2006 MCS Divisional Seminars & Colloquia

The global ocean state estimated over the last decade as an optimization/optimal control problem

   Patrick Heimbach/Chris Hill

  Hosted by  Jean Utke

10:30 AM, March 8, 2006
Building 221,  Room A216

The Consortium "Estimating the Circulation and Climate of the Ocean (ECCO) has as its overall goal the production of estimates of the global ocean circulation, over extended periods, employing as much of the diverse oceanographic data sets as is practical. The data sets are necessarily sparse, extremely diverse, and of non-uniform sampling. A general circulation model is required to synthesize the results into a unifying picture which is both dynamically and statistically consistent and useful for understanding the ocean in all its aspects.

Although what ECCO does is sometimes called "data assimilation" because of its analogues in numerical weather prediction, we distinguish the actual practice from data assimilation by using the label "state estimation", which is borrowed from control theory. The reason for the distinction lies primarily with the different goals and practice: ECCO is directed (at the present time), not at forecasting, but at monitoring, measuring and understanding. In control terminology, it addresses the "smoothing" problem rather than the "filtering" and prediction ones. As such, and in contrast to NWP, a premium is placed on avoiding trajectory jumps at the analysis times in NWP. Our preferred description is that of a gradient-based least-squares fit of the data sets to the model. We use the method of Lagrange multipliers (adjoint method, Pontryagin principle) to derive the gradient of the cost function with respect to a set of control variables. The adjoint model is derived via the automatic differentiation tool TAF. The optimization is solved iteratively via a quasi-Newton variable storage algorithm (M1QN3).

Solving the optimization problem is a formidable task. The model is a fully fledged non-linear general circulation model with a state space of 1E8 elements per time step, a control space made up of 3-dimensional distributions of initial temperature and salinity, and time-varying 2-dimensional buoynancy and momentum forcing field, yielding a 1E9-dimensional control space. One iteration of the current configuration at 1 degree horizontal resolution takes ~2 days of wall clock time on 60 processors of an SGI Altix.

Some of the issues involved are: (1) size of the problem, (2) conditioning of the gradient and ill-posedness of the optimization, (3) nonlinearity of the problem, exponential sensitivity growth (Liapunov exponents), and multiple minima, (4) non-differentiability of the numerical formulation (such as encountered in parameterization schemes) or oscillating derivatives.

Over the next five years we plan, as part of the new ECCO-2 project (http://ecco2.org), to increase the state space of the general circulation model by an order of magnitude. This will result in a more faithful representation of underlying physics, but also increases the non-linearity and degree of complexity of the optimization process.