scholarly journals A Maximum Likelihood Ensemble Filter via a Modified Cholesky Decomposition for Non-Gaussian Data Assimilation

Sensors ◽  
2020 ◽  
Vol 20 (3) ◽  
pp. 877 ◽  
Author(s):  
Elias David Nino-Ruiz ◽  
Alfonso Mancilla-Herrera ◽  
Santiago Lopez-Restrepo ◽  
Olga Quintero-Montoya
Keyword(s):  
Maximum Likelihood ◽  
General Circulation ◽  
Circulation Model ◽  
Cholesky Decomposition ◽  
Practical Implementation ◽  
Square Root ◽  
Background Error ◽  
Modified Cholesky Decomposition ◽  
Highly Nonlinear ◽  
Maximum Likelihood Ensemble Filter

This paper proposes an efficient and practical implementation of the Maximum Likelihood Ensemble Filter via a Modified Cholesky decomposition (MLEF-MC). The method works as follows: via an ensemble of model realizations, a well-conditioned and full-rank square-root approximation of the background error covariance matrix is obtained. This square-root approximation serves as a control space onto which analysis increments can be computed. These are calculated via Line-Search (LS) optimization. We theoretically prove the convergence of the MLEF-MC. Experimental simulations were performed using an Atmospheric General Circulation Model (AT-GCM) and a highly nonlinear observation operator. The results reveal that the proposed method can obtain posterior error estimates within reasonable accuracies in terms of ℓ − 2 error norms. Furthermore, our analysis estimates are similar to those of the MLEF with large ensemble sizes and full observational networks.

scholarly journals A Compensatory Approach of the Fixed Localization in EnKF

2014 ◽  
Vol 142 (10) ◽  
pp. 3713-3733 ◽  
Author(s):  
Xinrong Wu ◽  
Wei Li ◽  
Guijun Han ◽  
Shaoqing Zhang ◽  
Xidong Wang
Keyword(s):  
General Circulation ◽  
Computational Cost ◽  
Circulation Model ◽  
Multiple Scale ◽  
Long Distance ◽  
Background Error ◽  
Weather Forecasts ◽  
Error Covariance ◽  
Analysis Technique ◽  
Cutoff Distance

Abstract While fixed covariance localization can greatly increase the reliability of the background error covariance in filtering by suppressing the long-distance spurious correlations evaluated by a finite ensemble, it may degrade the assimilation quality in an ensemble Kalman filter (EnKF) as a result of restricted longwave information. Tuning an optimal cutoff distance is usually very expensive and time consuming, especially for a general circulation model (GCM). Here the authors present an approach to compensate the demerit in fixed localization. At each analysis step, after the standard EnKF is done, a multiple-scale analysis technique is used to extract longwave information from the observational residual (referred to the EnKF ensemble mean). Within a biased twin-experiment framework consisting of a global barotropical spectral model and an idealized observing system, the performance of the new method is examined. Compared to a standard EnKF, the hybrid method is superior when an overly small/large cutoff distance is used, and it has less dependence on cutoff distance. The new scheme is also able to improve short-term weather forecasts, especially when an overly large cutoff distance is used. Sensitivity studies show that caution should be taken when the new scheme is applied to a dense observing system with an overly small cutoff distance in filtering. In addition, the new scheme has a nearly equivalent computational cost to the standard EnKF; thus, it is particularly suitable for GCM applications.

scholarly journals Using Temperature–Salinity Relations in a Global Ocean Implementation of a Multivariate Data Assimilation Scheme

2007 ◽  
Vol 135 (11) ◽  
pp. 3785-3807 ◽  
Author(s):  
A. Bellucci ◽  
S. Masina ◽  
P. DiPietro ◽  
A. Navarra
Keyword(s):  
Data Assimilation ◽  
General Circulation ◽  
Circulation Model ◽  
Ocean General Circulation Model ◽  
Global Ocean ◽  
Special Focus ◽  
Background Error ◽  
Background Error Covariance ◽  
Error Covariance ◽  
The Impact

