The solution of the inverse gravity potential problem for cylindrical bodies

Summary. The inverse gravity potential problem consists in the determination of the form and the density of the body by its exterior gravity potential. We describe two similar classes of bodies for which this problem has a unique constructive solution.
(1) The first class contains the cylindrical bodies with finite length, arbitrary form of section and ρ( R , ø, z) =ρ₁( z )ρ₂( R , ø) density distribution, where z is the cylindrical coordinate; R , ø are the polar coordinates in a section plane. This class is important for prospecting geophysics in that it allows us to determine in a unique and constructive way, the function ρ₁( R , ø), the length, form and orientation of the cylinder if we know the function ρ₁( z ) and the exterior potential. The classical moment problem of functions is the basis for the solution of this problem.
(2) The analogous problem for the class of the spherical cylinders, or bodies bounded by arbitrary similar sections of two different concentric spheres and the radial lateral surface, appears when bodies of planetary size are studied. (An example of these bodies would be the Moon mascons.) The density distribution of these cylinders is ρ(τ, θ, ø) =ρ₁(τ)ρ₂(θ, ø) where τ, θ, ø are the spherical coordinates. The function ρ₁(θ, ø), length and form of spherical sections can be uniquely determined by exterior potential if we know the function ρ₁(τ). We propose a new constructive method for harmonic continuation of the gravity potential into the region containing the perturbing masses for the solution of the problem.     

Rigorous velocity bounds from soft τ (p) and X(p) data

Summary. The convergence of two methods of inferring bounds on seismic velocity in the Earth from finite sets of inexact observations of τ ( p ) and X( p ) are examined: the linear programming (LP) method of Garmany, Orcutt & Parker and the quadratic programming (QP) method of Stark & Parker. The LP method uses strict limits on the observations of τ and X as its data, while QP uses estimated means and variances of τ and X. The approaches are quite similar and involve only one inherent approximation: they use a finite-dimensional representation of seismic velocity within the Earth. Clearly, not every Earth model can be written this way. It is proved that this does not hinder the methods - they may be made as accurate as desired by increasing the number of dimensions in a specified way. It is shown how to get the highest accuracy with a given number of dimensions.     

Inverse kinematical problems of reflection seismology – I. Classical problems in laterally inhomogeneous media

Summary. The first non-trivial inverse problem for media with non-horizontal reflectors z = h ( x, y ) was set up for a model of the type: V = V ( z ), 0 ≤ z ≤ h ( x, y ), and the possibility of reconstructing the functions h ( x, y ) and V ( z ) at z ↦ (min h , max h ) was shown. In the alternative case, when h = constant and V = V ( x ) there is a unique solution. Only particular cases were considered for media with h = constant, v = V ( x, z ). In the second half of the 1970s, the conditional correctness of a number of inverse problems was proved and the important concept of a sufficient data system was proposed.
Over the last 20 yr much attention has been paid to layered homogeneous media with curved interfaces, which are reflectors and refractors at the same time. The task of continuing the eikonals second derivatives played a very important role in this problem. Using the connection between the second derivatives of the CDP travel-time curve and the eikonal from a phantom source at the base of the normal ray (V. Chernyak, S. Gritsenko, T. krey) there were obtained formulae of the Dix type.
Recently methods based on linearization using a small parameter were proposed for media with slightly curved interfaces. A number of iterative algorithms for optimization and inversion have been developed, which exploit advances in the solution of direct kinematical problems. The development of the theory of inverse problems and the statistical theory of interpretation has led to the creation of a general concept of multistep algorithms and their classification.     

On high-frequency spheroidal modes and the structure of the upper mantle

Summary. The asymptotic properties of spheroidal mode dispersion at high frequency for fixed phase velocity are related to the intercept times τ_β( p ) for P and S waves. If the mode eigenfrequency and the ratio of horizontal to vertical displacement at the surface for the mode are known τ_α( p ) and τ_β( p ) may be separately estimated. If discontinuities exist in the velocity model then 'solotone' effects occur, in frequency at fixed slowness, and in τ_α( p ), τ_β( p ) estimated from the mode dispersion as a function of slowness. The coupling of P and S waves in the spheroidal modes means that the interaction of P waves with upper-mantle discontinuities affects also the estimates of the S wave τ_β( p ) values for which the corresponding turning points lie in the lower mantle. The asymptotic formalism also shows that sharp pulses formed by superposition of spheroidal modes correspond to multiple PS reflections.
A study of τ_α( p ), τ_β( p ) estimates derived from spheroidal modes with periods from 45–50s, calculated for model 1066B, shows that even in the presence of strong upper-mantle discontinuities the errors in intercept time are only about one-tenth of a period. The asymptotic properties may there-for provide a useful means of estimating intercept times from modes with a few seconds period as a supplement to travel-time methods.     

