The application of particle filters in geophysical systems is reviewed. Some background on Bayesian filtering is provided, and the existing methods are discussed. The emphasis is on the methodology, and not so much on the applications themselves. It is shown that direct application of the basic particle filter (i.e., importance sampling using the prior as the importance density) does not work in high-dimensional systems, but several variants are shown to have potential. Approximations to the full problem that try to keep some aspects of the particle filter beyond the Gaussian approximation are also presented and discussed.

/pdf/particle-filtering-in-geophysical-systems-3xwqdf9hlh.pdf

Particle Filtering in Geophysical Systems

[1] Sequential importance resampling (SIR) filter, residual resampling filter (RR), and an ensemble Kalman (EnKF) filter that can handle dynamic nonlinear/non-Gaussian models are compared to correct erroneous model inputs and to obtain a rainfall-runoff update with a conceptual rainfall-runoff model HBV-96 for flood forecasting purposes. EnKF performs best with a low number of ensemble members. The RR filter performs best at intermediate and high number of particles, although differences are small. With all filters the rainfall error could be estimated during a synthetic experiment when the soil is not too dry and the measurement error on the discharge is not dominant. The temperature error could only be estimated when the temperature is close to 0� C. When applying these methods to a real case, good results are obtained. For low flows, EnKF outperforms both particle filters, because it is less sensitive to misspecification of the model and uncertainties. These methods are feasible and easy to implement in real flood forecasting systems. Further research on the assumptions on model uncertainties and measurement uncertainties is recommended.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2005WR004093

Particle filtering and ensemble Kalman filtering for state updating with hydrological conceptual rainfall-runoff models

/pdf/general-review-of-rainfall-runoff-modeling-model-calibration-1yy49mrn9x.pdf

General Review of Rainfall-Runoff Modeling: Model Calibration, Data Assimilation, and Uncertainty Analysis

