Implicit ensemble tangent linear models (IETLMs) for model differentiation

Abstract

Ideally, tangent linear models (TLMs) predict the difference between perturbed and unperturbed non‐linear forecasts of interest. The adjoint of a TLM gives the gradient of the non‐linear model and is used in 4DVar data assimilation and in adjoint‐based Forecast Sensitivity to Observation Impact (FSOI). The accuracy of the local ensemble TLM (LETLM) has been shown to be limited by its inability to account for implicit time stepping. Here we derive implicit ensemble TLMs (IETLMs) that, at most, require the number of independent ensemble members to be equal to the number of variables in the implicit computational stencil. The accuracy of the IETLM in the linear regime is confirmed using an implicitly time‐stepped Lorenz 96 model and a nine‐member ensemble. IETLMs feature two sparse matrices: matrix N that operates on an initially unknown future time perturbation, and matrix L that operates on the current time perturbation. For ensemble perturbations in the non‐linear regime, we develop a diagonally robust (DR) IETLM that reduces the chances of N becoming ill‐conditioned. The performance of the DR IETLM was compared with traditional TLM performance using IETLM ensemble perturbations whose ‘Gilmour et al., 2001’ measure of non‐linearity ranged up to the non‐linearity of operational 32‐hr ensemble forecast perturbations. Over a wide range of non‐linearity, the DR IETLM performance was found to match that of the traditional TLM provided the initial standard deviation of the ensemble perturbations was ≤0.1$$ \le 0.1 $$ times the standard deviation of the test perturbations. Ideal FSOI requires the adjoint of a TLM that accurately predicts the known difference between corrected and uncorrected non‐linear forecasts. The DR IETLM was found to meet this FSOI accuracy requirement much more closely than the traditional TLM when the ensemble perturbations were created by subtracting the corrected forecast from ensemble members that were centred on the uncorrected forecast.

Toward a strongly coupled assimilation in the Earth System Prediction Capability system

“…As an example, Goodliff and Penny (2022) explored several approaches to developing a four‐dimensional variational (4DVar) SCDA with a quasi‐geostrophic model as a constraint in the optimization problem and found certain advantages in using the ensemble information in specifying both the background error covariances and approximations to the tangent linear and adjoint models. The latter approach to 4DVar DA (Frolov et al 2016) has been rapidly developing in recent years (Allen et al 2020; Yaremchuk et al 2020; Frolov et al 2021; Payne 2021; Bishop and Eizenberg 2022), indicating potential importance of the ensemble information within the hybrid assimilation systems in SCDA applications.…”