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In atmosphere/ocean models, unresolved scales of motion are typically filtered out of the prognostic equations by way of balance approximations, with their effects accounted for separately by parametrizations.
Balance approximations make the modeling problem tractable, and also imply statistical relationships between variables and model grid points, which are exploited in data assimilation in order to optimize the use of observations.
These statistical relationships are typically contained in an error covariance field or matrix, which is a linearization and discretization of statistical relationships, and determines how information from observations is mapped to a forecast state.
In data assimilation, if the balance between (for example) wind and height fields is not sufficiently represented by the error covariance field, the insertion of observations can excite spurious gravity waves. At the same time, gravity waves which exist in the true state (and therefore appear, to some extent, in the observed state) can be wrongfully interpreted, if their presence is not properly accounted for in the assimilation. In physical regimes where gravity waves and vortical modes are not clearly separated in timescale (such as the tropics), it becomes very difficult to distinguish different types of motions, and to recover unobserved fields from observations. Balance relationships are difficult to capture within a covariance matrix, because they by nature are approximate, flow-dependent, and usually nonlinear and asymptotic. Spurious gravity waves can be avoided by adding a balance constraint to the analysis computation, but this is only useful insofar as the balance approximation used is accurate and the true state is actually balanced. In regimes where gravity waves comprise a significant part of the motion (such as the mesosphere) or where timescales are not clearly separated (such as the tropics), initialization can move the analysis further from the truth, and reject useful observations. Four-dimensional data assimilation (4DDA) schemes are designed to evolve the covariance model within the assimilation cycle itself, by evolving forecast errors in time and space, and then updating them with information brought in from observations. Because of this, 4D schemes should, in theory, allow us to forego the initialization step. In practice, however, assimilation schemes are built on approximations and assumptions which compromise the accuracy of the covariance model. Consequently, the twin problems of retaining balance and capturing imbalance remain. My Ph.D. research is concerned with using simplified models of balance dynamics to understand how these issues pan out in three fundamental 4DDA schemes: the Extended Kalman Filter (EKF), the Ensemble Kalman Filter (EnKF), and 4D Variational Assimilation (4DVAR). Low-order models are a good place to start examining the assimilation problem, because numerical results can be easily interpreted in a very transparent, analytical sense. This then makes the interpretation of realistic / operational systems easier. Simple models also allow us to run many experiments, customizing the physical regime and testing the sensitivity of the results to different parameters. Figure 1 shows average assimilation errors for a series of numerical experiments comparing the EKF, EnKF, and 4DVAR, as a function of the time between observations. Errors are divided into three components: total slow mode error, gravity wave magnitude error, and gravity wave phase error. As the time between observations, and thus the effective nonlinearity of the assimilation problem, increases, the relative performances of the three schemes change, and differently for each error component. This is interesting because the three algorithms are equal in the limits of perfect assumptions, but can perform very differently in practice. These plots show that the presence and gravity waves in particular is handled very differently by each algorithm. Theoretical reasons for this result and similar experiments are explored further in my Ph.D. thesis. |
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