It is a long road from deriving the formulae for the generalized inverse of a model and data to seeing results. First experiments (McIntosh and Bennett, 1984) involved a linear barotropic model separated in time, simple coarsely-resolved numerical approximations, a handful of pointwise measurements of sea level and a serial computer. Contemporary models of oceanic and atmospheric circulation involve nonlinear dynamics and parameterizations, advanced high-resolution numerical approximations, vast quantities of data often of a complex nature, and parallel computers. Chapter 3 introduces some general principles for travelling this long road of implementation.

The first principle is accelerating the representer algorithm by task decomposition, that is, by simultaneous computation of representers on parallel processors. The objective may be either the full representer matrix as required by the direct algorithm, or a partial matrix for preconditioning the indirect algorithm. The calculation of an individual representer, or indeed any backward or forward integration, may itself be accelerated by domain decomposition, but this is a common challenge in modern numerical computation (Chandra et al., 2001; Pacheco, 1996) and will not be addressed here. Even without considering the coarse grain of task decomposition or the fine grain of domain decomposition, the direct and indirect representer algorithms for linear inverses are highly intricate. Schematics are provided here in the form of “time charts”.

Dynamical errors and input errors may be correlated in space or in time or in both. Error covariances must be convolved with adjoint variables. […]