An introduction to the data assimilation concept can be found in Section 4.3. This Section focuses on the numerical characteristics of the DAS largely used in circulation modelling.
CHAPTER
COORDINATORS
COORDINATORS
Stefania Ciliberti
5.5
Data assimilation systems
5.5.1
Basic concepts
In ocean forecasting the objective is to produce an estimate xa of the true state xt of the ocean at initial time to initialise forecasts. Ide et al., 1997, De Mey-Frémaux et al. (1998), and Bouttier and Courtier (2002) provide an extensive introduction to DAS basic concepts and herein are recalled and summarised.
DA consists in calculating the «best» estimate of the state of a physical system, of its evolution in time, given observations and a prognostic numerical model.
The basic objective information that can be used to produce the analysis is a collection of observed values provided by observations of the true state. If the model state is overdetermined by the observations, then the analysis is reduced to an interpolation problem. In most cases the analysis problem is under-determined because data are sparse and only indirectly related to the model variables. In order to make it a well-posed problem, it is necessary to rely on some background information in the form of an a priori estimate of the model state.
A discrete model for the evolution of the ocean from ti to ti+1 is governed by the following Eq. 5.13:
Where x is the so-called state vector (velocities, temperature, salinity, etc., at model grid positions) and M is the corresponding dynamics operator. The state vector has dimension n. The dynamic operator in Eq. 5.13 is commonly non linear and deterministic, while the true ocean state may differ from Eq. 5.13 by random and systematic error.
Observations yo t at time ti are defined by Eq. 5.14:
where H is an observation operator and ϵ is a noise process. The observation vector has dimension pi . A major problem of data assimilation is that typically pi<<n. The observation operator H can be also non-linear like M and both can contain explicit time dependence in addition to the implicit dependence via the state vector xf i ≡ xf (ti). The noise process ϵ is commonly used to have zero mean and we denote its covariance matrix by R: it consists of instrumental and representativeness errors which covariance matrices are E and F, respectively, with R=E+F.
The key of the analysis is the use of discrepancies between observations and state vector:
When calculated with the background xb it is called innovations and with the analysis xa analysis residuals.
In the following, we present two data assimilation types of approaches: the sequential methods and the variational methods.
5.5.2
Sequential methods
Several schemes have been proven useful and implemented using a sequential-estimation approach including the Bluelink Ocean Data Assimilation System (BODAS) (Oke et al., 2008) and the Singular Evolutive Extended Kalman (SEEK) filter (Pham et al., 1998). An extensive literature is available on related methods, such as OI (Daley, 1991), EnOI, and EnKF (Evensen, 2003). Following Ide et al., (1997), the true ocean fluid xf is assumed to differ from that of the numerical model (Eq. 13) by stochastic perturbations:
Where η is a noise process with zero mean and covariance matrix Q. The EKF consists of a forecast step based on previously obtained state variables, which include previous assimilation steps, xf (ti+1) and an updated probability function described by Pf (ti):
The core of the Kalman Filter method is an update step in which the observations available at time i is blended with the previous information, taking account of their joint probability distributions and carried forward by the forecast step:
Where the observation residual or innovation vector is defined by:
The Kalman gain Ki is defined by:
The innovation vector di is evidently a displacement of the modelled forecast toward the observed data, scaled by the Kalman gain. The Kalman gain accounts for the weighting required by the joint probability function for the model and observation variability. In practice, various simplifications are introduced to describe P to overcome the computational burden involved in the matrix calculation (Oke et al., 2008; Pham et al., 1998).
Another example is the OI that is quite frequently used in oceanography and meteorology. It is a particular suboptimal filter, in which the EKF’s error covariance matrix Pf is replaced by an approximation, B, computed as a product of variances placed in the diagonal matrix D and of correlations placed in a matrix C with unit diagonal (Ghil and Malanotte-Rizzoli, 1991):
The state vector is still given by Eq. 5.13. The OI gain writes:
Where HiBf (ti)HT i is evaluated from the correlation model, and the state update is given by:
5.5.3
Variational methods
Several schemes have been implemented using variational methods such as 3D-Var, e.g. the Navy Coupled Ocean Data Assimilation (NCODA) (Cummings, 2005) and the Forecasting Ocean Assimilation Model (FOAM) (Martin et al., 2007). 4D-Var methods are used extensively in Numerical Weather Prediction and are one of the future directions for ocean prediction systems. The NEMOVAR system (Mogensen et al., 2012) is able to handle both categories of variational approaches for the NEMO modelling system.
Following Ide et al. (1997), 4D-Var minimises the objective function J that measures the weighted sum of distance Jb to the background state xb and Jo to the observation yo distributed over a time interval [t0 , tn ]:
Where yi ≡Hi[x(ti)]. Here B-1 is an a priori weight matrix, with B-1 meant to approximate the error covariance matrix xb, and a minimization is done with respect to the initial state x(to).
