Seismic Ray Theory Research Articles

On Einstein-reversible mth root Finsler metrics

The theory of mth root Finsler metrics has been applied to Biology, Ecology, Gravitation, Seismic ray theory, etc. It is regarded as a direct generalization of Riemannian metric in a sense, namely, the second root metric is a Riemannian metric. On the other hand, the Riemannian curvature faithfully reveals the local geometric properties of a Riemann–Finsler metric. The reversibility of Riemannian and Ricci curvatures of Finsler metrics is an essential concept in Finsler geometry. Here, we study the Riemannian curvature of the class of third and fourth root [Formula: see text]-metrics. Then, we find the necessary and sufficient condition under which a cubic and fourth root [Formula: see text]-metric be Einstein-reversible.

Ray theory for elastic wave propagation in graded metamaterials

We present a ray theory for modeling elastic wave propagation in spatially graded mechanical metamaterials. Wave propagation in periodic metamaterials has been well studied, motivated by their beneficial wave steering and bandgap properties. By contrast, comparably little work has explored wave propagation in spatially graded metamaterials despite the increased design opportunities, largely due to the lack of efficient modeling techniques. We develop a ray theory to model waves in graded metamaterials based on high-frequency asymptotics and the assumption of local periodicity. This work builds upon the well-developed ray theories that are fundamental in a wide range of fields, from optics to seismology. Our derivations produce a practical framework for computing approximate wave fields in graded metamaterials. Ray trajectories are computed by independently solving a system of ordinary differential equations for each ray, requiring only knowledge of local dispersion relations throughout the metamaterial, which vary smoothly in space due to grading. Equations for the wave amplitude along rays are also derived in the two-dimensional setting. We show that the form of the ray tracing equations are nearly identical to those for smooth solids in seismic ray theory, with the primary difference being the dispersion relations. A numerical framework for computing ray solutions is demonstrated on a mass–spring network with analytical dispersion relations as well as a truss metamaterial that requires the numerical evaluation of dispersion relations. Through these examples, we demonstrate that ray theory provides an efficient means of studying the fascinating behavior of waves in graded metamaterials such as wave guiding along curved trajectories.

A strange symbiosis: Seismic ray theory and Saturday touch football — Donald V. Helmberger as a graduate advisor

A strange symbiosis: Seismic ray theory and Saturday touch football — Donald V. Helmberger as a graduate advisor

On conformally flat fourth root (α,β)-metrics

The theory of 4-th root metrics plays an important role in Biology, General Relativity and Seismic Ray Theory. Using the Shen's sy-separating trick, we prove that every conformally flat weakly Einstein 4-th root (α,β)-metric on a manifold M of dimension n≥3 is either a Riemannian metric or a locally Minkowski metric. Then, we show that every conformally flat 4-th root (α,β)-metric of almost vanishing Ξ-curvature on a manifold M of dimension n≥3 reduces to a Riemannian metric or a locally Minkowski metric.

Four families of projectively flat Finsler metrics with $$\mathbf{K}=1$$ K = 1 and their non-Riemannian curvature properties

For the first time, Bryant introduced a locally projectively flat non-Riemannian Finsler metric with flag curvature $$\mathbf{K}=1$$ on $${\mathbb {S}}^2$$ . In this paper, we construct four new non-Riemannian families of projectively flat Finsler metrics with flag curvature $$\mathbf{K}=1$$ . These metrics belong to the class of generalized 4-th root metrics. The class of generalized 4-th root metrics is an extension of the class of 4-th root metrics which has close relations with General Relativity and Seismic Ray Theory. In this class, we characterize Finsler metrics of isotropic flag curvature. We find the necessary and sufficient condition under which a metric in this class be weakly Einstein. In this class, we characterize Finsler metrics of isotropic S-curvature and almost vanishing $$\Xi $$ -curvature. Finally, we find the necessary and sufficient condition under which a Finsler metric in this class has vanishing projective Ricci curvature.

Correction to “Possible crippling of the core dynamo of Mars by Borealis impact”

