scholarly journals DATA ASSIMILATION BY ENSEMBLE KALMAN FILTER WITH THE LORENZ EQUATIONS

2016 ◽  
Vol 38 ◽  
pp. 190
Author(s):  
Regis Sperotto de Quadros ◽  
Fabrício Pereira Harter ◽  
Daniela Buske ◽  
Larri Silveira Pereira
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Weather Forecasting ◽  
Initial Conditions ◽  
Background Field ◽  
Lorenz Equations ◽  
Integration Period ◽  
Numerical Weather Forecasting ◽  
Chaotic Nature

Data Assimilation is a procedure to get the initial condition as accurately as possible, through the statistical combination of collected observations and a background field, usually a short-range forecast. In this research a complete assimilation system for the Lorenz equations based on Ensemble Kalman Filter is presented and examined. The Lorenz model is chosen for its simplicity in structure and the dynamic similarities with primitive equations models, such as modern numerical weather forecasting. Based on results, was concluded that, in this implementation, 10 members is the best setting, because there is an overfitting for ensembles with 50 and 100 members. It was also examined if the EnKF is effective to track the control for 20% and 40% of error in the initial conditions. The results show a disagreement between the “truth” and the estimation, especially in the end of integration period, due the chaotic nature of the system.  It was also concluded that EnKF have to be performed sufficiently frequently in order to produce desirable results.

A Probabilistic Collocation-Based Kalman Filter for History Matching

SPE Journal ◽  
2011 ◽  
Vol 16 (02) ◽  
pp. 294-306 ◽  
Author(s):  
Lingzao Zeng ◽  
Haibin Chang ◽  
Dongxiao Zhang
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
History Matching ◽  
Initial Conditions ◽  
Collocation Point ◽  
Computational Effort ◽  
Production Data ◽  
Sampling Errors ◽  
Probabilistic Collocation

Summary The ensemble Kalman filter (EnKF) has been used widely for data assimilation. Because the EnKF is a Monte Carlo-based method, a large ensemble size is required to reduce the sampling errors. In this study, a probabilistic collocation-based Kalman filter (PCKF) is developed to adjust the reservoir parameters to honor the production data. It combines the advantages of the EnKF for dynamic data assimilation and the polynomial chaos expansion (PCE) for efficient uncertainty quantification. In this approach, all the system parameters and states and the production data are approximated by the PCE. The PCE coefficients are solved with the probabilistic collocation method (PCM). Collocation realizations are constructed by choosing collocation point sets in the random space. The simulation for each collocation realization is solved forward in time independently by means of an existing deterministic solver, as in the EnKF method. In the analysis step, the needed covariance is approximated by the PCE coefficients. In this study, a square-root filter is employed to update the PCE coefficients. After the analysis, new collocation realizations are constructed. With the parameter collocation realizations as the inputs and the state collocation realizations as initial conditions, respectively, the simulations are forwarded to the next analysis step. Synthetic 2D water/oil examples are used to demonstrate the applicability of the PCKF in history matching. The results are compared with those from the EnKF on the basis of the same analysis. It is shown that the estimations provided by the PCKF are comparable to those obtained from the EnKF. The biggest improvement of the PCKF comes from the leading PCE approximation, with which the computational burden of the PCKF can be greatly reduced by means of a smaller number of simulation runs, and the PCKF outperforms the EnKF for a similar computational effort. When the correlation ratio is much smaller, the PCKF still provides estimations with a better accuracy for a small computational effort.

scholarly journals Improvement of ozone forecast over Beijing based on ensemble Kalman filter with simultaneous adjustment of initial conditions and emissions

2011 ◽  
Vol 11 (24) ◽  
pp. 12901-12916 ◽  
Author(s):  
X. Tang ◽  
J. Zhu ◽  
Z. F. Wang ◽  
A. Gbaguidi
Keyword(s):  
Kalman Filter ◽  
Air Quality ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Initial Conditions ◽  
Urban Site ◽  
Ozone Forecast ◽  
Regional Air Quality ◽  
Simultaneous Adjustment ◽  
Mean Square Errors

Abstract. In order to improve the surface ozone forecast over Beijing and surrounding regions, data assimilation method integrated into a high-resolution regional air quality model and a regional air quality monitoring network are employed. Several advanced data assimilation strategies based on ensemble Kalman filter are designed to adjust O3 initial conditions, NOx initial conditions and emissions, VOCs initial conditions and emissions separately or jointly through assimilating ozone observations. As a result, adjusting precursor initial conditions demonstrates potential improvement of the 1-h ozone forecast almost as great as shown by adjusting precursor emissions. Nevertheless, either adjusting precursor initial conditions or emissions show deficiency in improving the short-term ozone forecast at suburban areas. Adjusting ozone initial values brings significant improvement to the 1-h ozone forecast, and its limitations lie in the difficulty in improving the 1-h forecast at some urban site. A simultaneous adjustment of the above five variables is found to be able to reduce these limitations and display an overall better performance in improving both the 1-h and 24-h ozone forecast over these areas. The root mean square errors of 1-h ozone forecast at urban sites and suburban sites decrease by 51% and 58% respectively compared with those in free run. Through these experiments, we found that assimilating local ozone observations is determinant for ozone forecast over the observational area, while assimilating remote ozone observations could reduce the uncertainty in regional transport ozone.

