### Abstract:

The time-averaged velocity field in the North Pacific was estimated in two
sets of inverse calculations. The planetary geostrophic equations were the basis for
dynamical models of the flow in each case. The inverse estimates of the circulation
were obtained by minimizing a positive-definite cost function, which measured the
inconsistency of the model's predictions against a set of observations comprised of
a large, high-quality hydrographic data set, and surface fluxes of heat, fresh water,
and momentum.
In the first part of this work, four solution methods for the generalized inverse
of a linear planetary geostrophic model of the North Pacific are compared.
A conjugate gradient solver applied to the equation for the generalized inverse,
expressed in terms of a representer expansion, was the most computationally efficient
solution method. The other methods, in order of decreasing efficiency, were,
a conjugate gradient descent solver (preconditioned with the inverse of the model
operators), a direct solver for the representer coefficients, and a second conjugate
gradient descent solver (preconditioned so that the diagonal elements of the cost
Redacted for Privacy
function Hessian were unity). All but the last method were successful at minimizing
the penalty function.
Inverse estimates of the circulation based on the linear planetary geostrophic
model were stable to perturbations in the data, and insensitive to assumptions
regarding the model forcing and boundary condition uncertainties. A large calculation,
which involved approximately 18,000 observations and 60,000 state variables,
indicated that the linear model is remarkably consistent with the observations.
The second part of this work describes an attempt to use a nonlinear planetary
geostrophic model (which included realistic bottom topography, lateral momentum
mixing, out-cropping layers, and air-sea fluxes of heat, freshwater, and
momentum) to assimilate the same hydrographic data set as above. Because of
the nonlinearity in the model, descent methods (rather than a representer-based
method) were used to solve the inverse problem. The nonlinearity of the model and
the poor conditioning of the cost function Hessian confounded the minimization
process. A solver for the tangent-linearization of the planetary geostrophic system
should be used as a preconditioner if calculations of this type are attempted in the
future.