Maximum Likelihood Ensemble Filter Research Articles

The maximum likelihood ensemble smoother for the Kuramoto–Sivashinsky equation

AbstractData assimilation (DA) aims to combine observations/data with a model to maximize the utility of information for obtaining the optimal estimate. The maximum likelihood ensemble filter (MLEF) is a sequential DA method or a filter-type method. Weaknesses of the filter method are assimilating time-integrated observations and estimating empirical parameter estimation. The reason is that the forward model is employed outside of the analysis procedure in this type of DA method. To overcome these weaknesses, the MLEF is now extended as a smoother and the novel maximum likelihood ensemble smoother (MLES) is proposed. The MLES is a smoothing method with variational-like qualities, specifically in the cost function. Rather than using the error information from a single temporal location to solve for the optimal analysis update as done by the MLEF, the MLES can include observations and the forward model within a chosen time window. The newly proposed DA method is first validated by a series of rigorous and thorough performance tests using the Lorenz 96 model. Then, as DA is known to be used extensively to increase the predictability of the commonly chaotic dynamical systems seen in meteorological applications, this study demonstrates the MLES with a model chaotic problem governed by the 1D Kuramoto–Sivashinky (KS) equation. Additionally, the MLES is shown to be an effective method in improving the estimate of uncertain empirical model parameters. The MLES and MLEF are then directly compared and it is shown that the performance of the MLES is adequate and that it is a good candidate for increasing the predictability of a chaotic dynamical system. Future work will focus on an extensive application of the MLES to highly turbulent flows.

Performance assessment of the maximum likelihood ensemble filter and the ensemble Kalman filters for nonlinear problems

Performance assessment of the maximum likelihood ensemble filter and the ensemble Kalman filters for nonlinear problems

The Maximum Likelihood Ensemble Filter with State Space Localization

AbstractA new method for ensemble data assimilation that incorporates state space covariance localization, global numerical optimization, and implied Bayesian inference is presented. The method is referred to as the MLEF with state space localization (MLEF-SSL) due to its similarity with the maximum likelihood ensemble filter (MLEF). One of the novelties introduced in MLEF-SSL is the calculation of a reduced-rank localized forecast error covariance using random projection. The Hessian preconditioning is accomplished via Cholesky decomposition of the Hessian matrix, accompanied with solving triangular system of equations instead of directly inverting matrices. For ensemble update, the MLEF-SSL system employs resampling of posterior perturbations. The MLEF-SSL was applied to Lorenz model II and compared to ensemble Kalman filter with state space localization and to MLEF with observation space localization. The observations include linear and nonlinear observation operators, each applied to integrated and point observations. Results indicate improved performance of MLEF-SSL, particularly in assimilation of integrated nonlinear observations. Resampling of posterior perturbations for an ensemble update also indicates a satisfactory performance. Additional experiments were conducted to examine the sensitivity of the method to the rank of random matrix and to compare it to truncated eigenvectors of the localization matrix. The two methods are comparable in application to low-dimensional Lorenz model, except that the new method outperforms the truncated eigenvector method in case of severe rank reduction. The random basis method is simple to implement and may be more promising for realistic high-dimensional applications.

The maximum likelihood ensemble filter for computational flame and fluid dynamics

Abstract The numerical solution of partial differential equations that govern fluid dynamics with turbulence and combustion is challenging due to the multiscale nature of the dynamical system and the need to resolve small-scale physical features. In addition, the uncertainties in the dynamical system, including those in the physical models and parameters, initial and boundary conditions and numerical methods, impact the computational fluid dynamics (CFD) prediction of turbulence and chemical reactions. To improve the CFD prediction, this study focuses on the development and application of a maximum likelihood ensemble filter (MLEF), an ensemble-based data assimilation (DA), for flows featuring combustion and/or turbulence. MLEF finds the optimal analysis and its uncertainty by maximizing the posterior probability density function. The novelty of the study lies in the combination of advanced DA and CFD methods for a new comprehensive application to predict engineering fluid dynamics. The study combines important aspects, including an ensemble-based DA with analysis and uncertainty estimation, an augmented control vector that simultaneously adjusts initial conditions and model empirical parameters and an application of DA to CFD modeling of combustion and flows with complex geometry. The DA performance is validated by a turbulent Couette flow. The new CFD–DA system is then applied to solve the time-evolving shear-layer mixing with methane-air combustion and the turbulent flow over a bluff-body geometry. Results demonstrate the improvement of estimates of model parameters and the uncertainty reduction in initial conditions (ICs) for CFD modeling of flames and flows by the MLEF method.

