Vertical Component Of Displacement Research Articles

Recovery of the elastic parameters of a layered half-space

In this article we begin The study of the problem of recovering the elastic parameters of a layered half-space from single component measurements of reflected waves, and applications to reflection seismology. We analyse the perturbational relationship between the elastic moduli (p, λ and µ and the vertical component of surface particle displacement due to a point impulsive traction. We show that this linearized problem has a unique solution and derive some stability estimates. The analysis of the linearized problem proceeds by way of the plane wave decomposition, and the uniqueness proof is constructive. To illustrate these ideas, we implement the construction numerically and recover parameter perturbations from data perturbations in a synthetic example. The solution of the linearized problem may be viewed as a step in a Newton-like scheme for solution of the full inverse problem. In considering the application of any such technique to field recorded data, however, one must confront problems such as model inadequacy, data error, source signature error, band limitation, etc. The extent to which these problems are resolvable has been explored for the simpler acoustic model using techniques compatible with those presented here. We plan to address such (practically crucial) stability considerations, and their implications for the design of useful algorithms, elsewhere. The focus of the present paper is thus the mathematical uniqueness question, which may be regarded as the most primitive stability question.

A NUMERICAL STUDY OF LAMB'S PROBLEM*

AbstractThis paper considers propagation of elastodynamic waves in an imperfectly elastic half‐space. Two different excitation modes are investigated: a buried source of compressional waves and a vertically directed areal load applied to the surface. Numerical integration of the analytical solution of the wave equation allows study of the vertical and horizontal components of displacement and/or particle velocity anywhere in the half‐space. One case of particular interest concerns the examination of particle displacement and velocity at the surface in a circular area above the source. In another application seismograms generated by an explosive buried source are contrasted with seismograms generated by the transient application of a vertically directed load to the free surface. Still another application of considerable practical interest concerns the study of the nongeometrical pS—wave, in particular its characteristics as functions of range and depth. Finally, in the last application the behavior of a rarely observed wave (denoted here by the letter U) is studied in both elastic and visco‐elastic half‐spaces.

Seismic moment tensor components of an earthquake source in the eastern Himalayas, 26.61°N, 97.03°E

An earthquake with epicentre located at 26.61°N, 97.03°E is studied for its moment tensor components, using vertical component Rayleigh wave records from a number of stations. As the moment components are sensitive to source geometry and earth model, determination of focal parameters on the basis of first motion P-wave data forms the first step in the analysis. Further, as an idealization, values of the phase and group velocities are assumed for a standard Gutenberg layered earth model. Rayleigh wave spectra corrected for instrument response and attenuation effects of the medium are used for the calculation of moment tensor components. The basis of this analysis is the theoretical relationship between spectra of the vertical component of displacement of Rayleigh waves with the space derivatives of the Green's function and seismic moment tensor components. This is discussed briefly in the text. Least square linear inversion is adopted to optimize the solution for moment tensor components and then to calculate the most likely focal depth. Total seismic moment, stress drop and average dislocation on the fault are calculated using spectra of body waves after applying the necessary corrections. It is concluded that the total moment and its components have characteristic lower values. Interpretation of the result is included.

Stable plane-wave decomposition and spherical-wave reconstruction: applications to converted S-mode separation and trace interpolation

Plane-wave decomposition of the vertical displacement component of a spherical-wave field corresponding to a compres-sional point source is solved as a set of inverse problems. The solution method utilizes the power and stability of Backus and Gilbert (smallest and flattest) model-construction techniques, and achieves computational efficiency through the use of analytical solutions to the involved integrals. The theory and algorithms developed in this work allow stable and efficient reconstruction of spherical-wave fields from a relatively sparse set of their plane-wave components. Comparison of the algorithms with discrete integration of the Hankel transform shows very little or no advantage for the transformation from the time-distance (t-x) domain to the intercept time angle of emergence (T-y) domain if the seismograms are equisampled spatially. However, when the observed seismograms are not equally spaced or the transformation r-y to t-x is performed, the proposed schemes are superior to the discrete integration of the Hankel transform. Applicability of the algorithms to reflection seismology is demonstrated by means of the solution of the problem of trace interpolation, and also that of the separation of converted S modes from other modes presented in common-source gathers. In both cases the application of the algorithms to a set of synthetic reflection seismograms yields satisfactory results.

