Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’

Lecture 4. Aspects of

Chemical Data Assimilation

Adrian Sandu Computational Science Laboratory

Virginia Polytechnic Institute and State University

Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.

Simulation of air pollution, E. Asia, 3 days in March 2001, provides motivation

Motivation: large-scale simulations of PDE-based models with dynamic space/time adaptivity

data distribution on processors

refinement – according to criteria that minimize errors for

target species

discretization

~0.5 million variables ~0.5 billion variables

processor distribution


Data assimilation fuses information from (1) prior, (2) model, (3) observations to obtain consistent description of a physical system

Optimal analysis state

Chemical kinetics

Aerosols

CTM

Transport Meteorology

Emissions

Observations Data

Assimilation

Targeted Observ.

Improved: •  forecasts •  science •  field experiment design •  models •  emission estimates


Data assimilation problems of practical interest are large and computationally intensive

Lars Isaksen (http://www.ecmwf.int)

How large are the models? Typically O(108) variables, and O(10) different

physical processes

How many observations are being assimilated? All data assimilated at ECMWF 1996-2010: O(107)

Aug. 16, 2011. Statistical Engineering Division, NIST.

Some conventional and remote data sources used at ECMWF for numerical weather prediction


SYNOP/METAR/SHIP: pres., wind, RH Aircraft: wind, temperature

Lars Isaksen (http://www.ecmwf.int)

13 Sounders: NOAA AMSU-A/B, HIRS, AIRS, … Geostationary, 4 IR and 5 winds

The most popular approaches for chemical data assimilation are 4D-Var and EnKF

4D-Var n  4D-Var treats all observations within one assimilation window

simultaneously (can use observations from frequently reporting stations) n  4D-Var generates analysis consistent with the system dynamics

n  4D-Var more complex: needs linearized perturbation forecast model and its adjoint to solve the cost function minimization problem efficiently

n  Analysis covariances difficult to estimate

EnKF n  EnKF is easier to implement (no adjoint) n  EnKF updates both state and error covariance n  EnKF analysis flow inconsistent with dynamics (see EnKS) n  Rank-deficient covariances require additional work to fix n  EnKF sampling error is large in large-scale models


Discretize-then-linearize leads to the discrete adjoint model (e.g., of mass balance equations)

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on

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ρρ

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],[

],[1

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tHOR

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tCHEM

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Δ+

++Δ+

=

+=

φλλ

Continuous forward model (PDE)

Discrete forward model (discretized PDE)

Discrete adjoint model (computes derivatives on the numerical solution)

discr

adj


Linearize-then-discretize leads to the continuous adjoint model (e.g., of mass balance equations)

( ) ( )

( ) ( )( ) ( )

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i

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Γ=∂

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Γ=∂

∂+

Γ=

≥≥=

−⋅−⎟⎟⎠

⎞⎜⎜⎝

⎛∇⋅⋅∇−⋅∇−=

on

on0

on0,

,,

)(

λρλ

ρ

ρλρλ

λ

λλ

φλρρλ

ρλλ

( ) ( ) ( ) ( ) ( )'*'*'*'*'*],[

'*

11],[

'*

tHOR

tVERT

tCHEM

tVERT

tHORttt

kkttt

k

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MΔΔΔΔΔ

Δ+

++Δ+

=

+=

φλλ

Continuous forward model (PDE) Adjoint PDE model (exact derivatives)

Continuous adjoint model (discretized adjoint PDE)

adj

discr


Continuous and discrete adjoints (of mass balance equations) lead to different computational models

( ) ( )

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iiiii

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nCK

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Γ=∂

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Γ=

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on

on0

on,,

,,

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ρρ

ρρ

( ) ( )

( ) ( )( ) ( )

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Γ=∂

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Γ=

≤≤=

++∇⋅∇+∇⋅−=

on

on0

on,,

,,

11

000

ρρ

ρρ

( ) ( )

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⎛∇⋅⋅∇−⋅∇−=

on

on0

on0,

,,

)(

λρλρ

ρλρλ

λλλ

φλρρλρλλ

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Γ=∂

∂+

Γ=

≥≥=

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⎞⎜⎜⎝

⎛∇⋅⋅∇−⋅∇−=

on

on0

on0,

,,

)(

λρλρ

ρλρλ

λλλ

φλρρλρλλ

Continuous forward model (PDE)

Discrete forward model (discretized PDE)

Adjoint PDE

Adjoint models ≠

adj

discr

adj

discr


( ) ( )

( ) ( )( )

( )

( ) GROUNDi

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∂

Γ=∂

∂+

Γ=

≥≥=

−⋅−⎟⎟⎠

⎞⎜⎜⎝

⎛∇⋅⋅∇−⋅∇−=

on

on0

on0,

,,

)(

λρλρ

ρλρλλ

λλ

φλρρλρλλ

What do each adjoint represents? The question is relevant for sensitivity analysis and inverse problems

Sensitivity analysis: how well does the derivative of the numerical solution approximate the continuous derivative?

