• Alves, J. O., , M. A. Balmaseda, , D. L. T. Anderson, , and T. N. Stockdale, 2002: Sensitivity of dynamical seasonal forecasts to ocean initial conditions. ECMWF Tech. Memo. 369, 24 pp. [Available online at http://www.ecmwf.int/publications/.].

    • Search Google Scholar
    • Export Citation
  • Baturin, N. G., , and P. P. Niiler, 1997: Effect of instability waves in the mixed layer of the equatorial Pacific. J. Geophys. Res., 102 , 2777127793.

    • Search Google Scholar
    • Export Citation
  • Behringer, D., , M. Ji, , and A. Leetma, 1998: An improved coupled model for ENSO prediction and implications for ocean initialization. Part I: The ocean data assimilation system. Mon. Wea. Rev., 126 , 10131021.

    • Search Google Scholar
    • Export Citation
  • Bennett, A. F., , B. S. Chua, , D. E. Harrison, , and M. J. McPhaden, 2000: Generalized inversion of Tropical Atmosphere–Ocean (TAO) data using a coupled model of the tropical Pacific. J. Climate, 13 , 27702785.

    • Search Google Scholar
    • Export Citation
  • Bloom, S. C., , L. L. Takacs, , A. M. Da Silva, , and D. Ledvina, 1996: Data assimilation using incremental analysis updates. Mon. Wea. Rev., 124 , 12561271.

    • Search Google Scholar
    • Export Citation
  • Bonekamp, H., , G. J. van Oldenborgh, , and G. Burgers, 2001: Variational assimilation of TAO and XBT data in the HOPE OGCM, adjusting the surface fluxes in the tropical ocean. J. Geophys. Res., 106 , 1669316709.

    • Search Google Scholar
    • Export Citation
  • Burgers, G., , M. A. Balmaseda, , F. C. Vossepoel, , G. J. van Oldenborgh, , and P. J. van Leeuwen, 2002: Balanced ocean-data assimilation near the equator. J. Phys. Oceanogr., 32 , 25092519.

    • Search Google Scholar
    • Export Citation
  • Daley, R., 1991: Atmospheric Data Analysis. Cambridge Atmospheric and Space Sciences Series, Vol. 2, Cambridge University Press, 457 pp.

    • Search Google Scholar
    • Export Citation
  • Daley, R., 1992: Estimating model-error covariances for application to atmospheric data assimilation. Mon. Wea. Rev., 120 , 17351746.

  • Derber, J. C., , and A. Rosati, 1989: A global oceanic data assimilation system. J. Phys. Oceanogr., 19 , 13331347.

  • Fisher, M., , and E. Andersson, 2001: Developments in 4DVAR and Kalman filtering. ECMWF Tech. Memo. 347, 36 pp. [Available online at http://www.ecmwf.int/publications/.].

    • Search Google Scholar
    • Export Citation
  • Gibson, J., , P. Kållberg, , S. Uppala, , A. Noumura, , A. Hernandez, , and E. Serrano, 1997: ERA description. ECMWF Re-Analysis Project Report Series, No. 1, 77 pp. [Available online at http://www.ecmwf.int/research/era/ERA-15/Report_Series/.].

    • Search Google Scholar
    • Export Citation
  • Grima, N., , A. Bentamy, , K. Katsaros, , Y. Quilfen, , P. Delecluse, , and C. Lévy, 1999: Sensitivity study of an oceanic general circulation model forced by satellite wind-stress fields. J. Geophys. Res., 104 , 79677989.

    • Search Google Scholar
    • Export Citation
  • Hollingsworth, A., , and P. Lönnberg, 1986: The statistical structure of short-range forecast errors as determined from radiosonde data. Part I: The wind field. Tellus, 38A , 111136.

    • Search Google Scholar
    • Export Citation
  • Ji, M., , and A. Leetma, 1997: Impact of data assimilation on ocean initialization and El Niño prediction. Mon. Wea. Rev., 125 , 742753.

    • Search Google Scholar
    • Export Citation
  • Kessler, W. S., , M. C. Spillane, , M. J. McPhaden, , and D. E. Harrison, 1996: Scales of variability in the equatorial Pacific inferred from the Tropical Atmosphere–Ocean buoy array. J. Climate, 9 , 29993024.

    • Search Google Scholar
    • Export Citation
  • Levitus, S., 1982: Climatological Atlas of the World Ocean. NOAA Prof. Paper 13, 173 pp. and 17 microfiche.

  • Lorenc, A. C., 1986: Analysis methods for numerical weather prediction. Quart. J. Roy. Meteor. Soc., 112 , 11771194.

  • Madec, G., , P. Delecluse, , M. Imbard, , and C. Levy, 1998: OPA, release 8.1: Ocean general circulation model reference manual. LODYC/IPSL Tech. Note 11, Paris, France, 91 pp. [Available online at http://www.lodyc.jussieu.fr/opa/.].

    • Search Google Scholar
    • Export Citation
  • McCreary, J. P., 1981: A linear stratified ocean model of the Equatorial Undercurrent. Philos. Trans. Roy. Soc. London, 298A , 603635.

    • Search Google Scholar
    • Export Citation
  • Meyers, G., , H. Phillips, , N. Smith, , and J. Sprintall, 1991: Space and time scales for optimal interpolation of temperature—Tropical Pacific Ocean. Progress in Oceanography, Vol. 28, Pergamon Press, 189–218.

    • Search Google Scholar
    • Export Citation
  • Moore, A. M., 1990: Linear equatorial wave mode initialization in a model of the tropical Pacific Ocean: An initialization scheme for tropical ocean models. J. Phys. Oceanogr., 20 , 423445.

    • Search Google Scholar
    • Export Citation
  • Moore, A. M., , and D. L. T. Anderson, 1989: The assimilation of XBT data into a layer model of the tropical Pacific Ocean. Dyn. Atmos. Oceans, 13 , 441464.

    • Search Google Scholar
    • Export Citation
  • Polavarapu, S., , M. Tanguay, , and L. Fillion, 2000: Four-dimensional variational data assimilation with digital filter initialization. Mon. Wea. Rev., 128 , 24912510.

    • Search Google Scholar
    • Export Citation
  • Reverdin, G., , E. Frankignoul, , E. Kestenare, , and M. J. McPhaden, 1994: Seasonal variability in the surface currents of the equatorial Pacific. J. Geophys. Res., 99 , 2032320344.

    • Search Google Scholar
    • Export Citation
  • Reynolds, R. W., , and T. M. Smith, 1994: Improved global sea surface temperature analyses using optimal interpolation. J. Climate, 7 , 929948.

    • Search Google Scholar
    • Export Citation
  • Rosati, A., , K. Miyakoda, , and R. Gudgel, 1997: The impact of ocean initial conditions on ENSO forecasting with a coupled model. Mon. Wea. Rev., 125 , 754772.

    • Search Google Scholar
    • Export Citation
  • Segschneider, J., , D. L. T. Anderson, , and T. N. Stockdale, 2000: Toward the use of altimetry for operational seasonal forecasting. J. Climate, 13 , 31163138.

    • Search Google Scholar
    • Export Citation
  • Segschneider, J., , D. L. T. Anderson, , J. Vialard, , M. A. Balmaseda, , and T. N. Stockdale, 2001: Initialization of seasonal forecasts assimilating sea level and temperature observations. J. Climate, 14 , 42924307.

    • Search Google Scholar
    • Export Citation
  • Smith, N. R., , J. E. Blomley, , and G. Meyers, 1991: A univariate statistical interpolation scheme for subsurface thermal analyses in the tropical oceans. Progress in Oceanography, Vol. 28, Pergamon Press, 219–256.

    • Search Google Scholar
    • Export Citation
  • Troccoli, A., , M. A. Balmaseda, , J. Segschneider, , J. Vialard, , D. L. T. Anderson, , T. N. Stockdale, , K. Haines, , and A. D. Fox, 2002: Salinity adjustments in the presence of temperature data assimilation. Mon. Wea. Rev., 130 , 89102.

    • Search Google Scholar
    • Export Citation
  • Vialard, J., , C. Menkes, , J-P. Boulanger, , P. Delecluse, , E. Guilyardi, , M. J. McPhaden, , and G. Madec, 2001: A model study of oceanic mechanisms affecting equatorial Pacific sea surface temperature during the 1997–98 El Niño. J. Phys. Oceanogr., 31 , 16491675.

    • Search Google Scholar
    • Export Citation
  • Vidard, P. A., 2001: Vers une prise en compte des erreurs modèle en assimilation de données 4D-variationnelle. Application à un modèle réaliste d'océan. Ph.D. thesis, Université Joseph Fourier, Grenoble, France, 196 pp.

    • Search Google Scholar
    • Export Citation
  • Weaver, A. T., , and P. Courtier, 2001: Correlation modelling on the sphere using a generalized diffusion equation. Quart. J. Roy. Meteor. Soc., 127 , 18151846.

    • Search Google Scholar
    • Export Citation
  • Weaver, A. T., , J. Vialard, , D. L. T. Anderson, , and P. Delecluse, 2002: Three- and four-dimensional variational assimilation with a general circulation model of the tropical Pacific Ocean. ECMWF Tech. Memo. 365, 74 pp. [Available online at http://www.ecmwf.int/publications/.].

