choice of control parameters. Specifically, modelGreens functions are here used to blend existing esti-mates of initial and surface boundary conditions and toestimate diffusion coefficients, critical Richardsonnumbers, and relaxation time scales.

The Greens function approach is described in sec-tion 2 using, where possible, the notation of Ide et al.(1997). The power of this approach is best illustrated byexample. For this purpose sections 35 discuss the ap-plication of the Greens function approach to improv-ing the estimates of a quasi-operational, global-oceancirculation analysis system. Summary and concludingremarks follow in section 6.

2. Greens function approach

In practice, the Greens function approach involvesthe computation of GCM sensitivity experiments fol-lowed by a recipe for constructing a solution that is thebest linear combination of these sensitivity experi-ments. Technically, Greens functions are used to lin-earize the GCM, and discrete inverse theory is used toestimate uncertain GCM parameters. The followingdiscussion assumes that the reader is familiar with dis-crete inverse theory and its application to geophysicaldata analysis. If not, a brief but excellent introduction is

found in Menke (1989).Algebraically, a GCM can be represented by a set of

rules for time stepping a state vector:

xfti1 Mixfti. 1

For the ocean GCM example, state vector xf(ti) in-cludes temperature, salinity, velocity, and sea surfaceheight on a predefined grid at discrete time ti. FunctionMirepresents the known GCM time-stepping rules, in-

cluding initial conditions, boundary conditions, empiri-cal mixing coefficients, etc.

The discretized dynamics of the true geofluid xt areassumed to differ from that of the numerical model (1)by a vector of stochastic perturbations:

xtti1 M

ixtt

i, , 2

where is a noise process, which is assumed to havezero mean and covariance matrix Q. Vector containsa set of uncertain parameters that can be used as con-trolsfor bringing the GCM simulation closer to obser-vations. For the ocean GCM example, vector includesterms that represent errors in the initial and boundaryconditions and in the empirical mixing coefficients.

The state estimation problem aims to estimate pa-rameters given a set of observations

yo H

xtt0

xttN

, 3

where vector yo represents all available observationsduring the estimation period, t0 ti tN, H is themeasurement function, and residual is a noise pro-cess, which is assumed to have zero mean and covari-ance matrix R. Vectorrepresents measurement errorsand all model errors that are not represented by in(2). For the ocean GCM example, includes variabilitydue to internal waves, mesoscale eddies, tides, etc.

For the Greens function approach, Eqs. (2) and (3)are combined, resulting in

yo G , 4

whereG is the convolution of measurement functionHwith GCM dynamics Mi. Control parameters can beestimated by minimizing a quadratic cost function

J TQ1 TR1, 5

where superscript T is the transpose operator. Equa-tions (4) and (5) are those of the familiar least squaresminimization problem. Complications arise because thedimensions of and of can be huge, because covari-ance matrices Q and R are usually not known, and be-cause functionGis nonlinear. Most practical estimationmethods assume that (4) can be usefully linearizedabout a particular GCM trajectory. If the linearizationassumption holds, (4) simplifies to

yd yo G0 G , 6

where 0 is the null vector, G(0) is the baseline GCMintegration sampled at the data locations, vector yd isthe model-data difference, and G is a matrix whosecolumns are the Greens functions of G. Specifically,thejth column of matrix Gis

gjGej G0

ej, 7

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where ej is a perturbation vector that is everywherezero except for elementj, which is set toej. That is, eachcolumn ofG can be computed using a GCM sensitivityexperiment. Matrix G is called the data kernel becauseit relates the data yd with model parameters . Theminimization of (5) given (6) is a discrete linear inverse

problem with solution

a PGTR1yd 8

and uncertainty covariance matrix

P Q1 GTR1G1. 9

Discrete linear inverse theory is the subject of a vastliterature, and many useful tools exist for deriving andfor analyzing the solutions (e.g., Menke 1989; Wunsch1996; and references therein).

The validity of the GCM linearization can be tested aposteriori by comparing the residual of Eqs. (4) and (6)for a. If the linearity assumption holds, then it is

expected that G(a

) G(0) G

a

Therefore a rea-sonable requirement is that

absGa G0 Ga diagR12, 10

where operator abs() returns a vector that contains theabsolute values of the input-vector elements, and op-erator diag() returns a vector that contains the diago-nal elements of the input matrix. If condition (10) is notsatisfied, it may be possible to further reduce cost func-tion (5) by using an iterative approach. Specifically, theGCM is relinearized about a instead of0, matrix G isrecomputed, and a new solution is sought.

3. Ocean state estimation exampleThe Greens function approach has been applied to

the calibration of a general circulation model, which isused for quasi-operational analysis of the time-evolvingocean circulation. This analysis is a product of the con-sortium for Estimating the Circulation and Climate ofthe Ocean (ECCO), it is maintained at the Jet Propul-sion Laboratory (JPL), it is updated approximatelyonce per week, it is freely available (http://ecco.jpl.nasa.gov), and it is being used for a variety of science appli-cations (e.g., Dickey et al. 2002; Fukumori et al. 2004;Gross et al. 2003; Lee and Fukumori 2003; McKinley etal. 2003). The discussion that follows is not meant to bethe definitive description of the ECCO/JPL ocean cir-culation analysis; it is only meant to provide a concreteexample for the application of the Greens function ap-proach.

a. Baseline 19912000 integration

The ECCO/JPL nearreal time analysis is based on aquasi-global configuration of the Massachusetts Insti-tute of Technology General Circulation Model (MITGCM; Marshall et al. 1997). The model grid has 360

224 horizontal grid cells. Zonal grid spacing is 1 oflongitude. Meridional grid spacing is 0.3 of latitudewithin 10of the equator and increases to 1latitudeoutside the Tropics, as shown on the left panel of Fig. 1.There are 46 vertical levels with thicknesses rangingfrom 10 to 400 m down to a maximum depth of 5815 m,

as shown on the right panel of Fig. 1. Figure 2 shows themodel bathymetry. Ocean regions north of 73N andsouth of 73S are not represented in order to permit a1-h integration time step. The model employs the K-Profile Parameterization (KPP) vertical mixing schemeof Large et al. (1994) and the isopycnal mixing schemesof Redi (1982) and Gent and McWilliams (1990) withsurface tapering as per Large et al. (1997). Laplaciandiffusion and friction are used except for horizontalfriction, which is biharmonic. Lateral boundary condi-tions are closed. No-slip bottom, free-slip lateral, andfree surface boundary conditions are employed. Sur-face freshwater fluxes are applied as virtual salt fluxes.

