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Initial-boundary compatibility for inverse regional models

In this paper we investigate the importance of compatibility conditions in inverse regional models. In forward models, the initial condition and boundary condition must be continuous at the space’time corner to ensure that the solution is continuous at the space’time corner and along the characteristic emanating from it. Furthermore, the initial condition, boundary condition and forcing must satisfy the partial differential equation at the space’time corner to ensure that the solution is continuously differentiable at the space’time corner and along the characteristic emanating from it. Assimilation of data into a model requires a null hypothesis regarding residuals in the dynamics (including, for example, forcing, boundary conditions and initial conditions) and data. In most published work, these errors are assumed to be uncorrelated, although Bogden does investigate the importance of correlation between boundary conditions and forcing, and shows that by generalizing the penalty functional to include this cross-correlation the open boundaries behave more like an interface with a true ocean. In this paper, the more general importance of cross-correlations and their impact on the smoothness of the optimal solution are discussed. It is shown that in the absence of physical diffusion, cross-correlations are necessary to obtain a smooth solution; if errors are assumed to be uncorrelated, discontinuities propagate along the characteristic emanating from the space’time corner, even when the initial condition, boundary condition and forcing are compatible, as described above. Although diffusion damps such discontinuities, it does not reproduce the inherently smooth solution achieved when cross-correlation of residuals are accounted for. Rather, the discontinuity is smoothed into a spurious front.