Data Assimilation and Coupling...Coupling Technologies for Earth System Modelling: Today and Tomorrow Toulouse, December 15th-17th 2010 IS-ENES Workshop p. 2 December 15th-17th 2010

IS-ENES Workshop December 15th-17th 2010 A. Piacentini

Data Assimilation and Coupling

Andrea Piacentini (CERFACS)

IS-ENES WorkshopCoupling Technologies for Earth System Modelling:

Today and TomorrowToulouse, December 15th-17th 2010

IS-ENES Workshop p. 2 December 15th-17th 2010 A. Piacentini

Overview

Basic facts Data Assimilation actors Data Assimilation implementattion Specific requirements for D.A. coupling Some examples Indications for D.A. in coupled models


Basic facts 1.

Some terminology based on everyday experience:The problem:


Basic facts 1.

Some terminology based on everyday experience:The problem: WHAT TIME IS IT?


Basic facts 1.

Some terminology based on everyday experience:The problem: WHAT TIME IS IT?I read my watch. This is a "first guess" for the answer.


Basic facts 1.

Some terminology based on everyday experience:The problem: WHAT TIME IS IT?I read my watch. This is a "first guess" for the answer.I look for more information to confirm or correct my first result.I want to add information upon the "background" given by my watch.I have a look to a second watch and I compute the "misfit" between the two sources of information.


Basic facts 2.

Neglecting any bias of the watches, before answering the question 'what time is it?' I consider the brand, the age, the conditions of each watch to express some statistical characterizations of the "background error" and of the "observation error" to give a relative weight to the different sources of information.


Basic facts 2.

Neglecting any bias of the watches, before answering the question 'what time is it?' I consider the brand, the age, the conditions of each watch to express some statistical characterizations of the "background error" and of the "observation error" to give a relative weight to the different sources of information.

After some "algebra" I end up with the so called "analysis" of my sources of information as a [hopefully] optimal answer to the stated problem, and, by an adjustment of my watch, it will provide a better "initial condition" for my watch (read "model") for the next time I'll need to know the hour.


Basic facts 3.

Let's introduce new information. My second source of information is a clock in N.Y. or the sun

height. They express the time of the day but not in a form directly

comparable with the time expressed by the watch. If we want to compute the misfit, we need a way to express

the two sources of information in the same space and reference, by the action of an "observation operator".

This could be a simple lag removal or a sundial.


Basic facts 4.We've introduced all the main elements but three that we

cannot introduce with the watch example and for which we do not really need an example.




1) If the modelled quantity is not a scalar, but a 3D field, what we called the "background error" cannot be expressed just as a 3D point-wise r.m.s. We also need to express how the errors at different locations are correlated at a given moment.




1) If the modelled quantity is not a scalar, but a 3D field, what we called the "background error" cannot be expressed just as a 3D point-wise r.m.s. We also need to express how the errors at different locations are correlated at a given moment. That's why the object expressing the "background error" is a dreadful 3Dx3D variance/covariance matrix. For real size problems, such an object cannot be explicitly represented and we'll rather use an operator to represent the "B matrix vector product".




1) If the modelled quantity is not a scalar, but a 3D field, what we called the "background error" cannot be expressed just as a 3D point-wise r.m.s. We also need to express how the errors at different locations are correlated at a given moment. That's why the object expressing the "background error" is a dreadful 3Dx3D variance/covariance matrix. For real size problems, such an object cannot be explicitly represented and we'll rather use an operator to represent the "B matrix vector product".

The same consideration apply for the "observation error", for which we introduce the "R matrix vector product" operator, even if most of the time the R matrix is simplified to be diagonal or block diagonal.


2) With a time evolving model, we can take into account observations collected on a time interval called "window". In such a case, some D.A. methods look for the optimal initial condition that should have be used for the best fit on the assimilation window.

In such a case we can use our data assimilation suite to provide the best initial condition for the successive forecast, taken as the last instant of the assimilation window, or to provide the most complete estimate of the modelled system on the given window. That's what's currently called a "reanalysis"

Basic facts 5.


Basic facts 6.

3) We do not always look for the best initial condition. We can modify some model parameters (or forcings). The quantity we are going to modify is usually called the "control". The observation operator maps the control space onto the observation space. We also need a 'way to go back' relying variations on the control space to variations in the observation space. This is the "adjoint" of the observation operator.

In the case of 4D methods or when controlling model parameters, the observation operator includes a model integration, therefore its adjoint includes the adjoint of the model. Some linearization or finite differences approximations can help us when the adjoint model is not available.


Main actors

From the previous explanation

First Guess/BackgroundObservationObservation operatorAdjoint observation operatorMisfitBackground error/B matrix productObservation error/R matrix productControlAlgebra[Re]AnalysisModelAdjoint model


Implementation

Integration vs. couplingPerformances vs. flexibility/min intrusiveness

Exactly as in climate coupling, for integration, everything has to be decided beforehand.Coupling allows for flexibility and modularity and if well designed the low intrusiveness eases the use of existing codes.

Furthermore, the coupling approach allows for some level of task parallelism without rewriting codes. E.g. the of overlap of the observation operator and of model integration.


Specific coupling

First situation, sequential pure 3D "inversion" schemes.Task chaining + cycling


Hydrology


MERCATOR

Complex algorithmHigh performances… as required!


Specific coupling


Second situation, variational pure 3D "optimisation" schemesLoop around tasks, driven by the minimizer. Conditional and repeated executions.Convergence criterion: number of iterates not known in advance.


Shallow Water


Specific coupling


Second situation, variational pure 3D "optimisation" schemesLoop around tasks, driven by the minimizer. Conditional and repeated executions.Convergence criterion: number of iterates not known in advance.

Third situation, variational 4D (time window) schemesModel driven by assimilation. Repeated integrations moving back and forth the initial integration time.

From the Mercator lesson, PALM as a parallel dynamic coupler


Valentina


Data assimilation in coupled modelsSubset of models undergoing DA: best trade-off between the computational burden and the risk of component incoherence. Cf. the use of ocean reanalyses (from a forced ocean model) to initialize coupled decadal forecast. Difference between coupling data assimilation systems and assimilating in a coupled model. Control : collective initial condition. Flux corrections, ... ?Observations : most probably coming from pre-treatments and/or re-analyses. Observation operator will most probably not include space changes.Coupling strategies:*) Include the data assimilation operators in a single overall

coupling*) Nested coupled instances: The "model task" in an

assimilation coupling triggers a climate coupling


Data assimilation in coupled modelsThe advantage of assimilating in a coupled model is that observations of a component lead to a correction on all the coupled components.

Nevertheless this “information transmission” partly depend on the way you model the background error correlations. Risk of non physical corrections.Need of a precise characterization of the coupled model errors, in particular of biases.

Observation bias correction techniques can benefit model bias estimates.

Need of some workaround to account for differences in time and space scales of the coupled components.

Data Assimilation and Coupling...Coupling Technologies for Earth System Modelling: Today and Tomorrow Toulouse, December 15th-17th 2010 IS-ENES Workshop p. 2 December 15th-17th 2010

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