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Project Description
Motivation
An
Integrative Data Assimilation for the Arctic System (IDAAS) has been
recommended for development by a special interagency research program
“A Study of Environmental Arctic Change“ (SEARCH, 2005).
While existing operational reanalyses assimilate only atmospheric
measurements, an IDAAS activity would include non-atmospheric components:
sea ice, oceanic, terrestrial geophysical and biogeochemical parameters
and human dimensions data. The IDAAS was recommended for development
“because recent global reanalyses of the atmosphere have received
widespread use by the research community and because they are regarded as
one of the major success stories of the past decade in atmospheric
research” (SEARCH, 2005). Atmospheric reanalysis products play a
major role in the arctic system studies and are used to force sea ice,
ocean and terrestrial models, and to analyze the climate system’s
variability and to explain and understand the interrelationships of the
system’s components and the causes of their change.
Motivated by this success and the major
goals and recommendations of SEARCH, we develop an integrated set of
assimilation procedures for the ice–ocean system that is able to
provide gridded data sets that are physically consistent and constrained
to the observations of sea ice and ocean parameters. Building on our past
research activities in sea ice and ocean data assimilation, we make some
first steps toward the creation of an Arctic Climate System Reanalysis
that uses modern four-dimensional variational (4D-Var, adjoint) data
assimilation methods. We employ sea ice and ocean models with new data assimilation
procedures to maximize the integration of model results with observations
and thus attempt to provide the arctic research community with complete
and accurate data sets, ultimately for at least the last three decades.
Goals
The
first project goal is to develop an integrated set of assimilation
procedures for the ice–ocean system that is able to provide gridded
data sets that are physically consistent and constrained to match all
available observations of sea ice and ocean parameters.
One part of our team (Lindsay and
Zhang) has already accomplished an extensive reanalysis of the
ice–ocean system using data assimilation methods for ice
concentration and ice velocity (Zhang
et al.,2003; Lindsay and Zhang,
2005, 2006) spanning the period 1948–2005. The coupled
ice–ocean model they have used is the Pan-Arctic Ice-Ocean Modeling and Assimilation
System (PIOMAS). However,
the assimilation methods were based on optimal interpolation of model
results and observations.
These sequential techniques are not entirely consistent with the
physics represented in the model.
A more physically consistent method is the 4D-Var approach, which
may be used to adjust initial and boundary conditions, or model
parameters in a manner that is entirely consistent with the model physics
and in a manner that optimizes the fit of the model results with
observations (Wunsch, 1996).
Our experience with this method in a coupled ice–ocean model
has shown that it is very difficult because of the large nonlinearities
in the coupled system.
However the methods of using the adjoint model for data
assimilation in the ocean system alone are well established.
The second part of our team (Nechaev
and Panteleev) is very well versed in the application of adjoint methods
to the ocean. This group has developed, tested, and used an ocean model (Semi-Implicit Ocean Model; SIOM; Nechaev et al.,
2005; Panteleev et al., 2006)
that is capable of data assimilation employing the 4D-Var techniques.
Unfortunately, this model does not include sea ice dynamics and
thermodynamics. It has so far been applied in regional studies of the
Arctic marginal seas for ice free periods or in studies that neglect the
presence of sea ice (Nechaev et al.,
2004, 2005; Panteleev et al.,
2006a,b,c).
The third part of our team
(Proshutinsky and Krishfield) is well experienced in modeling (e.g., Proshutinsky, 1993, 2003a,b;
Kowalik and Proshutinsky, 1994; Hakkinen and Proshutinsky, 2003),
observational (e.g., Proshutinsky
et al., 2004; Krishfield and Perovich, 2005), and integrative studies of
the Arctic Ocean climate system (Proshutinsky et al., 1999, 2001, 2002, 2005). Model development and
simulations represent a comprehensive level of synthesis because this
activity integrates the accomplishments of numerous disciplines (physics,
mathematics, atmospheric, oceanic, cryospheric, and related sciences)
with observational data, and allows testing of different hypotheses via
numerical experiments. In this context, the third group is responsible
for linking high -quality observational data with modeling, aiding in
understanding the observational errors and in interpreting results. This
group has access via on-going projects to an extensive collection of
hydrographic data from the Arctic Ocean obtained by the former Soviet
Union. This data is extremely
valuable resource for our reanalysis effort and offers substantial
additional constraints to the evolution of the simulated system not
previously available.
