The Azimuth Project
Multifractal system

Contents

Idea

from Wikipedia:

A multifractal system is a generalization of a fractal system in which a single exponent - the fractal dimension is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed.

Multifractal systems are common in nature, especially geophysics. They include fully developed turbulence, stock market time series, real world scenes, the Sun’s magnetic field time series, heartbeat dynamics, human gait, and natural luminosity time series. Embryogenesis is also multifractal system which represents a new type of physics, named fractal mechanics.

Details

Mathematics

One spanish research group has tried to test the viability of using Finite Sized Lyapunov Exponents:

Much of atmospheric and oceanic transport is associated with coherent structures. Lagrangian methods are emerging as optimal tools for their identication and analysis. An important Lagrangian technique which is starting to be widely used in oceanography is that of Finite-Size Lyapunov Exponents (FSLEs). Despite this growing relevance there are still many open questions concerning the reliability of the FSLEs in order to analyse the ocean dynamics. Here is a video showing the use of FSLE

In particular, it is still unclear how robust they are when confronted with real data. In this paper we analyze the effect on this Lagrangian technique of the two most important eects when facing real data, namely noise and dynamics of unsolved scales. Our results, using as a benchmarch data from a primitive numerical model of the Mediterranean Sea, show that even when some dynamics is missed the FSLEs results still give an accurate picture of the oceanic transport properties

Environmental applications

Used in climate modeling and analysis and in dynamics of geophysical systems, as a way to model the non-linear aspects like positive or negative feedback mechanisms in climate models particularly turbulence using wavelets.

References

Evidence of past climate variations are stored in polar ice caps and indicate glacial-interglacial cycles of 100 kyr. Using advanced scaling techniques we study the long-range correlation properties of temperature proxy records of four ice cores from Antarctica and Greenland.

These series are long-range correlated in the time scales of 1–100 kyr. We show that these time series are nonlinear for time scales of 1–100 kyr as expressed by temporal long-range correlations of magnitudes of temperature increments and by a broad multifractal spectrum. Our results suggest that temperature increments appear in clusters of big and small increments— a big ( positive or negative) climate change is most likely followed by a big ( positive or negative) climate change and a small climate change is most likely followed by a small climate change.

Abstract:

Using wavelets, statistically significant interannual and interdecadal oscillations that occurred haphazardly have been detected in southwestern (SW) Canadian seasonal precipitation anomalies. At interannual scales, station precipitation anomalies show unstable relations with large-scale climate anomalies such as the El Niño–Southern Oscillation ENSO, Pacific Decadal Oscillation (PDO), Pacific/North America (PNA), East Pacific (EP) and West Pacific (WP) patterns, and the Central North Pacific (CNP) index. Not all significant precipitation activities could be matched by similar activities in one or more climate anomalies considered. Inconsistent wavelet coherence and phase difference between the leading principal components (PC) of regional precipitation anomalies and climate indices as well as weak Pearson’s correlations between band-passed precipitation PCs and climate indices for the 2–3 year and 3–8 year scales provide supporting evidence for unstable precipitation climate relationships at the interannual scale. On the other hand, interdecadal precipitation variability is mainly associated with low-frequency variability in CNP, PDO and ENSO.

Composite analysis of winter precipitation shows that ENSO, PDO, PNA and WP offer better separation of positive and negative precipitation anomalies than EP and CNP. However, the effect of ENSO is found to be stronger than the others. Precipitation power spectrum plots mostly reveal two linear decay regions of different slopes separated by a breakpoint located approximately at 20 to 30 days, while empirical probability plots reveal power law behavior and hyperbolic intermittency in these data, whose correlation dimensions (D 2) are between 8 and 9. Different multifractal behaviors are observed among stations because the amount of different rainfall generating mechanisms vary from station to station, as reflected by the haphazard nature of oscillations detected in most precipitation data. Although the leading PCs of winter regional precipitation show modest correlations at zero- to three-season lead times with ENSO and PDO indices, the high D2 values and absence of consistent interannual precipitation activities suggest that prediction of SW Canadian seasonal precipitation by teleconnection with climate indices is likely limited. Adding other predictor fields such as sea surface temperature and/or sea pressure may be useful