Abstract In this paper results from the application of an ocean data assimilation (ODA) system, combining a multivariate reduced-order optimal interpolator (OI) scheme with a global ocean general circulation model (OGCM), are described. The present ODA system, designed to assimilate in situ temperature and salinity observations, has been used to produce ocean reanalyses for the 1962–2001 period. The impact of assimilating observed hydrographic data on the ocean mean state and temporal variability is evaluated. A special focus of this work is on the ODA system skill in reproducing a realistic ocean salinity state. Results from a hierarchy of different salinity reanalyses, using varying combinations of assimilated data and background error covariance structures, are described. The impact of the space and time resolution of the background error covariance parameterization on salinity is addressed.

scholarly journals Assessment of Snow Cover and Surface Albedo in the ECHAM5 General Circulation Model

2006 ◽  
Vol 19 (16) ◽  
pp. 3828-3843 ◽  
Author(s):  
Andreas Roesch ◽  
Erich Roeckner
Keyword(s):  
Snow Cover ◽  
Land Surface ◽  
General Circulation ◽  
Surface Albedo ◽  
Climate Model ◽  
Circulation Model ◽  
Masking Effect ◽  
Second Phase ◽  
Snow Cover Area ◽  
Highly Nonlinear

Abstract Land surface albedo, snow cover fraction (SCF), and snow depth (SD) from two versions of the ECHAM climate model are compared to available ground-based and remote-sensed climatologies. ECHAM5 accurately reproduces the annual cycle of SD and correctly captures the timing of the snowmelt. ECHAM4, in contrast, simulates an excessive Eurasian snow mass in spring due to a delayed snowmelt. Annual cycles of continental snow cover area (SCA) are captured fairly well in both ECHAM4 and ECHAM5. The negative SCA trend observed during the last two decades of the twentieth century is evident also in the ECHAM5 simulation but less pronounced. ECHAM5 captures the interannual variability of SCA reasonably well, which is in contrast with results that were reported earlier for second-phase Atmospheric Model Intercomparison Project (AMIP II) models. An error analysis revealed that, for studies on SCA, it is essential to test the data records for their homogeneity and trends. The second part of the paper compares simulated surface albedos with remote-sensed climatologies derived from PINKER and the Moderate Resolution Imaging Spectroradiometer (MODIS). ECHAM5 is in better agreement with observations in the Himalayan–Tibetan area than ECHAM4. In contrast, the positive surface albedo bias over boreal forests under snow conditions in ECHAM4 is even more pronounced in ECHAM5. This deficiency is mainly due to the neglect of the snow-masking effect of stems and branches after trees have lost their foliage. The analysis demonstrates that positive biases in the SCA are not necessarily related to positive albedo biases. Furthermore, an overestimation of the area-averaged SD is not always related to positive SCF anomalies since the relationship between SD and SCF is highly nonlinear.

scholarly journals An efficient ensemble Kalman Filter implementation via shrinkage covariance matrix estimation: exploiting prior knowledge

2021 ◽  
Vol 25 (3) ◽  
pp. 985-1003
Author(s):  
Santiago Lopez-Restrepo ◽  
Elias D. Nino-Ruiz ◽  
Luis G. Guzman-Reyes ◽  
Andres Yarce ◽  
O. L. Quintero ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Prior Knowledge ◽  
Covariance Matrix ◽  
Ensemble Kalman Filter ◽  
General Circulation ◽  
Circulation Model ◽  
Covariance Matrix Estimation ◽  
Practical Implementation ◽  
Matrix Estimation ◽  
The Impact

