Adjoint assimilation of altimetric, surface drifter, and hydrographic data in a quasi-geostrophic...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 100, NO. C12, PAGES 25,007-25,025, DECEMBER 15, 1995

Adjoint assimilation of altimetric, surface drifter, and hydrographic data in a quasi-geostrophic model of the Azores Current

Rosemary Morrow and Pierre De Mey

Unit6 Mixte de Recherche 39, Group½ De Recherche de G6od6sie Spatitle, Toulouse, France

Abstract. The flow characteristics in the region of the Azores Current are investigated by assimilating TOPEX/POSEIDON and ERS 1 altimeter data into the multilevel Harvard quasi- geostrophic (QG) model with open boundaries (Miller etal., 1983) using an adjoint variational scheme (Moore, 1991). The study site lies in the path of the Azores Current, where a branch retroflects to the south in the vicinity of the Madeira Rise. The region was the site of an intensive field program in 1993, SEMAPHORE. We had two main aims in this adjoint assimilation project. The first was to see whether the adjoint method could be applied locally to optimize an initial guess field, derived from the continous assimilation of altimetry data using optimal interpolation (OI). The second aim was to assimilate a variety of different data sets and evaluate their importance in constraining our QG model. The adjoint assimilation of surface data was effective in optimizing the initial conditions from OI. After 20 iterations the cost function was generally reduced by 50-80%, depending on the chosen data constraints. The primary adjustment process was via the barotropic mode. Altimetry proved to be a good constraint on the variable flow field, in particular, for constraining the barotropic field. The excellent data quality of the TOPEX/POSEIDON (T/P) altimeter data provided smooth and reliable forcing; but for our mesoscale study in a region of long decorrelation times 0(30 days), the spatial coverage from the combined T/P and ERS 1 data sets was more important for constraining the solution and providing stable flow at all levels. Surface drifters provided an excellent constraint on both the barotropic and baroclinic model fields. More importantly, the drifters provided a reliable measure of the mean field. Hydrographic data were also applied as a constraint; in general, hydrography provided a weak but effective constraint on the vertical Rossby modes in the model. Finally, forecasts run over a 2-month period indicate that the initial conditions optimized by the 20-day adjoint assimilation provide more stable, longer-term forecasts.

1. Introduction

Over the last decade the interest in data assimilation studies

has increased as a direct response to the large amount of oceanographic data now available to scientists, in particular, surface observations from remote sensing satellites and in situ hydrographic measurements from various coordinated international programs. We are still far from the global, synoptic data sets available to meteorologists, but satellite altimeters, such as TOPEX/POSEIDON (T/P), provide good spatial and temporal sampling of the large-scale ocean circulationcomplementary with ERS 1, which is less accurate but has better spatial coverage of the mesoscale. The challenge for oceanographers is how to combine the various data sets that are now available with our improved ocean circulation models.

Our particular interest is to use data assimilation techniques and combine various different data sets with a regional open- ocean model, so that we can better model and predict the ocean dynamics in the Azores Current region. Our study domain lies in the path of the Azores Current as a branch of it reflects to the south in the vicinity of the Madeira Rise

Copyright 1995 by the American Geophysical Union.

Paper number 95JC02315. 0148-0227/95/95JC-02315505.00

(Figure 1). The dynamics of the region are quite complex and have been documented in a number of studies [e.g., Gould, 1985; Klein and $eidler, 1989]; within the domain the predominantly zonal jet interacts with a series of seamounts before turning southward. The variability generated downstream is reasonably large for an eastern boundary region [Le Traon et al., 1990] and includes a large seasonal component and the influence of Rossby waves propagating from the east in winter [Le Traon and De Mey, 1994]. An additional complexity is that subsurface eddies formed from the Mediterranean outflow (meddies) are often located in the region [e.g., Armi and Zenk, 1984].

A subregion of our modeling domain was recently the site for the SEMAPHORE intensive field program in 1993 [Eymard et al., 1991], which included synoptic arrays of hydrographic measurements during three cruise periods and the deployment of surface and sub-surface (RAFOS) floats and long-term current meter arrays. These in situ data can first provide an independent test for our assimilation of surface altimeter data in the regional model, in particular, testing the model performance at depth. We already have some confidence in this, as previous studies have shown how surface altimetry can strongly constrain the eddy field, even at depth [Hurlhurt, 1986; De Mey and Robinson, 1987]. Secondly, we would like to investigate ways of assimilating the different SEMAPHORE data sets directly in our model. In particular, we wish to make use of the surface Lagrangian drifter data (which

25,007

25,008 MORROW AND DE MEY: ADJOINT ASSIMILATION IN THE AZORES CURRENT

40N

30N

20N

40W 30W 20W 10W

Figure 1. The model domain in the Azores Current region, showing the model bottom relief (contour is 200 m). The SEMAPHORE region discussed in the text is also delineated.

contain both the mean and eddy signals) and the subsurface hydrographic data to introduce data constraints at depth.

The dynamics are modeled using the Harvard multilevel quasi-geostrophic (QG) open-ocean model [Miller et al., 1983]. The simple QG level model formulation is suitable for modeling fine-resolution mesoscale processes over domains of O(100-1000 km) and away from regions of strong convection; the model has already been applied to the Gulf Stream region [DeMey and Robinson, 1987; Robinson et al., 1989; Moore, 1991] and to the Athena-88 region of the northeast Atlantic [Dombrowsy and De Mey, 1992]. Note that by choosing a quasi-geostrophic model, thermohaline forced cases, such as meddies, are not well represented. We will see the effect of this when assimilating hydrographic data through a meddy in section 5.

The present study consists of making two assimilation methods work together. The first is the optimal interpolation method [De Mey and Robinson, 1987; Dombrowsky and De Mey, 1992; De Mey, 1992] which is applied to the same model in the same region [De Mey, 1994]. Optimal interpolation (OI) yields robust convergence over long periods of time, but its short-term forecast capabilities are sometimes deceiving. One disadvantage of OI is that it takes

no account of the nature of the dynamics of the system in the interpolation step, which is purely statistical. As a consequence, one has to make sure that the included dynamics (the model integration), the statistical properties of the data, and the included statistics (the analysis step) are consistent with each other, so that the assimilation converges.

In this paper the assimilation is performed using an adjoint variational method, which is a special variational inverse method based on optimal control theory [e.g., Le Dimet and Talagrand, 1986; Thacker and Long, 1988; SchrSter et al., 1993]. Variational methods explicitly make use of the actual model dynamics; the individual data measurements are used to force the model dynamics directly, without needing a separate analysis of the data distribution nor its statistical properties. The adjoint method also works as a global, not a local inverse method, in that it makes use of all available data at once. This means the data forcing is consistent everywhere, but the added computational burden can put a limit on the space-time domain that can be resolved.

We will apply the adjoint method to improve on the OI analyses locally, especially whenever irregularly spaced, dense data sets are available, as in the case of cruises. We use

the adjoint model developed by Moore [1991] for the Harvard

MORROW AND DE MEY: ADJOINT ASSIMILATION IN THE AZORES CURRENT 25,009

QG model, with some additional features that apply to the Azores-Madeira region. These mainly concern incorporating bottom topography in the model, since the interaction with bottom relief is critical in the eastern North Atlantic [Barnier and Le Provost, 1989; Dombrowsky and De Mey, 1992], and this is not fully taken into account in the OI algorithm. Other practical concerns, such as the choice of initial vorticity and bottom density as our control variables for optimization and the appropriate scaling of these variables, are discussed in detail in section 2.

Section 2 also gives the formulation for the different data constraints as applied to the adjoint model for QG dynamics. These include the assimilation of altimeter sea surface

heights, described for the model by Moore [1991]; surface drifter velocities [e.g., Schr6ter et al., 1993]; and assimilating vertical pressure modes derived from the hydrographic data at each conductivity-temperature-depth (CTD) station. The numerical concerns are also discussed here, such as the choice of descent algorithm for finding a minimum solution, preconditioning the descent, and the different weights to be applied to a combination of data constraints. The model results for the three different types of data constraints are discussed separately in sections 3-5, and longer-term forecast runs are presented in section 6.

2. Formulation of the Assimilation Problem

2.1. Adjoint Assimilation

The adjoint variational method is based on ideas of optimal control theory [e.g., Le Dimet and Talagrand, 1986; Thacker and Long, 1988]. The adjoint method produces fields which minimize the distance to the observations and, at the same

time, satisfy the explicit dynamical constraints of the model equations. A detailed description and background on the adjoint method we use are given by Moore [1991].

In brief, our problem is to minimize the "distance" between the model and observation trajectories, which is normally referred to as the cost function J, by tuning our chosen control variables. Control variables can be the initial

or boundary conditions, or indeed any free tunable parameters of the model. We assume that the best estimates of the initial

and boundary conditions of the model are those which minimize the distance J between the two trajectories. The result will be a correction to the trajectory of the model; the new trajectory will also satisfy the same dynamical model equations as the initial trajectory.

This is a constrained minimization problem and solving it directly is a large and complex operation. Instead, the adjoint variational method is used to transform the constrained minimization problem in physical model space into an unconstrained problem in control variable space. The adjoint model is then used to find the gradient of the cost function J.

The variational method is solved as an iterative scheme;

we solve the direct model equations forward in time and calculate the cost function; then solve the adjoint model equations backward in time (subject to the appropriate boundary conditions) to give the adjoint variables and thus the gradient of J with respect to the control variables. Given J and its gradient, we can then use a general unconstrained minimization algorithm (or descent algorithm) to estimate the new control variables. These processes are repeated until we eventually converge toward the minimum in the cost function J.

