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20. Lilly, J. M. (2012). The unity of instantaneous spectral moments and physical moments. Submitted to Proceedings of the Royal Society of London A. [0.3 Mb pdf]
    
    Abstract. A modulated oscillation in two or three dimensions can be represented as the trajectory traced out in space by a particle orbiting an ellipse, the properties of which vary as a function of time. Generalizing ideas from signal analysis, the signal variability can be described in terms of kinematic quantities, the instantaneous moments, that formalize our intuitive notions of time-varying frequency and amplitude. On the other hand, if we observed an ellipse evolving in space we would seek to describe it in terms of its physical moments, such as angular momentum, moment of inertia, etc. The main result of this paper is to show that the two sets of moments are identical. Most significantly, an essential physical quantity—the circulation—is the same as the product of the two most important kinematic quantities, the instantaneous frequency and the squared instantaneous amplitude. In addition to providing a rich set of geometric tools for the analysis of nonstationary oscillations in two or three dimensions, this result also has implications for the practical problem of inferring physical ellipse parameters from the trajectory of a single particle on the ellipse periphery, as is frequently encountered in the study of vortex motions.
    
    
19. Bower, A. S., R. M. Hendry, D. E. Amrhein, and J. M. Lilly (2012). Direct observations of formation and propagation of subpolar eddies into the subtropical North Atlantic. Submitted to Deep-Sea Research. [12 Mb pdf]
    
    Abstract. Subsurface float and moored observations are presented to show for the first time the formation and propagation of anticyclonic submesoscale coherent vortices that transport relatively cold, fresh subpolar water to the interior subtropical North Atlantic. Acoustically tracked RAFOS floats released in the southward-flowing Western Boundary Current at the exit of the Labrador Sea reveal the formation of three of these eddies at the southern tip of the Grand Banks (42°N, 50°W). Using a recently developed method to detect eddies in float trajectories and estimate their kinematic properties, it was found that the eddies had average rotation periods of 5–7 days at radii of 1025 km, with mean rotation speeds of up to 0.3 m/s. One especially long-lived (5.1 months) eddy crossed under the Gulf Stream path and translated southwestward in the subtropical recirculation to at least 35°N, where it hit one of the Corner Rise Seamounts. Velocity, temperature and salinity measurements from a nine-month deployment of two moorings south of the Gulf Stream at 38°N, 50°W reveal the passage of at least two eddies with similar hydrographic and kinematic properties. The core temperature and salinity properties of the eddies imply their formation at intermediate levels of the Labrador Current south of the Tail of the Grand Banks. These observations confirm earlier speculation that eddies form in this region and transport anomalously cold, low-salinity water directly into the subtropical interior. Possible formation mechanisms and potential importance of these eddies to interior ventilation and the equatorward spreading of Labrador Sea Water are discussed.
    
    
18. Lilly, J. M., and S. C. Olhede (2012). Analysis of modulated multivariate oscillations. IEEE Transactions on Signal Processing, 60 (2), 600–612. [1.1 Mb pdf]
    
    Abstract. The concept of a common modulated oscillation spanning multiple time series is formalized, a method for the recovery of such a signal from potentially noisy observations is proposed, and the time-varying bias properties of the recovery method are derived. The method, an extension of wavelet ridge analysis to the multivariate case, identifies the common oscillation by seeking, at each point in time, a frequency for which a bandpassed version of the signal obtains a local maximum in power. The lowest-order bias is shown to involve a quantity, termed the instantaneous curvature, which measures the strength of local quadratic modulation of the signal after demodulation by the common oscillation frequency. The bias can be made to be small if the analysis filter, or wavelet, can be chosen such that the signal's instantaneous curvature changes little over the filter time scale. An application is presented to the detection of vortex motions in a set of freely-drifting oceanographic instruments tracking the ocean currents.
    
