296
Air Pollution and Turbulence: Modeling and Applications
performed at low resolution with the atmosphere divided into a small number of
boxes. The estimation technique was based on the Kalman fi lter (Gelb, 1974). This
approach was later extended to studies with full 3-D atmospheric transport models
(Hartley and Prinn, 1993; Haas-Laursen et al., 1996; Huang, 2000). As discussed in
Enting (2002), when using the Kalman fi lter formalism it is important to distinguish
the case of estimating a fi xed source (with the estimate evolving in time as additional
data are obtained) from estimating a changing source. These cases will differ in the
model that is used in the Kalman fi lter, the former corresponding to a fi xed state (so
that the estimation takes the form of recursive regression) while the latter requires a
prior stochastic representation of the evolving state.
11.3 STATISTICAL CHARACTERISTICS
11.3.1 ILL-CONDITIONING
Enting (2002) defi ned an inverse problem as one in which the direction of mathematical inference was opposite to that of real-world causality for the system in question.
Since many real-world processes are dissipative, information is lost or severely attenuated, particularly on small-scales. Consequently, the associated inverse problems,
in attempting to reconstruct causes from attenuated signals, are subject to severe
amplifi cation of both model error and data error. This characteristic is known as
ill-conditioning. Indeed the standard usage is to take such ill-conditioning as the
defi ning characteristic of inverse problems (i.e., the mathematical study of inverse
problems is effectively defi ned as the mathematical study of these computationally
challenging inverse problems.)
The ill-conditioning in trace gas inversions has been recognized from the very
fi rst of such calculations. Bolin and Keeling (1963) noted the problem and stated that
“no details of the sources and sinks are reliable.” Similarly, in reviewing the work
of Bolin and Keeling, Junge and Czeplak (1968) concluded that “It seems hardly
likely that detailed information could be obtained on the latitudinal dependence
of K and the CO 2 source function from atmospheric CO 2 observations, even if the
number and quality of the data were very considerably increased.” (K was a “diffusion coeffi cient” characterizing atmospheric transport). Less pessimistically, a 1980
WMO report (Pearman, 1980) estimated the observational requirements for various
objectives as
About three stations: Determine global inventories and trends
•
Additional 5–10 stations: Determine meridional transport of CO
•
2
Over 100 stations: Determine air-surface exchange within regions with
•
signifi cant anthropogenic infl uence
The diffi culty of the inversion can be quantifi ed, in part, by determining how
rapidly the error amplifi cation grows as the resolution increases. This has been
analyzed for the dependence on latitudinal wave-number n. If the forward problem
attenuates fl uxes of wave number n with an n −α decay, then the inverse problem
involves an error amplifi cation that grows as n α In their 2D numerical modeling,
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
performed at low resolution with the atmosphere divided into a small number of
boxes. The estimation technique was based on the Kalman fi lter (Gelb, 1974). This
approach was later extended to studies with full 3-D atmospheric transport models
(Hartley and Prinn, 1993; Haas-Laursen et al., 1996; Huang, 2000). As discussed in
Enting (2002), when using the Kalman fi lter formalism it is important to distinguish
the case of estimating a fi xed source (with the estimate evolving in time as additional
data are obtained) from estimating a changing source. These cases will differ in the
model that is used in the Kalman fi lter, the former corresponding to a fi xed state (so
that the estimation takes the form of recursive regression) while the latter requires a
prior stochastic representation of the evolving state.
11.3 STATISTICAL CHARACTERISTICS
11.3.1 ILL-CONDITIONING
Enting (2002) defi ned an inverse problem as one in which the direction of mathematical inference was opposite to that of real-world causality for the system in question.
Since many real-world processes are dissipative, information is lost or severely attenuated, particularly on small-scales. Consequently, the associated inverse problems,
in attempting to reconstruct causes from attenuated signals, are subject to severe
amplifi cation of both model error and data error. This characteristic is known as
ill-conditioning. Indeed the standard usage is to take such ill-conditioning as the
defi ning characteristic of inverse problems (i.e., the mathematical study of inverse
problems is effectively defi ned as the mathematical study of these computationally
challenging inverse problems.)
The ill-conditioning in trace gas inversions has been recognized from the very
fi rst of such calculations. Bolin and Keeling (1963) noted the problem and stated that
“no details of the sources and sinks are reliable.” Similarly, in reviewing the work
of Bolin and Keeling, Junge and Czeplak (1968) concluded that “It seems hardly
likely that detailed information could be obtained on the latitudinal dependence
of K and the CO 2 source function from atmospheric CO 2 observations, even if the
number and quality of the data were very considerably increased.” (K was a “diffusion coeffi cient” characterizing atmospheric transport). Less pessimistically, a 1980
WMO report (Pearman, 1980) estimated the observational requirements for various
objectives as
About three stations: Determine global inventories and trends
•
Additional 5–10 stations: Determine meridional transport of CO
•
2
Over 100 stations: Determine air-surface exchange within regions with
•
signifi cant anthropogenic infl uence
The diffi culty of the inversion can be quantifi ed, in part, by determining how
rapidly the error amplifi cation grows as the resolution increases. This has been
analyzed for the dependence on latitudinal wave-number n. If the forward problem
attenuates fl uxes of wave number n with an n −α decay, then the inverse problem
involves an error amplifi cation that grows as n α In their 2D numerical modeling,
© 2010 by Taylor and Francis Group, LLC