It is through the combination of
expertise on the team that we feel significant progress can be made to
create a physically consistent reanalysis of the ocean system. The coupled ice–ocean model
group (Zhang and Lindsay) will use conventional methods of sequential
assimilation to make a first guess at the state of the ice–ocean
system with PIOMAS and provide the surface and lateral boundary
conditions to the second group (Nechaev and Panteleev). This second group will determine
the 4D-Var optimal solution using SIOM and a large number of ocean
observations as constraints.
Depending on the method used, Zhang and Lindsay will then rerun
their coupled model with nudging terms to make the simulation more
consistent with the SIOM solution and determine new surface and lateral
boundary conditions. A few
iterations of this procedure will provide the best estimate of the state
of the system (see Panteleev et al.,
2004). The third part of our
team will be responsible for obtaining and processing the hydrographic
observations and in interpreting the results.
The second
project goal is to validate the system performance, assess the quality of
the major system products, and provide the community with gridded sea ice
and ocean parameters for three approximately seven-year periods
characterizing different arctic climate states.
The
circulation and hydrographic structure of the Arctic Ocean is highly
changeable and can be more accurately described using data assimilation
techniques to better understand this variability. Previous studies have
generated numerous possible circulation schemes (see AOMIP at http://www.whoi.edu/projects/AOMIP).
The historical data covering
this area in different seasons and different circulation regimes are not
sufficient to fully describe and explain the causes of this variability.
Our
final goal is to investigate arctic system variability and the processes
important for causing the observed changes based on the reanalysis
products.
Recent measurements in the Arctic Ocean
(Morison et al., 2000) show
that “the Arctic is in the midst of change extending from the stratosphere
to below 1000 m in the ocean.”
Such changes resonate with global climate modeling studies that
consistently show the Arctic to be one of the most sensitive regions to
climate change. In turn,
processes occurring in the Arctic Ocean appear influential to the
subpolar North Atlantic and possibly the global ocean circulation.
One can argue that the ocean hydrography and
circulation data are available from results of the global and regional
arctic models. Unfortunately, most global climate models do not
accurately simulate the mean state of the Arctic Ocean, sea ice, and
atmosphere (Dethloff et al.,
2005). For example, model atmospheric circulations are always
anticyclonic, and simulated sea-ice distributions tend to overestimate
ice thicknesses. The
literature concludes that regional Arctic Ocean models reproduce the
basic dynamics and thermodynamics of the Arctic reasonably well (Semtner,
1976; Hakkinen, 1993, 1995, 1999; Nazarenko, et al., 1998; Zhang, et
al., 1998a,b, 2000). But
the AOMIP found that striking differences existed across models in nearly
every parameter analyzed (Proshutinsky et al., 2001, 2005).
For example, significant differences between models and the
presently available climatology are evident in estimated freshwater and
heat content fields (AOMIP web site). Which, if any, of these descriptions
is correct? Unfortunately, we
lack an accurate high -resolution hydrographic and current database to
compare against the models, as well as detailed information to validate
model parameterizations.
4D-Var analysis (e.g., Wunsch, 1996, Stammer et al., 2002) provides dynamically
balanced information for all physical parameters. Results of this approach with data
reconstruction could be used for analysis and for validation of AOMIP
models, their calibration and their improvement. There are many publications
investigating different aspects of the Arctic Ocean (see publications at
AOMIP web site) but these dynamics have never been validated against
observations.
Approach
In this
section we discuss how the existing data assimilation systems described
above (PIOMAS and SIOM) could be merged efficiently to provide the
research community with sea ice and ocean products consistent with and
constrained reasonably well to observational data. We propose to apply
two approaches:
a) A conventional
4D-Var method will be implemented by employing our SIOM model to the
Arctic Ocean. In this case PIOMAS results will be used by SIOM as the
first approximation and to force SIOM at the surface and open boundaries.
Then SIOM will reconstruct the ocean circulation and hydrography applying
4D-Var techniques and assimilating oceanic observations. This approach
allows us to take into account sea ice processes influencing the ocean
indirectly via PIOMAS where sea ice was already assimilated from observations.
Formally, this approach solves the problem of data assimilation in an
ice–ocean system. This
approach is relatively straightforward but computationally expensive. The
quality of ocean reanalysis will depend on the capabilities of SIOM where
the ocean physics are simplified to satisfy applicability conditions of
the 4D-Var method.
b) An incremental approach (Courtier et al., 1994) will also be used to
reduce the computational burden of the 4D-Var data assimilation procedure
by combining SIOM directly with the state-of-the-art PIOMAS model. This
method allows the development of a very efficient data assimilation
system with performance comparable to other suboptimal data assimilation
methods. Properly designed incremental data assimilation systems have been
used for atmospheric reanalysis and oceanographic reconstructions
(Courtier et al., 1994; Veerse and Thepaut, 1998; Thompson et
al., 1999; Lu et al., 2001; Andersson et al. 2003).