Smooth profiles from τ (p) and X(p) data

Summary. We reduce the problem of constructing a smooth, 1-D, monotoni-cally increasing velocity profile consistent with discrete, inexact τ ( p ) and X( p ) data to a quadratic programming problem with linear inequality constraints. For a finite-dimensional realization of the problem it is possible to find a smooth velocity profile consistent with the data whenever such a profile exists. We introduce an unusual functional measure of roughness equivalent to the second central moment or 'Variance' of the derivative of depth with respect to velocity for smooth profiles, and we prove that its minimal value is unique. In our experience, solutions minimizing this functional are very smooth in the sense of the two-norm of the second derivative and can be constructed inexpensively by solving one quadratic programming problem. Still smoother models (in more traditional measures) may be generated iteratively with additional quadratic programs. All the resulting models satisfy the τ ( p ) and X( p ) data and reproduce travel-time data remarkably well, although sometimes τ ( p ) data alone are insufficient to ensure arrivals at large X; then an X( p ) datum must be included.     

Slowness—time mapping of near offset seismic reflection data

Summary. The transformation of a set of seismograms to the delay time-slowness, τ—p, domain is presented as a sequence of Fourier and Bessel transforms, For a horizontally layered medium, this sequence gives an exact cylindrical wave decomposition of the response to a point source; correctly compensating for the phase shifting and geometrical spreading associated with transmission through the Earth. The resultant τ—p map or 'slant stack' contains true amplitude and phase information. The spatial aliasing properties of the transformation, when applied to a dataset, are greatly improved by the use of only outgoing waves in the Bessel transform. This is equivalent to using Hankel functions rather than Bessel functions, and is justified by the absence of incoming waves from most datasets. The WKBJ approximation to the medium response enables predictions to be made about the shape and amplitude variation with slowness of truncation effects. Theoretically the τ—p transformation is reversible, thus the τ—p domain is a suitable one in which to perform filtering operations before seismogram reconstruction.     

Three-dimensional digital imaging of environmental data: selection of gridding parameters

Abstract. Three-dimensional imaging is a powerful technique for the visualization and interpretation of environmental data. The success of the process is linked to careful, technically-justifiable selection of variable parameters during the gridding and imaging process. The impacts of various approaches to gridding and possible setting of parameters on the final image and volume calculations were examined by generating alternative images for a very well characterized contaminated site in layered coastal plain sediments. To image properly scattered data collected at close intervals in wells from layered geological media, a higher grid density in the z direction is required along with a weighting factor to emphasize the influence of data in the x and y directions. For steeply-varying contaminant concentration data, the best results were obtained by gridding the log of the property value; an anti-log transformation is carried out to restore property values to the correct value before the visualization file is prepared. The techniques and recommendations made in this article were designed for modelling contaminant values with very steep gradients dispersed in a strongly anisotropic media. These recommendations may not apply directly to other sites but the process of selecting parameters should be similar.     

Determination of earthquake early warning parameters, τc and Pd, for southern California

We explore a practical approach to earthquake early warning in southern California by determining a ground-motion period parameter  τ _c   and a high-pass filtered displacement amplitude parameter Pd from the initial 3 s of the P waveforms recorded at the Southern California Seismic Network stations for earthquakes with M > 4.0. At a given site, we estimate the magnitude of an event from  τ _c   and the peak ground-motion velocity ( PGV ) from Pd . The incoming three-component signals are recursively converted to ground acceleration, velocity and displacement. The displacements are recursively filtered with a one-way Butterworth high-pass filter with a cut-off frequency of 0.075 Hz, and a P -wave trigger is constantly monitored. When a trigger occurs,  τ _c   and Pd are computed. We found the relationship between  τ _c   and magnitude ( M ) for southern California, and between Pd and PGV for both southern California and Taiwan. These two relationships can be used to detect the occurrence of a major earthquake and provide onsite warning in the area around the station where onset of strong ground motion is expected within seconds after the arrival of the P wave. When the station density is high, the methods can be applied to multistation data to increase the robustness of onsite early warning and to add the regional warning approach. In an ideal situation, such warnings would be available within 10 s of the origin time of a large earthquake whose subsequent ground motion may last for tens of seconds.     

