Background Error Analysis Testbed with BOXMOX

The Background Error Analysis Testbed with BOXMOX allows simple and fast investigation, comparison and evaluation of any system of Ordinary Differential Equations (ODE) evolving over time. By 'Background Error' we designate a general term for model error characterization. The BEATBOX framework consist of the 3 following elements:

and its structure is built upon Observing System Simulation Experiments (OSSE, Arnold and Dey, 1986). OSSEs are widely used in the field of numerical weather prediction (e.g. Kuo et al., 1998; Wang et al., 2008; Liu et al., 2009) as well as atmospheric composition and air quality predictions (e.g. Edwards et al., 2009; Claeyman et al., 2011; Barré et al., 2015, 2016). OSSEs allow to assess the benefit of a potential new type of instrument for environmental predictions using a data assimilation system. OSSEs are of crucial importance to define requirements of a given instrument. Also the model and data assimilation requirements should be assessed to meet a required predictive capability. The current BEATBOX framework with its box modeling approach avoids the space dimension problem (the dimensionality of the problem is set to time and variables only), hence the geometry, radiative specifications and spatial resolution of a new instrument type are non-relevant. What can be easily explored within BEATBOX is the type of data assimilation method used (e.g. background error calculations, localizations), the revisit time (or temporal sampling) of a measurement, or the type of observed variable and straightforward comparisons between set of ODE (chemical schemes in our case). To achieve this, several elements are required:

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The BEATBOX package provides a data assimilation tool that sample observations form the NR, with possibility of tuning the observation error parameters. The system is coded in python and shell script that allows easy implementation. The system cycles the assimilation windows: each analysis obtained from data assimilation is used as initial condition for the next forecast that will be corrected by observation using data assimilation and so on (see figures below)... With assimilating observations, different sensitivity analyses can be used, such as adjoint, ensemble or hybrid sensitivity are included in this version of BEATBOX.

Observation generator – synthetic observations

Generating observation consists of the following steps:

Because of the 0D (e.g. no latitude,longitue and altitude) nature of BOXMOX sampling the NR to simulate synthetic observations is straightforward. The observations could be assumed as perfect and hence no error has to be simulated and the observation value is exactly the same as the NR sampled value.

In case of non-perfect observation an error is simulated by perturbing the sampled NR value. In the current version of BEATBOX the observation perturbation is assumed to be gaussian, but other probability density functions (PDF) can be easily implemented.

Finally, an associated observation error is associated to the observation value. BEATBOX can simulated for bias (mean of the perturbation PDF) and/or precision (standard deviation of the PDF). Overestimating or underestimating observation error can be also tested in BEATBOX.

Adjoint sensitivity

To derive a sensitivity from a change to a given variable from another an adjoint model can be computed. We use the BOXMOX/KPP capability to output the Jacobian matrix of the ODE system evolution within the assimilation window and derive an adjoint sensitivity. The adjoint sensitivity uses the model equations directly.

Ensemble sensitivity

The ensemble method uses an ensemble of perturbed model realizations to obtain the sensitivity of the model (or the ODE system) to a change in a given variable. We need to a certain number of realization of perturbed models runs in order to have a statistically sound (i.e. reduce the sampling error) relationship between variables. Perturbations can be initially generated in the initial condition files of BOXMOX. Initial perturbations can be choosen to be on parameters of choice such as concentrations or temperature and be normal or log normal. During further assimilation cycles inflation techniques are also implemented to perturb the ensemble.

Ensemble of Adjoints (Hybrid)

In this framework we called hybrid sensitivity a combination of adjoint method and ensemble method to derive the sensitivity from a chemical compound to another. An ensemble of BOXMOX simulation are run in parallel and Jacobian of each ensemble members are used to calculate an ensemble adjoints. Then the analysis is computed by using the adjoint on each ensemble member. This last example of sensitivity shows what can be easily implemented in BEATBOX for exploring data assimilation techniques in reactive gas phase chemistry and highly non-linear systems in general.

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