inverse modeling of atmospheric composition data daniel j. jacob see my web site under...
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INVERSE MODELING OF ATMOSPHERIC COMPOSITION DATAINVERSE MODELING OF ATMOSPHERIC COMPOSITION DATA
Daniel J. Jacob
• See my web site under “educational materials” for lectures on • inverse modeling• atmospheric transport and chemistry models
• Notation and terminology in this lecture follows that of Rodgers (2000)
THE INVERSE MODELING PROBLEMTHE INVERSE MODELING PROBLEMOptimize values of an ensemble of variables (state vector x) using observations:
THREE MAIN APPLICATIONS FOR ATMOSPHERIC COMPOSITION:
1. Retrieve atmospheric concentrations (x) from observed atmospheric
radiances (y) using a radiative transfer model as forward model
2. Invert sources (x) from observed atmospheric concentrations (y) using a CTM as forward model
3. Construct a continuous field of concentrations (x) by assimilation of sparse
observations (y) using a forecast model (initial-value CTM) as forward model
a priori estimate
xa + a
observation vector yforward model
y = F(x) +
“MAP solution”“optimal estimate”
“retrieval”
ˆ ˆx + εBayes’theorem
BAYES’ THEOREM: FOUNDATION FOR INVERSE MODELSBAYES’ THEOREM: FOUNDATION FOR INVERSE MODELS
( ) ( )( )
( )
P PP
P
y | x xx | y
y
P(x) = probability distribution function (pdf) of xP(x,y) = pdf of (x,y)P(y|x) = pdf of y given x
a priori pdfobservation pdf
normalizing factor (unimportant)
a posteriori pdf
Maximum a posteriori (MAP) solution for x given y is defined by
( | )P x x y 0
max( ( | ))P x y
solve for
P(x,y)dxdy
( ) ( )
( ) ( )
P d P d
P d P d
x x y | x y
y y x | y x
Bayes’theorem
SIMPLE LINEAR INVERSE PROBLEM FOR A SCALARSIMPLE LINEAR INVERSE PROBLEM FOR A SCALARuse single measurement used to optimize a single sourceuse single measurement used to optimize a single source
a priori bottom-up estimate
xa a
Monitoring site measures
concentration yForward model gives y = kx
“Observational error”
• instrument• fwd model
y = kx
22
2 2
( )( )ln ( | ) ln ( | ) ln ( ) a
a
x xy kxP x y P y x P x
Max of P(x|y) is given by minimum of cost function 22
2 2
( )( )( ) a
a
x xy kxJ x
Solution: ˆ ( )a ax x g y kx where g is a gain factor
2
2 2 2a
a
kg
k
22 2 1( ( / ) )a k
Alternate expression of solution: (1 ) ay kx x ax a x g
where a = gk is an averaging kernel
solve for / 0dJ dx
Assume random Gaussian errors, let x be the true value. Bayes’ theorem:
Variance of solution:
GENERALIZATION:GENERALIZATION:CONSTRAINING CONSTRAINING nn SOURCES WITH SOURCES WITH mm OBSERVATIONS OBSERVATIONS
1
n
j ij ii
y k x
Linear forward model:
A cost function defined as , 1
1 2 21 1, ,
( )( )
( ,... )
n
j ij in mi a i i
ni ja i j
y k xx x
J x x
is generally not adequate because it does not account for correlation between sources or between observations. Need vector-matrix formalism:
1 1( ,... ) ( ,... ) T Tn mx x y y x y y Kx ε
Jacobian matrix
JACOBIAN MATRIX FOR FORWARD MODELJACOBIAN MATRIX FOR FORWARD MODEL
Consider a non-linear forward model y = F(x)
Use of vector-matrix formalism requires linearization of forward model
and linearize it about xa:
( ) ( ) (+ higher-order terms) a ay F x K x x
x
yK = F =
xis the Jacobian of F evaluated at xa
iij
j
yk
x
with elements
If forward model is non-linear, K must be recalculated iteratively for successive solutions
1 1 1
1
/ /
/ /
n
m m n
y x y x
y x y x
K
KT is the adjoint of the forward model (to be discussed later)
Construct Jacobian numerically column by column: perturb xa by xi,
run forward model to get corresponding y
GAUSSIAN PDFs FOR VECTORSGAUSSIAN PDFs FOR VECTORSA priori pdf for x:
Scalar x Vector
2
2
( )1( ) exp[ ]
22a
aa
x xP x
1 ,1 1 ,1 ,
1 ,1 , ,
var( ) cov( , )
cov( , ) var( )
a a n a n
a n a n n a n
x x x x x x
x x x x x x
aS
-112ln ( ) ( ) ( )TP c a a ax x x S x x
1/ 2/ 2
1 1( ) exp[
2(2 )T
na
P
-1a a ax (x - x ) S (x - x )]
S
1( ,... )Tnx xx
where Sa is the a priori error covariance matrix describing error statistics on (x-xa)
In log space:
OBSERVATIONAL ERROR COVARIANCE MATRIXOBSERVATIONAL ERROR COVARIANCE MATRIX
( ) i my F x ε + ε
observationtrue value
instrument error
fwd model error observational error
i mε = ε + ε
Observational error covariance matrix
1 1
1
var( ) cov( , )
cov( , ) var( )
n
n n
S
is the sum of the instrument and fwd model error covariance matrices:
i mε ε εS = S + S
How well can the observing system constrain the true value of x ?