    
17. Lilly, J. M., R. K. Scott, and S. C. Olhede (2011). Extracting waves and vortices from Lagrangian trajectories. Geophysical Research Letters, 38, L23605, 1–5. [2.6 Mb pdf] [33 Mb mov]
    
    Abstract. A method for extracting time-varying oscillatory motions from time series records is applied to Lagrangian trajectories from a numerical model of eddies generated by an unstable equivalent barotropic jet on a beta plane. An oscillation in a Lagrangian trajectory is represented mathematically as the signal traced out as a particle orbits a time-varying ellipse, a model which captures wavelike motions as well as the displacement signal of a particle trapped in an evolving vortex. Such oscillatory features can be separated from the turbulent background flow through an analysis founded upon a complex-valued wavelet transform of the trajectory. Application of the method to a set of one hundred modeled trajectories shows that the oscillatory motions of Lagrangian particles orbiting vortex cores appear to be extracted very well by the method, which depends upon only a handful of free parameters and which requires no operator intervention. Furthermore, vortex motions are clearly distinguished from wavelike meandering of the jet—the former are high frequency, nearly circular signals, while the latter are linear in polarization and at much lower frequencies. This suggests that the proposed method can be useful for identifying and studying vortex and wave properties in large Lagrangian datasets. In particular, the eccentricity of the oscillatory displacement signals, a quantity which is not normally considered in Lagrangian studies, emerges as an informative diagnostic for characterizing qualitatively different types of motion.
    
    
16. Lilly, J. M. (2011). Modulated oscillations in three dimensions. IEEE Transactions on Signal Processing, 59 (12), 5930–5943. [1.1 Mb pdf]
    
    Abstract. The analysis of the fully three-dimensional and time-varying polarization characteristics of a modulated trivariate, or three-component, oscillation is addressed. The use of the analytic operator enables the instantaneous three-dimensional polarization state of any square-integrable trivariate signal to be uniquely defined. Straightforward expressions are given which permit the ellipse parameters to be recovered from data. The notions of instantaneous frequency and instantaneous bandwidth, generalized to the trivariate case, are related to variations in the ellipse properties. Rates of change of the ellipse parameters are found to be intimately linked to the first few moments of the signal's spectrum, averaged over the three signal components. In particular, the trivariate instantaneous bandwidth---a measure of the instantaneous departure of the signal from a single pure sinusoidal oscillation---is found to contain five contributions: three essentially two-dimensional effects due to the motion of the ellipse within a fixed plane, and two effects due to the motion of the plane containing the ellipse. The resulting analysis method an informative means of describing nonstationary trivariate signals, as is illustrated with an application to a seismic record.
    
    
15. Lilly, J. M., and S. C. Olhede (2010). On the analytic wavelet transform. IEEE Transactions on Information Theory, 56 (8), 4135–4156. [1.3 Mb pdf]
    
    Abstract. An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of non-negligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at increasingly higher orders, the local departures of the signal from a uniform sinusoidal oscillation. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing an hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias.
    
    
14. Lilly, J. M., and S. C. Olhede (2010). Bivariate instantaneous frequency and bandwidth. IEEE Transactions on Signal Processing, 58 (2), 591–603. [0.7 Mb pdf]
    
    Abstract. The generalizations of instantaneous frequency and instantaneous bandwidth to a bivariate signal are derived. These are uniquely defined whether the signal is represented as a pair of real-valued signals, or as one analytic and one anti-analytic signal. A nonstationary but oscillatory bivariate signal has a natural representation as an ellipse whose properties evolve in time, and this representation provides a simple geometric interpretation for the bivariate instantaneous moments. The bivariate bandwidth is shown to consists of three terms measuring the degree of instability of the time-varying ellipse: amplitude modulation with fixed eccentricity, eccentricity modulation, and orientation modulation or precession. An application to the analysis of data from a free-drifting oceanographic float is presented and discussed.
    
    
13. Lilly, J. M., and S. C. Olhede (2009). Wavelet ridge estimation of jointly modulated multivariate oscillations. 43rd Asilomar Conference on Signals, Systems, and Computers, 452–456. Refereed conference proceedings paper. [0.7 Mb pdf]
    
    Abstract. Wavelet ridge analysis is a technique for estimating the time-varying properties of a modulated oscillatory signal from a contaminated observation. Here this technique is extended to the multivariate case, that is, to a set of N real-valued time series. The bivariate case is illustrated with an application to a set of freely-drifting oceanographic floats. A freely distributed software package implementing this algorithm is available online at http://www.jmlilly.net.
    
