Carbon flux bias estimation employing Maximum Likelihood Ensemble Filter (MLEF)

We evaluate the capability of an ensemble based data assimilation approach, referred to as Maximum Likelihood Ensemble Filter (MLEF), to estimate biases in the CO2 photosynthesis and respiration fluxes. We employ an off-line Lagrangian Particle Dispersion Model (LPDM), which is driven by the carbon fluxes, obtained from the Simple Biosphere - Regional Atmospheric Modeling System (SiB-RAMS). The SiB-RAMS carbon fluxes are assumed to have errors in the form of multiplicative biases. Our goal is to estimate and reduce these biases and also to assign reliable posterior uncertainties to the estimated biases. Experiments of this study are performed using simulated CO2 observations, which resemble real CO2 concentrations from the Ring of Towers in northern Wisconsin. We evaluate the MLEF results with respect to the 'truth' and the Kalman Filter (KF) solution. The KF solution is considered theoretically optimal for the problem of this study, which is a linear data assimilation problem involving Gaussian errors. We also evaluate the impact of forecast error covariance localization based on a new 'distance' defined in the space of information measures. Experimental results are encouraging, indicating that the MLEF can successfully estimate carbon flux biases and their uncertainties. As expected, the estimated biases are closer to the 'true' biases in the experiments with more ensemble members and more observations. The data assimilation algorithm has a stable performance and converges smoothly to the KF solution when the ensemble size approaches the size of the model state vector (i.e., the control variable of the data assimilation problem

Estimating parameters in stochastic systems:a variational Bayesian approach

This work is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational Bayesian algorithm. The paper begins by showing that the new extension of this variational algorithm can be used for state estimation (smoothing) and converges to the original algorithm. However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new approach is validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein–Uhlenbeck process, the exact likelihood of which can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz ’63 (3D model). As a special case the algorithm is also applied to the 40 dimensional stochastic Lorenz ’96 system. In our investigation we compare this new approach with a variety of other well known methods, such as the hybrid Monte Carlo, dual unscented Kalman filter, full weak-constraint 4D-Var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases. In particular we show that we are able to estimate parameters in both the drift (deterministic) and the diffusion (stochastic) part of the model evolution equations using our new methods

An ensemble data assimilation system to estimate CO2 surface fluxes from atmospheric trace gas observations

We present a data assimilation system to estimate surface fluxes of CO2 and other trace gases from observations of their atmospheric abundances. The system is based on ensemble data assimilation methods under development for Numerical Weather Prediction (NWP) and is the first of its kind to be used for CO2 flux estimation. The system was developed to overcome computational limitations encountered when a large number of observations are used to estimate a large number of unknown surface fluxes. The ensemble data assimilation approach is attractive because it returns an approximation of the covariance, does not need an adjoint model or other linearization of the observation operator, and offers the possibility to optimize fluxes of chemically active trace gases (e.g., CH4, CO) in the same framework. We assess the performance of this new system in a pseudodata experiment that resembles the real problem we will apply this system to. The sensitivity of the method to the choice of several parameters such as the assimilation window size and the number of ensemble members is investigated. We conclude that the system is able to provide satisfactory flux estimates for the relatively large scales resolved by our current observing network and that the loss of information in the approximated covariances is an acceptable price to pay for the efficient computation of a large number of surface fluxes. The full potential of this data assimilation system will be used for near–real time operational estimates of North American CO2 fluxes. This will take advantage of the large amounts of atmospheric data that will be collected by NOAA-CMDL in conjunction with the implementation of the North American Carbon Program (NACP)

Initiation of Ensemble Data Assimilation

A specification of the initial ensemble in ensemble data is addressed. The presented work examines the impact of ensemble initiation in the Maximum Likelihood Ensemble Filter (MLEF) framework, but it is applicable to other ensemble data assimilation algorithms as well. Two new methods are considered: first, based on the use of the Kardar-Parisi-Zhang (KPZ) equation to form sparse random perturbations, followed by spatial smoothing to enforce desired correlation structure, and second, based on spatial smoothing of initially uncorrelated random perturbations. Data assimilation experiments are conducted using a global shallow-water model and simulated observations. The two proposed methods are compared to the commonly used method of uncorrelated random perturbations. The results indicate that the impact of the initial correlations in ensemble data assimilation is beneficial. The root-mean-square error rate of convergence of data assimilation is improved, and the positive impact of initial correlations is noticeable throughout the data assimilation cycles. The sensitivity to the choice of the correlation length scale exists, although it is not very high. The implied computational savings and improvement of the results may be important in future realistic applications of ensemble data assimilation