The baseline integration spans 1991 to 2000 and is

forced at the surface with 12-hourly wind stress andwith daily heat and freshwater fluxes from the NationalCenters for Environmental Prediction (NCEP) meteo-rological reanalysis (Kistler et al. 2001) with the follow-ing modifications:

1) The 198097 time-mean NCEP fluxes are subtractedand replaced with the 194593 time-mean Compre-hensive OceanAtmosphere Data Set (COADS)fluxes (Woodruff et al. 1998).

2) The 194593 time-mean COADS heat and freshwa-ter fluxes have further been adjusted so that thespatial integral is zero over the model domain.

3) Model sea surface temperature (SST) is relaxed to

FIG. 1. The ocean GCM has 360 zonal by 224 meridional by 46vertical grid cells. Zonal grid spacing is 1. (left) Meridional gridspacing as a function of latitude, and (right) level thickness as afunction of depth.



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NCEP SST using thedQ/dTformulation of Barnieret al. (1995), where Q is heat flux and Tis SST.

4) Shortwave radiation is depth-penetrating using theformula of Paulson and Simpson (1977).

5) Any model temperature that becomes less than1.8C is reset to 1.8C in order to simulate thefreezing of seawater.

6) Sea surface salinity (SSS) is relaxed to monthlymean SSS from the National Oceanographic DataC e nt e r ( N OD C ) W o rl d O c e an A t la s 1 9 9 8(WOA98) with a relaxation constant of 60 days.

Isopycnal diffusivity and isopycnal thickness diffusiv-ity is 500 m2 s2. Hereinafter, isopycnal diffusivity alsorefers to isopycnal thickness diffusivity, which is set tothe same value. Vertical diffusivity is 5 106 m2 s2.Horizontal and vertical viscosity are 1013 m4 s1 and104 m2 s2, respectively. The model is initialized fromrest and from the WOA98 temperature and salinity cli-matology and integrated for 10 yr using the 198097mean NCEP seasonal cycle. It is then integrated fromJanuary 1980 to December 1990 using real-time fluxes

to obtain January 1991 initial conditions for the base-line integration. These particular choices need not befurther justified here, since they are superseded later inthis manuscript using the Greens function approach.Suffice to say that they were the result of dozens oftrial-and-error experiments, over the course of severalyears, by a handful of experienced physical oceanogra-phers.

b. Data used to constrain the baseline integration

The data that are used to constrain the baseline in-tegration are observations of sea surface height vari-ability and a collection of vertical temperature profiles.

Sea surface height data are from the National Aero-nautics and Space Administration Goddard SpaceFlight Center (NASA GSFC) Pathfinder TopographicOcean Experiment (TOPEX)/Poseidon Altimetry Ver-sion 9.1 (http://podaac.jpl.nasa.gov). Specifically, colin-ear sea surface height data are used, which are georef-erenced to a specific ground track and are given at 1-sintervals, approximately every 6 km along each track.The data are corrected for all known geophysical, me-dia, and instrument effects, including tides and atmo-

spheric loading. The Pathfinder data are further binaveraged along each track, consistent with the modelresolution.

Vertical temperature profile data from expendablebathythermograph (XBT) and from the Tropical At-mosphere Ocean (TAO) array are processed, quality

checked, and made available by D. Behringer (2002,personal communication). These data are comple-mented with temperature profiles from the WorldOcean Circulation Experiment (WOCE), from theHawaii Ocean Time Series (HOTS), from the Ber-muda Atlantic Time Series (BATS), and from Profil-ing Autonomous Lagrangian Circulation Explorer(PALACE) floats. Figures 3 and 4 show, respectively,the horizontal and vertical data distributions. For thisstudy, the temperature data are bin averaged insideeach model grid box and for 10-day intervals spanning1 January 1992 to 31 December 2000. In all there are498 277 vertical temperature profiles, which are bin av-eraged into 5 227 445 spacetime bins.

c. Sequential smoother and adjoint method

The baseline integration described in section 3a wasinitially constrained with the data of section 3b usingthe partitioned sequential smoother of Fukumori(2002). As currently implemented, the Fukumori (2002)smoother is used to estimate adiabatic corrections dueto errors in the time-varying surface wind stress. Butthe smoother has not yet been extended to handlemodel biases or to correct errors in surface heat andfreshwater fluxes and in diabatic processes.

A powerful methodology for removing model biasesand for correcting errors in surface heat and freshwaterfluxes and in diabatic processes is provided by the ad-

joint model (e.g., Stammer et al. 2003). But at the timethat this work was carried out, the available computerresources were insufficient for a complete 19912000adjoint-model optimization using the model configura-tion just described. Based on the experience of Stam-mer et al. (2003), a complete adjoint-method optimiza-tion may have required the equivalent of some 500 for-ward-model integrations over the 19912000 estimation

FIG. 2. Model bathymetry in km. The ocean domain spans 73 Sto 73N and excludes the Arctic Ocean.

FIG. 3. Horizontal distribution of temperature profiles. Blackdots indicate locations of XBT and TAO profiles. Blue dots in-dicate locations of WOCE profiles; red dots indicate locations ofPALACE profiles; and green and magenta dots indicate locationsof HOTS and BATS profiles, respectively.



