A reduced-dynamics variational approach for the assimilation of altimeter data into eddy-resolving ocean models

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Abstract

A new method of assimilating sea surface height (SSH) data into ocean models is introduced and tested. Many features observable by satellite altimetry are approximated by the first baroclinic mode over much of the ocean, especially in the lower (but non-equatorial) and mid latitude regions. Based on this dynamical trait, a reduced-dynamics adjoint technique is developed and implemented with a three-dimensional model using vertical normal mode decomposition. To reduce the complexity of the variational data assimilation problem, the adjoint equations are based on a one-active-layer reduced-gravity model, which approximates the first baroclinic mode, as opposed to the full three-dimensional model equations. The reduced dimensionality of the adjoint model leads to lower computational cost than a traditional variational data assimilation algorithm. The technique is applicable to regions of the ocean where the SSH variability is dominated by the first baroclinic mode. The adjustment of the first baroclinic mode model fields dynamically transfers the SSH information to the deep ocean layers. The technique is developed in a modular fashion that can be readily implemented with many three-dimensional ocean models. For this study, the method is tested with the Navy Coastal Ocean Model (NCOM) configured to simulate the Gulf of Mexico.

Introduction

Measurements of the ocean environment are difficult and expensive, especially within the deep ocean. As a result, oceanic data are usually sparse and non-uniformly sampled in time and space. The launch of many ocean observing satellites over recent decades has yielded a vast amount of data for the ocean’s surface. In order to deepen and broaden our understanding of ocean circulation, it is very important to optimize the use of these expanded valuable datasets. Data assimilation is a powerful tool for extracting the maximum amount of information from observations, blending present observations with the theoretical knowledge from past observations (Le Dimet and Navon, 1988, Gill and Malanotte-Rizzoli, 1991, Wunsch, 1996), and has been extensively used in numerical oceanic modeling. In order to assimilate observations into numerical ocean models, there exist a variety of different methods, among which the variational method (Le Dimet and Talagrand, 1986, Courtier and Talagrand, 1990, Yu and O’Brien, 1991, Chua and Bennett, 2001, Bennett, 2002, Wunsch and Heimbach, 2007) and Kalman filter (Ghil et al., 1981, Evensen, 1994, Fukumori and Malanotte-Rizzoli, 1995, Fukumori, 2002) are the two most advanced approaches. Variational methods estimate a system state that fits a dynamical model and available observations in a least-square sense. The four-dimensional variational data assimilation (4D-VAR) minimizes a cost function, the distance between the observations and their model counterpart, by adjusting some chosen model parameters (control variables), such as initial conditions, while the set of model dynamics acts as a strong constraint.

It is very resource-demanding to solve a complicated 4D-VAR problem in meteorology and oceanography, especially with a high-resolution eddy-permitting model. Reduced-space approximations, resulting in fewer degrees of freedom than those formally in the models, have been applied to reduce the computational cost. Primarily, these techniques reduce the order of the problem by projecting the state space onto a linear subspace. One commonly used approach is to construct a low dimension subspace based on the first few empirical orthogonal functions (EOF) (Blayo et al., 2003, Robert et al., 2005, Hoteit, 2008). Hoteit et al. (2005) introduced a decomposition of the velocities at the open boundaries to deal with very strong sensitivities of the model sea surface height (SSH) to the barotropic component of the inflow. Particularly, Kohl and Willebrand (2002) improved the state estimate of a dynamical system by the assimilation of statistical moments of the model state.

The purpose of this study is to develop a computationally efficient variational assimilation approach by using an adjoint model with a reduced state space. Different from the above-mentioned techniques, the subspace in this new method is constructed based on the local ocean dynamics. The adjoint model is simplified by approximating the ocean model variable fields by their first baroclinic mode counterparts. Vertical normal mode decomposition and reconstruction is employed to link the different forward and backward models with different dynamics. Because of the near-surface intensification of baroclinic modes, sea level perturbations measured by altimeters primarily reflect the first baroclinic mode, and thus the motion of the main thermocline (Wunsch, 1996). The application of the method here assimilates satellite altimeter data and provides a mechanism for transferring the information vertically from the ocean surface to deep layers, leading to an improved estimate of the ocean state throughout the water column. Complementing the assimilation method is an application of an interpolation algorithm to map the inhomogeneously sampled SSH to horizontally propagate the information from observation locations to data void areas and thus expedite the assimilation procedure. Several assumptions are made in this reduced-dynamics variational data assimilation technique, which lead to a more efficient way of assimilating the observations into the model system at a cost of an approximated gradient, and hence a compromised solution.

