A numerical integration-based Kalman filter for moderately nonlinear systems

Figures & data

Fig. 1. The L² error in the estimated (a) EnKF mean (3.15) for k = 5, (b) EnKF mean for k = 10, (c) EnKF mean for k = 100, and (d) AGR filter mean (3.6) for increasing values of c₂ (horizontal axis) and c₄ (vertical axis). Note that the color scales are different between the first two plots (a) & (b) and the second two (c) & (d).

Fig. 2. The L² error in (a) EnKF prior covariance estimate for k = 100 and (b) the error in the AGR filter prior covariance estimate for increasing values of c₂ (horizontal axis) and c₄ (vertical axis).

Fig. 3. Contour plot of the wave amplitude over the domain (vertical axis) of the KdV equation over time (horizontal axis).

Fig. 4. The 512 sorted eigenvalues of the initial background error covariance $P_0^b$ created from a 35000 member climatological ensemble. The horizontal axis is the eigenvalue number and the vertical axis is the magnitude of each eigenvalue.

Fig. 5. (a) The L₂ error (vertical axis) in the estimate of the prior mean for the EnKF with k = 500, the AGR1 with m = 250, and AGR2 with m = 250 for time step length t₀ (horizontal axis). (b)The error in the Frobenius norm of the corresponding covariance estimates.

Fig. 6. (a) The L₂ error (vertical axis) in the estimate of the prior mean for the EnKF with k = 40, the AGR1 with m = 20, and AGR2 with m = 20 for time step length t₀ (horizontal axis). (b) The corresponding error in the Forbenius norm for the prior covariance estimates.

Fig. 7. The error in the Frobenius norm (vertical axis) of the prior covariance estimates in the AGR2 filter for m = 20 with $P_0^b$ computed using $k_e=40,80,160,35000$ ensemble members for time step length t₀ (horizontal axis).

Fig. 8. The L² error (vertical axis) averaged over the assimilation window for increasing t₀ (horizontal axis), the time between cycles, for the EnKF and the AGR2 filter.

Fig. 9. The true solution of (5.1) at time t = 15000, or 200 cycles of 75 seconds, for (a) vorticity and (b) temperature. The vertical axis is the height and the horizontal axis is the distance.

Fig. 10. The averaged L² error (vertical axis) across the data assimilation window in the mean estimates for particular t₀ (horizontal axis) for (a) vorticity and (b) temperature.