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12. Lilly, J. M. and S. C. Olhede (2008). On the design of optimal analytic wavelets. Submitted to IEEE Transactions on Signal Processing. arXiv:0802.2377v1. [0.5 Mb pdf]
    
    Abstract. Properties determining optimal behavior of continuous analytic wavelet functions are investigated, and wavelet functions nearly obtaining such behavior are identified. This is accomplished through detailed investigation of the generalized Morse wavelets, a highly flexible family of exactly analytic continuous wavelets. The degree of time/frequency localization, the existence of a unique interpretation of scale as frequency, and the extent of bias involved in estimating properties of modulated oscillatory signals, are proposed as important considerations. Optimal wavelet behavior is then found to be achieved by minimizing third central moments in both the frequency and the time domains. A particular subset of the generalized Morse wavelets, recognized as deriving from an inhomogeneous Airy function, are shown to be nearly optimal. These “Airy wavelets” are expected to substantially outperform the only approximately analytic Morlet wavelets for high time localization. Further investigation of the generalized Morse wavelets reveals a remarkably broad range of behavior, and suggests that these wavelets can be considered as a generic analytic wavelet family appropriate for a variety of applications.
    
    
11. Lilly, J. M. and S. C. Olhede (2007). On the analytic wavelet transform. Submitted to IEEE Transactions on Information Theory. arXiv:0711.3834v1. [0.4 Mb pdf]
    
    Abstract. An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed. The analytic signal is first locally represented as a modulated oscillation, its demodulate is Taylor-expanded everywhere, and this expansion is given a novel representation in terms of the Bell polynomials. Time-domain moments of the demodulated signal are then seen to interact with frequency-domain derivatives of the wavelet, inducing an hierarchy of departures of the analytic wavelet transform from the analytic signal. Assumptions regarding the joint properties of the signal and wavelet then permit considerable simplification of the expressions for the time and scale-varying form of these deviations. Using this result one may quantify the time-varying bias associated with signal estimation via wavelet ridge analysis for both amplitude and phase definitions of the ridges. An improved estimator which eliminates the lowest-order bias contribution is proposed.
    
    
10. Lilly, J. M. and E. E. Lettvin (2007). Identity of two formulations of gravity-capillary wave interactions. Submitted to Nonlinear Processes in Geophysics. [0.2 Mb pdf]
    
    Abstract. It is shown that two seemingly different formulations of weakly nonlinear gravity–capillary wave interactions are in fact dynamically identical, resolving a long-standing controversy. The two apparently different solutions express a single physical wave field evaluated at two different vertical levels. The complicated algebraic relationship between the two solutions is a consequence of the simple fact that the three-dimensional velocity potential is a harmonic function. The identity of these two approaches becomes clear after the presentation of considerably simplified expressions for the interaction coefficients.
    
    
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. [1.9 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.