Seismic waves in a stratified half space

Summary. For a buried source in a stratified elastic half space, the surface displacements are calculated by numerical integration of the Fourier–Bessel transform of the response. In the transform space this response is conveniently represented in terms of the reflection and transmission properties of the half space. For a layered medium this procedure avoids all problems associated with growing exponential terms in the evanescent regime. A slightly attenuative medium is assumed, so that the surface wave poles are shifted off the real slowness axis and thus a contour of integration along this axis may be employed. A general point source is represented by an arbitrary moment tensor.
The procedure is illustrated by calculations of three component seismograms including all P , SV and SH contributions for body and surface waves at moderate ranges. For local earthquakes we illustrate the striking effect of focal depth and also show the effect of sedimentary cover on strong ground motion.     

Recurrence relations for computing complete P and SV seismograms

Summary. A set of recurrence relations which are computationally more efficient than those of the reflection matrix method of Kennett & Kerry is presented for P - and SV -wave generation in a ( n + 1) layered medium. The recurrence relations contain no growing terms and thus provide a stable algorithm for computing complete P and SV synthetic seismograms. Our algorithm requires a fewer algebraic operations for computing the reflectivity and transmissivity coefficients, ranging from 15 per cent less for a source in the half-space to 30 per cent less for a source in the top layer, than the reflection matrix method.     

Seismic waves in stratified anisotropic media

Summary. The response of a structure composed of anisotropic strata can be built up from the reflection and transmission properties of individual interfaces using a slightly modified version of the recursion scheme of Kennett. This scheme is conveniently described in terms of scatterer operators and scatterer products. The effects of a free surface and the introduction of a simple point source at any depth can be accommodated in a manner directly analogous to the treatment for isotropic structures. As in the isotropic case the results so obtained are stable to arbitrary wavenumbers. For isotropic media, synthetic seismograms can be constructed by computing the structure response as a function of frequency and radial wavenumber, then performing the appropriate Fourier and Hankel transforms to obtain the wavefield in time-distance space. Such a scheme is convenient for any system with cylindrical symmétry (including transverse isotropy). Azimuthally anisotropic structures, however, do not display cylindrical symmétry; for these the transverse component of the wavenumber vector will, in general, be non-zero, with the result that phase, group, and energy velocities may all diverge. The problem is then much more conveniently addressed in Cartesian coordinates, with the frequency-wavenumber to time-distance transformation accomplished by 3-D Fourier transform.     

Seismic moment tensors and kinematic source parameters

Summary. Parameters pertaining to the kinematics of a finite source are usually estimated by fitting specific fault models to the data. On the other hand, these parameters, including source location, are also contained in the moment tensors of higher degree. In this paper, the seismic response is represented in terms of 20 source parameters which are related to components of the moment tensors; they are also related to the parameters of fault models, as will be demonstrated for a number of 'classical' models. A linearized inversion for the moment tensor shows that with real data, or with realistic synthetic data, the results are not necessarily physically meaningful, unless constraints are imposed. The constraints are precisely those appearing as a priori assumptions in the conventional methods of source analysis; it is thus possible to investigate the impact of these assumptions. We will discuss in particular the assumption of a general deviatoric point source (not necessarily a double couple) versus that of a plane fault in finite sources. Although at this stage experience with practical performance of the new method is limited, it is suggested that in the appropriate circumstances constrained inversion for the seismic moment tensors offers a viable alternative to estimate kinematic source parameters.     

