Robert P. Lowe

Localization of the complex spectrum: the S transform

The S transform, which is introduced in the present correspondence, is an extension of the ideas of the continuous wavelet transform (CWT) and is based on a moving and scalable localizing Gaussian window. It is shown to have some desirable characteristics that are absent in the continuous wavelet transform. The S transform is unique in that it provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum. These advantages of the S transform are due to the fact that the modulating sinusoids are fixed with respect to the time axis, whereas the localizing scalable Gaussian window dilates and translates.

Review of mesospheric temperature trends

In recent times it has become increasingly clear that releases of trace gases from human activity have a potential for causing change in the upper atmosphere. However, our knowledge of systematic changes and trends in the temperature of the mesosphere and lower thermosphere is relatively limited compared to the Earths lower atmosphere, and not much effort has been made to synthesize these results so far. In this article, a comprehensive review of long-term trends in the temperature of the region from 50 to 100 km is made on the basis of the available up-to-date understanding of measurements and model calculations. An objective evaluation of the available data sets is attempted, and important uncertainly factors are discussed. Some natural variability factors, which are likely to play a role in modulating temperature trends, are also briefly touched upon. There are a growing number of experimental results centered on, or consistent with, zero temperature trend in the mesopause region (80–100 km). The most reliable data sets show no significant trend but an uncertainty of at least 2 K/decade. On the other hand, a majority of studies indicate negative trends in the lower and middle mesosphere with an amplitude of a few degrees (2–3 K) per decade. In tropical latitudes the cooling trend increases in the upper mesosphere. The most recent general circulation models indicate increased cooling closer to both poles in the middle mesosphere and a decrease in cooling toward the summer pole in the upper mesosphere. Quantitatively, the simulated cooling trend in the middle mesosphere produced only by CO 2 increase is usually below the observed level. However, including other greenhouse gases and taking into account a “thermal shrinking” of the upper atmosphere result in a cooling of a few degrees per decade. This is close to the lower limit of the observed nonzero trends. In the mesopause region, recent model simulations produce trends, usually below 1 K/decade, that appear to be consistent with most observations in this region

Pattern analysis with two-dimensional spectral localisation: Applications of two-dimensional S transforms

Abstract An image is a function, f(x, y) , of the independent space variables x and y . The global Fourier spectrum of the image is a complex function F(k x , k y ) of the wave numbers k x and k y . The global spectrum may be viewed as a construct of the spectra of an arbitrary number of segments of f(x, y) , leading to the concept of a local spectrum at every point of f(x, y) . The two-dimensional S transform is introduced here as a method of computation of the local spectrum at every point of an image. In addition to the variables x and y , the 2-D S transform retains the variables k x and k y , being a complex function of four variables. Visualisation of a function of four variables is difficult. We skirt around this by removing one degree of freedom, through examination of ‘slices’. Each slice of the 2-D S transform would then be a complex function of three variables, with separate amplitude and phase components. By ranging through judiciously chosen slice locations the entire S transform can be examined. Images with strictly periodic patterns are best analysed with a global Fourier spectrum. On the other hand, the 2-D S transform would be more useful in spectral characterisation of aperiodic or random patterns.

Local S Spectrum Analysis of 1-D and 2-D Data

Abstract The local changes of the spectrum with time are often more interesting than the spectrum of the whole time series. For example, there is an apparent drift in the nominal 28 day fluctuations of sunspot numbers over the period of the sunspot cycle, averaging ∼ 11.1 years. This time-local change in spectrum is due to a combination of Sporers Law and the differential rotation of the sun. Similarly, the space-local variations in the 2-D spectrum on an image conveys visual information on textures, boundaries and shapes. In this paper we use the recently developed S -transform to analyse two segments of the Wolf Sunspot series, a seismogram, and a synthetic 2-D image as examples of applications of the S -transform for time-local and space-local spectral analysis.

The Wind Imaging Interferometer (WINDII) on the Upper Atmosphere Research Satellite: A 20 year perspective

The Wind Imaging Interferometer (WINDII) was launched on the NASAs Upper Atmosphere Research Satellite on 12 September 1991 and operated until 2003. Its role in the mission was to measure vector winds in the Earths atmosphere from 80 to 110 km, but its measurements extended to nearly 300 km. The approach employed was to measure Doppler shifts from a suite of visible region airglow lines emitted over this altitude range. These included atomic oxygen O(1S) and O(1D) lines, as well as lines in the OH Meinel (8,3) and O2 Atmospheric (0,0) bands. The instrument employed was a Doppler Michelson Interferometer (DMI) that measured the Doppler shift as a phase shift of the cosinusoidal interferogram generated by single airglow lines. An extensive validation program was conducted after launch to confirm the accuracy of the measurements. The dominant wind field, the first one observed by WINDII, was that of the migrating diurnal tide at the equator. The overall most notable WINDII contribution followed from this; determining the influence of dynamics on the transport of atmospheric species. Currently, non-migrating tides are being studied in the thermosphere at both equatorial and high latitudes. Other aspects investigated included solar and geomagnetic influences, temperatures from atmospheric scale heights, nitric oxide concentrations and the occurrence of polar mesospheric clouds. The results of these observations are reviewed from a perspective of twenty years. A future perspective is then projected, involving more recently developed concepts. It is intended that this description will be helpful for those planning future missions.