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period, that is, approximately 100 forward- and adjoint-model integrations, with each adjoint-model integra-tion requiring approximately 4 times as much time tocomplete as a forward-model integration. Also thecomputer memory and disk storage requirements for anadjoint-model optimization are larger, typically by afactor of 10, or more, than those of the forward inte-gration. This is because of the need to store intermedi-ary model variables in order to reduce recomputations(Heimbach et al. 2002).

It should also be pointed out that the particular GCMconfiguration, which is used to carry out the work de-

scribed here, does not have a well-defined tangent lin-ear for periods longer than about 10 days. This is be-cause of sensitivity issues related to the vertical andisopycnal mixing parameterizations. Therefore it is notpossible to directly apply the adjoint-model method:some modifications or simplifications of the GCM codeare required. The Greens function approach is morerobust, as is demonstrated below, because it relies on anapproximate linearization of the GCM, not on the exacttangent-linear model.

Finally, the development of a partitioned smoother

or of an adjoint-method optimization requires substan-tial manpower and expertise. By comparison, the com-putation of model Greens functions is straightforward.A model Greens function is derived by perturbing amodel parameter relative to the baseline integrationand then integrating the model forward from 1991 to

2000. That is, the computation of a model Greens func-tion is equivalent to the computation of a model sensi-tivity experiment.

All the above reasons motivated the development ofthe Greens function approach, which is described next,as a way to remove model biases and to correct errorsin surface heat and freshwater fluxes and in diabaticprocesses for the ECCO/JPL ocean circulation analysis.

4. A first Greens function optimization

A first test of the Greens function approach is car-ried out using six sensitivity experiments. For experi-ments 13 the baseline 19912000 integration of section3a is repeated with perturbed vertical diffusivity, verti-

cal viscosity, and isopycnal diffusivity coefficients, asindicated in Table 1. For experiment 4, the time-meanwind stress of the baseline integration is replaced by atime-mean wind stress derived from NASA quick scat-terometer (QuikSCAT) data (W. Tang 2002, personalcommunication). For experiment 5, a temperature per-turbation is generated by optimal interpolation (OI) ofthe observed model-data difference and added to the1991 initial conditions. For experiments 6, the model isreinitialized in 1991 from the January WOA98 tem-perature and salinity climatology. In terms of the nota-tion of section 2,G(0)in (7) corresponds to the baseline19912000 integration sampled at the locations andtimes of the temperature data; G(ej) in (7) represents

the six sensitivity experiments of Table 1, also sampledat the locations and times of the temperature data. Thedata kernel matrix, G in (6), (8), and (9), is a tall, skinnymatrix, with six columns and a number of rows equal tothe number of data, 104 randomly selected observationsout of the total of 5 106 bin-averaged temperatureobservations.

a. Cost function

An important step for optimization studies is thedefinition of cost functionJin (5) and, in particular, the

FIG. 4. Vertical distribution of available temperature data.

TABLE1. List of sensitivity experiments for the first Greens function optimization. Column 3 lists the baseline parameters. Column4 lists the perturbed parameters for each of six sensitivity experiments. Columns 57 list the optimized parameters and uncertainty forthree different cost functions. For experiments 46, the optimized parameters are indicated in terms of fractional perturbation QSCAT-COADS, SPINUP-OI, and SPINUP-WOA98, respectively.

Expt Parameter Baseline Perturbation Case 1 Case 2 Case 3

1 Vertical diffusivity (106 m2 s2) 5 10 15.1 2 15.2 0.8 15.4 0.82 Vertical viscosity (106 m2 s2) 100 200 68 60 59 22 46 283 Isopycnal diffusivity (m2 s2) 500 400 605 48 592 18 572 264 Time-mean wind stress COADS QSCAT 0.22 0.18 0.21 0.06 0.43 0.085 Initial temperature SPINUP OI 0.10 0.14 0.17 0.06 0.11 0.086 Initial temperature and salt SPINUP WOA98 0.75 0.14 0.67 0.06 0.71 0.08



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specification of prior error covariance matrices Q and R(e.g., Menemenlis and Chechelnitsky 2000). For theGreens function approach, the number of observationsis generally much larger than the number of parametersbeing estimated, which simplifies this task. First, thesmall number of control parameters limits the solu-

tions degrees of freedom; therefore the choice ofQ andofR, if they are reasonable, is not expected to changethe solution much. Second, the data kernel matrix G issmall enough to be defined explicitly; therefore manyinteresting properties of the solutionfor example, themodel resolution matrixcan be derived and evalu-ated. Third, the solution of (8) and (9), once the kernelmatrix G has been derived, is trivial; therefore it is pos-sible to test the impact of particular choices ofQ and ofR, as is done next.

For the first Greens function optimization we testthree different cost functions. In all three cases the formof the cost functions is

Ji

yio xi

i 2

, 11

where yoi represents temperature data, xi is the modelestimate, 2i is the assumed data error variance, andsubscripti represents a specific location and time. Costfunction (11) implies that the data-error covariancematrixR is diagonal and that there is no a priori infor-mation about the parameters to be optimized, that is,Q1 0. The assumption of diagonal R is justified be-

cause the temperature data, which are already bin av-eraged inside each model grid box and for 10-day in-tervals, are further decimated so that each optimization

is carried out using 104

randomly selected observationsout of the total of 5 106 bin-averaged temperatureobservations. The agnostic assumption thatQ1 0 haslittle impact on the solution because the minimizationproblem is highly overdetermined.

