Section: Research Program

Multiscale description in terms of multiplicative cascade

GEOSTAT is studying complex signals under the point of view of methods developed in statistical physics to study complex systems, with a strong emphasis on multiresolution analysis. Linear methods in signal processing refer to the standard point of view under which operators are expressed by simple convolutions with impulse responses. Linear methods in signal processing are widely used, from least-square deconvolution methods in adaptive optics to source-filter models in speech processing. Because of the absence of localization of the Fourier transform, linear methods are not successful to unlock the multiscale structures and cascading properties of variables which are of primary importance as stated by the physics of the phenomena. This is the reason why new approaches, such as DFA (Detrented Fluctuation Analysis), Time-frequency analysis, variations on curvelets  [45] etc. have appeared during the last decades. Recent advances in dimensionality reduction, and notably in Compressive Sensing, go beyond the Nyquist rate in sampling theory using nonlinear reconstruction, but data reduction occur at random places, independently of geometric localization of information content, which can be very useful for acquisition purposes, but of lower impact in signal analysis. One important result obtained in GEOSTAT is the effective use of multiresolution analysis associated to optimal inference along the scales of a complex system. The multiresolution analysis is performed on dimensionless quantities given by the singularity exponents which encode properly the geometrical structures associated to multiscale organization. This is applied successfully in the derivation of high resolution ocean dynamics, or the high resolution mapping of gaseous exchanges between the ocean and the atmosphere; the latter is of primary importance for a quantitative evaluation of global warming. Understanding the dynamics of complex systems is recognized as a new discipline, which makes use of theoretical and methodological foundations coming from nonlinear physics, the study of dynamical systems and many aspects of computer science. One of the challenges is related to the question of emergence in complex systems: large-scale effects measurable macroscopically from a system made of huge numbers of interactive agents [36], [33], [50], [40]. Some quantities related to nonlinearity, such as Lyapunov exponents, Kolmogorov-Sinai entropy etc. can be computed at least in the phase space  [34]. Consequently, knowledge from acquisitions of complex systems (which include complex signals) could be obtained from information about the phase space. A result from F. Takens  [46] about strange attractors in turbulence has motivated the determination of discrete dynamical systems associated to time series  [38], and consequently the theoretical determination of nonlinear characteristics associated to complex acquisitions. Emergence phenomena can also be traced inside complex signals themselves, by trying to localize information content geometrically. Fundamentally, in the nonlinear analysis of complex signals there are broadly two approaches: characterization by attractors (embedding and bifurcation) and time-frequency, multiscale/multiresolution approaches. Time-frequency analysis  [35] and multiscale/multiresolution are the subjects of intense research and are profoundly reshaping the analysis of complex signals by nonlinear approaches  [32], [37]. In real situations, the phase space associated to the acquisition of a complex phenomenon is unknown. It is however possible to relate, inside the signal's domain, local predictability to local reconstruction and deduce from that singularity exponents [11]   [7]. We are working on:

• the determination of quantities related to universality classses,

• the geometric localization of multiscale properties in complex signals,

• cascading characteristics of physical variables.

The alternative approach taken in GEOSTAT is microscopical, or geometrical: the multiscale structures which have their "fingerprint" in complex signals are being isolated in a single realization of the complex system, i.e. using the data of the signal itself, as opposed to the consideration of grand ensembles or a wide set of realizations. This is much harder than the ergodic approaches, but it is possible because a reconstruction formula such as the one derived in  [47] is local and reconstruction in the signal's domain is related to predictability. This approach is analogous to the consideration of "microcanonical ensembles" in statistical mechanics.

A multiscale organization is a fundamental feature of a complex system, it can be for example related to the cascading properties in turbulent systems. We make use of this kind of description when analyzing turbulent signals: intermittency is observed within the inertial range and is related to the fact that, in the case of FDT, symmetry is restored only in a statistical sense, a fact that has consequences on the quality of any nonlinear signal representation by frames or dictionaries.