122ln ( ) ( ) ( )TP c
y | x y Kx S y KxCorresponding pdf, in log space:
MAXIMUM A POSTERIORI (MAP) SOLUTIONMAXIMUM A POSTERIORI (MAP) SOLUTION-1
12ln ( ) ( ) ( )TP c a a ax x x S x x1
22ln ( ) ( ) ( )TP c y | x y Kx S y Kx
-1 132ln ( ) ( ) ( ) ( ) ( )T TP c
a a ax | y x x S x x y Kx S y Kx
Bayes’ theorem:
MAP solution: ( | ) P x x y 0 miminize cost function J:
-1 1( ) ( ) ( ) ( ) ( )T TJ a a ax x x S x x y Kx S y Kx
Solve for -1 1( ) 2 ( ) 2 ( )TJ x a ax S x x K S Kx - y 0
ˆ ( ) a ax x G y Kx
Analytical solution:
1( )T T
a aG S K KS K S
1ˆ ( )T 1 1
aS K S K S
with gain matrix
bottom-up constraint top-down constraint
PARALLEL BETWEEN VECTOR-MATRIX AND SCALAR SOLUTIONS:PARALLEL BETWEEN VECTOR-MATRIX AND SCALAR SOLUTIONS:
Scalar problem Vector-matrix problem
2
2 2 2
2 22 1
ˆ ( )
( / )
ˆ (1 )
a a
a
a
a
a
x x g y kx
kg
k
k
x ax a x g
a gk
MAP solution:
1
1
ˆ ( )
( )
ˆ ( )
ˆ ( )
T T
T
a a
a a
1 1a
n a
x x G y Kx
G S K KS K S
S K S K S
x Ax I A x Gε
A GK
Gain factor:
A posteriori error:
Averaging kernel:
Jacobian matrix
ˆ G x/ y
K = y/ x sensitivity of observations to true state
Gain matrix sensitivity of retrieval to observations
Averaging kernel matrix ˆ A x/ x sensitivity of retrieval to true state
A LITTLE MORE ON THE AVERAGING KERNEL MATRIXA LITTLE MORE ON THE AVERAGING KERNEL MATRIX
A describes the sensitivity of the retrieval to the true state
1 1 1
1
ˆ ˆ/ /ˆ
ˆ ˆ/ /
n
n n n
x x x x
x x x x
xA
x
ˆ ( ) n ax Ax I A x Gεand hence the smoothing of the solution:
smoothing error retrieval error
MAP retrieval gives A as part of the retrieval:
1( )T T
a aA = GK = S K KS K S K
Other retrieval methods (e.g., neural network, adjoint method) do not provide A
# pieces of info in a retrieval = degrees of freedom for signal (DOFS) = trace(A)
APPLICATION TO SATELLITE RETRIEVALSAPPLICATION TO SATELLITE RETRIEVALS
Here y is the vector of wavelength-dependent radiances (radiance spectrum);
x is the state vector of concentrations;
forward model y = F(x) is the radiative transfer model
Illustrative MOPITT averaging kernel matrix for CO retrieval
MOPITT retrieves concentrations at 7 pressure levels; lines are the corresponding columns of the averaging kernel matrix
trace(A) = 1.5 in this case;1.5 pieces of information
INVERSE ANALYSIS OF MOPITT AND TRACE-P (AIRCRAFT) DATA INVERSE ANALYSIS OF MOPITT AND TRACE-P (AIRCRAFT) DATA TO CONSTRAIN ASIAN SOURCES OF COTO CONSTRAIN ASIAN SOURCES OF CO
TRACE-P CO DATA(G.W. Sachse)Bottom-up
emissions(customized for TRACE-P)
Fossil and biofuel Daily biomass burning(satellite fire counts)
GEOS-Chem CTM
MOPITT CO
Inverse analysis
validation
chemicalforecasts
top-downconstraints
OPTIMIZATION OF SOURCES
Streets et al. [2003] Heald et al. [2003a]
COMPARE TRACE-P OBSERVATIONS COMPARE TRACE-P OBSERVATIONS WITH CTM RESULTS USING WITH CTM RESULTS USING A PRIORI A PRIORI SOURCESSOURCES
Model is low north of 30oN: suggests Chinese source is lowModel is high in free trop. south of 30oN: suggests biomass burning source is high
• Assume that Relative Residual Error (RRE) after bias is removed describes the observational error variance (20-30%)
• Assume that the difference between successive GEOS-Chem CO forecasts during TRACE-P (to+48h and to + 24 h) describes the covariant error structure (“NMC method”)
Palmer et al. [2003], Jones et al. [2003]
CHARACTERIZING THE OBSERVATIONAL ERROR CHARACTERIZING THE OBSERVATIONAL ERROR COVARIANCE MATRIX FOR MOPITT CO COLUMNSCOVARIANCE MATRIX FOR MOPITT CO COLUMNS
Diagonal elements (error variances) obtained by residual relative error method
Add covariant structure from NMC method
Heald et al. [2004]