We plan to also employ this method for our reconstructions to increase the
quality of the reconstructed fields by relying on the more realistic
physics implemented in PIOMAS.
Rrealistic variational reanalysis of the sea
ice and ocean circulation in the Arctic is a challenging task because of
the relative sparseness of the observations, methodological complexity,
and high computational requirements of the 4D-Var method. The conventional 4D-Var method is
more robust and, in principle, should be able to provide a better fit to
the available observations. But utilization of this method in the PIOMAS
model requires development of adjoint code, which has known convergence
problems in a highly nonlinear system with complicated dynamics such as
sea ice. The incremental approach allows us to perform variational
assimilation of data into the PIOMAS model, but the incremental methods
are not guaranteed to converge to the optimal solution. There are several
publications (e.g., Lawless et al., 2005) where the
convergence of the incremental approach was investigated for
non-tangent-linear simple models. Even properly designed incremental
methods show convergence of the solution of the complex models toward the
observations only for the first several iterations (typically 5–7)
of the assimilation procedure.
We propose to implement and compare the two
variational methods to reveal which property of the data assimilation
algorithm is more important for the Arctic reanalysis system:
(a)
application of the more advanced data assimilation technique to
the model with simplified (but still reasonably realistic) physics (SIOM
solution) or
(b)
application of the suboptimal data assimilation method to the
system with state-of-the-art coupled ice–ocean dynamics (PIOMAS
solution).
The application and comparison of the two data
assimilation methods will not require double the resources for the
proposed work. The common tasks for both methods include setting up the
same modeling systems, processing the same data sets, development of the
same procedures for estimation of the cost functions, and validation of
the results. Below, we first describe the SIOM and PIOMAS systems,
briefly outline major features of the data assimilation methods to be
employed, and then pay more attention to the data we plan to use in this
project.
3.1. Model
descriptions
PIOMAS: PIOMAS was developed at the Polar
Science Center, University of Washington. This is a coupled parallel
ocean and sea ice model capable of assimilating sea ice concentration and
velocity data. It consists of the thickness and enthalpy distribution
(TED) sea-ice model (Zhang and Rothrock, 2001; 2003) and the Parallel
Ocean Program (POP) developed at the Los Alamos National Laboratory. The
TED sea-ice model is a dynamic thermodynamic model that also explicitly
simulates sea-ice ridging. It has 12 categories each for ice thickness,
ice enthalpy, and snow. It employs a teardrop viscous-plastic ice
rheology that determines the relationship between ice internal stress and
ice deformation (Zhang and Rothrock, 2005), a mechanical redistribution
function that determines ice ridging (Thorndike et al. 1975; Rothrock,
1975; Hibler, 1980), and an efficient numerical method to solve the ice
motion equation (Zhang and Hibler, 1997). PIOMAS is configured to cover
the region north of 43oN. The model grid is based on a
generalized orthogonal curvilinear coordinate system with the northern
grid pole displaced into Greenland. This allows the model to have good
resolution in the connections between the Arctic Ocean and the Atlantic
Ocean. The mean horizontal resolution is 22 km
for the Arctic, Barents, and GIN (Greenland-Iceland-Norwegian) seas, and
Baffin Bay. The model is one-way nested to a Global Ice-Ocean Modeling
and Assimilation System which consists of similar sea ice and ocean
models. Output from this model will be specified along the southern
boundaries of POIMAS (43oN)
as open boundary conditions.
