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Gerd Bürger

Abstract

In a recent paper, Maraun describes the adverse effects of quantile mapping on downscaling. He argues that when large-scale GCM variables are rescaled directly to small-scale fields or even station data, genuine small-scale covariability is lost and replaced by uniform variability inherited from the larger scales. This leads to a misrepresentation mainly of areal means and long-term trends. This comment acknowledges the former point, although the argument is relatively old, but disagrees with the latter, showing that grid-size long-term trends can be different from local trends. Finally, because it is partly incorrectly addressed, some clarification is added regarding the inflation issue, stressing that neither randomization nor inflation is free of unverified assumptions.

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Gerd Bürger

Abstract

Complex principal oscillation pattern (CPOP) analysis is introduced as an extension of conventional POP analysis. Both are intended to resolve regular evolving patterns from processes with many degrees of freedom. While POP analysis, like many other techniques, deals with the concept of the system state as a real vector, it is argued that this notion be extended into the complex domain. The approach used here results from a critical review of the theory of linear systems of first order. It turns out that these systems cannot appropriately model standing oscillations. The notion of the traveling rate of a mode is defined, and it is demonstrated that the mode's frequency and traveling rate are directly coupled via the system matrix. One consequence is that clean standing oscillations cannot be modeled by linear systems of first order.

CPOP analysis introduces a new vector of state. By defining the complex state “state + i · momentum,” both the conventional state itself and its momentum are simultaneously described. The method is capable of resolving oscillatory patterns of any given traveling rate from a stationary process. First experiments show that the CP0Ps evolve more regularly and with less noise than corresponding P0Ps. A prediction scheme that is appropriate for CP0Ps is defined by introducing a transformation technique that can be considered as a causal Hilbert transform. With this scheme prediction skills that are significantly stronger than those of the POP model are gained.

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Gerd Bürger

Abstract

Indices of oscillatory behavior are conveniently obtained by projecting the fields in question into a phase space of a few (mostly just two) dimensions; empirical orthogonal functions (EOFs) or other, more dynamical, modes are typically used for the projection. If sufficiently coherent and in quadrature, the projected variables simply describe a rotating vector in the phase space, which then serves as the basis for predictions. Using the boreal summer intraseasonal oscillation (BSISO) as a test case, an alternative procedure is introduced: it augments the original fields with their Hilbert transform (HT) to form a complex series and projects it onto its (single) dominant EOF. The real and imaginary parts of the corresponding complex pattern and index are compared with those of the original (real) EOF. The new index explains slightly less variance of the physical fields than the original, but it is much more coherent, partly from its use of future information by the HT. Because the latter is in the way of real-time monitoring, the index can only be used in cases with predicted physical fields, for which it promises to be superior. By developing a causal approximation of the HT, a real-time variant of the index is obtained whose coherency is comparable to the noncausal version, but with smaller explained variance of the physical fields. In test cases the new index compares well to other indices of BSISO. The potential for using both indices as an alternative is discussed.

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Gerd Bürger, Stephen E. Zebiak, and Mark A. Cane

Abstract

In Part II of this study on the application of the interactive Kalman filter to higher-dimensional systems, a modification suited to periodically forced systems is introduced. As in Part I, the object of study here is the ENSO model of Zebiak and Cane, but here the technique of quasi-fixed points is applied to certain Poincare maps of that system that are related to the forcing period of 1 year. As a result, it is possible to search the model systematically for possible periodic orbits, no matter whether they are stable or unstable. An unstable 4-year cycle is found in the model, and it is argued that this cycle can be traced back to a 4-year limit cycle, which is known to exist under weak atmosphere–ocean coupling. All other quasi-fixed points are related to orbits that do not appear to be periodic. The findings are applied to the modified version of the interactive Kalman filter, which deals with cycle as regimes. Comparing these results with the findings in Part I, it is found that the filter performances improve using, in the following order, the extended filter, the interactive filter with cycles, a seasonal average film, and the original interactive Kalman filter from Part I.

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Gerd Bürger, Stephen E. Zebiak, and Mark A. Cane

Abstract

In an effort to apply the interactive Kalman filter to higher-dimensional systems, the concept of a quasi-fixed point is introduced. This is defined to be a system state where the tendency, in a suitable reduced space, is at a minimum. It allows one to use conventional search algorithms for the detection of quasi-fixed points. In Part I quasi-fixed points of the ENSO model of Zebiak and Cane are found when run in a permanent monthly mode, the reduced space being defined via a multiple EOP projection. The stability characteristics of the quasi-fixed points are analyzed, and it is shown that they are significantly different from the (in)stabilities of the average monthly models. With these quasi-fixed points, assimilation experiments are carried out with the interactive Kalman filter for the Zebiak–Cane model in the reduced space. It is demonstrated that the results are superior to both a seasonal Kalman filter and the extended Kalman filter.

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Hans von Storch, Gerd Bürger, Reiner Schnur, and Jin-Song von Storch

Abstract

The principal oscillation pattern (POP) analysis is a technique used to simultaneously infer the characteristic patterns and timescales of a vector time series. The POPs may be seen as the normal modes of a linearized system whose system matrix is estimated from data.

The concept of POP analysis is reviewed. Examples are used to illustrate the potential of the POP technique. The best defined POPs of tropospheric day-to-day variability coincide with the most unstable modes derived from linearized theory. POPs can be derived even from a space-time subset of data. POPs are successful in identifying two independent modes with similar timescales in the same dataset.

The POP method can also produce forecasts that may potentially be used as a reference for other forecast models.

The conventional POP analysis technique has been generalized in various ways. In the cyclostationary POP analysis, the estimated system matrix is allowed to vary deterministically with an externally forced cycle. In the complex POP analysis, not only the state of the system but also its “momentum” is modeled.

Associated correlation patterns are a useful tool to describe the appearance of a signal previously identified by a POP analysis in other parameters.

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