The three cost functions that are tested are labeledcases 13. For case 1, the a priori error variance, 2i in(11), is assumed horizontally homogeneous and equalto the data variance at each depth, as shown in Fig. 5.This assumption is a conservative upper bound for dataand model representation errors. For case 2 the a priorierror variance is assumed horizontally homogeneousbut equal to the variance of the model-data differenceat each depth, also shown in Fig. 5. Finally, for case 3

the mean a priori error variance at each depth is alsoproportional to the variance of the model-data differ-ence, as is that of case 2, but it is scaled horizontally bythe sea level anomaly variance observed by TOPEX/Poseidon (Fig. 6). For case 3, the global-mean a priorierror variance is further scaled using the following adhoc approach. Two hundred different estimates are ob-tained using two hundred randomly sampled subsets ofthe data. The global-mean a priori error variance isthen adjusted so that the a posteriori uncertainty vari-

ance of the estimatesthat is, the diagonal elements ofmatrixP in (9)are approximately equal to the diago-nal elements of cov(a), the sample covariance of thetwo hundred estimates.

Columns 57 in Table 1 list the optimized parametersand the uncertainties for the three different a prioriassumptions described above. Uncertainty here refersto twice the square root of the diagonal elements ofmatrix Pin (9), the 95% confidence level if the errors

are normally distributed. Admittedly, all three a priorierror variance estimates are ad hoc. What matters forthe present discussion is that the three cases are differ-ent, ranging from likely upper bound (case 1), lowerbound (case 2), and something in between (case 3). Yetall three cases give similar estimates; the error barsoverlap at the 95% confidence level. In particular, theoptimized estimates of vertical viscosity and diffusivityand also of initial temperature and salinity conditions

FIG. 6. Variance of sea level anomaly in cm2 observed by theTOPEX/Poseidon altimeter during the 19932000 period. Thismap provides horizontal scaling for the case-3 a priori error vari-ance, which is used for weighting the data errors in the cost func-tion.

FIG. 5. Assumed vertical profiles of a priori error variance. Case1 is the mean data variance at each depth. Case 2 is the meanvariance of the modeldata difference at each depth. Case 3 isproportional to the variance of the modeldata difference but hasfurther been scaled as described in the text.



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are radically different from those that are used in thebaseline integration.

Because of coarse resolution, artificial northernboundary conditions, and lack of an interactive sea icemodel, the present model configuration is not expectedto be very realistic at high latitudes. Therefore, for the

remainder of this article, we use the spatially varyingdefinition for the a priori error variancethat of case 3,which downweighs the high latitudes.

b. Linear approximation

The fundamental assumption that underlies theGreens function approach is that the estimation prob-lem can be linearized relative to the baseline integra-tion. That is, the optimal solution can be obtained as alinear combination of the baseline and sensitivity ex-periments. The extent to which this assumption is validcan be evaluated by comparing the optimal linear com-bination of the baseline and sensitivity experimentswith a new model integration, which is carried out using

the optimized parameters.Assuming linearity, the expected cost function reduc-

tion relative to the baseline integration is 30% for thecase-3 parameters, those of column 7 in Table 1. Theactual cost function reduction, when the case-3 param-eters are used to carry out a new model integration, is33%. This is 3% better than what would be expectedfor a perfectly linear problem. While in general we donot expect such substantial cost function reduction, thispreliminary optimization demonstrates that exact lin-earity is not required for the Greens function approachto work and that the optimization of a small number ofcarefully chosen parameters can have a large positiveimpact on the solution.

On average, for the first Greens function optimiza-tion, the errors due to nonlinearity are approximately25% of the assumed a priori errors in the data and inthe model; that is, the right-hand side of (10) is approxi-mately 4 times larger than the left-hand side. Thereforethe linear approximation is satisfied and no further it-erations are needed in order to optimize the six param-eters listed in Table 1.

c. Linear dependence

Once the linear approximation has been validated,the kernel matrix G, which is explicitly computed in the

Greens function approach, can be used to ask manyinteresting and importantwhat ifquestions. This ca-pability is a key advantage of the Greens function ap-proach. Below we use G to determine the consequencesof estimating the parameters of Table 1 one at a time,to determine the relative contribution of each param-

eter to cost function reduction, and to infer the robust-ness of the estimates that have been obtained.

Table 2 lists estimates from one-at-a-time optimiza-tions and compares the results to those of case 3. Thetable shows that the one-at-a-time estimates differ sub-stantially from those of case 3. This is because the pa-rameter estimates are linearly dependent on eachother. Therefore the parameters cannot be estimatedindependently. Note that the largest impact on costfunction reduction comes first from the vertical diffu-sivity parameter and second from the initial conditions.This will be explained in sections 5f and 5g as resultingprimarily from reduction of drift in the upper pycno-cline and from compensation of model bias accumu-

lated in that same region during model spinup.To gauge the relative contribution of each parameter

to cost function reduction, additional optimizations arecarried out using only five out of the six possible pa-rameters. The results of these optimizations are sum-marized in Table 3. The table shows that by optimizingonly five of the six parameters, the cost function reduc-tion ranges from 19.8% to 29.7% as compared to 29.8%for case 3, in which all six parameters are optimized. Inorder of decreasing importance for cost function reduc-tion, the parameters are 1) vertical diffusivity, 2) initialconditions, 3) time-mean wind stress, 4) isopycnal dif-fusivity, and 5) vertical viscosity.

The optimizations summarized in Tables 2 and 3 can

also be used to gauge the likely impact of increasing thenumber of control parameters, that is, the number ofdegrees of freedom of the optimization. For example,one may infer that the estimate of vertical diffusivity isrelatively robust since its range is limited: 15.0 106

to 17.4 106 m2 s2. By comparison the estimate ofvertical viscosity is not very robust since it ranges from6.0 106 to 348 106 m2 s2.

5. A second Greens function optimization

The encouraging results from the six-parameter op-timization discussed above motivated the computation

TABLE2. Optimized parameters for case 3 (Table 1) are compared to parameters estimated one at a time. The last row displays thecost function reduction in percent assuming that the problem is linear. Because the parameter estimates are linearly dependent, theone-at-a-time estimates differ substantially from those of case 3.