    
12. T. Rykova, F. Straneo, J. M. Lilly, and I. Yashayaev, (2009). Irminger Current anticyclones in the Labrador Sea observed in the hydrographic record, 1990–2004. Journal of Marine Research, 67 (3), 361–384. [16 Mb pdf]
    
    Abstract. A significant fraction of the lateral heat transport into the Labrador Sea's interior, needed to balance the net heat loss to the atmosphere, is attributed to the Irminger Current Anticyclones. These mesoscale eddies advect warm, salty boundary current water, of subtropical origin, from the boundary current to the interior---but when or how they release their anomalous heat content has not been previously investigated. In this study, we discuss the seasonal and interannual evolution of these anticyclones as inferred from the analysis of hydrographic data from the Labrador Sea from 1990 to 2004. The 29 identified anticyclones fall into two categories, which we refer to as unconvected and convected. Unconvected anticyclones have properties that are close to those of the boundary current, including a fresh surface layer, and they are found near the boundaries and never observed in winter. Convected anticyclones, on the other hand, contain a mixed layer, lack a freshwater cap and are observed throughout the year. Using a one-dimensional mixing model, it is shown that the convected eddies are those Irminger Current Anticyclones that have been modified by the large winter buoyancy loss of the region. This provides evidence that such eddies can survive the strong winter buoyancy loss in the Labrador Sea and that their anomalous heat and salt content is not trivially mixed into the Sea's interior. Finally, we observe a clear trend in the eddies' properties toward warmer and saltier conditions after 1997 reflecting changes in the source waters and the reduced atmospheric forcing over the Labrador Sea.
    
    
11. Lilly, J. M., and S. C. Olhede (2009). Higher-order properties of analytic wavelets. IEEE Transactions on Signal Processing, 57 (1), 146–160. [1.8 Mb pdf]
    
    Abstract. The influence of higher-order wavelet properties on the analytic wavelet transform behavior is investigated, and wavelet functions offering advantageous performance are identified. This is accomplished through detailed investigation of the generalized Morse wavelets, a two-parameter family of exactly analytic continuous wavelets. The degree of time/frequency localization, the existence of a mapping between scale and frequency, and the bias involved in estimating properties of modulated oscillatory signals, are proposed as important considerations. Wavelet behavior is found to be strongly impacted by the degree of asymmetry of the wavelet in both the frequency and the time domain, as quantified by the third central moments. A particular subset of the generalized Morse wavelets, recognized as deriving from an inhomogeneous Airy function, emerge as having particularly desirable properties. These "Airy wavelets" substantially outperform the only approximately analytic Morlet wavelets for high time localization. Special cases of the generalized Morse wavelets are examined, revealing a broad range of behaviors which can be matched to the characteristics of a signal.
    
    
10. Lagerloef, G, F. R. Colomb, D. Le Vine, F. Wentz, S. Yueh, C. Ruf, J. Lilly, J. Gunn, Y. Chao, A. deCharon, G. Feldman, and C. Swift (2008). The Aquarius/SAC-D mission: Designed to meet the salinity remote-sensing challenge. Oceanography, 21 (1), 69–81. [4.2 Mb pdf]
    
    Abstract. In an Oceanography article published 13 years ago, three of us identified salinity measurement from satellites as the next ocean remote-sensing challenge. We argued that this represented the next "zeroth order" contribution to oceanography (Lagerloef et al., 1995) because salinity variations form part of the interaction between ocean circulation and the global water cycle, which in turn affects the ocean's capacity to store and transport heat and regulate Earth's climate. Now, we are pleased to report that a new satellite program scheduled for launch in the near future will provide data to reveal how the ocean responds to the combined effects of evaporation, precipitation, ice melt, and river runoff on seasonal and interannual time scales. These measurements can be used, for example, to close the marine hydrologic budget, constrain coupled climate models, monitor mode water formation, investigate the upper-ocean response to precipitation variability in the tropical convergence zones, and provide early detection of low-salinity intrusions in the subpolar Atlantic and Southern oceans. Sea-surface salinity (SSS) and sea-surface temperature (SST) determine sea-surface density, which controls the formation of water masses and regulates three-dimensional ocean circulation.
    
    
9. Rilling, G., P. Flandrin, P. Gonçalves, and J. M. Lilly (2007). Bivariate empirical mode decomposition. IEEE Signal Processing Letters, 14 (12), 936–939. [0.3 Mb pdf]
   
   Abstract. The empirical mode decomposition (EMD) has been introduced quite recently to adaptively decompose nonstationary and/or nonlinear time series. The method being initially limited to real-valued time series, we propose here an extension to bivariate (or complex-valued) time series that generalizes the rationale underlying the EMD to the bivariate framework. Where the EMD extracts zero-mean oscillating components, the proposed bivariate extension is designed to extract zero-mean rotating components. The method is illustrated on a real-world signal and properties of the output components are discussed. Free Matlab/C codes are available at http://perso.ens-lyon.fr/patrick.flandrin.
   
