fastopt a road map to an adjoint analysis of the summer 2007 low in arctic sea-ice area frank...

FastOpt

A road map to an adjoint analysis of the summer 2007 low in Arctic sea-ice area

Frank Kauker, Thomas Kaminski, Michael Karcher, Ralf Giering, Rüdiger Gerdes and Michael Vossbeck

OASys, FastOpt and AWI

http://FastOpt.com http://OASys-Research.de

http://awi.de

FastOpt

Goals ADNOASIM and NAOSIMDAS

Construct the adjoint of the coupled sea ice-ocean NAOSIM (ADNAOSIM) by means of the compiler tool TAF (Giering and Kaminski, 1998)

NAOSIM was developed at the AWI (Gerdes et al. 2003, Karcher et al. 2005, Kauker et al, 2005)

Calculate adjoint sensitivities with ADNOASIM

Build a data assimilation system around ADNAOSIM - NAOSIMDAS

Perform 4DVAR data assimilation for two time periods:

– 1979 to 1981 “high” ice cover, start of remote-sensing data

– most recent 2006(7) to 2008(9) “low “ ice cover

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Test configuration

• Integration time: 1 hour to 1 year

• Time step: 1 hour

• 2 x 2 degree horizontal resolution

• 11 vertical layers

• Model domain: north of about 50N

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“Wish” configuration

• Integration time: up to 3 years

• Time step: 1/2 hour

• 0.5 x 0.5 degree horizontal resolution

• 20 vertical layers

• Model domain: north of about 50N

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Forward integration: Ice extent summer 2007

NSIDC 09/2007 NAOSIM 09/2007

NAOSIM 1/12°forced with dailyNCEP reanalysis

(Gerdes et al., 2008)

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NSIDC data includes the Seaof Okhotsk and Bering Sea -model data NOT


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f1 f2

Statements in codedefine elementary functionssuch as +, / , **, sin , exp …

fN ……

Numerical Model

fN-1

target quantity(of physical or societal relevance)

vector of parameters,initial and boundaryconditions

Sensitivities via AD

m y

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Sensitivities via AD: Tangent

Df1 Df2 DfN ……

Tangent Linear Model

DfN-1

Applies chain rule in forward direction

δ m

Derivatives of elementary functions are simple,they define local Jacobians

Cost of gradient evaluation proportional to length of control vector :One run of TLM per gradient component

∇ y⋅δ m

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Sensitivities via AD: Adjoint

Adjoint Model

Df1 Df2 DfN ……DfN-1

Applies chain rule in reverse direction

Cost of gradient evaluation independent of length of control vector:One run of adjoint for entire gradientBut: Reversal of control flow complicates coding

∇ y δy= 1

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Compiler Tool TAF

•TAF (http://FastOpt.com) is a commercial tool for automatic differentiation

•It accepts a model code in Fortran 77-95 and generates a derivative code

•Can generate tangent linear or adjoint code

•Can do vector mode (many perturbations at a time)

•Higher order derivative (e.g., Hessian) code by invoking TAF recursively

•Command line tool with many options

•Generated code is structured and well-readable

•Can handle black box routines (via TAF flow directives)

•Supports parallelisation (MPI/OpenMP)

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Adjoint sensitivity: Ice area summer 2007

•target variable y : total ice area A at the end of the integration (tend

) August 2007

•input x: all sbc (monthly), model parameter, initial state

•initialize the model at 1. Mar. 2007

•run ADNAOSIM for 6 months

•output: dA(tend

)/dx

•calculate monthly sbc and initial state anomalies for 2007 rel. to long-term mean

•<anom_sbc,sens_sbc> gives you the impact of the sbc anomaly on the ice area

Results will be presented at the EGU (session OS7)Here adjoint sensitivities for 1979 shown!

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wind stress [10^4 km2/(N/m2)] 2mT [10^4 km2/K]

Surface boundary conditions:

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Sensitivities via AD: Test gradient

adjoint gradient

tangent linear gradient

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Show movie!