SIOM: SIOM was designed specifically for the
implementation of 4D-Var methods into regional models controlled by
currents at the open boundaries and by surface fluxes. SIOM is a modification of the C-grid, z-coordinate OGCM developed at
the Laboratoire d'Oceanographie Dynamique et de Climatologie, (Madec et al., 1999). The model is
semi-implicit both for barotropic and baroclinic modes permitting
simulations with relatively large time steps. The tangent-linear model
was obtained by direct differentiation of the forward model code. The
adjoint code of the model was built analytically by transposition of the
operator of the tangent-linear model, linearized in the vicinity of the
given solution of the forward model (Wunsch, 1996). The original version
of SIOM does not have a sea ice component. In this project, we plan to
improve SIOM physical parameterizations and numerical methods. We plan to
estimate values of eddy diffusion coefficients for SIOM by analyzing the
fields of these coefficients and the Reynolds stresses in the PIOMAS
experiments and will keep these coefficients unchanged in the data
assimilation experiments with SIOM. We will also modify SIOM’s
numerical code and incorporate a Flux Corrected Transport (FCT) scheme
into a tracer advection scheme and introduce orthogonal curvilinear
coordinates to improve the solution near the coastlines. The SIOM will be configured for
the area of PIOMAS with its southern open boundary along 60ºN (Figure
1). The SIOM 4D-Var data assimilation system has been implemented
successfully for the reconstruction of the summer circulation in the
Barents, Bering and Kara seas (Panteleev et al., 2006a,b,c), and for the variational hindcast of the
circulation in the Tsushima Strait (Nechaev et al., 2005).
3.2. Data assimilation
procedures
3.2.1 Optimal interpolation data
assimilation procedures for sea ice observations
POIMAS can assimilate
both ice motion and concentration data (Figure 1, right). Assimilation of
ice motion data is based on a two-dimensional optimal interpolation (OI)
technique that blends sparse velocity measurements (with known error
characteristics) and model velocity estimates (Zhang et al., 2003). The velocity observations include raw
buoy displacements from the International Arctic Buoy Program (IABP) and
satellite-derived ice velocities based on images from the Scanning
Multichannel Microwave Radiometer (SMMR) and the Special Sensor
Microwave/Imager (SSMI), available from the National Snow and Ice Data
Center (NSIDC). The OI assimilation of observed ice velocities not only
improves the modeled ice motion, but also the ice thickness estimates
when compared to measurements taken by submarines. This procedure also
improves the modeled ice deformation when compared to RADARSAT
Geophysical Processor System (RGPS) measurements (Lindsay et al., 2003). Assimilation of
satellite ice concentration data is based on an innovative assimilation
procedure (Lindsay and Zhang, 2006a) that nudges the model estimate of
ice concentration toward the observed concentration in a manner that
emphasizes the ice extent and minimizes the effect of observational
errors in the interior of the ice pack. This is a relatively simple yet
effective assimilation scheme that is computationally affordable for
long-term integrations and experiments. In addition to improving the
simulated ice edge, comparisons to observed ice thickness measurements in
the Arctic indicate that the assimilation of ice concentration also
improves the simulated ice thickness.
3.2.2 Conventional 4D-Var method
The conventional
variational data assimilation method realized in SIOM is similar to a
traditional least squares problem (Le Dimet and Talagrand, 1986; Thacker
and Long, 1988). The optimal solution of the model is found through
constrained minimization of a quadratic cost function, where the cost
function measures squared weighted distances between the model solution
and data and can incorporate other constraints such as the smoothness of
the solution. Under a statistical interpretation the optimal solution is
the most probable model state for the given data and prior error
statistics where the cost function weights are the inverse covariances of
the corresponding data errors (Thacker, 1989; Wunsch, 1996).
.
 
Figure 1. Left: Region of project studies. Right: Data flow chart
for the data assimilation procedure “a”.
To
determine the optimal solution of the forward model, the cost function is
minimized on the space of control vectors for the model. To improve the
controllability of the data assimilation algorithm we will implement the
weak constraint version of the 4D-Var method, which accounts for the
model errors. The control vector of the model includes “free”
model parameters, such as the grid values of the initial conditions, open
boundary conditions, and surface fluxes. Additionally the control vector
will include correction terms (model errors) in the regions and during
the time intervals where, as we expect, the model may have significant dynamical
errors (e.g. due to known flaws in the turbulent closure scheme). Such
corrections might be necessary during the late fall/early winter period,
i.e. during the events of strong winter cooling and vertical mixing. The
absolute values of the restoring terms will be physically reasonable and
will be controlled by the incorporation the quadratic norm of these terms
into the cost function. That should minimize the violation of the model
dynamic equations in SIOM caused by the correction terms.
The
application of the conventional 4D-Var data assimilation approach
involves (i) running the forward model starting with some prior estimate
(the so-called first guess) of the model control to estimate the cost
function and the model-data misfits; (ii) running the adjoint model
backward in time to compute the gradient of the cost function with
respect to the control vector, and (iii) application of a descent
algorithm to find updated values of the control vector components. The
procedure is repeated for the updated model control vector until some
convergence criterion is satisfied. Typically the minimization of the
cost function requires hundreds of forward and adjoint model runs.