Parameter Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9

Vertical diffusivity (106 m2 s2) 15.4 17.4 Vertical viscosity (106 m2 s2) 46 348 Isopycnal diffusivity (m2 s2) 572 399 Time-mean wind stress 0.43 0.72 Initial temperature 0.11 0.60 Initial temperature and salt 0.71 2.5Cost function reduction (%) 29.8 19.4 0.58 0.14 5.42 6.46 14.2



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of 20 additional model sensitivity experiments. Theseadditional experiments are summarized in Table 4.Note that these 20 new sensitivity experiments werecomputed relative to the case-3 solution of the firstGreens function optimization and that they include arepeat of all six sensitivity experiments listed in Table 1.The end result of this second optimization is a further10% cost function reduction, as indicated in Table 5.

The resulting estimates of vertical mixing coefficients,surface heat and freshwater fluxes, isopycnal diffusivity,surface wind stress, and initial conditions are discussedbelow, followed by an analysis of improvements in bias,drift, and explained variance relative to earlier solu-tions and to data.

a. Vertical mixing

Sensitivity experiments 14 in Table 4 pertain to therepresentation of vertical mixing in the model. Noticethat the background vertical diffusivity, which had beendeemed a relatively robust estimate in the earlier dis-cussion, remains unchanged with a value of (15.1 0.1)106 m2 s2. This value is consistent with inferencesfrom microstructure and tracer studies (e.g., Kelley andVan Scoy 1999, and references therein).

The estimate of background vertical viscosity is (18 3) 106 m2 s2, which is approximately 6 timessmaller than the value of 104 m2 s2, which is oftenused for ocean modeling (e.g., Large et al. 2001). Apossible explanation for this difference is that the op-timal background vertical viscosity is strongly depen-dent on the values of other model variables, in particu-lar on the values of vertical and isopycnal diffusivity.

Two additional parameters of the Large et al. (1994)KPP scheme, Ric and Ri0, have been estimated; Ric isthe critical bulk Richardson number, which sets thedepth of the oceanic boundary layer. The estimate of0.354 0.004 is 18% larger than the value suggested byLarge et al. (1994). This compensates, in part, for shal-low boundary layers depths in the baseline integrationrelative to the data; Ri0 is a threshold gradient Rich-

ardson number for shear instability vertical mixing,which is especially important for equatorial dynamics.The estimated value of 0.699 0.008 is the same as thatsuggested by Large et al. (1994).

b. Surface heat and freshwater fluxes

Experiments 5 and 6 are used to adjust the surfacesalinity and temperature relaxation terms. The esti-mates shown in Table 4 indicate that the baseline valuesof the relaxation coefficients are too weak for salinityand too strong for temperature. Figure 7 compares themean and standard deviation of the resulting estimatesof surface heat and freshwater fluxes with those fromthe NCEP reanalysis. The corrections to the time-meansurface fluxes are substantial, up to 100 W m2 for heatand 2 m yr1 for freshwater, which are values compa-rable to the time-mean fields themselves.

It is interesting to compare the estimated time-meansurface flux corrections (Figs. 7b and 7f) to the esti-mates obtained independently by Stammer et al. (2004,their Fig. 3) using the adjoint method. Except for theequatorial Pacific, the similarities of the two estimatesboth in pattern and in magnitude are striking. The prin-cipal differences between the two estimates occur near

TABLE3. Optimized parameters for case 3 (Table 1) are compared to estimates for optimizations where one of the six parametersis not used. The last row displays the cost function reduction in percent assuming that the problem is linear.

Parameter Case 3 Case 10 Case 11 Case 12 Case 13 Case 14 Case 15

Vertical diffusivity (106 m2 s2) 15.4 15.0 15.2 16.4 15.5 16.9Vertical viscosity (106 m2 s2) 46 115 6 54 47 41Isopycnal diffusivity (m2 s2) 572 540 599 579 569 571

Time-mean wind stress 0.43 0.64 0.42 0.44 0.43 0.38Initial temperature 0.11 0.23 0.10 0.08 0.13 0.42Initial temperature and salt 0.72 0.90 0.72 0.71 0.69 0.76 Cost function reduction (%) 29.8 19.8 29.5 29.3 27.9 29.7 24.5

TABLE4. List of sensitivity experiments and optimized parameters for the second Greens function optimization. For experiment 6,the optimized parameter is indicated as a factor multiplying the Q/T fields of Barnier et al. (1995).

Expt Parameter Baseline Optimized

1 Vertical diffusivity (106 m2 s2) 5 15.1 122 Vertical viscosity (106 m2 s2) 100 17.7 3.03 Ric, boundary layer depth 0.300 0.354 0.0044 Ri0, shear instability 0.700 0.699 0.0085 Salinity relaxation (days) 60 44.5 1.26 Temperature relaxation (Q/T) 1.000 1.630 .008

710 Isopycnal diffusivity (m2 s2) 500 Linear combination1114 Surface wind stress NCEP/COADS Linear combination1520 Initial conditions SPINUP Linear combination



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the equator where the meridional grid spacing of thepresent study is higher than that of Stammer et al.(2004), that is, 35 km instead of 110 km. A detailedcomparison of the two solutions is in preparation. Boththe present results and those of Stammer et al. (2004)suggest that NCEP reanalysis heat and freshwaterfluxes are suboptimal surface boundary conditions forocean modeling.

c. Isopycnal diffusivity

Isopycnal diffusivity is estimated as the linear com-bination of four sensitivity experiments. The objectiveis to obtain a crude estimate of the time-independenthorizontal and vertical variations of this parameter.

