Fig. 1
(a) Data assimilation with a forecast spread adjustment of η. (b) Standard data assimilation with R o inflated by η 2. The two data assimilation cycles depicted above are initialised during the forecast phase, that is, .
Fig. 2
(a) The data assimilation cycle on P b and P a including a forecast spread adjustment of η and inflation of P b with ρ. (b) The data assimilation cycle on P b and P a including inflation of P b with ρ b and inflation of P a with ρ a.
Fig. 3
Sample Model II output with observations for 60 grid points, half of which are observed (*), observation error σ=1, smoothing parameter K=2 and forcing constant F=12.
Fig. 4
Tuning η (grey) and the effect of η on ensemble spread during the forecast phase (red) and analysis phase (blue) for our default parameters: k=10, F=14 (F t=12), σ=1. Comparison of the ensemble spread during the analysis phase (blue) to the actual RMSE for various values of η (grey). The solid grey curve shows that η tunes to 2.5. We tuned η using the tuned value of ρ for η=1.
Fig. 5
Tuning ρ and the effect of ρ on ensemble spread for our default parameters with η=1. The dotted lines correspond to the background and the solid lines correspond to the analysis. The spread is indicated in black and the RMSE in grey.
Table 1. Tuned values of ρ for LETKF with various parameters
Default parameters101411.20Vary k51411.20101411.20201411.17401411.15Vary F10911.19101011.17101111.13101211.06101311.13101411.20101511.22Vary σ10140.51.25101411.20101421.14
Fig. 6
(a–d) The analysis RMSE as a function of η for: (a) our default parameters:an ensemble with k=10 members, model error due to the forcing constant F=14 where the true forcing is F t=12, and observation error of 1; (b) various ensemble sizes: k=5, 10, 20 and 40 members; (c) different amounts of model error: F=9, 10, 11, 12, 13, 14 and 15 with F t=12; (d) different amounts of observation error: σ=0.5, 1 and 2. In each of (b), (c) and (d), one of the parameters from (a) (k=10, F=14, σ=1) is varied, keeping the other two parameters at their default values; the curve graphed in (a) appears in (b), (c) and (d) as well. We restrict the range of the y-axis in (a) to better show the structure of the curve. (e–h) Comparing the RMSE when η=1 to the minimum RMSE when η is allowed to vary. Subplot (e) shows the percent improvement (of the optimal η versus η=1) in the RMSE with our default parameters, that is, setting η=2.5 gives results with an RMSE, that is, 14% smaller than when η=1. Also shown are the percent improvements versus (f) various ensemble sizes, (g) different forcing constants and (h) different amounts of observation error. Each right hand figure (e), (f), (g) and (h) is derived from the same data as the figure immediately to its left, that is, (a), (b), (c) and (d), respectively.
Table 2. Comparison between the XKF RMSE and the ηETKF RMSEs for and 10−6 and k=40, 60 and 80 with our default parameters
η=10.880.770.76η=1/20.950.820.81η=1/40.970.830.82η=1/80.970.830.83η=10−60.960.830.83XKF0.83
Fig. 7
RMSE of the mean of a 48-hour η-adjusted ensemble forecast versus the truth. Panels (a), (b), (c) and (d) correspond to the same parameters as in Fig. 6.