Inversion of seismic refraction data in planar dipping structure

Summary. A new method is presented for the direct inversion of seismic refraction data in dipping planar structure. Three recording geometries, each consisting of two common-shot profiles, are considered: reversed, split, and roll-along profiles. Inversion is achieved via slant stacking the common-shot wavefield to obtain a delay time–slowness (tau– p ) wavefield. The tau– p curves from two shotpoints describing the critical raypath of refracted and post-critically reflected arrivals are automatically picked using coherency measurements and the two curves are jointly used to calculate velocity and dip of isovelocity lines iteratively, thereby obtaining the final two-dimensional velocity model.
This procedure has been successfully applied to synthetic seismograms calculated for a dipping structure and to field data from central California. The results indicate that direct inversion of closely-spaced refraction/wide-aperture reflection data can practically be achieved in laterally inhomogeneous structures.     

Generalized Radon transforms and slant stacks

Summary. The Radon transform or slant stack is becoming a widely used technique for analysing high-quality reflection and refraction data. The transform normally used is applicable to data from a line source in a plane model, that is, one Cartesian coordinate. The theoretical basis for the Radon transform pair for one Cartesian coordinate has appeared in the seismological literature. For a point source in plane or spherical geometry, or a line source in cylindrical geometry only the Radon transform for the direct problem (computation of synthetic seismograms) has been published. To analyse data an approximate inverse transform has been used. In this research note, the exact forms of the generalized Radon transform pairs are completed for a point source in plane or spherical geometry, and for a line source in cylindrical geometry. The differences will be important if the waveforms are being interpreted, and are most significant for near-vertical reflections—the type of data most commonly slant stacked.     

Evaluation of stress and displacement fields due to an elliptical plane shear crack

Summary. Following the classic work of Eshelby, the slip and stress distributions due to an elliptical plane shear crack are evaluated. The relation between average (or maximum) slip on the crack and the (constant) static stress drop, for faults of different aspect ratios, is found. The slip vector is not parallel to the applied stress but makes a small angle to it, except when the stress is applied along the major or minor axis of the ellipse. The stress -distribution around the crack shows that in addition to the expected stress concentration along the crack edge, there are broad regions of stress increase off the crack plane for circular and elliptical cracks, similar to those known to exist for in-plane but not for antiplane shear cracks. Whether the stress- intensity factor at the end of one axis is greater or less than that at the end of the other axis ( k_a ≶ k_b ), depends on the condition: √ b/a ≶ (1 − v ) where a and b are the semi-axes of the ellipse, k_a and k_b are the stress-intensity factors at the end of the a- and b -axes and v is Poisson's ratio. The total stress-intensity factor varies smoothly along the edge of the ellipse from one axis to the other and it is found that this variation is rather small.     

Transient and impulse responses of a one-dimensional linearly attenuating medium—II. A parametric study

Summary. We investigate one-dimensional waves in a standard linear solid for geophysically relevant ranges of the parameters. The critical parameters are shown to be T*= t_u/Q_m where t _u is the travel time and Q_m the quality factor in the absorption band, and τ⁻¹ _m , the high-frequency cut-off of the relaxation spectrum. The visual onset time, rise time, peak time, and peak amplitude are studied as functions of T* and τ _m. For very small τ _m , this model is shown to be very similar to previously proposed attenuation models. As τ _m grows past a critical value which depends on T* , the character of the attenuated pulse changes. Seismological implications of this model may be inferred by comparing body wave travel times with a'one second'earth model derived from long-period observations and corrected for attenuation effects assuming a frequency independent Q over the seismic band. From such a comparison we speculate that there may be a gap in the relaxation spectrum of the Earth's mantle for relaxation times shorter than about one second. However, observational constraints from the attenuation of body waves suggest that such a gap might in fact occur at higher frequencies. Such a hypothesis would imply a frequency dependence of Q in the Earth's mantle for short-period body waves.     