Non-linear data assimilation via trust region optimization

In this paper, we propose an efficient and practical implementation of the ensemble Kalman filter based on trust region method for non-Gaussian data assimilation. The proposed method works as follows: the empirical moments of an ensemble of model realizations are employed to estimate the parameters of the background error distribution, then an iterative method is proposed to, among iterations, build quadratic surrogate models of the three-dimensional variational (3D-Var) cost function, these models are utilized to approximate solutions of the 3D-Var optimization problem, and the quality of solutions are assessed using the trust region framework. Furthermore, the global convergence of the proposed trust region implementation is theoretically proven. Experimental tests are carried out making use the Lorenz 96 model. Even more, two different observation operators are considered during the experiments: a linear observation operator, and a non-convex and discontinuous one. Different ensemble sizes as well as number of observations (and their time frequencies) are employed to enrich the numerical results. The Maximum Likelihood Ensemble Filter (MLEF) is utilized as a reference filter to estimate the performance of our proposed filter implementation. Ten experiments (repetitions) are performed for each configuration of parameters. The results reveal that, in average, the proposed filter outperforms the MLEF in terms of root mean square errors and even more, it converges to posterior modes of error distributions in cases wherein filter divergence can be possible in the MLEF context.

Uncertainty in solid precipitation and snow depth prediction for Siberia using the Noah and Noah-MP land surface models

In this study, we investigate the uncertainties associated with land surface processes in an ensemble predication context. Specifically, we compare the uncertainties produced by a coupled atmosphere–land modeling system with two different land surface models, the Noah- MP land surface model (LSM) and the Noah LSM, by using the Maximum Likelihood Ensemble Filter (MLEF) data assimilation system as a platform for ensemble prediction. We carried out 24-hour prediction simulations in Siberia with 32 ensemble members beginning at 00:00 UTC on 5 March 2013. We then compared the model prediction uncertainty of snow depth and solid precipitation with observation-based research products and evaluated the standard deviation of the ensemble spread. The prediction skill and ensemble spread exhibited high positive correlation for both LSMs, indicating a realistic uncertainty estimation. The inclusion of a multiple snowlayer model in the Noah-MP LSM was beneficial for reducing the uncertainties of snow depth and snow depth change compared to the Noah LSM, but the uncertainty in daily solid precipitation showed minimal difference between the two LSMs. The impact of LSM choice in reducing temperature uncertainty was limited to surface layers of the atmosphere. In summary, we found that the more sophisticated Noah-MP LSM reduces uncertainties associated with land surface processes compared to the Noah LSM. Thus, using prediction models with improved skill implies improved predictability and greater certainty of prediction.

Maximum Likelihood Ensemble Filter State Estimation for Power Systems

Maximum likelihood ensemble filter (MLEF) is an ensemble-based deterministic filtering method. It optimizes a nonlinear cost function through maximum likelihood and utilizes low-dimensional ensemble space on the calculation of Hessian preconditioning of the cost function. This paper implements the MLEF as a state estimation tool for the estimation of the states of a power system, and presents the first MLEF application study on a power system state estimation. The MLEF methodology is introduced into power systems and the simulations are implemented for a three-node benchmark power system and 68-bus test system which have been employed in several previous studies to address a discontinuous problem where derivative is not defined. This is in contrast to gradient-based methods in the literature that needs gradient and Hessian information which is not defined in jumps. The performance of the filter on the presented problem is analyzed and the results are presented. Results indicate that the estimation convergence is achieved with the MLEF method.

Impact of the OMI aerosol optical depth on analysis increments through coupled meteorology–aerosol data assimilation for an Asian dust storm