Guided wave propagation in laterally varying media - I. Theoretical development

Summary. The propagation of surface waves in a laterally varying medium can be described by representing the wavetrain as a superposition of modal contributions for a reference structure. As the guided waves propagate through a heterogeneous zone the modal coefficients needed to describe the wavetrain vary with position, leading to interconversions between modes and reflection into backward travelling modes. The evolution of the modal terms may be described by a set of first-order differential equations which allow for coupling to both forward and backward travelling waves; the coefficients in these equations depend on the differences between the actual structure and the reference structure. This system is established using the orthogonality properties of the modal eigenfunctions and is valid for SH -waves, P - SV -waves and full anisotropy.
The reflected and transmitted wavefields for a region of heterogeneity can be related to the incident wave by introducing reflection and transmission matrices which connect the modal coefficients in these fields to those in the incident wavetrain. By considering a sequence of models with increasing width of heterogeneity we are able to derive a set of Ricatti equations for the reflection and transmission matrices which may be solved by initial value techniques. This avoids an awkward two-point boundary value problem for a large number of coupled equations. The method is demonstrated for 1 Hz Lg - and Sn -waves in a multilayered model for which there are 19 coupled modes.
The method is applicable to three-dimensional heterogeneity, and we are able to show that the interconversion between Love and Rayleigh waves, in the presence of gradients in seismic properties transverse to the propagation path, leads to a net rate of increase of the transverse components of the seismogram at the expense of the other components.     

Inversion of a normally incident reflection seismogram by the Gopinath–Sondhi integral equation

Summary. The one-dimensional acoustic wave equation has been transformed to two coupled first-order equations whose inverse solution is obtained through application of the Gopinath and Sondhi integral equation. A scattering solution of the Schrödinger wave equation for an explosive source leads us to express the kernel of the Gopinath–Sondhi integral equation in terms of a seismic reflection response. A numerical solution of the integral equation obtained by a trapezoidal rule yields a continuous impedance profile whose derivative has step-like discontinuities. The method is illustrated with computer model studies.     

Effects of topography and crustal heterogeneities on the source estimation of LP event at Kilauea volcano

The main goal of this study is to improve the modelling of the source mechanism associated with the generation of long period (LP) signals in volcanic areas. Our intent is to evaluate the effects that detailed structural features of the volcanic models play in the generation of LP signal and the consequent retrieval of LP source characteristics. In particular, effects associated with the presence of topography and crustal heterogeneities are here studied in detail. We focus our study on a LP event observed at Kilauea volcano, Hawaii, in 2001 May. A detailed analysis of this event and its source modelling is accompanied by a set of synthetic tests, which aim to evaluate the effects of topography and the presence of low velocity shallow layers in the source region. The forward problem of Green's function generation is solved numerically following a pseudo-spectral approach, assuming different 3-D models. The inversion is done in the frequency domain and the resulting source mechanism is represented by the sum of two time-dependent terms: a full moment tensor and a single force. Synthetic tests show how characteristic velocity structures, associated with shallow sources, may be partially responsible for the generation of the observed long-lasting ringing waveforms. When applying the inversion technique to Kilauea LP data set, inversions carried out for different crustal models led to very similar source geometries, indicating a subhorizontal cracks. On the other hand, the source time function and its duration are significantly different for different models. These results support the indication of a strong influence of crustal layering on the generation of the LP signal, while the assumption of homogeneous velocity model may bring to misleading results.     

Frequency response modelling of seismic waves using finite difference time domain with phase sensitive detection (TD–PSD)

This paper describes an efficient approach for computing the frequency response of seismic waves propagating in 2- and 3-D earth models within which the magnitude and phase are required at many locations. The approach consists of running an explicit finite difference time domain (TD) code with a time harmonic source out to steady-state. The magnitudes and phases at locations in the model are computed using phase sensitive detection (PSD). PSD does not require storage of time-series (unlike a fast Fourier transform), reducing its memory requirements. Additionally, the response from multiple sources can be obtained from a single finite difference run by encoding each source with a different frequency. For 2-D models with many sources, this time domain phase sensitive detection (TD–PSD) approach has a higher arithmetic complexity than direct solution of the finite difference frequency domain (FD) equations using nested dissection re-ordering (FD–ND). The storage requirements for 2-D finite difference TD–PSD are lower than FD–ND. For 3-D finite difference models, TD–PSD has significantly lower arithmetic complexity and storage requirements than FD–ND, and therefore, may prove useful for computing the frequency response of large 3-D earth models.     