The first sensitivity experiment is a constant perturba-tion, similar to experiment 3 in the first Green s func-tion optimization. The second experiment is a verticallyhomogeneous but spatially varying perturbation. Fol-lowing the suggestion of Holloway (1986) the spatialvariation of this perturbation is proportional to gh/ | f | ,whereg is the acceleration of gravity, h is the standarddeviation of observed sea surface height variations afterremoving tidal effects and the seasonal cycle, and f2 sin is the Coriolis parameter, where is the

FIG. 7. Comparison of estimated surface heat and freshwater fluxes with NCEP reanalysis for the 19932000 period: (a) estimate ofmean heat flux entering the ocean; (b) (a) minus NCEP; (c) standard deviation of estimated surface heat flux; (d) standard deviationof difference with NCEP; (e) estimate of mean evaporation minus precipitation minus runoff; (f) (e) minus evaporation plus precipi-tation from NCEP; (g) standard deviation of the estimated evaporation minus precipitation minus runoff; (h) standard deviation of thedifference with NCEP. Units are W m2 for heat and m yr1 for freshwater.

TABLE5. Cost function reduction relative to baselineintegration.

Integration Cost function Reduction (%)

Baseline 5606 Optimization 1 3773 32.7Optimization 2 3191 43.1



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earths rotation rate and is latitude; in the Tropics,23S to 23N, |f| is set equal to 5.7 105 s1 for thiscomputation. The third and fourth sensitivity experi-ments are also spatially varying as per Holloway (1986)but with exponentially decaying amplitude in the ver-tical, exp(z/500) and exp(z/1000), respectively,

wherez is the depth in meters.Figures 8 and 9 display, respectively, a horizontal

map of estimated isopycnal diffusivity at 1000-m depthand vertical profiles of minimum and maximum diffu-sivity. The estimates range from 550 to 1350 m2 s2 andstraddle the value of 800 m2 s2 suggested by Large etal. (1997) but are considerably lower than the 1500 to4000 m2 s2 range that had been inferred by Holloway(1986) using satellite altimeter data. In the vertical, theestimates exhibit a 5%35% decrease near the surface.This is contrary to the a priori expectation that theestimated isopycnal diffusivity coefficient would belarger near the surface where the eddy kinetic energy ishigher. The estimated decrease in near-surface isopyc-

nal diffusivity is in addition to the Large et al. (1997)surface-tapering scheme, which has also been applied inthis study.

d. Surface wind stress

Surface wind stress is estimated as a linear combina-tion of four sensitivity experiments. The first two ex-periments perturb the time-mean wind stress while pre-serving the variability of the NCEP reanalysis. The nexttwo experiments perturb the time-variable wind stress.Note that in the baseline integration, the time-meanNCEP wind stress has already been replaced with atime-mean wind stress derived from the COADS cli-

matology, as discussed in section 3a.Specifically, the first sensitivity experiment, labeledQSCAT, replaces the time-mean wind stress of thebaseline integration with a time-mean wind stress de-rived from QuikSCAT data (W. Tang 2002, personalcommunication). The second sensitivity experiment, la-beled ERSMEAN, replaces the time-mean wind stresswith a wind product derived from European RemoteSensing (ERS) satellites and obtained from the ERSProcessing and Archiving Facility (CERSAT) at theFrench Research Institute for Exploration of the Sea(IFREMER). The third sensitivity experiment, labeled

ERS, includes both the time mean and the time vari-ability of the CERSAT wind product. Finally, thefourth sensitivity experiment, labeled SM, replaces thetime-variable winds with those estimated by the ap-proximate smoother described in section 3c.

The optimal surface wind stress estimate is

r,t r r,t, 12

where

r 0.55 COADS 0.56 ERSMEAN

0.11 QSCAT 13

is the time-mean wind stress,

r,t 1.02 SM 0.41 ERS 0.43 NCEP 14

is the time-variable wind stress, and r and tare spaceand time coordinates, respectively. The wind stress es-timates are compared to the NCEP reanalysis in Fig. 10.The wind stress estimates can also be compared tothose obtained independently by Stammer et al. (2004;Fig. 9) using the adjoint method. In the large scale, bothestimates show an increase in the trade winds over thetropical Pacific and a weakening of the midlatitudewinds, especially above the Southern Ocean. In termsof meridional wind stress changes, both estimates indi-

cate a smaller poleward component at latitudes higherthan 30N. The principal differences between the twoestimates is that the adjoint-model solution containsmany small-scale wind stress correctionsespecially inwestern boundary current regions and above the Ant-arctic Circumpolar Currentthat are not present in theGreens function solution.

The estimated time variability of the surface windstress is very similar to that estimated by the smoother,SM in (14). This is an indication of the consistency and

FIG. 8. Estimated isopycnal diffusivity in m2 s1 at the1000-m depth.

FIG. 9. Vertical profile of estimated isopycnal diffusivity.



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quality of that estimate. But there nevertheless aresome small corrections to the SM wind stress variabilitythat improve the explained sea level variance of theGreens function solution relative to the smoother so-lution, as will be shown in section 3h.

e. Initial conditions

Initial conditions are estimated as a linear combina-tion of the 20-yr spinup integration, labeled SPINUPand described in section 3a, and of six sensitivity ex-periments. The objective is to remove model bias whileminimizing model drift relative to the data. The firsttwo experiments are a repeat of the OI and climato-logical (WOA98) initial condition experiments dis-cussed in section 4, but using the diffusivity, viscosity,

and time-mean wind stress estimated by the firstGreens function optimization, case 3 in Table 1. Athird experiment, labeled WOCE, is initialized from atemperature and salinity climatology derived fromWOCE data (Gouretski and Koltermann 2004).

The most substantial drift during spinup, when the

model is initialized from a climatology, occurs at highlatitudes. In an attempt to minimize this drift, whilepreserving realistic initial conditions in the Tropics, afourth sensitivity experiment, labeled BLEND, is ini-tialized from a blend of WOA98 and SPINUP. Be-tween 20S and 20N, temperature and salinity is set toWOA98 January values. Poleward of 30S and 30N,SPINUP initial conditions are used. There is a gradual,sinusoidal transition between WOA98 and SPINUP inthe latitude bands of 2030N and 2030S.