This
algorithm will be utilized as the approach (a) in our plan. The data flow
and algorithm structure are shown in Figure 1. In this case POIMAS
provides SIOM with the first guess boundary and initial conditions and
mixing coefficients. SIOM assimilates oceanic data through minimization
of the following cost function:
 where
 ;  ;  .
Here y denotes vector of the SIOM
solution, y* is the
vector of ocean observations, H is
an operator for projecting y onto
data locations, c is the vector
of control parameters of the SIOM, and cfg represents the first guess values of these
parameters. The vector of observations y* will include the water temperature, salinity,
velocity, and the sea surface height all measured at many times and
locations. For any given year y*
may contain on the order of 105-106 elements. The
cost function term Jdata
attracts the model solution toward the data, Jsmooth insures a reasonably smooth solution. The
term Jcntr penalizes the amplitude of the
control vector changes during the minimization procedure and makes the
data assimilation problem formally well-posed. The cost function weights
will be represented by diagonal matrices. The elements of the matrix Wdata are specified as
the estimates of error variance of the corresponding data, Wcntr represent the
prior estimates of the first guess solution error variance at the
locations of the SIOM control variables. The matrix Wsm will be estimated from the analysis of typical
spatial and temporal scales of variability of the solution y (Panteleev et al., 2000; Panteleev et
al., 2006a).
Due to
the high computational cost of the 4D-Var data assimilation, the
optimization of the SIOM solution will be performed sequentially by
assimilation of the oceanic data over several one-year time
intervals. We specify a
one-year interval because we are quite confident that we are able to
perform multiple one-year-long 4D-Var data assimilation runs with SIOM
given the currently available computational resources. Initial conditions
will be used as control parameters only during the first data
assimilation interval.
Daily
reanalysis products from SIOM will be stored in the project archives and
will be available to users by request. The monthly reanalysis data will
be posted at the project web site and will be available for consumers
without restrictions.
3.2.3 Incremental data assimilation approach
The incremental
approach was proposed by Courtier et
al., (1994) to reduce the computational burden of the 4D-VAR data
assimilation. This approach can be considered as an approximate way to
minimize the cost function constrained by the state-of-the-art
(“complex”, in our case PIOMAS) model through a series of
quadratic cost function minimizations under simpler linear dynamical
constraints (“simple” model, in our case SIOM, which
describes the evolution of small perturbations to the complex model
solution). The complex forward model is used to estimate the cost
function and the misfits between the complex model solution and
observations. The corrections to the control vector of the complex model
are calculated as the control vector of the simple model producing the
perturbations, which reduce the complex model-data misfits in the least
squares sense.
The formal best choice of the linear model for
the incremental approach is the exact tangent-linear
“complex” model. In this case the incremental approach
becomes equivalent to the conventional 4D-Var data assimilation
algorithm, though no gain in computational cost is achieved. In practice,
the incremental approach utilizes an approximation to the tangent-linear
model with reduced dimension of the control vector, and/or simplified
dynamics, and/or coarser grid resolution.
For optimization of the PIOMAS solution with
respect to the oceanic observations we propose to implement the
incremental approach with the SIOM tangent-linear code. A consistent
description of the perturbation dynamics in SIOM and convergence of the
incremental approach will be ensured by similarly configured spatial grid
and bottom topography, by linearization of the SIOM in the vicinity of
PIOMAS solution, and by utilization of the time and space varying eddy
diffusion coefficients obtained from PIOMAS. The 4D-Var assimilation of
the misfits between the PIOMAS solution and data into the tangentlinear
SIOM will provide us with the corrections. These will then be introduced into
the PIOMAS fields to improve the PIOMAS solution. In other words, the function
of the 4D-Var assimilation with the SIOM tangent-linear model will be to
redistribute observational information among all model variables and to
project this information from data locations onto the boundary of the
SIOM domain. In the traditional incremental approach, the complex and
simple models are supposed to have common control variables (that is
– the common open boundaries). We propose the following
modification of the traditional procedure:
- All simulations
with PIOMAS will be performed in the model’s
“native” region with the boundary at approximately 43°N, while SIOM
calculations will be performed in a smaller domain (north from
60°N).