In this study, we investigate the possibility of using a coupled meteorology–aerosol data assimilation (DA) to improve the forecast of a dust storm. We interfaced a coupled meteorology–chemistry model (WRF-Chem) with a hybrid variational-ensemble DA system — the Maximum Likelihood Ensemble Filter (MLEF). The system is strongly coupled, in the sense that cross-covariances between meteorological and aerosol control variables are used in DA. Experiments are conducted for an Asian dust storm case that occurred over the Korean Peninsula on 13 May 2011, by assimilating both meteorological and aerosol observations. Meteorological observations include the conventional ones and the Microwave Humidity Sounder (MHS) satellite radiances, whereas aerosol observations contain aerosol optical depth (AOD) at 500nm from the Ozone Monitoring Instrument (OMI). In the control experiment (CTNL), we assimilated only meteorological observations. In the AODDA experiment, we assimilated both AOD and meteorological observations. Control variables of DA include both meteorological and aerosol ones. Our results indicate that meteorological observations can partially contribute to the change of aerosol initial conditions and vice versa, implying a cross-component impact of observations, and hence improving the utility of observations in coupled DA. The forecast comparison indicates a consistently improved fit to AOD observations after 24 h of DA. In addition, uncertainty of the 24-hour aerosol forecast in AODDA decreases significantly in local regions — by up to ∼21.6% in terms of innovations, and up to ∼81.4% in terms of square root forecast error covariance. These results are encouraging for future applications of ensemble DA with strongly coupled meteorology–chemistry modeling system.

A case study involving single observation experiments performed over snowy Siberia using a coupled atmosphere‐land modelling system

AbstractThrough a series of single observation experiments, we investigate forecast error covariance and correlation structures by applying the Maximum Likelihood Ensemble Filter (MLEF) data assimilation method with a coupled atmosphere–land surface model. We consider a coupled approach that includes the full complexity of the cross‐component covariance and correlation structures, specifically between 2‐m air temperature and snow temperature. We show that the error covariance and correlation are complex and are strongly dependent on both weather conditions and the land surface scheme, indicating a need for further investigation of the relevance of flow‐dependent ensemble covariance in coupled systems. We also demonstrate that the use of coupled error covariance methods improves the efficiency of information transfer between the atmosphere and the land surface by allowing the well‐observed atmosphere to influence land surface variables.

Ensemble data assimilation of total column ozone using a coupled meteorology–chemistry model and its impact on the structure of Typhoon Nabi (2005)

Abstract. Ozone (O3) plays an important role in chemical reactions and is usually incorporated in chemical data assimilation (DA). In tropical cyclones (TCs), O3 usually shows a lower concentration inside the eyewall and an elevated concentration around the eye, impacting meteorological as well as chemical variables. To identify the impact of O3 observations on TC structure, including meteorological and chemical information, we developed a coupled meteorology–chemistry DA system by employing the Weather Research and Forecasting model coupled with Chemistry (WRF-Chem) and an ensemble-based DA algorithm – the maximum likelihood ensemble filter (MLEF). For a TC case that occurred over East Asia, Typhoon Nabi (2005), our results indicate that the ensemble forecast is reasonable, accompanied with larger background state uncertainty over the TC, and also over eastern China. Similarly, the assimilation of O3 observations impacts meteorological and chemical variables near the TC and over eastern China. The strongest impact on air quality in the lower troposphere was over China, likely due to the pollution advection. In the vicinity of the TC, however, the strongest impact on chemical variables adjustment was at higher levels. The impact on meteorological variables was similar in both over China and near the TC. The analysis results are verified using several measures that include the cost function, root mean square (RMS) error with respect to observations, and degrees of freedom for signal (DFS). All measures indicate a positive impact of DA on the analysis – the cost function and RMS error have decreased by 16.9 and 8.87 %, respectively. In particular, the DFS indicates a strong positive impact of observations in the TC area, with a weaker maximum over northeastern China.

Improving water quality forecasting via data assimilation – Application of maximum likelihood ensemble filter to HSPF

Summary An ensemble data assimilation (DA) procedure is developed and evaluated for the Hydrologic Simulation Program – Fortran (HSPF), a widely used watershed water quality model. The procedure aims at improving the accuracy of short-range water quality prediction by updating the model initial conditions (IC) based on real-time observations of hydrologic and water quality variables. The observations assimilated include streamflow, biochemical oxygen demand (BOD), dissolved oxygen (DO), chlorophyll a (CHL-a), nitrate (NO 3 ), phosphate (PO 4 ) and water temperature (TW). The DA procedure uses the maximum likelihood ensemble filter (MLEF), which is capable of handling both nonlinear model dynamics and nonlinear observation equations, in a fixed-lag smoother formulation. For evaluation, the DA procedure was applied to the Kumho Catchment of the Nakdong River Basin in the Republic of Korea. A set of performance measures was used to evaluate analysis and prediction of streamflow and water quality variables. To remove systematic errors in the model simulation originating from structural and parametric errors, a parsimonious bias correction procedure is incorporated into the observation equation. The results show that the DA procedure substantially improves predictive skill for most variables; reduction in root mean square error ranges from 11% to 60% for Day-1 through 3 predictions for all observed variables except DO. It is seen that MLEF handles highly nonlinear hydrologic and biochemical observation equations very well, and that it is an effective DA technique for water quality forecasting.