COMPARATIVE INVERSE ANALYSIS OF ASIAN CO SOURCES COMPARATIVE INVERSE ANALYSIS OF ASIAN CO SOURCES USING DAILY MOPITT AND TRACE-P DATA USING DAILY MOPITT AND TRACE-P DATA
• MOPITT has higher information content than TRACE-P because it observes source regions and Indian outflow
• Don’t trust a posteriori error covariance matrix; ensemble modeling indicates 10-40% error on a posteriori sources
Heald et al. [2004]
CO observations from Spring 2001, GEOS-Chem CTM as forward model
TRACE-P Aircraft CO MOPITT CO Columns
4 degreesof freedom
10 degreesof freedom
(from validation)
“Ensemble modeling”: repeat inversion with different values of forward model and covariance parameters to span uncertainty range
Analytical solution to inverse problemAnalytical solution to inverse problem
1ˆ ( ) ( )T T
a a a ax x S K KS K S y Kx
requires (iterative) numerical construction of the Jacobian matrix K and matrix operations of dimension (mxn); this limits the size of n, i.e., the number of variables that you can optimize
Address this limitation with Kalman filter (for time-dependent x) or with adjoint method
-1 1( ) 2 ( ) 2 ( )TJ x a ax S x x K S Kx - y 0
BASIC KALMAN FILTERBASIC KALMAN FILTERto optimize time-dependent state vectorto optimize time-dependent state vector
a priori xa,0 ± Sa,0
to
time observations state vector
y0ˆˆ 0 0x S
Evolution model M for [t0, t1]:
ˆ ˆ
ˆ T
a,1 0 M
a,1 0 M
x = Mx ± S
S = MS M S
t1 y11 1
ˆˆ x S
Apply evolution model for [t1, t2]…etc.
ADJOINT INVERSION (4-D VAR)ADJOINT INVERSION (4-D VAR)
°
°
°
°
a
2
1
3
x2
x1
x3
xa
Minimum of cost function J
-1 1( ) 2 ( ) 2 ( ( ) )TJ x a ax S x x K S F x - y 0
Solve
numerically rather than analytically
1. Starting from a priori xa, calculate ( )Jx ax
2. Using an optimization algorithm (BFGS),
get next guess x1
3. Calculate ( )Jx 1x , get next guess x2
4. Iterate until convergence
NUMERICAL CALCULATION OF COST FUNCTION GRADIENTNUMERICAL CALCULATION OF COST FUNCTION GRADIENT
-1 1( ) 2 ( ) 2 ( ( ) ))TJ x a ax S x x K S F x - y
Adjoint model is applied to error-weighted difference between model and obs
…but we want to avoid explicit construction of K
( ) ( ) ( -1) (1) (0)
(0) ( -1) ( -2) (0) (0)
...i i i
i i
y y y y y
Kx y y y x
( ) ( -1) (1) (0) (0) (1) ( -1) ( )
( -1) ( -2) (0) (0) (0) (0) ( -2) ( -1)
... ...
T T T T T
i i n nT
i i n n
y y y y y y y yK
y y y x x y y y
and since (AB)T = BTAT,
Apply transpose of tangent linear model to the adjoint forcings; for time interval [t0, tn], start from observations at tn and work backward in time until t0, picking up new observations (adjoint forcings) along the way.
Construct tangent linear model ( ) ( 1)/i i y ydescribing evolution of concentration field over time interval [ti-1, ti]
Sensitivity of y(i) to x(0) at time t0 can then be written
of forward model
adjoint “adjoint forcing”
APPLICATION OF GEOS-Chem ADJOINT TO CONSTRAIN ASIAN SOURCES APPLICATION OF GEOS-Chem ADJOINT TO CONSTRAIN ASIAN SOURCES WITH HIGH RESOLUTION USING MOPITT DATAWITH HIGH RESOLUTION USING MOPITT DATA
MOPITT daily CO columns (Mar-Apr 2001)
Correction factors to a priori emissions with 2ox2.5o resolution
A priori knowledge of emissions
Kopacz et al. [2007]
GEOS-Chem adjointwith 2ox2.5o resolution
Fossil fuel
Biomass burning
20000
22000
24000
26000
28000
30000
0 5 10 15 20iteration number (n )
cost
fu
nct
ion
J(xn)
number of observations
Iterative approachto minimum of cost function
COMPARE ADJOINT AND ANALYTICAL INVERSIONSCOMPARE ADJOINT AND ANALYTICAL INVERSIONSOF SAME MOPITT CO DATAOF SAME MOPITT CO DATA
Adjoint solution reveals aggregation errors, checkerboard pattern in analytical solution
[Heald et al., 2004]
Kopacz et al. [2007]