scholarly journals Beating the Uncertainties: Ensemble Forecasting and Ensemble-Based Data Assimilation in Modern Numerical Weather Prediction

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Hailing Zhang ◽  
Zhaoxia Pu
Keyword(s):  
Data Assimilation ◽  
Numerical Weather Prediction ◽  
Weather Forecasting ◽  
Weather Prediction ◽  
Ensemble Forecasting ◽  
Atmospheric Flow ◽  
Numerical Weather ◽  
Recent Developments ◽  
Numerical Weather Forecasting ◽  
Chaotic Nature

Accurate numerical weather forecasting is of great importance. Due to inadequate observations, our limited understanding of the physical processes of the atmosphere, and the chaotic nature of atmospheric flow, uncertainties always exist in modern numerical weather prediction (NWP). Recent developments in ensemble forecasting and ensemble-based data assimilation have proved that there are promising ways to beat the forecast uncertainties in NWP. This paper gives a brief overview of fundamental problems and recent progress associated with ensemble forecasting and ensemble-based data assimilation. The usefulness of these methods in improving high-impact weather forecasting is also discussed.

scholarly journals Pseudo-Orbit Data Assimilation. Part I: The Perfect Model Scenario

2014 ◽  
Vol 71 (2) ◽  
pp. 469-482 ◽  
Author(s):  
Hailiang Du ◽  
Leonard A. Smith
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
State Estimation ◽  
Ensemble Kalman Filter ◽  
Weather Forecasting ◽  
Weather Prediction ◽  
Attractive Alternative ◽  
Perfect Model ◽  
Improved Performance ◽  
Orbit Data

Abstract State estimation lies at the heart of many meteorological tasks. Pseudo-orbit-based data assimilation provides an attractive alternative approach to data assimilation in nonlinear systems such as weather forecasting models. In the perfect model scenario, noisy observations prevent a precise estimate of the current state. In this setting, ensemble Kalman filter approaches are hampered by their foundational assumptions of dynamical linearity, while variational approaches may fail in practice owing to local minima in their cost function. The pseudo-orbit data assimilation approach improves state estimation by enhancing the balance between the information derived from the dynamic equations and that derived from the observations. The potential use of this approach for numerical weather prediction is explored in the perfect model scenario within two deterministic chaotic systems: the two-dimensional Ikeda map and 18-dimensional Lorenz96 flow. Empirical results demonstrate improved performance over that of the two most common traditional approaches of data assimilation (ensemble Kalman filter and four-dimensional variational assimilation).

An Iterative Ensemble Kalman Filter for Multiphase Fluid Flow Data Assimilation

SPE Journal ◽  
2007 ◽  
Vol 12 (04) ◽  
pp. 438-446 ◽  
Author(s):  
Yaqing Gu ◽  
Dean S. Oliver
Keyword(s):  
Porous Media ◽  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Weather Forecasting ◽  
Weather Prediction ◽  
Flow In Porous Media ◽  
Petroleum Reservoir ◽  
State Variables ◽  
Highly Nonlinear