Inverse Methods in Physical Oceanography: Bibliography

The ensemble Kalman filter (EnKF) has been used for history matching a simulation model of a North Sea reservoir. Parameters such as initial fluid contacts, vertical transmissivity multipliers and fault transmissivity multipliers have been estimated as well as 3D fields of porosity and permeability. It is shown that for several of the parameters a large initial uncertainty is reduced to an acceptable level by the assimilation of well-log measurements and production rates of oil, gas and water. The result is an ensemble of history matched realizations which can be used to predict the uncertainty in future production. It is also shown that the formulation used in the EnKF reduces a nonlinear minimization problem in a huge parameter space, involving the minimization of an objective function with multiple local minima, to a statistical minimization problem in the ensemble space. Thus, by searching for the mean rather than the mode of the posterior pdf, the method avoids getting trapped in local minima and is thus promising for history matching reservoir simulation models. Furthermore, the EnKF provides an ideal setting for operational reservoir monitoring and prediction, including proper representation and prediction of uncertainty. Introduction Recently, there has been a growing interest in more mathematical and statistical methods for history matching. These involve both brute force direct minimization techniques and gradient methods based on the use of adjoints. Common for these is that they have all considered a pure parameter estimation problem. This differs from the combined parameter and state estimation problem which is considered when using the Ensemble Kalman Filter (EnKF) introduced by Evensen (1994, 2006). The EnKF has recently been taken into use with simulation models for oil and gas reservoirs, with the purpose of estimating poorly known parameters and to improve the predictive capability of the models. Traditional methods for assisted history matching minimize a cost function which measures the difference between simulated and observed production rates. The methods use the following loop: (1) The flow simulator is run for the complete production period; (2) the cost function is evaluated based on the difference between historical and simulated production rates; (3) the static parameters are updated, and the and the simulator is rerun. These methods solve a so called strong constraint formulation where the model errors are assumed to be accounted for by the set of parameters included in the cost function. The search for the solution is conducted in a space with dimension equal to the number of parameters and the problem becomes highly nonlinear leading to a cost function which typically will contain many local minima. This effectively limits the number of parameters which can be included in the optimization. Consequently the parameterization used becomes critical and the major uncertainty in the model must be represented by as few parameters as possible (Evensen, 2006). The use of EnKF for history matching reservoir simulation models was first proposed by Nævdal et al. (2003). Using the EnKF, the reservoir model state and parameters are updated sequentially in time, based on the information contained in pressure and rate measurements from production wells. There are now several publications discussing the potential of using the EnKF for parameter estimation. We refer to Evensen (2003, 2006) which outlines how the 2 Using the EnKF for Assisted History Matching of a North Sea Reservoir Model SPE 106184 model state can be augmented with a set of poorly known parameters, and the joint model state and parameters are then updated simultaneously. Applications of the EnKF for parameter estimation in reservoir simulation models include the following papers: Nævdal et al. (2002) used the EnKF in a reservoir application to estimate model permeability. They showed that there could be a great benefit of using the EnKF to improve the model through parameter estimation, and that this could lead to improved predictions. Nævdal et al. (2003) continued the development of the EnKF for estimation of permeability in the whole reservoir, and again showed promising results. Gu and Oliver (2004) examined the EnKF for combined parameter and state estimation in a standardized reservoir test case. They obtained promising results using a fairly small ensemble size but also pointed out several issues for further investigation. Gao and Reynolds (2005) compared the EnKF with another method named randomized maximum likelihood. They used the same reservoir example as Gu and Oliver (2004), and pointed out certain similarities between the methods. Liu and Oliver (2005a,b) examined the EnKF for facies estimation in a reservoir simulation model. This is a highly nonlinear problem where the reservoir consists of sand and shale classes of vastly different porosity and permeability. Thus, the pdf for the petrophysical parameters will be multi-modal, and there is currently no general way to treat this in the EnKF. In this particular example the facies distribution for each ensemble member is represented by two normal Gaussian fields, using a method named truncated pluri-Gaussian simulation (Lantuéjoul , 2002), and an improved facies distribution was obtained. Wen and Chen (2005) provided another discussion on the use of EnKF for estimation of the permeability field in a two dimensional reservoir simulation model, and they also examined the effect of ensemble size in their experiments. Zafari and Reynolds (2005) used simple but highly nonlinear models to examine the validity of the linear update scheme used in the EnKF. They concluded that the EnKF has problems with multi-modal distributions where the mean is not a good estimator, but on the other hand obtained reasonable results with a less nonlinear but more realistic reservoir model. They also showed that the rerun algorithm proposed by Wen and Chen (2005) is inconsistent and should not be used. Skjervheim et al. (2005) used the EnKF to assimilate 4D seismic data. It was shown that the EnKF could handle the large data sets and that a positive impact could be found despite the high noise level in the data. Other related publications on parameter estimation using ensemble methods are Lorentzen et al. (2003), Kivman (2003), Annan and Hargreaves (2004), Annan et al. (2005), Moradkhani et al. (2005) and Lorentzen et al. (2005). Below we will discuss the implementation and use of the EnKF in a North Sea field model. First the methodology is discussed where we focus on the parameterization used in this particular model as well as the general EnKF methodology and its properties. We then present and discuss the results from the field experiments and point out strengths and weaknesses of the methodology.

Using the EnKF for Assisted History Matching of a North Sea Reservoir Model

. The quality of the prediction of dynamical system evolution is determined by the accuracy to which initial conditions and forcing are known. Availability of future observations permits reducing the effects of errors in assessment the external model parameters by means of a filtering algorithm. Usually, uncertainties in specifying internal model parameters describing the inner system dynamics are neglected. Since they are characterized by strongly non-Gaussian distributions (parameters are positive, as a rule), traditional Kalman filtering schemes are badly suited to reducing the contribution of this type of uncertainties to the forecast errors. An extension of the Sequential Importance Resampling filter (SIR) is proposed to this aim. The filter is verified against the Ensemble Kalman filter (EnKF) in application to the stochastic Lorenz system. It is shown that the SIR is capable of estimating the system parameters and to predict the evolution of the system with a remarkably better accuracy than the EnKF. This highlights a severe drawback of any Kalman filtering scheme: due to utilizing only first two statistical moments in the analysis step it is unable to deal with probability density functions badly approximated by the normal distribution.

/pdf/sequential-parameter-estimation-for-stochastic-systems-9k3u39upx4.pdf

G. A. Kivman

Papers

Sequential parameter estimation for stochastic systems