The example of FDT as a standard "template" for developing general methods that apply to a vast class of complex systems and signals is of fundamental interest because, in FDT, the existence of a multiscale hierarchy ${ℱ}_h$ which is of multifractal nature and geometrically localized can be derived from physical considerations. This geometric hierarchy of sets is responsible for the shape of the computed singularity spectra, which in turn is related to the statistical organization of information content in a signal. It explains scale invariance, a characteristic feature of complex signals. The analogy from statistical physics comes from the fact that singularity exponents are direct generalizations of critical exponents which explain the macroscopic properties of a system around critical points, and the quantitative characterization of universality classes, which allow the definition of methods and algorithms that apply to general complex signals and systems, and not only turbulent signals: signals which belong to a same universality class share common statistical organization. In GEOSTAT, the approach to singularity exponents is done within a microcanonical setting, which can interestingly be compared with other approaches such that wavelet leaders, WTMM or DFA. During the past decades, classical approaches (here called "canonical" because they use the analogy taken from the consideration of "canonical ensembles" in statistical mechanics) permitted the development of a well-established analogy taken from thermodynamics in the analysis of complex signals: if $ℱ$ is the free energy, $𝒯$ the temperature measured in energy units, $𝒰$ the internal energy per volume unit $𝒮$ the entropy and $\widehat{\beta }=1/𝒯$, then the scaling exponents associated to moments of intensive variables $p\to {\tau }_p$ corresponds to $\widehat{\beta }ℱ$, $𝒰\left(\widehat{\beta }\right)$ corresponds to the singularity exponents values, and $𝒮\left(𝒰\right)$ to the singularity spectrum.

The singularity exponents belong to a universality class, independently of microscopic properties in the phase space of various complex systems, and beyond the particular case of turbulent data (where the existence of a multiscale hierarchy, of multifractal nature, can be inferred directly from physical considerations). They describe common multiscale statistical organizations in different complex systems  [44], and this is why GEOSTAT is working on nonlinear signal processing tools that are applied to very different types of signals.

For example we give some insight about the collaboration with LEGOS Dynbio team (http://www.legos.obs-mip.fr/recherches/equipes/dynbio.) about high-resolution ocean dynamics from microcanonical formulations in nonlinear complex signal analysis. Indeed, synoptic determination of ocean circulation using data acquired from space, with a coherent depiction of its turbulent characteristics remains a fundamental challenge in oceanography. This determination has the potential of revealing all aspects of the ocean dynamic variability on a wide range of spatio-temporal scales and will enhance our understanding of ocean-atmosphere exchanges at super resolution, as required in the present context of climate change. We show that the determination of a multiresolution analysis associated to the multiplicative cascade of a typical physical variable like the Sea Surface Temperature permits an optimal inference of oceanic motion field across the scales, resulting in a new method for deriving super resolution oceanic motion from lower resolution altimetry data; the resulting oceanic motion field is validated at super resolution with the use of Lagrangian buoy data available from the Global Drifter Program (http://www.aoml.noaa.gov/phod/dac/index.php.). In FDT, singularity exponents range in a bounded interval: $]h_{\infty },h_{max}[$ with $h_{\infty }<0$ being the most singular exponent. Points $𝐫$ for which $h\left(𝐫\right)<0$ localize the stongest transitions in the turbulent fluid, where an intensive physical variable like sea surface temperature behaves like $1/{𝐫}^{\left|h\right(𝐫\left)\right|}$. The links between the geometricaly localized singularity exponents, the scaling exponents of structure functions, the multiplicative cascade and the multiscale hierarchy ${ℱ}_h$ is the following:

Let $𝔖\left(𝐱\right)$ be the bidimensionnal signal recording, for each sample point $𝐱$ representing a pixel on the surface of the ocean of given resolution, the sea surface temperature (sst). To this signal we associate a measure $\mu$ whose density w.r.t Lebesgue measure is the signal's gradient norm, and from which the singularity exponents are computed [6]. It is fundamental to notice here that, contrary to other types of exponents computed in Oceanography, such as Finite Size Lyapunov exponents, singularity exponents are computed at instantaneous time, and do not need time series.

