state and its uncertainties, as well as insight into
missing model physics and uncertain model parameters, such as mixing. Considerable experience is
needed in assessing the realism of ocean models
and in evaluating the adequacy and design of climate-observing systems. Both areas benefit greatly
from data analyses and ocean state estimation during the WOCE synthesis phase, as will be described
below.
Although the field of ocean state estimation is
still in its early stage of development, the substantial progress achieved over the last years has given
rise to great expectations of what can be attained
in the near future. Several attempts are now underway to estimate the time-evolving ocean state for
up to a decade from basin to global scales. Examples include synthesis activities in the Indian,
Pacific and Atlantic Oceans, with many accompanied by efforts on the global scale.
7.1.4.1 The estimation problem
Thorough treatments of ocean state estimation can
be found in the text books by Bennett (1992) and
Wunsch (1996) and recent applications are summarized in Malanotte-Rizzoli (1996). Marotzke
and Willebrand (1996) discuss details of a variety
of ocean inverse models. Procedurally, ocean state
estimation interpolates and extrapolates data
information in space, time, and among different
variables exploiting the model equations. In the
process, the information is combined with other
data, which further improves the description of the
oceanic state. In essence, state estimation is a
dynamic extrapolation as well as a synthesis and
averaging process.
Figures 7.1.5 and 7.1.6 (see Plates 7.1.5 and
7.1.6, p. 588) illustrate the extrapolation and averaging effects of data assimilation. Figure 7.1.5 (see
Plate) shows sea-level anomalies as a function of
longitude and time in the tropical Pacific Ocean
(Fukumori, 1995). The figure illustrates how
assimilation (b) corrects the model (c) to be consistent with TOPEX/POSEIDON observations (a),
including the magnitude of seasonal changes and
the westward-propagating waves. At the same
time, the assimilation eliminates noise from the
data and interpolates over data dropouts (grey
area in the left panel). Figure 7.1.6 (see Plate 7.1.6,
p. 588) illustrates how assimilation extrapolates
data information (sea level) into improvements of
different properties of the ocean, such as subsurface
temperature (a) and subsurface zonal velocity (b).
Sea-level information (TOPEX/ POSEIDON)
corrects the model (red) into an estimate (blue)
that is in closer agreement with independent insitu measurements (black), consistent with formal
error estimates (bars) (Fukumori et al., 1999).
As opposed to numerical simulations, ocean
state estimation is mathematically an inverse problem. That is, instead of computing properties of
the ocean (e.g. model equivalent of data) given the
state (numerical simulation), assimilation estimates
the state based on observations. The difference
between state estimation and simpler inversions
described in Section 7.1.2.1 is in the underlying
model dynamics. Instead of simple balances and
conservation statements applied over large regions
or with a steady-state assumption, data assimilation employs the full, time-dependent, non-linear
physics embodied in numerical ocean circulation
models at every model gridpoint. As such, the
unknowns (x) are independent variables of the
model, such as temperature, salinity and velocity,
over the entire model domain, and various model
parameters, as well as uncertainties in external
forcing and boundary conditions.
From a practical standpoint, the distinguishing
property of data assimilation in relation to other
inverse problems is its enormous dimensionality.
Typical ocean models contain on the order of several million independent variables at any particular instant. The size of the problem precludes any
direct method of solution and, as a result, devising
methods of assimilation has been one of the central issues in ocean state estimation.
Many of the so-called ‘advanced’ assimilation
methods originate in estimation and control theories (e.g. Bryson and Ho, 1975), which in turn are
based on ‘classic’ inverse methods. These include
the adjoint, representer, Kalman filter and related
smoothers, and Green’s function methods. These
techniques are characterized by their explicit
assumptions under which the inverse problem is
solved consistently. The assumptions include, for
example, the data and model weights used in the
inversion and specific statistical criteria in choosing particular ‘optimal’ solutions, such as leastsquares, minimum error variance and maximum
likelihood. As with ‘classic’ inverse methods, these
assimilation schemes are equivalent to each other
and result in the same solution so long as the
assumptions are the same. Using specific weights
7.1 Towards a WOCE Synthesis
537
Talley, Stammer and Fukumori
missing model physics and uncertain model parameters, such as mixing. Considerable experience is
needed in assessing the realism of ocean models
and in evaluating the adequacy and design of climate-observing systems. Both areas benefit greatly
from data analyses and ocean state estimation during the WOCE synthesis phase, as will be described
below.