The effect of a second-order velocity discontinuity on elastic waves near their turning point

Summary. The rather abrupt changes in velocity gradient which have sometimes been proposed, notably in the upper mantle and near the base of the mantle, have an effect equivalent to that of one or more second-order discontinuities, where partial reflection occurs due to a change in curvature of the wavefront across these discontinuities. The effect is ignored in the classical WKBJ approximation to the wave functions, but it can be explicitly demonstrated by applying the extended WKBJ method (Langer's approximation) to a piecewise smooth layered model. For the purpose of this study it is convenient to represent the response of such a modelby a generalized reflection coefficient. For a model of one or a system of several second-order discontinuities (approximating a change in velocity gradient over a finite depth interval), the reflection coefficient can be perhaps surprisingly large for long-period waves near their turning point. It is shown that this effect can significantly alter the amplitude decay of SH waves diffracted around the core, in models where a change in velocity gradient near the core—mantle boundary constitutes a low-velocity zone at the base of the mantle; such models have recently been proposed. With the same velocity gradients, the effect on P diffraction is less important. The results for SH diffraction in these models support the conclusion that a small amplitude decay must be explained by a velocity decrease with depth, i.e. a low-velocity zone at the base of the mantle.     

The Rise and Fall of the Davisian Cycle of Erosion: Prelude,Fugue, Coda,and Sequel

The Cycle of Erosion formulated by the American geographer William Morris Davis in the 1880s remained the dominant paradigm in geomorphology well into the 20th century, before it waned in response to improved understanding of Earth's crustal and surface behavior. The Davisian model sought to explain landforms in terms of structure, process, and stage. Following initial rapid tectonic uplift, landforms were presumed to evolve on a quiescent crust through stages of youth, maturity, and old age, to culminate in a peneplain. A new cycle would be initiated by landform rejuvenation in response to a changing base level of erosion. This model was a reflection of its time, of the cycle mania of the 19th century, which in turn was founded on Hutton's limitless "succession of worlds" and dissatisfaction with earlier notions of landscape origins constrained by limited Earth time. Davis's model was derived from ideas regarding orogenic cycles favored by Dana and Le Conte, and of prolonged subaerial denudation toward base-level observed by Powell and Dutton. The model's supremacy was challenged from time to time, notably by the Pencks (father and son) and by alternative cyclic denudation models that invoked pediplanation and etchplanation rather than peneplanation. The relevance of the Davisian model declined after 1940 in response to a growing awareness of Earth's crustal mobility, changing climates and geomorphic processes, and refined dating of geologic time. The subsequent quantitative revolution in geomorphology, with its emphasis on measurement of form and process aided by rapidly improving technologies, and based in part on lingering antecedents, sounded the death knell for the Davisian model but also triggered something of a theoretical hiatus. In recent years, resurrection of the concept of isostasy, defined by Dutton but ignored by Davis, has led to the formulation of a more realistic but more complex model, briefly introduced here, in which landforms may be viewed as responses to more-or-less continuous interaction between tectonic activity, subaerial denudation, and isostatic adjustment.     

Amplification of ground motion and waveform complexity in fault zones: examples from the San Andreas and Calaveras Faults

