First-order asymptotics for the eigenfrequencies of the Earth and application to the retrieval of large-scale lateral variations of structure

Summary . We present a variety of examples, showing systematic fluctuations as a function of angular order of measured eigenfrequencies for given normal modes of the Earth. The data are single station measurements from the GEOSCOPE network. Such fluctuations are attributed to departures from the lowest order asymptotic expression of the geometrical optics approximation. We derive first-order asymptotic expressions for the location parameter for all three components of the Earth's motion, by a method based on the stationary phase approximation and geometric relations on the unit sphere.
We illustrate the sensitivity of the fluctuations to the different parameters involved (source parameters, epicentral distance, laterally heterogeneous earth model) with synthetic examples corresponding to GEOSCOPE observations. Finally, we show the results of first attempts at inversion, which indicate that, when the fluctuations are taken into account, more accurate estimates of the great circle average eigenfrequencies can be obtained, and additional constraints put on the structure in the neighbourhood of the great circle.     

A vortex-tube model of eddies in the inertial range

Abstract

A vortex-tube geometry of the cascade of energy to small-scale eddies, in the inertial range of fully-developed turbulence, is proposed. The model is a special case of the beta model of Frisch, Sulem and Nelkin (1978). We require that the cascade conserve the principal invariants of inviscid, incompressible flow, namely volume, topological knottedness, circulation, and, at discrete times marking the termination of steps in the cascade, energy. The process terminates in a finite time, as in any beta model, leaving behind a self-similar network of “inactive” tubes. We associate a self-similar scaling dimension D with the structure, equal to the Hausdorff dimension of the set of “active” tubes at the termination of the cascade. Because circulation Λ plays a key role in the analysis of the cascade, we refer to these vortex-tube geometries as “gamma models”. The viewpoint throughout is entirely deterministic.

We describe two examples of gamma models. In the ring geometry, an eddy is a vortex ring, and the cascade produces “rings upon rings”, so we allow cutting and fusing of tubes while conserving total helicity. In the preferred helical model, no cutting is needed, and the cascade produces an infinite progression of braided “coils upon coils”. We suggest that latter geometry as a candidate for the topology of a singularity of the inviscid limit of a Navier-Stokes flow, when modeled by discrete vortex tubes.

A crucial ingredient of a gamma model, not explicitly present in a beta model, is the possibility of “splitting” a vortex tube into sub-tubes carrying smaller circulation. We suggest a dynamical basis for this process, as an instability of tubes whose cores violate the Rayleigh criterion.

The parameters describing a gamma model are not uniquely determined by our study, but there is a “simplest” helical gamma model, involving minimal splitting and distortion of tubes. The dimension D of the structure is 13/5, with a scale factor Λ = 2^?5/4. This value of D agrees with that suggested by Hentschel and Procaccia (1982), by analogy with established results for certain branched polymers.     

局部和全局边界条件的组合

时域局部人工边界是以行波概念为基础的,原则上不能用于模拟零频分量（消减运动）。Ｗｏｌｆ提出了能同时模拟零频分量和行波的双渐近－多向透射边界。本文指出,在这一组合中的双渐近边界的高频渐近项是不合理的,还查能降低模拟精度。据此,本文提出一种概念清楚、简便实用的组合方案－多次射透－静力边界,并通过数值试验对此方案进行检验。     

Self-Similar Fold Evolution Under Prescribed End-Shortening

The evolution of single-layer folds under prescribed end-shortening conditions displays folds of varying wavelength. We investigate a simple model of this kind and characterize the long-term behaviour of fold profiles. In particular we determine the evolution of the axial load and the variation of the wavelength, and we show that fold profiles are highly self-similar.     

Continuous surface representation and approximation: spatial analytical implications

Field-based continuous representation in a geographical information system (GIS) has long been important for describing complex spatially distributed phenomena, such as population, precipitation, air pollution, temperature elevation and land cover. Though theoretical knowledge and properties of continuous distributions can be employed, such surfaces are generally approximated or abstracted in practice due to a lack of complete information. That is, such surfaces are based on finite spatial samples, which is a practical necessity with regard to the infinite underlying attribute variability. These approximated surfaces are then used in various spatial analyses, yet impacts are not well understood. This article will examine theoretical properties and errors that result in practice when approximated continuous surfaces are relied on in spatial analysis.     

Application of the Maslov Seismogram Method in Three Dimensions

Asymptotic methods provide an efficient way to compute seismograms in heterogeneous media. However, zeroth-order ray theory, the simplest of the asymptotic methods, often fails because of the presence of caustics. Maslov theory is an extension of zeroth-order ray theory, which gives a uniformly valid expression of the wavefield everywhere, including the caustics. This result is given in terms of an integral of ray data over one or two ray parameters. It is shown in this paper how geometrical arrivals are constructed in the one and two-parameter Maslov integrals.In practice Maslov seismograms have been computed using only one ray parameter. However, in three-dimensional media two parameters are needed to uniquely define a ray. In this paper we present an efficient algorithm to compute two-parameter Maslov integrals. The Maslov integral is evaluated by computing the frequency-to-time Fourier transform prior to integration over the ray parameters. The wavefield is then discretized by smoothing with a boxcar function. The resulting expression, which only requires the results of ordinary kinematic and dynamic ray tracing, cen be computed efficiently and robustly. A numerical example is given that illustrates the use of this algorithm.