This paper addresses a bias problem in the estimate of wavelet power spectra for atmospheric and oceanic datasets. For a time series comprised of sine waves with the same amplitude at different frequencies the conventionally adopted wavelet method does not produce a spectrum with identical peaks, in contrast to a Fourier analysis. The wavelet power spectrum in this definition, that is, the transform coefficient squared (to within a constant factor), is equivalent to the integration of energy (in physical space) over the influence period (time scale) the series spans. Thus, a physically consistent definition of energy for the wavelet power spectrum should be the transform coefficient squared divided by the scale it associates. Such adjusted wavelet power spectrum results in a substantial improvement in the spectral estimate, allowing for a comparison of the spectral peaks across scales. The improvement is validated with an artificial time series and a real coastal sea level record. Also examined is the previous example of the wavelet analysis of the Niño-3 SST data.

Given two time series, can one faithfully tell, in a rigorous and quantitative way, the cause and effect between them? Based on a recently rigorized physical notion, namely, information flow, we solve an inverse problem and give this important and challenging question, which is of interest in a wide variety of disciplines, a positive answer. Here causality is measured by the time rate of information flowing from one series to the other. The resulting formula is tight in form, involving only commonly used statistics, namely, sample covariances; an immediate corollary is that causation implies correlation, but correlation does not imply causation. It has been validated with touchstone linear and nonlinear series, purportedly generated with one-way causality that evades the traditional approaches. It has also been applied successfully to the investigation of real-world problems; an example presented here is the cause-and-effect relation between the two climate modes, El Niño and the Indian Ocean Dipole (IOD), which have been linked to hazards in far-flung regions of the globe. In general, the two modes are mutually causal, but the causality is asymmetric: El Niño tends to stabilize IOD, while IOD functions to make El Niño more uncertain. To El Niño, the information flowing from IOD manifests itself as a propagation of uncertainty from the Indian Ocean.

Information flow or information transfer the widely applicable general physics notion can be rigorously derived from first principles, rather than axiomatically proposed as an ansatz. Its logical association with causality is firmly rooted in the dynamical system that lies beneath. The principle of nil causality that reads, an event is not causal to another if the evolution of the latter is independent of the former, which transfer entropy analysis and Granger causality test fail to verify in many situations, turns out to be a proven theorem here. Established in this study are the information flows among the components of time-discrete mappings and time-continuous dynamical systems, both deterministic and stochastic. They have been obtained explicitly in closed form, and put to applications with the benchmark systems such as the Kaplan-Yorke map, Rössler system, baker transformation, Hénon map, and stochastic potential flow. Besides unraveling the causal relations as expected from the respective systems, some of the applications show that the information flow structure underlying a complex trajectory pattern could be tractable. For linear systems, the resulting remarkably concise formula asserts analytically that causation implies correlation, while correlation does not imply causation, providing a mathematical basis for the long-standing philosophical debate over causation versus correlation.

The past years have seen the success of a novel multiscale energetics formalism in a variety of ocean and engineering fluid applications. In a self-contained way, this study introduces it to the atmospheric dynamical diagnostics, with important theoretical updates. Multiscale energy equations are derived using a new analysis apparatus, namely, multiscale window transform, with respect to both the primitive equation and quasi-geostrophic models. A reconstruction of the "atomic" energy fluxes on the multiple scale windows allows for a natural and unique separation of the in-scale transports and cross-scale transfers from the intertwined nonlinear processes. The resulting energy transfers bear a Lie bracket form, reminiscent of the Poisson bracket in Hamiltonian mechanics; we hence would call them "canonical". A canonical transfer process is a mere redistribution of energy among scale windows, without generating or destroying energy as a whole. By classification, a multiscale energetic cycle comprises of available potential energy (APE) transport, kinetic energy (KE) transport, pressure work, buoyancy conversion, work done by external forcing and friction, and the cross-scale canonical transfers of APE and KE which correspond respectively to the baroclinic and barotropic instabilities, among others, in geophysical fluid dynamics. A buoyancy conversion takes place in an individual window only, bridging the two types of energy namely KE and APE; it does not involve any processes among different scale windows, and is hence basically not related to instabilities. This formalism is exemplified with a preliminary application to the Madden-Julian Oscillation study.

Recently, a rigorous yet concise formula was derived to evaluate information flow, and hence the causality in a quantitative sense, between time series. To assess the importance of a resulting causality, it needs to be normalized. The normalization is achieved through distinguishing a Lyapunov exponent-like, one-dimensional phase-space stretching rate and a noise-to-signal ratio from the rate of information flow in the balance of the marginal entropy evolution of the flow recipient. It is verified with autoregressive models and applied to a real financial analysis problem. An unusually strong one-way causality is identified from IBM (International Business Machines Corporation) to GE (General Electric Company) in their early era, revealing to us an old story, which has almost faded into oblivion, about "Seven Dwarfs" competing with a giant for the mainframe computer market.

Information flow or information transfer is an important concept in dynamical systems which has applications in a wide variety of scientific disciplines. In this study, we show that a rigorous formalism can be established in the context of a generic stochastic dynamical system. The resulting measure of of information transfer possesses a property of transfer asymmetry and, when the stochastic perturbation to the receiving component does not rely on the giving component, has a form same as that for the corresponding deterministic system. An application with a twodimensional system is presented, and the resulting transfers are just as expected. A remarkable observation is that, for two highly correlated time series, there could be no information transfer from one certain series, say x 2 , to the other (x 1 ). That is to say, the evolution of x 1 may have nothing to do with x 2 , even though x 1 and x 2 are highly correlated. Information transfer analysis thus extends the traditional notion of correlation analysis by providing a quantitative measure of causality between time series.PACS numbers: 05.45.-a, 89.70.+c, 89.75.-k, 02.50.-r

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