Equation 5.25 reflects the imposition of a strong constraint (Sasaki, 1970). Alternative formulations based on a weak constraint are given by Bennett (1992) and by Menard and Daley (1996). Efficient methods for performing the minimization of J require its partial derivatives with respect to the elements of x(to) given by:
Where:
This follows from:
M(ti+1,ti)T is usually called adjoint model and HT i is the adjoint observation operator. 4D-Var reduces to three-dimensional variational assimilation (3D-Var) if the time dimension is taken out.
Figure 5.9 shows an example of 4D-Var capacity: xa is used as the initial state for a forecast, then by construction of 4D-Var one is sure that the forecast will be completely consistent with the model equations and the 4D distribution of observations until the end of the 4D-Var time interval n (the cutoff time).
Figure 5.9. Example of 4D-Var intermittent assimilation in a numerical forecasting system (adapted from Bouttier and Courtier 2002).
5.5.4
Modelling errors
As reported in Bouttier and Courtier (2002), to represent the fact that there is some uncertainty in the background and in the analysis, we need to assume some model of the errors between these vectors.
Given a background field xb just before doing an analysis, we define the vector of errors that separates it from the true state:
If we are able to repeat each analysis experiment a large number of times, under exactly same conditions but with different realisation of errors generated by unknown causes, b would be different every time. It can be represented through a probability density function (PDF), able to provide all statistics, including the average (or expectation) -b and the variances. A popular model of scalar pdf is the Gaussian function, that can be generalised to a multivariate PDF.
The errors in the background and in the observations are modelled as follows:
- Background errors. They are the estimation errors of the background state, given by the difference between the background state vector and its true value;
- Observation errors. They contain errors in the observation process (i.e instrumental errors), errors in the design of the operator H, and representativeness errors (i.e. discretization errors which prevent x t from being a perfect image of the true state;
- Analysis error. Defined through the trace of the covariance matrix A:
They are the estimation errors of the analysis state, which is what we want to minimize. In a scalar system, the background error covariance is the variance:
In a multidimensional system, B is a square symmetric matrix with n×n dimension. The diagonal of B contains the variances, while the off-diagonal contains the cross-covariances between a pair of variables in the model. The off-diagonal terms can be transformed into error correlations:
The error statistics are functions of the physical processes governing the meteorological or the ocean state and the observing network. They depend on a priori knowledge of the errors: variances reflect our uncertainty in features of the background or the observations. To estimate statistics, it is necessary to assume that they are stationary over a period and uniform over a domain, so that one can take a number of error realisations and make empirical statistics.
In setting a DAS, the estimated statistics is very difficult and we can rely on diagnostics of an existing data assimilation system using the observational method.
5.5.5
Overview of current data assimilation systems in operational forecasting
Data assimilation techniques, schematically introduced in previous paragraphs and that are widely documented in Daley (1991), Evensen (2003) and Zaron (2011), represent the baseline of the modelling framework with general circulation models for operational forecasting and reanalysis. At international level, the GODAE’s OceanView (Bell et al., 2015) and OceanPredict initiatives have promoted strong synergies with GOOS, ETOOFS and GEO BluePlanet contributing to a value chain from observations, data, information systems, predictions, and scientific assessments to end users, with the aim to promote the use and impact of observations and ocean predictions for societal benefit, and increasing visibility of operational oceanography advances.
Martin et al. (2015) presents an overviewofthe main characteristics of the data assimilation used in each GODAE OceanView systems; this is a list of the adopted numerical techniques:
- Bluelink (Oke et al., 2013) adopts MOM4 ocean model and EnOI algorithm;
- GIOPS (Smith et al., 2016) uses NEMO ocean model and SEEK (with fixed basis) for the ocean component, and 3DVar for assimilation in the ice component;
- ECMWF (Balmaseda et al., 2013) uses NEMO ocean model and 3DVar for the assimilation component (+ bias correction technique);
- FOAM (Waters et al., 2014) uses NEMO ocean model and 3DVar for the assimilation component (+ bias correction technique);
- GOFS (Cummings and Smedstad, 2013) uses HYCOM ocean model with 3DVar data assimilation scheme;
- Mercator Ocean (Lellouche et al., 2013) uses NEMO ocean model with SEEK-FGAT (with fixed basis) and 3DVar bias correction;
- MOVE (Usui et al., 2006) uses MRI COM model and 3DVar data assimilation scheme;
- TOPAZ (Sakov et al., 2012) uses HYCOM with EnKF techniques.
Description of the operational initiatives is also provided at GODAE OceanView website (🔗2 ) and OceanPredict website (🔗3 ).
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