- The corrections
to the PIOMAS fields will be introduced through the restoring terms
in the momentum and thermodynamic equations associated with the
control parameters of the SIOM model used in the conventional 4D-Var
method. For example, we plan to incorporate additional terms into
the PIOMAS code within a 5-10 grid cell buffer zone around the SIOM
open boundary and in the vicinity of the major rivers. In the
prognostic equation for PIOMAS variables the restoring terms will be
defined as r(x-xb)c’(xb)/Tinc, where c’(xb) is the optimal correction computed by
SIOM on the open boundary with coordinates xb , r(x-xb) is the shape
function distributing the corrections over the buffer zone. The time
scale Tinc for the restoring terms will be the same in
all equations and will be defined through several numerical
experiments. Our preliminary estimate of Tinc based upon
prior estimate of the data density and PIOMAS solution error
variance is in the range of 1 week to 1 month.
Figure 2. Data flow chart for the data assimilation
procedure “b” employing an incremental data assimilation
method.
The applicability of the tangent-linear code is
limited to corrections with sufficiently small amplitude. If application
of the incremental approach results in high amplitude corrections and the
inconsistency of the tangent-linear model slows down the convergence of
the PIOMAS solution to the oceanic observations, we plan to test an
alternative approach to nest the SIOM and PIOMAS assimilation systems. We
will initialize the full non-linear SIOM model using the results of the
POIMAS simulations and we will look for the solution of the SIOM model
which minimizes the model-data misfits. The difference between the PIOMAS
results and the solution of the SIOM 4D-Var data assimilation problem
will then be used to set up the nudging terms in the PIOMAS model as
described above.
The optimization of the PIOMAS solution will be performed sequentially
by assimilation of the oceanic data over several one-year time
intervals. Note that
sequential optimization of the PIOMAS solution on discrete time intervals
still produces continuous in time solution over the entire period of
integration, because the optimization of PIOMAS solution does not involve
any re-initialization of the PIOMAS solution at the beginning of the new
data assimilation interval.
3.2.4. Coupled reanalysis algorithm
We propose the following reanalysis
algorithm (procedure “b”) by applying the incremental 4D-Var
data assimilation approach (Courtier et
al., 1994) based on PIOMAS and SIOM. In this algorithm, the
optimization of the PIOMAS solution will be performed every year of the
system reanalysis through the following procedure:
1. Initially PIOMAS will
be integrated for at least 20 years prior to the analysis period with the
observed atmospheric forcing, with assimilation of ice concentration and
velocity, and with no interaction between POIMAS and SIOM in this initial
phase. Starting from this state, the PIOMAS integration will continue to
cover the reanalysis period, using two-way communication with SIOM.
2. The results of the first
guess PIOMAS run for the first period will be used to build one-day
averages and will be stored for every day of the model integration. These
daily fields will be used to estimate the misfits q between the PIOMAS solution and various oceanic
observations q = H ypiomas - y* where ypiomas denotes the PIOMAS daily fields, y* is the vector of
observations, and H is an
operator for projecting ypiomas
onto data locations. The
initial value of the cost function for the PIOMAS solution is Jpiomas
3. The
SIOM will be linearized in the vicinity of the PIOMAS solution ypiomas (which will be
represented by piece-wise linear functions in time) and will utilize the
eddy-diffusion coefficients derived from the PIOMAS run. We will search
for the solution y’ of
the tangent-linear SIOM which minimizes the following cost function:
 where
 ;  ;  .
The cost function J’ is minimized on the space
of the control vectors c’ of the tangent-linear
SIOM. The control vector c’ will include the same
physical variables as the control of the full SIOM model described in the
section 3.2.2. The cost function terms Jdata, J’data
attract the model solution toward the data, J’smooth insures a reasonably smooth
solution. The term J’cntr
penalizes the amplitude of the control vector c’ of the tangent-linear
model. The cost function weights will be represented by diagonal matrices
as described in the section 3.2.2. The daily averaged fields of optimal
control vector c’ will be
stored for every day of the SIOM model integration.
4. To
introduce the data-induced corrections into the PIOMAS model we will
re-run the model in the sea ice data assimilation mode with three groups
of restoring terms. The first group will include corrections r(x- xb)(c’(xb,t)/Tinc concentrated within a buffer zone
(5-10 grid cells) around the open boundary of SIOM and along the Siberian, Greenland and
American coasts (2-3 grid cells). The second group will introduce
corrections to PIOMAS surface momentum, heat, and salt flux values
according to the optimized estimates of the corresponding c’ components.
The third group of corrections will compensate for PIOMAS errors
in the in the interior of the SIOM domain. These restoring terms will
force PIOMAS during the 2-3 month winter period.
5. The updated PIOMAS solution will be low-pass
filtered to compute new estimates of model-data misfits and an updated
value of the PIOMAS cost function, which will be used to evaluate the
algorithm convergence. Steps two, three, four and five will be repeated
until a convergence criterion is satisfied, or the reduction of the
PIOMAS cost function slows down significantly.
