INVERSE MODELING OF ATMOSPHERIC COMPOSITION DATA

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INVERSE 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)

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THE INVERSE MODELING PROBLEM
Optimize values of an ensemble of variables
(state vector x) using observations
a priori estimate xa ea
MAP solution optimal estimate retrieval
Bayes theorem
forward model y F(x) e
observation vector y

• THREE MAIN APPLICATIONS FOR ATMOSPHERIC
  COMPOSITION
• Retrieve atmospheric concentrations (x) from
  observed atmospheric radiances (y) using a
  radiative transfer model as forward model
• Invert sources (x) from observed atmospheric
  concentrations (y) using a CTM as forward model
• Construct a continuous field of concentrations
  (x) by assimilation of sparse observations (y)
  using a forecast model (initial-value CTM) as
  forward model

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BAYES THEOREM FOUNDATION FOR INVERSE MODELS
P(x) probability distribution function (pdf) of
x P(x,y) pdf of (x,y) P(yx) pdf of y given x
P(x,y)dxdy
observation pdf
a priori pdf
a posteriori pdf
Bayes theorem
a
normalizing factor (unimportant)
Maximum a posteriori (MAP) solution for x given y
is defined by
a
solve for
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SIMPLE LINEAR INVERSE PROBLEM FOR A SCALARuse
single measurement used to optimize a single
source
Monitoring site measures concentration y
Forward model gives y kx

• a priori bottom-up estimate
• xa sa

• instrument
• fwd model

Observational error se
y kx se
Assume random Gaussian errors, let x be the true
value. Bayes theorem
Max of P(xy) is given by minimum of cost
function
g solve for
where g is a gain factor
Solution
Variance of solution
Alternate expression of solution
where a gk is an averaging kernel
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GENERALIZATIONCONSTRAINING n SOURCES WITH m
OBSERVATIONS
Linear forward model
A cost function defined as
is generally not adequate because it does not
account for correlation between sources or
between observations. Need vector-matrix
formalism
Jacobian matrix
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JACOBIAN MATRIX FOR FORWARD MODEL
Use of vector-matrix formalism requires
linearization of forward model
Consider a non-linear forward model y F(x)
and linearize it about xa
is the Jacobian of F evaluated at xa
with elements
Construct Jacobian numerically column by column
perturb xa by Dxi, run forward model to get
corresponding Dy
If forward model is non-linear, K must be
recalculated iteratively for successive solutions
KT is the adjoint of the forward model (to be
discussed later)
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GAUSSIAN PDFs FOR VECTORS
A priori pdf for x
Scalar x
Vector
where Sa is the a priori error covariance matrix
describing error statistics on (x-xa)
In log space
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OBSERVATIONAL ERROR COVARIANCE MATRIX
How well can the observing system constrain the
true value of x ?
observation
fwd model error
observational error
true value
instrument error
Observational error covariance matrix
is the sum of the instrument and fwd model error
covariance matrices
Corresponding pdf, in log space
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MAXIMUM A POSTERIORI (MAP) SOLUTION
Bayes theorem
bottom-up constraint top-down
constraint
miminize cost function J
MAP solution
Solve for
Analytical solution
with gain matrix
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PARALLEL BETWEEN VECTOR-MATRIX AND SCALAR
SOLUTIONS
Scalar problem
Vector-matrix problem
MAP solution

Gain factor
A posteriori error
Averaging kernel
Jacobian matrix
sensitivity of observations to true state
sensitivity of retrieval to observations
Gain matrix
sensitivity of retrieval to true state
Averaging kernel matrix
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A LITTLE MORE ON THE AVERAGING KERNEL MATRIX
A describes the sensitivity of the retrieval to
the true state
and hence the smoothing of the solution
smoothing error retrieval error
MAP retrieval gives A as part of the retrieval
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)
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APPLICATION 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
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INVERSE ANALYSIS OF MOPITT AND TRACE-P (AIRCRAFT)
DATA TO CONSTRAIN ASIAN SOURCES OF CO
Daily biomass burning (satellite fire counts)
Fossil and biofuel
TRACE-P CO DATA (G.W. Sachse)
Heald et al. 2003a
Streets et al. 2003
Bottom-up emissions (customized for TRACE-P)
chemical forecasts
GEOS-Chem CTM
top-down constraints
validation
Inverse analysis
OPTIMIZATION OF SOURCES
MOPITT CO
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COMPARE TRACE-P OBSERVATIONS WITH CTM RESULTS
USING A PRIORI SOURCES
Model is low north of 30oN suggests Chinese
source is low Model 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 (to48h and
  to 24 h) describes the covariant error
  structure (NMC method)

Palmer et al. 2003, Jones et al. 2003
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CHARACTERIZING THE OBSERVATIONAL ERROR COVARIANCE
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
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COMPARATIVE INVERSE ANALYSIS OF ASIAN CO SOURCES
USING DAILY MOPITT AND TRACE-P DATA
CO observations from Spring 2001, GEOS-Chem CTM
as forward model
TRACE-P Aircraft CO
MOPITT CO Columns
(from validation)
4 degrees of freedom
10 degrees of freedom
Heald et al. 2004

• MOPITT has higher information content than
  TRACE-P because it observes source regions and
  Indian outflow
• Dont trust a posteriori error covariance matrix
  ensemble modeling indicates 10-40 error on a
  posteriori sources

Ensemble modeling repeat inversion with
different values of forward model and covariance
parameters to span uncertainty range
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Analytical solution to inverse problem
a
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
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BASIC KALMAN FILTERto optimize time-dependent
state vector
time observations
state vector
a priori xa,0 Sa,0
to
y0
Evolution model M for t0, t1
t1
y1
Apply evolution model for t1, t2
etc.
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ADJOINT INVERSION (4-D VAR)
?a
xa

?1
Solve
?2

x1

x2
numerically rather than analytically
1. Starting from a priori xa, calculate
2. Using an optimization algorithm (BFGS),
get next guess x1 3. Calculate
?3
x3

, get next guess x2
4. Iterate until convergence
Minimum of cost function J
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NUMERICAL CALCULATION OF COST FUNCTION GRADIENT
adjoint forcing
adjoint
Adjoint model is applied to error-weighted
difference between model and obs but we want to
avoid explicit construction of K
Construct tangent linear model
of forward model
describing evolution of concentration field over
time interval ti-1, ti
Sensitivity of y(i) to x(0) at time t0 can then
be written
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.
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APPLICATION OF GEOS-Chem ADJOINT TO CONSTRAIN
ASIAN SOURCES WITH 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
Fossil fuel
GEOS-Chem adjoint with 2ox2.5o resolution
Biomass burning
Kopacz et al. 2007
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COMPARE ADJOINT AND ANALYTICAL INVERSIONSOF SAME
MOPITT CO DATA
Adjoint solution reveals aggregation errors,
checkerboard pattern in analytical solution
Kopacz et al. 2007