For the minimization process, numerous algorithms can be used. We have chosen to use the Polak and Ribiere [1969] conjugate gradient method which gives a good rate of convergence toward the minimum of a function, given its gradient. The conjugate gradient method provides the new direction of the minimization, which is conjugate to all previous directions. We then use a line minimization technique to find the optimal distance or step size which will minimize the cost function. Since we only have discrete estimates of the cost function J, we cannot minimize this function using standard procedures for continuous functions. However, Thacker and Long [1988] show that we can calculate the optimal step size from the change in the cost function, albeit, this method requires an additional run of the forward model.

2.2. The Quasi-Geostrophic Model

The numerical model used for this study is the finite element, open boundary, quasi-geostrophic model [Dornbrowsky and De Mey, 1992] based on the original Harvard model of Miller et al. [1983]. The model integrates the quasi-geostrophic equation for the conservation of potential vorticity on the 13 plane, allowing multiple levels in the vertical. The nondimensional QG equations in continuous form are

•+ •)+fl =0 (1) & 8x

(2)

where ¾r is the QG streamfunction, • is the QG dynamic

vorticity, _ or(z)-•.•No 2 / N 2 (z)] is the vertical stratification (N is the Brunt-V•iis•ila frequency and No is its typical scale), and the dimensionless scaling parameters os,•,F 2 are described in detail in Table 1. J(•,•) is the Jacobian operator (•x•y - •y•x)' Equation (1) describes the temporal evolution of the vorticity field. The "diagnostic" equation (2) allows streamfunction to be calculated instantaneously at all levels in the vertical (once the vorticity has been integrated in time) by means of the dynamic Rossby modes.

Discretization in time is by means of a three-stage, second- order Adams-Bashforth time stepping scheme; the equations are discretized horizontally onto a finite element Arakawa B grid [Haidvogel et al., 1980] and vertically by solving a set of Helmholtz equations after projection onto discretized Rossby modes [Miller et al., 1983]. These vertical modes will be used in assimilating hydrographic data (section 5), and De Mey [1994] describes the choice of vertical modes for our level model. Our model domain is an 1125 km x 800 km box

centered at 32øN, 23.5øW (Figure 1). The model parameters and scaling are summarized in Table 1.

Boundary conditions. Top and bottom boundary conditions can be included if wind forcing or bottom topography are important. Local wind forcing in the model domain is not applied directly, rather, the indirect geostrophic response to the large-scale wind forcing (including propagating Rossby waves) is assumed to be included via the assimilation of surface altimeter data and the

specification of the mean climatology. In the eastern Atlantic, bottom topography has been shown to be an important forcing term [Barnier and Le Provost, 1989], so it will be included in our model. The bottom boundary forcing is

25,010 MORROW AND DE MEY: ADJOINT ASSIMB_,ATION IN THE AZORES CURRENT

Table 1. Parameters and Scaling Used for the Harvard Quasi-Geostrophic Level Model

Parameter Value ,

Planetary vorticitYf0 0.771 x 10 -4 s -1 Grad of planetary vorticity ]•0 0.194 x 10 -10 s -1 m -1 Timescale t o 8.0 days

Integration time step 1.5h= 0.0625 days

Horizontal scale D 69.12 km

Horizontal grid spacing 12.5 km x 12.5 km

Horizontal velocity scale V 0 0.1 ms -1 Vertical scale H 900.0 m

Number of vertical levels 5

Midlevel of model variables •, • 200, 625, 125, 1950, 3650 m

Water depth 4800.0 m

Brunt-V/iis•il•i frequency scale, N O 10 -3 s- 1 Bottom friction timescale, /C-1 200.0 days

Shapiro filter (dissipation) order 8 (triharmonic)

Rossby number Ro = (foto) - 1 0.188 x 10-1 ct = Voto D-1 1.0 (ct scaling)

[• = •0t0 D 0.927

r 2 =fo2D2No-2tF 2 35.038

calculated by integrating the nondimensional top and bottom density equations, which have a similar form to (1), i.e., in continuous form

ø•z -osFgcrJ(•, U/z)-J(u/,h)= 0 z=-H (3) -F 2 cr where h is the nondimensional bottom relief.

The open boundary conditions of Charney et al. [1958] are applied to our rectangular domain; i.e., streamfunction is specified everywhere on the boundary, while vorticity and bottom density fluctuations are specified only at inflow points, although they can leave the domain via the radiative boundary conditions. The boundary values are obtained from a previous model run, where mapped altimetric data have been assimilated every 14 days using OI techniques [De Mey, 1992]. Forecast fields are produced every 7 and 14 days from these OI runs; from these boundary values are then interpolated at each model time step. Thus the boundary values will be consistent with the model dynamics and include some additional information from the OI assimilation of

altimeter data.

Choice of control variables. Following Moore [1991], we choose the initial vorticity field as our control variable. Moore found that initial streamfunction was a good choice of control variable when there is adequate data coverage and that it produced a rapid decrease in the cost function. However, when observational data are sparse, excessive vorticity instabilities can develop in the model runs. When initial vorticity is chosen as the control variable, the gradient of the cost function tends to be smoother, since

the streamfunction forcing term is integrated twice in three- dimensional space [Moore, 1991]. This slows the convergence rate of J, but both the streamfunction and vorticity fields are numerically more stable. This is of primary '•nportance in our modeling domain with a full range of spatial scales, so initial vorticity is chosen as our main control variable.

Bottom density anomalies are also integrated in our model in a way akin to vorticity at intermediate levels (compare (1) and (3)); in addition, bottom density anomaly drives the bottom boundary condition, as it contains the nonseparable part of the streamfunction and vorticity terms. As a consequence, we choose to include the initial bottom density anomaly as another control variable; that is, the control vector used in the minimization process will include the initial vorticity at all model levels plus the initial bottom density anomaly field.

Choice of initial conditions. The initial

conditions are of particular importance in our case, since they are our control variables. A good set of initial conditions, which are internally consistent and consistent with the model dynamics, can provide a more stable initial forward model run and can also lead to a faster convergence of the conjugate gradient algorithm to the minimum in J. Luong [1995] has shown that a simple application of the adjoint technique is less efficient in identifying the initial conditions but provides good final conditions for forecasting, hence the need for good initial conditions.

We choose to start the adjoint assimilation from a "first guess" field, obtained from the continuous assimilation of altimeter data using OI techniques [Dombrowslcy and De Mey, 1992]. The OI assimilation of T/P and ERS 1 data was applied to the same model, so the initial conditions will be dynamically consistent with our model run. Furthermore, the initial fields have been obtained after assimilating more than a year of altimeter data using the OI technique. The resulting streamfunction and vorticity fields from OI have stabilized in the deeper levels and show more small-scale structure than can be resolved from the large-scale climatology, such as the Robinson et a/.[1979] altlas for example (hereinafter referred to as RBS).

We use the 7-day forecast fields from the OI assimilation and extract the initial vorticity and bottom density fields. The OI forecast fields are also used to provide boundary conditions at the appropriate times for the forward QG model run. With initial vorticity and bottom density, the boundary conditions, and the diagnostic equation (3) we are able to calculate the streamfunction fields; all fields are necessary for the initialization of the Adams-Bashforth time-stepping scheme. The error variances associated with the OI analysis can also be used as a weighting for the background constraint (see section 2.6).

Adjoint model. The derivation of the adjoint model equations is described in detail in Moore [1991], so it will not be repeated here. However, how the boundary conditions are optimized by the adjoint model is a critical problem and requires some discussion.

The adjoint variables are set to zero outside the time domain of the assimilation period. This ensures that we start from zero adjoint fields at t=tma x before the backward time integration. Also, the "optimal" character of our result pertains only to the model-data misfit fit over our chosen space-time domain. Adjoint boundary conditions for our


open-ocean domain are chosen consistent with the QG model and such that the boundary integrals vanish. Following Moore [1991] we specify (1) diagnostic adjoint variables to be zero on the boundaries and (2) prognostic adjoint variables are zero at QG model outflow points and unconstrained at QG model inflow points. Our gradient with respect to the control variables u at t=0, VuJ, depends on the (integrated) prognostic adjoint variables; so the boundaries of the optimized initial fields have been updated at the (integrated) inflow points. Although the initial boundary conditions are updated at inflow points, the boundary conditions in the forward QG model remain fixed from the OI fields, which imposes a limit on the length of time of our forward integration [Fukornori et al., 1993]. For long integration periods the open boundaries can be included as control variables in the optimization process; this is discussed for the adjoint of a QG model by Seller [1993].

Integration period. In the following case studies we have chosen to integrate the model over a 20-day period, since we are primarily concerned with optimization for short, forward integrations of the QG model, for example, when cruise data are available. In principle, we are also interested in improving our longer-term forecasting capabilities, and a good forecast depends on reducing the errors in the final conditions. A long integration period allows more data to be assimilated, which can be important in constraining the initial conditions in our strongly underdetermined problem; but we are also limited by the predictability timescale of the flow, given the nonlinear dynamics. It has been shown that the number of local minima in J for a nonlinear model will

progressively increase as the period of assimilation increases, so that convergence to a unique solution will become more difficult. This risk is increased in our case by the fact that the boundary conditions are not optimized in our forward QG model runs. In addition, Luong [1995] has shown that there is little improvement to the initial conditions if the integration time is longer than the temporal decorrelation scales for the flow regime. -The temporal decorrelation time for our region is around 1 month [Le Traon and De Mey, 1994], and it is generally believed that the Eulerian predictability timescale for eddy motions in QG dynamics is also O(1 month). This is important, since we have a fixed model domain on an evolving flow, with mean advection by the Azores Current which strongly controls the 1-month Eulerian predictability limit.

