Variational data assimilation: examination of results obtained by different combinations of

numerical algorithms and splitting procedures

Zahari Zlatev and Jørgen BrandtZahari Zlatev and Jørgen BrandtNational Environmental Research InstituteNational Environmental Research Institute

Frederiksborgvej 399, P. O. Box 358Frederiksborgvej 399, P. O. Box 358DK-4000 Roskilde, DenmarkDK-4000 Roskilde, Denmarkzz@dmu.dk, jrb@dmu.dkzz@dmu.dk, jrb@dmu.dk

CONTENTS

Some Some basicbasic ideas ideas Optimization issues: calculation of the Optimization issues: calculation of the gradientgradient of the of the

object functionobject function AlgorithmicAlgorithmic representation of the variational data representation of the variational data

assimilationassimilation Tools:Tools: optimization methods, numerical algorithms and optimization methods, numerical algorithms and

splitting proceduressplitting procedures Performance of the Performance of the combinationcombination of the tools of the tools Some conclusionsSome conclusions

BASIC IDEAS - 1

An attempt to adjust globally the results of the model

to the complete set of available observations (Talagrand and Courtier, 1987)

Consistency between the dynamics of the model and

the final results of the assimilation.(Talagrand and Courtier, 1987)

BASIC IDEAS - 2 Assumption:Assumption: the variational data the variational data

assimilation is used to improve the initial assimilation is used to improve the initial values of the resolved problem values of the resolved problem

N

n

obs

nn

obs

nnn cccctWcJ0

0,

21

Variational data assimilation can be used for Variational data assimilation can be used for several other purposes (as, for example, to several other purposes (as, for example, to improve the quality of the emission fields).improve the quality of the emission fields).

Calculation of the gradient

Assume that Assume that fivefive fields (i.e. N=5) of observations are fields (i.e. N=5) of observations areavailableavailable

43210ccccc

Calculation of the gradient

obsccq

ccccc

111

43210

Calculation of the gradient

obs

occqq

ccccc

111

1

43210

Calculation of the gradient

obs

obs

o

ccqq

ccqq

ccccc

222

2

0

111

1

43210

||

Calculation of the gradient

obs

obs

obs

o

ccqq

ccqq

ccqq

ccccc

333

3

0

222

2

0

111

1

43210

||

||||

Calculation of the gradientFinishing the forward-backward calculationsFinishing the forward-backward calculations

obs

obs

obs

obs

o

ccqq

ccqq

ccqq

ccqq

ccccc

444

4

0

333

3

0

222

2

0

111

1

43210

||

||||

||||||

Calculation of the gradient

obs

obs

obs

obs

o

ccqq

ccqq

ccqq

ccqq

ccccc

444

4

0

333

3

0

222

2

0

111

1

43210

||

||||

||||||

N

n

nqJGRADIENT0

0)(

Adjoint equations

cxt,Atc

qx,tA

tq T

obsccq

)c,x,t(Btc

c)c,x,t('B

t)c(

q)c,x,t('Btq T

1. Linear operators1. Linear operators

2. Non-linear operators2. Non-linear operators

Algorithmic representationINITIALIZE SCALAR VARIABLES, VECTORS AND ARRAYS; SET THE GRADIENT TO ZERO

DO ITERATIONS = 1, MAX_ITERATIONS

DO LARGE_STEPS = 1, P_STEP

DO FORWARD_STEPS = (LARGE_STEPS – 1)*P_LENGTH + 1, LARGE_STEPS*P_LENGTH

Perform a forward step with the model END DO FORWARD_STEPS

DO BACKWARD_STEPS = LARGE_STEPS*P_LENGTH, 1, -1 Perform a backward step with the adjoint equation END DO BACKWARD_STEPS

UPDATE THE GRADIENT; COMPUTE THE VALUE OF THE OBJECT FUNCTION

END DO LARGE_STEPS

COMPUTE AN APPROXIMATION OF PARAMETER RHO UPDATE THE INITIAL VALUE FIELD (NEW FIELD = OLD FIELD - RHO*GRADIENT) CHECK THE STOPPING CRITERIA; IF SATISFIED EXIT FROM LOOP DO ITERATIONS