Having computed the singularity exponents at each point of a SST signal, a microcanonical version of the multiplicative cascade associated to the scaling properties of the sst become available. The idea of the existence of a geometrically localized multiplicative cascade goes back to  [43]. The multiplicative cascade, written pointwise, introduces random variables ${\eta }_{l^{\text{'}}/l}\left(𝐱\right)$ for $0<l^{\text{'}}<l$ such that

in which the equality is valid pointwise and not only in distribution. Any mother wavelet $\psi$ such that the process ${\eta }_{l^{\text{'}}/l}\left(𝐱\right)$ is independant of ${𝒯}_{\psi }\mu (𝐱,l^{\text{'}})$ is called an optimal wavelet: it optimizes inference of physical variables across the scales and consequently describes the multiplicative cascade at each point $𝐱$ in the signal domain. The injection variables ${\eta }_{l^{\text{'}}/l}\left(𝐱\right)$ are indefinitely divisible: ${\eta }_k\left(𝐱\right){\eta }_{k^{\text{'}}}\left(𝐱\right)\dot{=}{\eta }_{k{k}^{\text{'}}}\left(𝐱\right)$. It is possible to optimize cross-scale inference of physical variables by considering a multiresolution analysis associated to a discrete covering of the "space-frequency" domain. Denoting as usual ${\left(V_j\right)}_{j\in \mathbb{Z}}$ and ${\left(W_j\right)}_{j\in \mathbb{Z}}$ the discrete sequence of approximation and detail spaces associated to a given scaling function, and denoting by $\psi \in L^2\left({ℝ}^2\right)$ a wavelet which generates an Hilbertian basis on each detail space $W_j$, it is known that the detail spaces encode borders and transition information, which is ideally described in the case of turbulent signals by the singularity exponents $𝐡\left(𝐱\right)$. Consequently, a novel idea for super-resolution consists in computing a multiresolution analysis on the signal of singularity exponents $𝐡\left(𝐱\right)$, and to consider that the detail information coming from spaces $W_j$ is given the signal $𝐡\left(𝐱\right)$. The associated orthogonal projection ${\pi }_j:L^2\left({ℝ}^2\right)\to W_j$ defined by ${\displaystyle {\pi }_j\left(𝐡\right)=\sum _{n\in \mathbb{Z}}\langle 𝐡|{\psi }_{j,n}\rangle {\psi }_{j,n}}$ is then used in the reconstruction formula for retrieving a physical variable at higher resolution from its low resolution counterpart. If $𝔖\left(𝐱\right)$ is such a variable, we use a reconstruction formula: $A_{j-1}𝔖=A_j𝔖+{\pi }_j\left(𝐡\right)$ with $A_j:L^2\left({ℝ}^2\right)\to V_j$ is the orthogonal projection on the space $V_j$ (approximation operator) and ${\pi }_j$ is the orthogonal projection on the detail spaces $W_j$ associated to the signal of singularity exponents $𝐡\left(𝐱\right)$. Validation is performed using Lagrangian buoy data with very good results [10]. We have realized a demonstration movie showing the turbulent ocean dynamics at an SST resolution of 4 km computed from the SST microcanonical cascade and the low-resolution GEKCO product for the year 2006 over the southwestern part of the Indian Ocean. We replace the missing data in the SST MODIS product (clouds and satellite swath) by the corresponding data available from the Operational SST and Sea Ice Analysis (OSTIA) provided by the Group for High-Resolution SST Project [11], which, however, is of lower quality. Two images per day are generated for the whole year of 2006. The resulting images show the norm of the vector field in the background rendered using the line integral convolution algorithm. In the foreground, we show the resulting vector field in a linear gray-scale color map. See link to movie (size: 800 Mo).