A Hessian-Based Method for Uncertainty Quantification in Global Ocean State Estimation

Abstract

Derivative-based methods are developed for uncertainty quantification (UQ) in large-scale ocean state estimation. The estimation system is based on the adjoint method for solving a least-squares optimization problem, whereby the state-of-the-art MIT general circulation model (MITgcm) is fit to observations. The UQ framework is applied to quantify Drake Passage transport uncertainties in a global idealized barotropic configuration of the MITgcm. Large error covariance matrices are evaluated by inverting the Hessian of the misfit function using matrix-free numerical linear algebra algorithms. The covariances are projected onto target output quantities of the model (here Drake Passage transport) by Jacobian transformations. First and second derivative codes of the MITgcm are generated by means of algorithmic differentiation (AD). Transpose of the chain rule product of Jacobians of elementary forward model operations implements a computationally efficient adjoint code. Computational complexity of the Hessian code is reduced via forward-over-reverse mode AD, which preserves the efficiency of adjoint checkpointing schemes in the second derivative calculation. A Lanczos algorithm is applied to extract the leading eigenvectors and eigenvalues of the Hessian matrix, representing the constrained uncertainty patterns and the inverse of the corresponding uncertainties. The dimensionality of the misfit Hessian inversion is reduced by omitting its nullspace (as an alternative to suppressing it by regularization), excluding from the computation the uncertainty subspace unconstrained by the observations. Inverse and forward uncertainty propagation schemes are designed for assimilating observation and control variable uncertainties and for projecting these uncertainties onto oceanographic target quantities.