Fig. 1
Top: drifter paths for the centre case (left) and the saddle case (right). Note: in both cases a snapshot of the corresponding velocity field is plotted at t=0. Bottom: height field evaluated along drifter paths corresponding to the fields and trajectories shown directly above.
Fig. 2
Plotted above are marginal prior and posterior distributions for the saddle case, ‘truth’. Blue dots: projected samples of the prior distribution into the and planes. Pink–black dots: discrete values of posterior distribution, location in plane is state value while colour is weight (pink=0 to black=1). Top: posterior distributions from assimilating standard Lagrangian DA. Bottom: posterior distributions from assimilating en-route observations in addition to standard Lagrangian DA.
Fig. 3
In all figures, solid curves are the mean of assimilating 500 different sets of observations for the same initial condition cases – left column is the centre case, right column is the saddle case. In all cases, solid curves include en-route observations collected along the drifter's path, while dashed lines represent standard Lagrangian data assimilation. Top: ratio of determinants of the posterior distribution to the prior distribution plotted against time. Bottom: as a function of time.
Table 1. The averages of and at t=5 and fractions of failure rate for 500 sets of data for the centre and the saddle case, with high and low observational noise levels for using sequential Monte Carlo
d a , K=02.9282.9262.8912.890d a , K=92.9532.9612.9082.920d a , K=992.9542.9612.9112.9211.26×10−51.34×10−52.98×10−52.95×10−53.46×10−61.70×10−61.82×10−59.77×10−63.34×10−61.62×10−61.66×10−59.30×10−6Posterior err, K=00.270.260.110.09Posterior err, K=90.140.140.040.03Posterior err, K=990.150.150.050.03
Fig. 4
In both figures, all curves are the mean of RMS error of the height field calculated for 500 different sets of observations for the same initial condition cases – left is the centre case, right is the saddle case. The RMS error compares the difference between the true height field and a ‘median’ height field found through assimilation. In both cases, solid curves correspond to assimilation of en-route observations collected along the drifter's path while dashed lines represent standard Lagrangian data assimilation. For reference, the lightly shaded curves represent RMS error if the 5% percentile height field is compared to the true field.
Fig. 5
Plotted above are marginal prior and posterior distributions for the saddle case: , , , is the ‘truth’. In this case, each sample has equal weight and the posterior is simply proportional to the density of samples. Blue dots correspond to the prior, red for pure Lagrangian data, green for K=9 height observations, magenta for K=99, and black for K=999. The bottom-right panel shows the non-convergence rate of the MCMC.
Fig. 6
This figure shows the same metrics as in Fig. 3 as a function of time, but for the case of the short trajectory and using the smoothing method. The top row is for the three variables with low noise in height observations while the middle row is for the three variables with high height observation noise: . The bottom row shows the low noise case but for the posterior on all the six variables: . The two lines for each of the cases correspond to two different realisations of the MCMC chain.