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A practical quadratic ensemble filter

Le Duc (JAMSTEC)

The ensemble Kalman filter is constructed from the linear filter using a finite sample to estimate the first and second moments of the prior and posterior distributions. The linear filter assumes a linear relationship between the estimated state and the innovation vector y-h[x] through an operator. The operator at which the second moment of the estimated state attains the minimum value is called the Kalman gain. As a result, the Kalman gain and the posterior distribution are estimated from the first two moments of the prior distribution.

In the quadratic filter, the innovation vector is augmented to include the second order tensor product of itself (y-h[x])?(y-h[x]). This enables the additional use of the third and fourth moments of the prior distribution in estimation of the posterior distribution, which is preferable for non-Gaussian distributions. However, this augmentation increases the size of the Kalman gain by a factor equivalent to the size of the innovation vector, thus increasing the computational cost considerably. To reduce the computational cost, we simplify the quadratic filter by using only the diagonal terms of the tensor product (y-h[x])?(y-h[x]) in its estimation. This simplified quadratic filter can be adapted to the ensemble form similar to the ensemble Kalman filter. Experiments with the JMA-NHM model using real observations will be shown.