Elementary Seismograms of Seismic Body Waves in Dissipative Media

Approximate expressions for elementary seismograms of seismic body waves propagating in media with small causal absorption are derived. Special attention is devoted to modulated signals with a smooth envelope, for which especially simple formulae were obtained. The derived expressions give a good picture of all important effects of causal absorption, viz., the frequency dependent exponential decrease of amplitude, the velocity dispersion related to absorption, and the decrease of the prevailing frequency.     

Effects of causal absorption on seismic body waves

Summary The effects of the causal absorption on seismic wave fields are discussed. Several computer programs for the evaluation of seismic wave fields have been modified to include the causal absorption. The programs are based mostly on matrix and ray methods. The main attention is paid to ray methods. A simple plane wave algorithm is described, which can be used along the ray to compute elementary seismograms in dissipative media. For a certain class of signals, commonly used in seismology, its application does not require any convolution or frequency-domain computations, and is only slightly slower than the evaluation of elementary seismograms in non-dissipative media. The derived algorithm even describes the velocity dispersion related to absorption and the decrease of the prevailing frequency of the signal well. Numerical examples are presented, both exact and approximate.     

Tangent Vectors of Isochron Rays and Velocity Rays Expressed in Global Cartesian Coordinates

This paper provides Cartesian expressions for tangent vectors of isochron rays and velocity rays previously derived in so-called isochron-orthonormal coordinates. The Cartesian expressions are simpler and easier to implement than the expressions in isochron-orthonormal coordinates. It is shown that the expressions in both coordinate systems are equivalent.     

OH radical concentration in the stratosphere

The consumption of both methane and carbon monoxide in the lower stratosphere is due predominantly to reaction with OH radicals. The possibility of deriving OH concentration from measurements of the decrease of CH₄ and CO mixing ratios above the tropopause is explored. The observations and the basic chemistry are briefly summarized. Simple one-dimensional diffusion models are employed to derive expressions for the decrease of CH₄ and CO mixing ratios with altitude above the tropopause, and the influence of important parameters is discussed. Vertical air velocities resulting from large-scale organized mean motion and from synoptic variations are shown to distort the concentration altitude profiles of methane and carbon monoxide, respectively. Suitable averaging of observational data is required to eliminate the effects due to vertical motion. Then a reliable value for the effective OH number density should be obtainable. At present an estimate of 4×10⁶ molecules/cm³ is derived.     

High angle prestack depth migration with absorption compensation

The absorption effect of actual subsurface media can weaken wavefield energy, decrease the dominating frequency, and further lead to reduced resolution. In migration, some actions can be taken to compensate for the absorption effect and enhance the resolution. In this paper, we derive a one-way wave equation with an attenuation term based on the timespace domain high angle one-way wave equation. A complicated geological model is then designed and synthetic shot gathers are simulated with acoustic wave equations without and with an absorbing term. The derived one-way wave equation is applied to the migration of the synthetic gathers without and with attenuation compensation for the simulated shot gathers. Three migration profiles are obtained. The first and second profiles are from the shot gathers without and with attenuation using the migration method without compensation, the third one is from the shot gathers with attenuation using the migration method with compensation. The first and third profiles are almost the same, and the second profile is different from the others below the absorptive layers. The amplitudes of the interfaces below the absorptive layers are weak because of their absorption. This method is also applied to field data. It is concluded from the migration examples that the migration method discussed in this paper is feasible.     

Apparent resistivity over a three layer earth with anisotropic interstratum

Summary The present work aims at investigating the effects of anisotropy on the apparent resistivity measurements over the surface of a three layer Earth. An appropriate model with the anisotropic layer embedded between two isotropic media is chosen for study. The point current source is assumed to be present at the surface of such a model. After writing the values of the potentials in each layer, expressions for apparent resistivities for Wenner and Schlumberger configurations are derived. Also, the expressions for potential for the limiting cases in which the bottom layer is infinitely resistive or infinitely conducting are derived. It is shown that these expressions can be recast into simpler forms corresponding to isotropic layer (s) on an isotropic half-space.     

On characteristic frequencies of a five-component model of magnetospheric plasma

Summary The author gives more general relations derived for characteristic frequencies of a more complex plasma model. One may assume that expressions derived herein, represent a more appropriate approximation of reality, especially in regions of ionospheric and magnetospheric plasma with a higher concentration of heavy ions.     

Dynamic characteristics of non-classically damped structures

The equations of motion of a structure in undamped modal coordinates may have non-zero off-diagonal terms in the damping matrix. Although these terms are commonly neglected, studies have shown that they may have a significant influence on the response to dynamic loads. In this paper, two independent criteria are developed to determine when these damping terms will affect the structure's modal properties and response. It is found that even small off-diagonal damping values can be significant if the structure has closely spaced natural frequencies. To quantify and understand the influence of these damping terms, closed-form analytical expressions are derived for the modal properties and harmonic and stochastic response of structures with closely spaced natural frequencies. One conclusion is that off-diagonal damping terms will decrease a modal damping ratio for each pair of closely spaced modes. This is significant, since a response analysis performed by neglecting these off-diagonal terms will underestimate the true response.     

