Robert G. Stockwell

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.

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.

Airglow imaging of gravity waves: 1. Results from a small network of OH nightglow scanning imagers

Images of the variations of OH nightglow intensities have been recorded by a three-station network of scanning radiometers in southwestern Ontario during the period from May until September 1998. Each radiometer performs a 16 by 16 point raster scan of the night sky every minute, turning itself on and off when the solar depression angle is <6°. The horizontal phase velocity, horizontal wavelength, and frequency of the internal gravity waves are derived from the OH intensity measurements using joint time-frequency analysis techniques, including the S transform, localized cross-spectral analysis, and generalized instantaneous frequency. The ensemble of gravity waves seen in all three instruments are analyzed and observed to have a strong dependence in their propagation direction with the majority of waves seen at each of the three sites having a horizontal wave vector in the northeast direction. The most common parameters observed for horizontal phase speed is 45 m/s, horizontal wavelength is 25 km, and period is 10 min.

Gravity waves and momentum fluxes in the mesosphere and lower thermosphere using 430 MHz dual‐beam measurements at Arecibo: 2. Frequency spectra, momentum fluxes, and variability

[1] Janches et al. (2006) described a new dual-beam use of the 430 MHz incoherent scatter radar at the Arecibo Observatory in Puerto Rico. We found the technique to define the radial wind field in the mesosphere and lower thermosphere with sufficient accuracy to characterize gravity waves occurring at high frequencies and small spatial scales over an extended altitude range. The coplanar, dual-beam experiment was also designed to test the ability of the system to measure gravity wave momentum fluxes and their frequency distributions, and we report here on those results. Initial measurements were of limited duration and necessarily represent a case study, but they demonstrate the value of such measurements for studies of GW variability and large-scale interactions. Radial velocity variances reveal preferential eastward propagation for most intervals and altitudes, with the greatest propagation bias at lower altitudes and later times on 11 September when strong westward mean winds favor strong gravity filtering. The momentum fluxes observed during this experiment had � 50-min averages that were often near zero, occasionally achieved amplitudes of � 20 to 50 m 2 s � 2 , displayed significant consistency in altitude, and exhibited an approximate anticorrelation with the zonal wind field in cases with significant momentum fluxes. Frequency spectra defined the major contributions to the momentum fluxes, while S transforms were employed to examine the temporal variability of the GWs and momentum fluxes in greater detail.

Generation of aquifer heterogeneity maps using two-dimensional spectral texture segmentation techniques

Numerical models that solve the governing equations for subsurface fluid flow and transport require detailed quantitative maps of spatially variable hydraulic properties. Recently, there has been great interest in methods that can map the spatial variability of hydraulic properties such as porosity and hydraulic conductivity (permeability). Presently, only limited data on natural permeability spatial structure are available. These data are often based on extensive discrete sampling in outcrops or boreholes. Then methods are used to interpolate between data values to map aquifer heterogeneity. Interpolation methods often mask critical local or intermediate scale heterogeneities. As sediment texture is directly correlated with many hydraulic properties we developed two new texture segmentation algorithms based on a space-local two-dimensional wavenumber spectral method known as the S-Transform. Existing texture segmentation algorithms could not delineate the subtle and continuous texture variations that exist in natural sediments. The S-Transform algorithms successfully delineated geologic structures and grain size patterns in photographs of outcrops in a glacial fluvial deposit; thus, no interpolation methods were required to produce continuous two-dimensional maps of texture facies. The S-Transform method is robust and is insensitive to changes in light intensity, and moisture variations. This makes the algorithm particularly applicable to natural sedimentary outcrops. The effectiveness of our methods are tested by correlating measured relative grain sizes in the images with actual grain size measurements taken from the sedimentary outcrops.

Recent results with an MF radar at McMurdo, Antarctica: Characteristics and variability of motions near 12‐hour period in the mesosphere

We present an analysis of the first wind measurements obtained with an MF (medium frequency) radar installed at McMurdo Station, Antarctica in January 1996. The largest amplitude motions are the quasi-12-hr motions, with additional periodicities at 24 hr, ∼10 hr and 2 to 10 days. The quasi-12-hr motions occur at discrete periods distinct from 12 hr and exhibit pronounced amplitude and vertical wavenumber modulations at planetary wave periodicities. During the months analyzed in this report (mid-January through April 1996) the 12-hr wave is by far the dominant motion the mesosphere. The source of this wave is unknown, but our results provide additional support for the suggestion that such motions arise through nonlinear interactions involving the migrating semidiurnal tide and low-frequency planetary wave motions.

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.

Airglow imaging of gravity waves: 2. Critical layer filtering

Images of the variations of OH intensities have been recorded by a three-station network of scanning radiometers in southwestern Ontario during the period from May until September 1998. The ensemble of gravity waves seen in all three instruments is analyzed and observed to have a strong dependence in their propagation direction with the majority of waves seen at each of the three sites having a horizontal wave vector in the northeast direction. In order to understand these results a model of the transmission of gravity waves through the atmosphere was employed. The gravity waves observed by the OH Scanning Radiometers (OHSCRs) are assumed to originate in the troposphere with an isotropic spectrum, meaning that the average wave amplitudes are equal in all directions. The gravity waves propagate upward where they will undergo a selection process owing to the effect of the “background winds.” Using the horizontal winds from the Horizontal Wind Model (HWM-93), the transmission of gravity waves from a tropospheric source is modeled and shown to be consistent with the observations. The strong zonal winds account for the summer eastward and winter westward propagation, but it is the tidal components of the meridional wind that account for the northward tendency of wave propagation (and the corresponding relative absence of southward propagating waves). In order to further validate the transmission, model winter measurements from the OHSCR data set are examined and shown to be consistent.

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.