Sensitivity analysis in an assimilative System

The sensitivity of a receptor to changes in a source can be studied using first order adjoint methods on the model describing the source-receptor relationship. When some inputs of the model are the solution of a data assimilation problem, the first order adjoint is no longer sufficient. In this study, we derive the second order adjoint method for the sensitivity study in an assimilative pollution model. Theoretical differences with the first order adjoint methods are derived and show that the sensitivity in an assimilative system must be carried out on the optimality system of the Data Assimilation.

The method is illustrated using a coupled Burgers and convection-diffusion model. The pollutant is transported by a velocity field that evolves according to the Burgers model. The emission of the pollutant is defined by a source function. Obervations of the transport velocity and of the concentration of the pollutant are used in a Variational Data Assimilation process to determine the optimal trajectory of the transport velocity and of the concentration of the pollutant. A response region is defined and used to compute the ensitivity of a response function with respect to the source function. Numerical experiments illustrate the feasability of the second order adjoint method and confirm the theoretical analysis. The first order adjoint substancially underestimates the order of magnitude of the sensitivity and the sources of the pollutant.