Inversion of Atmospheric CO 2 Concentrations
293
In terms of computational effi ciency, Gx − z is easy to evaluate, since Gx is obtained
by a single model integration with sources x. Multiplying this vector by X is a simple
matrix-times-vector operation in general, and specifi c cases are often much simpler
if X has a block-diagonal (or even diagonal) structure due to independence of various
subsets of data. The diffi culty comes from multiplying this vector by G T . The direct
approach, used in synthesis inversion, is to calculate the full matrix G by integrating the model with a set of basis functions. The number of such integrations is the
dimensionality of x. For high-resolution inversions, this becomes computationally
infeasible.
The alternative to such “brute-force” is to use what is known as an adjoint model.
This is a model whose operation corresponds to the effect of G T . This adjoint model
is then run with X[Gx − z] as its input. There are software tools (Giering, 2000;
Griewank, 2000) that take the computer code (that formally computes Gx for arbitrary x) and automatically generates “adjoint” code that calculates G T z for arbitrary z. Adjoint techniques apply to linearized calculations as well as the fully-linear
case described here.
These adjoint techniques have been the basis for obtaining high-resolution
inversions, regularized, at least in part, by smoothness constraints rather than relying on fi xing the spatial structure of low-resolution basis functions (Rödenbeck
et al., 2003).
11.2.4 PROCESS INVERSIONS
The inversions described in Section 2.2 obtain estimates of fl uxes, given observations
of concentrations. Much of the interest in such fl ux estimates comes from trying to
identify the processes responsible for the spatial and temporal variability in these
fl uxes. This immediately suggests the desirability of inversions that directly estimate
characteristics of processes. Some aspects of process inversion have been reviewed
(Enting, 2002, Section 12.4), mainly in the context of the likely need for nonlinear
estimation in such studies. This review noted some “precursor” studies that involved
aspects of process inversion including (a) the estimation of hydroxyl (OH) concentrations from studies of halogenated compounds and (b) the use of satellite data for
prescribing the timing of seasonal vegetation fl uxes in a study of the seasonal cycle
of CO 2 (Fung et al., 1987).
In the last few years, there have been several studies that could be classed as
“genuine” process inversions. These have taken the form of estimating parameters in
terrestrial models (Knorr and Heimann, 1995; Kaminski et al., 2002; Rayner et al.,
2005). If one has fl uxes as functions x(p) of parameters, p, then the gradient of the
cost function is given by
∇ Θ = ∇
−
p
p
1
(
)
( ) ]
2
T
x G X[Gx p z
(11.9)
These calculations have been performed using the adjoint of the combined terrestrial +
transport model, that is (∇ p x)G T , to obtain the gradients used in the iterative minimization of the cost function. (In this paper, the superscript T denotes the matrix
© 2010 by Taylor and Francis Group, LLC
293
In terms of computational effi ciency, Gx − z is easy to evaluate, since Gx is obtained
by a single model integration with sources x. Multiplying this vector by X is a simple
matrix-times-vector operation in general, and specifi c cases are often much simpler
if X has a block-diagonal (or even diagonal) structure due to independence of various
subsets of data. The diffi culty comes from multiplying this vector by G T . The direct
approach, used in synthesis inversion, is to calculate the full matrix G by integrating the model with a set of basis functions. The number of such integrations is the
dimensionality of x. For high-resolution inversions, this becomes computationally
infeasible.
The alternative to such “brute-force” is to use what is known as an adjoint model.
This is a model whose operation corresponds to the effect of G T . This adjoint model
is then run with X[Gx − z] as its input. There are software tools (Giering, 2000;
Griewank, 2000) that take the computer code (that formally computes Gx for arbitrary x) and automatically generates “adjoint” code that calculates G T z for arbitrary z. Adjoint techniques apply to linearized calculations as well as the fully-linear
case described here.
These adjoint techniques have been the basis for obtaining high-resolution
inversions, regularized, at least in part, by smoothness constraints rather than relying on fi xing the spatial structure of low-resolution basis functions (Rödenbeck
et al., 2003).
11.2.4 PROCESS INVERSIONS
The inversions described in Section 2.2 obtain estimates of fl uxes, given observations
of concentrations. Much of the interest in such fl ux estimates comes from trying to
identify the processes responsible for the spatial and temporal variability in these
fl uxes. This immediately suggests the desirability of inversions that directly estimate
characteristics of processes. Some aspects of process inversion have been reviewed
(Enting, 2002, Section 12.4), mainly in the context of the likely need for nonlinear
estimation in such studies. This review noted some “precursor” studies that involved
aspects of process inversion including (a) the estimation of hydroxyl (OH) concentrations from studies of halogenated compounds and (b) the use of satellite data for
prescribing the timing of seasonal vegetation fl uxes in a study of the seasonal cycle
of CO 2 (Fung et al., 1987).
In the last few years, there have been several studies that could be classed as
“genuine” process inversions. These have taken the form of estimating parameters in
terrestrial models (Knorr and Heimann, 1995; Kaminski et al., 2002; Rayner et al.,
2005). If one has fl uxes as functions x(p) of parameters, p, then the gradient of the
cost function is given by
∇ Θ = ∇
−
p
p
1
(
)
( ) ]
2
T
x G X[Gx p z
(11.9)
These calculations have been performed using the adjoint of the combined terrestrial +
transport model, that is (∇ p x)G T , to obtain the gradients used in the iterative minimization of the cost function. (In this paper, the superscript T denotes the matrix
© 2010 by Taylor and Francis Group, LLC