However, we show that augmenting the state vector with the data results in a correct procedure for sampling the pdf if at every data assimilation step, the predicted data vector is a linear function of the combined unaugmented state vector and the average predicted data vector is equal to the predicted data evaluated at the average of the predicted combined state vector. Without these assumptions, we know of no way to show EnKF samples correctly.
For completeness, in Appendix C, we show that each ensemble member of model parameters obtained at each step of EnKF is a linear combination of the initial ensemble, which emphasizes the importance of obtaining a sufficiently large initial ensemble. Introduction The ensemble Kalman filter EnKF was introduced by Evensen in the context of ocean dynamics literature as a Monte Carlo approximation of the extended Kalman filter and has been extensively discussed in the weather prediction literature.
EnKF was recently introduced into the petroleum engineering literature Naevdal et al. Since its introduction into the petroleum engineering literature, EnKF has been investigated by a variety of researchers including Gu and Oliver ; Skjervheim et al. The method has also recently been applied successfully to a true field case Evensen et al.
As shown in Gao et al.
For the most part, EnKF has performed well for reservoir characterization examples. However, it is relatively easy to generate toy problems with multimodal conditional pdf's for which EnKF samples very poorly and hence provides a poor assessment of uncertainty Zafari ; Zafari and Reynolds b ; Reynolds et al.
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Reynolds et al. Because of this result, it not surprising that it may be necessary to use an iterative procedure to obtain an acceptable match of data for highly nonlinear problems. Here, we present a detailed derivation of our current version of the Reynolds et al. However, we iterate using a gradient based algorithm to obtain a better match of data than can be obtained by EnKF.
To iterate with either of these two methods, we compute at each iteration the gradient of an objective function using the adjoint procedure Li et al. To avoid the forward and adjoint solution from time zero, we formulate an iterative scheme, IENKF 3 , which at each iteration, simply requires a forward run from the previous data assimilation time to the current data assimilation time and an adjoint solution from the current data assimilation time back to the previous data assimilation time.
While this last iterative scheme is far more efficient than the first two, it requires iteratively updating the primary variables of the reservoir simulator at the previous data assimilation time. It is conceivable that this could lead to highly non-physical values which could introduce errors into our estimates. Our limited experiments show that this highly efficient method gives reasonable results, but far more testing is needed. For ease of writing, define. Let , which defines the analysis increment in observation space, and thus:.
As such the the analysis ensemble can then be constructed by:.
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For to be the mean of the analysis ensemble, the sum across the columns of X a must be zero, i. By the definition of X b ,. Therefore for it is required that is an eigenvector of W a. Wang et al. It can be seen that all the matrix operations take place in ensemble space i.
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The prediction step of the classic Kalman filter Eqs. The forecast model which moves the state from one time step to the next is replaced by a not necessarily linear model which moves each ensemble member forward in time. First the background perturbation matrices in both model and observation space are calculated globally. Then for each grid point the observation vector and associated background perturbation matrix mapped into observation space Y b are restricted to only include observations from a region around that point Fig.
Similarly the perturbation matrix and associated mean vector are restricted to only include model variables from the grid point. The output of the equations for each region only updates the centre grid point. For each grid point green dots, selected point is blue a localisation region is drawn blue circle and only those observations red triangles which are in the localised region are included in the analysis.
Cartoon based on that by Kalnay At each grid point an optimum combination of ensemble members are calculated compared to the data. These represent local multidimensional unstable manifolds Szunyogh, The global analysis is constructed by smoothly joining each of these together, i. Szunyogh describes how this combination can be made using a model propagator; however in the LETKF this combination relies upon the choice of the matrix square root in the construction of the analysis Wang et al.
The choice of the symmetric square root transformation forces consistency between adjacent local analyses since the transformation is a continuous function of the analysis error covariance matrix Hunt et al. This consistency does not occur if the transformation suggested in Bishop et al. As long as subsets of observations used for neighbouring grid points overlap heavily, the local weight vectors for the grid points are very similar. When weight vectors do not change much, the analysis ensemble members are roughly linear combinations of the ensemble members, and so the changes between neighbouring grid points remain reasonably physical.
The effectiveness of this smoothing as compared to the global approach i. The regions around each model grid point should be chosen such that their intersection is sufficiently large in comparison to the number of observations in each region. One issue with all EnKFs is undersampling, whereby the ensemble is too small to statistically represent the error covariance matrix.
Consequences of undersampling are spurious covariances and filter divergence Anderson, Spurious covariances between variables are ones where the variables are not physically related. These may be in parts of the state vector that are physically a large distance apart.
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Filter divergence is the term used to describe when the assimilation stops improving the analysis state Anderson, This can occur if the model variances become too large or too small in relation to the observation variance. The localisation component of the the LETKF helps to address filter divergence by improving the specification of the covariance matrix. One of the underlying assumptions of ensemble modelling is that the set of ensemble members span the space of the model variables. In reality the whole space is often not fully spanned, but it is necessary that it is a reasonable approximation.
By reducing the ensemble analysis to local regions, the ensemble members only need to span the region within those regions. The dimensionality of the reduced region is significantly less than the whole space and the approximation that the ensemble spans the space is a better one.
Hunt et al. The results showed that the global ETKF required an ensemble size of half the model dimension to be effective. In contrast doubling the dimension of the model had little effect on the performance of the LETKF, and a good performance was obtained for ensemble sizes greater than or equal to However the most abundant data source in the upper atmosphere is an integrated measurement of electron density called the total electron content TEC described in Sect. The optimal number of ensemble members balancing performance and computational cost is still to be determined.
Benchmarking test results have shown that using 32 ensemble members provides good agreement with independent observations. At each time step the continuity, energy, and momentum equations are solved for neutral and ion species using a fourth-order, centred finite difference scheme Roble et al.
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The model is driven by the daily F These values are related to Kp via Neale, :. The other species considered by the model also have their grids extended to the same height to ensure the altitude grid is the same across all species ; however all values above the TIE-GCM maximum altitude are set to zero. It has been designed to have continuously integrable vertical profiles which allows for rapid calculation of the total electron content for trans-ionospheric propagation applications.
As well as time and location, the model is also driven by F A semi-Epstein layer represents the model topside with a height-dependent thickness parameter that has been empirically determined Nava et al. The spread of the ensemble members should be sufficient to estimate the background covariance matrix Sect. The F By construction the mean of the F These values have been chosen based on the estimated correlation lengths in the ionosphere from McNamara Table 1.
Table 1 Ionospheric F region correlation lengths km from McNamara Any EnKF depends on the accuracy of the estimated background and observation error covariances and errors can effect the optimality of the data assimilation scheme Liang et al. Unfortunately, using the ensemble to estimate the covariance matrices can result in sampling errors as the number of ensemble members is limited.
Specifically the ensemble members become too similar.
Inria - Data assimilation with the Weighted Ensemble Kalman Filter
To mitigate this covariance inflation has been used. First proposed by Anderson and Anderson covariance inflation artificially increases the uncertainty in the background model. The required inflation value to use depends on the specific case both domain and ensemble size Hamill et al. Whilst there are various strategies for implementing covariance inflation e. Multiplicative inflation is more suited to the ionosphere-thermosphere problem than other inflation schemes due to the dynamic range of values involved. Since this is equivalent to multiplying X b by.
This can be achieved efficiently by replacing with in Equation 15 Hunt et al. It is well known that physics-based general circulation models GCMs suffer in data assimilation schemes when sporadically forced by data Baker et al. The sporadic nature of the assimilated data be it temporally or spatially can upset the intrinsic balance of the equations Ham et al.