Revisiting prior distributions,Part II: Implications of the physical prior in maximum entropy analysis

The well-known “Maximum Entropy Formalism” offers a powerful framework for deriving probability density functions given a relevant knowledge base and an adequate prior. The majority of results based on this approach have been derived assuming a flat uninformative prior, but this assumption is to a large extent arbitrary (any one-to-one transformation of the random variable will change the flat uninformative prior into some non-constant function). In a companion paper we introduced the notion of a natural reference point for dimensional physical variables, and used this notion to derive a class of physical priors that are form-invariant to changes in the system of dimensional units. The present paper studies effects of these priors on the probability density functions derived using the maximum entropy formalism. Analysis of real data shows that when the maximum entropy formalism uses the physical prior it yields significantly better results than when it is based on the commonly used flat uninformative prior. This improvement reflects the significance of the incorporating additional information (contained in physical priors), which is ignored when flat priors are used in the standard form of the maximum entropy formalism. A potentially serious limitation of the maximum entropy formalism is the assumption that sample moments are available. This is not the case in many macroscopic real-world problems, where the knowledge base available is a finite sample rather than population moments. As a result, the maximum entropy formalism generates a family of “nested models” parameterized by the unknown values of the population parameters. In this work we combine this formalism with a model selection scheme based on Akaike’s information criterion to derive the maximum entropy model that is most consistent with the available sample. This combination establishes a general inference framework of wide applicability in scientific/engineering problems.     

On the theory of translationally invariant magnetohydrodynamic equilibria with anisotropic pressure and magnetic shear

We present an improved formalism for translationally invariant magnetohydrodynamic equilibria with anisotropic pressure and currents with a field aligned component. The derivation of a Grad-Shafranov type equation is given along with a constraint which links the shear field to the parallel pressure. The difficulties of the formalism are discussed and various methods of circumventing these difficulties are given. A simple example is then used to highlight the methods and difficulties involved.     

On the sampling errors from raingauges and microwave attenuation measurements

The sampling error formalism by North and Nakamoto (1989) has been widely referenced in research papers on sampling using space-borne sensors or ground-borne sensors. However, their formalism is found to not only underestimate the sampling error, especially for the raingauge network case, but also not be applicable for the cases of using a line of raingauges or microwave attenuation measurements. In this paper, the sampling error formalism has been revised and applied to the same sampling design and the same rainrate model as in North and Nakamoto (1989) for the comparison. The sampling error estimated using the revised formula was found to be more than 50% higher than that by North and Nakamoto (1989). For the case of using a line of raingauges we found that the sampling error converges to a certain value, not zero as in North and Nakamoto formalism, as the number of gauges increases. The microwave attenuation measurements case, which is the same as the case of using a line of infinite raingauges, also gives non-zero sampling errors. Finally, the combined sampling using both satellite and ground-borne sensors (e.g., raingauge network, a line of raingauges, or microwave attenuation measurements) was reviewed to check their design orthogonality and estimated the sampling errors for the combination of satellite and raingauge network case to see its behavior depending on various settings of these two different measurements.     

Theory of anisotropic dynamic ray tracing in ray-centred coordinates

A 4×4-propagator matrix formalism is presented for anisotropic dynamic ray tracing, including the propagation across curved interfaces. The computations are organised in the same way as in ervený's well-known isotropic propagator matrix formalism. Attention is paid to cases where double eigenvalues of the Christoffel matrix result in unstable expressions in the dynamic ray tracing system, but where geometrical spreading is well-behaved.     

Thermodynamics of liquid silicates, a preliminary report

We propose a model for the calculation of the thermodynamic properties of silicate liquids. In our model we make use of the Flory-Huggins regular solution formalism for liquid solutions. In this preliminary paper we have limited ourselves to demonstrating that the Flory-Huggins formalism describes satisfactorily the thermodynamic properties of simple liquid silicate solutions, i.e. systems in which only end-member components or solid solutions thereof occur as solid phases and where there is only one well-behaved liquid phase. To compare theoretical results with laboratory observations we need thermodynamic data for liquid silicates. For this reason we have included in this report a review of the available information on the latent heat of fusion of silicate minerals.     

Excitation of the Slichter mode by collision with a meteoroid or pressure variations at the surface and core boundaries

We use a normal-mode formalism to compute the response of a spherical, self-gravitating anelastic PREM-like Earth model to various excitation sources at the Slichter mode period. The formalism makes use of the theory of the Earth’s free oscillations based upon an eigenfunction expansion methodology. We determine the complete response in the form of Green’s function obtained from a generalization of Betti’s reciprocity theorem. Surficial (surface load, fluid core pressure), internal (earthquakes, explosions) and external (object impact) sources of excitation are investigated to show that the translational motion of the inner-core would be best excited by a pressure acting at the core boundaries at time-scales shorter than the Slichter eigenperiods.     

