Inversion of Atmospheric CO 2 Concentrations
297
Enting and Mansbridge found a n 1 growth in the extent to which errors with latitudinal wave-number n were amplifi ed. In contrast, the Bolin and Keeling analysis
implied n 2 growth, fully justifying the pessimistic assessments quoted above. Enting
and Mansbridge (incorrectly) attributed the difference to their use of advective transport as opposed to the diffusive transport approximation used by Bolin and Keeling.
The real reason for the difference between the n 2 and n 1 growth was identifi ed by
Newsam and Enting (1988) as having the n 2 growth appear as an artifact of approximating the boundary-value problem as a vertically-averaged problem. Numerical
studies with the GISS 3D transport model (Enting, 2000a) showed that this case
also exhibits the n 1 growth in error amplifi cation. Thus, although the concerns of
Bolin and Keeling were overstated, the degree of ill-conditioning is an important
limitation on the global-scale trace gas inversion problem.
11.3.2 CHARACTERIZING RESOLUTION
Since ill-conditioned inverse problems are typically characterized by increasing
diffi culty in estimating successively smaller scale detail, an important theme of inversion studies, across all areas of science, has been the analysis of what features can
be validly estimated (Backus and Gilbert, 1968; Jackson, 1972). The main trade-off
is between resolution and variance in the estimates. Below some limiting resolution,
there can be an explosive growth in variance.
The principle, identifi ed by Wunsch and Minster (1982), is that only a small
number of modes can be resolved and that these specifi c modes will be defi ned by
the data. Failure to recognize this, with a small number of modes selected for estimation without reference to the resolution implied by the data, was one of the failings
in the analysis by Fan et al. (1998) discussed in Section 11.4.1.
While the resolution of inverse problems requires a careful analysis for each
particular case, qualitative characterizations may be possible. A set of characteristic
numbers has been defi ned (Enting, 2002, Section 8.1), defi ning such things as the
number of signal components about the noise level, the number of effectively independent data values, and the computational resolution. This can lead to signifi cant
“truncation error,” which is often neglected. An appropriate formulation is given by
Trampert and Sneider (1996) in the context of seismology. The applicability to CO 2
inversions was confi rmed by Kaminski et al. (2001). A schematic of the formalism
for assessing truncation error has been given in Figure 8.3 of Enting (2002), and
Enting (2008) has used examples from digital fi ltering of time series to illustrate how
various forms of noise are affected by truncation.
The conditions for working with reduced computational resolution are either
A The small-scale variability is known to be small or both of
B1 The small-scale variability is not of interest, and
B2 The small-scale variability can be separated from the larger-scale
variability.
A generic mathematical technique for tackling inverse problems is known as
regularization (Tikhonov, 1963). Various regularizations have been applied to CO 2
© 2010 by Taylor and Francis Group, LLC
297
Enting and Mansbridge found a n 1 growth in the extent to which errors with latitudinal wave-number n were amplifi ed. In contrast, the Bolin and Keeling analysis
implied n 2 growth, fully justifying the pessimistic assessments quoted above. Enting
and Mansbridge (incorrectly) attributed the difference to their use of advective transport as opposed to the diffusive transport approximation used by Bolin and Keeling.
The real reason for the difference between the n 2 and n 1 growth was identifi ed by
Newsam and Enting (1988) as having the n 2 growth appear as an artifact of approximating the boundary-value problem as a vertically-averaged problem. Numerical
studies with the GISS 3D transport model (Enting, 2000a) showed that this case
also exhibits the n 1 growth in error amplifi cation. Thus, although the concerns of
Bolin and Keeling were overstated, the degree of ill-conditioning is an important
limitation on the global-scale trace gas inversion problem.
11.3.2 CHARACTERIZING RESOLUTION
Since ill-conditioned inverse problems are typically characterized by increasing
diffi culty in estimating successively smaller scale detail, an important theme of inversion studies, across all areas of science, has been the analysis of what features can
be validly estimated (Backus and Gilbert, 1968; Jackson, 1972). The main trade-off
is between resolution and variance in the estimates. Below some limiting resolution,
there can be an explosive growth in variance.
The principle, identifi ed by Wunsch and Minster (1982), is that only a small
number of modes can be resolved and that these specifi c modes will be defi ned by
the data. Failure to recognize this, with a small number of modes selected for estimation without reference to the resolution implied by the data, was one of the failings
in the analysis by Fan et al. (1998) discussed in Section 11.4.1.
While the resolution of inverse problems requires a careful analysis for each
particular case, qualitative characterizations may be possible. A set of characteristic
numbers has been defi ned (Enting, 2002, Section 8.1), defi ning such things as the
number of signal components about the noise level, the number of effectively independent data values, and the computational resolution. This can lead to signifi cant
“truncation error,” which is often neglected. An appropriate formulation is given by
Trampert and Sneider (1996) in the context of seismology. The applicability to CO 2
inversions was confi rmed by Kaminski et al. (2001). A schematic of the formalism
for assessing truncation error has been given in Figure 8.3 of Enting (2002), and
Enting (2008) has used examples from digital fi ltering of time series to illustrate how
various forms of noise are affected by truncation.
The conditions for working with reduced computational resolution are either
A The small-scale variability is known to be small or both of
B1 The small-scale variability is not of interest, and
B2 The small-scale variability can be separated from the larger-scale
variability.
A generic mathematical technique for tackling inverse problems is known as
regularization (Tikhonov, 1963). Various regularizations have been applied to CO 2
© 2010 by Taylor and Francis Group, LLC