    • Search Google Scholar
    • Export Citation
  • Weaver, A. T., , J. Vialard, , and D. L. T. Anderson, 2003: Three- and four-dimensional variational assimilation with a general circulation model of the tropical Pacific Ocean. Part I: Formulation, internal diagnostics, and consistency checks. Mon. Wea. Rev., 131 , 13601378.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    Meridional section along 140°W of the 1993–96 average temperature in (a) EX4D, (b) EX3D, and (c) EXCL. The contour interval is 1°C

  • View in gallery

    The 1993–96 average temperature profile in (a) Niño-4 and (c) Niño-3. EX4D is displayed with a solid curve, EX3D with a dashed–dotted curve, and EXCL with a dashed curve. (b), (d) Average (thin curve) and standard deviation (thick curve) of the increments in EX4D and EX3D

  • View in gallery

    The 1993–96 average of the depth of the 20°C isotherm from (a) EX4D, (b) EX3D, and (c) EXCL. The contour interval is 20 m, and the shaded regions indicate values shallower than 140 m

  • View in gallery

    Rms value of the 0–300 m vertically averaged temperature increments along 5°N for EX4D (solid curve) and EX3D (dashed–dotted curve). The thick vertical lines indicate the positions of the TAO moorings

  • View in gallery

    Surface zonal current climatologies. (a) The Reverdin et al. (1994) climatology is representative of the Jan 1987–Apr 1992 period. The climatologies from (b) EX4D, (c) EX3D, and (d) EXCL are averaged over the 1993–96 period. The contour interval is 0.1 m s−1, and the shaded regions indicate eastward currents

  • View in gallery

    The 1993–96 average vertical currents along the equator from (a) EX4D, (b) EX3D, and (c) EXCL. The contour interval is 0.5 m day−1. Dashed contours indicate downwelling

  • View in gallery

    Vertical profiles of zonal currents at the equator from TAO (thick solid curve), EX4D (thin solid curve), EX3D (dashed–dotted curve), and EXCL (dashed curve). This figure is based on a 1993–98 average of the monthly currents with the model being sampled in the same way as the data (i.e., when there are gaps in the data, no model values are used)

  • View in gallery

    Vertical profile of the logarithm of the 1993–96 average vertical mixing coefficient in Niño-3. The solid curve corresponds to EX4D, the dashed–dotted curve to EX3D, and the dashed curve to EXCL

  • View in gallery

    (a) Averaged temperature analysis for EX4D (solid curve), EX3D (dashed–dotted curve), and EXCL (dashed curve). The average is taken over the first 300 m in the TAO region. The thick horizontal line indicates the Levitus climatological value. (b) The instantaneous (thin curve) and time-accumulated (thick curve) temperature analysis increment from EX4D, averaged as in (a). (c) As in (b) but for EX3D. Note the difference in vertical scale between (b) and (c)

  • View in gallery

    (a) Mean and (b) rms of the difference between the average temperature of three experiments without data assimilation with that of EX4D. EX4D is considered here as our best estimate of the true thermal state of the ocean. The average is taken over the first 300 m of the TAO region. The experiments without data assimilation are EXCL (dashed curve) and two data retention experiments, REX3D (dashed–dotted curve) and REX4D (solid curve), initiated from the respective EX3D and EX4D analyses on 26 Dec 1993

  • View in gallery

    (a) Vertical section along the equator of the difference between the temperature in the data retention experiment REX4D (a free integration of the model starting from the EX4D analysis on 26 Dec 1993) and that of EX4D after 1 month. (b) The same quantity for REX3D (a free integration initiated from the EX3D analysis on 26 Dec 1993)

  • View in gallery

    Time series of the daily averaged D20 during 1994 at 5°N, 140°W. The thick curve is the (assimilated) TAO in situ data. The solid curve corresponds to EX4D, the dashed–dotted curve to EX3D, and the dashed curve to EXCL. All time series have been smoothed using a 5-day sliding average. The crosses indicate the D20 of EX3D computed at the observation points from the linear (static) analysis wb(t0) + δwa(t0). The dashed–dotted curve corresponds to the nonlinear analysis trajectory wa(ti) obtained by applying the analysis increment gradually over a 10-day assimilation window

  • View in gallery

    Time series of the high-pass-filtered daily averaged D20 during 1994 at 5°N, 140°W. The high-pass filter retains all timescales shorter than 15 days and thus isolates equatorially trapped internal gravity waves. The thick solid curve corresponds to TAO, the thin solid curve to EX4D, the dashed–dotted curve to EX3D, and the dashed curve to EXCL. For clarity, the curves for EX3D, EX4D, TAO, and EXCL have been shifted by −15, −5, 5, and 15 m, respectively

  • View in gallery

    Time series of the monthly sea level interannual anomaly from TOPEX/Poseidon data (thick curve), and dynamic height from the model referenced to 2000 m in (a) Niño-4 and (b) Niño-3. EX4D is the solid curve, EX3D the dashed–dotted curve, and EXCL the dashed curve. A (0.25, 0.5, 0.25) filter was applied to each curve

  • View in gallery

    (a) The averaged salinity analysis for EX4D (solid curve), EX3D (dashed–dotted curve), and EXCL (dashed curve). The average is taken over the first 300 m in the TAO region (defined as 10°S–10°N, 160°E–80°W). The thick horizontal line indicates the Levitus climatological value. (b) The instantaneous (solid curve) and time-accumulated (thick curve) salinity analysis increment from EX4D [averaged as in (a)]

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 33 33 4
PDF Downloads 18 18 0

Three- and Four-Dimensional Variational Assimilation with a General Circulation Model of the Tropical Pacific Ocean. Part II: Physical Validation

View More View Less
  • 1 European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom, and Laboratoire d'Oceanographie Dynamique et de Climatologie/CNRS/IRD/UPMC/MNHN, Paris, France
  • | 2 Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique/SUC URA 1875, Toulouse, France
  • | 3 European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom
  • | 4 Laboratoire d'Océanographie Dynamique et de Climatologie/CNRS/IRD/UPMC/MNHN, Paris, France
© Get Permissions
Full access

Abstract

Three- and four-dimensional variational assimilation (3DVAR and 4DVAR) systems have been developed for the Océan Parallélisé (OPA) ocean general circulation model of the Laboratoire d'Océanographie Dynamique et de Climatologie. They have been applied to a tropical Pacific version of OPA and cycled over the period 1993–98 using in situ temperature observations from the Global Temperature and Salinity Pilot Programme. The assimilation system is described in detail in Part I of this paper. In this paper, an evaluation of the physical properties of the analyses is undertaken. Experiments performed with a univariate optimal interpolation (OI) scheme give similar results to those obtained with the univariate 3DVAR and are thus not discussed in detail. For the 3DVAR and 4DVAR, it is shown that both the mean state and interannual variability of the thermal field are improved by the assimilation. The fit to the assimilated data in 4DVAR is also very good at timescales comparable to or shorter than the 30-day assimilation window (e.g., at the timescale of tropical instability waves), which demonstrates the effectiveness of the linearized ocean dynamics in carrying information through time. Comparisons with data that are not assimilated are also presented. The intensity of the North Equatorial Counter Current is increased (and improved) in both assimilation experiments. A large eastward bias in the surface currents appears in the eastern Pacific in the 3DVAR analyses, but not in those of 4DVAR. The large current bias is related to a spurious vertical circulation cell that develops along the equatorial strip in 3DVAR. In 4DVAR, the surface current variability is moderately improved. The salinity displays a drift in both experiments but is less accentuated in 4DVAR than in 3DVAR. The better performance of 4DVAR is attributed to multivariate aspects of the 4DVAR analysis coming from the use of the linearized ocean dynamics as a constraint. Even in 4DVAR, however, additional constraints seem necessary to provide better control of the analysis of currents and salinity when observations of those variables are not directly assimilated. Improvements to the analysis can be expected in the future with the inclusion of a multivariate background-error covariance matrix. This and other possible ways of improving the analysis system are discussed.

Corresponding author address: Dr. Jérôme Vialard, LODYC, Case 100, 4 Place Jussieu, 75252 Paris Cedex 05, France. Email: jv@lodyc.jussieu.fr

Abstract

Three- and four-dimensional variational assimilation (3DVAR and 4DVAR) systems have been developed for the Océan Parallélisé (OPA) ocean general circulation model of the Laboratoire d'Océanographie Dynamique et de Climatologie. They have been applied to a tropical Pacific version of OPA and cycled over the period 1993–98 using in situ temperature observations from the Global Temperature and Salinity Pilot Programme. The assimilation system is described in detail in Part I of this paper. In this paper, an evaluation of the physical properties of the analyses is undertaken. Experiments performed with a univariate optimal interpolation (OI) scheme give similar results to those obtained with the univariate 3DVAR and are thus not discussed in detail. For the 3DVAR and 4DVAR, it is shown that both the mean state and interannual variability of the thermal field are improved by the assimilation. The fit to the assimilated data in 4DVAR is also very good at timescales comparable to or shorter than the 30-day assimilation window (e.g., at the timescale of tropical instability waves), which demonstrates the effectiveness of the linearized ocean dynamics in carrying information through time. Comparisons with data that are not assimilated are also presented. The intensity of the North Equatorial Counter Current is increased (and improved) in both assimilation experiments. A large eastward bias in the surface currents appears in the eastern Pacific in the 3DVAR analyses, but not in those of 4DVAR. The large current bias is related to a spurious vertical circulation cell that develops along the equatorial strip in 3DVAR. In 4DVAR, the surface current variability is moderately improved. The salinity displays a drift in both experiments but is less accentuated in 4DVAR than in 3DVAR. The better performance of 4DVAR is attributed to multivariate aspects of the 4DVAR analysis coming from the use of the linearized ocean dynamics as a constraint. Even in 4DVAR, however, additional constraints seem necessary to provide better control of the analysis of currents and salinity when observations of those variables are not directly assimilated. Improvements to the analysis can be expected in the future with the inclusion of a multivariate background-error covariance matrix. This and other possible ways of improving the analysis system are discussed.

Corresponding author address: Dr. Jérôme Vialard, LODYC, Case 100, 4 Place Jussieu, 75252 Paris Cedex 05, France. Email: jv@lodyc.jussieu.fr

1. Introduction

The goal of ocean data assimilation is to combine sparse observations with the output of a numerical model, taking into account their respective error statistics, to produce a more accurate estimate of the ocean state than could be achieved by using either the data or model alone. Improved estimates of the ocean state can be used as initial conditions in coupled ocean–atmosphere models to produce better forecasts of seasonal to interannual climate variability associated with, for example, the El Niño–Southern Oscillation (ENSO) phenomenon (Ji and Leetma 1997; Rosati et al. 1997; Alves et al. 2002; Segschneider et al. 2000, 2001). Combining available observations and an ocean model into a dynamically consistent picture of the ocean state can also help to provide better insight into the processes determining ocean variability at various timescales.

In Part I of this paper, Weaver et al. (2003) described incremental three- and four-dimensional variational assimilation (3DVAR and 4DVAR) systems that have been developed for the Océan Parallélisé (OPA) ocean general circulation model (OGCM) of the Laboratoire d'Océanographie Dynamique et de Climatologie (LODYC). In both systems, a correction (increment) to the model background state (the result of a previous model integration) is sought, which minimizes a cost function measuring the statistically weighted squared differences between the observational information and their model equivalent over a specified time window. In 3DVAR, the approximation is made that changes in the model state at the initial time persist for the duration of the assimilation window, while in 4DVAR, changes at the initial time are translated to changes at later times in the assimilation window by the tangent-linear (TL) model of the OGCM. One of the goals of this study is to assess the benefits of including this dynamical constraint directly in the assimilation problem.