( ) JJJ

J

J

∇−∇⋅∇≤−

=

=

hopt

2opt

hopt

hhopt

opt

)(cond

argmin

argmin

xxx

x

x

0

0

x

x

Inverse problems: how well does the discrete optimum approximate the continuous optimum?

Compute to represent


!#↑ℎ  !#

The construction of adjoint models is work-intensive and error-prone. Automatic implementation attractive


Chemical kinetics

Aerosols

CTM


Emissions

Observations Data

Assimilation

Targeted Observ.



Discrete adjoints of advection numerical schemes can become inconsistent with the adjoint PDE

Change of forward scheme pattern: •  Change of upwinding direction •  Sources/sinks •  Inflow boundaries scheme Example: 3rd order upwind FD [Liu and Sandu, 2005]

Active forward limiters act as pseudo-sources in adjoint

Example: minmod


discretization change

upwind direction change



Chemical kinetics

Aerosols

CTM


Emissions

Observations Data

Assimilation

Targeted Observ.



KPP automatically generates simulation and direct/adjoint sensitivity code for chemistry

#INCLUDE atoms #DEFVAR O = O; O1D = O; O3 = O + O + O; NO = N + O; NO2 = N + O + O; #DEFFIX O2 = O + O; M = ignore; #EQUATIONS { Small Stratospheric } O2 + hv = 2O : 2.6E-10*S; O + O2 = O3 : 8.0E-17; O3 + hv = O + O2 : 6.1E-04*S; O + O3 = 2O2 : 1.5E-15; O3 + hv = O1D + O2 : 1.0E-03*S; O1D + M = O + M : 7.1E-11; O1D + O3 = 2O2 : 1.2E-10; NO + O3 = NO2 + O2 : 6.0E-15; NO2 + O = NO + O2 : 1.0E-11; NO2 + hv = NO + O : 1.2E-02*S;

SUBROUTINE FunVar ( V, F, RCT, DV ) INCLUDE 'small.h' REAL*8 V(NVAR), F(NFIX) REAL*8 RCT(NREACT), DV(NVAR) C A - rate for each equation REAL*8 A(NREACT) C Computation of equation rates A(1) = RCT(1)*F(2) A(2) = RCT(2)*V(2)*F(2) A(3) = RCT(3)*V(3) A(4) = RCT(4)*V(2)*V(3) A(5) = RCT(5)*V(3) A(6) = RCT(6)*V(1)*F(1) A(7) = RCT(7)*V(1)*V(3) A(8) = RCT(8)*V(3)*V(4) A(9) = RCT(9)*V(2)*V(5) A(10) = RCT(10)*V(5) C Aggregate function DV(1) = A(5)-A(6)-A(7) DV(2) = 2*A(1)-A(2)+A(3)-A(4)+A(6)-&A(9)+A(10) DV(3) = A(2)-A(3)-A(4)-A(5)-A(7)-A(8) DV(4) = -A(8)+A(9)+A(10) DV(5) = A(8)-A(9)-A(10) END

K P P

[Damian et.al., 1996; Sandu et.al., 2002]

Chemical mechanism Simulation code


Sparse Jacobians , Hessians, and sparse linear algebra routines are automatically generated by KPP


Methods available in the KPP numerical library

n  FIRK 3-stage: Radau-2A (ord.5), Radau-1A (ord.5), Lobatto-3C (ord.4), Gauss (ord.6)

n  SDIRK: 2a, 2b (2s, ord.2), 3a (3s, ord.2), 4a, 4b (5s, ord.4) n  Rosenbrock: Ros2, Ros3, Ros4, Rodas3, Rodas4.