6. Daily
reanalysis products from PIOMAS will be stored in the project archives
and will be available to users by request. The monthly reanalysis data
will be posted at the project web site and will be available for
consumers without restrictions. To obtain to the next year of reanalysis
the process will start again with the first guess PIOMAS solution (return
to step one). The subsequent PIOMAS runs will use the optimal fields
obtained during the previous year’s reanalysis as initial
conditions.
3.2.5 Validation of the data assimilation
results
The major complication of the weak constraint
method is related to the specification of the model error statistics. We
plan to conduct a model study aimed specifically at the quantification of
the PIOMAS and SIOM errors and understanding of their physical origin. We
believe that such a study is critical for the consistent formulation of
the data assimilation procedure and may allow us to impose robust
statistical and physical constraints on the model errors to reduce the
number of degrees of freedom introduced by the additional control vector
components. For example, the errors in the mixing rates should result in
the corrections of the PIOMAS and SIOM fields with zero vertical
integral.
Verification of statistical hypothesis: Routine analysis of
data assimilation results commonly involves several tests of the
hypothesis underlying the framing of the variational problem. For
example, the model-data misfits in the optimal solution exceeding the
statistically significant data-error-confidence limits would mean that
the model or data assimilation algorithm requires improvement, or that
prior estimates of the data error or model error variances are incorrect.
The statistical properties of the model-data misfits should not contradict
the hypothesis that the errors in the model and data are Gaussian random
variables with zero mean, which is usually tested by comparing the
statistics of the weighted squared model-data misfits with the c2 distribution. To
analyze the overall performance of the assimilation algorithm we will
estimate what fraction of the low-frequency data variability is explained
by the optimal solution vs. by the first guess solution. A high
percentage indicates a good formulation of the data assimilation
procedures. We will also estimate what part of the data variability can
be explained by the adjustment of the inflow parameters along open
boundaries.
Comparison with independent measurements: To validate the results of the
Arctic Ocean circulation reconstruction we will use a conventional test
of data assimilation procedures by excluding some data points from the
assimilation and comparing the optimal solutions with these data.
4. Data sources
PIOMAS will be driven
by atmospheric forcing applied to the ocean and ice surface and will
assimilate sea ice concentration and drift. In addition, within the
reanalysis algorithm this model will receive data flows from SIOM
providing links between the two data assimilation system components.
SIOM will use all available ocean hydrography
and current data for assimilation and will be driven by data from PIOMAS
at the ocean-ice surface. The major sources of data needed for model
forcing and assimilation are outlined here (see also Figures 1 and 2).
Atmospheric
forcing data: Atmospheric forcing data is needed for both PIOMAS and SIOM.
They will be taken from the ERA-40 Reanalysis through 2001 and the ECMWF
operational analysis after that.
The fields we need are daily averages of the 10-m wind vector, the
2-m air temperature and humidity, the sea level pressure, and the
downwelling long- and shortwave radiative fluxes. The reason we select the
ERA-40/ECMWF products over the NCEP reanalysis is that the downwelling
radiative fluxes in the NCEP products are known to have large errors (Serreze
et al., 1998). Previous simulations often have
used climatological cloud fractions and parameterized downwelling fluxes,
but by using the ERA-40 fluxes we are able to include realistic
interannual variability in the radiative fluxes. The ERA-40 downwelling fluxes
compare very well to those measured during SHEBA (Liu et al., 2005). Our analysis of the
wind and temperatures of these products show there is no significant jump
in their bias after 2001 when compared to NCEP products. Some inhomogeniety in the products
may exist because of atmospheric model changes after 2001, but the
changing mix of available observations during the entire reanalysis
period also adds unavoidable inhomogeneities in the results.
Figure 3. Spatial distribution of CTD stations from
Western and Russian data archives for the decades of 1950s, 1960s, 1970s,
1980s, 1990s, and 2000s. Not all data from 2000s have been incorporated
in our data archive at this time.
Surface
data:
PIOMAS will assimilate ice concentration (IC), ice velocity (IV), and
wet-ocean sea-surface temperature (SST). The IC and SST data will be
obtained from the ERA-40/ECMWF data sets to insure that the air
temperature, IC, and SST fields are mutually consistent. The source data from these fields
are: 1) the monthly mean HadISST data set from the UKMO Hadley Centre for
1956-1981; and 2) the weekly NCEP 2D-VAR data for 1982-present (Reynolds et al., 2002). Both data sets are
based on satellite and conventional SST/IC observations. The principal
reason for the higher quality of these source data sets is the use of a
common consensus IC and a common IC-SST relationship in the sea ice
margins. The most recent
ECMWF SST fields are from new daily analyses made at NCEP. The IV will be taken from the
optimally interpolated ice velocity fields produced by Chuck Fowler and
archived as a Polar Pathfinder dataset at NSIDC. They are derived from buoy, AVHRR,
and passive microwave estimates of the ice velocity.