Both the 3DVAR and 4DVAR systems were applied to the tropical Pacific and cycled over the period 1993–98 using in situ temperature observations from the Global Temperature and Salinity Pilot Programme (GTSPP). A control experiment in which no data were assimilated was used as a reference. A 10-day window was used in 3DVAR and a 30-day window in 4DVAR. Persistence was shown to be a reasonable assumption over 10 days, and the TL model was shown to provide a good description of the large-scale oceanic state over several months. Single-observation experiments were performed to illustrate the effect of the TL dynamics in modifying the prior estimates of the background-error variances. The TL dynamics were shown to modify the variances in a physically sensible way. The variances were diminished in the mixed layer, and the maximum value of the variance in the profile was generally increased and displaced to the level of the background thermocline, where thermal variability (and background error) is greatest.

In Part I, certain key diagnostics were used to assess the overall performance of the assimilation systems. A detailed examination of the fit of the different analyses to the assimilated data revealed a large bias below the thermocline in the control analyses, which was strongly reduced in the 3DVAR analyses and almost entirely absent from the 4DVAR analyses. The root-mean-square (rms) difference between the analyses and observations was also very much reduced within the thermocline region. Whereas the rms difference with Tropical Atmosphere Ocean (TAO) array data was about 2.8°C in the control, it was reduced to 1.1°C in 3DVAR and to below 0.5°C in 4DVAR, which was less than the specified standard deviation of (TAO) observation error. Over the TAO region, the rms difference between the observations and the 3DVAR and 4DVAR analyses was shown to be stationary in time (close to 0.9°C in 3DVAR and 0.4°C in 4DVAR) and, in particular, did not exhibit any dependence on the number of observations or interannual variability. Two underlying assumptions in the formulation of the 3DVAR and 4DVAR problems are that the model is perfect and that the background state is unbiased. While these assumptions are not strictly valid, computed statistics on the difference between the analysis and the assimilated data suggested that they were not a major source of problems for the assimilation windows (10 and 30 days) used here.

The purpose of this paper is to extend the evaluation of the assimilation systems by examining the physical properties of the analyses and by comparing the analyses to independent data. The organization of the paper is as follows. Section 2 provides a brief review of the assimilation systems and experiments. In section 3, the 1993–96 average of the analyses is presented and discussed, both for the thermal state and for currents and salinity (for which no observations are used in the assimilation system). In section 4, the variability of the analyzed fields, both at interannual and intraseasonal timescales, is discussed. A summary and discussion are given in section 5.

2. Description of the assimilation systems and experiments

a. The assimilation systems

A short description of the assimilation systems is given here for completeness. A more thorough description can be found in Part I.

1) The incremental variational formulation

An incremental formulation is adopted for both 3DVAR and 4DVAR. Consider a time sequence of observations yoi distributed over a time window t0titn. Let wb = wb(t0) denote the model background state at the beginning of the window. The solution of the incremental variational assimilation problem is then defined by the increment δwa = δwa(t0) to wb that minimizes the quadratic cost function:
i1520-0493-131-7-1379-e1
where di = yoiGi(wb) is the innovation vector, Gi being a nonlinear operator relating wb to the observed quantity, and 𝗚i a linear operator defined such that Gi(w) ≈ Gi(wb) + 𝗚iδw. The matrices 𝗕 and 𝗥i contain estimates of the background- and observation-error covariances, respectively. The Jo term measures the sum of the statistically weighted squared differences between the observations yoi and their linearized model equivalent Gi(wb) + 𝗚iδw, while the Jb term penalizes the size of the increment (i.e., measures the fit to the background state wb).

In both 3DVAR and 4DVAR, Gi contains an integration of the nonlinear model from t0 to ti, followed by a linear interpolation to the observation point, so that the background state can be compared with the observations at their appropriate measurement time and location. The linearized model operator, which is used in 𝗚i to transport the increment forward in time, is fundamentally different in 3DVAR and 4DVAR, however. In 4DVAR, the increment is propagated by the TL operator (i.e., a linearization of the nonlinear model about the time-dependent background-state trajectory),1 whereas in 3DVAR, the increment is transported by the identity matrix (i.e., persistence). This particular version of incremental 3DVAR is known in meteorology as 3D-FGAT, for first guess at the appropriate time (Fisher and Andersson 2001), to distinguish it from the more classical approach to 3DVAR that involves assimilating time-distributed observations simultaneously at the center of the time window. In addition to the advantages brought by the FGAT feature, this incremental 3DVAR provides a conceptually and practically straightforward pathway to incremental 4DVAR.

In practice, the cost function is minimized approximately using an iterative gradient descent method. On each iteration, the gradient of the cost function with respect to the increment is required in order to generate an improved estimate of the increment. This gradient is computed efficiently using the adjoint of the linear operator 𝗚i (the adjoint of the TL model in the case of 4DVAR). A feedback between the linear and nonlinear operators is introduced by allowing the basic-state trajectory of the linear operator to be regularly updated with the most recent estimate of the state trajectory obtained during minimization. The updates are performed on an outer loop of the assimilation algorithm, while the iterations of the actual minimization are performed on a quadratic cost function within an inner loop. The outer loop provides a practical way of accounting for nonlinearities in the assimilation algorithm while retaining the computational advantages of a quadratic minimization problem. In particular, as illustrated in Part I, with several outer iterations, incremental 4DVAR with the linearized model Gi(wb) + 𝗚iδw is able to provide an excellent approximation to the complete 4DVAR problem, which employs the full nonlinear model Gi(wb + δw).

The way the analysis increment δwa is used to correct the trajectory of the nonlinear model differs in our 3DVAR and 4DVAR systems. In 4DVAR, δwa is added directly to the background initial state and the nonlinear model is integrated forward in time to obtain an “analyzed” state trajectory wa(ti) throughout the assimilation window. This is possible since, although the background-error covariance matrix used here is univariate and only thermal data are assimilated, the 4DVAR will also generate increments in currents and salinity because the solution is constrained to verify the linearized ocean dynamics. Because of this constraint, the increment δwa is partly balanced and thus spurious adjustment problems are diminished. A different procedure was necessary in 3DVAR because it generates an analysis increment for temperature only. Adding the temperature analysis increment directly to the background state, as in 4DVAR, produced unrealistic oscillations and large biases in the model fields. These problems were largely alleviated by applying the increment progressively over the assimilation window through a constant three-dimensional forcing term in the prognostic model equation for temperature.

2) The nonlinear, tangent-linear, and adjoint models

A tropical Pacific, rigid-lid version of the OPA OGCM (Madec et al. 1998) is used in this study. This version of the model is described in detail in Vialard et al. (2001). The model is forced using wind stress products from the European Remote Sensing (ERS) satellite's scatterometer (Grima et al. 1999) and a daily heat and freshwater flux climatology computed from the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalyses (ERA-15 reanalysis; Gibson et al. 1997). A relaxation to weekly analyses of sea surface temperature (Reynolds and Smith 1994) is also applied with a −40W m−2 K−1 relaxation coefficient.

The tangent-linear and adjoint models were derived directly from the numerical code of the nonlinear model by applying standard, hand-coding techniques. The only significant approximation that was made in deriving these models was to neglect a vertical mixing term associated with first-order variations of the vertical mixing coefficients.

3) The background-error covariance matrix

A univariate background-error covariance matrix (𝗕) is used in this study. In practice, 𝗕 is defined implicitly through a sequence of operators that are applied within a preconditioning transformation (and its adjoint). Horizontal and vertical autocorrelation functions for the background errors of temperature, salinity, and horizontal velocity are modeled using a diffusion equation (Weaver and Courtier 2001; Part I). Cross correlations between these variables are not accounted for in our current (univariate) 𝗕. The autocorrelation functions implicit in the diffusion operator are approximately Gaussian. A tensor has been introduced in the Laplacian operator of the diffusion equation so that the correlations can be varied spatially and made anisotropic. The horizontal length scales are taken to be a function of latitude and symmetric about the equator, with values close to the climatological observation statistics of Meyers et al. (1991) and Kessler et al. (1996) and broadly similar to those used in previous ocean data assimilation studies of the tropical Pacific (Smith et al. 1991; Behringer et al. 1998; Segschneider et al. 2001). The zonal and meridional length scales are 8° and 2°, respectively, at the equator, and 4° poleward of 20°N/S, with a linear transition between these values within the equatorial strip. The vertical length scales are taken to be a function of depth, being twice the model's vertical resolution to provide adequate smoothing between model levels; the vertical scale is thus 20 m down to about 100 m and increases at greater depths.

The variances of the model state variables have been computed with respect to the climatological mean state of the control integration. These variances have been used at the beginning of each assimilation cycle as a rough approximation to the true background-error variances.

4) Observations

The assimilation dataset consists of in situ temperature observations from the GTSPP of the National Oceanographic Data Center. This includes data from TAO moorings, XBTs, and from a limited number of conductivity–temperature–depth (CTD) casts and drifting buoys. A manual quality control procedure was used to remove suspect data (Alves et al. 2002). Surface data were not used, in order to avoid redundancy with the observed SST (Reynolds and Smith 1994) used in the relaxation term. The observation-error covariances are assumed to be uncorrelated in space and time. The error variances are set to (0.5°C)2 for TAO data and (1.0°C)2 for all other data.

b. Experimental setup

Both the 3DVAR and 4DVAR have been cycled over the period 1 January 1993–30 December 1998 using a 10- and 30-day assimilation window, respectively. A total of 60 (inner) iterations of the minimization were performed per cycle, with an outer iteration performed every 10 inner iterations in 4DVAR. No outer iterations were performed in 3DVAR since the increment model (persistence) is independent of the basic state of the nonlinear model. Two additional experiments were performed to provide a reference for evaluating the 3DVAR and 4DVAR: a control experiment in which no data were assimilated, and an assimilation experiment using a univariate optimal interpolation (OI) scheme (Smith et al. 1991), which is also employed in the operational seasonal forecasting system at ECMWF (Alves et al. 2002). For the OI, the background- and observation-error statistics were chosen to match as closely as possible those used in 3DVAR. In all experiments (hereafter referred to as EX3D, EX4D, EXCL, and EXOI), the initial conditions on 1 January 1993 were generated from a 4-yr spinup of the model starting from rest and from Levitus's (1982) climatological temperature and salinity. The wind stress forcing used for the first 3 yr of the spinup was a climatology computed from the ERS wind stress products. The final year of the spinup was a transition year between ERS climatological and year 1992 products.