A note on the fast computation of response spectra estimates

An algorithm is presented for fast computation of estimates of response spectra. It is based on the formalism of random vibration theory. Numerical results show that estimates are reliable and useful for practical applications.     

A Lagrangian-trajectory study of a gradually mixed estuary

When modelling is used for investigating estuarine systems, a choice generally has to be made between applying simple mass-balance considerations or using a process-resolving three-dimensional (3-D) numerical circulation model. In the present investigation of the Gulf of Finland, a gradually mixed estuary in the Baltic Sea, it is demonstrated how Lagrangian-trajectory analysis applied to the output from a 3-D model minimizes the disadvantages associated with both of the modelling techniques referred to above. This formalism made it possible to demonstrate that the main part of the Gulf is dominated by water originating from the Baltic proper, and that the most pronounced mixing with fresh water from the river Neva takes place over a limited zone in the inner part of the Gulf. Dynamical insights were furthermore obtained by using the Lagrangian formalism to construct overturning stream-functions for the two source waters.     

On the magnetic flux linkage of an electrically-conducting fluid: A treatment of the relativistic case using the exterior calculus formalism

Abstract

Hide's (1979) theorem on the magnetic flux linkage of an electrically-conducting fluid is extended to the fully general relativistic case by rederiving the theorem in the elegant and succinct formalism of Cartan's exterior calculus.     

A Bayesian Assessment of Seismic Semi-Periodicity Forecasts

Among the schemes for earthquake forecasting, the search for semi-periodicity during large earthquakes in a given seismogenic region plays an important role. When considering earthquake forecasts based on semi-periodic sequence identification, the Bayesian formalism is a useful tool for: (1) assessing how well a given earthquake satisfies a previously made forecast; (2) re-evaluating the semi-periodic sequence probability; and (3) testing other prior estimations of the sequence probability. A comparison of Bayesian estimates with updated estimates of semi-periodic sequences that incorporate new data not used in the original estimates shows extremely good agreement, indicating that: (1) the probability that a semi-periodic sequence is not due to chance is an appropriate estimate for the prior sequence probability estimate; and (2) the Bayesian formalism does a very good job of estimating corrected semi-periodicity probabilities, using slightly less data than that used for updated estimates. The Bayesian approach is exemplified explicitly by its application to the Parkfield semi-periodic forecast, and results are given for its application to other forecasts in Japan and Venezuela.     

A platform for Kirchhoff data mapping in scalar models of data acquisition

Kirchhoff data mapping (KDM) is a procedure for transforming data from a given input source/receiver configuration and background earth model to data corresponding to a different output source/receiver configuration and background model. The generalization of NMO/DMO, datuming and offset continuation are three examples of KDM applications. This paper describes a 'platform' for KDM for scalar wavefields. The word, platform, indicates that no calculations are carried out in this paper that would adapt the derived formula to any one of a list of KDMs that are presented in the text. Platform formulae are presented in 3D and in 2.5D. For the latter, the validity of the platform equation is verified — within the constraints of high-frequency asymptotics — by applying it to a Kirchhoff approximate representation of the upward scattered data from a single reflector and for an arbitrary source/receiver configuration. The KDM formalism is shown to map this Kirchhoff model data in the input source/receiver configuration to Kirchhoff data in the output source/receiver configuration, with one exception. The method does not map the reflection coefficient. Thus, we verify that, asymptotically, the ray theoretical geometrical spreading effects due to propagation and reflection (including reflector curvature) are mapped by this formalism, consistent with the input and output modelling parameters, while the input reflection coefficient is preserved. In this sense, this is a 'true-amplitude' formalism. As with earlier Kirchhoff inversion, a slight modification of the kernel of KDM provides alternative integral operators for estimating the specular reflection angle, both in the input configuration and in the output configuration, thereby providing a basis for amplitude-versus-angle analysis of the data.     

E.—NON-GOVERNMENTAL ORGANIZATIONS / E.—ORGANISATIONS NON GOUVERNEMENTALES

Abstract

It is shown that a thermodynamic analogy can be set up for structurally cyclic as well as for noncyclic river nets that satisfy Horton's law of stream numbers. In the latter case, the microcanonical formalism of statistical thermodynamics has to be used.     