AbstractIn this paper, we propose an efficient and practical implementation of the ensemble Kalman filter via shrinkage covariance matrix estimation. Our filter implementation combines information brought by an ensemble of model realizations, and that based on our prior knowledge about the dynamical system of interest. We perform the combination of both sources of information via optimal shrinkage factors. The method exploits the rank-deficiency of ensemble covariance matrices to provide an efficient and practical implementation of the analysis step in EnKF based formulations. Localization and inflation aspects are discussed, as well. Experimental tests are performed to assess the accuracy of our proposed filter implementation by employing an Advection Diffusion Model and an Atmospheric General Circulation Model. The experimental results reveal that the use of our proposed filter implementation can mitigate the impact of sampling noise, and even more, it can avoid the impact of spurious correlations during assimilation steps.

scholarly journals Non-Gaussian statistics in global atmospheric dynamics: a study with a 10 240-member ensemble Kalman filter using an intermediate AGCM

2018 ◽  
Author(s):  
Keiichi Kondo ◽  
Takemasa Miyoshi
Keyword(s):  
Kalman Filter ◽  
General Circulation ◽  
Circulation Model ◽  
Tropical Regions ◽  
Highly Nonlinear ◽  
Gaussian Statistics ◽  
Ensemble Transform Kalman Filter ◽  
Analysis Error ◽  
Non Gaussian ◽  
Analysis Errors

Abstract. We previously performed local ensemble transform Kalman filter experiments with up to 10 240 ensemble members using an intermediate atmospheric general circulation model. While the previous study focused on the localization impact on the analysis accuracy, the present study focuses on the probability density functions (PDFs) represented by the 10 240-member ensemble. The 10 240-member ensemble can resolve the detailed structures of the PDFs and indicates that the non-Gaussian PDF is caused by multimodality and outliers. The results show that the spatial patterns of the analysis errors correspond well with the non-Gaussianity. While the outliers appear randomly, large multimodality corresponds well with large analysis error, mainly in the tropical regions where highly nonlinear convective processes appear frequently. Therefore, we further investigate the lifecycle of multimodal PDFs, and show that the multimodal PDFs are generated by the on-off switch of convective parameterization and disappear naturally. Sensitivity to the ensemble size suggests that approximately 1000 ensemble members be necessary to capture the detailed structures of the non-Gaussian PDF.

Application of Coupled Bred Vectors to Seasonal-to-Interannual Forecasting and Ocean Data Assimilation

2009 ◽  
Vol 22 (11) ◽  
pp. 2850-2870 ◽  
Author(s):  
Shu-Chih Yang ◽  
Christian Keppenne ◽  
Michele Rienecker ◽  
Eugenia Kalnay
Keyword(s):  
Data Assimilation ◽  
General Circulation ◽  
Coupled Model ◽  
Circulation Model ◽  
Model Errors ◽  
Background Error ◽  
Salinity Field ◽  
Ocean Data Assimilation ◽  
Bred Vectors ◽  
The Impact

Abstract Coupled bred vectors (BVs) generated from the NASA Global Modeling and Assimilation Office (GMAO) coupled general circulation model are designed to capture the uncertainties related to slowly varying coupled instabilities. Two applications of the BVs are investigated in this study. First, the coupled BVs are used as initial perturbations for ensemble-forecasting purposes. Results show that the seasonal-to-interannual variability forecast skill can be improved when the oceanic and atmospheric perturbations are initialized with coupled BVs. The impact is particularly significant when the forecasts are initialized from the cold phase of tropical Pacific SST (e.g., August and November), because at these times the early coupled model errors, not accounted for in the BVs, are small. Second, the structure of the BVs is applied to construct hybrid background error covariances carrying flow-dependent information for the ocean data assimilation. Results show that the accuracy of the ocean analyses is improved when Gaussian background covariances are supplemented with a term obtained from the BVs. The improvement is especially noticeable for the salinity field.

scholarly journals Strongly Coupled Data Assimilation Experiments with Linearized Ocean–Atmosphere Balance Relationships

