## General Considerations

Inferring the distribution of sources and sinks of CO2 and their temporal variations in a consistent way from atmospheric concentration observations by means of a model of atmospheric transport constitutes an inverse problem of considerable complexity. Early attempts to address this problem were made as soon as the first reliable atmospheric CO2 observations became available (Bolin and Keeling 1963). At that time, however, comprehensive meteorological observations of the tropospheric transport were not available, hence the atmospheric concentration measurements were also used to deduce the large-scale strength of meridional atmospheric mixing, in addition to the carbon sources.

With the increasing density of the observation network after 1980 and the emergence of three-dimensional atmospheric transport models based on realistic meteorological data from climate models or weather forecast models (Fung et al. 1983), it became possible to address the inversion problem in more detail. A comprehensive description of the mathematical methods involved has been given by Enting (2002). In general, the domain of interest is split into a series of spatiotemporal source patterns. Using the atmospheric transport model, a "base function" is computed for each of these source patterns, defined as the atmospheric response at the observation locations and times, subsequent to a unit input emitted from the source pattern. Finally, a linear combination of the base functions is determined that optimally matches the measurements at the observation points. The determined weights of the base functions then constitute the source strengths of the different source patterns.

The inversion problem raises a series of important difficulties:

1. Current atmospheric transport models are not perfect.

2. The observational network is very sparse—that is, there exist only =100 monitoring stations worldwide, a small number compared with the heterogeneity of the terrestrial or oceanic carbon sources. Furthermore, at some stations the sampling frequency is low, and there are often gaps in the observations.

3. Technically, the "inversion" of the atmospheric transport model is not trivial and requires much larger computing resources than running the model in the forward mode.

4. Individual measurements are often not representative of the appropriate temporal and spatial scale of the transport model.

5. Individual concentration observations are of limited accuracy and precision, and observations from different monitoring networks are often not easily comparable because of differences in measurement techniques and uses of different standards.

The most serious limitation of the top-down approach follows from the limited number of observations and the need to adequately represent sources and sinks with a relatively high spatial and temporal resolution. Indeed, a very large number of possible surface source-sink configurations are in principle consistent with the atmospheric observations. An example is provided by Kaminski and Heimann (2001), which demonstrates that an unrealistic sink of 2 PgC y-1 over Europe may be perfectly compatible with the entire observations from the global monitoring station network. The atmospheric observations alone are thus not sufficient to uniquely determine the sources at the surface of the Earth.

Formally, the atmospheric inversion problem constitutes an ill-posed mathematical problem, which must be regularized by means of a priori information on spatial and temporal variability of sources and sinks including a priori estimates of their uncertainty. This prior information may be used to define a smaller number of spatiotemporally more complex base functions in order to solve for fewer unknowns compared with the number of observations (Enting et al. 1995; Fan et al. 1998; Rayner et al. 1999; Bousquet et al. 2000; Gurney et al. 2002). Alternatively, one may perform a high-resolution inversion by using the prior information in a Bayesian approach to regularize the problem (Kaminski et al. 1999; Rödenbeck et al. 2003).

To assess the impact of the different atmospheric transport models and the various possible methodologies on the determination of sources and sinks of CO2, the international project TRANSCOM was established with support by the Global Analysis,

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Figure 8.2. Annual mean latitudinal land-ocean breakdown of non-fossil-fuel carbon sources as determined in the TRANSCOM inversion intercomparison study of Gurney et al. 2002. Gray boxes: a priori fluxes and uncertainties. Crosses: inferred mean fluxes, average of 16 models or model variants. Vertical error bars indicate between-model variations (1 sigma); open circles show the mean estimated uncertainty across all models. Fluxes are representative for the five-year interval 1992—1996, determined from 76 atmospheric stations from the Cooperative Atmospheric Data Integration Project (2000) database.

Figure 8.2. Annual mean latitudinal land-ocean breakdown of non-fossil-fuel carbon sources as determined in the TRANSCOM inversion intercomparison study of Gurney et al. 2002. Gray boxes: a priori fluxes and uncertainties. Crosses: inferred mean fluxes, average of 16 models or model variants. Vertical error bars indicate between-model variations (1 sigma); open circles show the mean estimated uncertainty across all models. Fluxes are representative for the five-year interval 1992—1996, determined from 76 atmospheric stations from the Cooperative Atmospheric Data Integration Project (2000) database.

Integration and Modeling task force of the International Geosphere-Biosphere Program. Several carefully defined model intercomparison studies have been conducted in TRANSCOM, including forward simulations of the terrestrial seasonal cycle and the fossil-fuel CO2, (Law et al. 1996) and of the inert, anthropogenic tracer SF6 (Denning et al. 1999). Later intercomparison experiments have explored the impact of different transport models based on annual mean global inversions (Gurney et al. 2002). As an illustration, Figure 8.2 shows the meridional land-ocean breakdown of the net non-fossil-fuel CO2 sources aggregated in three global latitude bands separated at 30°N and 30°S, inferred by the 16 transport models or model variants included in the study of Gurney et al. (2002). These fluxes are representative for the five-year period 1992— 1996. The light gray boxes show the specified a priori sources and their uncertainty (± 1 standard deviation). The crosses indicate the mean of the different model inversions performed using the observations over 1992—1996 from 76 stations included in the GL0BALVIEW-2000 data set (Cooperative Atmospheric Data Integration Project 2000). The vertical error bars reflect one standard deviation of the spread of the model averages ("between-model" error), while the circles indicate the average of the individual posterior source errors estimated by the different models (average "within-model" error). The Northern Hemisphere estimates exhibit substantial CO2 uptake, especially on land, corroborating the existence of the northern extra-tropical CO2 sink inferred by the earlier studies. The tropics, in particular the tropical land areas, are much less well constrained by the inversions, reflecting the poor atmospheric station coverage, but also a relatively larger spread of the simulated transport by the models in these regions as shown by the larger "between-model" error. Overall, tropical land and oceans appear as CO2 sources in the inversion. The inferred estimates of the Southern Ocean fluxes are consistently less negative than the specified prior fluxes. This finding indicates a bias in air-sea flux field ofTakahashi et al. (2002), which was used as prior ocean flux estimate in the inversion. The study of Gurney et al. (2002) represents the most comprehensive assessment of the impact of different atmospheric transport models on atmospheric inversions. On the other hand, the employed methodology using a time-independent mean inversion with a limited number of large-scale regions severely reduces the number of possible inversion solutions, as already discussed (see also the discussion of the aggregation error in global inversions by Kaminski et al. 2001). Therefore the error estimates shown in Figure 8.2 most likely are too optimistic.