Perturbation theory in low rate seismology

Summary In the case of low frequency motions in an elastic continuum, first-order perturbation theory can be used to estimate the effect of real-world deviations from the ideal model. Although the effects are often small, some interesting results can be found fairly easily. In spite of the simplicity in the structure of the formalism, nontrivial applications lead to remarkably complicated results.     

Forecasting tsunamis in Poverty Bay,New Zealand,with deep-ocean gauges

The response/transfer function of a coastal site to a remote open-ocean point is introduced, with the intent to directly convert open-ocean measurements into the wave time history at the site. We show that the tsunami wave at the site can be predicted as the wave is measured in the open ocean as far as 1,000+ km away from the site, with a straightforward computation which can be performed almost instantaneously. The suggested formalism is demonstrated for the purpose of tsunami forecasting in Poverty Bay, in the Gisborne region of New Zealand. Directional sensitivity of the site response due to different conditions for the excitation of the shelf and the bay’s normal modes is investigated and used to explain tsunami observations. The suggested response function formalism is validated with available records of the 2010 Chilean tsunami at Gisborne tide gauge and at the nearby deep-ocean assessment and reporting of tsunamis (DART) station 54401. The suggested technique is also demonstrated by hindcasting the 2011 Tohoku tsunami and 2012 Haida Gwaii tsunami at Monterey Bay, CA, using an offshore record of each tsunami at DART station 46411.

     

Forecasting tsunamis in Poverty Bay, New Zealand, with deep-ocean gauges

The response/transfer function of a coastal site to a remote open-ocean point is introduced, with the intent to directly convert open-ocean measurements into the wave time history at the site. We show that the tsunami wave at the site can be predicted as the wave is measured in the open ocean as far as 1,000+ km away from the site, with a straightforward computation which can be performed almost instantaneously. The suggested formalism is demonstrated for the purpose of tsunami forecasting in Poverty Bay, in the Gisborne region of New Zealand. Directional sensitivity of the site response due to different conditions for the excitation of the shelf and the bay’s normal modes is investigated and used to explain tsunami observations. The suggested response function formalism is validated with available records of the 2010 Chilean tsunami at Gisborne tide gauge and at the nearby deep-ocean assessment and reporting of tsunamis (DART) station 54401. The suggested technique is also demonstrated by hindcasting the 2011 Tohoku tsunami and 2012 Haida Gwaii tsunami at Monterey Bay, CA, using an offshore record of each tsunami at DART station 46411.     

Is the Fisher Distribution Additive?

The Probability Density Function (PDF) of the Fisher distribution is expressed by the expansion in spherical harmonics in order to prove that the product of two Fisher-distributed random rotations has a distribution of different type, which, however, may be well approximated by the Fisher one. The introduced formalism provides the pattern of how to cope with various types of the distribution on the sphere being combined as rotations.     

Green's functions and response functions in geochemical modelling

The use of Green's functions and response functions in modelling atmospheric and oceanic tracers is discussed. It is shown that low-resolution Green's function representations can describe tracer distributions more accurately than low-resolution models. The Green's function formalism is also used to discuss the problem of determining surface CO₂ sources from observations of surface CO₂ concentrations.     

Nonequilibrium effects in models of three-phase flow in porous media

In this paper we extend to three-phase flow the nonequilibrium formalism proposed by Barenblatt and co-workers for two-phase porous media flow. The underlying idea is to include nonequilibrium effects by introducing a pair of effective water and gas saturations, which are linked to the actual saturations by a local evolution equation. We illustrate and analyze how nonequilibrium effects lead to qualitative and quantitative differences in the solution of the three-phase flow equations.     

Field theoretical techniques in statistical fluid dynamics: With application to nonlinear wave dynamics

Abstract

A derivation of two-point Markovian closure is presented in classical statistical field theory formalism. It is emphasized that the procedures used in this derivation are equivalent to those employed in the quantum statistical field theory derivation of the Boltzmann equation. Application of these techniques to the study of two-dimensional flow on a β-plane yields a quasi-homogeneous, quasi-stationary transport equation and a renormalized dispersion relation for Rossby waves     

Contribution of the functional formulation of the problem of turbulence to the process of its algebraization

Summary The progressive development of the theory of turbulence is based on the formalism of characteristic functionals. Drawing on an incomplete analogy between the Hopf equation for these functionals and the equation for the quantized boson field, a system of fundamental principles of the theory of turbulence is proposed. This represents a set of statements which can be used in an attempt to classify the theory in the sense of its algebraization and, therefore, also its axiomatization.