Behavior of the iterative ensemble-based variational method in nonlinear problems

Abstract

The behavior of the iterative ensemble-based data assimilation algorithm is discussed. The ensemble-based method for variational data assimilation problems, referred to as the 4-dimensional ensemble variational method (4DEnVar), is a useful tool for data assimilation problems. Although the 4DEnVar is derived based on a linear approximation, highly uncertain problems, where system nonlinearity is significant, are solved by applying this method iteratively. However, it is not necessarily trivial how the algorithm works in highly uncertain problems where nonlinearity is not negligible. In the present study, an 5 ensemble-based iterative algorithm is reformulated to allow us to analyze its behavior in nonlinear problems. The conditions for monotonic convergence to a local maximum of the objective function are discussed in nonlinear context. The findings as the results of the present study were also experimentally supported. https://doi.org/10.5194/npg-2020-9 Preprint. Discussion started: 17 April 2020 c Author(s) 2020. CC BY 4.0 License.Several studies have suggested that estimation in nonlinear problems can be improved by iterative algorithms in which 25 the ensemble is repeatedly updated in each iteration (e.g., Gu and Oliver, 2007;Kalnay and Yang, 2010; Chen and Oliver, 2012; Sakov, 2013, 2014;Raanes et al., 2019). These iterative algorithms can be regarded as a variant of the 4DEnVar method based on an approximation of the Gauss-Newton method or the Levenberg-Marquardt method. The Gauss-Newton and the Levenberg-Marquardt methods are variants of the Newton-Raphson method for solving nonlinear least squares problems by using the Jacobian of a nonlinear function. Thus, when the Gauss-Newton or the Levenberg-Marquardt framework 30 is strictly applied to data assimilation problems, the tangent linear of the system model is required. Indeed, if the tangent linear of the system model is obtained, 4-dimensional variational data assimilation problems can be solved with the incremental formulation (Courtier et al., 1994) which can be regarded as an instance of the Gauss-Newton framework (Lawless et al., 2005). The ensemble-based methods avoid computing the Jacobian of a nonlinear system model by a low-rank linear approximation using the ensemble. This ensemble-based approximation is justified if linearity can be assumed over the range where the 35 ensemble members are distributed. However, it has not necessarily been clarified how the algorithm works in highly uncertain problems where nonlinearity is not negligible.The present study aims to reformulate an ensemble-based iterative algorithm in order to analyze its behavior in nonlinear problems. We then explore the conditions for achieving monotonic convergence to a local maximum of the objective function in nonlinear context. The monotonic convergence means that the discrepancies between estimates and observations are reduced 40 in each iteration. It is thus ensured that the algorithm would attain a satisfactory result in nonlinear problems. This study is originally moti...

A candidate secular variation model for IGRF-13 based on MHD dynamo simulation and 4DEnVar data assimilation

“…(8), which is similar to the iterative ensemble Kalman smoother algorithm (Gu and Oliver 2007;Bocquet and Sakov 2013), minimizes Eq. (2) in the subspace spanned by the ensemble members (Nakano 2020). After obtaining x 0,m+1 it is necessary to perform MHD dynamo simulations with a set of initial conditions x…”

“…This strategy seems reasonable when both assimilation window and forecast period are shorter than the timescale of nonlinearity of the system. We illustrate that this approach is feasible with our iterative algorithm based on 4DEnVar (Nakano 2020) for generation of a candidate SV model for IGRF, with numerical experiments using the real geomagnetic data from 2004.50 to 2014.25 and with comparison of the forecast performances with that of The remainder of this paper is organized as follows: we first explain our assimilation method ("Method" section). Secondly, we report results of numerical experiments using the past datasets ("Numerical experiments" section), and then describe details of estimation of our candidate model for IGRF-13 SV ("The SV candidate model for IGRF-13" section).…”