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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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
Discretize-then-linearize leads to the discrete adjoint model (e.g., of mass balance equations)
( ) ( )
( ) ( )( ) ( )
GROUNDii
DEPi
i
OUTi
ININii
Fii
iiiii
QCVnCK
nCK
xtCxtCtttxCxtC
ECfCKCudtdC
Γ−=∂
∂
Γ=∂
∂
Γ=
≤≤=
++∇⋅∇+∇⋅−=
on
on0
on,,
,,
11
000
ρρ
ρρ
tHOR
tVERT
tCHEM
tVERT
tHORttt
kttt
k
TTRTTN
CNCΔΔΔΔΔ
Δ+
Δ++
=
=
],[
],[1
( ) ( ) ( ) ( ) ( )'*'*'*'*'*],[
'*
11],[
'*
tHOR
tVERT
tCHEM
tVERT
tHORttt
kkttt
k
TTRTTN
NΔΔΔΔΔ
Δ+
++Δ+
=
+=
φλλ
Continuous forward model (PDE)
Discrete forward model (discretized PDE)
Discrete adjoint model (computes derivatives on the numerical solution)
discr
adj
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
Linearize-then-discretize leads to the continuous adjoint model (e.g., of mass balance equations)
( ) ( )
( ) ( )( ) ( )
GROUNDii
DEPi
i
OUTi
ININii
Fii
iiiii
QCVnCK
nCK
xtCxtCtttxCxtC
ECfCKCudtdC
Γ−=∂
∂
Γ=∂
∂
Γ=
≤≤=
++∇⋅∇+∇⋅−=
on
on0
on,,
,,
11
000
ρρ
ρρ
( ) ( )
( ) ( )( )
( )
( ) GROUNDi
DEPi
i
OUTii
INi
oFFi
Fi
iiTi
ii
Vn
K
nKu
xttttxxt
CFKudtd
Γ=∂
∂
Γ=∂
∂+
Γ=
≥≥=
−⋅−⎟⎟⎠
⎞⎜⎜⎝
⎛∇⋅⋅∇−⋅∇−=
on
on0
on0,
,,
)(
λρλ
ρ
ρλρλ
λ
λλ
φλρρλ
ρλλ
( ) ( ) ( ) ( ) ( )'*'*'*'*'*],[
'*
11],[
'*
tHOR
tVERT
tCHEM
tVERT
tHORttt
kkttt
k
TTRTTM
MΔΔΔΔΔ
Δ+
++Δ+
=
+=
φλλ
Continuous forward model (PDE) Adjoint PDE model (exact derivatives)
Continuous adjoint model (discretized adjoint PDE)
adj
discr
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
Continuous and discrete adjoints (of mass balance equations) lead to different computational models
( ) ( )
( ) ( )( ) ( )
GROUNDii
DEPi
i
OUTi
ININii
Fii
iiiii
QCVnCK
nCK
xtCxtCtttxCxtC
ECfCKCudtdC
Γ−=∂
∂
Γ=∂
∂
Γ=
≤≤=
++∇⋅∇+∇⋅−=
on
on0
on,,
,,
11
000
ρρ
ρρ
( ) ( )
( ) ( )( ) ( )
GROUNDii
DEPi
i
OUTi
ININii
Fii
iiiii
QCVnCK
nCK
xtCxtCtttxCxtC
ECfCKCudtdC
Γ−=∂
∂
Γ=∂
∂
Γ=
≤≤=
++∇⋅∇+∇⋅−=
on
on0
on,,
,,
11
000
ρρ
ρρ
( ) ( )
( ) ( )( )
( )
( ) GROUNDi
DEPi
i
OUTii
INi
oFFi
Fi
iiTi
ii
Vn
K
nKu
xttttxxt
CFKudtd
Γ=∂
∂
Γ=∂
∂+
Γ=
≥≥=
−⋅−⎟⎟⎠
⎞⎜⎜⎝
⎛∇⋅⋅∇−⋅∇−=
on
on0
on0,
,,
)(
λρλρ
ρλρλ
λλλ
φλρρλρλλ
( ) ( )
( ) ( )( )
( )
( ) GROUNDi
DEPi
i
OUTii
INi
oFFi
Fi
iiTi
ii
Vn
K
nKu
xttttxxt
CFKudtd
Γ=∂
∂
Γ=∂
∂+
Γ=
≥≥=
−⋅−⎟⎟⎠
⎞⎜⎜⎝
⎛∇⋅⋅∇−⋅∇−=
on
on0
on0,
,,
)(
λρλρ
ρλρλ
λλλ
φλρρλρλλ
Continuous forward model (PDE)
Discrete forward model (discretized PDE)
Adjoint PDE
Adjoint models ≠
adj
discr
adj
discr
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
( ) ( )
( ) ( )( )
( )
( ) GROUNDi
DEPi
i
OUTii
INi
oFFi
Fi
iiTi
ii
Vn
K
nKu
xttttxxt
CFKudtd
Γ=∂
∂
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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
!#↑ℎ !#
The construction of adjoint models is work-intensive and error-prone. Automatic implementation attractive
Optimal analysis state
Chemical kinetics
Aerosols
CTM
Transport Meteorology
Emissions
Observations Data
Assimilation
Targeted Observ.