Stable plane‐wave decomposition and spherical‐wave reconstruction: Applications to converted S-mode separation and trace interpolation

Plane‐wave decomposition of the vertical displacement component of a spherical‐wave field corresponding to a compressional point source is solved as a set of inverse problems. The solution method utilizes the power and stability of Backus and Gilbert (smallest and flattest) model‐construction techniques, and achieves computational efficiency through the use of analytical solutions to the involved integrals. The theory and algorithms developed in this work allow stable and efficient reconstruction of spherical‐wave fields from a relatively sparse set of their plane‐wave components. Comparison of the algorithms with discrete integration of the Hankel transform shows very little or no advantage for the transformation from the time‐distance (t-x) domain to the intercept time‐angle of emergence (τ-γ) domain if the seismograms are equisampled spatially. However, when the observed seismograms are not equally spaced or the transformation τ-γ to t-x is performed, the proposed schemes are superior to the discrete integration of the Hankel transform. Applicability of the algorithms to reflection seismology is demonstrated by means of the solution of the problem of trace interpolation, and also that of the separation of converted S modes from other modes presented in common‐source gathers. In both cases the application of the algorithms to a set of synthetic reflection seismograms yields satisfactory results.

Magma reservoir subsidence mechanics: Theoretical summary and application to Kilauea Volcano, Hawaii

An analytic model has been developed for the prediction of the three‐dimensional deformation field generated by the withdrawal of magma from a sill‐like storage compartment during an intrusion or eruption cycle. The model is based on the work of Berry and Sales (1961, 1962) and predicts the vertical displacement components over the areal plane. Model parameters are the depth of burial h, the intrusion half width a, the intrusion half length b, the thickness of the magmatic interior at the moment of melt withdrawal tm, and the planform aspect ratio ξ = a/b. The products of the model include areal deformation maps. Systematic variation in model parameters within the context of Kilauea Volcano, Hawaii, have revealed that circular and elliptical deformation patterns result from the collapse of draining rectilinear intrusions at depth. Moreover, the geometric parameters of a storage compartment may interact in complex ways to produce similar deformation patterns. The model has been applied to Kilauea Volcano for three periods of pronounced summit subsidence: (1) 1921–1927 (bracketing the steamblast eruptive phases of 1924); (2) June 1972 to December 1972, and (3) December 1972 to May 1973. Application of the model requires the simultaneous optimization of five predicted deformation features with respect to field measurements and the derivative deformation maps: (1) the vertical displacement maxima; (2) the vertical displacement gradients over the areal plane, (3) the lateral extent of the deformation field, (4) the aspect ratio of the subsidence pattern, and (5) the strike of the major axis of the deformation field. The constrained geometries and volumes of the inferred collapsed storage cavities for each period are (1) 1921–1927: depth ≅ 3 km, a ≅ 1500 m, b ≅ 4500 m, tm ≅ 20 m, V 540×106 m3, (2) June 1972 to December 1972: depth ≅ 3.3 km, a ≅ 600 m, b ≅ 2000 m, tm ≅ 1 m, V ≅ 4.8×106 m3, and (3) December 1972 to May 1973: depth ≅ 2.2 km, a ≅ 500 m, b ≅ 1612 m, tm ≅ 1 m, V ≅ 3.2×106 m3. For 2 and 3, calculated magmatic thicknesses tm happen to be in the range (3.48–0.15 m) of measurements for sill‐like bodies in deeply dissected Hawaiian shield volcanoes. The fits obtained between calculated and observed deformation patterns allow quantification of the location, overall dimensions, orientation, and volume of the discrete, still molten, interior of sill‐like compartments from which magma is tapped during eruption or intrusion.

