## 1. Introduction

### 1.1. The resolution issue

Researchers using inverse modelling techniques in atmospheric chemistry have faced the so-called ‘resolution problem’.

A first example is given by the gridded emission inventories which are multidimensional fields and key components of the models. Unfortunately, the uncertainty of these fields is quite high (of the order of 40% for the ozone precursors in air quality at continental scale, for instance). Observations could help to constrain the emission fields through inverse modelling and reduce this uncertainty, e.g. Elbern *et** al.* (2007) for an application to the precursors of ozone, or Davoine and Bocquet (2007) for an application to an accidental release of radionuclides. Both the model equations and the control space of the emission field need to be discretised at some predefined space and time resolution. The space and time resolutions of the discretised control space are not necessarily the same as those of state space. There is a non-trivial choice of resolution to be made. Furthermore, inventories are built at a given resolution, the model runs at another, and the data assimilation scheme injects the information of all observations into the system at still another scale depending on the nature of the instruments: ground-based, satellite, radar, lidar, etc. Therefore, the system should ideally be considered multiscale.

Another example pertains to the inverse modelling of greenhouse gases. Early carbon flux inversions relied on a partition of the globe (the control space of fluxes) into about 20 sub-domains representing several types of continental or ocean exchange with the atmosphere, with an annual or monthly time resolution (e.g. Fan *et al.*, 1998; Bousquet *et al.*, 2000). This was necessary because of the limited computational power together with a limited number of precise observations of CO_{2} concentration. However, such gross partitioning led to severe aggregation errors (Kaminski *et** al.* 2001; Trampert and Sneider 1996). Thus it is tempting to increase the space and time resolutions of control space. But the total number of variables could dramatically exceed the total number of observations. Besides, because of the nature of transport and dispersion, the inverse modelling problem is ill-posed. Therefore a regularisation is needed (Rödenbeck *et** al.* 2003), which can be written as a Tikhonov regularising term, as is usually done in geophysical data assimilation. This regularisation, which spatially and temporally correlates the errors, may stem from real physical correlations due, for instance, to similar ecosystems (Chevallier *et** al.* 2006). But it may also be artificial and correspond to a smooth aggregate of variables. Note that this distinction is not always made clear in the literature.

In both cases, there is a difficult choice to be made on the resolution of control space. To make the problem worse, Bocquet (2005) has shown that, for atmospheric dispersion problems, the source estimation of atmospheric pollutant from inversions using pointwise measurements depends strongly on the control space resolution, even when using a proper classical Tikhonov regularisation (background-error term of quadratic form in the cost function).

### 1.2. Multiscale approach

To partially solve this resolution issue, a multiscale framework for such inversions was proposed (Bocquet, 2009). It is at the crossroads between a coarse partitioning of control space subject to aggregation errors and a highly resolved control space where regularisation is decisive. The method consists of constructing an adaptive grid of control space (also called a *representation* of control space in the following). This adaptive grid is optimal in the sense that it is designed to optimally capture the information carried by the observations and inject into control space through a model and the assimilation system. This is achieved by maximizing an objective function that measures the reduction of uncertainty granted by the observations on a space of all potential adaptive grids (later called a *dictionary* or *class*).

The method quantifies how observational information is propagated into control space. It diagnoses poorly observed areas. It informs how space- and time-scales should be related for the problem at hand. Also, it has strong algorithmic implication. Indeed, the method shows how to devise adaptive grids of control space that have significantly fewer grid cells than the original finest regular grid, but which can still capture most of the information content of observations. Such an adaptive grid was built and tested on the European Tracer Experiment (ETEX; Nodop *et al.*, 1998). The inversion of the source term of this dispersion event was performed much faster with an optimization over about 100 times fewer independent variables in this adaptive grid, with results very similar to those obtained with a regular fine grid.

The method also offers a starting point for a general conceptual and mathematical framework for multiscale data assimilation in atmospheric chemistry, or in other areas of geophysics.

This two-part article aims to continue and improve the potential of this formalism and prepare for large-scale applications. The first part explores a few essential questions still unanswered, such as

Can the Bayesian approach that is currently used in geophysical data assimilation be made consistent with the multiscale framework of the method?

Can a non-diagonal background-error covariance matrix be taken into account in this formalism? Such matrices are often used in air quality, greenhouse gas flux inversions and, more generally, in data assimilation schemes for geophysical forecasting systems.

Can scale-dependent errors be accounted for?

Can one use other grid optimization objective functions, such as DFS, or observation-dependent criteria?

Can one perform the optimization within a simpler or more economical dictionary of adaptive grids than the so-called

*tiling*dictionary introduced by Bocquet (2009)?

The results are obtained with a view to applications in atmospheric chemistry and air quality, but most of the findings are more general and could be applied outside this scope whenever the choice of control space is complex and decisive.

### 1.3. Outline

The conceptual and mathematical framework will be presented in section 2. The multiscale description of control space is made consistent with the assimilation of observations using Bayesian principles.

Section 3 deals with errors which may enter the inversions, and which are scale-dependent. Of particular interest are the aggregation errors occurring when grid cells are merged. They lead to representativeness errors.

The construction of optimal representations of control space requires the definition of a criterion that ranks adaptive grids in a given dictionary of representations. In addition to the so-called Fisher criterion introduced by Bocquet (2009), we add two new criteria in section 4. One is based on the DFS which measures the theoretical information gain in the analysis. A third criterion is defined with an objective function that not only depends on the prior statistics but also on the observations themselves.

In section 5, most of the developments will be illustrated on two test cases: the ETEX-I dispersion event using real measurements and realistic physics (from a chemistry and transport model), and another demonstration case based on a simplified European CO_{2} network (CarboEurope-IP).

In section 6, the dictionary of tilings is compared to a dictionary of quaternary tree structures (later called *qtrees*). Although suggested in Bocquet (2009), the quaternary tree structure was not tested and studied there.

Finally, in section 7, we summarise the results. We discuss its connection with other multiscale formalisms introduced very recently in data assimilation. We also discuss the scope of the method and its extension to nonlinear models. Elements that justify the need for Part II (Bocquet *et** al.* 2011) of this work are explained.