Summary. P -wave seismograms at ranges less than 10 km are synthesized by asymptotic ray theory and by summation of Gaussian beams for point sources located in a low-velocity wedge surrounding a fault. The computations are performed using models of the wedge inferred from the analysis of reflection and refraction experiments across the San Andreas and Hayward-Calaveras faults. Calculations in these models show that the 10–20Hz vertical displacements of earthquakes located at 3–10km depth are amplified by up to an order of magnitude in a 1–2km wide region centred on the fault trace compared to displacements predicted by laterally homogeneous models of the crust. This amplification is not cancelled by high attentuation in the fault zone and compensates for the reduction in amplitudes directly above the source predicted from the radiation pattern of a strike-slip earthquake. Depending on the source depth of the earthquake and the structure and velocity contrast of the wedge, multiple triplications in the travel-time curve of direct P - and S -waves will occur at stations in the fault zone. A wedge model successfully predicts the triplications observed in the P waveforms of aftershocks of the Coyote Lake earthquake recorded in the fault zone, showing that body waves from microearthquakes can be used to determine the three-dimensional velocity structure of the fault zone. The amplification, waveform complexity, and distortion of ray paths introduced by the low- velocity wedge suggest that its effects should be included in the interpretation of strong ground motions and travel times observed in the fault zone. For realistic models of the wedge, asymptotically approximate methods of calculating the body waveforms are strictly valid for frequencies greater than 20Hz. Numerical methods may be necessary to calculate accurately the wavefield at lower frequencies.     

The PWaves From the Amchitka Island Explosions

Summary. There is evidence that the equivalent seismic sources of the Amchitka Island explosions — Longshot, Milrow and Cannikin — depart significantly from the simple model of a point compressional-source in a layered elastic-medium. Consequently modelling the observed seismograms using standard source-models may not be the most efficient method of determining source properties. Here an alternative to modelling is used to obtain information on the seismic sources due to the explosions. Broad-band (BB) estimates of the P signals are obtained from the short-period (SP) seismograms, corrected for attenuation, and interpreted in terms of P, pP and radiation from secondary sources. the main conclusions are:
(i) BB estimates of the radiated displacement from the explosions can be obtained with only a small reduction in the signal-to-noise ratio seen on SP seismograms;
(ii) observations of differences in pulse amplitudes and spectra are not necessarily due to differences in anelastic attenuation;
(iii) P and pP at a given station may differ in shape so that notches in the signal spectrum may not be related to source depth;
(iv) there is evidence of arrivals that others have identified as due to slap-down but which could be interpreted as an overshoot to pP;
(v) direct interpretation of the estimated ground displacement is a better procedure for determining the seismic source properties of explosions than modelling SP seismograms using idealised models as a starting point.     

Generalized Born inversion of seismic reflection data

Summary . Born inverse methods give accurate and stable results when the source wavelet is impulsive. However, in many practical applications (reflection seismology) an impulsive source cannot be realized and the inversion needs to be generalized to include an arbitrary source function. In this paper, we present a Born solution to the seismic inverse problem which can accommodate an arbitrary source function and give accurate and stable results. It is shown that the form of the generalized inversion algorithm reduces to a Wiener shaping ***filter, which is solved efficiently using a Levinson recursion algorithm. Numerical examples of synthetic and real field data illustrate the validity of our method.     

The fine structure of the normal incidence synthetic seismogram

Summary . The spectral function of a perfectly elastic, horizontally stratified medium has been demonstrated previously to provide an attractive formulation to describe the properties of the one-dimensional synthetic seismogram (Robinson & Treitel). Here we examine the mathematical framework of the Model in still greater detail. Knowledge of this fine structure of the synthetic seismogram leads to the solution of two particular seismic inverse problems. First, we consider a layered medium with an arbitrary surface reflection coefficient c _o, where | c _o|<1, and which contains an impulsive source immediately above the surface. Given the corresponding synthetic seismogram, we develop an inverse, or backward recursion formalism which recovers the entire series of original reflection coefficients. Second, we consider a similar problem for an impulsive source located just below the surface. Both inversion procedures constitute a continuation of the work of Goupillaud and of Sherwood & Trorey, and represent a generalization of the classical technique originally proposed by Kunetz which, however, only holds for the marine case, c_o =±1. The present approach is not so constrained and thereby becomes applicable to land seismograms as well.
If products of third or higher order in the reflection coefficients can be neglected, significant simplifications arise in the theory. In that event the usual representation of the synthetic seismogram as a ratio of two polynomials in the complex variable z becomes particularly revealing. The numerator polynomial is then approximately equal to the z transform of the reflection coefficient series, while the denominator polynomial is approximately equal to the z transform of the autocorrelation of these reflection coefficients. The resulting simplified theory affords important computational savings in the appropriate backward recursion algorithms.     