Forward TLM (DDM) Discrete ADJ


Discrete Runge-Kutta adjoints can be regarded as “numerical methods” applied to the adjoint ODE

€

λn = λn+1 + θ i

i=1

s

∑

θ i = h JT Yi( )⋅ bi λn+1 + a j,iθj

j=1

s

∑⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

yn+1 = yn + h bif Yi( )

i=1

s

∑ ,

Yi = yn + h ai,j f Yj( )

i=1

s

∑

RK Method Discrete RK Adjoint [Hager, 2000]


Error analysis: The discrete adjoint (of order p RK method) converges with order p to the solution of the adjoint ODE. [Sandu, 2005]

Discrete Runge-Kutta adjoints: error analysis Local error analysis: The discrete adjoint of RK method of order p is an order p discretization of the adjoint equation. [Sandu, 2005] . This: - works for both explicit and implicit methods - true for arbitrary orders p


Global error analysis: The discrete adjoint (of a RK method convergent with order p) converges with order p to the solution of the adjoint ODE. [Sandu, 2005] The analysis accounts for: 1.  the truncation error at each step, and 2.  the different trajectories about which the continuous and the discrete

adjoints are defined

Stiff case: Consider a stiffly accurate Runge Kutta method of order p with invertible coefficient matrix A. The discrete adjoint provides: 1.  an order p discretization of the adjoint of nonstiff variable 2.  an order min(p,q+1,r+1) of the adjoint of stiff variable [Sandu, 2005]

Adjoint differentiation of adaptive time step mechanism can result in O(1) perturbation

€

yn+1 =Θ yn , hn , t n( )hn+1 = Γ yn, hn , t n( )t n+1 = hn + t n

€

λ(y )n =Θy

T ⋅ λ(y )n+1 + Γy

T ⋅ λ(h )n+1

λ(h )n =Θh

T ⋅ λ(y )n+1 + Γh

T ⋅ λ(h )n+1 + λ( t )

n+1

λ( t )n =Θ t

T ⋅ λ(y )n+1 + Γt

T ⋅ λ(h )n+1 + λ( t )

n+1

Analysis (for one-step methods) uses extended state

Adjoints of step λ(h) influence adjoints of state

Prothero-Robinson example (DOPRI-5) with and without

adjoint correction




Chemical kinetics

Aerosols

CTM


Emissions

Observations Data

Assimilation

Targeted Observ.



June 20, 2005

Chemical D.A. and Data Needs

for Air Quality Forecasting

Populations of aerosols (particles in the atmosphere) are described by their mass density

( ) ( )

( ) ( )( ) ( ) .0,,0,0

,1,,

''),'()',(

''),'(),'()','(

00

0

0

=∞===

≤≤==

+−+∂

∂−+

−

−

−−=

∂

∂

∫

∫∞

tmqtmqnimqttmq

qRLqSmmHqm

qH

dmmtmqmmq

dmmmtmmqtmqmmm

tq

ii

ii

iiiii

i

m

ii

β

β

m+m’ m m’

Coagulation

m imH

Growth

Aerosol dynamic equation - IPDE

June 20, 2005



Adjoint aerosol dynamic models are needed to solve inverse problems

( ) ( ) ( ) ( )),,(),,()( 11

112

1121 k

nkk

N

kk

Tkn

kkBTB qqhyRqqhyppBppp −−+−−= ∑=

−−ψ

( ) [ ]

[ ] ( )

( ) ( )( ) ( ) ( ) ( ) .0,0,,,

)(),(),(,0,

'),'(),(),'(),'(

'),'(),(),'(')',(

100

1

10 1

1

1

0

1

=−+=

−+==

−−−+−

≤≤+−+−=∂

∂

−+

−+−

=

∞

=

−

−−

+

∞−

∑∫ ∑

∫

tppBptmtm

qhyRhtmtmtm

mHHdmtmqtmtmmmmm

tttLdmtmqtmtmmmmmt

iBT

qii

kkk

Tq

ki

ki

Ni

mi

n

jjj

n

jjjj

kkiii

i

i

i

λλλ

λλλ

λλλλβ

λλλβλ

[Sandu et. al., 2004; Henze et. al., 2004]

( )( ) ( ) ( ))(1

1

111 kkk

Tq

n

j

kj

Tki

Tkii

TkTki

ki qhyRhAqBLAqBGt

i−+⎥

⎦

⎤⎢⎣

⎡×+⋅−×+⋅Δ+= −

=

+−+−+ ∑λλλλ

Discrete adjoint equation

Continuous adjoint equation

Cost functional

June 20, 2005



Data assimilation simultaneously recovers the aerosol initial distribution and model parameters

[Sandu et. al., 2004; Henze et. al., 2004]

(Growth rate) (Coag. kernel)

Complete info: Observations of density in each bin allow complete recovery

(Growth rate) (Coag. kernel)

Optical info: Observations of total surface density allow partial recovery



Chemical kinetics

Aerosols

CTM


Emissions

Observations Data

Assimilation

Targeted Observ.





