The experiments have been performed on a Sun Fire 6800 server with 24 proces-sors, each having a frequency of 1050 MHz. The experiments in section 9.4 used the solver PILUT while the experiments in section 9.5 used PETSc. This different choice was motivated by the fact that the use of PILUT resulted in inferior speedup values than PETSc. In contrast to this, the assimilation experiments with the PILUT solver provided a better filtering performance than those using PETSc. Since this work is not aimed at the optimization of the model, the solver was chosen depending on the best results either in terms of filtering performance or in terms of speedup.
EnKF algorithm reproduces accurately the shape of the true ζ. The locations of the minimum and the maximum are well estimated. The amplitudes are underestimated by about 10%. In contrast to this, the sea surface height without assimilation deviates strongly from both the true and SEIK-estimated ζ.
The velocity componentsu and v are updated via the estimated cross correlations between the sea surface height and the velocity components. Despite this, the relative estimation errors of the meridional velocity component u are of comparable size to
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Figure 9.4: Time dependence of the relative estimate errors E1 for experiments with N = 60. Shown isE1 separately for the sea surface heightζ (top left), the temperature field T (top right), and the two components u, v (respectively on the left and right hand sides of the bottom row) of the velocity fields.
those of the sea surface height in the case of EnKF and SEIK. This relation shows, that the cross covariances are estimated quite well by the nonlinear ensemble forecast. In contrast to this, the linearized forecast performed in SEEK yields much worse estimates of the cross covariances. This can be deduced from the much larger estimation errors for u obtained with SEEK.
The estimate of the zonal velocity component v is less precise than the estimate of u for all three filters. After the first analysis phase, the estimation error of both velocity components is of comparable size. While the estimation error foru decreases during the course of the assimilation experiment, the estimation error for v remains at a level of about 0.4 when using the EnKF or the SEIK filter. Thus, the cross co-variances are not estimated sufficiently precise to further decrease the error level for this velocity component. During some analysis updates, e.g. at day 25, the estimation error increases. In this case the estimated cross covariances have the wrong sign.
The relative estimation error of the temperature field T shows a behavior distinct from the other model fields. The error reduction at the first analysis update is smaller forT than for the other fields. For the EnKF and SEIK filters, the relative estimation error of the temperature field increases immediately after the first analysis update.
Further, the estimation error remains almost unchanged during the analysis update.
Thus, no useful estimates of the cross correlations are available after the first analy-sis update. The estimates of variances and correlations within some model field are typically much more precise than estimated cross correlations. Thus, even a limited number of temperature measurements would enhance the estimation quality of the temperature field for all three filters.
9.4.2 Estimation of 3-dimensional Fields
To examine the ability of the filter algorithms to estimate the 3-dimensional model fields by assimilating only surface measurements profiles of the relative estimation errors at the end of the assimilation period are shown in figure 9.5. The values displayed in the diagrams are the normalized rms estimation errors computed over single levels of the model.
The profiles for the two velocity components u and v, displayed on the left and middle panels, show a small relative estimation error from the surface to -1000m depth.
Below -3000m the estimation error is a also small, but it increases toward the bottom.
At the depth of -2000m the estimation error shows a maximum. For the experiments with SEIK and EnKF, this maximum is even larger than one. The estimation errors obtained with SEEK are of similar size to those achieved by the EnKF and SEIK filters. They are, however, larger at all depths, except at -2000m. For all three filters, the relative estimation errors are smaller for the meridional velocity componentuthan for the zonal velocity v.
The peak in the relative estimation error at the depth of -2000m is due to the normalization by the estimation error of the evolution without assimilation. As has been described in section 9.3, the temperature anomalies generate a counterclockwise rotation in the upper levels and a clockwise rotation in the lower levels. The turning
point of these rotations is approximately at the depth of -2000m. Due to this, the velocities are minimal at this depth in the true state, the free state and the assimilated states. This causes minimal rms deviations of the velocities of the free evolution from the velocities of the true evolution. Without normalization, the estimation errors of the assimilated velocities are of comparable size to those of the non-assimilated velocities at -2000m depth. Due to the normalization, the estimation errors appear larger than their absolute value.
The increase of the relative estimation error below -3000m is not due to the normal-ization, as the absolute estimation errors also increase below -2000m depth. Thus, the quality of covariances between the sea surface height and the velocity fields is worse in the deep ocean than for the upper levels. Overall, all three filters show good abilities to reduce the estimation error of the velocity field also in the lower levels of the model.
The level -2000m appears to be a rather pathological situation which the algorithms cannot handle well.
The profile of the relative estimation errors of the temperature field, shown on the right hand side of figure 9.5, exhibits a different dependence on depth than the estimation errors of the velocity field. In the uppermost levels the estimation error of
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Figure 9.5: Profiles of the rms estimation errors of single layers normalized by the corresponding rms deviation of the free state from the true state for N = 60. Shown are the two components u, vof the velocity fields and the values for the temperature field T at the end of the assimilation period.
the temperature field is not reduced by the SEIK and EnKF algorithms. In contrast to this, the relative estimation error is decreased to a level of about 0.8 when the SEEK filter is applied. Between -100m and -2000m all three filters reduce the estimation error to similar level of about 0.85. Below -2000m the relative estimation error increases for all three filter algorithms to a level around unity.
The large relative estimation errors in the uppermost 100 meters are misleading.
This becomes apparent from the panels on the right hand side of figure 9.6. The uppermost panel shows the true temperature field at a depth of -50m. The panel in the middle shows the temperature field as estimated by the EnKF with N = 60.
For comparison, the free temperature field is displayed in the lowermost panel. The shape of the estimate from the EnKF reproduces the shape of the true temperature field rather well. The amplitude of the positive temperature spot is, however, over-estimated. The free temperature field is distinct by showing only a single positive temperature anomaly.
In the level at -500m and below the temperatures are generally over-estimated by about 0.1◦C. This is displayed in figure 9.7 which shows the temperature fields analogous to the right hand side of figure 9.6 for the levels at -1000m and -3800m.
While the shape of the estimated temperature field is still reasonable at -1000m, this is no more the case for the level at -3800m. Here, the estimate resembles the shape of the free temperature field which is obtained from the evolution of the state estimate without assimilating observations. The assimilation has only a small influence on the temperature field at -3800m. Namely, the warm area with temperatures above 6.3◦C is shifted further to the north-east. In addition, the temperature is decreased around (44◦N, 7◦E).
Overall, the three filter algorithms show a very limited ability to estimate the tem-perature correctly when only measurements of the sea surface height are assimilated.
The shape of the temperature field is reproduced by the estimates in the upper 1000 meters. However, there is a bias in the temperature estimates. Due to this, addi-tional temperature measurements, also in the depth, would be useful to obtain better estimates of the temperature field.