The OI analyses are presented in Weaver et al. (2002) and are very similar to those of 3DVAR. This result may be expected from a theoretical point of view since OI and 3DVAR are just different numerical algorithms for solving the same statistical analysis problem (Lorenc 1986). In the present case, however, there are some important differences between the two systems (e.g., in the way the vertical analysis problem is solved, and in the observation-error statistics) that justified the comparison. Because of the great similarity of results between the OI and 3DVAR, the OI analyses will not be discussed further in this paper.

3. Evaluation of 3DVAR and 4DVAR: Climatological mean

In this section, the extended set of analyses (3DVAR, 4DVAR, and control) is examined to assess the impact of the assimilation of in situ data on the model's mean state. The model mean thermal field will be considered first. The climatology of variables for which no direct observations are assimilated (currents and salinity) will then be discussed. In the analyses below, the mean has been computed with respect to the period 1993–96 in order to exclude the very strong 1997/98 El Niño event.

a. Temperature analysis

A vertical section of the 1993–96 average temperature at 140°W is shown in Fig. 1 for EXCL, EX3D, and EX4D. In Part I, it was shown that the 4DVAR analysis (EX4D) is very close to observations in both an average and rms difference sense. The thermal field climatology from EX4D will thus be considered as our best estimate of the true thermal field climatology. As noted by Vialard et al. (2001), the thermocline structure in the control experiment (EXCL) is too diffuse compared to observations. To compensate for this problem, the assimilation of in situ temperature data, in both EX3D and EX4D, produces a large mean correction to the thermal field, which results in a colder thermocline with increased stratification. It can also be seen that the tilt of the isotherms in the North Equatorial Counter Current (NECC) region between 5° and 10°N is considerably reinforced in both data assimilation experiments. The thermocline tightness and tilt is stronger in EX4D than in EX3D (cf. Figs. 1a and 1b near 7°N). This is consistent with the fact that, compared to 3DVAR, 4DVAR produces a closer fit to the assimilated data as seen in Part I.

Figure 2 shows the average profile and the standard deviation of the 4DVAR temperature analysis increment in the Niño-3 (5°S–5°N, 150°–90°W) and Niño-4 (5°S–5°N, 160°E–150°W) regions. First, it can be noted that, while the mean thermal profiles in EX3D and EX4D are very similar, the assimilation increments used to produce them can be very different. Furthermore, the increments do not have an intuitive effect; for example, they do not necessarily tend to cool the model in the region where the control experiment is too warm. This is particularly true for EX3D in Niño-4, where the increments warm the deep ocean even though the control is too warm, and cool the upper ocean, which is too cold in the control. In contrast, in both regions in EX4D, the analysis increments generally act systematically to heat the upper part of the thermocline and cool its lower part, thus counteracting the tendency of the model to spread the thermocline. In Niño-3, the analysis increments in EX3D cool rather strongly the entire thermocline. The quite different vertical structure of the mean analysis increments in EX3D compared to those in EX4D is linked to a spurious circulation cell that develops in the 3DVAR analysis but not in the 4DVAR analysis. As a result, the analysis increments in 3DVAR tend to correct for a bias created by the 3DVAR analysis itself, whereas in 4DVAR they tend to correct for the bias present in the control. This point will be considered further later in this section.

Figure 1 provides a strong indication that the assimilation largely acts to correct for a bias in the model's subsurface temperature structure. One of the assumptions implicit in the present formulation of the assimilation problem is that the increment is unbiased. The long-term average of the temperature increment should be zero and the increment should therefore act to reduce random errors rather than systematic errors. It can be seen in Figs. 2b and 2d that, in both regions, the absolute value of the increment average is significantly smaller than its standard deviation, suggesting that the assumption that the increment is unbiased holds reasonably well in EX4D. In EX3D, however, this assumption seems to breakdown in Niño-3, where the average of the increment reaches −0.6°C at 75-m depth compared to a standard deviation of 1.3°C. This is a large systematic increment, especially so when considering that increments are applied every 10 days in EX3D (thus corresponding to a −0.06°C day−1 systematic correction in EX3D compared to less than 0.01°C day−1 in EX4D). As will be seen later, this strong systematic increment appears primarily to counteract the effects of the spurious vertical downwelling that develops close to the equator in the eastern Pacific in EX3D. Finally, it can be noted that the standard deviation of the temperature corrections is largely concentrated at the level of the thermocline (around 160 m in Niño-4 and 80 m in Niño-3), where the temperature variability is strongest.

Figure 3 shows the 1993–96 average depth of the 20°C isotherm (hereafter D20) for the various experiments. The D20 provides another standard diagnostic for evaluating the impact of the assimilation on the mean thermal field. Once again, EXCL displays an excessively warm thermocline and underestimated meridional gradients between the equator and 8°N, when compared to the other experiments. EX3D exhibits a trough of D20 in the eastern Pacific, near 2°S, which is absent from both the EX4D analysis and TAO data. This spurious trough in EX3D will be discussed in more detail below. EX4D displays undulations in the D20 field near 8°N in the eastern Pacific, which are absent from the other analyses. These oscillations are spurious and are the result of a noisy analysis at locations near 5°–8°N where there is little data in between TAO moorings. This problem also appears in EX3D, but there it is weaker and is not seen clearly in the mean D20. It might be the result of overfitting the data, which can result in unrealistic oscillations being generated between observation points (Daley 1991). Indeed, the rms of the 0–300-m vertically averaged temperature increments along 5°N displays a maximum between the locations of TAO moorings (Fig. 4), with peak-to-trough differences largest in EX4D, which is consistent with Fig. 3. It also appears that the rms is smaller when the TAO moorings have shorter longitudinal separation (10° rather than 15°).

b. Currents and salinity analysis

Up to this point, the impact of the data assimilation on the assimilated field (subsurface temperature) has been examined. How the assimilation of data for this field influences the climatology of other fields will now be investigated. Figure 5 compares the 1993–96 average surface zonal velocity from the various experiments to the Reverdin et al. (1994) climatology. This climatology was derived from drifting buoys and TAO current meters over the January 1987–April 1992 period, so only a qualitative comparison can be made. As noted by Vialard et al. (2001), the control experiment reproduces the main features of the equatorial circulation: the two branches of the westward Southern Equatorial Current (SEC) near 2°N and 3°S and the eastward NECC near 7°N. The NECC and southern branch of the SEC, however, are underestimated in the control experiment (Fig. 5). The intensity of the NECC is clearly improved in both EX3D and EX4D (from 0.1 m s−1 in EXCL to more than 0.2 m s−1 in EX3D and EX4D, compared to 0.3 m s−1 in the Reverdin climatology). This improvement of the NECC intensity can be linked to the steepening of the meridional temperature gradient between 5° and 10°N as noted in Figs. 1b,c, which results in a stronger geostrophic zonal current. Likewise, the southern branch of the SEC is improved in EX3D.

More striking is the large error in EX3D in the eastern equatorial Pacific, which is associated with a surfacing of the Equatorial Undercurrent (EUC) (figure not shown) and unrealistic eastward currents of 0.3 m s−1. This feature is linked through geostrophy to the spurious D20 trough south of the equator in EX3D that was discussed earlier (Fig. 3). It also appears in other univariate data assimilation systems. Bell et al. (2002, manuscript submitted to Quart. J. Roy. Meteor. Soc., hereafter BMN) and Burgers et al. (2002) argue that this feature is associated with a disruption to the main dynamical balance along the equator between the wind stress and the pressure gradient, caused when ocean mass field data are assimilated in order to correct for the effects of wind stress error. In particular, BMN show that this results in a spurious downwelling at depth in the eastern Pacific. This downwelling also appears in EX3D (Fig. 6b), but not in EX4D and EXCL (Figs. 6a,c). It tends to warm the thermocline region in the eastern Pacific in EX3D and probably explains the large negative increments needed there to keep the analysis close to the observations (Fig. 2d). Figure 6b also indicates too strong upwelling in the central/western Pacific, suggesting that this feature, together with the eastward surface currents and unrealistic downwelling in the east, are part of a false circulation cell in the equatorial zone in EX3D. Burgers et al. (2002) have shown that these unrealistic currents in the assimilation experiments can be eliminated by applying velocity corrections in geostrophic balance with the mass field corrections (the meridional derivative of the geostrophic relation being applied close to the equator). In 4DVAR, the analysis increment is constrained to satisfy the dynamics of the TL model so that the velocity and temperature corrections generated by 4DVAR will already be balanced to some extent. This possibly explains why EX4D does not display the false circulation cell and related reversal of the surface currents.

At the equator, the impact of the assimilation on the climatology of the subsurface currents can be assessed by comparing with current meter data from TAO. Figure 7 shows the 1993–98 average zonal current profiles, at three different locations, from TAO data, and from the various experiments. The zonal currents from the control experiment are already quite close to observations in the west but the intensity of the EUC is underestimated in the central and eastern Pacific (Vialard et al. 2001). In the western Pacific (Fig. 7a), the currents in EX3D, with an eastward bias of up to 0.2 m s−1 between 50 and 150 m, are farther from the observations than the currents in EXCL and EX4D, which are quite similar. In the central and eastern Pacific, close to the surface, the currents in EX4D have a 0.1–0.2 m s−1 westward bias, while those in EX3D have a 0.25–0.3 m s−1 eastward bias. The intensity of the EUC maximum is much closer to the observed values in EX3D than in either EXCL or EX4D. However, below the EUC maximum in the eastern Pacific, EX3D produces eastward currents that are far too strong, probably because of the downward advection of momentum by the spurious vertical circulation evident in Fig. 6b. It is not clear why the EUC intensity in EX4D is degraded relative to EXCL. The EUC depends on both the zonal pressure gradient and the vertical distribution of zonal wind stress through vertical mixing processes (McCreary 1981). Since the temperature field fits the observations, the zonal pressure gradient near the equator should be in close agreement with the true gradient. The underestimation of the EUC in EX4D thus possibly stems either from too strong easterlies or from a too deep penetration of the wind momentum in the ocean.

Figure 8 shows the logarithm of the average vertical diffusion coefficient in the whole TAO region (defined here as 10°S–10°N, 160°E–80°W) for EX4D, EX3D, and EXCL. While EXCL has a physically sensible structure, with the vertical mixing coefficient decreasing toward very low values below the surface mixed layer and shear zone of the EUC, both EX3D and EX4D display far too strong values of vertical mixing below 100 m. Closer inspection reveals that this behavior is intimately linked to the background-error vertical correlation scales. The vertical scales that were chosen (locally set to twice the thickness of the model level) are sufficient to provide a smooth analysis in the vertical when the TAO profiles are complete, but when data are missing at some levels, the smoothing provided by the vertical covariances is not sufficient to prevent local instabilities from generating strong spurious mixing over a timescale of a few days. The missing data are frequent enough to increase significantly the climatological value of the mixing at depth. This is one illustration of the critical importance of the background-error covariances in both 3DVAR and 4DVAR.