Although the field of ocean state estimation is
still in its early stage of development, the substantial progress achieved over the last years has given
rise to great expectations of what can be attained
in the near future. Several attempts are now underway to estimate the time-evolving ocean state for
up to a decade from basin to global scales. Examples include synthesis activities in the Indian,
Pacific and Atlantic Oceans, with many accompanied by efforts on the global scale.
7.1.4.1 The estimation problem
Thorough treatments of ocean state estimation can
be found in the text books by Bennett (1992) and
Wunsch (1996) and recent applications are summarized in Malanotte-Rizzoli (1996). Marotzke
and Willebrand (1996) discuss details of a variety
of ocean inverse models. Procedurally, ocean state
estimation interpolates and extrapolates data
information in space, time, and among different
variables exploiting the model equations. In the
process, the information is combined with other
data, which further improves the description of the
oceanic state. In essence, state estimation is a
dynamic extrapolation as well as a synthesis and
averaging process.
Figures 7.1.5 and 7.1.6 (see Plates 7.1.5 and
7.1.6, p. 588) illustrate the extrapolation and averaging effects of data assimilation. Figure 7.1.5 (see
Plate) shows sea-level anomalies as a function of
longitude and time in the tropical Pacific Ocean
(Fukumori, 1995). The figure illustrates how
assimilation (b) corrects the model (c) to be consistent with TOPEX/POSEIDON observations (a),
including the magnitude of seasonal changes and
the westward-propagating waves. At the same
time, the assimilation eliminates noise from the
data and interpolates over data dropouts (grey
area in the left panel). Figure 7.1.6 (see Plate 7.1.6,
p. 588) illustrates how assimilation extrapolates
data information (sea level) into improvements of
different properties of the ocean, such as subsurface
temperature (a) and subsurface zonal velocity (b).
Sea-level information (TOPEX/ POSEIDON)
corrects the model (red) into an estimate (blue)
that is in closer agreement with independent insitu measurements (black), consistent with formal
error estimates (bars) (Fukumori et al., 1999).
As opposed to numerical simulations, ocean
state estimation is mathematically an inverse problem. That is, instead of computing properties of
the ocean (e.g. model equivalent of data) given the
state (numerical simulation), assimilation estimates
the state based on observations. The difference
between state estimation and simpler inversions
described in Section 7.1.2.1 is in the underlying
model dynamics. Instead of simple balances and
conservation statements applied over large regions
or with a steady-state assumption, data assimilation employs the full, time-dependent, non-linear
physics embodied in numerical ocean circulation
models at every model gridpoint. As such, the
unknowns (x) are independent variables of the
model, such as temperature, salinity and velocity,
over the entire model domain, and various model
parameters, as well as uncertainties in external
forcing and boundary conditions.
From a practical standpoint, the distinguishing
property of data assimilation in relation to other
inverse problems is its enormous dimensionality.
Typical ocean models contain on the order of several million independent variables at any particular instant. The size of the problem precludes any
direct method of solution and, as a result, devising
methods of assimilation has been one of the central issues in ocean state estimation.
Many of the so-called ‘advanced’ assimilation
methods originate in estimation and control theories (e.g. Bryson and Ho, 1975), which in turn are
based on ‘classic’ inverse methods. These include
the adjoint, representer, Kalman filter and related
smoothers, and Green’s function methods. These
techniques are characterized by their explicit
assumptions under which the inverse problem is
solved consistently. The assumptions include, for
example, the data and model weights used in the
inversion and specific statistical criteria in choosing particular ‘optimal’ solutions, such as leastsquares, minimum error variance and maximum
likelihood. As with ‘classic’ inverse methods, these
assimilation schemes are equivalent to each other
and result in the same solution so long as the
assumptions are the same. Using specific weights
7.1 Towards a WOCE Synthesis
537
Talley, Stammer and Fukumori