Improved: • forecasts • science • field experiment design • models • emission estimates
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
discretization change
upwind direction change
The construction of adjoint models is work-intensive and error-prone. Automatic implementation attractive
Optimal analysis state
Chemical kinetics
Aerosols
CTM
Transport Meteorology
Emissions
Observations Data
Assimilation
Targeted Observ.
Improved: • forecasts • science • field experiment design • models • emission estimates
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
Sparse Jacobians , Hessians, and sparse linear algebra routines are automatically generated by KPP
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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]
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
The construction of adjoint models is work-intensive and error-prone. Automatic implementation attractive
Optimal analysis state
Chemical kinetics
Aerosols
CTM
Transport Meteorology
Emissions
Observations Data
Assimilation
Targeted Observ.
Improved: • forecasts • science • field experiment design • models • emission estimates
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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
Chemical D.A. and Data Needs
for Air Quality Forecasting
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
Chemical D.A. and Data Needs
for Air Quality Forecasting
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
The construction of adjoint models is work-intensive and error-prone. Automatic implementation attractive
Optimal analysis state
Chemical kinetics
Aerosols
CTM
Transport Meteorology
Emissions
Observations Data
Assimilation
Targeted Observ.
Improved: • forecasts • science • field experiment design • models • emission estimates
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
The construction of adjoint models is work-intensive and error-prone. Automatic implementation attractive
Optimal analysis state
Chemical kinetics
Aerosols
CTM
Transport Meteorology
Emissions
Observations Data
Assimilation
Targeted Observ.
Improved: • forecasts • science • field experiment design • models • emission estimates
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
Adjoint variables for an ozone sensor located on Cheju Island at the end of the three days period
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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]
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
Ensemble-based chemical data assimilation is an alternative to variational techniques
Optimal analysis state
Chemical kinetics
Aerosols
CTM
Transport Meteorology
Emissions
Observations Ensemble
Data Assimilation
Targeted Observ.
Improved: • forecasts • science • field experiment design • models • emission estimates
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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,
Nov. 30, 2011. Lecture 1: Data assimilation.
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
Nov. 30, 2011. Lecture 1: Data assimilation.
Parallelization important to speed up 4D-Var
[Sandu et.al., 2003-2008]
Chemistry w/ abstract vectors
Transport on accelerators
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
The use of adjoints in large scale simulations: atmospheric chemical transport models
Optimal analysis state
Chemical kinetics
Aerosols
CTM
Transport Meteorology
Emissions
Observations 4D-Var
Data Assimilation
Targeted Observ.
Improved: • forecasts • science • field experiment design • models • emission estimates
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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]
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
Assimilation of ozonesonde observations for July 20, 2004, show importance of vertical information
[Chai et al., 2006]
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
Assimilation of NOAA P3 in-situ and DC-8 lidar observations for July 20, 2004
[Chai et al., 2006]
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
The model results show a considerably improved agreement with observations after data assimilation
q-q modeled O3 vs observations
[Chai et al., 2006]
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
[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]
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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( )
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
[Sandu et. al., 2007]
Hessian singular vectors describe directions of maximum error growth in finite time
vvMAM ⋅∇⋅=⋅⋅⋅ ψγ 2''*
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
[Sandu et. al., 2007]
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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
[Sandu, 2006]
Assimilation of TES ozone column observations from Aug. 2006. IONS-6 ozonesonde data for validation.
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
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)
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
[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
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
Thank you for the great visit, and for the opportunity to discuss some data assimilation ideas …
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.