Comparative evaluation of maximum likelihood ensemble filter and ensemble Kalman filter for real-time assimilation of streamflow data into operational hydrologic models

Various data assimilation (DA) methods have been used and are being explored for use in operational streamflow forecasting. For ensemble forecasting, ensemble Kalman filter (EnKF) is an appealing candidate for familiarity and relative simplicity. EnKF, however, is optimal in the second-order sense, only if the observation equation is linear. As such, without an iterative approach, EnKF may not be appropriate for assimilating streamflow data for updating soil moisture states due to the strong nonlinear relationships between the two. Maximum likelihood ensemble filter (MLEF), on the other hand, is not subject to the above limitation. Being an ensemble extension of variational assimilation (VAR), MLEF also offers a strong connection with the traditional single-valued forecast process through the control, or the maximum likelihood, solution. In this work, we apply MLEF and EnKF as a fixed lag smoother to the Sacramento (SAC) soil moisture accounting model and unit hydrograph (UH) for assimilation of streamflow, mean areal precipitation (MAP) and potential evaporation (MAPE) data for updating soil moisture states. For comparative evaluation, three experiments were carried out. Comparison between homoscedastic vs. heteroscedastic modeling of selected statistical parameters for DA indicates that heteroscedastic modeling does not improve over homoscedastic modeling, and that homoscedastic error modeling with sensitivity analysis may suffice for application of MLEF for soil moisture updating using streamflow data. Comparative evaluation with respect to the model errors associated with soil moisture dynamics, the ensemble size and the number of streamflow observations assimilated per cycle showed that, in general, MLEF outperformed EnKF under varying conditions of observation and model errors, and ensemble size, and that MLEF performed well with an ensemble size as small as 5 while EnKF required a much larger ensemble size to perform closely to MLEF. Also, MLEF was not very sensitive to the uncertainty parameters and performed reasonably well over relatively wide ranges of parameter settings, an attribute desirable for operational applications where accurate estimation of such parameters is often difficult.

Joint estimation of soil moisture profile and hydraulic parameters by ground‐penetrating radar data assimilation with maximum likelihood ensemble filter

AbstractGround‐Penetrating Radar (GPR) has recently become a powerful geophysical technique to characterize soil moisture at the field scale. We developed a data assimilation scheme to simultaneously estimate the vertical soil moisture profile and hydraulic parameters from time‐lapse GPR measurements. The assimilation scheme includes a soil hydrodynamic model to simulate the soil moisture dynamics, a full‐wave electromagnetic wave propagation model, and petrophysical relationship to link the state variable with the GPR data and a maximum likelihood ensemble assimilation algorithm. The hydraulic parameters are estimated jointly with the soil moisture using a state augmentation technique. The approach allows for the direct assimilation of GPR data, thus maximizing the use of the information. The proposed approach was validated by numerical experiments assuming wrong initial conditions and hydraulic parameters. The synthetic soil moisture profiles were generated by the Hydrus‐1D model, which then were used by the electromagnetic model and petrophysical relationship to create “observed” GPR data. The results show that the data assimilation significantly improves the accuracy of the hydrodynamic model prediction. Compared with the surface soil moisture assimilation, the GPR data assimilation better estimates the soil moisture profile and hydraulic parameters. The results also show that the estimated soil moisture profile in the loamy sand and silt soils converge to the “true” state more rapidly than in the clay one. Of the three unknown parameters of the Mualem‐van Genuchten model, the estimation of n is more accurate than that of α and Ks. The approach shows a great promise to use GPR measurements for the soil moisture profile and hydraulic parameter estimation at the field scale.

Assimilating AMSU-A Radiances in the TC Core Area with NOAA Operational HWRF (2011) and a Hybrid Data Assimilation System: Danielle (2010)