Pan-African Granite Intrusion in Response to Tectonic Volume Changes in a Ductile Shear Zone from Northern Saudi Arabia

The Ajjaj shear zone of the Njad fault system of northwestern Saudi Arabia was a conduit for Pan-African (500-700 m.y.) calc-alkalic and peralkalic magmas. Ductility contrasts between the wall rocks and the deformed rocks of the shear zone were responsible for differences in size, intrusion mechanism, and degree of deformation of the granites. A quantitative strain analysis across the Ajjaj shear zone reveals a finite oblique strike-slip displacement of with a vertical displacement component of . Granite diapirs were intruded within the region of volume gain; between 83-90% of the volume of the Ash Sha'b diapir was produced by expansion of the granite. Changes in late Pan-African granite magmatism, especially abrupt variations in alkalic granite distribution, can be indirectly linked with the vertical displacement components of the crustal blocks flanking the Ajjaj shear belt.

Tectonic strain and drift

The writer's (1958) determination of the PHS is critically examined; while holding for dominantly normal, transcurrent and reverse faults, it can be in error by up to 11.6 for hybrid faults with equal vertical and horizontal components of displacement. A corrected formula is presented, which shows the tectonic strain axis to be consistent throughout New Zealand within and between different tectonic regimes. In any area the tectonic strain axis and its rate are closely similar to the strain axes and rates derived from geodetic and seismic studies, provided the area has not experienced a major earthquake during the monitoring interval. Other anomalies can be attributed to the precursory phase heralding a major earthquake. The formula presented will also determine the subsurface dip of the fault plane. Finally the horizontal component of the Tectonic Drift is presented and is suggested to be a useful contribution to the study of plate tectonics on land.

Discussion--Nomograms for Estimating Components of Fault Displacement from Measured Height of Fault Scarp

The accompanying nomographs permit the determination of actual vertical, horizontal, and dip-slip components of displacement on a fault from field measurements of fault scarp height, dip of fault and slope of original surface displaced.

On contact of the crack edges

There is considered an infinite elastic plane with a symmetric crack that can be mapped into a circle by a rational function. It is assumed that opposite edges of the crack come into contact because of loading at infinity. it is also assumed that the vertical displacement component is known on the crack contour in the contact range; there is no tangential stress because of symmetry, and there are no stresses on the free part of the boundary. By means of a suitable selection of two piecewise-meromorphic functions, the boundary conditions result in a Riemann boundary value problem for the vector functions holomorphic on a plane slit along pairs of circular arcs, and having finite order at infinity. A solution is given in quadratures for the problem formulated. Conditions are written down which define the unknown coefficients of the general solution. A specific example is considered when the crack has the profile of a prolate ellipse.

Holocene displacement along branch and secondary faults in the San Andreas fault zone, southern California