FIG. 10. Comparison of estimated surface wind stress with NCEP reanalysis for the 19932000 period: (a) estimate of mean zonal windstress; (b) (a) minus NCEP; (c) standard deviation of the estimated zonal wind stress; (d) standard deviation of the difference withNCEP; (e) estimate of mean meridional wind stress; (f) (e) minus NCEP; (g) standard deviation of the estimated meridional wind stress;(h) standard deviation of the difference with NCEP. Units are N m2. Positive values are eastward for zonal wind stress and northwardfor meridional wind stress.



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The fifth sensitivity experiment, labeled SPINUP2, isinitialized from the final conditions of the fourth sen-sitivity experiment and the sixth sensitivity experiment,labeled SPINUP3, is initialized from the final condi-tions of the fifth sensitivity experiment. This procedureprovides some additional degrees of freedom from

which the Greens function minimization can choosesuitable initial conditions.

The optimal initial conditions are

IC 0.53 WOCE 0.30 WOA98 0.18 SPINUP2

0.07 SPINUP 0.05 OI 0.05 SPINUP3

0.08 BLEND. 15

The estimated initial temperature and salinity condi-tions at the 156- and 626-m depths are shown in Fig. 11and are compared to the WOA98 January climatology.

Except in some isolated regions, for example, thenorthern edge of the Antarctic Circumpolar Current,the estimated January 1991 temperature is generallywarmer than WOA98 in the upper ocean. This warmingrelative to WOA98 is most pronounced in the centralequatorial Pacific, almost 2C warmer than WOA98,

and also in the Gulf Stream and Kuroshio regions.The estimates of initial salinity also show some large

differences, up to 0.2 psu, relative to the JanuaryWOA98 climatology. Particularly striking is the plumeof increased salinity at 600-m depth in the Atlantic,flowing out of Gibraltar Strait.

f. Bias

Overall, the Greens function optimization substan-tially improves the time mean, the trend, and the vari-ability of the solution relative to earlier estimates and

FIG. 11. Comparison of estimated 1991 initial conditions with WOA98 Jan climatology: (a) estimate of temperature at 156 m; (b)estimate of temperature at 626 m; (c) (a) minus WOA98; (d) (b) minus WOA98; (e) estimate of salinity at 156 m; (f) estimate of salinityat 626 m; (g) (e) minus WOA98; (h) (f) minus WOA98. Units are C for potential temperature and psu for salinity.



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to data (see Figs. 1215). The following methodology isused to analyze the temperature data, which are sparseand irregularly sampled in space and in time. Themodel estimates are first sampled at the exact locations

and times of the temperature data. The temperaturedata and the model estimates are then binned and ana-lyzed in 20 zonal by 10 meridional grid cells. Globalaverages are weighted by area and are obtained by av-eraging the results of all grid cells that contain morethan 100 temperature samples.

Figure 12a shows that the Greens function optimi-zation has reduced the bias of the previous solutionsrelative to data throughout the entire water column.Notice that although the smoother solution corrects thetemporal variability, it nevertheless has a measurableimpact on the time-mean temperature profile.

The bias reduction of the Greens function solution ismost significant at the base of the equatorial ther-mocline as can be seen by comparing Figs. 13c and 13e.To a large extent this is the result of vertical diffusivitybeing too weak in the baseline and in the smootherintegrations, hence resulting in a thermocline that is toosharp and too shallow relative to data.

Although the bias of the Greens function solution

relative to data is decreased on a global average whencompared to earlier solutions, there are some regionswhere the bias remains significant. One of these regionsis the Indian Ocean, which is too warm by about 1Cin the Greens function solution at 200-m depth. Theseresidual discrepancies contain information about re-

FIG. 12. Comparison of baseline integration (Base), smoothersolution (SM), and Greens function solution (GF) to observedtemperature profiles: (a) global root-mean-square (rms) differ-ence relative to data; (b) global rms drift relative to data; and (c)percent explained variance of the baselinedata difference.

FIG. 13. Time-mean potential temperature, 19932000: (a) Greens function estimate at the equator down to 500-m depth; (b) Greensfunction estimate at the 156-m depth; (c) smoother bias relative to data at the equator; (d) smoother bias relative to data at the 156-mdepth; (e) Greens function bias relative to data at the equator; and (f) Greens function bias relative to data at the 156-m depth. UnitsareC.



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maining model parameterization and boundary con-dition errors. Therefore these discrepancies can guidefuture model-parameterization improvements and/or

the selection of additional model sensitivity experi-ments.

g. Drift

Given the large changes in 1991 initial conditionsrelative to those obtained from model spinup, an im-portant question is whether the bias reduction has

FIG. 15. Percent explained variance of the baseline-data residual for (a) sea surface height variability of the smoother solution, (b)sea surface height variability of the Greens function solution, (c) temperature variability at 156 m of the smoother solution, and (d)temperature variability at 156 m of the Green s function solution.

FIG. 14. Potential temperature trend, 19932000: (a) Greens function estimate at the equator down to 500-m depth; (b) Greensfunction estimate at the 156-m depth; (c) smoother drift relative to data at the equator; (d) smoother drift relative to data at the 156-mdepth; (e) Greens function drift relative to data at the equator; and (f) Greens function drift relative to data at the 156-m depth. UnitsareC yr1.



