4DVar Balance Operator

[arango]

The 4DVar data assimilation algorithms have now a fully working balance operator. This can be activated with option BALANCE_OPERATOR. The algorithm follows the methodology of Weaver et al. (2005). The balance operator is used in 4DVar data assimilation algorithms to constraint the error covariance. It is a multivariate approach to estimate the off-diagonal terms of the error covariance matrix. This allows the unobserved state variables information to be extracted from directly observed quantities. The ocean state vector is split between balanced and unbalanced components, except for temperature which is used as the starting point to estimate the balanced part of the other variables. The multivariate formulation is obtained by establishing linear balance relationships between temperature and other state variables using T-S empirical formulas, the linear equation of state, hydrostatic balance, and geostrophic balance.

The balanced free-surface increment can be computed by solving an elliptical equation as in Fukomuri et al. (1998). This can be activated with option ZETA_ELLIPTIC. Otherwise, a it may computed from the hydrostatic balance.

This is a very difficult and complex algorithm. Specially the solution of the elliptic equation with the biconjugate gradient algorithm. We needed the tangent linear and adjoint versions of the biconjugate gradient algorithm. This was very very tricky... Many thanks to Andy Moore for coding and testing this difficult algorithm. We cannot follow the adjoint recipe roules here since the biconjugate gradient method operate on the transpose matrix.

This is the last major peice of ROMS 4DVar data assimilation algorithms. It has been a very torturous and difficult path that took several years and lot of patience. Many thanks to the ROMS adjoint group for their great help and to ONR for funding us to develop these algorithms. Again, many thanks to Andy Moore for his unvaluable cooperation... Also many thanks to Anthony Weaver for his work and advice.