In contrast, assimilating over short periods can give excellent results in the surface conditions, but there is also

the issue of the propagation of surface information to depth. Results from the adjoint assimilation of altimetry in a regional QG model with closed boundaries by Luong [1995] have shown that 2-3 months are necessary to constrain the bottom flow when starting from scratch. However, in our case starting from OI fields, which are already consistent with the bottom relief, we do not require such a long vertical propagation time. Tests were carried out using different integration periods of 10-, 20-, 35- and 60-day integration periods. In general, the 10-day integration period had too few data points and incomplete surface coverage to adequately constrain the initial conditions and still provide stable forecasts. The 20-, 35- and 60-day integrations all provided reliable optimized initial conditions, stable forecasts; and qualitatively little difference in the 35-day and 60-day initial fields. Our examples presented here are based on the 20-day

assimilation period, which is the normal duration of our cruises and is long enough to include two complete cycles of TOPEX data and half an ERS 1 cycle. A final factor is the computational cost; 20 iterations of the adjoint model with a 20-day integration period takes 3.5 CPU hours on a CRAY YMP-92, while a 60-day integration takes 20 CPU hours.

2.3 Altimetric Constraint Formulation

The adjoint scheme aims to minimize a cost function, which can include one or many data constraints (the data- model misfit) and/or a background constraint (the background-model misfit). In this section we start from the general formulation for the altimeter data constraint; specific concerns for assimilating T/P data, ERS 1 data, or a combination of both will be addressed where necessary.

The cost function depends only on the data-model misfit, i.e.,

1 Z )T J=•( i•-HkX k Fffl(zk-H•X•) (4) where Xk is the state vector containing the model variables and Zk is the vector of observations (altimetric streamfunction). Theoretically, F k is the covariance matrix of the observation error. In our case the full covariance

matrix is unknown. Instead, the error for the processed altimeter data in the model domain is estimated as 5% of the

model surface streamfunction variance (O(100 dynamic cm2)) calculated over the model domain. Not all the altimeter

observations are fully independent. We assume every third point is uncorrelated along-track (21 km) and that each ground track is spatially independent (with 100 km separation). This scaling factor for the effective degrees of freedom is incorporated in Fk, which is also normalized by the number of observations.

H k is a mathematical operator applied to the model variables X in order to compare the model variables with the observations. We have a nonlinear observation operator H k for our nonlinear QG model. Theoretically, H k contains the time integration from the initial conditions to the observation time plus a scheme that converts the model variables at the observation time into the equivalent observed quantity. In our case we simply extract from the QG model the

Figure 2. The three-dimensional structure of the prognostic adjoint variable after assimilation of one ground track of data. This represents the sum of Green's functions (as many as data points) after the adjoint model has been integrated backward in time to a time step with data (approximately 0.5 days).


surface streamfunction value which is closest to the along- track altimeter data point in space and time (the resolution of the 12.5-km model grid is close to the 7-km altimeter resolution), without needing analyzed maps of the data. The data-model misfit requires the total surface streamfunction field to be assimilated. Altimeter data only give the residual component of the sea surface height; a mean sea surface needs to be provided for the adjoint assimilation, essentially to be removed from the model fields (see section 3.1). The total altimeter sea surface heights are then transformed using the geostrophic relation into an equivalent surface streamfunction, which can be compared directly with the model results.

The observations are assimilated into the adjoint model as the forcing (or source) terms in the adjoint equations. These are the derivatives of the cost function with respect to the model variables (8•, 8•) and are nonzero only at observation points, i.e.,

gJ_ &, _-Z ø - (s) i

These data forcing terms behave as finite delta functions. Solutions of the adjoint equation that are forced by delta' impulse functions 8 are the adjoint Green's functions. Physically, the observational information is propagated backwards in time along wave characteristics to find the contribution from all surrounding grid points to the data point, i.e., the inverse of an analysis by Green's functions. Figure 2 shows this three-dimensional response (in x,y,z ) of the adjoint variables as one ground track of surface altimeter data is assimilated. The isosurfaces depend on the amplitude of the eddies. Consider a purely diffusive process with an initial point disturbance which creates a conelike influence in phase space. The adjoint solutions are the same, only with time reversed. The larger space scales at depth are a consequence of the integration time. Note in our QG formulation the data assimilation will occur instantaneously via the adjoint diagnostic equation, so the fields are updated without any transient stage or phase lag.

2.4 Drifter Constraint

An extra term is added to the cost function to account for

the assimilation of the surface drifter data [Schr6ter, 1994]:

Jdrift •(Z•-HkXk) T D-1 = F k (Z•-HkXk) (6) where Z• D = (u, v) contains the drifter velocity vectors at each observation position k . The weights of the covariance matrixF• D are approximated by the model surface streamfunction variance, scaled by a factor of 10% as an estimate of the error in the surface velocity variance [Hernandez, 1995]. The effective degrees of freedom depends on the temporal decorrelation scale for the drifters. The drifter time series data have been filtered to remove inertial motions

with a 3-day cut off period; this is also chosen as our minimum decorrelation time. So, with 3-hour drifter sampling, every 1 in 24 observations is independent,

, (7)

The derivative of the drifter constraint also enters as an

additional forcing term to the adjoint equations, i.e.,

where u j, vj are the east, north model velocity components, specifically,

Oq/(k, 1 ] - t) -• (-gt(k, / +1) + gt(k,/- 1)) •(k,l) - t•½ (k,l) (•(k+l,/)- •(k-l,/)) ] -77x cxx (9)

Here 8x, 8y are the grid spacing and k,l are the position coordinates on the grid. As with the altimetric assimilation, we assimilate the observed drifter velocities directly in space and time to the closest model grid point. Note that the finite difference calculation of velocity at this grid point can introduce some minor error to the model velocities.

2.5 Hydrographic Constraint

Density information derived from hydrographic data can be an important constraint on ocean models, not least of all because it can constrain the mass field at depth. The effectiveness of assimilating temperature data has been demonstrated in primitive equation models [e.g., Derber and Rosati, 1992] and shallow-water models [e.g., Greiner and Perigaud, 1994]; but in QG models the assimilation is not so simple, since density is not calculated directly as a model variable but is the scaled vertical derivative of

streamfunction. In our level QG model formulation the vertical problem is solved as a projection onto the Rossby (pressure) modes. So we choose to project the hydrographic density data onto similar vertical density modes, then use the corresponding spectrum in pressure space to evaluate the baroclinic pressure modes. We can then calculate the misfit of the pressure mode amplitude with its model equivalent and force the adjoint model in Rossby mode space.

The processing procedure for the hydrographic data is as follows. The mean density modes Om D are calculated from the Kiel hydrographic data in the region and are the same Rossby density modes as those used in the QG model. The mean static density profile is calculated based on the long-term RBS climatology and is removed from the individual density profiles at each hydrographic station to yield a density anomaly p'/•. The density profiles are then projected onto the mean density modes using a similar orthogonality relation as described in the QG model [Miller et al., 1983], i.e.,

n 1 am = Omk P'k Az (10)

D where a m are the density mode amplitudes, n is the number of standard levels, N 2 is the Brunt-V•iis•ila frequency, and Az is the (constant) thickness of standard levels. The (nondimensional) pressure mode amplitudes am P are then calculated as

p_ 1 ( g D/ am V'• Ampøf am (11) where VO, d ,and PO are the appropriate nondimensional scaling factors; f is the Coriolis parameter; g is gravity; and Am is the root-mean-square sum of the density modes. The first four pressure mode amplitudes derived from the hydrographic data are then removed from the equivalent four


baroclinic model modes, extracted at the same location in

horizontal space and time. The first model mode is barotropic and cannot be resolved from hydrographic data. Note that here we project the hydrographic data to the closest model grid point, which is slightly different from fitting the model to cruise data, as in the work by Nechaev and Yaremchuk [1994].

The extra term added to the cost function for the

assimilation of the hydrographic pressure modes is

Jhydro:--•(Z•-HkXk)TF)-l(z•-HkXk) (12) Z• contains the hydrographic pressure mode amplitudes, and HkX k represents the equivalent baroclinic model mode. At each station the modes are independent, and each station is assumed independent when calculating the effective degrees of freedom. The error covariance matrix F• h is unknown for the hydrographic measurements, although the errors are assumed to be small. F• h is approximated as a scalar factor of (0.4 times the number of observations) -1, which gives an initial

(a) Level 1 30.ow 28.ow 26.ow 24.ow 22.ow 20.ow

36.øN

34.øN

32.øN

30.øN

' ' ' i .... i .... i .... i ' , , , i , , , , ,

Cont Lev=.005

(b) Level 5 30.øW 28.øW 26.øW 24.øW 22.øW 20.øW

36.øN

32 .øN 30.øN

....... I .... I ....

Cont Lev=.002

(c) Bottom Density 30.øW 28.øW 26.øW 24.øW 22.øW 20.øW

36.ON ,,, • .... • .... , .... • .... , .... , .

34.øN

32.øN

- ' ' ' I .... I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I '

Cont Lev=.002

01 Model Error Variances: July 6, 1995

Figure 3. Error variance matrices from the optimal interpolation analysis fields in the domain in nondimensional units for (a) surface vorticity, (b) bottom vorticity, and (c) bottom density anomaly.

misfit of equivalent magnitude O(1) to the altimeter and drifter cost functions. In physical terms this error weighting of 35% of the model surface streamfunction variance is large, as the expected errors are around 20% of the signal variance for expendable bathythermographs (XBT) and 5% for CTDs [Hernandez et al., this issue]. We have chosen to maintain equivalent magnitudes for the different data misfits in order to give equal weighting to the gradient function. Giving different weights to the hydrographic data makes little change in the rate of decrease of the total cost function.

