To extend the examination of filtering performance presented in chapter 4 and to study the parallel efficiency of the filter algorithms, identical twin experiments are performed with the idealized configuration of FEOM. Synthetic observations only of the sea surface height are assimilated. The physical process which is simulated in the assimilation experiments is the propagation of interacting baroclinic Rossby waves. The waves are initialized with two horizontally localized columnar temperature anomalies of the same amplitude but opposite sign. This initialization is shown in figure 9.2.
Propagating westward, the anomalies become deformed. They tilt toward each other via the induced velocity field. That is, a negative temperature anomaly produces a counterclockwise rotation in the upper levels and a clockwise rotation in the lower levels. The rotation of a positive temperature anomaly is vice versa. These opposing rotations introduce non-linearity which is necessary to test the filtering performance of the error subspace Kalman filters.
The data assimilation experiments are conducted over a period of 40 days. The interval between subsequent analyses is set to 2.5 days. For the twin experiments the “true” state trajectory is generated by integrating the initialization displayed in figure 9.2 over a period of 45 days. To generate synthetic observations of the sea surface height, Gaussian noise with constant variance of 0.01 m2 is added at each time step to the sea surface height field of the true state sequence. The amplitude of the temperature anomalies, and thus of the sea surface height, decreases over time.
This is caused by diffusion. Hence, the relative noise amplitude of the observations increases during the assimilation period. Initially the standard deviation of the noise in the observations is at about 20 percent of the amplitude of the true surface height.
After 45 days, the errors in the observations increased to about the same level of the surface height itself. The generated observations are used with an offset of 5 days in model time. Assimilating only observation of the sea surface height, the dimension of the observation vector amounts to m = 961. Figure 9.3 compares the observed sea surface height field with the true one at the initial time of the experiments. The observation errors are clearly visible, but also the observed information is apparent.
To initialize the filter in the twin experiments, the covariance matrix of 2268 state vectors is computed. These vectors are generated by 28 model forecasts using different initial locations of the temperature anomalies. Further, an additional variance of the
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Figure 9.1: FEOM model grid used for the data assimilation experiments. It consists of 10571 nodes. Vertical levels are at the surface and in the following depths: 7.5, 20, 50, 100, 500, 1000, 2000, 3000, 3800, and 4000 meters. The coloring shows the linear temperature stratification.
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Figure 9.2: Cut into the model grid showing the temperature anomalies.
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Figure 9.3: Comparison of the true (left) and the observed (right) sea surface height field ζ at the initial analysis update.
sea surface height fields of 0.1m is assumed. The obtained covariance matrix, which describes the temporal variations and correlations of the model fields, is used as the initial error estimate in the filter experiments. The initial state estimate for the twin experiments is chosen as the mean state of the 28 model runs. The generation of the state ensembles for SEIK and EnKF and the initialization of the mode matrix for SEEK is performed as described for the experiments with the shallow-water-equation model in chapter 4. To examine the abilities of the filter algorithms to estimate the true state from the chosen initial state, an evolution of the initial state estimate is performed without assimilating observations. This state sequence is denoted the “free” state trajectory.
To simulate model errors in the application of the EnKF and SEIK filters, a wind forcing field of two gyres is applied whose shape and amplitude are controlled by two parameters. To obtain a stochastic forcing, these parameters are initialized by ran-dom numbers. Each ensemble member was forced by a different wind field which was constant over the forecast period. To retain comparability, the SEEK filter was used without a forgetting factor, since this could be applied to all three filters, or explicit treatment of a model error covariance matrix. Thus, the twin experiments using SEEK are performed without consideration of model errors.
Most of the computation time is spent in evolving the model states. Since the computation time is usually a limiting factor in data assimilation problems, results for assimilation experiments are compared in which all filters perform the same number of model evaluations. This configuration provides also comparable execution times for assessing the parallel efficiency of the three filter algorithms. To obtain configurations with equal numbers of model evaluations, the rank r used in SEEK and SEIK is set to r=N −1 where N is the ensemble size of the EnKF.
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.