In this section, it was shown that the assimilation of in situ temperature data impacted significantly the ocean mean state. The thermal structure is improved in the data assimilation experiments (tighter thermocline), with 4DVAR producing the closest fit to the assimilated data. However, the improvement of the climatology of the assimilated variable relative to the control is not necessarily associated with an improvement in all model parameters. In 3DVAR, the structure of the NECC and of the southern branch of the SEC are improved. On the other hand, there is a clear degradation of the surface zonal currents in the eastern Pacific, which is related to a spurious vertical circulation cell that develops along the equatorial strip. In 3DVAR, the analysis increments must continually counteract the effects of this spurious circulation to keep the temperature field close to observations. As a result, one of the main impacts of the 3DVAR is to correct a bias of its own making rather than to correct a bias present in the control. In 4DVAR, the NECC is also improved, the currents in the western Pacific are similar to those in the control, and the EUC is underestimated as in the control. Both 3DVAR and 4DVAR display excessive vertical mixing below the thermocline. The overall better performance of 4DVAR is probably linked to the multivariate aspects of the 4DVAR analysis. This point will be further discussed in section 5.

4. Evaluation of 3DVAR and 4DVAR: Variability

In the previous section, the climatological mean structure of the analyses from the different experiments was investigated. In this section, the variability of the analyses will be evaluated, first in terms of the thermal field and then in terms of other variables (sea level, salinity, and currents).

a. Subsurface temperature variability

1) Interannual variability

Figure 9a shows the time evolution of the monthly averaged temperature in the upper 300 m (hereafter T300) of the TAO region for the various experiments. It clearly shows that the heat content (which is proportional to T300) in EXCL is too high when compared, for example, to the Levitus climatology. This is consistent with the results of the previous section, which showed that the lower thermocline is too warm in EXCL. The analysis increment acts to cool the upper ocean and restore the T300 to the level of the Levitus climatology. While this takes about 2 months in EX3D (i.e., six assimilation cycles), this is achieved after the first 30-day assimilation cycle in EX4D (in January 1993).

Figure 9b shows the temperature analysis increment from EX4D averaged over the same region, together with its time-accumulated value. The analysis increment at the beginning of the first cycle cools the T300 by about 1.5°C over the TAO region, which is approximately equal to the difference between the Levitus climatology and the control at the beginning of 1993. After this abrupt initial cooling, the increment has a tendency in EX4D to warm the ocean from 1993 to the end of 1996, clearly shown by the time-accumulated increment in Fig. 9b. This means that, while the T300 of the model without assimilation has a tendency to be too warm, the T300 at the end of each 4DVAR assimilation window has a slight tendency to be too cold. To compensate for this tendency, a warm increment is then generated at the beginning of the next cycle. This illustrates how the assimilation can change the basic balances in the model by changing the mean circulation and mixing. For example, the increased vertical mixing in EX4D (Fig. 8) could be responsible for the tendency of EX4D to become too cold, by increasing the exchanges of the upper ocean with the colder deep ocean. In EX3D, the increments act to cool systematically the background, as shown in Fig. 9c (note the difference in vertical scale in Figs. 9b and 9c). It is largely due to the spurious downwelling in the eastern Pacific (Fig. 6b), which tends to warm the thermocline. The assimilation increments compensate for this by systematically removing heat (the assimilation increment has contributed to cool the T300 of the TAO region by 3.5°C by the end of EX3D). The tendencies of the increments in EX4D to warm the model and in EX3D to cool it are interannually dependent, as can be seen during the El Niño event in 1997–98.

In EX4D (and in EX3D, after February 1993), the T300 time series in Fig. 9a looks similar to that of EXCL with a constant offset. This illustrates that, despite the large bias, the model is able to reproduce the phase of the observed heat content variability from the specification of the observed wind. Closer examination, however, shows that the variability has a larger amplitude in EX3D and EX4D than in EXCL (e.g., the T300 change between the peak in November 1997 and the trough in August 1998 is 1°C in EXCL, compared to 1.6°C in the assimilation experiments). This shows that the assimilation of in situ temperature data not only has an effect on the mean state but also acts to correct the interannual variability of the heat content. This effect is, however, more modest than the correction to the bias (e.g., the change in the variability is less than 0.6°C, while the change in the T300 bias is about 1.4°C).

The assimilation thus mainly acts to correct for a bias in the temperature field that develops in the model because of systematic model and/or forcing error. Of particular interest is the characteristic time it takes for this bias to develop. To determine this, experiments were performed in which the model was integrated freely (i.e., with the assimilation “turned off”) starting from initial conditions given by the 3DVAR and 4DVAR analyses on 26 December 1993. These will be referred to hereafter as data retention experiments and denoted REX3D and REX4D, respectively. Figure 10 shows the evolution of the mean and rms of the difference between the T300 of REX3D, REX4D, and EXCL, with that of EX4D over the TAO region. EX4D is taken as reference here, because it is the experiment that fits the observed thermal data best (see Part I). For the first 2 months, the mean T300 of REX3D and REX4D stays very close to that of EX4D. However, the rms difference with EX4D shows significant differences in their spatial patterns after the first month. These differences are stronger in REX3D (about 1.3°C) than in REX4D (about 0.9°C).

Figures 11a and 11b show a monthly average of the difference between the temperature field of the first month of REX3D and REX4D, with that of EX4D. This shows clearly that, in REX4D, the model has not drifted far from the observed equatorial thermal state. On the other hand, REX3D displays much larger errors in its thermal state after 1 month (differences larger than 4°C). This is partly due to differences between the initial thermal states of REX3D and REX4D, but mainly to the spurious circulation cell (downwelling in the eastern Pacific, weaker but broader upwelling in the central/western Pacific) present in the initial conditions of REX3D and that, during the early stages of integration, contributes significantly to warm the upper ocean in the east and cool it in the west. After 6 months, REX4D has a mean error in the T300 that is slightly larger than that of REX3D (Fig. 10a), but an rms error that is smaller (Fig. 10b). After 1994, the mean error in T300 stabilizes at about 0.5°C for the period 1995–97. Associated with the 1998 La Niña, there is again a rise of the mean T300 error up to 1°C. The rms error displays a steady rise from about 1.3°C at the beginning of 1995 to about 1.9°C at the end of 1998, but it is always below that of EXCL (between 2.1° and 2.7°C). The model is thus able to retain information about the improved initial thermal state for more than 5 yr. This suggests that, after the initial adjustment, the growth of the error in the retention experiments is linked to slow processes (e.g., horizontal and vertical mixing). After several months, experiments REX4D and REX3D develop biases that look progressively more and more like those of the control, which illustrates the tendency of the model to return to its own mean state when the assimilation is stopped.

2) Higher-frequency variability

We have seen that the assimilation improves the thermal mean state and interannual variability. We will now investigate how the assimilation affects higher-frequency phenomena. Tropical instability waves (TIWs) are the largest contributor to variability in the tropical Pacific at intraseasonal timescales [e.g., see Baturin and Niiler (1997) for a description of TIWs in observations]. The impact of TIWs on the thermal field is strongest north of the equator in the central and eastern Pacific. Figure 12 shows the daily D20 at 5°N, 140°W from TAO data and from the various experiments. The TIWs manifest themselves as large oscillations (up to 60 m from peak to trough) at a 30–40-day timescale. There is no TIW activity during the April–June period.

EXCL does not exhibit clear TIW variability at this location. The TIW variability is slightly stronger in EX3D, but the amplitude is underestimated and there is a phase lag behind observations. In EX4D, however, the TIWs are reproduced with the correct phase and amplitude. The only significant misfit between EX4D and the data is observed in early February and lasts only a few days. This implies that 4DVAR is able to fit variability in the data having a timescale shorter than the 30-day assimilation window. At first, this may seem to contradict the results of Part I, which suggested that the TL model did a poor job of describing TIWs. However, it may point again to the important role of the outer iterations in providing a feedback mechanism between the TL and nonlinear models so that the analyzed trajectory can eventually achieve a very close fit to the data.

The D20 trajectory of EX4D is smooth in the sense that there are no significant “jumps” at analysis time when the increment is applied. This is also true of EX3D, because of the gradual way in which the increments are applied in 3DVAR. Since this procedure acts as a low-pass time filter (Bloom et al. 1996), the smoothness of the 3DVAR analysis is achieved at the expense of accurately resolving TIW activity as illustrated in Fig. 12. Whereas the (static) linear analysis wb(t0) + δwa(t0) [cf. Eq. (1)] of EX3D reproduces quite well the D20 variations associated with the TIWs, the resulting nonlinear “analysis” trajectory wa(ti) does not fit these variations as well.

Figure 13 shows the 15-day high-pass-filtered time series of the D20 from the various experiments and TAO at the same location as in Fig. 12. The TAO D20 exhibits clear variability at a timescale close to 5 days. These are most likely equatorially trapped internal gravity waves (IGWs), with a peak-to-trough amplitude of about 10 m. The amplitude of these waves is very weak in both EXCL and EX3D. On the other hand, EX4D reproduces these waves with both the correct amplitude and phase (the correlation of EX4D with the TAO time series is 0.76 compared to 0.11 for EX3D and −0.04 for EXCL). With a 30-day assimilation window, 4DVAR is thus able to fit phenomena at an approximately 5-day timescale. This is an even more impressive demonstration of the ability of the TL and adjoint models to carry information accurately through time. Such a close fit is not necessarily a good feature, however. Indeed, as seen earlier, overfitting the data in this region causes spurious undulations in the analyzed D20. Furthermore, energetic IGWs generally have small spatial scales and are, thus, not resolved correctly by the observation network. It may then be desirable to filter them out of the analysis. This point will be discussed further in section 5.

b. Comparison with independent observations

We now turn our attention to a comparison of the analyses with independent observations. First, time series of monthly averaged equatorial surface currents from TAO moorings and the different experiments at locations in the western (165°E), central (140°W), and eastern (110°W) Pacific (not shown) indicate that all experiments have an interannual variability that is in broad overall agreement with the observed one. This is true even in EX3D, which, as mentioned earlier, displays a large systematic eastward bias in the central and eastern Pacific. A more quantitative estimate of this agreement is provided in Table 1, which shows the correlation and rms difference between the analyzed currents and TAO. The statistics show that all three experiments behave similarly in the western Pacific and are in good agreement with TAO, with correlation and rms differences of about 0.9 and 0.14 m s−1, respectively. In the central Pacific, the strong eastward bias in EX3D shows up in the large rms difference (0.47 m s−1). On the other hand, the currents in EX4D are improved with respect to EXCL, with a correlation of 0.85 compared to 0.71. In the eastern Pacific, the currents in EX3D display a strong bias that results in a large rms difference (0.47 m s−1). EX4D and EXCL behave similarly in the eastern Pacific.