2.6 Background Constraint

Although we start our assimilation runs using a good estimate of the initial conditions from the OI analysis, this information on the background dynamical regime can be quickly lost when we only use a surface data contraint. Surface fluctuations in the data may be unable to constrain the mean flow at depth, and this becomes important in our limited-area domain with large topographic features. We found the surface data constraint alone can produce stable bottom flows in some (but not all) cases and then only with good data coverage and after 30-50 iterations, which is computationally expensive. Our test runs have evidenced that some information on the

dynamical regime should be added in order to get a more realistic solution, especially at depth. It is also necessary mathematically to have some form of background constraint, since the problem is underdetermined.

Since initial vorticity and bottom density are our control variables, we choose to minimize their departures from the background or initial guess field at every model grid point at the initial time. We believe the first guess field derived from the OI assimilation runs provides a good estimate of the initial conditions, especially at depth. This background constraint is similar to penalizing departures from the background potential vorticity field q since, effectively, 3• = 3q. Although one expects large variations in q at the surface, below the thermocline the potential vorticity remains fairly constant; this property of holding q constant at depth has already been explored as an assimilation constraint [Haines, 1991].

The constraint is formulated in the standard way for meteorology, following the 4DVAR variational principle; that is we wish to minimize the cost function

_ 2 (•3)

2

with respect to the initial conditions Xk, where E and F are the background and data error covariances respectively, and Z and H are the observation vector and the "observation

operator" respectively. The first term is the "background constraint"; the initial conditions are minimized with respect to the "first guess" initial vorticity fields over all model levels and the initial bottom density field X•' the second term is the original "data constraint" which is the sum of all aforementioned constraints. The derivative of the background penalty also enters as an additional forcing term in the adjoint model.

For the background constraint, the initial vorticity and bottom density fields X• are taken from 7-day OI forecast fields; the background error covariance matrix E is taken from the OI error variance maps produced at every OI time step. The


error variances for the surface and bottom vorticity field and the bottom density field for July 29, 1993, in the model domain are shown in Figure 3. These first guess fields are derived from the OI assimilation of T/P and ERS 1 altimeter

data, so the error variances show the pattern of the two satellite ground tracks quite clearly. that is, the error is smallest along the ground tracks and larger in the diamond regions in between (this is a result of the propagation scheme for errors in OI). Since the inverse E -1 is applied in the cost function, the background constraint is therefore stronger along the ground tracks. Thus, in our adjoint model, the strongest forcing from both the data and background constraints is along the satellite groundtracks. The magnitude of the error correlation derived from the OI analysis varies with depth (for error covariances this correlation value is multipied by the variance of the function at each level). Table 2 shows the average values of inverse error covariance E -1 and variance for initial vorticity and initial bottom density at each model level. Finally, as for the data misfits, the background error variances are scaled by the number of background data points (91x65 horizontal grid points and six levels = 35,490).

2.7 Preconditioning

We have chosen the weights for each different constraint by estimating their associated errors, either from the OI analysis in the case of the background constraint or from estimates of the "noise" in each of the data sets. This gives our best estimate of the relative errors, but the combined constraints can produce a complicated structure in the gradient of the cost function when integrated over the adjoint model run. In addition, when more than one control variable is

involved (i.e., vorticity over five levels and bottom density anomaly), the magnitude of one parameter may dominate the descent algorithm, leading to a poor estimate of the other parameters, and in many cases, a slow convergence. Geometrically, this corresponds to a flattening of the cost function in certain directions.

A solution is to "precondition" or scale the gradient vector before the descent algorithm is applied. Appropriate scaling of the control parameters can effectively provide steeper directions. The best preconditioning is to scale the gradient vector by the inverse of the Hessian matrix Vu2J [Thacker and Long, 1988]. However calculating the full Hessian matrix is computationally expensive for systems such as ours with a large number of unknown control variables.

Instead, we choose to multiply the gradient of the cost function by the forecast error covariance matrix [e.g., Derber and Rosati, 1992]; in our case E from the OI analysis is the

Table 2. Average Inverse Error Variances for June 29, 1993

Control Variable Average E -1 Variance Level 1 0.01 106.4

Level 2 0.05 19.9

Level 3 0.09 11.0

Level 4 0.12 8.5

Level 5 26.4 0.04

Bottom Density 2823.6 3.5 E -4

Levels refer to vorticity. E-1 is inverse error variance.

variance fields. (In our test runs this choice of scaling gave similar results to the method of Moore [1991] using a diagonal matrix whose elements containing the mean square variance of the unknown control variables at each model

level). One can see from Table 2 that preconditioning the gradient by the matrix E essentially gives larger weight to the surface perturbations in the gradient, forced by both the data misfit and the surface background misfit at t=0. This has the advantage of a rapid initial decrease in the cost function to reduce the large data-model misfit but with some perturbations in the bottom flow. Note that these bottom perturbations cannot grow unchecked; if their amplitude becomes too large and they begin to dominate the descent space, there is a necessary adjustment in the minimization process via the background constraint.

3. Assimilating A!timetric Data

Our first test of the adjoint assimilation scheme is to assimilate only surface altimetric data from either T/P or ERS 1 or the combined data set. Our discussions will be based on a

20-day assimilation period during the first phase of the SEMAPHORE campaign, starting on June 29,1993. This period is chosen to coincide with maximum in situ data coverage; in this first section the in situ data are used as an independent test of the altimeter assimilation. The two altimeter data sets are processed differently due to their different error structures, detailed below.

3.1 TOPEX/POSEIDON and ERS 1 Data Processing

During the period of the SEMAPHORE campaign, the European Space Agency (ESA) ERS 1 satellite was in its 35- day repeat orbit, with ground tracks spaced by 80 km at the equator. In earlier test runs we had applied the ESA fast delivery products (FDPs), but for this evaluation study we used the more precise ESA off-line ocean products (OPRs) distributed by F-PAF CERSAT. Most of the corrections supplied with the OPRs were applied, including the improved orbits, fides, and atmospheric and electromagnetic (EM) bias corrections as detailed by Hernandez et al. [this issue]. Corrected sea surface heights were resampled along-track every 7 km by cubic spline, and sea level anomalies (SLAs) were calculated for all passes within our region using collinear analysis to remove the 21-month mean.

The T/P altimeter has a 10-day repeat, implying a much coarser spatial sampling than ERS 1, with a distance of 315 km between neighboring ground tracks at the equator. Thus the T/P data may miss a significant fraction of the energy at smaller spatial scales, though the temporal resolution is better than ERS 1. We used the merged geophysical data records (GDR-Ms) distributed by AVISO, with 21 months of total data (October 4, 1992 to July 19, 1994) to determine the mean. Processing of the T/P residuals is again described by Hernandez et al. [this issue]. Data were selected over the same area and for the same period as the OPRs, and, in general, the same processing applied.

For assimilation in the adjoint model the mean sea surface is added to the residuals to get the total surface streamfunction (this is equivalent to removing the mean from the model fields, since the adjoint is forced by the misfits). We obtain an independent estimate of the mean sea surface based on the large-scale RBS climatology atlas but corrected by the 2-year mean Reynolds stresses from Geosat data assimilation which


introduces smaller-scale features [De Mey, 1994]. (Note this adjustment is minor in comparison to the variance of the residuals). This Geosat-adjusted climatological mean is then removed from the model fields. The T/P and ERS 1 residuals,

which have been filtered with a cutoff of 70 km wavelength to reduce background noise, are assimilated directly in the adjoint model at their observed position and time.

To see the main features in both data sets, 10-day maps of the surface streamfunction are presented for the assimilation period from June 29 - July 19, 1993 for T/P data (Figures 4a and 4c)' and for ERS 1 data (Figures 4b and 4d), referenced to the climatological mean. The differences in the plots are partly due to the different spatial and temporal sampling of the two altimeters; for example, the T/P data show strong meanders of the mean jet west of 33øN, 28øW, whereas the finer spatial resolution of ERS 1 detects a more complex zonal structure. Note these maps are for display purposes only; the adjoint method assimilates the data measurement direc0y at the closest space-time position in the model. The adjoint model will attempt to combine all of the available data with the QG dynamics in an optimal way; but this simple comparison of two altimeter data sets illustrates that we need to have a good estimate of the data error in order to give correct weights to multiple data constraints.

3.2. QG Model Run With No Optimization

As discussed earlier, the initial conditions for all our

assimilation runs are derived from a 7-day forecast field based on the OI assimilation of the same T/P and ERS 1 data sets, i.e., GDRMs and OPRs. The surface streamfunction field for the OI forecast of June 29, 1993, is shown in Figure 5a (left); the QG model is then integrated forward over 20 days giving the "predicted" flow on July 19, 1993 (Figure 5a; right). Note there is no assimilation during this run. The OI forecast field at level 1 is fairly smooth, with a clear zonal jet and weak meandering. Over the forward integration period the meandering increases and there is some westward propagation of the eddies. The intensification of the cyclonic eddy at

TOPEX/POSEIDON - June 29-July 9, 1993 b) ERS1 - June 29-July 9, 1993

c) TOPEX/POSEIDON - July 9-19• 1993 3o.oow 25.0ow 2o.oow

Figure 4. Observed surface streamfunction in nondimensional units from (a) T/P and (b) ERS 1 groundtracks over the first 10 days of our assimilation period from June 29- July 9, 1993; and for (c) T/P and (d) ERS 1 for the second 10

a) 01 Forecast M35TSW

34

-30 -25 -20

b) Optimised by TOPEX/POSEIDON 36 -• • .:•.....,.•\ •:•.:•.) '.....'---- •

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-30 -25 -20

c) Optimised by ERS1

34

-30 -25 -20

d) Optimised by WP + ERS1

34 32

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-30 -25 -20 29 Jun 1993

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34 32

3O

-30 -25 -20

-30 -25 -20 19 Jul 1993

-5 -4 -3 -2 -1 0 I 2 3 4 5

Surface Streamfunction (non-dim units)

Figur• 5. Surface streamfunction at (left) the beginning and (right) end of the integration period in nondimensional units for (a) 7-day OI forecast fields, (b) initial conditions (IC) optimized by T/P data, c) ICs optimized by ERS 1 data, and d) ICs optimized by the combined T/P and ERS 1 data.