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Model parameters:

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Initial state: oceanSST [10^2 km2/K] Temp 60m [10^2 km2/K]

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Initial state: iceice thickness [10^2 km2/m] ice concentration [10^2 km2]

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Variational Data Assimilation

Notation:s : state vector (ocean: u’,v’,s,tpot,ψ ; ice: h,a,age,hsn)t : time d : vector of observationsσ : vector observational uncertainties

Principle: •define vector of control variables x, e.g.,

•initial state (s0)•forcing/boundary conditions (f)•internal model parameters (p)

•define quality of fit by cost function:•minimise J(x) by variation of x

d1 (obs. 1)

t

s

uncertainty for obs. term uncertainty for prior term

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MinimisationEfficient minimisation algorithms use J(x) and

the gradient of J(x) in an iterative procedure.

Typically the prior value is used as starting point of the iteration.

The gradient is helpful as it always points uphill.

The adjoint is used to provide the gradient efficiently.

Example: Newton algorithm for minimisation

Gradient: g(x) = dJ/dx(x)

Hessian: H(x) = dg/dx(x) = d2J/dx2(x)

At the minimum, xmin

: g(xmin

) = 0, hence:

g(x) = g(x) – g(xmin

) ~ H (x) (x-xmin

)

rearranging yields:

(xmin

- x) ~ - H-1(x) g(x)

Smart gradient algorithms use an approximation of H(x)

Figure: Tarantola (1987)

Figure: Fischer (1996)

very high dimensional space

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Next Steps:

•Adjoint Sensitivities auf summer 2007 ice

•Speed up adjoint (storing on tape vs. in RAM)

•Check pointing

•Make code “smoother”

•Determine the “assimilation window”

•First assimilation in coarse resolution of 1979-81 (this summer)

•Gather data for most recent period

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Next Steps:

Data streams recent period:

We plan to use almost allDAMOCLES sea ice-oceandata which will be available.

AON data are very welcome!

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Next Steps:

Data streams recent period:

•We indent to assimilate ice drift products (met.no)

•We are gathering ARGO profiles (http://www.coriolis.eu.org)

ARGO profile distribution for 2006

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S4D - 4DVar Coupled Ice-Ocean Working Group:

The DAMOCLES partners FastOpt and OASys, as well as the US institutes MIT and JPL intend to collaborate on the development of variational data assimilation methods in a working group on polar regions coupled sea ice-ocean model data assimilation and adjoint sensitivities.

On the US side, the main variational data assimilation system (built around the MITgcm) is operated by the ECCO consortium. Their experts for the development and maintenance of the assimilation system's coupled sea ice/ocean component are Patrick Heimbach (MIT) and Dimitris Menemenlis (JPL).

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S4D - 4DVar Coupled Ice-Ocean Working Group:Major questions of the working group are:

●What is the range of validity for the linearization of the coupled sea ice-ocean system, i.e., for how large a perturbation is the linearization useful?●How does this range depend on factors such as the length of the integration period, the spatial and temporal resolution, and details of implementation?●What are the remedies in an optimization environment (see Koehl and Willebrand, 2000) and for pure for sensitivity calculations?●Are the adjoint sensitivities robust, i.e., what are the differences between the sensitivities from the MITgcm and those of NAOSIM?●Which data sets (remotely sensed and in-situ ocean and sea-ice data) are most useful for data assimilation? ●Are XBT ocean data sets usable or do we have to neglect them (see Gouretzki et al. 2007)?●Is it useful to preprocess data sets?●How can a quality control be implemented?●How do we systematically assign weights to the observations and other constraints?●How do we handle correlated uncertainties?●How do we quantify representation error?●Are there common tasks, e.g., in terms of preparing data sets for assimilation, than can be shared between the ECCO and DAMOCLES efforts?

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S4D - 4DVar Coupled Ice-Ocean Working Group:

The MIT/JPL/FastOpt/OASys working group is building the collaboration on two events, one one in Boston in autumn 2007 and in Hamburg 2008.

●The Working Visit Boston 2007 focussed on: Presentation of NAOSIM-DAS setup and progress, identification of key sensitivities to be compared

●The Working Visit Hamburg 2008: The WG plans a working session of 2-3 weeks in summer 2008, in Hamburg, Germany. Participants from the US side are Dimitris Menemenlis (JPL) and Patrick Heimbach (MIT). From the EU side, Ralf Giering , Thomas Kaminski, Michael Vossbeck (FastOpt), Michael Karcher, and Frank Kauker (OASys) will participate, The meeting focus will be on comparison and exchange of algorithms for handling of data streams esp. for new data types (glider data, ITP data, high resolution ice-drift), predictability issues

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Fini!

fastopt a road map to an adjoint analysis of the summer 2007 low in arctic sea-ice area frank...

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