Detailed geologic mapping of the San Andreas fault zone in Los Angeles County since 1972 has revealed evidence for diverse histories of displacement on branch and secondary faults near Palmdale. The main trace of the San Andreas fault is well defined by a variety of physiographic features. The geologic record supports the concept of many kilometers of lateral displacement on the main trace and on some secondary faults, especially when dealing with pre-Quaternary rocks. However, the distribution of upper Pleistocene rocks along branch and secondary faults suggests a strong vertical component of displacement and, in many locations, Holocene displacement appears to be primarily vertical. The most recent movement on many secondary and some branch faults has been either high-angle (reverse and normal) or thrust. This is in contrast to the abundant evidence for lateral movement seen along the main San Andreas fault. We suggest that this change in the sense of displacement is more common than has been previously recognized. The branch and secondary faults described here have geomorphic features along them that are as fresh as similar features visible along the most recent trace of the San Andreas fault. From this we infer that surface rupture occurred on these faults in 1857, as it did on the main San Andreas fault. Branch faults commonly form “Riedel” and “thrust” shear configurations adjacent to the main San Andreas fault and affect a zone less than a few hundred meters wide. Holocene and upper Pleistocene deposits have been repeatedly offset along faults that also separate contrasting older rocks. Secondary faults are located up to 1500 m on either side of the San Andreas fault and trend subparallel to it. Moreover, our mapping indicates that some portions of these secondary faults appear to have been “inactive” throughout much of Quaternary time, even though Holocene and upper Pleistocene deposits have been repeatedly offset along other parts of these same faults. For example, near 37th Street E. and Barrel Springs Road, a limited stretch of the Nadeau fault has a very fresh normal scarp, in one place as much as 3 m high, which breaks upper Pleistocene or Holocene deposits. This scarp has two bevelled surfaces, the upper surface sloping significantly less than the lower, suggesting at least two periods of recent movement. Other exposures along this fault show undisturbed Quaternary deposits overlying the fault. The Cemetery and Little Rock faults also exhibit selected reactivation of isolated segments separated by “inactive” stretches. Activity on branch and secondary faults, as outlined above, is presumed to be the result of sympathetic movement on limited segments of older faults in response to major movement on the San Andreas fault. The recognition that Holocene activity is possible on faults where much of the evidence suggests prolonged inactivity emphasizes the need for regional, as well as detailed site studies to evaluate adequately the hazard of any fault trace in a major fault zone. Similar problems may be encountered when geodetic or other studies, Which depend on stable sites, are conducted in the vicinity of major faults.

Candelaria and other left-oblique slip faults of the Candelaria region, Nevada

Research Article| February 01, 1979 Candelaria and other left-oblique slip faults of the Candelaria region, Nevada R. C. SPEED; R. C. SPEED 1Department of Geological Sciences, Northwestern University, Evanston, Illinois 60201 Search for other works by this author on: GSW Google Scholar ALLEN H. COGBILL ALLEN H. COGBILL 1Department of Geological Sciences, Northwestern University, Evanston, Illinois 60201 Search for other works by this author on: GSW Google Scholar GSA Bulletin (1979) 90 (2): 149–163. https://doi.org/10.1130/0016-7606(1979)90<149:CAOLSF>2.0.CO;2 Article history first online: 01 Jun 2017 Cite View This Citation Add to Citation Manager Share Icon Share Facebook Twitter LinkedIn MailTo Tools Icon Tools Get Permissions Search Site Citation R. C. SPEED, ALLEN H. COGBILL; Candelaria and other left-oblique slip faults of the Candelaria region, Nevada. GSA Bulletin 1979;; 90 (2): 149–163. doi: https://doi.org/10.1130/0016-7606(1979)90<149:CAOLSF>2.0.CO;2 Download citation file: Ris (Zotero) Refmanager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentBy SocietyGSA Bulletin Search Advanced Search Abstract The region comprising the Candelaria Hills and Excelsior Mountains, Nevada, is structurally anomalous with respect to surrounding tracts of the Basin and Range Province. Its major faults strike east, and some, if not all of them, have undergone significant left slip. Ratios of lateral to vertical displacement components on strands of such faults vary from 2 to 6.5. The region of left-oblique slip faulting seems to constitute the intermediate leg of a gigantic Z pattern of Basin and Range faults.One of these faults, the Candelaria fault system, has been studied closely in pursuit of kinematic details that might characterize the regional faulting. The Candelaria system consists of three en echelon faults of 5 to 10 km trace length joined by connector zones that possess compressional features. The amount of lateral slip transferred from one en echelon fault to another is uncertain; analysis of a complex of faults in one connector zone indicates at least some degree of integrated motion. The same connector zone is the site of a bulge that has an uplift estimated to have been 275 m since 2.8 m.y. B.P. On the basis of fault-plane striae, the extension direction is ∼ N82°W. We propose that the Candelaria fault system and a major asymmetric trough in its northern wall were created in Oligocene time by a regional left-lateral shear system oriented in the extension direction. The connector zones may be kinematic impediments which, at shallow levels at least, have prevented the system from maturing into a single plane fault. The existence of the bulge and the estimated rate of slip on part of the Candelaria fault system suggest that the system is still active. This content is PDF only. Please click on the PDF icon to access. First Page Preview Close Modal You do not have access to this content, please speak to your institutional administrator if you feel you should have access.