33øN, 31øW may be a model boundary effect. There is a strong baroclinic vertical structure in the depth profiles of the initial conditions (not shown), which remain stable at depth over the forward integration period due to the long decorrelation times at depth imposed by the QG model dynamics.

Some test runs have been performed starting the adjoint assimilation from different initial conditions on June 29,

1993. In particular, we have compared the results using forecast fields from the OI assimilation of T/P data alone and

of ERS 1 data alone. We have chosen first guess fields based on the combined assimilation of T/P and ERS 1 data, since it

gives a better coverage of the small-scale features at the surface and provides the most stable bottom flow throughout the model runs. The 7-day OI forecast fields are used rather than the OI analysis fields, as they are already in a stable transition for the forward model runs, without risk of

introducing initial transients. In general, we found the results were more strongly constrained by the surface data constraints than by the choice of initial conditions which are carried by the background constraint.

3.3 Assimilating Altimeter Data

Assimilating TOPEX/POSEIDON data. Our first case considers a simple cost function: assimilating only altimeter surface heights but also including the background

days from July 9-19, 1993. Time differences are not taken . constraint. The adjoint model is iterated 20 times, into account in the mapping. A corrected Robinson et al. assimilating the two cycles of T/P sea surface heights, which [1979] climatology has been added to the residual heights. are mapped in Figure 4 for reference. The main features


sampled by the T/P altimeter are the prominent meander at 29øW (which is less prominent in the initial conditions from OI and in the ERS 1 data over this period) and an increased meandering of the mean jet to the east at the end of the 20-day period. These T/P data features are now present in the optimized initial conditions (Figure 5b; left) and are strongly evident at the end of the 20-day forward integration period (Figure 5b; right).

Although the main features present in the altimeter data have been incorporated in the optimized solution, the initial conditions have also retained some of the features evident in

the 7-day OI forecasts. This is due to the background constraint and demonstrates how the prior knowledge can be incorporated with the new data and again be consistent with the model dynamics. For example, the western boundary region is only sampled by T/P around 32.5øS, yet most of the features along the entire boundary region are consistent with the background constraint. So, in data sparse areas, the features in the OI 7-day forecasts which are used as our background constraint are more evident.

The surface fields appear consistent with the strong data forcing, but we have not reached a stable model solution at depth (Figure 6). Clearly, there is a poor identification of the controls at depth, despite the background constraint. After

a) 01 Forecast M35TSW

•.::•.• •.' .............. • .... ß 32

30 "'? ,.

-30 -25 -20

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b) Optimised by TOPEX/POSEIDON

36 - • L '" 36 o•i•o.,• • .: ........ -.. LO 34 •.' ." ,• 34

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Bottom Streamfunction (non-dim units)

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30 '.• •• .....'" ::.•.:';' p ( -30 -25 -20

-30 -25 -20 19 Jul 1993

-40 -20 0 20 40

Bottom Density Anomaly (non-dim units)

Figure 6. Bottom streamfunction at the (left) beginning and (right) end of the integration period in nondimensional units for (a) 7-day OI forecast fields, (b) ICs optimized by T/P data and bottom density anomaly for (c) the 7-day OI forecast fields and (d) ICs optimized by T/P data.

optimization some change is still noticeable over the forward integration of the QG model, suggesting the initial regime is wrong. Bottom topographic steering in the QG model physics is strong, so that by the end of the forward QG run we have regained more realistic amplitudes and features in the bottom flow. The strongest change is around 31øN, 25øW in both the surface fields and the fields at depth, a region which falls between the T/P ground tracks so it has no data constraint. There is very little change in the optimized bottom density anomaly fields due to the chosen weighting (Figures 6c and d), which is typical for all the data constraints. The conjugate gradient descent needed to be restarred during this integration, due to a loss of conjugacy from the nonlinearities of the model. Thus although the optimized solution is closer to the data, we have not reached a minimum solution with only 20 iterations. It is also possible that the wide spatial sampling of T/P in this region is not sufficient to constrain the smaller-scale mesoscale features of

the flow.

Assimilating ERS 1 data. The ERS 1 data in Figure 4 show a slightly stronger zonal jet with less meandering; this is also noted in the optimized initial conditions after 20 adjoint iterations (Figure 5c; left). The meandering of the zonal jet between 23øW and 26øW is measured by ERS 1 in the second half of the assimilation period and thus is included toward the end of the optimized model runs (Figure 5c; fight). The cyclonic eddy at 21øW, 33øN measured by ERS 1 in the first part of the assimilation period falls between the T/P ground tracks and has an opposite sign in the T/P optimized solution. Its presence in the ERS 1 optimized solution has a weaker amplitude, modified by the background constraint. During the forward QG model integration there is much less change in the optimized surface fields than for the case when T/P data alone were assimilated and virtually no change at depth. This reinforces a strength of the adjoint assimilation: the initial conditions have been optimized by the data forcing but using the dynamics of the model, and the resulting fields are more stable in the forward model runs.

Assimilating both TOPEX/POSEIDON and ERS1 data. We now assimilate all the available T/P and ERS 1

altimeter data over the 20-day integration period. The optimized initial fields after 20 iterations (Figure 5d; left) now show a blend of all the features present in the two data sets (Figure 4), since the data weights are equal. In particular, the region of positive dynamic height south of the jet extends farther east, as defined by the finer-resolution ERS 1 data. Again, the streamfunction fields show only minor change over the integration period, indicating we are close to a stable solution.

The cyclonic eddy at 24øW, 36øN has a temperature/salinity signature with traces of Mediterannean water (see section 5); this "meddy" has a consistent signal throughout each assimilation period. In this model run the meddy penetrates to the third model level (1125 m); its vertical structure is investigated further in section 5, when we introduce hydrographic data at depth as an additional constraint.

The improvement from the assimilation of T/P and ERS 1 data can be quantified for the three cases by the reduction in the total cost function (Figure 7a). Clearly, altimeter data act as an important constraint on the QG model dynamics, with the altimeter-model cost function reduced by 60 - 75%. The assimilation of each altimeter alone is more efficient than the

combined T/P and ERS 1 assimilation. This is because the

MORROW AND DE MEY: ADJOINT ASSIMILATION 1N THE AZORES CURRENT 25,017

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d) Gradient J' (T/P + ERSl)

I

0.0 20.0

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iterations

Figure 7. (a) The altimetric cost function, (b) drifter cost function, (c) hydrographic cost function,(d) gradient of the total cost (altimetric plus background), shown for each of the three cases, optimized by T/P data, ERS 1 data, and the combined ERS 1 and T/P data.

solution of the combined data set is a balance between the two

(sometimes conflicting) data sources and the QG model dynamics. This rapid reduction in the cost over this "short" 20-day integration period is aided by the fact that the errors have no time to propagate between observations in phase space and produce a contradiction. Differences between altimetric measurements at crossover points (or on repeating tracks) can also introduce measurement errors, if they are inconsistent with QG dynamics in the intervening time. Thus ERS 1 with fewer crossover/repeating points shows less errors, the combined T/P and ERS 1 assimilation has the

largest number of crossover points and the highest model-data misfit. Note that the chosen weights keep the background constraint reasonably small in all three cases, and the reduction in the total cost is dominated by the stronger data

constraint. The norm of the gradient of the cost tends to zero in all cases (Figure 7d), indicating we are closer to a minimum solution, though not necessarily the global minimum. However, the reduction in the gradient is not smooth, and the oscillating gradient suggests a complex descent topology; the solution may be trapped in a local minimum.

Finally, we can use the available in-situ data as an independent test of our altimeter assimilation. We have available surface drifter velocities (see section 2.4) to compare with surface geostrophic velocities derived from the model surface streamfunction. Figure 7b shows the misfit between the drifter velocities and the model-derived velocities

for the three cases assimilating different altimeter data sets. Note here the drifter data are not assimilated or used to

minimize this "cost"; this is an independent test. The rapid decrease in the cost function due to the T/P assimilation

(Figure 7a) initially takes our solution farther from the independent drifter data. The situation improves with more iterations, but the T/P optimized solution remains the farthest from the independent data. The best results are for the ERS 1 optimized solution: here we have reduced the misfit between the model and the independent drifter data by 15%. The combined T/P and ERS 1 optimized solution is also reasonable, reducing the drifter misfit by 10%. The efficiency of the ERS 1 data in constraining the solution is mostly due to its fine spatial resolution, which is close to the dense drifter resolution, but also because the variability in the Azores has a long temporal decorrelation of 0(30 days), consistent with the ERS 1 repeat sampling.

