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Incremental 4D-Var is a data assimilation algorithm used routinely at operational numerical weather predictions centers worldwide.This paper implements a new method for parallelizing incremental 4D-Var, the Randomized Incremental Optimal Technique (RIOT), which replaces the traditional sequential conjugate gradient (CG) iterations in the inner-loop of the minimization with fully parallel randomized singular value decomposition (RSVD) of the preconditioned Hessian of the cost function. RIOT is tested using the standard Lorenz-96 model (L-96) as well as two realistic high-dimensional atmospheric source inversion problems based on aircraft observations of black carbon concentrations. A new outer-loop preconditioning technique tailored to RSVD was introduced to improve convergence stability and performance. Results obtained with the L-96 system show that the performance improvement from RIOT compared to standard CG algorithms increases significantly with non-linearities. Overall, in the realistic black carbon source inversion experiments, RIOT reduces the wall-time of the 4D-Var minimization by a factor 2-3, at the cost of a factor 4-10 increase in energy cost due to the large number of parallel cores used. Furthermore, RIOT enables reduction of the wall-time computation of the analysis error covariance matrix by a factor 40 compared to a standard iterative Lanczos approach. Finally, as evidenced in this study, implementation of RIOT in an operational numerical weather prediction system will require a better understanding of its convergence properties as a function of the Hessian characteristics and, in particular, the degree of freedom for signal (DOFs) of the inverse problem.

Variational methods are widely used to solve geophysical inverse problems. Although gradient-based minimization algorithms are available for high-dimensional problems (dimension >106), they do not provide an estimate of the errors in the optimal solution. In this study, we assess the performance of several numerical methods to approximate the analysis-error covariance matrix, assuming reasonably linear models. The evaluation is performed for a CO2 flux estimation problem using synthetic remote-sensing observations of CO2 columns. A low-dimensional experiment is considered in order to compare the analysis error approximations to a full-rank finite-difference inverse Hessian estimate, followed by a realistic high-dimensional application. Two stochastic approaches, a Monte-Carlo simulation and a method based on random gradients of the cost function, produced analysis error variances with a relative error 120%), a new preconditioner that efficiently accumulates information on the diagonal of the inverse Hessian dramatically improves the results (relative error