### Abstract:

The generalized inverse is constructed for a nonlinear, single-layer quasigeostrophic model, together with initial conditions and a finite number of interior data. With the exception of doubly Periodic boundary condition all constraints are weak. The inverse minimizes a penalty functional that is quadratic in the errors in prior estimates of the model forcing, the initial conditions, and the measurements. The quadratic form consists of the inverses of the prior error covariances. The nonlinear Euler-Lagrange equations are solved iteratively. Each iterate is a linear Euler-Lagrange problem, which in turn is solved in terms of a prior streamfunction estimate, plus a finite-linear combination of representers (one for each linear measurement functional). The sequence of inverse streamfunction estimates is bounded and so has at least one limit Point. The sequence of representer matrices is uniformly positive definite and so the limiting inverse is a minimum, rather than simply an extremum of the penalty functional. The streamfunction estimated with the weak constraint inverse is compared. with the result of a strong constraint inverse, which is identical in every respect, except that the prior estimate of the model forcing is assumed to be exact.
A statistical linearization about the limiting inverse yields the inverse or posterior error covariance. Similarly, the reduced penalty functional is a χ² variable with as many effective degrees of freedom as the observing system.
The algorithm is easily implemented on multiple-processor computers. The parameter experiments reported here were Performed on a 128-processor Intel iPSC/860 and also on a single-processor workstation. The formal number of degrees of freedom exceeded a quarter of a million. The representer algorithm is tractable and stable since it identifies the five number of degrees of freedom, namely, the number M of measurements. Here M = 128.