Uncertainty quantification in ocean state estimation
Citable URI
https://hdl.handle.net/1912/5818Location
Drake Passage transportDOI
10.1575/1912/5818Keyword
Adjoint model uncertainty; Sensitivity; Posterior error reduction; Reduced rank Hessian matrix; Automatic Differentiation; Ocean state estimation; Barotropic modelAbstract
Quantifying uncertainty and error bounds is a key outstanding challenge in ocean state
estimation and climate research. It is particularly difficult due to the large dimensionality
of this nonlinear estimation problem and the number of uncertain variables involved. The
“Estimating the Circulation and Climate of the Oceans” (ECCO) consortium has
developed a scalable system for dynamically consistent estimation of global timeevolving
ocean state by optimal combination of ocean general circulation model (GCM)
with diverse ocean observations. The estimation system is based on the "adjoint method"
solution of an unconstrained leastsquares optimization problem formulated with the
method of Lagrange multipliers for fitting the dynamical ocean model to observations.
The dynamical consistency requirement of ocean state estimation necessitates this
approach over sequential data assimilation and reanalysis smoothing techniques. In
addition, it is computationally advantageous because calculation and storage of large
covariance matrices is not required. However, this is also a drawback of the adjoint
method, which lacks a native formalism for error propagation and quantification of
assimilated uncertainty. The objective of this dissertation is to resolve that limitation by
developing a feasible computational methodology for uncertainty analysis in dynamically
consistent state estimation, applicable to the large dimensionality of global ocean models.
Hessian (second derivativebased) methodology is developed for Uncertainty
Quantification (UQ) in largescale ocean state estimation, extending the gradientbased
adjoint method to employ the second order geometry information of the modeldata
misfit function in a highdimensional control space. Large error covariance matrices are
evaluated by inverting the Hessian matrix with the developed scalable matrixfree
numerical linear algebra algorithms. Hessianvector product and Jacobian derivative
codes of the MIT general circulation model (MITgcm) are generated by means of
algorithmic differentiation (AD). Computational complexity of the Hessian code is
reduced by tangent linear differentiation of the adjoint code, which preserves the speedup
of adjoint checkpointing schemes in the second derivative calculation. A Lanczos
algorithm is applied for extracting the leading rank eigenvectors and eigenvalues of the
Hessian matrix. The eigenvectors represent the constrained uncertainty patterns. The
inverse eigenvalues are the corresponding uncertainties. The dimensionality of UQ
calculations is reduced by eliminating the uncertainty nullspace unconstrained by the
supplied 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. Two versions of these schemes are
developed: one evaluates reduction of prior uncertainties, while another does not require
prior assumptions. The analysis of uncertainty propagation in the ocean model is timeresolving.
It captures the dynamics of uncertainty evolution and reveals transient and
stationary uncertainty regimes.
The system is applied to quantifying uncertainties of Antarctic Circumpolar Current
(ACC) transport in a global barotropic configuration of the MITgcm. The model is
constrained by synthetic observations of sea surface height and velocities. The control
space consists of twodimensional maps of initial and boundary conditions and model
parameters. The size of the Hessian matrix is O(1010) elements, which would require
O(60GB) of uncompressed storage. It is demonstrated how the choice of observations
and their geographic coverage determines the reduction in uncertainties of the estimated
transport. The system also yields information on how well the control fields are
constrained by the observations. The effects of controls uncertainty reduction due to
decrease of diagonal covariance terms are compared to dynamical coupling of controls
through offdiagonal covariance terms. The correlations of controls introduced by
observation uncertainty assimilation are found to dominate the reduction of uncertainty of
transport. An idealized analytical model of ACC guides a detailed timeresolving
understanding of uncertainty dynamics.
Description
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2013
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