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Presented here is the general framework and the initial results of a joint effort to derive novel research tools and easy to use software to analyze and model turbulence and transition. Given here is a brief review of the issues, a summary of some basic properties of wavelets, and preliminary results. Technical aspects of the implementation, the physical conclusions reached at this time, and current developments are discussed.
Double-parameter expansions, leading to an instantaneous multiscale analysis, are used to obtain time-frequency graphs from time-series data for interpretation of scatterometer response to sea surface geometry. This method is used to define, as well as possible, one-dimensional signal input (elevation and slope). However, some important fundamental parameters, such as the asymmetric development of short scales in crests and troughs, the long-wave amplitude modulation, the different zero-crossings, and so forth, are easily extracted from the wavelet images.
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We have developed a new technique based on a wavelet transform analysis to quantify the small-scale (less than a few arcminutes) X-ray structure of clusters of galaxies. We apply this technique to the ROSAT position sensitive proportional counter (PSPC) and Einstein high-resolution imager (HRI) images of the central region of the cluster Abell 1367 to detect sources embedded within the diffuse intracluster medium. In addition to detecting sources and determining their fluxes and positions, we show that the wavelet analysis allows a characterization of the sources extents. In particular, the wavelet scale at which a given source achieves a maximum signal-to-noise ratio in the wavelet images provides an estimate of the angular extent of the source. To account for the widely varying point response of the ROSAT PSPC as a function of off-axis angle requires a quantitative measurement of the source size and a comparison to a calibration derived from the analysis of a Deep Survey image. Therefore, we assume that each source could be described as an isotropic two-dimensional Gaussian and used the wavelet amplitudes, at different scales, to determine the equivalent Gaussian Full Width Half-Maximum (FWHM) (and its uncertainty) appropriate for each source. In our analysis of the ROSAT PSPC image, we detect 31 X-ray sources above the diffuse cluster emission (within a radius of 24 min), 16 of which are apparently associated with cluster galaxies and two with serendipitous, background quasars. We find that the angular extents of 11 sources exceed the nominal width of the PSPC point-spread function. Four of these extended sources were previously detected by Bechtold et al. (1983) as 1 sec scale features using the Einstein HRI. The same wavelet analysis technique was applied to the Einstein HRI image. We detect 28 sources in the HRI image, of which nine are extended. Eight of the extended sources correspond to sources previously detected by Bechtold et al. Overall, using both the PSPC and the HRI observations, we detect 16 extended features, of which nine have galaxies coincided with the X-ray-measured positions (within the positional error circles). These extended sources have luminosities lying in the range (3 - 30) x 10(exp 40) ergs/s and gas masses of approximately (1 - 30) x 10(exp 9) solar mass, if the X-rays are of thermal origin. We confirm the presence of extended features in A1367 first reported by Bechtold et al. (1983). The nature of these systems remains uncertain. The luminosities are large if the emission is attributed to single galaxies, and several of the extended features have no associated galaxy counterparts. The extended features may be associated with galaxy groups, as suggested by Canizares, Fabbiano, & Trinchieri (1987), although the number required is large.
The asteroid family determination has been analysis method dependent for a longtime. A new cluster analysis based on the wavelet transform has allowed an automatic definition of families with a degree of significance versus randomness. Actually this method is rather general and can be applied to any kind of structural analysis. We will rather concentrate on the main features of the method. The analysis has been performed on the set of 4100 asteroid proper elements computed by Milani and Knezevic (see Milani and Knezevic 1990). Twenty one families have been found and influence of the chosen metric has been tested. The results have beem compared to Zappala et al.'s ones (see Zappala et al 1990) obtained by the use of a completely different method applied to the same set of data. For the first time, a good overlapping has been found between both method results, not only for the big well known families but also for the smallest ones.
External Tank Cable Tray vibration data for three successive Space Shuttle flights were analyzed to assess response to buffet and the effect of removal of the Protuberance Air Loads (PAL) ramp. Waveform integration, spectral analysis, cross-correlation analysis and wavelet analysis were employed to estimate vibration modes and temporal development of vibration motion from a sparse array of accelerometers and an on-board system that acquired 16 channels of data for approximately the first two minutes of each flight. The flight data indicated that PAL ramp removal had minimal effect on the fluctuating loads on the cable tray. The measured vibration frequencies and modes agreed well with predicted structural response.
External Tank Cable Tray vibration data for three successive Space Shuttle flights were analyzed to assess response to buffet and the effect of removal of the Protuberance Air Loads (PAL) ramp. Waveform integration, spectral analysis, cross-correlation analysis and wavelet analysis were employed to estimate vibration modes and temporal development of vibration motion from a sparse array of accelerometers and an on-board system that acquired 16 channels of data for approximately the first 2 min of each flight. The flight data indicated that PAL ramp removal had minimal effect on the fluctuating loads on the cable tray. The measured vibration frequencies and modes agreed well with predicted structural response.
