![Page 1: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/1.jpg)
Lecture 4. Aspects of
Chemical Data Assimilation
Adrian Sandu Computational Science Laboratory
Virginia Polytechnic Institute and State University
![Page 2: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/2.jpg)
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
Simulation of air pollution, E. Asia, 3 days in March 2001, provides motivation
![Page 3: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/3.jpg)
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.
![Page 4: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/4.jpg)
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.
![Page 5: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/5.jpg)
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.
![Page 6: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/6.jpg)
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
![Page 7: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/7.jpg)
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.
![Page 8: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/8.jpg)
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.
![Page 9: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/9.jpg)
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.
![Page 10: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/10.jpg)
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
![Page 11: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/11.jpg)
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 !#
![Page 12: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/12.jpg)
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.
![Page 13: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/13.jpg)
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
![Page 14: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/14.jpg)
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.
![Page 15: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/15.jpg)
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.
![Page 16: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/16.jpg)
Sparse Jacobians , Hessians, and sparse linear algebra routines are automatically generated by KPP
Lecture given at EISDA 2012. Seoul, Korea, Aug. 2012.
![Page 17: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/17.jpg)
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.
![Page 18: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/18.jpg)
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]
![Page 19: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/19.jpg)
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]
![Page 20: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/20.jpg)
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.
![Page 21: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/21.jpg)
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.
![Page 22: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/22.jpg)
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
![Page 23: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/23.jpg)
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
![Page 24: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/24.jpg)
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
![Page 25: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/25.jpg)
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.
![Page 26: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/26.jpg)
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.
![Page 27: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/27.jpg)
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.
![Page 28: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/28.jpg)
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.
![Page 29: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/29.jpg)
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.
![Page 30: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/30.jpg)
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.
![Page 31: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/31.jpg)
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.
![Page 32: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/32.jpg)
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.
![Page 33: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/33.jpg)
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.
![Page 34: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/34.jpg)
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.
![Page 35: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/35.jpg)
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.
![Page 36: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/36.jpg)
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.
![Page 37: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/37.jpg)
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.
![Page 38: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/38.jpg)
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.
![Page 39: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/39.jpg)
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]
![Page 40: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/40.jpg)
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.
![Page 41: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/41.jpg)
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]
![Page 42: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/42.jpg)
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]
![Page 43: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/43.jpg)
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]
![Page 44: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/44.jpg)
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
![Page 45: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/45.jpg)
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]
![Page 46: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/46.jpg)
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.
![Page 47: Lecture 4. Aspects of Chemical Data Assimilationpeople.cs.vt.edu/~asandu/Public/EISDA2012/Lectures/Ewha_L04_chemistry.pdfuKFC dt d = Γ ∂ ∂ =Γ ∂ ∂ + =Γ = ≥ && ⋅− ’](https://reader034.vdocument.in/reader034/viewer/2022052002/60150904e8c738284e523734/html5/thumbnails/47.jpg)
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.