Summary The dynamical equations for multiphase flow in porous media are highly nonlinear and the number of variables required to characterize the medium is usually large, often two or more variables per simulator gridblock. Neither the extended Kalman filter nor the ensemble Kalman filter is suitable for assimilating data or for characterizing uncertainty for this type of problem. Although the ensemble Kalman filter handles the nonlinear dynamics correctly during the forecast step, it sometimes fails badly in the analysis (or updating) of saturations. This paper focuses on the use of an iterative ensemble Kalman filter for data assimilation in nonlinear problems, especially of the type related to multiphase ow in porous media. Two issues are key:iteration to enforce constraints andensuring that the resulting ensemble is representative of the conditional pdf (i.e., that the uncertainty quantification is correct). The new algorithm is compared to the ensemble Kalman filter on several highly nonlinear example problems, and shown to be superior in the prediction of uncertainty. Introduction For linear problems, the Kalman filter is optimal for assimilating measurements to continuously update the estimate of state variables. Kalman filters have occasionally been applied to the problem of estimating values of petroleum reservoir variables (Eisenmann et al. 1994; Corser et al. 2000), but they are most appropriate when the problems are characterized by a small number of variables and when the variables to be estimated are linearly related to the observations. Most data assimilation problems in petroleum reservoir engineering are highly nonlinear and are characterized by many variables, often two or more variables per simulator gridblock. The problem of weather forecasting is in many respects similar to the problem of predicting future petroleum reservoir performance. The economic impact of inaccurate predictions is substantial in both cases, as is the difficulty of assimilating very large data sets and updating very large numerical models. One method that has been recently developed for assimilating data in weather forecasting is ensemble Kalman filtering (Evensen 1994; Houtekamer and Mitchell 1998; Anderson and Anderson 1999; Hamill et al. 2000; Houtekamer and Mitchell 2001; Evensen 2003). It has been demonstrated to be useful for weather prediction over the North Atlantic. The method is now beginning to be applied for data assimilation in groundwater hydrology (Reichle et al. 2002; Chen and Zhang 2006) and in petroleum engineering (Nævdal et al. 2002, 2005; Gu and Oliver 2005; Liu and Oliver 2005a; Wen and Chen 2006, 2007; Zafari and Reynolds 2007; Gao et al. 2006; Lorentzen et al. 2005; Skjervheim et al. 2007; Dong et al. 2006), but the applications to state variables whose density functions are bimodal has proved problematic (Gu and Oliver 2006). For applications to nonlinear assimilation problems, it is useful to think of the ensemble Kalman filter as a least squares method that obtains an averaged gradient for minimization not from a variational approach but from an empirical correlation between model variables (Anderson 2003; Zafari et al. 2006). In addition to providing a mean estimate of the variables, a Monte Carlo estimate of uncertainty can be obtained directly from the variability in the ensemble.

Atmospheric Data Assimilation with an Ensemble Kalman Filter: Results with Real Observations

2005 ◽  
Vol 133 (3) ◽  
pp. 604-620 ◽  
Author(s):  
P. L. Houtekamer ◽  
Herschel L. Mitchell ◽  
Gérard Pellerin ◽  
Mark Buehner ◽  
Martin Charron ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Initial Conditions ◽  
Forecast Error ◽  
Lower Troposphere ◽  
Model Error ◽  
Atmospheric Data ◽  
Ensemble Mean ◽  
Parameterized Model

Abstract An ensemble Kalman filter (EnKF) has been implemented for atmospheric data assimilation. It assimilates observations from a fairly complete observational network with a forecast model that includes a standard operational set of physical parameterizations. To obtain reasonable results with a limited number of ensemble members, severe horizontal and vertical covariance localizations have been used. It is observed that the error growth in the data assimilation cycle is mainly due to model error. An isotropic parameterization, similar to the forecast-error parameterization in variational algorithms, is used to represent model error. After some adjustment, it is possible to obtain innovation statistics that agree with the ensemble-based estimate of the innovation amplitudes for winds and temperature. Currently, no model error is added for the humidity variable, and, consequently, the ensemble spread for humidity is too small. After about 5 days of cycling, fairly stable global filter statistics are obtained with no sign of filter divergence. The quality of the ensemble mean background field, as verified using radiosonde observations, is similar to that obtained using a 3D variational procedure. In part, this is likely due to the form chosen for the parameterized model error. Nevertheless, the degree of similarity is surprising given that the background-error statistics used by the two procedures are rather different, with generally larger background errors being used by the variational scheme. A set of 5-day integrations has been started from the ensemble of initial conditions provided by the EnKF. For the middle and lower troposphere, the growth rates of the perturbations are somewhat smaller than the growth rate of the actual ensemble mean error. For the upper levels, the perturbation patterns decay for about 3 days as a consequence of diffusive model dynamics. These decaying perturbations tend to severely underestimate the actual error that grows rapidly near the model top.

scholarly journals Improvement of ozone forecast over Beijing based on ensemble Kalman filter with simultaneous adjustment of initial conditions and emissions

2011 ◽  
Vol 11 (3) ◽  
pp. 7811-7849 ◽  
Author(s):  
X. Tang ◽  
J. Zhu ◽  
Z. F. Wang ◽  
A. Gbaguidi
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Initial Conditions ◽  
Pollution Monitoring ◽  
Ozone Forecast ◽  
Ozone Data ◽  
Surrounding Areas ◽  
The Impact ◽  
Urban Sites

