The Local Ensemble Tangent Linear Model: an enabler for coupled model 4D‐Var

Abstract

A leading Data Assimilation (DA) technique in meteorology is 4D-Var which relies on the Tangent Linear Model (TLM) of the nonlinear model and its adjoint. The difficulty of building and maintaining traditional TLMs and adjoints of coupled ocean-wave-atmosphere-etc. models is daunting. On the other hand, coupled model ensemble forecasts are readily available. Here, we show how an ensemble forecast can be used to construct an accurate Local Ensemble TLM (LETLM) and adjoint of the entire coupled system. The method features a local influence region containing all the variables that could possibly influence the time evolution of some target variable(s) near the centre of the region. We prove that high accuracy is guaranteed provided that (i) the ensemble perturbations are governed by linear dynamics, and (ii) the number of ensemble members exceeds the number of variables in the influence region. The approach is illustrated in a simple coupled model. This idealized coupled model has some realistic features including reasonable predictability limits in the upper atmosphere, lower atmosphere, upper ocean and lower ocean of 10, 96, 160 and 335 days, respectively. In addition, the length-scale of eddies in the ocean is about one fifth of those in the atmosphere. The easy manner in which the adjoint is obtained from the LETLM is also described and illustrated by demonstrating how the LETLM adjoint predicts the high sensitivity of oceanic boundary-layer evolution to changes in the atmosphere. Finally, the feasibility of LETLMs for 4D-Var is demonstrated. Specifically, a case is considered with a 5-day data assimilation window in which nonlinear terms play a significant role in the evolution of forecast error; it is shown that the posterior mode delivered by 4D-Var with an LETLM, its adjoint and ten outer loops approximately recovers the true state in spite of a spatially sparse observational network.

Mathematical foundations of hybrid data assimilation from a synchronization perspective

“…For the application to large-scale systems, candidate methods for this investigation are the breeding on the data assimilation system (BDAS) of Trevisan and Uboldi (2004) and Carrassi et al (2007) and the use of the Local Ensemble Tangent Linear Model (LETLM) introduced by Bishop et al (2017) and Frolov and Bishop (2016) to evaluate the linearized dynamics. More recent investigations of time-delay methods (Pazo et al, 2016 andAn et al, 2017) may also provide useful supplements to the current state-of-the-art data assimilation methodologies, particularly for sparsely observed systems.…”

On the ensemble‐based linearization of numerical models

“…In recent years, several attempts have been made to extract linearized representation of the model operator from the information contained in the ensemble of model trajectories (Frolov and Bishop, ; Allen et al ., ; Bishop et al ., ; Frolov et al ., ). The approach is different from the ones mentioned above because it does not involve projection of the gradient on the range of the (localized) background‐error covariance or on other subspaces predetermined by the model run, but attempts to directly employ the ensemble statistics for reconstructing (an approximation to) the TLM matrix.…”

“…The approach is different from the ones mentioned above because it does not involve projection of the gradient on the range of the (localized) background‐error covariance or on other subspaces predetermined by the model run, but attempts to directly employ the ensemble statistics for reconstructing (an approximation to) the TLM matrix. The underlying assumption of the approach is that the number of ensemble members is comparable to the grid‐point stencil size of the matrix representation of the linearized model operator (Bishop et al ., ). Although numerical experiments with a shallow‐water model performed by Allen et al .…”

Challenges of Increased Resolution for the Local Ensemble Tangent Linear Model

“…The latter proved that when the time evolution of each model variable over a single LETLM time step Dt depends only on variables within the local influence volume, then the LETLM is guaranteed to be accurate when the ensemble size exceeds the size of the ''computational stencil,'' which we define here as the number of grid points contained within the local influence volume multiplied by the number of variables used for the LETLM. Bishop et al (2017) also demonstrated how the LETLM and its adjoint can be applied to 4DVar in strongly nonlinear regimes where multiple outer loops are required to achieve convergence. Allen et al (2017) then demonstrated an accurate LETLM-based hybrid 4DVar system using a global shallow-water model.…”

“…One alternative approach to conventional TLMs is the local ensemble tangent linear model (LETLM), in which the forecast is determined from nonlinear ensemble forecasts within a ''local influence volume,'' which we define as a specified geometric shape surrounding a grid point. This statistical approach has been tested in a hierarchy of cases, starting with simple models in Frolov and Bishop (2016) and Bishop et al (2017). The latter proved that when the time evolution of each model variable over a single LETLM time step Dt depends only on variables within the local influence volume, then the LETLM is guaranteed to be accurate when the ensemble size exceeds the size of the ''computational stencil,'' which we define here as the number of grid points contained within the local influence volume multiplied by the number of variables used for the LETLM.…”