A consistent formalism for a Bayesian design of control space for an optimal assimilation of observations was proposed in Part I of this two-part article. This optimal discretization of control space leads to an efficient data assimilation scheme implementation. However, the construction of the grid itself, prior to its use for data assimilation, requires an optimization that may be challenging for high-dimensional systems. This paper derives analytical solutions for these optimal grids in the limit where the discretization of control space has a large number of grid cells. Analytical solutions for the density of grid cells are obtained for the so-called tilings, qtrees and ftrees, that represent different types of adaptive grids, with more or fewer degrees of freedom. These analytical solutions are explicit and the algorithms that allow densities to be converted into discrete adaptive grids are costless.
The approach is tested with a simplified physics in the Jacobian matrix in a tracer dispersion context in which radionuclides are monitored by the global observation network operated by the Comprehensive Nuclear Test Ban Treaty Organisation of the United Nations. The asymptotic solutions are then compared to the optimal grids obtained from the methodology perfected in Part I. In this example, and using qtree representations, the discrepancy between the approximate solution and the exact solution almost vanishes when the number of grid cells represents as few as 1% of the total number of grid cells in the finest grid. This opens the way to the application of this multiscale data assimilation framework to computationally challenging problems. Copyright © 2011 Royal Meteorological Society