Abstract A regional hybrid variational–ensemble data assimilation system (HVEDAS), the maximum likelihood ensemble filter (MLEF), is applied to the 2011 version of the NOAA operational Hurricane Weather Research and Forecasting (HWRF) model to evaluate the impact of direct assimilation of cloud-affected Advanced Microwave Sounding Unit-A (AMSU-A) radiances in tropical cyclone (TC) core areas. The forward components of both the gridpoint statistical interpolation (GSI) analysis system and the Community Radiative Transfer Model (CRTM) are utilized to process and simulate satellite radiances. The central strategies to allow the use of cloud-affected radiances are (i) to augment the control variables to include clouds and (ii) to add the model cloud representations in the observation forward models to simulate the microwave radiances. The cloudy AMSU-A radiance assimilation in Hurricane Danielle's (2010) core area has produced encouraging results with respect to the operational cloud-cleared radiance preprocessing procedures used in this study. Through the use of the HVEDAS, ensemble covariance statistics for a pseudo-AMSU-A observation in Danielle's core area show physically meaningful error covariances and statistical couplings with hydrometeor variables (i.e., the total-column condensate in Ferrier microphysics). The cloudy radiance assimilation in the TC core region (i.e., ASR experiment) consistently reduced the root-mean-square errors of the background departures, and also generally improved the forecasts of Danielle's intensity as well as the quantitative cloud analysis and prediction. It is also indicated that an entropy-based information content quantification process provides a useful metric for evaluating the utility of satellite observations in hybrid data assimilation.

Improving soil moisture profile reconstruction from ground-penetrating radar data: a maximum likelihood ensemble filter approach

Abstract. The vertical profile of shallow unsaturated zone soil moisture plays a key role in many hydro-meteorological and agricultural applications. We propose a closed-loop data assimilation procedure based on the maximum likelihood ensemble filter algorithm to update the vertical soil moisture profile from time-lapse ground-penetrating radar (GPR) data. A hydrodynamic model is used to propagate the system state in time and a radar electromagnetic model and petrophysical relationships to link the state variable with the observation data, which enables us to directly assimilate the GPR data. Instead of using the surface soil moisture only, the approach allows to use the information of the whole soil moisture profile for the assimilation. We validated our approach through a synthetic study. We constructed a synthetic soil column with a depth of 80 cm and analyzed the effects of the soil type on the data assimilation by considering 3 soil types, namely, loamy sand, silt and clay. The assimilation of GPR data was performed to solve the problem of unknown initial conditions. The numerical soil moisture profiles generated by the Hydrus-1D model were used by the GPR model to produce the "observed" GPR data. The results show that the soil moisture profile obtained by assimilating the GPR data is much better than that of an open-loop forecast. Compared to the loamy sand and silt, the updated soil moisture profile of the clay soil converges to the true state much more slowly. Decreasing the update interval from 60 down to 10 h only slightly improves the effectiveness of the GPR data assimilation for the loamy sand but significantly for the clay soil. The proposed approach appears to be promising to improve real-time prediction of the soil moisture profiles as well as to provide effective estimates of the unsaturated hydraulic properties at the field scale from time-lapse GPR measurements.

Impact of non‐smooth observation operators on variational and sequential data assimilation for a limited‐area shallow‐water equation model

AbstractWe investigate the issue of variational and sequential data assimilation with nonlinear and non‐smooth observation operators using a two‐dimensional limited‐area shallow‐water equation model and its adjoint. The performance of the four‐dimensional variational approach (4D‐Var: two dimensions plus time) compared with that of the maximum‐likelihood ensemble filter (MLEF), a hybrid ensemble/variational method, is tested in the presence of non‐smooth observation operators.Following the work of Lewis & Overton and Karmitsa, we investigate minimization of the data‐assimilation cost functional using the limited‐memory Broyden–Fletcher–Goldfarb–Shanno (L‐BFGS) quasi‐Newton algorithm originally intended for smooth optimization and the limited‐memory bundle method (LMBM) algorithm specifically designed to address large‐scale non‐smooth minimization problems.Numerical results obtained for the MLEF method show that the LMBM algorithm yields results superior to the L‐BFGS method. Results for 4D‐Var suggest that L‐BFGS performs well when the non‐smoothness is not extreme, but fails for non‐smooth functions with large Lipschitz constants. The LMBM method is found to be a suitable choice for large‐scale non‐smooth optimization, although additional work is needed to improve its numerical stability. Finally, the results and methodologies of 4D‐Var and MLEF are compared and contrasted. Copyright © 2011 Royal Meteorological Society

Assimilating synthetic GOES-R radiances in cloudy conditions using an ensemble-based method

The weather research and forecasting (WRF) model and the maximum likelihood ensemble filter (MLEF) data assimilation approach are used to examine the potential impact of observations from the future Geostationary Operational Environmental Satellite, generation R (GOES-R) on improving our knowledge about clouds. Synthetic radiances are assimilated from the 10.35 μm channel of the GOES-R advanced baseline imager (ABI) employing a ‘non-identical twins’ experimental setup. The experimental results are examined for an extratropical cyclone named Kyrill that produced unusually strong winds, widespread damage and fatalities in Western Europe in January 2007. The data assimilation problem is especially challenging for this case, as there is a large error in the model-simulated radiances resulting from incorrect cloud location. Although this problem is difficult to eliminate, data assimilation results indicate the potential of GOES-R data to significantly reduce these errors.