Computer simulation is an essential part of the design and development of jet engines for the aeropropulsion industry. Engineers concerned with calculating the flow in jet engine components, such as compressors and turbines, need simple engineering models that accurately describe the complex flow of air and gases and that allow them to quickly estimate loads, losses, temperatures, and other design parameters. In this ongoing collaborative project, advanced wavelet analysis techniques are being used to gain insight into the complex flow phenomena. These insights, which cannot be achieved by commonly used methods, are being used to develop innovative new flow models and to improve existing ones. Wavelet techniques are very suitable for analyzing the complex turbulent and transitional flows pervasive in jet engines. These flows are characterized by intermittency and a multitude of scales. Wavelet analysis results in information about these scales and their locations. The distribution of scales is equivalent to the frequency spectrum provided by commonly used Fourier analysis techniques; however, no localization information is provided by Fourier analysis. In addition, wavelet techniques allow conditional sampling analyses of the individual scales, which is not possible by Fourier methods.
Estimates of solar normal mode frequencies from helioseismic observations can be improved by using Multitaper Spectral Analysis (MTSA) to estimate spectra from the time series, then using wavelet denoising of the log spectra. MTSA leads to a power spectrum estimate with reduced variance and better leakage properties than the conventional periodogram. Under the assumption of stationarity and mild regularity conditions, the log multitaper spectrum has a statistical distribution that is approximately Gaussian, so wavelet denoising is asymptotically an optimal method to reduce the noise in the estimated spectra. We find that a single m-upsilon spectrum benefits greatly from MTSA followed by wavelet denoising, and that wavelet denoising by itself can be used to improve m-averaged spectra. We compare estimates using two different 5-taper estimates (Stepian and sine tapers) and the periodogram estimate, for GONG time series at selected angular degrees l. We compare those three spectra with and without wavelet-denoising, both visually, and in terms of the mode parameters estimated from the pre-processed spectra using the GONG peak-fitting algorithm. The two multitaper estimates give equivalent results. The number of modes fitted well by the GONG algorithm is 20% to 60% larger (depending on l and the temporal frequency) when applied to the multitaper estimates than when applied to the periodogram. The estimated mode parameters (frequency, amplitude and width) are comparable for the three power spectrum estimates, except for modes with very small mode widths (a few frequency bins), where the multitaper spectra broadened the modest compared with the periodogram. We tested the influence of the number of tapers used and found that narrow modes at low n values are broadened to the extent that they can no longer be fit if the number of tapers is too large. For helioseismic time series of this length and temporal resolution, the optimal number of tapers is less than 10.
We have further analysed data from an experiment performed in satellite orbit, in Spacelab-1. In micro-gravity the hypocotyls of Helianthus annuus, cv. "Teddy Bear", showed short period circumnutations (periods around 30 minutes) as well as the already reported long period nutations (with an average period of about 115 minutes). We applied various types of signal analysis (Fourier and wavelet analysis) to the data series. The long period circumnutations have a larger amplitude than the short term circumnutations. Both short and long period circumnutations exist in one and the same hypocotyl. (This is in contrast to our ground control experiments, where were found only the long-period nutations.) The period of the nutations changed throughout the experiment. These results are extending the conclusions drawn after the Spacelab experiment (Brown et al. 1990). In particular they emphasize the existence of both short- and long-period circumnutations in micro-gravity.
There has been an increased interest in the quantification of pattern in ecological systems over the past years. This interest is motivated by the desire to construct valid models which extend across many scales. Spatial methods must quantify pattern, discriminate types of pattern, and relate hierarchical phenomena across scales. Wavelet analysis is introduced as a method to identify spatial structure in ecological transect data. The main advantage of the wavelet transform over other methods is its ability to preserve and display hierarchical information while allowing for pattern decomposition. Two applications of wavelet analysis are illustrated, as a means to: (1) quantify known spatial patterns in Douglas-fir forests at several scales, and (2) construct spatially-explicit hypotheses regarding pattern generating mechanisms. Application of the wavelet variance, derived from the wavelet transform, is developed for forest ecosystem analysis to obtain additional insight into spatially-explicit data. Specifically, the resolution capabilities of the wavelet variance are compared to the semi-variogram and Fourier power spectra for the description of spatial data using a set of one-dimensional stationary and non-stationary processes. The wavelet cross-covariance function is derived from the wavelet transform and introduced as a alternative method for the analysis of multivariate spatial data of understory vegetation and canopy in Douglas-fir forests of the western Cascades of Oregon.
The purpose of this grant was to introduce a broad group of NASA researchers and administrators to wavelet technology and to determine its future role in research and development at NASA JSC. The activities of several briefings held between NASA JSC scientists and Rice University researchers are discussed. An attached paper, 'Recent Advances in Wavelet Technology', summarizes some aspects of these briefings. Two proposals submitted to NASA reflect the primary areas of common interest. They are image analysis and numerical solutions of partial differential equations arising in computational fluid dynamics and structural mechanics.