Three numerical experiments are conducted with a Gulf of Mexico (GoM) configuration of the Navy Coastal Ocean Model (NCOM) for testing purposes. The first experiment assimilates the full SSH field from a numerical model solution as “observations” into a model with a perturbed initial condition in attempt to recover the original initial conditions. This experiment is designed to test the reduced-dynamics assimilation method, show that it can handle a poorly chosen initial guess of the ocean state, and demonstrate the assimilation of surface fields into the ocean model’s three-dimensional state variables. A similar experiment, but with the inclusion of a mapping technique, is tested in the second trial. This experiment simulates a more realistic scenario by sampling the unperturbed model SSH as observations in a similar manner as the satellite altimeters (along satellite ground tracks). The sampled SSH data are mapped to a regular grid offline using a mapping technique before the assimilation procedures. The mapping technique horizontally brings the information from observation locations to the native model grid, and thus expedites the horizontal propagation of the observation information in the assimilation. The last experiment is a real-world implementation of the technique using observations from TOPEX/Poseidon (T/P) and Jason-1 satellite altimeters.

Section snippets

Methodology and data

The assimilation system consists of several components, combined in a modular fashion that can be applied to a number of three-dimensional ocean models. For this research, the NCOM is used as the “forward” model for testing the method. The adjoint, often called the “backward model” because it is integrated backward in time, is developed from a simplified model, a one-active-layer reduced-gravity model. The forward and backward models are connected through vertical normal mode decomposition and

Experiments and results

The reduced-dynamics variational data assimilation method, described in Chapter 2, is applied to an NCOM simulation of the GoM for testing and demonstration. Three numerical experiments are run. The first is an idealized experiment in that the SSH data to be assimilated are from a model run. The data are then assimilated into a model that has started from a different initial state and the goal is to adjust the model initial state toward the original one from which the assimilated SSH data are

Discussion

It is computationally expensive to implement all but the simplest oceanic data assimilation systems due to the complexity and grid sizes of ocean models, especially high-resolution eddy-permitting models. Previous studies have shown success in reducing the dimension of control variables by projecting the state space onto a linear subspace based on statistical decomposition, such as EOF analysis. Due to the much lower degree of freedom in the search space, the computational cost is greatly

Acknowledgements

Suggestions and comments from the two anonymous reviewers are greatly appreciated. The authors thank Drs. Paul Martin and Alan Wallcraft for assistance with the NCOM development, and Drs. I. Michael Navon and Xiaolei Zou for their advice. Funding for this project was provided by the Office of Naval Research, the NOAA Office of Global Programs, and the NASA Physical Oceanography Program.

References (38)

  • B. Chua et al.

    An inverse ocean modeling system

    Ocean Modell.

    (2001)
  • C. Robert et al.

    A reduced-order strategy for 4D-Var data assimilation

    J. Mar. Syst.

    (2005)
  • C. Wunsch et al.

    Practical global oceanic state estimation

    Physica D

    (2007)
  • T.P. Barnett

    Interaction of the monsoon and Pacific trade wind system at interannual time scales. Part I: the equatorial zone

    Mon. Weather Rev.

    (1983)
  • A.F. Bennett

    Inverse Modeling of the Ocean and Atmosphere

    (2002)
  • E. Blayo et al.

    Reduced order strategies for variational data assimilation in oceanic models

  • M. Cooper et al.

    Altimetric assimilation with water property conservation

    J. Geophys. Res.

    (1996)
  • P. Courtier et al.

    Variational assimilation of meteorological observations with the direct and adjoint shallow-water equation

    Tellus

    (1990)
  • DaSilva, A., Young, A.C., Levitus, S., 1994. Atlas of Surface Marine Data 1994, Algorithms and Procedures, NOAA Atlas...
  • W.K. Dewar et al.

    On the propagation of baroclinic waves in the general circulation

    J. Phys. Oceanogr.

    (2000)
  • W.K. Dewar et al.

    Adjustment of the ventilated thermocline

    J. Phys. Oceanogr.

    (2001)
  • G. Evensen

    Sequential data assimilation with a nonlinear quasi-geostropic model using Monte Carlo methods to forecast error statistics

    J. Geophys. Res.

    (1994)
  • I. Fukumori

    A partitioned Kalman filter and smoother

    Mon. Weather Rev.

    (2002)
  • I. Fukumori et al.

    An approximate Kalman filter for ocean data assimilation: an example with an idealized Gulf Stream model

    J. Geophys. Res.

    (1995)
  • M. Gill et al.

    Data assimilation in meteorology and oceanograph

    Adv. Geophys.

    (1991)
  • M. Ghil et al.

    Application of estimation theory to numerical weather prediction

  • Hoteit, I., 2008. A reduced-order simulated annealing approach for four-dimensional variational data assimilation in...
  • I. Hoteit et al.

    Treating strong adjoint sensitivities in tropical eddy-permitting variational data assimilation

    Q.J.R. Meteorol. Soc.

    (2005)
  • A. Kohl et al.

    An adjoint method for the assimilation of statistical characteristics into eddy-resolving ocean models

    Tellus

    (2002)
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      Powell et al. (2008) implemented the ROMS-4DVAR assimilation system constraining the model with SSH/SST and in situ current data. A reduced-order variational approach has been also tested by Yu et al. (2009) using the Navy Coastal ocean Model (NCOM). All the described GoM assimilation systems are either OI- or variational-based systems that are generally based on time-invariant background error covariance.

    1

    Present address: Jet Propulsion Laboratory, California Institute of Technology, M/S 300-323, 4800 Oak Grove Drive, Pasadena, CA 91109, USA.

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