Noise reduction and detection of weak, coherent signals through phase-weighted stacks

We present a new tool for efficient incoherent noise reduction for array data employing complex trace analysis. An amplitude-unbiased coherency measure is designed based on the instantaneous phase, which is used to weight the samples of an ordinary, linear stack. The result is called the phase-weighted stack (PWS) and is cleaned from incoherent noise. PWS thus permits detection of weak but coherent arrivals. The method presented can easily be extended to phase-weighted cross-correlations or be applied in the τ p domain. We illustrate and discuss the advantages and disadvantages of PWS in comparison with other coherency measures and present examples. We further show that our non-linear stacking technique enables us to detect a weak lower-mantle P -to- S conversion from a depth of approximately 840 km on array data. Hints of an 840 km discontinuity have been reported; however, such a discontinuity is not yet established due to the lack of further evidence.     

Receiver functions at the stations of the German Regional Seismic Network (GRSN)

Several years of broad-band teleseismic data from the GRSN stations have been analysed for crustal structure using P -to- S converted waves at the crustal discontinuities. An inversion technique was developed which applies the Thomson-Haskell formalism for plane waves without slowness integration. The main phases observed are Moho conversions, their multiples in the crust, and conversions at the base of the sediments. The crustal thickness derived from these data is in good agreement with results from other studies. For the Gräfenberg stations, we have made a more detailed comparison of our model with a previously published model obtained from refraction seismic experiments. The refraction seismic model contains boundaries with strong velocity contrasts and a significant low-velocity zone, resulting in teleseismic waveforms that are too complicated as compared to the observed simple waveforms. The comparison suggests that a significant low-velocity zone is not required and that internal crustal boundaries are rather smooth.     

Reflection—refraction of general P-and type-I S-waves in elastic and anelastic solids

Summary. The reflection and refraction of general (homogeneous or inhomo-geneous) plane P and type-I S ( SV ) body waves incident on plane boundaries are considered for general linear viscoelastic solids. Reflection—refraction laws, physical characteristics of the waves, and the nature of critical angles are examined in detail at welded boundaries and a free surface. General visco-elasticity with no low-loss approximations predicts that contrasts in intrinsic absorption at boundaries give rise to inhomogeneous reflected and refracted waves with elliptical particle motions, velocities and maximum attenuations that vary with frequency and angle of incidence, energy propagation at speeds and directions different from phase propagation, phase propagation that in general is parallel to the boundary for at most one angle of incidence, and reflection—transmission coefficients dependent on energy flow due to wave interaction. None of these physical characteristics are predicted for waves incident on boundaries that respond instantaneously.     

Polarization analysis and polarization filtering of three-component signals with the time–frequency S transform

From basic Fourier theory, a one-component signal can be expressed as a superposition of sinusoidal oscillations in time, with the Fourier amplitude and phase spectra describing the contribution of each sinusoid to the total signal. By extension, three-component signals can be thought of as superpositions of sinusoids oscillating in the x -, y -, and z -directions, which, when considered one frequency at a time, trace out elliptical motion in three-space. Thus the total three-component signal can be thought of as a superposition of ellipses. The information contained in the Fourier spectra of the x -, y -, and z -components of the signal can then be re-expressed as Fourier spectra of the elements of these ellipses, namely: the lengths of their semi-major and semi-minor axes, the strike and dip of each ellipse plane, the pitch of the major axis, and the phase of the particle motion at each frequency. The same type of reasoning can be used with windowed Fourier transforms (such as the S transform), to give time-varying spectra of the elliptical elements. These can be used to design signal-adaptive polarization filters that reject signal components with specific polarization properties. Filters of this type are not restricted to reducing the whole amplitude of any particular ellipse; for example, the 'linear' part of the ellipse can be retained while the 'circular' part is rejected. This paper describes the mathematics behind this technique, and presents three examples: an earthquake seismogram that is first separated into linear and circular parts, and is later filtered specifically to remove the Rayleigh wave; and two shot gathers, to which similar Rayleigh-wave filters have been applied on a trace-by-trace basis.     

A new derivation of the ratio Q(wobble)/Qμ (mantle)

Summary An extension of the Love-Larmor theory to a low-loss unelastic earth model, leads to the surprisingly simple approximation
   
where τ_s= 447.4 sidereal day is the static wobble period, τ_R= 306 sidereal day is the rigid-earth wobble period and τ_w= 433 sidereal day is the observed Chandler period. Q _W, Q _μ are the respective average Q values of the wobble and the Earth's mantle at τ_W. The known numerical factor F is only slightly dependent on the Earth structure.     

