294
Air Pollution and Turbulence: Modeling and Applications
transpose (or adjoint in the case of operators) and the notation does not distinguish
between row and column vectors, relying on the context to indicate the appropriate
case.)
However, given the linearity of the transport, if only a small number of parameters, p α , are involved, an alternative way of calculating gradients would be to obtain
the sensitivities of the terrestrial model (i.e., from
α
∂ ∂
/ p
x
, the tangent linear model
for the terrestrial component) and use these sensitivities as basis functions in an
integration of the transport model. Equation 11.9 becomes
α
α
∂Θ
∂
=
−
∂
∂
1
[ ( ) ]
2 p
p
T
x G X Gx p z
(11.10)
which can be evaluated by calculating G
α
∂ ∂
/ p
x
, that is, running the forward transport model with surface forcing taken as the tangent linear sensitivities of the fl ux
model and then taking the scalar product of this vector with the vector X[Gx(p) − z].
While Equation 11.10 is formally less effi cient than Equation 11.9, since the evaluation of Equation 11.10 needs to be repeated for each of the parameters p α , it avoids
the need to have adjoints of the transport model, the terrestrial model, or the combined terrestrial + transport model.
This simplifi cation may be useful in some cases, but extensions to more complex
problems may require the full adjoint:
Consideration of spatially varying uncertainties in the functional rela•
tionships in the terrestrial model means that the sensitivities will require
full spatial resolution rather than being represented by a small set of basis
functions.
Cases where the terrestrial model is “forced” by functions obtained by the
•
assimilation of proxy data (e.g., NPP derived from the model with satellite data as a constraint) may again lead to a gradient relation that requires
the full adjoint (∇ p x)G T in order to achieve acceptable computational
effi ciency.
In looking at future directions, just as fl ux estimates are a step toward understanding
processes, process estimates are a step toward understanding the role of these processes in the coupled earth system. In this context, inversion studies of CO 2 are being
subsumed into a more general framework termed “model-data fusion” (Raupach
et al., 2005). CO 2 inversions have recently been reviewed from this perspective by
Wang et al. (2009).
11.2.5 OTHER TRACERS
While much of the development of trace gas inversions has been in the context of
CO 2 , other compounds have been analyzed in this way. A number of studies are
reviewed in Enting (2002). The discussion in the present section is mainly focussed
on methodology, identifying aspects where the techniques differ from the practice of CO 2 inversions. Such generalizations may be useful in future CO 2 studies,
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
transpose (or adjoint in the case of operators) and the notation does not distinguish
between row and column vectors, relying on the context to indicate the appropriate
case.)
However, given the linearity of the transport, if only a small number of parameters, p α , are involved, an alternative way of calculating gradients would be to obtain
the sensitivities of the terrestrial model (i.e., from
α
∂ ∂
/ p
x
, the tangent linear model
for the terrestrial component) and use these sensitivities as basis functions in an
integration of the transport model. Equation 11.9 becomes
α
α
∂Θ
∂
=
−
∂
∂
1
[ ( ) ]
2 p
p
T
x G X Gx p z
(11.10)
which can be evaluated by calculating G
α
∂ ∂
/ p
x
, that is, running the forward transport model with surface forcing taken as the tangent linear sensitivities of the fl ux
model and then taking the scalar product of this vector with the vector X[Gx(p) − z].
While Equation 11.10 is formally less effi cient than Equation 11.9, since the evaluation of Equation 11.10 needs to be repeated for each of the parameters p α , it avoids
the need to have adjoints of the transport model, the terrestrial model, or the combined terrestrial + transport model.
This simplifi cation may be useful in some cases, but extensions to more complex
problems may require the full adjoint:
Consideration of spatially varying uncertainties in the functional rela•
tionships in the terrestrial model means that the sensitivities will require
full spatial resolution rather than being represented by a small set of basis
functions.
Cases where the terrestrial model is “forced” by functions obtained by the
•
assimilation of proxy data (e.g., NPP derived from the model with satellite data as a constraint) may again lead to a gradient relation that requires
the full adjoint (∇ p x)G T in order to achieve acceptable computational
effi ciency.
In looking at future directions, just as fl ux estimates are a step toward understanding
processes, process estimates are a step toward understanding the role of these processes in the coupled earth system. In this context, inversion studies of CO 2 are being
subsumed into a more general framework termed “model-data fusion” (Raupach
et al., 2005). CO 2 inversions have recently been reviewed from this perspective by
Wang et al. (2009).
11.2.5 OTHER TRACERS
While much of the development of trace gas inversions has been in the context of
CO 2 , other compounds have been analyzed in this way. A number of studies are
reviewed in Enting (2002). The discussion in the present section is mainly focussed
on methodology, identifying aspects where the techniques differ from the practice of CO 2 inversions. Such generalizations may be useful in future CO 2 studies,
© 2010 by Taylor and Francis Group, LLC