END DO ITERATIONS

PERFORM OUTPUT OPERATIONS AND STOP THE COMPUTATIONS

Applying splitting procedures

ModelModel

)c,x,t(Btc

Model splittingModel splitting

)c,x,t(Btc

111

)c,x,t(Btc

222

Adjoint equationAdjoint equation

qc,x,t'Btq T

Adjoint splittingAdjoint splitting

1T

1'1

1 qc,x,tBt

q

2T

2'2

2 qc,x,tBt

q

Computational toolsA data assimilation code can be considered as a A data assimilation code can be considered as a combination combination of of threethreekinds of computational tools:kinds of computational tools:

optimization methods,optimization methods, numerical algorithms andnumerical algorithms and splitting procedures.splitting procedures.

It is important to understand that the choice of one of the tools is It is important to understand that the choice of one of the tools is notnotindependent of the choice of the others and the choice independent of the choice of the others and the choice dependsdepends also on also onwhat is wanted in the particular study.what is wanted in the particular study.

Finding the Finding the optimaloptimal (or even only a good) combination of computational (or even only a good) combination of computationaltools is a great challenge.tools is a great challenge.

Splitting versus numerical errorsSplitting Eulerb Mid-point RK6

No splitting 1 2 6Sequential 1 1 1Symmetric 1 2 2

Weigted sequential 1 2+ 2Weighted symmetric 1 2 2

Order of the numerical method: pOrder of the numerical method: pOrder of the splitting procedure: qOrder of the splitting procedure: q

Order of the combined methodsOrder of the combined methods: : r = min(p,q)r = min(p,q)

Synchronizing the choice of splitting procedures Synchronizing the choice of splitting procedures and numerical methodsand numerical methods

Atmospheric chemical schemeAtmospheric chemical scheme

A chemical scheme containing 56 species has been used A chemical scheme containing 56 species has been used in the experimentin the experiment

Species SO2, SO4, O3, NO, NO3, HNO3 PAN, NH3, Species SO2, SO4, O3, NO, NO3, HNO3 PAN, NH3, NH4, OH and many hydrocarbonsNH4, OH and many hydrocarbons

Mathematical description: Mathematical description: dc/dt = f(t,c)dc/dt = f(t,c) Properties:Properties:

stiffstiff badly scaledbadly scaled the solution varies in very wide rangethe solution varies in very wide range

Treatment of the chemical scheme

Six numerical methodsSix numerical methods Backward EulerBackward Euler Implicit Mid-point RuleImplicit Mid-point Rule Two-stage Runge-Kutta Two-stage Runge-Kutta Three-stage Runge-KuttaThree-stage Runge-Kutta Two-stage RosenbrockTwo-stage Rosenbrock Trapezoidal RuleTrapezoidal Rule

Five splittingsFive splittings No splittingNo splitting Sequential splittingSequential splitting Symmetric SplittingSymmetric Splitting Weighted sequential splittingWeighted sequential splitting Weighted symmetric splittingWeighted symmetric splitting

Some conclusions from the runs The Backward Euler method is very robust for such problems, but The Backward Euler method is very robust for such problems, but

might be expensive because its order of accuracy is only one.might be expensive because its order of accuracy is only one. The Implicit Mid-point Rule, the Trapezoidal Rule and the Three-stage The Implicit Mid-point Rule, the Trapezoidal Rule and the Three-stage

Runge-Kutta method have difficulties.Runge-Kutta method have difficulties. The Two-stage Runge-Kutta Method and the Two-stage Rosenbrock The Two-stage Runge-Kutta Method and the Two-stage Rosenbrock

Method are robust, but perform as first-order methods in spite of the Method are robust, but perform as first-order methods in spite of the fact that their actual order is two.fact that their actual order is two.