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taken place at the expense of increased drift in theoptimized solution. Figure 12b shows that, overall, theGreens function optimization has also reduced thedrift of the solution relative to data. The largest driftreduction compared to the prior smoother solution isonce again at the base of the equatorial thermocline, as

can be seen by comparing Figs. 14c and 14e. There are,however, localized regions where model drift relative todata is larger than that of the baseline and of thesmoother integrations, for example, in the northernNorth Atlantic below 1000-m depth. Again these dis-crepancies can guide future improvements of the solu-tion, for example, in the representation of high-latitudeprocesses and in the formation of deep water masses.

h. Explained variance

Owing to the improved estimate of the time-meanstate and to the combination of solutions, the Greensfunction optimization also improves model variabilityof temperature and of sea surface height relative todata compared to both the baseline and to the priorsmoother integrations. Figure 12c shows that overallthe Greens function solution results in a 10%30%increase in explained temperature variance comparedto the earlier solutions. Explained variance is here de-fined as one minus the variance of the analysis-datadifference divided by the variance of the baseline-datadifference:

explained variance 1 varGa yo

varGb yo. 16

The spatial pattern of the explained variance relativeto the baseline integration is shown in Fig. 15 first for

sea level and second for temperature at the 156-mdepth. The smoother solution, Fig. 15a, already ex-plains a large fraction, up to 50%, of the baseline-datadifference for sea level variability. But overall it de-grades the temperature variability, as shown in Fig. 15c.Even though the smoother solution has more realisticheaving of the water column than the baseline integra-tion, the resulting temperature variability is degradedcompared to observations because the time-mean ver-tical temperature gradient is inaccurate. By compari-son, the Greens function solution improves both thetemperature and the sea level variability (Figs. 15b and15d) even though altimetric data have not been used asconstraints.

6. Summary and concluding remarks

The work discussed hereinabove demonstrates thatGreens functions provide a simple yet effectivemethod to test and to calibrate GCM parameteriza-tions, to study and to quantify model and data errors, tocorrect model biases and trends, and to blend estimatesfrom different solutions and data products.

The Greens function method was applied to an

ocean GCM, resulting in substantial improvements ofthe solution relative to observations as compared toprior estimates: overall model bias and drift are re-duced, and there is a 10%30% increase in explainedvariance. Within the context of this optimization, thefollowing new estimates for commonly used ocean

GCM parameters have been obtained. Backgroundvertical diffusivity is (15.1 0.1) 106 m2 s2. Back-ground vertical viscosity is (18 3) 106 m2 s2. Thecritical bulk Richardson number, which sets boundarylayer depth, is Ric 0.354 0.004. The threshold gra-dient Richardson number for shear instability verticalmixing is Ri0 0.699 0.008. The estimated isopycnaldiffusivity coefficient ranges from 550 to 1350 m2 s2,with the largest values occurring at depth in regions ofincreased mesoscale eddy activity. Surprisingly, the es-timated isopycnal diffusivity exhibits a 5%35% de-crease near the surface. Improved estimates of initialand boundary conditions were also obtained. Theabove estimates are the backbone of a quasi-operation-

al, global-ocean circulation analysis system whose prod-ucts are freely available and are being used for a varietyof science applications (http://ecco.jpl.nasa.gov).

There remain many aspects of the above solutionthat can be improved, for example, the warm bias in theIndian Ocean at the 200-m depth, the drift of the solu-tion below 1000 m in the northern North Atlantic, andin general the poor representation of high-latitude pro-cesses and of deep water mass formation rates. Thepuzzling estimates of isopycnal diffusivity are also anoutstanding research issue; additional control param-eters may be required, for example, depth-varyingbackground vertical viscosity and diffusivity to accountfor increased dissipation rates near rough topography

and separate estimates of isopycnal and isopycnal thick-ness diffusivities. The Greens function approach pro-vides a powerful mechanism for addressing the abovequestions and for identifying and for reducing residualmodel-data discrepancies in a physically consistentmanner.

Compared to other methods, the key advantages ofthe Greens function approach are simplicity of imple-mentation, robustness in the presence of nonlinearities,and the explicit computation of the data kernel matrix.While the application of an adjoint model or an ap-proximate smoother require substantial additionalmodel-development and coding efforts, all that is re-quired for applying the Greens function approach is

the computation of GCM sensitivity experiments. Fur-thermore, while the adjoint method requires that theexact tangent linear of the GCM be well behaved,Greens functions provide an approximate lineariza-tion, which can be used to reduce the cost function evenwhen the adjoint model is ill behaved, as is the case inthe above ocean GCM example. Finally, the explicitcomputation of the data kernel matrix makes availablea vast array of tools from discrete linear inverse theoryfor deriving and for analyzing the solutions.



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The key drawback of the Greens function approachis that computational cost increases linearly with thenumber of control parameters. Therefore the method isonly applicable to situations where a small number ofcontrol parameters need to be estimated. Neverthelessthe present work demonstrates that a small number of

carefully chosen control parameters can result in sub-stantial improvement of the solution. For example, onlysix control parameters were used in the first test opti-mization and yet the cost function was reduced by 33%.While in general we do not expect such substantial costfunction reduction to be possible for all problems, it isclear that the Greens function approach is an ex-tremely powerful tool in the repertoire of ocean stateestimation.

What distinguishes the present work from previousapplications of Greens functions to ocean state estima-tion is the choice of control parameters. Previous ap-plications used model Greens functions to obtain acoarse-scale representation of ocean GCM dynamics,

for example, the GCM response to large-scale, geo-strophically adjusted density or sea surface height per-turbations. The breakthrough here is that rather thanusing Greens functions to approximate GCM dynam-ics, Greens functions are instead used to calibrate asmall number of key GCM parameters and to blendestimates from existing solutions and data products.This new approach has the advantage of permitting arelatively large impact on the solution from a smallnumber of control variables. Additionally, the repre-sentation of GCM dynamics is implicit and exact ratherthan explicit and approximate.

Work is underway to apply the Greens function ap-proach to a global, eddy-permitting GCM configuration

that includes the Arctic Ocean and an interactive seaice model. The Greens function approach is also beingapplied to the calibration of an atmospheric GCM andof a coupled oceanatmosphere climate GCM.

Acknowledgments.This is a contribution of the Con-sortium for Estimating the Circulation and Climate ofthe Ocean (ECCO) funded by the National Oceano-graphic Partnership Program. We are indebted to C.Wunsch for teaching us about inverse methods and forsuggesting application of Greens functions to theocean circulation inverse problem.

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Using Green's Functions to Calibrate an Ocean General Circulation Model

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