We can also compare vertical modes derived from hydrographic data (section 2.5) and the optimized model results (Figure 7c). Clearly, there is little reduction in the hydrographic cost function from the assimilation of altimeter data. This is mostly because hydrography only measures the baroclinic modes, whereas most of the rapid adjustment in the altimeter assimilation is via the barotropic mode. Figure 8 shows the variance of the model mode amplitudes throughout the final model integration after optimization by T/P and ERS 1 data (note that the mean over the 20-day integration has been removed). The partitioning between the modes is specified by the QG formulation; the amplitude of the barotropic mode is initially half the first baroclinic mode. We can see that the adjustment from the surface altimetric assimilation is projected vertically but mostly via the barotropic mode in the model interior, with litre change to the baroclinic flow field. Note that the variance of the

barotropic mode is specified small at the boundaries Thus the large initial barotropic perturbations are damped by the low amplitude at the boundaries, which can quickly penetrate into the interior within the 20-day assimilation period. In the following sections we will discuss this further, in particular, when assimilating hydrographic data to constrain the baroclinic field directly.

•mdatmg Lagrangian Drifters

During the first SEMAPHORE experiment, 29 Surdrift surface drifters were released in the intensive SEMAPHORE

domain starting from July 10, 1993 [Hernandez et al., this issue]. The Surdrift surface floats were drogued at 150 m, and provided near-real-time position measurements. Velocity measurements were calculated from the buoy positions, and the time series were filtered using a cosine-lanczos filter with


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otropic

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Model Time Step (doys)

Optimized by T/P+ERS-1 (29 June - 19 July, 1993)

Figure 8. Variance of the barotropic and first baroclinic modes throughout the final model integration after optimization by T/P and ERS 1 data for the a) total field; b) boundaries; c) interior field.

a 3-day cutoff period. The filtering removed inertial oscillations in the buoy trajectories, and the resulting

velocity time series (subsampled at 3-hour intervals) was closer to quasi-geostrophic timescales. The drifter velocity misfits at 150 m were then assimilated directly into the surface (200 m) level of the adjoint model.

The drifter positions are plotted in Figure 9 for the same 20-day period (June 29 to July 19, 1993) used in the previous analyses. All the drifters had just been deployed within the SEMAPHORE domain; most remain within the bounds of the

domain throughout this period, indicating weak advection. The adjoint model is integrated over this 20-day period, starting from the 7-day OI forecast fields based on T/P and ERS 1 data; these are the same initial conditions used in

section 3. We now consider the following two different data constraints (the background constraint is always applied):

assimilating the drifter data alone and assimilating the combined drifter and T/P altimeter data.

The optimized initial conditions and 20-day forecasts for the two cases are shown in Figure 10 after 20 model iterations and should be compared with the initial guess from the OI forecast fields in Figure 5a. The drifter data are concentrated in the SEMAPHORE domain east of 27øW and north of 30øN.

The most striking effect of assimilating the drifter data (Figure 10a) is the large-amplitude anticyclonic "eddy" at 23øW, 33øN which is measured by ERS 1 and T/P but with weaker amplitude (Figure 4). There is also a strong mean jet which continues in a series of meanders out to 21-22øW before

weakening and turning to the south, again more consistent with the ERS 1 runs in section 3. The drifter data are

obviously providing a much stronger E-W mean flow in the SEMAPHORE domain. There are a number of points to consider here. The drifters may be overestimating the flow at 150 m, if they have lost their drogues or are subject to windage. This was of some concern during the SEMAPHORE experiment, though unlikely so soon after the deployment. Alternatively, the difference may be due to the relatively weak climatological mean that is added to both altimetry data sets (and which is assimilated in the OI forecast fields); in reality, the mean flow added to the 18-month altimetric time series

may be stronger. Thus the altimeter sea surface height data may not be a strong constraint on the position or strength of the mean flow, although we have clearly seen how well it constrains the eddy field. In contrast, assimilating surface drifter velocities strongly constrains the mean.

The surface fields optimized with drifter data alone show some upstream adjustment, especially for the meander around 29øW, 33øN. This is expected, since the adjoint dynamics propagate information backward in time (i.e., upstream); but there is virtually no adjustment of the field at the western boundary, either in the surface field or the bottom field (not shown). The chain of seamounts at 29øW may act as a barrier to the upstream propagation of information by the adjoint model in the absence of upstream data forcing. A more likely explanation is that with weak advection over the 20-day period the mean "countercurrent" in the adjoint may be too slow to carry the drifter information very far upstream.

When T/P and ERS 1 data are also assimilated (Figure 10b), the optimized surface initial conditions show a much stronger recirculation pattern in the southwest part of the model domain. This feature is present only in the altimetry data, and so the combination of altimetry and drifters produces a surface field which is consistent with both data sets and has

not strayed too far from the background constraint. When drifter data are assimilated alone, there is an initial, rapid adjustment in the surface fields which produces an anomalous cyclonic bottom flow in the southeastern corner (not shown). This feature appears unphysical, as it undergoes a rapid change to disappear by the end of the 20-day QG model integration. When altimeter data are also assimilated, the anomaly is weaker and limited in spatial extent. Including the additional altimeter data forcing over the entire region appears important not only in correcting the surface fields in the west of the domain, but also to maintain realistic bottom flOW.

The adjoint assimilation is very efficient in reducing the drifter-model misfit, in preference to the altimeter-model misfit. For drifters alone there is a 80% reduction in the

drifter-model cost function, but the altimeter-model misfit


30.øW

Surface Drifters (29 June - 19 July, 1993) Figure 9. Data distribution of Surdrift surface drifters (crosses) in the SEMAPHORE domain; referenced to the model surface streamfunction in nondimensional units in the model domain.

increases; that is, the solution is not consistent with the

altimeter data. However, the hydrographic data misfit is also reduced by 25%, indicating that the drifter data add extra information to the model which is consistent with the vertical

baroclinic structure measured by hydrography. It may also be an indication that the mean is wrong. Including the three different constraints (altimetry, drifters, and background) on the model slows down the reduction of the total cost function

(Figure 11) from the earlier cases with altimetry or drifter data alone. However, the three constraints provide a consistent improvement on all the data-model misfits, including the independent hydrographic data.

Assimilating Hydrographic Data

During the first SEMAPHORE campaign a total of 43 CTD measurements were collected, with 37 of these in the 20-day period between June 29 and July 19, 1993, [Hernandez, 1995]. Pressure modes were calculated at each hydrographic station from the vertical profiles, as described in section 2.5. The amplitudes of the first two baroclinic pressure modes are shown in Figure 12 for all stations within the 20-day assimilation period. The first baroclinic mode has the largest amplitude and shows the distinctive features of the flow in the SEMAPHORE domain during this period as follows: a strong mean E-W jet, with a positive pressure ridge to the south at 33øN, with a series of cyclonic eddies; a trough centered at 35øN north of the jet; and the trace of a meddy at the northern boundary at 24øW. The E-W ridge-trough system is the main feature apparent in the second baroclinic mode (although with reduced amplitude); the meddy is also evident in this mode. The influence of the meddy is clearly depicted by its temperature-salinity characteristics and is best represented by a combination of the farst three pressure modes.

The hydrographic data were assimilated using all four pressure modes to best depict the vertical structure, especially

for the meddy. In fact, our optimized results were almost identical when we used only the first baroclinic mode, which has the largest amplitude. Test runs showed that hydrographic data cannot be applied as the sole constraint in the adjoint assimilation, since the barotropic mode is unconstrained and the coverage is too local (the baroclinic modes are also unconstrained in the areas untouched by hydrographic measurements, except for some slow upstream propagation). We have already seen that most of the rapid adjustment over our 20-day integration is via the barotropic mode. Thus, for our QG formulation, the hydrographic constraint must always be applied with other data constraints which include some barotropic information (such as drifters and alfimetry).

a) Optimised by Drifters

I :: '•'•,' ."'" .: ' • " ""•" 3o.:.•:,.;..:•:•::-•.• ........... • "•,;•-:-•':...i' 3o.• ,..;: •..."-:-.•..'•.., . -30 -25 -20 -30 -25 -20

b) Optimised by T/P+ERSI+Dri•ers

36•••--• • 36_ ••• • % ¾--•.%. --- . ..... ....: •,'".• • • ..... •'•:•':-•::-• •'• ½L• "' •;: ........ '•'"'"'•;:;,•'-• "• "'•' ., - ...... :•...•.•.•.•.• %•t:• ..

.............. ß ß '.-:.,.:.:-:.•! .... •4•%: ":::::f,•J.•:•i• ..... :•. •:•:g:-•?•(...".'.::.:' :.. ..... -30 -25 -20 -30 -25 -20

29 Jun 1993 19 Jul 1993

:•.. •::,•:: ..........

Surface Streamfunction (non-dim units)

Figure 10. Optimized initial surface streamfunction fields (left) in nondimensional units after 20 iterations of adjoint assimilation of (a) drifter data alone and (b) combined drifters and TOPEX/POSEIDON data. The forecast fields after a 20 day model integration with no further assimilation are also shown (right) for each case.


c 10-- O '

_

._

o 0.8-- c- --

• 0.6-- _

-,-' 0.4--

o (3 0.2--

0.0

0.0

a) COST (Altimeter- Model)

/ ----- D

•.. ___TP+E+D+H

BACK

4.0 8.0 12.0 1 6.0 20.0

iterations

b) COST (Drifter- Model)

o 0.8-- ._

o c 0.6-- D --

•- 0.4-- (D --

o 0.2-- o

_

0.0

0.0

TP+E

\•- _.,:• TP + E + D TP+E+D+H

4.0 8.0 12.0 16.0 20.0

iterations

o 0.9-- ._

o _

•- 0.7--

O -- O

0.5

c) COST (Hydro - Model)

TP+E

TP+E+D+H

0.0 4.0 8.0 12.0 16.0 20.0

iterations

Figure 11. (a) The altimetric cost function, (b) drifter cost function, and (c) hydrographic cost function for each of three cases: optimized by altimetry data, drifter data, and the combined T/P plus ERS 1 plus drifters. The background error is also shown for the altimeter case.