Chemical kinetics

Aerosols

CTM


Emissions

Observations Data

Assimilation

Targeted Observ.



Adjoint variables for an ozone sensor located on Cheju Island at the end of the three days period


Adjoint sensitivity analysis of non-attainment metrics can help guide policy decisions

Estimated contributions by state to violating U.S. ozone NAAQS

in July 2004

[Hakami et al., 2005]


Ensemble-based chemical data assimilation is an alternative to variational techniques


Chemical kinetics

Aerosols

CTM


Emissions

Observations Ensemble

Data Assimilation

Targeted Observ.



Assimilation of ozone data from the ICARTT field campaign in Eastern U.S., July 2004

[Chai et al., 2006]

Nov. 30, 2011. Lecture 1: Data assimilation.

The Ensemble Kalman Filter (EnKF) popular in NWP but not extensively used before with CTMs

Specify initial ensemble (sample B) Covariance inflation: Prevents filter

divergence (additive, multiplicative, model-specific)

Covariance localization (limit long-distance spurious correlations)

Correction localization (limit increments away from observations)

Ozonesonde S2 (18 EDT, July 20, 2004) [Constantinescu et al., 2007]

( )( ) ( )kfk

kobs

Tk

kfkk

Tk

kf

kf

ka

ka

kkf tM

yHzHPHRHPyy

yy

−++=

=−

−−

1

11,


Ground level ozone at 2pm EDT, July 20, 2004, better matches observations after LEnKF data assimilation

Forecast (R2=0.24/0.28) Analysis (R2=0.88/0.32)

Observations: circles, color coded by O3 mixing ratio


Parallelization important to speed up 4D-Var

[Sandu et.al., 2003-2008]

Chemistry w/ abstract vectors

Transport on accelerators


The use of adjoints in large scale simulations: atmospheric chemical transport models


Chemical kinetics

Aerosols

CTM


Emissions

Observations 4D-Var

Data Assimilation

Targeted Observ.



STEM: Assimilation adjusts O3 predictions considerably at 4pm EDT on July 20, 2004

Observations: circles, color coded by O3 mixing ratio

Ground O3 (forecast) Ground O3 (analysis)

[Chai et al., 2006]


Assimilation of ozonesonde observations for July 20, 2004, show importance of vertical information

[Chai et al., 2006]


Assimilation of NOAA P3 in-situ and DC-8 lidar observations for July 20, 2004

[Chai et al., 2006]


The model results show a considerably improved agreement with observations after data assimilation

q-q modeled O3 vs observations

[Chai et al., 2006]


The inversion procedure can be extended to emissions, boundary conditions, etc.

NO2 emission

corrections

Texas: 4am CST July 16 to 8pm CST on July 17, 2004.

HCHO emission

corrections

O3 AirNow

NO2 Schiamacy


[Zhang, Sandu et al., 2006]

Second order adjoints provide Hessian-vector products useful in optimization and analysis

( ) ( ) ( ) 1cov −∇≈⋅∇=∇= JJJ 2x

002x

0x

0000 xxσλ δ

[Sandu et. al., 2007]


Smallest Hessian eigenvalues (vectors) approximate the principal aposteriori error components

(a) 3D view (5ppb)

(b) East view

(c) Top view

€

∇ y 0 ,y 02 Ψ( )−1 ≈ cov y opt( )



Hessian singular vectors describe directions of maximum error growth in finite time

vvMAM ⋅∇⋅=⋅⋅⋅ ψγ 2''*



Best observation locations are different for different chemical species

∑≥

=1

22max

2

kksT k

σ

σ(criterion based on SVs)

Verification: Korea, ground O3

0 GMT, Mar/4/2001

O3 NO2 HCHO


[Sandu, 2006]

Assimilation of TES ozone column observations from Aug. 2006. IONS-6 ozonesonde data for validation.


TES is one of four instruments on the NASA EOS Aura platform,

launched July 14 2004

Quality of TES ozone column assimilation results for several DA methods (August 1-15, 2006)


[Singh, Sandu et. al., 2010]

Dynamic integration of data and models is an important, growing field

n  Important challenges n  Size of the problem (models & data) n  Representation of uncertainty in data, models, priors

n  Most widely used methods: n  4D-Var, 3D-Var n  Suboptimal and ensemble filters


Thank you for the great visit, and for the opportunity to discuss some data assimilation ideas …


Documents

Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’