Institute of Geophysics, University of TehranJournal of the Earth and Space Physics2538-371X39120130421A comparative study of the Rossby-wave packet detection algorithmsA comparative study of the Rossby-wave packet detection algorithms1751903195410.22059/jesphys.2013.31954FAAlirezaMohebalhojehAssociate Professor, Space Physics Department, Institute of Geophysics, University of Tehran, IranSaraKaramiM.Sc. Graduate, Space Physics Department, Institute of Geophysics, University of Tehran, IranJournal Article20120522 <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /><sup>*</sup>نگارنده رابط: تلفن: 61118325-021 دورنگار: 88009560-021 E-mail:amoheb@ut.ac.ir <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br />Previous studies have shown that Rossby waves tend to be organized in the form of wave packets, especially in the upper troposphere. It is easier to track the wave packets than the individual troughs and ridges. The most common way to analyze wave packets is to determine their envelopes. In this study, we derive the envelope of wave packets using the three algorithms of complex demodulation, one-dimensional Hilbert transform, and the Hilbert transform with a meridional Fourier filtering. The complex demodulation and the one-dimensional Hilbert transform have previously been used in several studies. While addressing the limitation of the complex demodulation to a fixed, pre-assigned zonal wavenumber, the one-dimensional Hilbert transform may lead to erroneous results in cases when there is significant non-zonal wave propagation. A manifestation of the erroneous results is the appearance of two separate envelopes at nearby latitudes when in fact there is only one wave packet. In the literature an extension of the one-dimensional Hilbert transform has been developed to deal with such cases. The resulting algorithm, however, assumes that the waves propagate along streamlines, which is not generally the case. Since the short, meridional waves are believed to be responsible for the erroneous appearance of two nearby wave packets, it is shown that to avoid generation of such fictitious features, it suffices to augment the one-dimensional Hilbert transform by a meridional filter. The meridional filter is constructed using double Fourier transform on the sphere and its cut-off wavenumber is chosen in such a way as to achieve the desirable properties. <br />The data from the Global Forecast System (GFS) for the winter season covering December 2004, January and February 2005 are used. The envelope is computed for the perturbation meridional velocity at 300 hPa where perturbation is taken to be deviation from the seasonal mean. Based on their propagation characteristics, two particular wave packets are selected. <br />Crossing the North America on the 17<sup>th</sup> of Jan., the first wave packet propagates across the North Atlantic where its amplitude increases due to diabatic processes. Weakening over the continent, this wave packet reaches Asia and splits in two branches: whereas the northern branch is located over Russia (north of the Mediterranean and Caspian seas) with its maximum amplitude in, the southern branch crosses the North Africa and then the southern Asia with its maximum amplitude in. The southern branch remains coherent over a longer time and propagates into the North Pacific. Such splitting is consistent with the previous results published in the literature. Due to the action of baroclinic instability, the waves over the North pacific attain much higher frequency than the waves over the southern Asia. This case provides a clear example of a wave packet that can propagate across the whole Atlantic storm track, reach and then seed the Pacific storm track. First detected over the North Pacific on the 18<sup>th</sup> of Dec., the second wave packet crosses the North America and enters into the North Atlantic where its amplitude is increased substantially. This wave packet then reaches the North Africa and subsequently the southern Asia where undergoes gradual weakening and actually disappears on the 4<sup>th</sup> of Jan., making its life time 16 days. For the latter two wave packets as well as in terms of statistics for the whole winter season, the envelopes derived from each of the three wave-packet detection algorithms are compared qualitatively and quantitatively. <br />Comparison of the algorithms shows that because of the use of a single wave number in complex demodulation, the envelopes derived from complex demodulation are stronger than those derived from the one-dimensional Hilbert transform. This exhibits itself in positive values of the seasonal and latitudinal mean difference between the complex demodulation and the one-dimensional Hilbert transform. There are cases where using the complex demodulation and the one-dimensional Hilbert transform algorithms two separate envelopes are derived near to each other that actually belong in one wave packet when use is made of the Hilbert transform with meridional filtering. With a cut-off wavenumber of 9 for meridional filtering, some of the details are lost because of the coarse filtering. Comparison of the envelopes derived from the Hilbert transform with meridional filtering and the one-dimensional Hilbert transform shows that the seasonal and latitudinal mean differences are very small in all latitudes, if a cut-off wavenumber of 12 is used for meridional filtering leading to the best algorithm for detecting the envelopes. <br />As a final remark, it is worth mentioning that the general form of a two-dimensional Hilbert transform on the sphere has been introduced by Fleischmann et al. )2010(. It remains to be seen if the implementation and application of such transform can lead to an improvement over the one-dimensional Hilbert transform with meridional filtering. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /><sup>*</sup>نگارنده رابط: تلفن: 61118325-021 دورنگار: 88009560-021 E-mail:amoheb@ut.ac.ir <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br />Previous studies have shown that Rossby waves tend to be organized in the form of wave packets, especially in the upper troposphere. It is easier to track the wave packets than the individual troughs and ridges. The most common way to analyze wave packets is to determine their envelopes. In this study, we derive the envelope of wave packets using the three algorithms of complex demodulation, one-dimensional Hilbert transform, and the Hilbert transform with a meridional Fourier filtering. The complex demodulation and the one-dimensional Hilbert transform have previously been used in several studies. While addressing the limitation of the complex demodulation to a fixed, pre-assigned zonal wavenumber, the one-dimensional Hilbert transform may lead to erroneous results in cases when there is significant non-zonal wave propagation. A manifestation of the erroneous results is the appearance of two separate envelopes at nearby latitudes when in fact there is only one wave packet. In the literature an extension of the one-dimensional Hilbert transform has been developed to deal with such cases. The resulting algorithm, however, assumes that the waves propagate along streamlines, which is not generally the case. Since the short, meridional waves are believed to be responsible for the erroneous appearance of two nearby wave packets, it is shown that to avoid generation of such fictitious features, it suffices to augment the one-dimensional Hilbert transform by a meridional filter. The meridional filter is constructed using double Fourier transform on the sphere and its cut-off wavenumber is chosen in such a way as to achieve the desirable properties. <br />The data from the Global Forecast System (GFS) for the winter season covering December 2004, January and February 2005 are used. The envelope is computed for the perturbation meridional velocity at 300 hPa where perturbation is taken to be deviation from the seasonal mean. Based on their propagation characteristics, two particular wave packets are selected. <br />Crossing the North America on the 17<sup>th</sup> of Jan., the first wave packet propagates across the North Atlantic where its amplitude increases due to diabatic processes. Weakening over the continent, this wave packet reaches Asia and splits in two branches: whereas the northern branch is located over Russia (north of the Mediterranean and Caspian seas) with its maximum amplitude in, the southern branch crosses the North Africa and then the southern Asia with its maximum amplitude in. The southern branch remains coherent over a longer time and propagates into the North Pacific. Such splitting is consistent with the previous results published in the literature. Due to the action of baroclinic instability, the waves over the North pacific attain much higher frequency than the waves over the southern Asia. This case provides a clear example of a wave packet that can propagate across the whole Atlantic storm track, reach and then seed the Pacific storm track. First detected over the North Pacific on the 18<sup>th</sup> of Dec., the second wave packet crosses the North America and enters into the North Atlantic where its amplitude is increased substantially. This wave packet then reaches the North Africa and subsequently the southern Asia where undergoes gradual weakening and actually disappears on the 4<sup>th</sup> of Jan., making its life time 16 days. For the latter two wave packets as well as in terms of statistics for the whole winter season, the envelopes derived from each of the three wave-packet detection algorithms are compared qualitatively and quantitatively. <br />Comparison of the algorithms shows that because of the use of a single wave number in complex demodulation, the envelopes derived from complex demodulation are stronger than those derived from the one-dimensional Hilbert transform. This exhibits itself in positive values of the seasonal and latitudinal mean difference between the complex demodulation and the one-dimensional Hilbert transform. There are cases where using the complex demodulation and the one-dimensional Hilbert transform algorithms two separate envelopes are derived near to each other that actually belong in one wave packet when use is made of the Hilbert transform with meridional filtering. With a cut-off wavenumber of 9 for meridional filtering, some of the details are lost because of the coarse filtering. Comparison of the envelopes derived from the Hilbert transform with meridional filtering and the one-dimensional Hilbert transform shows that the seasonal and latitudinal mean differences are very small in all latitudes, if a cut-off wavenumber of 12 is used for meridional filtering leading to the best algorithm for detecting the envelopes. <br />As a final remark, it is worth mentioning that the general form of a two-dimensional Hilbert transform on the sphere has been introduced by Fleischmann et al. )2010(. It remains to be seen if the implementation and application of such transform can lead to an improvement over the one-dimensional Hilbert transform with meridional filtering. https://jesphys.ut.ac.ir/article_31954_52c49fe0149cbe8d7fd19fc7d3caf5ef.pdf