Major conclusion: It is not sufficient to have numerical methods of Major conclusion: It is not sufficient to have numerical methods of high order, it is also necessary to select methods with good high order, it is also necessary to select methods with good stability properties (L-stability is actually needed).stability properties (L-stability is actually needed).

Need for good stability properties Euler Mid-pointEuler Mid-pointTime-steps Error Rate Error RateTime-steps Error Rate Error Rate 144 9.0E-2 - 1.6E-2 -144 9.0E-2 - 1.6E-2 - 288 4.5E-2 2.0 6.1E-3 2.6288 4.5E-2 2.0 6.1E-3 2.6 576 2.3E-2 2.0 2.0E-3 3.0576 2.3E-2 2.0 2.0E-3 3.0 1152 1.1E-2 2.0 6.2E-4 3.31152 1.1E-2 2.0 6.2E-4 3.3 2304 5.6E-3 2.0 1.7E-4 3.62304 5.6E-3 2.0 1.7E-4 3.6 4608 2.8E-3 2.0 4.4E-5 3.84608 2.8E-3 2.0 4.4E-5 3.8 Assimilation window: 6:00 - 12:00, observations at the end of every hourAssimilation window: 6:00 - 12:00, observations at the end of every hour Random perturbations of 25% are introduced in the ozone concentrations.Random perturbations of 25% are introduced in the ozone concentrations. Data assimilation was applied using “exact” solutions as observations.Data assimilation was applied using “exact” solutions as observations. After the data assimilation procedure a forward run with the improved solution After the data assimilation procedure a forward run with the improved solution

was performed on the interval was performed on the interval from 6:00 to 12:00from 6:00 to 12:00

0.1,cmax

ccmaxERROR ref

in

refinin

N,...,1n;q,...,1i

Need for good stability properties-cont. Euler Mid-pointEuler Mid-pointTime-steps Error Rate Error RateTime-steps Error Rate Error Rate 1008 2.3E-1 - 3.2E-1 -1008 2.3E-1 - 3.2E-1 - 2016 1.1E-1 2.0 4.5E-2 6.52016 1.1E-1 2.0 4.5E-2 6.5 4032 5.5E-2 2.0 8.8E-3 5.74032 5.5E-2 2.0 8.8E-3 5.7 8064 2.7E-2 2.0 8064 2.7E-2 2.0 1.9E-2 0.51.9E-2 0.5 16128 1.4E-2 2.0 16128 1.4E-2 2.0 2.2E-2 0.92.2E-2 0.9 32256 6.8E-3 2.0 32256 6.8E-3 2.0 2.0E-2 1.42.0E-2 1.4 Assimilation window: 6:00 - 12:00, observations at the end of every hourAssimilation window: 6:00 - 12:00, observations at the end of every hour Random perturbations of 25% are introduced in the ozone concentrations.Random perturbations of 25% are introduced in the ozone concentrations. Data assimilation was applied using “exact” solutions as observations.Data assimilation was applied using “exact” solutions as observations. After the data assimilation procedure a forward run with the improved solution After the data assimilation procedure a forward run with the improved solution

was performed on the interval was performed on the interval from 6:00 to 48:00from 6:00 to 48:00

0.1,cmax

ccmaxERROR ref

in

refinin

N,...,1n;q,...,1i

General conclusions When will the variational data assimilation not work?When will the variational data assimilation not work?

the discretization of the model is crudethe discretization of the model is crude the number of observations is very small (???)the number of observations is very small (???)

When will the variational data assimilation work well?When will the variational data assimilation work well? there are sufficiently many observationsthere are sufficiently many observations interpolation rules (or other similar devices) are usedinterpolation rules (or other similar devices) are used the number of time-points at which observations are available the number of time-points at which observations are available

is perhaps not very importantis perhaps not very important How to continue this research?How to continue this research?

two-dimensional and three dimensional problemstwo-dimensional and three dimensional problems more about the effect of splitting on data assimilationmore about the effect of splitting on data assimilation order of accuracy of the data assimilation routinesorder of accuracy of the data assimilation routines interplay between ensembles and data assimilationinterplay between ensembles and data assimilation