平面冰层覆盖下水中声传播损失特性分析

理论计算了冰层的声波反射系数,分析了平面冰层覆盖下水中声传播损失特性。将冰层视作平面覆盖层,建立了"空气-冰层-水"结构的冰层反射物理模型,结合冰中的物理声学参数,得到冰层反射系数;利用Bellhop海洋声学传播模型得到冰层覆盖下声场的传播损失,对比分析了传播损失与声波频率、声源深度的关系;结合冰层反射系数计算结果,解释了冰下信道具有频率选择性的原因。结果表明,冰层覆盖下声场传播具有明显的频率选择性,这种频率选择性是由声波频率和冰层厚度共同决定的。     

Weak-contrast reflection/transmission coefficients in weakly anisotropic elastic media: P-wave incidence

We present approximate displacement and energy PP and PS reflection/transmission coefficients for weak-contrast interfaces in general weakly anisotropic elastic media. The coefficients were obtained by applying first-order perturbation theory and then expressed in a compact and relatively simple form. The formulae can be used for arbitrary orientations of the incidence plane and interface, without the need to transform the elasticity parameters to a local Cartesian coordinate system. The accuracy of the approximate formulae is illustrated for the PS reflection coefficient for two synthetic models. For these models, we also study the possibility of using the approximate PP reflection coefficient in the inverse problem.     

Long-period seismic source inversions using global tomographic models

We investigate the effect of laterally varying earth structure on centroid moment tensor inversions using fundamental mode mantle waves. Theoretical seismograms are calculated using a full formulation of surface wave ray theory. Calculations are made using a variety of global tomographic earth models. Results are compared with those obtained using the so-called great-circle approximation, which assumes that phase corrections are given in terms of mean phase slowness along the great circle, and which neglects amplitude effects of heterogeneity. Synthetic tests suggest that even source parameters which fit the data very well may have large errors due to incomplete knowledge of lateral heterogeneity. The method is applied to 31 shallow, large earthquakes. For a given earthquake, the focal mechanisms calculated using different earth models and different forward modelling techniques can significantly vary. We provide a range of selected solutions based on the fit to the data, rather than one single solution. Difficulties in constraining the dip-slip components of the seismic moment tensor often produce overestimates of seismic moment, leading to near vertical dip-slip mechanisms. This happens more commonly for earth models not fitting the data well, confirming that more accurate modelling of lateral heterogeneity can help to constrain the dip-slip components of the seismic moment tensor.     

Macrogeomorphic evolution of the post-Triassic Appalachian mountains determined by deconvolution of the offshore basin sedimentary record