   
8. Lilly, J. M., and J.-C. Gascard (2006). Wavelet ridge diagnosis of time-varying elliptical signals with application to an oceanic eddy. Nonlinear Processes in Geophysics 13, 467–483. [0.6 Mb pdf]
   
   Abstract. A method for diagnosing the physical properties of a time-varying ellipse is presented. This essentially involves extending the notion of instantaneous frequency to the bivariate case. New complications, and possibilities, arise from the fact that there are several meaningful forms in which a time-varying ellipse may be represented. A perturbation analysis valid for the near-circular case clarifies these issues. Diagnosis of the ellipse properties may then be performed using wavelet ridge analysis, and slowly-varying changes in the ellipse structure may be decoupled from the fast orbital motion through the use of elliptic integrals, without the need for additional explicit filtering. The theory is presented in parallel with an application to a position time series of a drifting subsurface float trapped in an oceanic eddy.
   
   
7. Bunge, L., C. Provost, J. M. Lilly, M. D'Orgeville, A. Kartavtseff, and J.-L. Mélice (2006). Variability of the horizontal velocity structure in the Upper 1600 m of the water column on the Equator at 10° W. Journal of Physical Oceanography 36, 1287–1304. [2.4 Mb pdf]
   
   Abstract. This paper presents initial results from new velocity observations in the eastern part of the equatorial Atlantic Ocean from a moored current-meter array. During the "EQUALANT" program (1999--2000), a mooring array was deployed around the equator near 10°W that recorded one year of measurements at various depths. Horizontal velocities were obtained in the upper 60 m from an upward-looking acoustic Doppler current profiler (ADCP) and at 13 deeper levels from current meters between 745 and 1525 m. To analyze the quasiperiodic variability observed in these records, a wavelet-based technique was used. Quasiperiodic oscillations having periods between 5 and 100 days were separated into four bands: 5--10, 10--20, 20--40, and 40--100 days. The variability shows (i) a strong seasonality (the first half of the series is dominated by larger periods than the second one) and (ii) a strong dependence with depth (some oscillations are present in the entire water column while others are only present at certain depths). For the oscillations that are present in the entire water column the origin of the forcing can be traced to the surface, while for the others the question of their origin remains open. Phase shifts at different depths generate vertical shears in the horizontal velocity component with relatively short vertical scales. This is especially visible in long- duration events (> 100 days) of the zonal velocity component. Comparison with a simultaneous lowered acoustic Doppler current profiler (LADCP) section suggests that some of these flows may be identified with equatorial deep jets. A striking feature is a strong vertical shear lasting about 7 months between 745 and 1000 m. These deep current-meter observations would then imply a few months of duration for the jets in this region.
   
   
6. Lilly, J. M., and E. E. Lettvin (2004). The "switch-on" problem for linear time-invariant operators. Signal Processing 84, 763–784. [0.5 Mb pdf]
   
   Abstract. The model of a linear system with stochastic input is a fundamental building block of physics. For such systems, Fourier analysis permits compact and intuitively appealing solutions. Here we extend the Fourier-based solution to include the "switch-on" problem, in which the forcing is suddenly turned on at a certain time. The modification of the solution due to a finite forcing history resembles the signal processing problem of estimating a spectrum from a finite sample. The asymptotic response of the system may then be found by considering the limiting behavior of familiar frequency-domain operators, an approach which has some advantages over the usual contour-integration method. The simple harmonic oscillator is studied in particular. The instantaneous variance is found to be proportional to the periodogram at the natural frequencies, and recovers the asymptotic behavior reported by Hasselmann (J. Fluid Mech. 12 (1962) 481--500) as the periodogram converges to the true spectrum. A simple approximation is presented which isolates the "resonant" growth of variance, and allows an useful classication according to whether this growth is linear or quadratic. Application to the physical problem of nonlinear wave-wave interaction is discussed.
   