Seismic wave velocities of rare gas solids through elastic properties in Earth’s lower mantle

The expressions for second (SOE) and third order elastic (TOE) constants for rare gas solids are derived for comparative study of elastic behavior within the framework of many body potentials including the effect of pressure. The derived expressions are used to obtain the relations for pressure derivatives of bulk and shear moduli of RGS solids. The values of SOE, TOE constants and pressure derivative of bulk and shear modulus for Ne up to 100 GPa, Ar up to 75 GPa, for Kr up to 136 GPa and Xe up to 53.4 GPa...     

Frequency domain expressions for surface and borehole gravity potential due to two- and three-dimensional mass models

Frequency domain expressions for surface and borehole gravity potential due to selected two- and three-dimensional mass models with linear variation of density are derived using conjugate complex variables formulation. The mass models include elliptical bodies, vertical and horizontal polygonal prisms, polyhedral bodies and bodies with rotational symmetry. Such analytic expressions are essential for development of interpretative theory for surface and borehole gravity data in the Fourier domain.     

On cartographical representation of geophysical fields

Summary A method is developed for rapid gravity interpretation of a sedimentary basin using a digital computer. Assuming a linear variable density contrast (both vertically and horizontally), expressions have been derived for the gravitational attraction of a rectangular strip. An interpretation of the structure of Godavari Valley (India) has been made using these expressions.     

Probabilistic estimation of irrigation requirement under climate uncertainty using dichotomous and marked renewal processes

This study addresses estimation of net irrigation requirement over a growing season under climate uncertainty. An ecohydrological model, building upon the stochastic differential equation of soil moisture dynamics, is employed as a basis to derive new analytical expressions for estimating seasonal net irrigation requirement probabilistically. Two distinct irrigation technologies are considered. For micro irrigation technology, probability density function of seasonal net irrigation depth (SNID) is derived assessing transient behavior of a stochastic process which is time integral of dichotomous Markov process. Probability mass function of SNID which is a discrete random variable for traditional irrigation technology is also presented using a marked renewal process with quasi-exponentially-distributed time intervals. Comparing the results obtained from the presented models with those resulted from a Monte Carlo approach verified the significance of the probabilistic expressions derived and assumptions made.     

Improved expressions for the ellipsoidal harmonic series representing the Earth gravitational potential and its first and second derivatives on and outside the reference ellipsoid of revolution

The Legendre functions of the second kind, renormalized by Jekeli, are considered in the external space on a set of ellipsoids of revolution which are confocal with respect to the normal ellipsoid. Among these ellipsoids a reference one is chosen which bounds the Earth. New expressions for the first and second order derivatives of the Legendre functions are derived. They depend on two very quickly convergent Gauss hypergeometric series which are obtained by transforming the slowly convergent initial hypergeometric series. The derived expressions are applied for constructing the ellipsoidal harmonic series for the Earth disturbing gravitational potential and its derivatives of the first and second orders. Since outside the chosen reference ellipsoid there are no Earth masses (as compared to the normal ellipsoid) then it is more appropriate for constructing the boundary-value equation and solving it on the basis of surface gravity data reduced to this ellipsoid.     

DESIGN OF SLIT DAMS FOR CONTROLLING STONY DEBRIS FLOWS

1 INTRODUCTION Stony debris flows are natural, highly concentrated water-sediment mixture, which forms wherever the simultaneous availability of water, debris material and an adequate slope, steeper than o10 are satisfied (Gregoretti, 2000). In mountainous regions of Taiwan, due to vast development and utilization of hills, stony debris flows are important from the point of disaster prevention, since they occur frequently and often bring about heavy loss of lives and properties. Therefo…     

Time‐domain response of linear hysteretic systems to deterministic and random excitations

The causal and physically realizable Biot hysteretic model proves to be the simplest linear model able to describe the nearly rate‐independent behaviour of engineering materials. In this paper, the performance of the Biot hysteretic model is analysed and compared with those of the ideal and causal hysteretic models. The Laguerre polynomial approximation (LPA) method, recently proposed for the time‐domain analysis of linear viscoelastic systems, is then summarized and applied to the prediction of the dynamic response of linear hysteretic systems to deterministic and random excitations. The parameters of the LPA model generally need to be computed through numerical integrals; however, when this model is used to approximate the Biot hysteretic model, closed‐form expressions can be found. Effective step‐by‐step procedures are also provided in the paper, which prove to be accurate also for high levels of damping. Finally, the method is applied to the dynamic analysis of a highway embankment excited by deterministic and random ground motions. The results show that in some cases the inaccuracy associated with the use of an equivalent viscous damping model is too large. Copyright © 2005 John Wiley & Sons, Ltd.     