Double‐peaked hydroxyl airglow profiles observed from WINDII/UARS

The WINDII instrument onboard the UARS satellite made many thousands of measurements of the apparent brightness of the hydroxyl airglow as a function of tangent height during the night. The measurements were mainly at latitudes between 42°S and 42°N. Most of the profiles of volume emission rate indicate a simple layer with a peak between 85 and 90 km. However, from 5 to 25% of the measured profiles have a more complex structure for the layer profile, most frequently with two peaks separated by 4 to 6 km. The horizontal extent varies from 200 to several thousand kilometers, and the duration may be as long as 1½ hours. Here the WINDII data from November 1991 to April 1995 are used to study the frequency of occurrence of the double-peaked layers as a function of latitude, season, and local time and to explore possible causes of the unusual profiles. Early in the night the double-peaked layers occur preferentially at low to medium latitudes, migrating toward the equator by midnight where they remain for the rest of the night. There is a pronounced asymmetry between the hemispheres which reverses between the March/April equinox and the September/October equinox. The solstice behavior is similar, though not so pronounced. Through simulation three possible causes to the observed OH* double-peaked layers were identified. Gravity waves are a likely source of isolated small-scale events but the large horizontal scale, long-duration structures are more likely to result from temperature inversion layers or mixing events or a combination of the two.

The nocturnal behavior of the hydroxyl airglow at the equatorial and low latitudes as observed by WINDII: Comparison with ground‐based measurements

Measurements of the hydroxyl nightglow from the Wind Imaging Interferometer (WINDII) satellite-borne experiment and from ground-based photometers are compared and combined in order to characterize the hydroxyl airglow nocturnal behavior at 23°S and the equatorial region. Ground-based observations at Cachoeira Paulista (23°S, 45°W) and at Fortaleza (4°S, 38°W) from 1987 to 1994 and WINDII data from March 1992 to January 1994 are selected, and two periods are studied: the March/April equinox and the December/January solstice. The WINDII results for the equinox/solstice difference and for the nocturnal behavior of the vertically integrated emission rate of the hydroxyl nightglow are found to be in very good agreement with the ground-based observations. The WINDII measurements revealed two general characteristics of the hydroxyl nightglow altitude variation at the latitudes above. The hydroxyl nocturnal variations are predominantly occurring below 90 km, and the vertically integrated emission rate and the peak height of the volume emission rate show an inverse relationship with seasonal, latitudinal, and inter annual effects. The interpretation of these results in terms of atmospheric tides is then discussed.

Review of applications of 1D and 2D S transforms

Gaussian modulated sinusoids are used in S-transform to extract time-local and space-local spectral information. Similar data sets recorded at neighboring spatial locations may be used with cross spectral analysis to determine frequency localized velocity spectrum. The 2D S-transform is used in image analysis for space localized wavenumber spectra. Local changes in the image spectrum are used to define textural boundaries on images. This paper summarizes several of the research projects involving S-transforms currently in progress at the University of Western Ontario including the application of the 2D S-transform to texture analysis, recognition, and the classification of images.

Instantaneous wave-vector analysis

2D local spectral information can be obtained form an image using Instantaneous Wavevector (IW). This 2D function is a vector quantity found by taking the gradient of the phase of the analytic image. Several synthetic images will be presented to illustrate the utility of IW analysis, and its application to OH airglow images will be discussed. The IW of an image gives us the dominant wavevector present at any point in the image. The amplitude of the analytic image gives us the magnitude of this component, and the phase differences of the analytic image between successive images allow us to infer the velocity of these waves. This method is used to determine phase velocities of internal waves from Hydroxyl airglow data. The instrument used, UWOSCR, is a scanning radiometer in the near infra-red, taking an 256 pixel image of the OH airglow every minute.

GWIM: a satellite instrument for gravity wave characterization

Gravity waves are ubiquitous in the Earths atmosphere transporting energy and momentum between regions. GWIM is a satellite instrument that will use airglow intensity variations to measure gravity wave parameters. A major interest lies in the correlation of the presence of gravity waves with their sources near the surface or in the lower atmosphere. The specific airglow emissions were chosen because of their high radiance and their low susceptibility to contamination by moonlight scattered from clouds. GWIM consists of four nadir-looking imagers, one each for signal and background for each of the two emissions. The field of each imager will be 175 km cross-track by 80 km along the track. An integration time of about ten seconds is required to achieve the required signal-to-noise ratio. To prevent image smearing due to the effects of satellite motion and the rotation of the earth, GWIM will operate in snapshot mode with an exposure time of about 0.25 second followed by on-board shift-and-add. GWIM is being considered for flight as part of the payload of EQUARS, a Brazilian satellite planned for launch into a near equatorial orbit in 2007.