Altimeter data provide another independent verification dataset for assessing the variability in the different experiments. They give a measure not only of heat content but also of salinity content and thus will provide complementary information to that in Fig. 9a. Figure 14 and Table 2 show a comparison of TOPEX/Poseidon (T/P) sea level data with the surface dynamic height field (relative to 2000 m) computed from the different analyses. The comparison is made on interannual anomalies (with respect to the 1993–96 seasonal cycle) averaged over the Niño-3 and Niño-4 regions. The statistics in Table 2 show that, in both regions, EX4D and EX3D compare well with T/P in the western Pacific, with a correlation around 0.9 and an rms difference smaller than 2.5 cm (compared to 0.54 and 4.8 cm, respectively, in EXCL). In the eastern Pacific, it is EX4D that performs best with an rms difference as low as 1.3 cm, compared to 2.1 cm for EX3D and 2.6 cm for EXCL. In Niño-3, EX4D is in very good agreement with the observed sea level for the whole period, while EX3D overestimates the sea level rise during the 1997 El Niño and EXCL underestimates the sea level fall during the 1998 La Niña. In Niño-4, EXCL and EX4D tend to produce an overestimated sea level anomaly in 1997–98 while EX3D does not. Since Fig. 14 displays anomalies with respect to the 1993–96 period, these discrepancies might be associated with a drift of the analyzed dynamic height over the 1993–98 period. In EXCL, these discrepancies are not surprising since Fig. 9b already showed that the heat content variability in EXCL was not fully consistent with one constrained by in situ temperature observations. For EX3D and EX4D, whose heat content is well constrained by the observations in the TAO region, the inconsistency between the dynamic height and T/P sea level anomalies is largely due to inaccuracies in the salinity.

Figure 15a shows the evolution of the average salinity in the top 300 m (hereafter S300) of the TAO region in the three experiments. All start from a value close to the Levitus climatology in January 1993. EXCL drifts slowly toward fresher values (about 0.1 psu freshening after 6 yr), probably due to a combination of model and forcing errors. This freshening is strongly accentuated in EX3D. In a separate study, using a different OGCM, Troccoli et al. (2002) observed a similar freshening of the upper ocean in analyses produced with a univariate OI scheme (the same one briefly discussed in section 2b and used as a reference in Weaver et al. 2002). They attributed this feature to spurious vertical mixing and convection, which was generated in regions where the density profile was destabilized by making corrections for temperature while leaving the salinity field unchanged. The fact that our 3DVAR system is also univariate might suggest that the aforementioned problem is also responsible for the excessive freshening seen in Fig. 15a. In the 4DVAR system, on the other hand, salinity corrections are induced via the TL model constraint. The time-accumulated salinity increment in EX4D, averaged over the top 300 m of the TAO region, is shown in Fig. 15b. During 1995, the 4DVAR increment adds an average of 0.2 psu in the top 300 m of the TAO region, which is partially reflected in the average salinity time series in Fig. 15a. The tendency of the model to drift in EXCL seems to be partially corrected for in EX4D, except after mid-1997 when EX4D exhibits a dramatic drop in the average salinity. This drop (0.2 psu of the S300 in 1 yr, which would correspond to 5 mm day−1 of precipitation over the whole TAO region) is too large to be realistic, and is largely associated with the 4DVAR salinity increments. This suggests that the constraints introduced by the TL dynamics are probably not sufficient to constrain the salinity field fully and that additional information may be necessary (e.g., direct observations of salinity and/or TS relationships).

In this section, we have shown that 3DVAR and 4DVAR improve the variability of the heat content; for example, the amplitude of the differences between the 1997 El Niño and 1998 La Niña are better reproduced in EX4D and EX3D than in EXCL. Here, 4DVAR is able to fit phenomena, such as TIWs and IGWs, which have characteristic timescales comparable to or much shorter than the width of the assimilation window. The sea level variability is also improved since it is very closely linked to the heat content variability. This improvement is most noticeable in EX4D. The improvement of the variability of other variables (surface currents and salinity) is less spectacular, however. The surface current variability is improved in EX4D in the central Pacific but very little in the eastern and western Pacific. A salinity drift is introduced in both 3DVAR and 4DVAR, suggesting that additional constraints on nonassimilated variables (such as salinity) are necessary in both 3DVAR and 4DVAR.

5. Summary and discussion

a. Summary

Three- and four-dimensional variational assimilation systems have been developed for the rigid-lid version of the OPA OGCM (Madec et al. 1998). The assimilation systems were described in detail in Part I (Weaver et al. 2003). Both systems have been applied to the tropical Pacific Ocean and cycled over the period 1993–98 using in situ temperature observations from the GTSPP. A 10-day window was used in 3DVAR and a 30-day window in 4DVAR. A control experiment without data assimilation was used as the main reference for evaluating the performance of the assimilation systems. An additional reference experiment using the ECMWF OI assimilation system (Alves et al. 2002) was also performed with the OPA model but the results were found to be very similar to those of the 3DVAR and thus were not presented in this paper. [Comparisons with the OI are discussed in Weaver et al. (2002).]

The 1993–96 climatologies of the model temperature field showed that the overall effect of the assimilation is to greatly reduce a model bias by lifting and tightening the thermocline relative to that of the control. Experiments initiated with 3DVAR and 4DVAR analyses were performed to determine how long these improvements in the heat content could be preserved when the assimilation is switched off. After 1 month, the experiment starting from the 4DVAR analysis was still close to the observed thermal state, whereas the one starting from the 3DVAR analysis degraded more quickly especially in the eastern Pacific. At longer lead times the two experiments displayed similar behavior, with a significant bias developing in the thermocline in both after about 6 months. Even so, improvement of the heat content is retained for more than 5 yr. This result is in marked contrast to the earlier studies of Moore and Anderson (1989) and Derber and Rosati (1989) where information obtained by assimilation was lost quickly through the action of internal Kelvin waves.

The assimilation improved the mean analyzed thermal state, which in turn resulted in an improvement in the intensity of the NECC. The assimilation did not result, however, in a global improvement of the current field. In 4DVAR, as in the control, the EUC was underestimated by 0.4 m s−1. The EUC was better reproduced in 3DVAR and OI but in those experiments the surface currents exhibited a large eastward bias in the eastern Pacific and a spurious vertical circulation.

Not only was the mean thermal state improved by the assimilation, but also the variability, with the 3DVAR and 4DVAR analyses showing a larger, more realistic interannual variability in heat content and sea level than the control analyses. The fit of the 4DVAR analyses to the assimilated thermal data was also very close at timescales comparable to or much shorter than the 30-day width of the assimilation window. The 4DVAR (nonlinear) analysis trajectory closely followed changes in the thermal state associated with TIWs (30–40-day timescale) and IGWs (5–7-day timescale). The 3DVAR static (linear) analysis also captured variability associated with TIWs. The 3DVAR (nonlinear) analysis trajectory, however, did not reproduce well this variability because of the temporal filtering properties of the procedure used to merge the analysis increment into the model. The variability of the surface currents was also investigated. The surface current variability in 4DVAR was better than the control in the central Pacific and as good in other regions. The salinity field displayed a strong drift toward fresher values during the whole 1993–98 period in the 3DVAR analyses, and at the end of the experiment in the 4DVAR analyses.

b. Discussion

In this paper, we have given a physical assessment of the ocean analyses generated by our incremental 3DVAR and 4DVAR systems. The strengths and limitations of the assimilation systems are discussed below.

In 4DVAR, the model analysis trajectory is able to reproduce variability in the data with timescales comparable to or much shorter than the 30-day window used in the experiments. In particular, TIWs with periods near 30 days and IGWs with periods near 5 days are both well represented. While fitting observations at the timescale of IGWs is not necessarily a good feature (a point discussed further later), this is nonetheless an impressive illustration of the ability of 4DVAR to adjust a model trajectory on a 30-day window. The 3DVAR analysis trajectory does not achieve such a close fit, mainly because of the gradual way in which the analysis increment is fed into the model, which was done out of necessity to reduce the excitation of unrealistic adjustment processes. The overall effect is that in 4DVAR the fit to the data is within the specified standard error, whereas in 3DVAR it is not. In this respect, the introduction of dynamical constraints is very positive.

As recently pointed out by Fisher and Andersson (2001), it is not obvious if the generally better performance of 4DVAR over 3DVAR can be attributed exclusively to the implicit flow dependency of the background-error covariances in 4DVAR. The close fit to the data obtained in 4DVAR can partly be explained by the single-observation experiments in Part I, which demonstrated that the TL dynamics act to decrease the weight given to the background state near the thermocline by implicitly increasing the background-error variances there. The close fit is possibly also the result of another desirable property of 4DVAR, however. Consider the exact minimizing solution of the cost function (1), which can be written as
δwa−1T−1−1T−1d
where the generalized quantities d = (… , dTi, …)T, 𝗚 = (… , 𝗚Ti, …)T, and 𝗥 = diag(… , 𝗥i, …) have been introduced to simplify the notation (Lorenc 1986). In (2), we note that the linearized model dynamics can influence the 4DVAR analysis in two separate ways. First, through the adjoint model, which is contained in 𝗚T and which is used to propagate the weighted innovation 𝗥−1d back to the analysis time at t0, and second, in the weighting factor 𝗣a = (𝗕−1 + 𝗚T𝗥−1𝗚)−1, which corresponds to the analysis-error covariance matrix. In 4DVAR, the action of the TL and adjoint models is to introduce a flow dependency in the term 𝗚T𝗥−1𝗚 of 𝗣a. Fisher and Andersson (2001) present a simple example in which 4DVAR produces superior results to 3DVAR even when the background-, analysis- and forecast-error covariance matrices are all stationary. This superiority is attributed to the interpolation properties of the adjoint model, which is used to propagate the weighted innovations as mentioned above. They provide further experimental evidence from the ECMWF meteorological 4DVAR system to suggest that covariance evolution is not the dominating effect. In our study, the ability of 4DVAR to act as an accurate propagator of information in time is demonstrated by the close fit to phenomena at the 5-day timescale (IGWs). This would suggest then that the interpolation properties are also an important feature here. To what extent, however, these effects are responsible for the main differences observed between our ocean 3DVAR and 4DVAR experiments is a matter for further research.