Structure and Function of the Elasmobranch Auditory System

Behavioral evidence indicates that sharks detect underwater sound at frequencies up to 1000 Hz, and that certain low frequency signals attract sharks from large distances. It appears that the adequate stimulus for “sound detecting” systems of the shark is panicle motion, as opposed to fluctuations in sound pressure. The elasmobranch ear consists of the three semi-circular canals for detecting angular accelerations, and otolith organs for detecting linear motion and accelerations due to gravity. Two of these organs, the sacculus and macula neglecta, have been shown to be responsive to vibratory motion, with the macula neglecta having best sensitivity to vertical movements. A direct vibrational pathway exists to the macula neglecta from the parietal fossa of the dorsal chondrocranium. It is not clear at present, however, whether it is the inner ear or the lateral line system which is responsible for hearing. Both detection systems are theoretically capable of providing information to the brain about sound source location using non-parallel arrays of directionally sensitive hair-cell receptors. Recent theories of underwater sound localization by fishes and sharks suggest that the ability to detect a vertical displacement component of an acoustic signal (e. g., via the macula neglecta) is necessary for instantaneous location decisions. It is not known, however, whether the sharks localize by processing information about various aspects of the sound field simultaneously (in parallel), or whether the sound field is sampled successively at different points in space. Clearly, more experimental work on the physiology of elasmobranch acoustic behavior is called for.

Seismic noise at the international gravity point in Prague-Ruzyně (Czechoslovakia)

The vertical component of ground displacement was measured at the Prague - Ruzyně International Gravity Point in the frequency range of 1–300 Hz. The permanent noise, the vibrations caused by the observers during gravimetric observations and by the wind, as well as those due to normal operations at the airport, display maximum peak-to-peak amplitudes of 0.06 µm in the frequency range of 1–50 Hz; with a CG-2 gravimeter this is not detrimental to the accuracy of the observations. The taxiing of turbo-jet and jet aircraft and engine tests of aircraft generate vibrations in frequency ranges of 75–90 and 190–270 Hz. Their amplitudes, according to the results of laboratory tests published for various types of gravity meters (CG-2, GAK-PT, GVP-3, KVG), are of magnitudes which generate errors in tenths of mgl.

Some numerical solutions for Lamb's problem

abstract We consider a version of Lamb's Problem in which a vertical time-dependent point force acts on the surface of a uniform half-space. The resulting surface disturbance is computed as vertical and horizontal components of displacement, particle velocity, acceleration, and strain. The goal is to provide numerical solutions appropriate to a comparison with observed wave forms produced by impacts onto granite and onto soil. Solutions for step- and delta-function sources are not physically realistic but represent limiting cases. They show a clear P arrival (larger on horizontal than vertical components) and an obscure S arrival. The Rayleigh pulse includes a singularity at the theoretical arrival time. All of the energy buildup appears on the vertical components and all of the energy decay, on the horizontal components. The effects of Poisson's ratio upon vertical displacements for a step-function source are shown. For fixed shear velocity, an increase of Poisson's ratio produces a P pulse which is larger, faster, and more gradually emergent, an S pulse with more clear-cut beginning, and a much narrower Rayleigh pulse. For a source-time function given by cos2(πt/T), −T/2 ≦ T/2, a × 10 reduction in pulse width at fixed pulse height yields an increase in P and Rayleigh-wave amplitudes by factors of 1, 10, and 100 for displacement, velocity and strain, and acceleration, respectively. The observed wave forms appear somewhat oscillatory, with widths proportional to the source pulse width. The Rayleigh pulse appears as emergent positive on vertical components and as sharp negative on horizontal components. We show a theoretical seismic profile for granite, with source pulse width of 10 µsec and detectors at 10, 20, 30, 40, and 50 cm. Pulse amplitude decays as r−1 for P wave and r−12 for Rayleigh wave. Pulse width broadens slightly with distance but the wave form character remains essentially unchanged.