The model barotropic mode and the first baroclinic pressure mode are shown in Figure 13a for the initial OI fields based on the combined T/P and ERS I data. The first baroclinic mode

has the larger variance, and most of the features noted earlier in the surface streamfunction fields are dominated by the baroclinic mode; that is, the E-W jet is still centered around 34øN and begins to meander southward by 22øW. In the hydrographic data the meddy has a strong anticyclonic center in the first baroclinic mode, whereas the model shows only cyclonic flow at the meddy location for the baroclinic mode and virtually no barotropic component. Will this be altered by assimilating the hydrographic data at depth?

To consider this question, we look at the modal structure of the initial conditions, optimized by the assimilation of various data sets (this is different from earlier plots of surface streamfunction, which are a summation of the five modes). Figures 13b-13d show the barotropic and first baroclinic modes from the model on June 29, 1993, after optimization by combinations of altimetry and drifters. Figure 13e shows the the modal structure when the four hydrographic data modes are also assimilated. As we found earlier, the assimilation of

altimeter data mainly affects the barotropic mode, with only minor adjustment to the first baroclinic mode. Interestingly, the altimetry data add the correct anticyclonic circulation at

the location of the meddy, but it appears in the barotropic mode rather than the baroclinic mode. Perhaps there is a barotropic component to the flow at this location, which is not measured by hydrography.

There are no surface drifter measurements in the region of the meddy, so there is no change from the background constraint at this location when the model is optimized by drifters; but the strengthening of the jet and the series of anticyclonic eddies at 33øN are clearly depicted in both the barotropic and baroclinic modes. Although drifters and altimetry are both measuring the surface regime, drifter measurements appear more effective in constraining the baroclinic structure of the flow. Adding the hydrographic constraint has a similar effect as for the drifter constraint, mostly because both data sets have dense measurements in the same SEMAPHORE domain and both introduce information on

the baroclinic field. However, the hydrographic constraint is fairly weak due to its low number of degrees of freedom; where there are differences ,the other data sets tend to dominate. For

example, the hydrographic data show the meddy with an anticyclonic structure in the first baroclinic mode, but the optimized solution keeps the anticyclonic feature in the barotropic mode and cyclonic in the baroclinic mode, which is more consistent with the other data constraints, such as

altimetry, drifters, and background (first guess field). The integral of this large combination of constraints can

produce a complex topology of the gradient and thus a difficult function to minimize. This is partly due to the combination of different data errors over the short integration time. Indeed, the descent algorithm needed a number of restarts during the 20 iterations, which were not necessary with longer, 35-day and 60-day model integrations. However, the main surface features discussed for the individual data sets

are now present in the optimized initial conditions; the amplitude of the eddies and the position of the mean are all consistent. Furthermore, the fields at all levels remain stable

during the 20-day forward model integration, indicating we are closer to a minimum solution. This is borne out by the reduction in the cost function for each type of data constraint and the consistent reduction in the total gradient.

The separation into barotropic and baroclinic modes gives some information about the different vertical structures that

result when assimilating different data types. We can also examine this using vertical profiles through our model domain. Figure 14 shows the zonal velocities through a meridional section at 23øW for the different assimilation

cases. This N-S section cuts through the zonal jet around 34øN and also the meddy north of 35.5øN. In the OI fields and those optimized by alfimetry the mean jet is mainly contained in the upper layer, with weaker penetration to the lower layers. The meddy just entering the domain is associated with increased eastward velocities in the upper layer structure for all fields, except the OI initial fields. As expected, the surface drifters increase the strength of the zonal jet, which has a strong baroclinic structure. Including the hydrographic constraint increases the vertical penetration of the zonal velocity field.

6. Forecast Tests

In sections 3-5, the reduction in the cost function gives a quantitative test of the improvement in our model runs during the assimilation period. Another question is whether the

MORROW AND DE MEY: ADJO]]N• ASSIM•ATION IN THE AZORES CURRENT 25,021

2nd Boroclinic

24.øW 22.øW

36.ON .,[ [ I i [ [ , ! • ! • ' [ • ß ,. ß + • • , i ! i

/ i + + •,• • + +l I + + + i ii %. i ii

/ /

54.ON --

26.øW 20.øW

I I .

52.ON -

++ + + + + + + ++ + + + + +

+

+ + + +++ + + + ,+ +

++ + ++ +

+ + ++

+ +

".,.- _ / t /

'+ / • + +

-*'' ..... 0.30" +

+ + +

+ / ;i + + + t x /

t +% x +

ß

+ •+ +

+

+

i + x x

/

i + x + +

I I I' I [ I I I

29 June - 19 July 95 CI = 0.2

Figure 12. The (top) first and (bottom) second baroclinic pressure modes in nondimensional units derived from all available hydrographic stations within the 20-day assimilation period June 29 to July 19, 1993.

improvement remains valid for longer time periods in "pure" forecast mode. To test this, we have run the model foreward

over a 2-month forecast period, starting from both the unoptimized 7-day forecast fields from OI and the optimized initial conditions from the adjoint assimilation of different data sets over the 20-day assimilation period. The boundary conditions for the 60-day run are again taken from OI forecast fields during this period. Figure 15a shows the ratio of the rms forecast error to the persistence error for each run, which indicates how successful the model is relative to the

persistence forecast. The forecast error measures the difference between the forecast streamfunction and a reference

field derived from the 7- and 14-day OI forecasts. The persistence error measures the difference between the initial streamfunction field and the reference field. The unoptimized fields show an initial decrease over the first 10 days, but thereafter the forecast field maintains a level slightly better

than persistance. In contrast, the fields optimized by.: altimetry show an initial increase where the forecast is worse than persistence, but after 10 days they have reached a stable level of around 0.7. Combining drifters or hydrography with altimetry gives a similar improvement in the results. Assimilating drifters alone gives a minor improvement in the long-term forecast field, but clearly, the excellent spatial coverage from altimetry is necessary for longer-term stability.

At the deepest model level (Figure 15a) the optimized solutions show the same pattern of improvement, but the unoptimized solution quickly deteriorates and remains much worse than persistence throughout the 60-day forecast. Note that the unoptimized solution also includes some information on the observed surface fields from the previous OI assimilation of altimetric data; but clearly, the vertical projection of the data via the variational assimilation (which


a) OI Forecast

..... • ....

..•.....:• . .. •....

-30 -25 -20 -30 -25 -20

b) Opt by Altimetw

• •.:•;• . . ......,.• • ........ ...% :-. ..... ----½, .

, ......... ? ...... = ......... : ............ . -30 -25 -20

c) Opt by Drifters

36 •:.---:: • • - :.::.:::!:-'--.'•----. ß--:•.••

,9. ø 32 ";:"" '; :" :'" m 3O

-3O -25 -2O

d) Opt by AIt+Drft

-30 -25 -20

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e) Opt by AIt+Hyd

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-30 -25 -20 -30 -25 -20

..• .

-2 -1 0 i' ...... :' 2 Mode Amplitude (non-dim units)

Figure 1:3. The model (left) barotropic mode and (right) first baroclinic pressure mode in nondimensional units at June 29, 1993; derived from (a) the 7-day OI forecast field based on assimilated T/P and ERS 1 data; (b) optimized by altimetry; (c) optimized by surface drifters; (d) optimized by altimetry and drifters; and (e) optimized by altimetry and hydrography.

uses the actual model dynamics rather than statistics) is providing better and more stable long-term forecasts which are consistently better than persistence.

These results are also confirmed by the forecast tendency correlation, which indicates the accuracy of the model in predicting feature propagation, relative to those observed. Again, the forecast tendency over a 60-day forecast period is consistently higher for the optimized solutions at the surface and bottom, with values of 0.7 to 0.8 (Figure 15b); whereas the unoptimized solution stays around 0.4 for both the surface and bottom levels. Thus the adjoint assimilation not only reduces the cost function within the 20-day assimilation period, but also can imrpove the long-term forecasts over a period of 2 months.

7. Discussion

The aim of this assimilation project was to try and minimize the difference between the observed ocean and our

model ocean using available data over a limited period to optimize our initial model fields. We have found that the surface assimilation of both altimetry and drifters acts as a

strong constraint on the QG model. The dynamics quickly adjust to the surface forcing in order to minimize the cost function. Most of this adjustment occurs in the first few iterations via the barotropic mode, which has the fastest adjustment over our 20-day integration period. In contrast, there is little improvement in the hydrographic data-model misfit, which only contains the slowly adjusting baroclinic information. For longer model integrations the hydrographic data may be beneficial, but for our short integration periods we find it is more important to constrain the barotropic field.

The barotropic adjustment quickly reduces the surface cost, but at the same time, this can induce large-scale perturbations throughout the water column. These large barotropic perturbations are temporary (the magnitude reaches a maximum after about 5 iterations); but they are not consistent with the chosen vertical modes structure in our QG model, where the baroclinic component is twice as large as the

ß

barotropic component [De Mey, 1994]. This has a number of implications. In the first few iterations the optimized initial conditions (with the large barotropic component) have to undergo a rapid transition during the forward (strongly

A) lnitial Conditionsfrom OI 0 ........ • .:. . __• %

•_-2000 •' -4000

29 30 31 32 33 34 35 36

B) Optimised by Altimetw

.... .... -2000

_4000 / • 29 30 31 32 33 34 35 36

C) Optimised by Driftem

o k.

•.-2000 -4000

29 30 31 32 33 34 35 36

-000 i i 5 29 30

D) Optimised by Altimetry + Drifters

31 32 33 34 35

-2000

-4000

29

36

E) Optimised by Altimetry + Hydro

30 31 32 33 34 35 36 Latitude

: :•'•;,.: "•.:•: ' .