[1] In the paper “Possible crippling of the core dynamo of Mars by Borealis impact” by Jafar Arkani‐Hamed (Journal of Geophysical Research, 115, E12021, doi:10.1029/ 2010JE003602, 2010) the impact heating of Mars by the Borealis impact, which likely created the northern lowland basin, was calculated. Using the crater scaling laws [Holsapple, 1993; Melosh, 1989] the basin diameter was related to the size of the impactor. Following the impact, a nearly uniform shock pressurewas generated inside an isobaric region in the mantle, outside which the shock pressure decreased exponentially with distance from the isobaric center following the average model of Pierazzo et al. [1997]. After crossing the core‐mantle boundary, the shock wave travelled in the core where the shock pressure again decayed with distance travelled from the isobaric center according to Pierazzo et al.’s average model but with the physical parameters of the core. The corresponding impact temperature increase is obtained using the Foundering model of Watters et al. [2009]. [2] The impact‐heated core stratified within a few tens of years and the possible pre‐existing core dynamo decayed with the characteristic time of 10–15 kyr [Arkani‐Hamed and Olson, 2010]. Shortly after the stratification, convection started in a thin layer in the outer core, the thickness of which increased in time as convection penetrated to deeper parts of the core until the entire core convected on a global scale. It took ∼60 Myr for the core to develop global convection and power a new core dynamo. [3] Here I describe a correction required in the calculation of the shock pressure distribution in the core. This correction is important because it increases the crippling time of the Martian core dynamo by a factor of larger than 4. The source of the required correction is the transmission coefficient of the shock wave across the core‐mantle boundary (CMB), the ratio of the shock pressure at the top of the core to that at the base of the mantle. The transmission coefficient was determined on the basis of seismic ray theory [Aki and Richards, 2002, section 5.2]. It resulted in a sudden drop of shock pressure as passing from the mantle to the core, in contrast to the sudden jump of the pressure according to recent Hydrocode models [Ivanov et al., 2010; Bierhaus et al., 2011].

Finsler geometry of seismic ray path in anisotropic media

The seismic ray theory in anisotropic inhomogeneous media is studied based on non-Euclidean geometry called Finsler geometry. For a two-dimensional ray path, the seismic wavefront in anisotropic media can be geometrically expressed by Finslerian parameters. By using elasticity constants of a real rock, the Finslerian parameters are estimated from a wavefront propagating in the rock. As a result, the anisotropic parameters indicate that the shape of wavefront is expressed not by a circle but by a convex curve called a superellipse. This deviation from the circle as an isotropic wavefront can be characterized by a roughness of wavefront. The roughness parameter of the real rock shows that the shape of the wavefront is expressed by a fractal curve. From an orthogonality of the wavefront and the ray, the seismic wavefront in anisotropic media relates to a fractal structure of the ray path.

Asymptotic Theory of Electroseismic Prospecting

In a porous medium such as the earth's subsurface, electromagnetic (EM) waves and mechanical waves are coupled through the phenomenon of electrokinetics, for which a complete set of partial differential equations was derived by S. Pride. In this paper, we derive from Pride's equations an asymptotic theory that enables forward modeling of the seismic response to an EM source in fully three-dimensional geometries on a scale that is relevant to exploration. For simplicity, we consider piecewise homogeneous media separated by interfaces which are curved surfaces in three dimensions. The following physical picture emerges: An EM source excites an EM wave which propagates into the earth, stirring up local mechanical movement. At an interface, EM energy is converted to seismic waves, which may be described by ray theory. Instantly, on the seismic time scale, every interface becomes a wavefront for both compressional and shear waves; that is, seismic P- and S-waves explode from both sides of each interface, at every point on it. The rays for these waves leave the interface in the orthogonal direction and propagate up and down into the homogeneous media on both sides of the surface. We derive formulas for the initial amplitudes of these waves. Conventional seismic ray theory then describes propagation of the P- and S-waves, including reflection, transmission, and mode conversion at any other interfaces that they may encounter. Thus, three-dimensional electroseismic modeling may be accomplished with conventional EM and conventional seismic modeling tools, using the present theory to provide the link between them.

The ‘gap’ between seismic ray theory and ‘full’ wavefield extrapolation

A derivation is given for the factorization of the generally anisotropic, elastic wave equation into two operators which are first order with respect to a preferred direction of propagation. One factor gives the ‘one-way’ equation for forward-going body waves. A 3 × 3 matrix notation is used which emphasizes the role of the Christoffel equation (or what might be called its ‘preferred coordinate projection’) familiar from ray theory. To recognize this it helps to obtain first an exact factorization of the wave equation for homogeneous anisotropic media. When inhomogeneities exist it is necessary to use a Fourier representation for differential operators with respect to the transverse coordinates, but still the Christoffel equation is the key. The so-called ‘square root operator’ required in the factorization becomes more correctly the ‘root of a matrix operator ratic equation’ intimately connected to the algebraic Christoffel equation. The factorization does not assume narrow-angle propagation or involve an a priori reference phase. It includes the P and two S waves and the forward coupling between them. It remains valid at slowness-surface singularities such as conical points (also known as acoustic axes—a most stringent test) and for wave fronts which fold. In the frequency domain this wide-angle one-way equation involves Fourier synthesis with respect to transverse position/slowness and its time-domain analogue involves 2-D slant stacking (Radon transformation), multiplication by a 3 × 3 ‘propagator’ matrix and an inverse 2-D slant stack. The one-way equation can be solved ‘analytically’ by using ‘path integrals’. These in turn may be reduced by stationary-phase arguments to standard ray theory, Maslov and Kirchhoff representations. In this sense, the ‘gap’ between ray theory and the numerical solution of the full wave equation is bridged or filled in, at least for forward propagation. Alternatively, the one-way wave equation can be solved by numerical ‘forward-stepping’ algorithms and here again a number of choices are available. Narrow-angle Taylor or rational approximations to the matrix propagator in the Fourier/slant-stack representation lead to the anisotropic analogues of the classic 15° and 45° partial differential equations. The integral wide-angle form is also reminiscent of the ‘split-step’and ‘elastic phase screen’ approaches of Tappert, Wu and others, suggesting other implementations.