2018 ◽  
Vol 146 (4) ◽  
pp. 1233-1257 ◽  
Author(s):  
Andrea Storto ◽  
Matthew J. Martin ◽  
Bruno Deremble ◽  
Simona Masina
Keyword(s):  
Data Assimilation ◽  
General Circulation ◽  
Circulation Model ◽  
Variational Data Assimilation ◽  
Global Ocean ◽  
Background Error ◽  
Near Surface ◽  
Strongly Coupled ◽  
Coupled Data Assimilation ◽  
The Impact

Coupled data assimilation is emerging as a target approach for Earth system prediction and reanalysis systems. Coupled data assimilation may be indeed able to minimize unbalanced air–sea initialization and maximize the intermedium propagation of observations. Here, we use a simplified framework where a global ocean general circulation model (NEMO) is coupled to an atmospheric boundary layer model [Cheap Atmospheric Mixed Layer (CheapAML)], which includes prognostic prediction of near-surface air temperature and moisture and allows for thermodynamic but not dynamic air–sea coupling. The control vector of an ocean variational data assimilation system is augmented to include 2-m atmospheric parameters. Cross-medium balances are formulated either through statistical cross covariances from monthly anomalies or through the application of linearized air–sea flux relationships derived from the tangent linear approximation of bulk formulas, which represents a novel solution to the coupled assimilation problem. As a proof of concept, the methodology is first applied to study the impact of in situ ocean observing networks on the near-surface atmospheric analyses and later to the complementary study of the impact of 2-m air observations on sea surface parameters, to assess benefits of strongly versus weakly coupled data assimilation. Several forecast experiments have been conducted for the period from June to December 2011. We find that especially after day 2 of the forecasts, strongly coupled data assimilation provides a beneficial impact, particularly in the tropical oceans. In most areas, the use of linearized air–sea balances outperforms the statistical relationships used, providing a motivation for implementing coupled tangent linear trajectories in four-dimensional variational data assimilation systems. Further impacts of strongly coupled data assimilation might be found by retuning the background error covariances.

scholarly journals An Ensemble Approach for the Estimation of Observational Error Illustrated for a Nominal 1° Global Ocean Model

2016 ◽  
Vol 144 (5) ◽  
pp. 1713-1728 ◽  
Author(s):  
Alicia R. Karspeck
Keyword(s):  
Least Squares ◽  
General Circulation Model ◽  
General Circulation ◽  
Circulation Model ◽  
Ocean General Circulation Model ◽  
Global Ocean ◽  
State Variables ◽  
Observational Error ◽  
Background Error ◽  
Ocean General Circulation

Least squares algorithms for data assimilation require estimates of both background error covariances and observational error covariances. The specification of these errors is an essential part of designing an assimilation system; the relative sizes of these uncertainties determine the extent to which the state variables are drawn toward the observational information. Observational error covariances are typically computed as the sum of measurement/instrumental errors and “representativeness error.” In a coarse-resolution ocean general circulation model the errors of representation are the dominant contribution to observational error covariance over large portions of the globe, and the size of these errors will vary by the type of observation and the geographic region. They may also vary from model to model. A straightforward approach for estimating model-dependent, spatially varying observational error variances that are suitable for least squares ocean data assimilating systems is presented here. The author proposes an ensemble-based estimator of the true observational error variance and outlines the assumptions necessary for the estimator to be unbiased. The author also presents the variance (or uncertainty) associated with the estimator under certain conditions. The analytic expressions for the expected value and variance of the estimator are validated with a simple autoregressive model and illustrated for the nominal 1° resolution POP2 global ocean general circulation model.

scholarly journals An Adaptive Compensatory Approach of the Fixed Localization in the EnKF

2015 ◽  
Vol 143 (11) ◽  
pp. 4714-4735 ◽  
Author(s):  
Xinrong Wu ◽  
Wei Li ◽  
Guijun Han ◽  
Lianxin Zhang ◽  
Caixia Shao ◽  
...  
Keyword(s):  
Multiscale Analysis ◽  
General Circulation ◽  
Computational Cost ◽  
Circulation Model ◽  
Adaptive Method ◽  
Background Error ◽  
Error Covariance ◽  
Incremental Improvement ◽  
Better Than