-0.2 -0.1 0 0.1 0.2 Zonal Velocities at 24W (cm/s)

Velocity sections in a meridional cut through Figure 14. 23øW in the model domain, passing through the mean jet at 34øN and the meddy at 35.5øN, (a) the 7-day OI forecast field based on assimilated T/P and ERS 1 data; (b) optimized by altimetry; (c) optimized by altimetry and surface drifters; (d) optimized by altimetry and hydrography; and (e) optimized by . altimetry and drifters and hydrography.


1.50

1.30

'- 1.10

• 0.90

0.70

0,50

a) rms Forecast/Persist Error I I I I I I

I I I

15890 15910 15950

Time (CNES days)

1.00

0.80 --

'-- 0.60-- Q) --

_• 0.40-

0.20 -

0.00

b) Forecast Tendency Correlation I I I I I I

-.• _ •+H

No Opt 15890 15910 15930

Time (CNES days)

2.80

2.40

2.00 1.60

1.20 0.80

0.40

0.00

I I I I 1.00

0.80 -

0.60 -

0.40 -

0.20 -

0.00 -

-0.20

I I I I

. .•.z•.• A+,....D A+H•~ D A

I I I I I I I I I I I I 15890 15910 15950 15890 15910 15950

Time (CNES days) Time (CNES days)

Figure 15. (a) Ratio of rms forecast error to rms persistence error and (b) forecast tendency error for the different model runs, including runs without optimization ("no opt") and with optimization by T/P and ERS1 altimetry (A), surface drifters (D) and hydrography (H). The vertical line denotes the end of the 20-day assimilation period for the optimized runs. Diagnostics are shown for both the (top) surface streamfunction and (bottom) bottom level streamfunction.

baroclinic) QG model run. This transition over the 20-day period occurs at all levels but is especially apparent in the deep layers, and it suggests that the initial regime is wrong. Indeed, after further iterations we reach a stage where the amplitude of the surface data misfit has been sufficiently reduced, so we no longer strongly force the barotropic mode. At this stage the predominant baroclinic partitioning in the QG model formulation (and its adjoint) begins to take effect on the optimized initial conditions, with a gradual reduction in the barotropic mode. The process may be enhanced by the fact that our boundary conditions for the QG model are not updated but are derived from the OI forecast fields which are predominantly baroclinic. Thus the large initial barotopic preturbations may be damped by the low amplitudes specified at the boundaries, which can quickly penetrate into the interior within the 20-day assimilation period.

The excitation of the barotropic mode may be a necessary part of the adjustment process to force the model onto another trajectory which is closer to the data. So we do not want to overconstrain the barotropic mode, at least not initially. However, our best model runs are those which eventually reduce the barotropic mode after 10-20 iterations, and this is most effectively achieved with the good spatial coverage of the altimeter data. (In contrast, the hydrographic data do not constrain the barotropic mode at all, so in our model they cannot be used as the sole data constraint). It is not possible to give more or less weight to a given mode in the data constraints (except hydrography) because altimeter heights and drifter velocities are a sum of modes. However, in further

work we may add an extra constraint to penalize large modal amplitudes and/or to act on the preconditioning.

One part of this assimilation project was the problem of combining different data contraints. We have attempted to estimate the errors and weight the data sets accordingly, as

well as normalising by the effective degrees of freedom. As expected, the model adjustment is slower with a combination of data constraints than when a single constraint is used; generally, more iterations are required to sufficiently decrease the cost function and its gradient. This is even true when we have two altimeter data sets, which should be essentially measuring the same surface dynamics, albeit at different positions and times. The larger number of data points should not slow the dynamical adjustment if they are measuring the same physical processes; but the measurement error inherent in the two data sets (different means removed, background noise, slightly different corrections applied) is not dynamically compatible, either between the two altimeters or with the QG model. Also, the combined noisy data add only a few extra degrees of freedom to our data coverage, but the errors are not statistically redundant, so adding more data adds more noisiness to the descent. However, the fastest

dynamical adjustment is not the final aim of this assimilation; including ERS 1 data does slow the adjustment, but the different sampling also incorporates more observed physical processes, helping produce more realistic surface flOWS.

Data quality is still a problem. Any noisy data after preprocessing will be distributed by the model dynamics in three-dimensional space and backward in time by the adjoint Green's function. In open ocean regions the model acts as a smoother; in sensitive regions, such as near the western boundary, this can produce large instabilities. Care must be taken in the rigorous processing of the data, especially for sparse data sets where one data point can strongly influence local dynamics.

In some model runs the minimization scheme failed after a

limited number of iterations, for example, when the optimal step size became negative. This results from a failure in the


descent algorithm to provide a downhill direction. This can arise when some criteria are in balance, e.g., an altimeter criterion can be downhill, whereas the drifter criterion is

uphill. IN this case (and especially with a coarse line search) the respective gradients cancel, resulting in a loss of conjugacy after some iterations. IN the future, more efficient minimization procedures may need to be implemented for large combinations of different data constraints (such as the limited-memory quasi-Newton method which provides an estimate of the inverse of the Hessian as a by-product).

The hydrographic measurements of the meddy provided an interesting check on how well the combined model and surface altimetry or drifters could predict this subsurface feature. A surface response to the meddy was clearly evident in the drifter and ERS 1 data and in the OI initial guess fields, confirming that their position can be detecting from surface fields [Stammer et al., 1991]. The edge of the meddy was sampled by both T/P and ERS 1 during the 20-day assimilation period, and its surface response suggests that some barotropic or geostrophic adjustment may be associated with the moving meddy. The subsurface response was not well represented by the QG dynamics, perhaps because four baroclinic modes were not sufficient to represent its vertical dynamical structure. The OI model projected most of the meddy's surface signature onto the barotropic mode, whereas the hydrographic data clearly showed the influence of the meddy in the first three baroclinic modes. The adjoint assimilation of the hydrographic data had a minor change on the partitioning of the modes, but the large barotropic adjustment from the surface altimetric forcing clearly dominated the solution. IN future work we will try weighing the modes differently in the constraints.

8. Conclusions

IN this paper we have demonstrated the capabilities of the adjoint method to assimilate a variety of different data sets. Altimetry proved to be a good constraint on the variable flow field and appeared particularly strong in both exciting and constraining the barotropic field. The excellent data quality of the TOPEX/POSEIDON altimeter data provided smooth and

assimilation period, but also can provide stable, longer-term forecasts. This results is made in comparison with the unoptimized initial conditions derived from 7-day forecasts from the OI assimilation of altimetric data. This is not a fair

comparison with the OI assimilation (since the data assimilation periods are different and favor the adjoint runs), but the comparison indicates that there are some basic differences in the statistical and dynamical data assimilation methods, in particular for the deep flow and longer-term forecasts.

9. Future Work

IN all our assimilation runs we have assumed that the

"mean" was perfect, i.e., that the misfits were unbiased. This is not necessarily the case, and we could add a penalty to minimize the bias, weighted by the error variances of the mean.

The difficulty with the hydrographic assimilation was to separate its effect from the stronger surface assimilation. IN the future we propose to combine the hydrographic constraint with other subsurface data constraints, such as the RAFOS floats and current meter data, which have the benefit of providing consistent baroclinic and barotropic fields at the deeper levels.

It is likely that the error growth in the model is mostly barotropic, particularly because the Azores Current is not baroclinically unstable in our model. One of the major conclusions of this paper is that we need to do further work in understanding the model error. We may also need to investigate additional regularization constraints, for example, constraints to penalize large modal amplitudes. This may come through the preconditioning. At present preconditioning is based on the total error variance at each level; we will look at preconditioning the different modal structure, especially for the barotropic mode. Finally a direct comparison between OI and adjoint assimilation is warranted to quantify their individual strengths and weaknesses for projects involving mesoscale dynamics.

Acknowledgments. We wish to thank Andy Moore for the adjoint reliable forcing. Except for mesoscale studies, the spatial • code and for the many helpful discussions. We also would like to thank coverage from the combined T/P and ERS 1 data sets was more Jens Schrtter, Eric Greiner, and an anonymous reviewer for their important for constraining the solution and providing stable flow at all levels. Because the adjoint model acts like a 3-D filter on data noise, the differences in the altimeter data sets due to different noise, corrections, or mean fields were minor in comparison to the benefits from the data coverage.

Surface drifters, when available, provided an excellent constraint on the both the barotropic and baroclinic components of the variable field. More importantly, the drifters provided a much stronger mean jet during our assimilation period, which extended down to 2000 m depth. Since the mean and variable components of the flow are a consistent measure with the drifters, this indicates that the

climatological mean field added to the altimeter data may be too weak, as we suspected. IN general, the vertical structure the model was only weakly improved by the hydrographic constraint on the vertical pressure modes. However, the mean jet was strengthened, consistent with the drifter measurements; and at the position of the meddy the first baroclinic mode was intensified.

The forecast runs over a 2-month period indicate that the adjoint assimilation not only reduces the cost within the

helpful comments and suggestions. The processed data sets were kindly provided by P.-Y. Le Traon (T/P data), Eric Dombrowsky (ERS 1 data) and Fabrice Hernandez (drifters). The hydrographic data collected during the SEMAPHORE campaign were provided by SHOM. The assimilation study was supported by DRET, under the QADRAN program, and by CNRS for P.D.M.

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(Received February 22, 1995; revised July 28,1995; accepted July 28, 1995.)

Adjoint assimilation of altimetric, surface drifter, and hydrographic data in a quasi-geostrophic...

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