Whereas the assimilation methods improve the fit to the assimilated data, they can have a negative effect on the other fields. This is particularly true for 3DVAR, which displayed mean eastward currents at the surface, unrealistic upwelling/downwelling in the western/eastern Pacific, and a salinity drift toward lower values. These problems can be traced back to the absence of multivariate balance constraints, for example, in the background-error covariance matrix (𝗕). Since only the thermal field is corrected in 3DVAR, and not salinity and currents, this can lead to adjustment problems. These problems are reduced in 4DVAR. Despite a univariate 𝗕, the dynamical constraint in 4DVAR results in salinity and current field corrections that are partially in balance with the thermal increments, leading to better surface currents and reduced salinity drift. However, there are still some areas where 4DVAR is not as good as the control. In particular, the EUC is underestimated in 4DVAR and the salinity content displays some unrealistic variations. This probably illustrates that, in 4DVAR, dynamical constraints alone are not sufficient to transfer information across model variables. A multivariate 𝗕 could improve the analysis of the unobserved variables. In 3DVAR, this could alleviate problems associated with the surfacing of the EUC in the eastern Pacific and with salinity drift (Burgers et al. 2002; Troccoli et al. 2002).

It has been shown that overfitting might be a problem in our 3DVAR and 4DVAR experiments. Overfitting can result in unrealistic oscillations being generated between observation points (e.g., see Daley 1991) and this feature typically appears between TAO moorings at about 5–8°N in both assimilation experiments. The 4DVAR analyses reproduce, at the observation points, the variability in the thermal field associated with IGWs. The TAO array is not dense enough, however, to resolve these waves properly so it may be wiser to filter them out altogether. For example, this could be done by treating them as representativeness error and modifying the observation-error covariance matrix (𝗥) accordingly. Furthermore, a better representation of 𝗥 would allow more information to be extracted from the background minus observation and analysis minus observation residuals for ultimately improving the estimate of 𝗕 (Hollingsworth and Lönnberg 1986; Daley 1992). A precise tuning of 𝗕 and 𝗥 is essential to avoid overfitting. The introduction of additional (weak) constraints in the assimilation method could also help to reduce the amount of “noise” in the analysis. Possibilities include the use of a normal mode initialization scheme to restrict the analysis to the slow manifold of the system (Moore 1990), or the inclusion of an additional term in the cost function to penalize short timescales (Polavarapu et al. 2000).

The assimilation window that has been used in the 4DVAR experiments in the present study is 30 days. In principle, we would like to use a longer assimilation window to allow greater dynamical propagation of information (e.g., it takes a first baroclinic mode Kelvin wave 2 months to cross the Pacific Ocean basin). However, the length of the assimilation window is limited by the validity of the TL model, and by the hypotheses that the model and forcing are perfect. We have seen that the TL approximation was accurate at the large scales, but somewhat limited in the small scales because of nonlinear processes associated with TIWs, convection and vertical mixing. However, there is scope for improving the physics in the TL model. The assumption that the model and the forcing fields are perfect is obviously not true, as illustrated by the bias that develops in the model when data assimilation is switched off. Diagnostics on the analysis increments and analysis minus observation residuals suggest, however, that model and forcing errors are not a major problem with a 30-day assimilation window.

The background- and observation-error covariance matrices used in this study were derived somewhat heuristically, with computational efficiency and simplicity being an important consideration. The structures are reasonable for a first estimate but the statistical parameters have not been tuned objectively. We are currently in the process of developing a more comprehensive background-error covariance matrix, including, for example, multivariate constraints between temperature and salinity, and between density and currents. The correlation model is also being extended to include non-Gaussian functions and flow-dependent coordinate surfaces (Weaver and Courtier 2001). Furthermore, we are currently using TAO data to develop an improved estimate of the observation-error covariance matrix that, in particular, accounts for IGWs as a source of representativeness error. The importance of model and forcing errors should also be assessed. For example, in an equatorial oceanographic context, where large spatial scales are deterministically related to the wind forcing, it might be desirable to control the forcing error (Bonekamp et al. 2001) or model error (Bennett et al. 2000) as well as the initial conditions. Some preliminary studies on these two aspects are currently being undertaken with the present system (Vidard 2001).

The present 3DVAR and 4DVAR systems have been applied to the tropical Pacific basin. Global versions for a free-surface version of OPA are being developed. This version will include some important new parameterizations to the physics such as isopycnal and eddy-induced diffusion. The use of improved physics might extend the period for which improvements brought by assimilation, such as a tightening of the thermocline, are retained. Sources of information other than in situ temperature data are available (salinity, current meter data, altimetry,  …). As a first step toward assimilating more observation types, we will be introducing sea level data from altimetry in the global systems. Finally, one of the main reasons for developing the assimilation systems is to provide improved ocean analyses for seasonal forecasting. The quality of the present analyses will thus have to be assessed in terms of forecast skill using a coupled ocean–atmosphere model.

Acknowledgments

Most of this work was pursued while the first author was working at ECMWF. We thank Oscar Alves for his help in implementing the OI in the OPA model. Erik Andersson, Magdalena Balmaseda, Mike Cullen, Mike Fisher, Andy Moore, Andrea Piacentini, and Philippe Rogel provided many useful suggestions for improving the manuscript. We are also grateful to three anonymous reviewers for their constructive and insightful remarks. The second author wishes to acknowledge funding from the French MERCATOR and EC-FP5 ENACT projects.

REFERENCES

  • Alves, J. O., , M. A. Balmaseda, , D. L. T. Anderson, , and T. N. Stockdale, 2002: Sensitivity of dynamical seasonal forecasts to ocean initial conditions. ECMWF Tech. Memo. 369, 24 pp. [Available online at http://www.ecmwf.int/publications/.].

    • Search Google Scholar
    • Export Citation
  • Baturin, N. G., , and P. P. Niiler, 1997: Effect of instability waves in the mixed layer of the equatorial Pacific. J. Geophys. Res., 102 , 2777127793.

    • Search Google Scholar
    • Export Citation
  • Behringer, D., , M. Ji, , and A. Leetma, 1998: An improved coupled model for ENSO prediction and implications for ocean initialization. Part I: The ocean data assimilation system. Mon. Wea. Rev., 126 , 10131021.

    • Search Google Scholar
    • Export Citation
  • Bennett, A. F., , B. S. Chua, , D. E. Harrison, , and M. J. McPhaden, 2000: Generalized inversion of Tropical Atmosphere–Ocean (TAO) data using a coupled model of the tropical Pacific. J. Climate, 13 , 27702785.

    • Search Google Scholar
    • Export Citation
  • Bloom, S. C., , L. L. Takacs, , A. M. Da Silva, , and D. Ledvina, 1996: Data assimilation using incremental analysis updates. Mon. Wea. Rev., 124 , 12561271.

    • Search Google Scholar
    • Export Citation
  • Bonekamp, H., , G. J. van Oldenborgh, , and G. Burgers, 2001: Variational assimilation of TAO and XBT data in the HOPE OGCM, adjusting the surface fluxes in the tropical ocean. J. Geophys. Res., 106 , 1669316709.

    • Search Google Scholar
    • Export Citation
  • Burgers, G., , M. A. Balmaseda, , F. C. Vossepoel, , G. J. van Oldenborgh, , and P. J. van Leeuwen, 2002: Balanced ocean-data assimilation near the equator. J. Phys. Oceanogr., 32 , 25092519.

    • Search Google Scholar
    • Export Citation
  • Daley, R., 1991: Atmospheric Data Analysis. Cambridge Atmospheric and Space Sciences Series, Vol. 2, Cambridge University Press, 457 pp.

    • Search Google Scholar
    • Export Citation
  • Daley, R., 1992: Estimating model-error covariances for application to atmospheric data assimilation. Mon. Wea. Rev., 120 , 17351746.

  • Derber, J. C., , and A. Rosati, 1989: A global oceanic data assimilation system. J. Phys. Oceanogr., 19 , 13331347.

  • Fisher, M., , and E. Andersson, 2001: Developments in 4DVAR and Kalman filtering. ECMWF Tech. Memo. 347, 36 pp. [Available online at http://www.ecmwf.int/publications/.].

    • Search Google Scholar
    • Export Citation
  • Gibson, J., , P. Kållberg, , S. Uppala, , A. Noumura, , A. Hernandez, , and E. Serrano, 1997: ERA description. ECMWF Re-Analysis Project Report Series, No. 1, 77 pp. [Available online at http://www.ecmwf.int/research/era/ERA-15/Report_Series/.].

    • Search Google Scholar
    • Export Citation
  • Grima, N., , A. Bentamy, , K. Katsaros, , Y. Quilfen, , P. Delecluse, , and C. Lévy, 1999: Sensitivity study of an oceanic general circulation model forced by satellite wind-stress fields. J. Geophys. Res., 104 , 79677989.

    • Search Google Scholar
    • Export Citation
  • Hollingsworth, A., , and P. Lönnberg, 1986: The statistical structure of short-range forecast errors as determined from radiosonde data. Part I: The wind field. Tellus, 38A , 111136.

    • Search Google Scholar
    • Export Citation
  • Ji, M., , and A. Leetma, 1997: Impact of data assimilation on ocean initialization and El Niño prediction. Mon. Wea. Rev., 125 , 742753.

    • Search Google Scholar
    • Export Citation
  • Kessler, W. S., , M. C. Spillane, , M. J. McPhaden, , and D. E. Harrison, 1996: Scales of variability in the equatorial Pacific inferred from the Tropical Atmosphere–Ocean buoy array. J. Climate, 9 , 29993024.

    • Search Google Scholar
    • Export Citation
  • Levitus, S., 1982: Climatological Atlas of the World Ocean. NOAA Prof. Paper 13, 173 pp. and 17 microfiche.

  • Lorenc, A. C., 1986: Analysis methods for numerical weather prediction. Quart. J. Roy. Meteor. Soc., 112 , 11771194.

  • Madec, G., , P. Delecluse, , M. Imbard, , and C. Levy, 1998: OPA, release 8.1: Ocean general circulation model reference manual. LODYC/IPSL Tech. Note 11, Paris, France, 91 pp. [Available online at http://www.lodyc.jussieu.fr/opa/.].

    • Search Google Scholar
    • Export Citation
  • McCreary, J. P., 1981: A linear stratified ocean model of the Equatorial Undercurrent. Philos. Trans. Roy. Soc. London, 298A , 603635.