Ocean data: The adjoint data assimilation procedures of
SIOM will use a variety of ocean data including salinity, temperature,
velocity, and sea surface height. The Arctic Ocean hydrographic data is
sparse in temporal and spatial coverage. Recently, the climatology has
expanded in two ways. First, historical hydrographic data have been
declassified and released by both Russian and western sources in the form
of smoothed, three-dimensionally gridded fields for summer and winter
[Environmental Working Group Atlas, EWG, 1997, 1998]. This represents a significant
advance but unfortunately, the data for these atlases were averaged for
the decades of the 1950s, 1960s, 1970s and 1980s, irregardless of
climatic regimes. Second, the arctic hydrography database has expanded
recently due to an increase in the number of high-latitude cruises and
the establishment of several long-term observational sites in key regions
of the Arctic Ocean including major ocean boundaries (Bering Strait, Fram
Strait, straits of the Canadian Archipelago, and in the central basin
such as observations conducted in the vicinity of the North Pole (North
Pole Environmental Observatory, NPEO, http://psc.apl.washington.edu/northpole)
and in the Western Arctic (Beaufort Gyre Observing System, BGOS, http://www.whoi.edu/beaufortgyre).
In addition, there has been at least one major expedition by either
icebreaker or submarine into the deep Arctic Ocean nearly every year
between 1992 and 2005 (information about existing arctic hydrographic
data is posted at the BGOS web site). Figure 3 shows the number of CTD
soundings currently available for different decades.
Other data include
current velocity measured at moorings in the major Arctic Ocean straits
and key regions of the deep basins. There are more than 900 months of
these observations available just from the Institute of Ocean Sciences,
Canada (Greg Holloway, personal communication). Other sources include the
Alfred Wegener Institute, the Polar Science Center, University of
Washington, and Ohio State University). Significant amounts of data have
already been incorporated into our data archives at WHOI. These data
include climatologic information from the EWG atlas and specially
selected and gridded T&S data provided by the scientists of the
Arctic and Antarctic Research Institute, Russia, for different
circulation regimes (http://www.whoi.edu/science/PO/arcticgroup/projects/andrey_project)
and for particular years. Completely new data are available from the NPEO
and BGOS observing systems. The mooring data from these observatories
also includes an upward-looking ADCP at the top mooring float to measure
ocean currents in the upper 50-m layer. For 2003/2004 we also collected
T&S data in the upper 50-m ocean layer from four ocean buoys. Since
2004 the BGOS archive includes data from a new instrument, the
Ice-Tethered Profiler (ITP), which repeatedly samples the properties in
the upper 800 m of the ocean at high vertical resolution over long time
periods. The instrument, its performance in the field, and examples of
the data returned from the system are presented at http://www.whoi.edu/itp. There are two
instruments operating in the Arctic Ocean now and in 2006 four more
instruments will be deployed in this area. These instruments in
combination with the Arctic Ocean observing activity and IPY studies will
recover an unprecedented amount of data from this region. To process and
utilize this huge amount of information and make it available for the
scientific community, methods of data assimilation need to be developed
and validated for the region.
Sea surface height data will be obtained from satellite altimetry
(C.K. Shum, Ohio State University) for the regions of ice free ocean and
from approximately 71 coastal tide gauges along the Northern Sea route (http://www.whoi.edu/science/PO/arcticsealevel).
There also are numerous other sources
of data containing water temperature and salinity fields; sea ice
thickness, concentration, and drift; sea level; and ocean currents. These
data are located in national data archives (NSIDC, ARCSS, NODC) and in
local archives of different institutions of the project PIs. Most of the
data archives are available publicly via the Internet. As part of this project
we will collect and reprocess all possible data from a variety of sources
for the period 1950-present and will prepare these data in a form
suitable for assimilation procedures. The pre-processing procedures will
include: quality control and preliminary data analysis; data unification
for data assimilation purposes including low-pass filtering and
interpolation to model grids; estimation of typical spatial and temporal
scales of variability; and obtaining physically meaningful estimates of
the data error variance.
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