Abstract Although the fixed covariance localization in the ensemble Kalman filter (EnKF) can significantly increase the reliability of background error covariance, it has been demonstrated that extreme impact radii can cause the EnKF to lose some useful information. Tuning an optimal impact radius, on the other hand, is always difficult for a general circulation model. The EnKF multiscale analysis (MSA) approach was presented to make up for the above-mentioned drawback of the fixed localization. As a follow-up, this study presents an adaptive compensatory approach to further improve the performance of the EnKF-MSA. The new method adaptively triggers a multigrid analysis (MGA) to extract multiscale information from the observational residual after the EnKF without inflation is completed at each analysis step. Within a biased twin experiment framework consisting of a barotropic spectral model and an idealized observing system, the performance of the adaptive method is examined. Results show that the MGA reduces the computational cost of the MSA by 93%. On the assimilation quality, the adaptive method has an incremental improvement over the EnKF-MSA. That is, the adaptive EnKF-MGA reduces to the EnKF without inflation, which is better than the EnKF-MSA, for moderate impact radii. The proposed scheme works for a broader range of impact radii than the standard EnKF (i.e., the EnKF with inflation). For extreme impact radii, the adaptive EnKF-MGA can produce smaller assimilation errors than the standard EnKF and shorten the spinup period by 53%. In addition, the computational cost of the MGA is negligible relative to that of the standard EnKF.

scholarly journals The local ensemble transform Kalman filter and the running-in-place algorithm applied to a global ocean general circulation model

2013 ◽  
Vol 20 (6) ◽  
pp. 1031-1046 ◽  
Author(s):  
S. G. Penny ◽  
E. Kalnay ◽  
J. A. Carton ◽  
B. R. Hunt ◽  
K. Ide ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
General Circulation ◽  
Initial Conditions ◽  
Circulation Model ◽  
Global Ocean ◽  
Observation Data ◽  
Background Error ◽  
Ensemble Transform Kalman Filter ◽  
Ensemble Transform

Abstract. The most widely used methods of data assimilation in large-scale oceanography, such as the Simple Ocean Data Assimilation (SODA) algorithm, specify the background error covariances and thus are unable to refine the weights in the assimilation as the circulation changes. In contrast, the more computationally expensive Ensemble Kalman Filters (EnKF) such as the Local Ensemble Transform Kalman Filter (LETKF) use an ensemble of model forecasts to predict changes in the background error covariances and thus should produce more accurate analyses. The EnKFs are based on the approximation that ensemble members reflect a Gaussian probability distribution that is transformed linearly during the forecast and analysis cycle. In the presence of nonlinearity, EnKFs can gain from replacing each analysis increment by a sequence of smaller increments obtained by recursively applying the forecast model and data assimilation procedure over a single analysis cycle. This has led to the development of the "running in place" (RIP) algorithm by Kalnay and Yang (2010) and Yang et al. (2012a,b) in which the weights computed at the end of each analysis cycle are used recursively to refine the ensemble at the beginning of the analysis cycle. To date, no studies have been carried out with RIP in a global domain with real observations. This paper provides a comparison of the aforementioned assimilation methods in a set of experiments spanning seven years (1997–2003) using identical forecast models, initial conditions, and observation data. While the emphasis is on understanding the similarities and differences between the assimilation methods, comparisons are also made to independent ocean station temperature, salinity, and velocity time series, as well as ocean transports, providing information about the absolute error of each. Comparisons to independent observations are similar for the assimilation methods but the observation-minus-background temperature differences are distinctly lower for LETKF and RIP. The results support the potential for LETKF to improve the quality of ocean analyses on the space and timescales of interest for seasonal prediction and for RIP to accelerate the spin up of the system.

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