    • Search Google Scholar
    • Export Citation
  • Meyers, G., , H. Phillips, , N. Smith, , and J. Sprintall, 1991: Space and time scales for optimal interpolation of temperature—Tropical Pacific Ocean. Progress in Oceanography, Vol. 28, Pergamon Press, 189–218.

    • Search Google Scholar
    • Export Citation
  • Moore, A. M., 1990: Linear equatorial wave mode initialization in a model of the tropical Pacific Ocean: An initialization scheme for tropical ocean models. J. Phys. Oceanogr., 20 , 423445.

    • Search Google Scholar
    • Export Citation
  • Moore, A. M., , and D. L. T. Anderson, 1989: The assimilation of XBT data into a layer model of the tropical Pacific Ocean. Dyn. Atmos. Oceans, 13 , 441464.

    • Search Google Scholar
    • Export Citation
  • Polavarapu, S., , M. Tanguay, , and L. Fillion, 2000: Four-dimensional variational data assimilation with digital filter initialization. Mon. Wea. Rev., 128 , 24912510.

    • Search Google Scholar
    • Export Citation
  • Reverdin, G., , E. Frankignoul, , E. Kestenare, , and M. J. McPhaden, 1994: Seasonal variability in the surface currents of the equatorial Pacific. J. Geophys. Res., 99 , 2032320344.

    • Search Google Scholar
    • Export Citation
  • Reynolds, R. W., , and T. M. Smith, 1994: Improved global sea surface temperature analyses using optimal interpolation. J. Climate, 7 , 929948.

    • Search Google Scholar
    • Export Citation
  • Rosati, A., , K. Miyakoda, , and R. Gudgel, 1997: The impact of ocean initial conditions on ENSO forecasting with a coupled model. Mon. Wea. Rev., 125 , 754772.

    • Search Google Scholar
    • Export Citation
  • Segschneider, J., , D. L. T. Anderson, , and T. N. Stockdale, 2000: Toward the use of altimetry for operational seasonal forecasting. J. Climate, 13 , 31163138.

    • Search Google Scholar
    • Export Citation
  • Segschneider, J., , D. L. T. Anderson, , J. Vialard, , M. A. Balmaseda, , and T. N. Stockdale, 2001: Initialization of seasonal forecasts assimilating sea level and temperature observations. J. Climate, 14 , 42924307.

    • Search Google Scholar
    • Export Citation
  • Smith, N. R., , J. E. Blomley, , and G. Meyers, 1991: A univariate statistical interpolation scheme for subsurface thermal analyses in the tropical oceans. Progress in Oceanography, Vol. 28, Pergamon Press, 219–256.

    • Search Google Scholar
    • Export Citation
  • Troccoli, A., , M. A. Balmaseda, , J. Segschneider, , J. Vialard, , D. L. T. Anderson, , T. N. Stockdale, , K. Haines, , and A. D. Fox, 2002: Salinity adjustments in the presence of temperature data assimilation. Mon. Wea. Rev., 130 , 89102.

    • Search Google Scholar
    • Export Citation
  • Vialard, J., , C. Menkes, , J-P. Boulanger, , P. Delecluse, , E. Guilyardi, , M. J. McPhaden, , and G. Madec, 2001: A model study of oceanic mechanisms affecting equatorial Pacific sea surface temperature during the 1997–98 El Niño. J. Phys. Oceanogr., 31 , 16491675.

    • Search Google Scholar
    • Export Citation
  • Vidard, P. A., 2001: Vers une prise en compte des erreurs modèle en assimilation de données 4D-variationnelle. Application à un modèle réaliste d'océan. Ph.D. thesis, Université Joseph Fourier, Grenoble, France, 196 pp.

    • Search Google Scholar
    • Export Citation
  • Weaver, A. T., , and P. Courtier, 2001: Correlation modelling on the sphere using a generalized diffusion equation. Quart. J. Roy. Meteor. Soc., 127 , 18151846.

    • Search Google Scholar
    • Export Citation
  • Weaver, A. T., , J. Vialard, , D. L. T. Anderson, , and P. Delecluse, 2002: Three- and four-dimensional variational assimilation with a general circulation model of the tropical Pacific Ocean. ECMWF Tech. Memo. 365, 74 pp. [Available online at http://www.ecmwf.int/publications/.].

    • Search Google Scholar
    • Export Citation
  • Weaver, A. T., , J. Vialard, , and D. L. T. Anderson, 2003: Three- and four-dimensional variational assimilation with a general circulation model of the tropical Pacific Ocean. Part I: Formulation, internal diagnostics, and consistency checks. Mon. Wea. Rev., 131 , 13601378.

    • Search Google Scholar
    • Export Citation

Fig. 1.
Fig. 1.

Meridional section along 140°W of the 1993–96 average temperature in (a) EX4D, (b) EX3D, and (c) EXCL. The contour interval is 1°C

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 2.
Fig. 2.

The 1993–96 average temperature profile in (a) Niño-4 and (c) Niño-3. EX4D is displayed with a solid curve, EX3D with a dashed–dotted curve, and EXCL with a dashed curve. (b), (d) Average (thin curve) and standard deviation (thick curve) of the increments in EX4D and EX3D

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 3.
Fig. 3.

The 1993–96 average of the depth of the 20°C isotherm from (a) EX4D, (b) EX3D, and (c) EXCL. The contour interval is 20 m, and the shaded regions indicate values shallower than 140 m

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 4.
Fig. 4.

Rms value of the 0–300 m vertically averaged temperature increments along 5°N for EX4D (solid curve) and EX3D (dashed–dotted curve). The thick vertical lines indicate the positions of the TAO moorings

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 5.
Fig. 5.

Surface zonal current climatologies. (a) The Reverdin et al. (1994) climatology is representative of the Jan 1987–Apr 1992 period. The climatologies from (b) EX4D, (c) EX3D, and (d) EXCL are averaged over the 1993–96 period. The contour interval is 0.1 m s−1, and the shaded regions indicate eastward currents

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 6.
Fig. 6.

The 1993–96 average vertical currents along the equator from (a) EX4D, (b) EX3D, and (c) EXCL. The contour interval is 0.5 m day−1. Dashed contours indicate downwelling

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 7.
Fig. 7.

Vertical profiles of zonal currents at the equator from TAO (thick solid curve), EX4D (thin solid curve), EX3D (dashed–dotted curve), and EXCL (dashed curve). This figure is based on a 1993–98 average of the monthly currents with the model being sampled in the same way as the data (i.e., when there are gaps in the data, no model values are used)

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 8.
Fig. 8.

Vertical profile of the logarithm of the 1993–96 average vertical mixing coefficient in Niño-3. The solid curve corresponds to EX4D, the dashed–dotted curve to EX3D, and the dashed curve to EXCL

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 9.
Fig. 9.

(a) Averaged temperature analysis for EX4D (solid curve), EX3D (dashed–dotted curve), and EXCL (dashed curve). The average is taken over the first 300 m in the TAO region. The thick horizontal line indicates the Levitus climatological value. (b) The instantaneous (thin curve) and time-accumulated (thick curve) temperature analysis increment from EX4D, averaged as in (a). (c) As in (b) but for EX3D. Note the difference in vertical scale between (b) and (c)

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 10.
Fig. 10.

(a) Mean and (b) rms of the difference between the average temperature of three experiments without data assimilation with that of EX4D. EX4D is considered here as our best estimate of the true thermal state of the ocean. The average is taken over the first 300 m of the TAO region. The experiments without data assimilation are EXCL (dashed curve) and two data retention experiments, REX3D (dashed–dotted curve) and REX4D (solid curve), initiated from the respective EX3D and EX4D analyses on 26 Dec 1993

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 11.
Fig. 11.

(a) Vertical section along the equator of the difference between the temperature in the data retention experiment REX4D (a free integration of the model starting from the EX4D analysis on 26 Dec 1993) and that of EX4D after 1 month. (b) The same quantity for REX3D (a free integration initiated from the EX3D analysis on 26 Dec 1993)

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 12.
Fig. 12.

Time series of the daily averaged D20 during 1994 at 5°N, 140°W. The thick curve is the (assimilated) TAO in situ data. The solid curve corresponds to EX4D, the dashed–dotted curve to EX3D, and the dashed curve to EXCL. All time series have been smoothed using a 5-day sliding average. The crosses indicate the D20 of EX3D computed at the observation points from the linear (static) analysis wb(t0) + δwa(t0). The dashed–dotted curve corresponds to the nonlinear analysis trajectory wa(ti) obtained by applying the analysis increment gradually over a 10-day assimilation window

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 13.
Fig. 13.

Time series of the high-pass-filtered daily averaged D20 during 1994 at 5°N, 140°W. The high-pass filter retains all timescales shorter than 15 days and thus isolates equatorially trapped internal gravity waves. The thick solid curve corresponds to TAO, the thin solid curve to EX4D, the dashed–dotted curve to EX3D, and the dashed curve to EXCL. For clarity, the curves for EX3D, EX4D, TAO, and EXCL have been shifted by −15, −5, 5, and 15 m, respectively

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 14.
Fig. 14.

Time series of the monthly sea level interannual anomaly from TOPEX/Poseidon data (thick curve), and dynamic height from the model referenced to 2000 m in (a) Niño-4 and (b) Niño-3. EX4D is the solid curve, EX3D the dashed–dotted curve, and EXCL the dashed curve. A (0.25, 0.5, 0.25) filter was applied to each curve

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Fig. 15.
Fig. 15.

(a) The averaged salinity analysis for EX4D (solid curve), EX3D (dashed–dotted curve), and EXCL (dashed curve). The average is taken over the first 300 m in the TAO region (defined as 10°S–10°N, 160°E–80°W). The thick horizontal line indicates the Levitus climatological value. (b) The instantaneous (solid curve) and time-accumulated (thick curve) salinity analysis increment from EX4D [averaged as in (a)]

Citation: Monthly Weather Review 131, 7; 10.1175/1520-0493(2003)131<1379:TAFVAW>2.0.CO;2

Table 1.

Correlation and rms difference (in m s−1) between monthly equatorial surface currents from TAO and the different analyses for 1993-98

Table 1.
Table 2.

Correlation and rms difference (cm) between monthly sea level from TOPEX/Poseidon and the model dynamic height from the different analyses for 1993-98

Table 2.

1

In practice, various simplifications have been introduced in the TL model (Weaver et al. 2003) so that, strictly speaking, it is not tangent to the full nonlinear model.

Save