Inversion description


Statistical methods are used to combine available information to estimate CO2 fluxes in an optimal way. In particular the following information is exploited in an inverse procedure (see figure 1 ):

A review article describing and comparing the CO2 inversions fluxes can be downloaded here. The CO2 flux inversion process follows the general scheme shown on figure 1.

The inversion products presented in this website are results from different laboratories. Details on the sepecifics of each product are available on the website of the data provider.


Figure 1 : General scheme of atmospheric inversion








figure 2 : Measurement Network

Atmospheric observations of GHG are performed all over the world by scientific networks generally operated by countries. Different types of observations are available. Flask sampling is historically the first method that was used, since 1958 at Mauna Loa observatory for CO2. Glass flasks are filled with air at a measurement point and analysed in a central laboratory. This method is still widely used today, with more than 150 sites all around the world. It has the advantages to be very precise and accurate, and provide measurements of several GHG with the same sample of air. However, it is discontinuous in time, and in space (large areas of the earth surface are not sampled). During the last 15 years, in-situ measurements have been developed to address the issue of time discontinuity. Instruments are placed in an observatory and measure GHG at this location. Depending on the technic used, instruments provide only measurements for one gas (e.g. infra-red non dispersive spectrometry) or for several GHG (e.g. gas chromatography). Such instruments can be placed at the surface or attached to a tall tower with several height of sampling in order to increase the fetch of the observatory. They give also very precise and accurate data. Since the late 1990s, aircraft measurement programs have been developed to sample the planetary boundary layer (first kilometers of the atmosphere). Punctual campaigns occured but also regular flights of small aircraft over precise locations, and commercial flights for a few routes around the world. More recently, Fourier transform infrared spectrometry (FTIR) has provided first retrievals of GHG columns in the atmosphere. Finally, satellite retrievals of atmospheric columns are already available for some gases (carbon monoxide, methane, ozone, nitrous oxide, formaldehyde) and will become available for CO2 in the next years. These data, although not as precise and accurate as surface measurements, will fill the spatial gaps the surface networks.


In inverse procedures, atmospheric observations can be used as means (yearly, monthly, weekly or daily) or as raw data for recent procedures. Uncertainties attached to these data should account for both observational error (generally small) and for representativeness error (possibly large). The latter is due to inabilities of atmospheric models to properly represent measurements mainly because of smoothed topography, transport error, and chemistry errors. Practically, representativeness error is difficult to calculate precisely and is often estimated by proxy approaches.


Figure 3 : Prior Fluxes

In most inverse procedures, a prior knowledge of the spatio-temporal distribution of the fluxes to estimate is necessary, directly or indirectly. This knowledge is given either by gridded inventories or by models . Inventories are generally used for anthropogenic emissions and are based on economic and trade data (e.g. fossil fuel emissions of CO2 from combustions). Natural sources and sinks often come from more or less sophisticated models representing involved processes (e.g. wetland emissions of CH4 calculated by a vegetation model). Uncertainties attached to prior fluxes can be a side-product of the methodologies employed to build the fluxes (inventories, models). However, this is not always available and prior flux uncertainties are often chosen (more or less arbitrarily) in order to be large enough not to nudge the flux estimate towards the prior knowledge. The relative weight of observations uncertainties and of prior flux uncertainties is a critical point of inverse procedures. Prior fluxes are optimize by the inverse procedure with a time resolution varying from one day to one year, most of the inversions performed so far being monthly inversions.



An atmospheric-transport model (ATM) solves the mass conservation equation to calculate the spatio-temporal distribution of an atmospheric trace gas. Surface sources and sinks are prescribed in ATMs and converted into atmospheric concentrations after being advected, convected and eventually chemically transformed (reactive species). In the case of CO2, a chemically passive tracer in the atmosphere, only transport models are used. ATMs need external forcings such as meteorological conditions, surface conditions over land and oceans, and initial conditions. These forcings are provided by meteorological centers (e.g. ECMWF), by general climate models (e.g. water fluxes) and by climatologies (e.g. SST). ATMs can be global or regional models. ATMs are used in a forward mode (emissions give concentrations) or in a backward mode (concentrations give areas from which emissions originate, see animation above). Inverse procedures require the calculation of the so-called response function that is the contribution of one given region to one given measurement. A region can be as small as a model grid-cell or can be a group of model grid cells.

Atmospheric-transport Model



An inverse procedure combines mathematically atmospheric observations, prior fluxes and a chemistry-transport model to produce an optimal estimate of fluxes together with their uncertainties. This is done through the minimization of a cost function J :

where :

y is the vector of observations
H(x) is the matrix of response functions (jacobian matrix)
x is the vector of fluxes (to be optimized)
  xb is the vector of prior fluxes
R is the variance-covariance matrix representing observation errors
B is the variance-covariance matrix representing flux errors

J is a quadratical form built as the sum of two terms, the first one figuring model versus observations differences weighted by R , and the second one figuring a distance between prior fluxes and optimized fluxes weighted by B (Bayesian term). As the inverse problem is ill-posed, the second term helps regularising the solution of the problem. This approach is based on the Bayes paradigm. J is minimized either analytically (for small problems) or numerically using, for instance, conjugate gradient formulations.

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