Samuel W. Stewart

Samuel W. Stewart, the father of real-time microearthquake locations, died December 27, 1996, after a battle with Parkinson's disease. Sam was known to his friends and colleagues as a very gentle and sincere person with steely determination and persistence. Sam graduated from Princeton University in 1955 with a degree in geology and mathematics and completed a Master's degree in 1956 at the University of Utah in Geophysics and Geology. He joined the U.S. Geological Survey in Menlo Park, California, studying earthquakes at the Nevada test site for a year before moving to Denver, Colorado to work on crustal studies. In 1962, he began studying at St. Louis University and earned a Ph.D. in 1966 on the application of seismic ray theory to refraction seismology. His study of the crustal structure of the midcontinent (Stewart, 1968) was one of the first to provide a detailed model of the central US and...

Inversion of waveform from the data of six moderate strong earthquakes in Chinese mainland to seismic moment tensors and source mechanism

Based on Generalized Seismic Ray Theory (Helmberger, 1968), a new quickly linear inversion method from the data of seismic waveform to seismic moment tensor and source mechanism for domestic earthquake is studied in this paper. Six moderately strong earthquakes which occurred in Chinese mainland in the past few years are studied. The seismic source parameters of these earthquakes, seismic moment tensors, scalar seismic moments, fault plane solutions and source time functionsetc, are obtained.

Interpretation of surface wave spectra in laterally inhomogeneous media

SUMMARY Surface waves are used for the investigation of sharp lateral boundaries. The spectra are interpreted in terms of seismic ray theory combined with the period-dependent transmission coefficients. Physical attenuation produces large transmission and reflection losses in this procedure. Some conclusions concerning the territory of the GDR are drawn. While the solution of the inverse problem for surface waves is nowadays routinely handled for lateral homogeneity, there seems to be a lack of satisfying procedures for laterally inhomogeneous media. As long as the deviations from homogeneity are small, surface wave tomography is successfully carried out using modern methods (Nolet 1987). The introduction of sharp boundaries into ray-tracing schemes is possible for body waves as well as surface waves. For the latter, however, this procedure is not trivial, because the reflection and transmission coefficients do not occur as a solution of an algebraic system of equations as for body waves (Malischewsky 1987). The combination of ray theory with the concept of sharp boundaries, with the aim to solve the tomographic problem in 3-D media with discontinuities, is connected with additional difficulties which are the main topic of Malischewsky (1989); in the following a brief outline of this paper is given. Whenever tomography is applied, we can work either in the reflection or in the transmission regime. In the case of surface waves the reflection regime is preferable when the main aim is the detection of unknown discontinuities, because interpretable special wave groups usually occur in the seismogram. In Fig. 1, we show seismograms of a near seismic extent recorded at the stations BerggieShiibel (BRG) and Pritzwalk (PRW) in the GDR. In the PRW record we recognize a Rayleigh wave group between 4 and ,a

Seismic ray theory for lithospheric structures with slight lateral variations

b In this paper, the method of small perturbations is applied to the ray and energy transport equations in an investigation of the effect of weak inhomogeneities on the propagation of seismic rays through a layer of fixed thickness. Just as Aki et al. have used travel-time residuals to infer first-order velocity perturbations in their block-modelling procedure, it is proposed that surface slowness and amplitude data may be used to give additional information about the structure of the velocity perturbations beneath the observer.

Seismic Ray Theory

Seismic ray theory is developed ab initio (apart from one or two standard elementary results) with special emphasis on the variables ζ and α, where ζ = d log ζ/d log r and α = 2/(i- ζ). Travel-time-distance relations are examined for a variety of types of velocity distribution, including the case α constant and cases where α and ζ change discontinuously, or rapidly but continuously, with increase of depth. The analysis is designed to provide an improved basis for working out in quantitative detail the effect on travel-times of the various types of velocity variation likely to be relevant to the Earth. In particular, it is hoped that the analysis will lead to the most effective use of ray theory in the current difficult problems of the structure of the Earth's outer mantle. Advantage has been taken of the opportunity to present a number of previous results involving ζ and α in revised form, as part of a wider logical development. Previous work on deriving seismic velocity distributions from travel-time data is generalized. The aim of the paper has been to set down a terse account of the basic theory, and no numerical applications have been included. References are given, however, to papers containing applications.