The nonlinear filtering problem occurs in many scientific areas. Sequential Monte Carlo solutions with the correct asymptotic behavior such as particle filters exist, but they are computationally too expensive when working with high-dimensional systems. The ensemble Kalman filter (EnKF) is a more robust method that has shown promising results with a small sample size, but the samples are not guaranteed to come from the true posterior distribution. By approximating the model error with a Gaussian distribution, one may represent the posterior distribution as a sum of Gaussian kernels. The resulting Gaussian mixture filter has the advantage of both a local Kalman type correction and the weighting/resampling step of a particle filter. The Gaussian mixture approximation relies on a bandwidth parameter which often has to be kept quite large in order to avoid a weight collapse in high dimensions. As a result, the Kalman correction is too large to capture highly non-Gaussian posterior distributions. In this paper, we have extended the Gaussian mixture filter (Hoteit et al., Mon Weather Rev 136:317-334, 2008) and also made the connection to particle filters more transparent. In particular, we introduce a tuning parameter for the importance weights. In the last part of the paper, we have performed a simulation experiment with the Lorenz40 model where our method has been A. S. Stordal (B) · G. Naevdal · B. Vallès IRIS, compared to the EnKF and a full implementation of a particle filter. The results clearly indicate that the new method has advantages compared to the standard EnKF.

Introduction and Background There has been great progress in data assimilation within atmospheric and oceanographic sciences during the last couple of decades. In data assimilation, one aims at merging the information from observations into a numerical model, typically of a geophysical system. A typical example where data assimilation is needed is in weather forecasting. Here, the atmospheric models must take into account the most recent observations of variables such as temperature and atmospheric pressure for better forecasting of the weather in the next time period. A major challenge for these models is that they contain very large numbers of variables. The progress in data assimilation is because of both increased computational power and the introduction of techniques that are capable of handling large amounts of data and more severe nonlinearities. The aim of this paper is to focus on one of these techniques, the ensemble Kalman filter (EnKF). The EnKF has been introduced to petroleum science recently (Lorentzen et al. 2001a) and, in particular, has attracted attention as a promising method for solving the history matching problem. The literature available on the EnKF is now rather overwhelming. We hope that this review will help researchers (and students) working on adapting the EnKF to petroleum applications to find valuable references and ideas, although the number of papers discussing the EnKF is too large to give a complete review. For practitioners, we have cited critical EnKF papers from weather and oceanography. We have also tried to review most of the papers dealing with the EnKF and updating of reservoir models available to the authors by the beginning of 2008. The EnKF is based on the simpler Kalman filter (Kalman 1960). We will start by introducing the Kalman filter. The Kalman filter is an efficient recursive filter that estimates the state of a linear dynamical system from a series of noisy measurements. The Kalman filter is based on a model equation, where the current state of the system is associated with an uncertainty (expressed by a covariance matrix) and an observation equation that relates a linear combination of the states to measurements. The measurements are also associated with uncertainty. The model equations are used to compute a forward step (Eqs. 1 and 2) where the state variables are computed forward in time with the current estimate of the state as initial condition. The observation equations are used in the analysis step (Eqs. 3 through 5) where the estimated value of the state and its uncertainty are corrected to take into account the most recent measurements See, e.g., Cohn (1997), Maybeck (1979), or Stengel (1994) for an introduction to the Kalman filter.

One of the key techniques towards energy efficiency and conservation is Non-Intrusive Load Monitoring (NILM) which lies in the domain of energy monitoring. Event detection is a core component of event-based NILM systems. This paper proposes two new low-complexity and computationally fast algorithms that detect the variations of load data and return the time occurrences of the corresponding events. The proposed algorithms are based on the phenomenon of a sliding window that tracks the statistical features of the acquired aggregated load data. The performance of the proposed algorithms is evaluated using real-world data and a comparative analysis has been carried out with one of the recently proposed event detection algorithms. Based on the simulations and sensitivity analysis it is shown that the proposed algorithm can provide the results of up to 93% and 88% in terms of recall and precision respectively.

During history match reservoir models are calibrated against production data to improve forecasts reliability. Often, the calibration ends up with a handful of matched models, sometime achieved without preserving the prior geological interpretation. This makes the outcome of many history matching projects unsuitable for a probabilistic approach to production forecast, then motivating the quest of methodologies casting history match in a stochastic framework. The Ensemble Kalman Filter (EnKF) has gained popularity as Monte-Carlo based methodology for history matching and real time updates of reservoir models. With EnKF an ensemble of models is updated whenever production data are available. The initial ensemble is generated according to the prior model, while the sequential updates lead to a sampling of the posterior probability function. This work is one of the first to successfully use EnKF to history match a real field reservoir model. It is, to our knowledge, the first paper showing how the EnKF can be used to evaluate the uncertainty in the production forecast for a given development plan for a real field model. The field at hand was an onshore saturated oil reservoir. Porosity distribution was one of the main uncertainties in the model, while permeability was considered a porosity function. According to the geological knowledge, the prior uncertainty was modeled using Sequential Gaussian Simulation and ensembles of porosity realizations were generated. Initial sensitivities indicated that conditioning porosity to available well data gives superior results in the history matching phase. Next, to achieve a compromise between accuracy and computational efficiency, the impact of the size of the ensemble on history matching, porosity distribution and uncertainty assessment was investigated. In the different ensembles the reduction of porosity uncertainty due to production data was noticed. Moreover, EnKF narrowed the production forecast confidence intervals with respect to estimate based on prior distribution.