   
5. Lilly, J. M., P. B. Rhines, F. Schott, K. Lavender, J. Lazier, U. Send, and E. D'Asaro (2003). Observations of the Labrador Sea eddy field. Progress in Oceanography 59, 75–176. [4.8 Mb pdf]
   
   Abstract. This paper is an observational study of small-scale coherent eddies in the Labrador Sea, a region of dense water formation thought to be of considerable importance to the North Atlantic overturning circulation. Numerical studies of deep convection emphasize coherent eddies as a mechanism for the lateral transport of heat, yet their small size has hindered observational progress. A large part of this paper is therefore devoted to developing new methods for identifying and describing coherent eddies in two observational platforms, current meter moorings and satellite altimetry. Details of the current and water mass structure of individual eddy events, as they are swept past by an advecting flow, can then be extracted from the mooring data. A transition is seen during mid-1997, with long-lived boundary current eddies dominating the central Labrador Sea year-round after this time, and convectively-formed eddies similar to those seen in deep convection modelling studies apparent prior to this time. The TOPEX / Poseidon altimeter covers the Labrador Sea with a loose "net" of observations, through which coherent eddies can seem to appear and disappear. By concentrating on locating and describing anomalous events in individual altimeter tracks, a portrait of the spatial and temporal variability of the underlying eddy field can be constructed. The altimeter results reveal an annual "pulsation" of energy and of coherent eddies originating during the late fall at a particular location in the boundary current, pinpointing the time and place of the boundary current eddy formation. The interannual variability seen at the mooring is reproduced, but the mooring site is found to be within a localized region of greatly enhanced eddy activity. Notably lacking in both the annual cycle and interannual variability is a clear relationship between the eddies or eddy energy and the intensity of wintertime cooling. These eddy observations, as well as hydrographic evidence, suggest an active role for the boundary current in shaping the energetics and heat content of the interior region.
   
   
4. Lilly, J. M., and P. B. Rhines (2002). Coherent eddies in the Labrador Sea observed from a mooring. Journal of Physical Oceanography 32, 585–598. [1.2 Mb pdf]
   
   Abstract. During June--November 1994, a mooring in the central Labrador Sea near the former Ocean Weather Station Bravo recorded a half-dozen anomalous events that prove to be two different types of coherent eddies. Comparisons with simple analytical models are used to classify these events as coherent eddies on the basis of their velocity signatures. The first clear examples of long-lived convectively generated eddies are reported. These four small (radius ~5--15 km) eddies are exclusively anticyclonic, with cold, fresh middepth potential temperature (θ) and salinity (S) cores surrounded by azimuthal currents of ~15 cm s^-1. Their θ/S properties identify them unambiguously as the products of wintertime deep convection in the interior Labrador Sea. Compared with eddies in other regions, these anticyclones are unusual for their strong surface expressions and composite θ/S cores. Two warm cyclones are also seen; these are larger (radius ~15 km) than the anticyclones and about as energetic (currents ~15 cm s^-1). Their θ/S and potential vorticity properties suggest that they are created by stretching a column of water from the Irminger Current, which surrounds the Labrador Sea on three sides.
   
   
3. Lilly, J. M., P. B. Rhines, M. Visbeck, R. E. Davis, J. R. N. Lazier, F. Schott and D. Farmer (1999). Observing deep convection in the Labrador Sea during winter 1994–95. Journal of Physical Oceanography 29, 2065–2098. [2.1 Mb pdf]
   
   Abstract. A 12-month mooring record (May 1994--June 1995), together with accompanying PALACE float data, is used to describe an annual cycle of deep convection and restratification in the Labrador Sea. The mooring is located at 56.75°N, 52.5°W, near the former site of Ocean Weather Station Bravo, in water of ~3500 m depth. This is a pilot experiment for climate monitoring, and also for studies of deep-convection dynamics. Mooring measurements include temperature (T), salinity (S), horizontal and vertical velocity, and acoustic measurement of surface winds. The floats made weekly temperature/salinity profiles between their drift level (near 1500 m) and the surface.
   
   With moderately strong cooling to the atmosphere (~300 W m^-2 averaged from November to March), wintertime convection penetrated from the surface to about 1750 m, overcoming the stabilizing effect of upper-ocean low-salinity water. The water column restratifies rapidly after brief vertical homogenization (in potential density, salinity, and potential temperature). Both the rapid restratification and the energetic high-frequency variations of T and S observed at the mooring are suggestive of a convection depth that varies greatly with location. Lateral variations in T and S exist down to very small scales, and these remnants of convection decay (with e-folding time ~170 day) after convection ceases. Lateral variability at the scale of 100 km is verified by PALACE profiles. The Eulerian mooring effectively samples the convection in a mesoscale region of ocean as eddies sweep past it; the Lagrangian PALACE floats are complementary in sampling the geography of deep convection more widely. This laterally variable convection leaves the water column with significant vertical gradients most of the year. Convection followed by lateral mixing gives vertical salinity profiles the (misleading) appearance that a one-dimensional diffusive process is fluxing freshwater downward.
   