Focal mechanisms of recent earthquakes in the Southern Korean Peninsula

We evaluate the stress field in and around the southern Korean Peninsula with focal mechanism solutions, using the data collected from 71 earthquakes ( M_L = 1.9–5.2) between 1999 and 2004. For this, the hypocentres were relocated and well-constrained fault plane solutions were obtained from the data set of 1270 clear P -wave polarities and 46 SH / P amplitude ratios. The focal mechanism solutions indicate that the prevailing faulting types in South Korea are strike-slip-dominant-oblique-slip faultings with minor reverse-slip component. The maximum principal stresses (σ₁) estimated from fault-slip inversion analysis of the focal mechanism solutions show a similar orientation with E–W trend (269°–275°) and low-angle plunge (10°–25°) for all tectonic provinces in South Korea, consistent with the E–W trending maximum horizontal stress (σ_Hmax) of the Amurian microplate reported from in situ stress measurements and earthquake focal mechanisms. The directions of the intermediate (σ₂) and minimum (σ₃) principal stresses of the Gyeongsang Basin are, however, about 90 deg off from those of the other tectonic provinces on a common σ₂–σ₃ plane, suggesting a permutation of σ₂ and σ₃. Our results incorporated with those from the kinematic studies of the Quaternary faults imply that NNW- to NE-striking faults (dextral strike-slip or oblique-slip with a reverse-slip component) are highly likely to generate earthquakes in South Korea.     

Diatoms and sockeye salmon (Oncorhynchus nerka) population dynamics: Reconstructions of salmon-derived nutrients over the past 2,200 years in two lakes from Kodiak Island, Alaska

The return of hundreds to millions of adult sockeye salmon (Oncorhynchus nerka), which have returned from the ocean to their natal nursery lake environment to spawn, can result in significant nutrient loading. By analyzing sedimentary diatom assemblages from nursery lakes, we demonstrated that a salmon-derived nutrient signal could be traced over time and be used to infer past sockeye salmon population dynamics. We conducted a 2,200 year paleolimnological study of two Alaskan sockeye salmon nursery lakes, Karluk and Frazer lakes. The two lakes are very similar, except that sockeye salmon were only introduced into Frazer Lake in 1951 (first spawners returned in 1956). In both lakes we found a strong correspondence between diatom assemblages and the number of adult salmon spawners recorded in the historical data (40 and 70 years for Frazer and Karluk lakes, respectively). Given this robust relationship, we then used our analyses of diatoms from Karluk Lake over the past 2,200 years to gain insight into salmon-derived nutrient loading changes (which are directly related to the number of sockeye salmon spawners). The diatom record from Karluk Lake recorded dramatic species changes on both decadal and century timescales, and was strongly correlated with an independent indicator of sockeye salmon abundances, ¹⁵N. Together, these data suggest pronounced variability in sockeye salmon abundances at Karluk Lake over the past 2,200 years. The direct impacts of regional environmental variability were not likely responsible for the patterns apparent in Karluk Lake, as the diatom and ¹⁵N profiles from Frazer Lake were relatively stable prior to the introduction of sockeye salmon. Application of total phosphorus transfer functions to the Karluk and Frazer lakes' diatom records revealed that sockeye salmon carcasses substantially increased the trophic status in these lakes, which has important implications for the health of juvenile salmon that rear in nursery lakes. Overall, this paper illustrates the potential use of diatoms in reconstructing past sockeye salmon population dynamics, which in turn can lead to a greater understanding of the mechanisms influencing abundances of sockeye salmon.     

Combined seismic reflection and refraction profiling in southwest Germany - detailed velocity mapping by the refraction survey

Summary. Within the framework of site survey studies of the Deep Drilling Program of the Fedral Republic of Germany (KTB), coincident deep-seismic reflection and refraction experiments in the Black Forest, southwest Germany, were carried out. The simultaneous interpretation of the reflection and the refraction data reveals in particular both a strong velocity reduction in the upper crust and a laterally varying laminated structure of the lower curst. Additional refraction lines result in a three-dimensional crustal model which shows two distinct crustal types of different seismic properties. These crustal types seem to correlate with the major geologic units of Southwest Germany. Variations of Poisson's ratio derived from clearly recorded shear wave data show a similar trend.