Jonathan M. Lilly

Bio: Jonathan M. Lilly is an academic researcher from University of Washington. The author has contributed to research in topics: Wavelet transform & Wavelet. The author has an hindex of 24, co-authored 57 publications receiving 2721 citations. Previous affiliations of Jonathan M. Lilly include Yale University.

Bivariate Empirical Mode Decomposition

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.

The Aquarius/SAC-D mission: Designed to meet the salinity remote-sensing challenge

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. The 1995 Oceanography paper evidently marked a turning point for salinity remote sensing. Until that time, there was no organized program for developing a satellite capability to measure this important yet sparsely observed ocean variable, in contrast to the established satellite programs to observe surface temperatures, winds, sea level, and ocean color. The paper demonstrated that newly evolving technology could provide global salinity measurements at a similar, scientifically useful accuracy and spatio-temporal resolution, and it came at a time of growing scientific awareness of the need for the data. Vigorous efforts begun in the late 1990s by the community of scientists interested in salinity measurements have led to the development of the Aquarius/SAC-D mission (Figure 1), now targeted for launch in mid 2010. This mission's principal scientific objective is to provide monthly global measurements of SSS. Here, we describe the Aquarius/SAC-D mission and how its capabilities are designed to meet the salinity remote-sensing challenge, providing the ocean research community with a preview of the measurement characteristics and scientific applications of this pioneering mission.

Generalized Morse Wavelets as a Superfamily of Analytic Wavelets

Abstract: The generalized Morse wavelets are shown to constitute a superfamily that essentially encompasses all other commonly used analytic wavelets, subsuming eight apparently distinct types of analysis filters into a single common form. This superfamily of analytic wavelets provides a framework for systematically investigating wavelet suitability for various applications. In addition to a parameter controlling the time-domain duration or Fourier-domain bandwidth, the wavelet shape with fixed bandwidth may be modified by varying a second parameter, called γ. For integer values of γ, the most symmetric, most nearly Gaussian, and generally most time-frequency concentrated member of the superfamily is found to occur for γ = 3. These wavelets, known as “Airy wavelets,” capture the essential idea of popular Morlet wavelet, while avoiding its deficiencies. They may be recommended as an ideal starting point for general purpose use.

Higher-Order Properties of Analytic Wavelets

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 ldquoAiry waveletsrdquo 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.

Forecasting, Structural Time Series Models and the Kalman Filter

Abstract: In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Table Of Integrals Series And Products

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The SMOS Mission: New Tool for Monitoring Key Elements ofthe Global Water Cycle

Abstract: It is now well understood that data on soil moisture and sea surface salinity (SSS) are required to improve meteorological and climate predictions. These two quantities are not yet available globally or with adequate temporal or spatial sampling. It is recognized that a spaceborne L-band radiometer with a suitable antenna is the most promising way of fulfilling this gap. With these scientific objectives and technical solution at the heart of a proposed mission concept the European Space Agency (ESA) selected the Soil Moisture and Ocean Salinity (SMOS) mission as its second Earth Explorer Opportunity Mission. The development of the SMOS mission was led by ESA in collaboration with the Centre National d'Etudes Spatiales (CNES) in France and the Centro para el Desarrollo Tecnologico Industrial (CDTI) in Spain. SMOS carries a single payload, an L-Band 2-D interferometric radiometer operating in the 1400-1427-MHz protected band . The instrument receives the radiation emitted from Earth's surface, which can then be related to the moisture content in the first few centimeters of soil over land, and to salinity in the surface waters of the oceans. SMOS will achieve an unprecedented maximum spatial resolution of 50 km at L-band over land (43 km on average over the field of view), providing multiangular dual polarized (or fully polarized) brightness temperatures over the globe. SMOS has a revisit time of less than 3 days so as to retrieve soil moisture and ocean salinity data, meeting the mission's science objectives. The caveat in relation to its sampling requirements is that SMOS will have a somewhat reduced sensitivity when compared to conventional radiometers. The SMOS satellite was launched successfully on November 2, 2009.