   During spring, summer, and fall the salinity, temperature, and buoyancy rise steadily with time throughout most of the water column. This is likely the result of mixing with the encircling boundary currents, compensating for the escape of Labrador Sea Water from the region. Low-salinity water mixes into the gyre only near the surface.
   
   The water-column heat balance is in satisfactory agreement with meteorological assimilation models. Directly observed subsurface calorimetry may be the more reliable indication of the annual-mean air/sea heat flux. Acoustic instrumentation on the mooring gave a surprisingly good time series of the vector surface wind.
   
   The three-dimensional velocity field consists of convective plumes of width ~200 to 1000 m, vertical velocities of 2 to 8 cm s^-1, and Rossby numbers of order unity, embedded in stronger (~20 cm s^-1) lateral currents associated with mesoscale eddies. Horizontal currents with timescales of several days to several months are strongly barotropic. They are suddenly energized as convection reaches great depth in early March, and develop toward a barotropic state, as also seen in models of convectively driven geostrophic turbulence in a weakly stratified, high-latitude ocean. Currents decay through the summer and autumn, apart from some persistent isolated eddies. These coherent, isolated, cold anticyclones carry cores of pure convected water long after the end of winter.
   
   Boundary currents nearby interact with the Labrador Sea gyre and provide an additional source of eddies in the interior Labrador Sea. An earlier study of the pulsation of the boundary currents is supported by observations of sudden ejection of floats from the central gyre into the boundary currents (and sudden ingestion of boundary current floats into the gyre interior), in what may be a mechanism for exchange between Labrador Sea Water and the World Ocean.
   
   
2. Lilly, J. M., and J. Park (1995). Multiwavelet spectral and polarization analysis. Geophysical Journal International 122, 1001–1021. [7.6 Mb pdf]
   
   Abstract. We present an algorithm, based on the wavelet transform and multiple taper spectral analysis, for providing a low-variance spectrum estimate of a non-stationary data process. The 'multiwavelet' algorithm uses, within each frequency band, a number of mutually orthogonal Slepian wavelets, optimaly concentrated in frequency. The sum of the squared wavelet transforms with the Slepian wavelets results in a spectrum estimate that is both low-variance and resistant to broad-band bias. The multiwavelet algorithm is used to estimate the time-varying spectral density matrix S(f,t) for two or more time series, in particular for three-component seismic data. Coherent three-component motion is described by motion along a single trajectory, with appropriate projections onto the three component axes. This trajectory is found by applying a singular value decomposition (SVD) to a matrix M(f,t) of the wavelet transform values. The normalized first singular value of the SVD determines whether a correlation among the three components of the seismogram is statistically significant. Where significant, coherent particle motion is reconstructed by a linear combination of the wavelets with coefficients specified by the first left-singular vector. The polarization of this motion with respect to the coordinate axes is given by the first right-singular vector. Where the wavelets are real-valued, the usefulness of this method is limited to cases in which the three components of the seismic record oscillate in phase with each other, as is often the case for seismic body waves. Elliptical polarization is handled by pairing even and odd Slepian wavelets into complex-valued wavelets, capable of detecting phase shifts between components. We demonstrate the multiwavelet spectrum and polarization estimators on seismic data from a large shallow earthquake in the Solomon Islands, and from the recent deep earthquake in Fiji (1994 March 9) and Bolivia (1994 June 9).
   
   
1. Bolton, E. W., K. A. Maasch and J. M. Lilly (1995). A wavelet analysis of Plio-Pleistocene climate indicators: A new view of periodicity evolution. Geophysical Research Letters 22, 2753–2756. [1.4 Mb pdf]
   
   Abstract. Wavelet analysis offers an alternative to Fourier based time-series analysis, and is particularly useful when the amplitudes and periods of dominant cycles are time dependent. We analyze climatic records derived from oxygen isotopic ratios of marine sediment cores with modified Morlet wavelets. We use a normalization of the Morlet wavelets which allows direct correspondence with Fourier analysis. This provides a direct view of the oscillations at various frequencies, and illustrates the nature of the time-dependence of the dominant cycles.