Wavelet transforms are used for on-line detection of aircraft loss of control. Wavelet transforms are compared with Fourier transform methods and shown to more rapidly detect changes in the vehicle dynamics. This faster response is due to a time window that decreases in length as the frequency increases. New wavelets are defined that further decrease the detection time by skewing the shape of the envelope. The wavelets are used for power spectrum and transfer function estimation. Smoothing is used to tradeoff the variance of the estimate with detection time. Wavelets are also used as front-end to the eigensystem reconstruction algorithm. Stability metrics are estimated from the frequency response and models, and it is these metrics that are used for loss of control detection. A Matlab toolbox was developed for post-processing simulation and flight data using the wavelet analysis methods. A subset of these methods was implemented in real time and named the Loss of Control Analysis Tool Set or LOCATS. A manual control experiment was conducted using a hardware-in-the-loop simulator for a large transport aircraft, in which the real time performance of LOCATS was demonstrated. The next step is to use these wavelet analysis tools for flight test support.
We present a study of horizontal winds in the mesosphere and lower thermosphere (MLT) during the boreal winters of 2009-2010 and 2012-2013 produced with a new high-altitude numerical weather prediction (NWP) system. This system is based on a modified version of the Navy Global Environmental Model (NAVGEM) with an extended vertical domain up to approximately 116 km altitude coupled with a hybrid four-dimensional variational (4DVAR) data assimilation system that assimilates both standard operational meteorological observations in the troposphere and satellite-based observations of temperature, ozone and water vapor in the stratosphere and mesosphere. NAVGEM-based MLT analyzed winds are validated using independent meteor radar wind observations from nine different sites ranging from 69 deg N-67 deg S latitude. Time-averaged NAVGEM zonal and meridional wind profiles between 75 and 95 km altitude show good qualitative and quantitative agreement with corresponding meteor radar wind profiles. Wavelet analysis finds that the 3-hourly NAVGEM and 1-hourly radar winds both exhibit semi-diurnal, diurnal, and quasi-diurnal variations whose vertical profiles of amplitude and phase are also in good agreement. Wavelet analysis also reveals common time-frequency behavior in both NAVGEM and radar winds throughout the Northern extra tropics around the times of major stratospheric sudden warmings (SSWs) in January 2010 and January 2013, with a reduction in semi-diurnal amplitudes beginning around the time of a mesospheric wind reversal at 60 deg N that precedes the SSW, followed by an amplification of semi-diurnal amplitudes that peaks 10-14 days following the onset of the mesospheric wind reversal. The initial results presented in this study demonstrate that the wind analyses produced by the high altitude NAVGEM system accurately capture key features in the observed MLT winds during these two boreal winter periods.
The Permian/Triassic (P/Tr) boundary (251.4 +/- 3 Myr) is marked by the most severe mass extinction in the geologic record. Recently, precise absolute dating has bracketed the marine extinctions and associated carbon-isotope anomaly within less than 1 Myr. We improve this resolution through high-resolution stratigraphy across the P/Tr boundary in the 331-m Gartnerkofel-1 core and nearby Reppwand outcrop section (Carnic Alps, Austria) utilizing FFT and wavelet timeseries analyses of cyclic components in down-hole core logs of density and natural gamma-ray intensity, and carbon-isotopic ratios of bulk samples. The wavelet analysis indicates continuity of deposition across the P/Tr boundary interval, and the timeseries analyses show evidence for persistent cycles in the ratio of approximately 40: 10: 4.7: 2.3 meters, correlated with Milankovitch-band orbital cycles of approximately 412: 100: 40: 20 kyr (eccentricity 1 and 2, obliquity, and precession), and giving a consistent average sedimentation rate of approximately 10 cm/1,000 yr. Milankovitch periods in delta C-13 and density in these shallow-water carbonates were most likely the result of climatically induced oscillations of sea level and climate, coupled with changes in ocean circulation and productivity, that affected sedimentation. Fluctuations in gamma radiation reflect varying input of clay minerals and the presence of shaly interbeds. Throughout the P/Tr boundary interval in the core, the 100,000-year eccentricity cycle seems to be dominant. Weaker obliquity and precession cycles are in line with the location of the Austrian section in the latest Permian, close to the Equator in the western bight of the Tethys, where obliquity and precessional effects on seasonal contrast might be subdued. Using the improved resolution provided by cycle analysis in the GK-1 core, we find that the dramatic change in the faunal record that marks the P/Tr boundary takes place over less than 6m, or less than 60,000 years. In the nearby Reppwand outcrop section, the same faunal changes occurs over only 0.8 m or about 8,000 years, close to the limit of time-resolution induced by bioturbation and reworking in these sediments. The sharp negative global carbon-isotope shift took place within less than or equal to 40,000 yr, and the isotope excursions persisted for approximately 480,000 yr into the Early Triassic. The results indicate that the severe marine faunal event that marks the P/Tr boundary was very sudden, perhaps less than the resolution window in the GK-1 core, and suggest a catastrophic cause. The wavelet-analysis approach to high-resolution cyclostratigraphy can be applied to other P/Tr boundary sections, and when combined with precise absolute dating and magnetostratigraphic methods promises a significant increase in resolution in determining the correlation and tempo of the end-Permian extinctions and related events worldwide.
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