Abstract. We performed ozone data assimilation by simultaneously adjusting the ozone initial conditions, precursor initial conditions and emissions based on the Ensemble Kalman Filter (EnKF) and assessed its impacts on ozone modeling and forecasting in Beijing and nearby regions. A high-resolution regional air quality model and a newly established regional monitoring network covering Beijing and its surrounding areas were employed. At each assimilation cycle, the forecast error covariance was sampled from a set of forecast ensembles that were generated by perturbing ozone precursor initial conditions, emissions, photolysis rates and deposition velocity. A model-error module and a local analysis scheme have been introduced to reduce the impact of filter divergence and spurious correlation that accompanied with EnKF. The results showed significant improvement of 1-hour ozone forecast in Beijing and its surrounding areas through separately adjusting ozone initial conditions, precursor initial conditions and emissions with ozone observations. However, adjustment of precursor initial conditions and emissions had minor effect on the 1-hour ozone forecast in suburban area. The best ozone forecast skill was obtained through jointly adjusting ozone initial conditions, NOx and VOC initial conditions and emissions. The root mean square errors of 1-hour ozone forecast at urban sites and suburban sites decreased by 54% and 59% respectively compared with those in free run. Furthermore, the specific impacts of observations from urban and suburban sites on ozone data assimilation were evaluated by implementing sensitivity experiments. Both urban and suburban sites were found to be very important for the improvement of regional ozone forecast. The importance of observational data at urban sites was particularly highlighted through its role in constraining the uncertainty of precursor initial conditions and emission rates. Further improvement of regional ozone forecast might therefore be expected with more routine regional air pollution monitoring stations.

scholarly journals A data assimilation method of the Ensemble Kalman Filter for use in severe dust storm forecasts over China

2007 ◽  
Vol 7 (6) ◽  
pp. 17511-17536
Author(s):  
C. Lin ◽  
Z. Wang ◽  
J. Zhu
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Dust Storm ◽  
Ensemble Kalman Filter ◽  
Transport Model ◽  
Initial Conditions ◽  
Dust Storms ◽  
Initial Time ◽  
Model Errors ◽  
Severe Dust Storm

Abstract. An Ensemble Kalman Filter (EnKF) data assimilation system was developed for a regional dust transport model. This paper applied the EnKF method to investigate modeling severe dust storm episodes occurred in March 2002 over China based on surface observations of dust concentrations to explore its impacts on forecast improvement. A series of sensitivity experiments using our system reveals that the EnKF is an advanced assimilation method to afford better initial conditions with surface observed PM10 in North China and lead to improved forecasts of dust storms, but forecast with large errors can be made by model errors. This result illustrates that it requires identifying and correcting model errors during the assimilation procedure in order to significantly improve forecasts. Results also show that the EnKF should use a large inflation parameter to obtain better model performance and forecast potential. Furthermore, the ensemble perturbations generated at the initial time should include enough ensemble spreads to represent the background error after several assimilation cycles.

Assessing Predictability with a Local Ensemble Kalman Filter

2007 ◽  
Vol 64 (4) ◽  
pp. 1116-1140 ◽  
Author(s):  
David Kuhl ◽  
Istvan Szunyogh ◽  
Eric J. Kostelich ◽  
Gyorgyi Gyarmati ◽  
D. J. Patil ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Initial Conditions ◽  
Forecast Error ◽  
Weather Prediction ◽  
Forecast Errors ◽  
Data Assimilation Scheme ◽  
Ensemble Mean ◽  
Assimilation Scheme

Abstract In this paper, the spatiotemporally changing nature of predictability is studied in a reduced-resolution version of the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS), a state-of-the-art numerical weather prediction model. Atmospheric predictability is assessed in the perfect model scenario for which forecast uncertainties are entirely due to uncertainties in the estimates of the initial states. Uncertain initial conditions (analyses) are obtained by assimilating simulated noisy vertical soundings of the “true” atmospheric states with the local ensemble Kalman filter (LEKF) data assimilation scheme. This data assimilation scheme provides an ensemble of initial conditions. The ensemble mean defines the initial condition of 5-day deterministic model forecasts, while the time-evolved members of the ensemble provide an estimate of the evolving forecast uncertainties. The observations are randomly distributed in space to ensure that the geographical distribution of the analysis and forecast errors reflect predictability limits due to the model dynamics and are not affected by inhomogeneities of the observational coverage. Analysis and forecast error statistics are calculated for the deterministic forecasts. It is found that short-term forecast errors tend to grow exponentially in the extratropics and linearly in the Tropics. The behavior of the ensemble is explained by using the ensemble dimension (E dimension), a spatiotemporally evolving measure of the evenness of the distribution of the variance between the principal components of the ensemble-based forecast error covariance matrix. It is shown that in the extratropics the largest forecast errors occur for the smallest E dimensions. Since a low value of the E dimension guarantees that the ensemble can capture a large portion of the forecast error, the larger the forecast error the more certain that the ensemble can fully capture the forecast error. In particular, in regions of low E dimension, ensemble averaging is an efficient error filter and the ensemble spread provides an accurate prediction of the upper bound of the error in the ensemble-mean forecast.

Data assimilation with the weighted ensemble Kalman filter

2010 ◽  
Author(s):  
Nicolas Papadakis ◽  
Etienne Mémin ◽  
Anne Cuzol ◽  
Nicolas Gengembre
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter
Sign in / Sign up

Export Citation Format

Share Document