Moment analysis for compounds undergoing sequential chain reactions with first-order decay

Temporal and spatial moment analysis of one-dimensional equations governing fate and transport of parent compounds along with their transformation products is useful for parameter estimation of model parameters, and for understanding the average attributes of contaminant behavior. The objective of this paper is to present analytical expressions for the lower order moments of members in a sequential chain reaction, where members undergo first-order decay to produce the next member in the chain. Specifically, moments up to second order are discussed for the first two members. For the case of purely advective transport (Peclet number tending to infinity), temporal moment expressions are provided for more members of the chain. The sensitivity of temporal moments is examined with respect to Peclet number and transformation rates. Spatial moments are derived by two methods—one using Fourier transforms and another using moment generating differential equations. The behavior of lower order moments for the first few members of a chain can be elucidated from their mathematical expressions. However, expressions for higher order moments tend to be very complicated especially for members further down the chain.     

Elastic-anelastic Regional Structures for the Iberian Peninsula Obtained from a Rayleigh Wave Tomography and a Causal Uncoupled Inversion

The elastic and anelastic structure of the lithosphere and asthenosphere of the Iberian Peninsula is derived by means of tomographic techniques applied to local phase and group velocities and local attenuation coefficients of Rayleigh wave fundamental mode. The database consists of surface wavetrains recorded at the broadband stations located in the Iberian Peninsula on the occasion of the ILIHA project. Path-averaged phase and group velocities and attenuation coefficients were previously obtained by standard filtering techniques of surface wavetrains and, subsequently, local dispersion curves were computed according to the Yanovskaya-Ditmar formulation. First, a principal component analysis (PCA) and the average linkage (AL) clustering algorithm are applied to these local values in order to classify the Iberian Peninsula in several rather homogeneous domains from the viewpoint of the similarity of the corresponding local dispersion curves, without previous seismotectonic constraints. Second, averaged phase and group velocities and attenuation coefficients representing each homogeneous region are used to derive the respective elastic and anelastic models of the lithosphere and asthenosphere. This purpose is achieved by using the uncoupled causal inversion of phase and group velocities and attenuation coefficients. The main features of the homogeneous regions are discussed by taking as reference the Hercynic, Alpine and Neogene domains of the Iberian Peninsula, and two questions affecting the reliability of the elastic-anelastic models are revised. First, the coherence of the shear-velocity and Q_β⁻¹ models obtained by causal uncoupled inversion for each region is analysed. Second, the influence of the causal phase and group velocities on the shear-velocity models is evaluated by comparing elastic and anelastic models derived from causal uncoupled inversion with those deduced from non-causal inversion.     

Asymptotic Expansions for the Calculation of the Transient Electromagnetic Fields Induced by a Vertical Magnetic Dipole Source Above a Conductive Halfspace

We have derived asymptotic expansions that can be used to estimate the impulse response of a conductive halfspace when excited by an airborne electromagnetic system. Two expressions are required to calculate the response, one is applicable in the late-time regime and a second is used for the early-time regime. Fortunately, there is a small overlap in the late- and early-time regimes. Asymptotic expansions are commonly used for interpreting ground EM data, but our expressions are the first ones applicable to the airborne situation. The expansions are simple to program, can be calculated extremely rapidly and will be useful for understanding and interpreting airborne electromagnetic data. As an example of the use of the expressions, we have calculated the response of seawater and compared this with the airborne electromagnetic response measured off the coast of Perth, Western Australia.     

A note on two-point paraxial travel times

Recently, several expressions for the two-point paraxial travel time in laterally varying, isotropic or anisotropic layered media were derived. The two-point paraxial travel time gives the travel time from point S′ to point R′, both these points being situated close to a known reference ray Ω, along which the ray-propagator matrix was calculated by dynamic ray tracing. The reference ray and the position of points S′ and R′ are specified in Cartesian coordinates. Two such expressions for the two-point paraxial travel time play an important role. The first is based on the 4 × 4 ray propagator matrix, computed by dynamic ray tracing along the reference ray in ray-centred coordinates. The second requires the knowledge of the 6 × 6 ray propagator matrix computed by dynamic ray tracing along the reference ray in Cartesian coordinates. Both expressions were derived fully independently, using different methods, and are expressed in quite different forms. In this paper we prove that the two expressions are fully equivalent and can be transformed into each other.     

Precision calculation of crustal deformation induced by radial steady laminellar flow of underground water from single well in multi-layered structural aquifers

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