The importance of S-wave precursors in shear-wave studies

Precursors to the S or shear wave can cause serious error in interpretation of seismograms, particularly if only the vertical component is available. A series of synthetic seismograms are constructed using either asymptotic ray theory or the Fourier transform of a crustal transfer function obtained from a Haskell-Thomson matrix formulation to show the contribution of S - to P -wave conversions as precursors to the shear wave. Errors of 1 to 6 sec may occur if only the vertical component of seismic displacement is available. Errors may occasionally occur if horizontal seismograms are used, particularly when seismic stations are situated on thick sedimentary basins. In this case, particle-motion studies and arrays of recording instruments are required to avoid errors which may be as large as 2 sec. The problem arises at all distances and is of particular importance when ts − tp times are used to determine epicentral distances, first-motion studies, and in the study of premonitory phenomena using the ratio ts / tp for earthquake prediction.

Dynamic Behavior of the Hyoid Bone, Mandible, Tongue, and Thyroid Cartilage During Speech Production

Using lateral-view cinettuorographic techniques, movement activities of the hyoid bone, mandible, tongue, and thyroid cartilage were observed. The speakers read 18 CVCV nonsense utterances imbedded in a carrier phrase. A stressed-unstressed/unstressed-stressed paradigm was employed within this disyllable utterance. Using a computerized frame-by-frame analysis technique [provided by the Speech Communications Research Laboratory, Inc., Santa Barbara, Calif.], movement characteristics were retrieved separately for the vertical and horizontal displacement components of each structure. From the retrieved data separate linear displacement plots for each structural component were produced as a function of time. The relations among simultaneous movements of the mandible, tongue, thyroid cartilage, and hyoid bone during productions of the vowels /i, a, u/ are reported as a function of adjacent consonant and syllable stress.

Seismic waves in a quarter plane

Abstract The equations for elastic wave propagation are solved by a finite difference scheme for the case of an elastic quarter plane. A point-source emitting a compressional pulse is located along the diagonal inside the quarter plane. Free-surface conditions are assumed on the boundary lines, so that the problem is nonseparable. Complete theoretical seismograms for the horizontal and vertical components of displacement are obtained. The effect of different finite difference formulations for the boundary conditions and the effect of different mesh sizes are studied. Various reflected volume and surface waves are identified, corner-generated surface waves are clearly seen in the seismograms and their particle motion is studied. The amplitude of the pulse observed at the corner is three times the amplitude of the initial pulse.

Propagation of Elastic Waves in layered media by finite difference methods

A finite difference equation formulation for the equations of elasticity is presented and applied to the problem of a layered half-space with a buried point source emitting a compressional pulse. Complete theoretical seismograms for the horizontal and vertical components of displacement are obtained. The results for a specific case are compared with those found by a completely different method in order to check the validity of the finite difference methods. The agreement is excellent. The effect of different mesh sizes on the theoretical seismograms is studied next and a suitable grid system selected for the applications that follow. The development of Rayleigh waves on the surface of a half-space and the change of the Rayleigh wave with depth and pulse width are examined. The problem of a layered half-space with a high velocity bottom is considered and the refraction arrivals on the surface and on the interface are studied. The problem of interface waves on the surface separating two semiinfinite media is also examined. Interface waves are found when the physical parameters lie both inside and outside the region determined by the Stoneley equation. Finally, a series of theoretical seismograms for a layered half-space showing the variation of the surface waves as a function of depth and of the density in the lower medium is presented.

On the propagation of certain types of surface waves in a non-homogeneous elastic layer

The propagation of a certain type of surface waves in a non-homogeneous elastic layer of finite thickness lying in welded contact with a semi-infinite homogeneous elastic medium has been investigated in this paper. The surface wave is characterised by the fact that the dilatation Δ and the vertical displacement component ω are both zero. It has been proved that such disturbance can propagate only when the phase velocity of the wave is greater than the shear wave velocity and the group velocity is inversely proportional to the phase velocity.