Practical global oceanic state estimation
Summary (2 min read)
1. Introduction
- In physical oceanography, the problem of combining observations with numerical models differs in a number of significant ways from its practice in the atmospheric sciences.
- It is these differences that lead us to use the terminology “state estimation” to distinguish the oceanographers’ problems and methods from those employed under the label “data assimilation” in numerical weather prediction.
- Estimation theory is a large subject with a wide range of methods available.
- Many of the problems involved in solving inverse problems in continuous formulations are generated by these high derivatives e.g., the singular behavior of an analytical Green function—see [5].
- If the discrete rendering of this and other terms is regarded as adequate, all issues of convergence, continuity, differentiability, and existence, essentially disappear.
2. The goal
- Meteorological data assimilation has been driven by the compelling need to forecast the weather.
- In the terminology of control theory, the meteorological problem has been primarily one of filtering and prediction ; the oceanographic one is primarily one of “smoothing” .
- Little is known of model error and an important unsolved mathematical problem concerns its representation as a function of time and space.
- (This description applies to the special case where the control vector consists of the initial conditions only.
- The MLM method, if a solution can be found, does not require the use of covariance matrices for the state vector— rather it is a whole domain method in which no averaging of interim solutions takes place—in the present context, that is its great attraction.
3. Making it work
- In some ways, the most interesting is the problem of finding the partial derivatives (9).
- AD tools permit the adjoint code to be updated and maintained in a practical way in the presence of continuing vigorous development and improvement of the parent nonlinear model.
- Part of Q(t) represents the covariances of the control vector, describing e.g., the extent to which the prior windfield is subject to modification during the calculation.
- The choice of the weights dominates the current effort.
- As noted above, because the relationship of the weight matrices to the true error covariances remains uncertain, the solutions described below should be regarded as least-squares solutions, rather than as maximum likelihood or minimum variance ones—much of the judgement as to acceptability lies with aesthetics, rather than with rigorous statistics.
4. Sample results
- A few papers have appeared (e.g., [27,21,9,28]) exploiting the adjoint model interpretation as the sensitivity of the 2 Because the control vector includes adjustments to the meteorological fields, and which can be regarded as in part “observed”, the distinction between state vector and controls and what is observed, is largely arbitrary.
- The misfit terms in J corresponding to the time-mean seasurface height (or mean dynamic topography, MDT) over the 13-year model run are displayed in Fig.
- The scientific issue is now whether these misfits represent (A) a mis-weighting of the data; (B) model error; or (C) simply that the iterative minimization has not had time to reduce these terms, or all of these things.
- Within the system itself, there is a very powerful test of the model against the data, as there is no guarantee that Au th or 's pe rs on al co py Fig.
- The solutions discussed here appear to be close to consistency with the data, but the optimization calculations are still incomplete.
5. Concluding discussion
- A practical system exists for solving a very large, globalscale, oceanographic least-squares problem using the method of Lagrange multipliers.
- As one example of that use, Fig. 4 displays the estimated mass, heat and salt fluxes in the North Atlantic across 26◦N as a function of time from the least-squares fit [56].
- Table 1, showing the main data sources, coverage, and some information about the originator.
- Parameterized in any easy or known way, and the estimation of their properties demands a model resolution that is impossible on a global scale.
- The major issue is that the sub-model boundaries must be taken as “open”, so as to permit exchange of fluid and information between the two models.
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Related Papers (5)
Frequently Asked Questions (11)
Q2. What is the way to solve the oceanic problem?
the oceanic problem is perhaps best regarded as a form of control problem—one seeks to find those controls (typically the wind, freshwater, and heat exchanges with the atmosphere) that drive the ocean (actually the model) through a trajectory within error bars of all of the observations.
Q3. What is the important unsolved mathematical problem?
Little is known of model error and an important unsolved mathematical problem concerns its representation as a function of time and space.
Q4. Why is the use of meteorological methods not appropriate?
Because much of oceanography has goals distinct from forecasting, the direct application of meteorological methods is often not appropriate.∗
Q5. What is the way to calculate the seasurface height?
In this particular case, calculation of the data errors involves the determination of Earth gravity (providing the reference gravitational surface called the “geoid”; see [58]), the myriad error terms in altimetric determination of seasurface topography (see e.g., [8]), and the expected temporal variability as a function of position and duration in the ocean [38].
Q6. What is the problem with the Kalman filter plus smoother?
In a sequential system such as the Kalman filter plus smoother, one is faced with the computation and storage of covariance matrices square of the dimension of [x(t), u(t)].
Q7. What is the purpose of the problem of using a stored data set over a finite?
The ECCO consortium thus has undertaken the smoothing problem of using a stored data set over a finite time interval, for the purpose of making best estimates of the oceanic state during that time in such a way that the final estimate would be dynamically and kinematically self-consistent, accompanied by an understanding of the structure of residual data/model misfits.
Q8. What is the way to solve the normal equations?
If L were a linear operator, one might contemplate simply solving the normal equations by Gaussian elimination or other algorithm.
Q9. What is the special case in control theory of a terminal constraint problem?
The special case in control theory of a terminal constraint problem (see e.g., [53]) is widely known as the “Pontryagin Principle”.
Q10. What is the well-known example of a biharmonic filter?
A particularly well-known one is Levinson’s [26] conversion of Norbert Wiener’s mathematically challenging filter theory—based on spectral factorization by Wiener–Hopf methods—to the leastsquares form widely exploited in exploration geophysics.)
Q11. What is the simplest way to describe the model error?
In practice, structures that are incorrectly rendered by a model are treated as data error with little understanding of the consequences.