A perplexing macrogeomorphic problem is the persistence of topography in mountain ranges that were initially formed by orogenic events hundreds of millions of years old. In this paper, we deconvolve the post-Triassic macrogeomorphic history of a portion of one of these ranges, the central and northern Appalachians, using a well-documented offshore isopach sedimentary record of the US Atlantic margin. Topography is an important signature of tectonic, eustatic and/or geomorphic processes that produces changes in the erodible thickness of the crust (ETC). We define ETC as the total thickness of crust that would have to be consumed by erosion to reduce the mean elevation of a landscape to sea level. We use the term ‘source flux’, designated by ν˙, to describe the rate of change in ETC attributed to deep-seated geological processes such as crustal thickening, crustal extension, magmatic intrusions or dynamic flow in the mantle. In a mountain belt, the rate of change of mean elevation with respect to a base level, designated by ? ′, can be represented as ? ′ = c(ν˙ ? k _d z ′ ?; ? _c ) ?& hairsp;l˙ , where k _d is a proportionality constant relating the mean mechanical erosion rate to mean elevation, ? _c is the mean chemcial erosion rate, c  is a compensation ratio (held constant for Airy isostasy at 0.21) and l˙  is the rate of eustatic sea-level change. This equation describes the sum of constructive source terms, two destructive erosion terms and a eustatic sea-level term. We use this simple linear equation to develop a landscape evolution model based on the concept of a unit response function. The unit response function is analogous to a unit hydrograph which describes the relationship between input (rainfall) and output (discharge) in a hydrological system. In our case, we solve for the general relationship between the source flux into the mountain belt and the erosional flux out of the belt. Offshore sediment volumes are used to determine the erosional flux. Drainage basin area is treated as either a constant (pinned drainage divide) or as increasing through time (migrating drainage divide). We use a third-order polynomial fit to a global sea-level curve to account for long-term eustatically driven changes in ETC and in drainage basin area. Chemical erosion is treated as a constant fixed at 5 m Myr^?1. We consider two end-member models. The first is a ‘tectonic’ model in which the source flux is allowed to vary while k _d is assumed to be constant over geological time and equal to its mean Pleistocene value of about 0.07 Myr^?1. The second is an ‘erodibility’ model in which k _d is allowed to vary, reflecting changes in climatic, climatic variables and rock-type erodibility, while the source flux is held constant at zero. The ‘tectonic’ model reveals four important increases in the source flux, ranging from 200 to 2000 m Myr^?1 that occur over short (<10 Myr) time spans, followed by a protracted period (>25 Myr) where ν˙ drops below zero to values of ?1000 to ?6000 m Myr^?1. The ‘erodibility’ model produces a topography that decays in a step-like fashion from an initial mean elevation that ranges between ～1800 and 2300 m. These models cannot unequivocally distinguish the relative importance of tectonic vs. climatic processes in the macrogeomorphic evolution of the post-rift Appalachians, but they do provide some first-order quantitative prediction about these two end-member options. In light of existing stratigraphic, geological and thermochronological data, we favour the tectonic model because most of the events correlate well in time and form with known syn- and post-rift magmatic events. Nevertheless, the most recent episode of increased sediment flux to the offshore basins during the Miocene remains difficult to explain by either model. Limited evidence suggests that this event may reflect asthenospheric flow-driven uplift and the development of dynamically supported topography at a time when mechanical erosion rates were increasing in response to a cooling terrestrial climate.     

The Fréchet derivative for the one-dimensional electromagnetic induction problem

Summary. From consideration of the higher order terms, it is shown that the magneto-telluric response is Fréchet differentiable with respect to conductivity; this result remains valid for discontinuous profiles, which is not so in the case of the corresponding free-oscdlation problem for the elastic earth. The remainder term in the Fréchet formula is shown to be O|δσ|² and a numerical estimate is made of the bounding constant for a restricted class of conductivity models.     

On the frequency contents of local events: source or path effect?

Local events which occurred in the Timanfaya volcanic field are analysed and their spectral contents are interpreted as being due to the effect of a finely layered medium with large impedance contrast, the presence of which is well known from well-logging data. By considering the layered medium as a filter, its spectral response (amplitude and phase) is computed from different randomly generated values of its parameters (total thickness, mean thickness of the individual layers and impedance contrast), showing that this stratigraphic filter acts as a low-pass filter. This effect can be correctly modelled by means of an exponential correlation function. However, in some cases resonant peaks appear in the spectrum superimposed on the low-pass filter effect, which can only be attributed to resonant scattering. An analysis of the phase and group delay associated with the amplitudes is made and strongly suggests that it is necessary to incorporate the theory of resonant scattering for the retrieval of medium properties. From the point of view of the amplitudes, the superposition of both effects, low-pass filter and resonant peaks, allows us to reproduce the gross features that are usually attributed to source effects. The need for a careful analysis of the local site response previous to any interpretation of the spectral features in terms of source characteristics is thus emphasized.