A Prototype WRF-Based Ensemble Data Assimilation System for Dynamically Downscaling Satellite Precipitation Observations

Abstract In the near future, the Global Precipitation Measurement (GPM) mission will provide precipitation observations with unprecedented accuracy and spatial/temporal coverage of the globe. For hydrological applications, the satellite observations need to be downscaled to the required finer-resolution precipitation fields. This paper explores a dynamic downscaling method using ensemble data assimilation techniques and cloud-resolving models. A prototype ensemble data assimilation system using the Weather Research and Forecasting Model (WRF) has been developed. A high-resolution regional WRF with multiple nesting grids is used to provide the first-guess and ensemble forecasts. An ensemble assimilation algorithm based on the maximum likelihood ensemble filter (MLEF) is used to perform the analysis. The forward observation operators from NOAA–NCEP’s gridpoint statistical interpolation (GSI) are incorporated for using NOAA–NCEP operational datastream, including conventional data and clear-sky satellite observations. Precipitation observation operators are developed with a combination of the cloud-resolving physics from NASA Goddard cumulus ensemble (GCE) model and the radiance transfer schemes from NASA Satellite Data Simulation Unit (SDSU). The prototype of the system is used as a test bed to optimally combine observations and model information to produce a dynamically downscaled precipitation analysis. A case study on Tropical Storm Erin (2007) is presented to investigate the ability of the prototype of the WRF Ensemble Data Assimilation System (WRF-EDAS) to ingest information from in situ and satellite observations including precipitation-affected radiance. The results show that the analyses and forecasts produced by the WRF-EDAS system are comparable to or better than those obtained with the WRF-GSI analysis scheme using the same set of observations. An experiment was also performed to examine how the analyses and short-term forecasts of microphysical variables and dynamical fields are influenced by the assimilation of precipitation-affected radiances. The results highlight critical issues to be addressed in the next stage of development such as model-predicted hydrometeor control variables and associated background error covariance, bias estimation, and correction in radiance space, as well as the observation error statistics. While further work is needed to optimize the performance of WRF-EDAS, this study establishes the viability of developing a cloud-scale ensemble data assimilation system that has the potential to provide a useful vehicle for downscaling satellite precipitation information to finer scales suitable for hydrological applications.

Uncertainty analysis using the WRF maximum likelihood ensemble filter system and comparison with dropwindsonde observations in Typhoon Sinlaku (2008)

In this study, the maximum likelihood ensemble filter (MLEF) is applied to a tropical cyclone case to identify the uncertainty areas in the context of targeting observations, using the WRF model. Typhoon Sinlaku (2008), from which dropwindsonde data are collected through THORPEX Pacific Asian Regional Campaign (TPARC), is selected for the case study. For the uncertainty analysis, a measurement called the deep layer mean (DLM) wind variance is employed. With assimilation of conventional rawinsonde data, the MLEF-WRF system demonstrated stable data assimilation performance over multiple data assimilation cycles and produced high uncertainties mostly in data-void areas, for the given tropical cyclone case. Dropwindsonde deployment through T-PARC turned out to occur inside or near the weak uncertainty areas that are identified through the MLEF-WRF system. The uncertainty analysis using the MLEF method can provide a guide for identifying more effective targeting observation areas.

The maximum likelihood ensemble filter performances in chaotic systems

The performance of the maximum likelihood ensemble filter (MLEF), is investigated in the context of generic systems featuring the essential ingredients of unstable dynamics and on a spatially extended system displaying chaos. The main objective is to clarify the response of the filter to different regimes of motion and highlighting features which may help its optimization in more realistic applications. It is found that, in view of the minimization procedure involved in the filter analysis update, the algorithm provides accurate estimates even in the presence of prominent non-linearities. Most importantly, the filter ensemble size can be designed in connection to the properties of the system attractor (Kaplan–Yorke dimension), thus facilitating the filter setup and limiting the computational cost by using an optimal ensemble. As a corollary, this latter finding indicates that the ensemble perturbations in the MLEF reflect the intrinsic system error dynamics rather than a sampling of realizations of an unknown error covariance.