Inverse Model and Error Analysis Routines
Brian Connor
21 November 2003
Subroutine perform_retrieval
Iteratively retrieves the state vector, from which XCO2 is calculated. For each iteration,
computes radiances and Jacobian, using do_forwardmodel and compute_pd.
Also computes retrieved radiances, the spectral goodness-of-fit, error estimates for the
state vector and XCO2, and averaging kernels. Current version uses optimal estimation,
which best reproduces the measured radiances while maintaining consistency with an a
priori estimate of the state vector.
The result of each retrieval will be described by a parameter “outcome”, which is an
integer with values 0-4, corresponding to the following:
0. Not yet determined
1. Convergence is reached and there is a good fit of measured and calculated spectra
2. Convergence is reached but the spectral fit is unacceptable
3. No convergence after a maximum number of iterations, max_iter
4. Continuing divergence for max_div iterations
For development and test purposes, all products will be calculated for all retrievals
attempted. However in production we may wish not to calculate all products for some
outcomes. So tests on the outcome parameter will control what is done.
One possibility: vectors of flags can be input parameters, with the index equal to the
outcome parameter. Two vectors will be sufficient, one to determine if the retrieved
radiances and the spectral goodness-of-fit should be calculated, and a corresponding one
for error estimates for the state vector and XCO2 and the averaging kernels. E.g. logical
array final_rad(0:4) could be initialized with values true, true, true, false, false. Then
radiances would be calculated for outcome = 0, 1 or 2, not for outcome = 3 or 4.
Meanwhile do_error(4) could be initialized true, true, false, false.
Code Outline:
Subroutine perform_retrieval
Do while (iter_flag==true)
iter = iter + 1
call do_forwardmodel
call compute_pd[optional]
call do_inversion
call test_convergence
do while(div_flag .eq. true .and. n_div .le. max_div)
call do_inversion
call test_convergence
enddo
if (final_rad(outcome)) call update_statestructure
EndDo
If (final_rad(outcome))
call do_forwardmodel
call test_goodness
if(do_error(outcome))
call compute_pd
call error_analysis
call compute_cwvmr
endif
endif
End
Subroutine do_inversion
Solves the equation for the state vector update dxi+1
( (1 + )Sa-1 + KiT Sε-1 Ki ) dxi+1 = [KiT Sε-1 (y - f(xi)) + Sa-1(xi - xa)]
also computes
di2 = dxi+1T Ŝ-1 dxi+1
using the relation
Ŝ-1 dxi+1  [KiT Sε-1 (y - f(xi)) + Sa-1(xi - xa)]
di2 is effectively the square of the state vector update in units of the solution variance,
and is used by test_convergence.
Subroutine test_convergence
Tests for convergence, divergence, and maximum iterations.
First, test for divergence: is di2 > di-12 ? If so, check for max_div iterations through the
divergence loop. If max_div, set outcome = 4 and exit. If not, increase  and exit with
flags set to not update state vector and to loop to do_inversion.
Then test for convergence. If di2 << n (the number of state vector elements),
convergence is reached. Set flags and output values to exit, updating state vector and
computing final radiances and spectral goodness-of-fit.
If convergence is not reached, has max_iter been reached? If so, set outcome = 3 and exit.
If not, exit with flags set to update state vector and iterate.
Subroutine test_goodness
Computes chi-squared, i.e. the ratio of the mean square of measured minus calculated
spectra to the mean square of the spectral errors, Χ2i, for each spectral band i.
Χ2i = mean(yi - fi( x̂ ))2/mean(2i)
If Χ2i < max_chi2  i, then the fit is acceptable, and set good = true.
Subroutine error_analysis
Given the Jacobian from the retrieved state vector, and the measurement and a priori
covariance matrices, this routine computes the retrieval uncertainties and averaging
kernels. From these it computes the quantities ACO2, ac, ŜCO2, Ŝii, and ~s 1, defined below,
which will be archived.
First computes the retrieval covariance matrix, Ŝ, and the averaging kernel matrix A. Ŝ is
given by
Ŝ = (KT S-1 K + Sa-1)-1
The averaging kernel matrix A is given by
A = Ŝ KT S-1 K
ACO2 and ŜCO2 are simply the square sub-matrices of A and Ŝ which correspond to the
CO2 profile. Ŝii is the diagonal of Ŝ.
~s 1 gives the covariance of XCO2 with the non-CO2 elements of the state vector. ac is the
column averaging kernel, giving the sensitivity of XCO2 to the state vector. They are
defines as follows.
The CO2 dry air mole fraction, XCO2, is obtained by averaging the CO2 profile, weighted
by the pressure weighting function, h, such that XCO2 = hT x̂ . The formal error variance
in the retrieved XCO2 is therefore given by σ2XCO2 = hTŜ h. Further, the sensitivity of XCO2
to the state vector is given by the column averaging kernel
ac = XCO2/x = hT A
An extension of this procedure can be used to calculate the covariance of XCO2 with the
non-CO2 elements of the state vector. Assume the CO2 profile occupies the first p
elements of the state vector, and define a matrix H with dimension n  (n  p  1) such
that
Hij =

hi , j  1
1, i  j  ( p  1), j  1
0, else
 XCO 2 


 xp  1 
 xp  2 
 and the covariance of H x̂ is given by HTŜ H. Denote the first row of
Thus H x̂ = 







 xn 


~
T
H Ŝ H by s 1 , where
~s 1 = ( σ2XCO2 ρXCO2,jσXCO2σj …) for j = p+1, n
~s 1 gives the covariance of XCO2 with the n – p non-CO2 elements of the state vector.
Subroutine compute_xco2
Calculates XCO2 and its uncertainty σXCO2. The CO2 dry air mole fraction, XCO2, is
obtained by averaging the CO2 profile, weighted by the pressure weighting function, h,
such that XCO2 = hT x̂ . The formal error variance in the retrieved XCO2 is therefore given
by σ2XCO2 = hTŜ h.