scholarly journals Total energy and energy spectral density of elastic wave radiation from propagating faults

1964 ◽  
Vol 54 (6A) ◽  
pp. 1811-1841 ◽  
Author(s):  
N. A. Haskell
Keyword(s):  
Total Energy ◽  
Fault Plane ◽  
Relative Displacement ◽  
Point Sources ◽  
Time Function ◽  
Displacement Discontinuity ◽  
Source Distribution ◽  
Wave Radiation ◽  
Short Period ◽  
Fault Displacement

abstract Starting with a Green's function representation of the solution of the elastic field equations for the case of a prescribed displacement discontinuity on a fault surface, it is shown that a shear fault (relative displacement parallel to the fault plane) is rigorously equivalent to a distribution of double-couple point sources over the fault plane. In the case of a tensile fault (relative displacement normal to the fault plane) the equivalent point source distribution is composed of force dipoles normal to the fault plane with a superimposed purely compressional component. Assuming that the fault break propagates in one direction along the long axis of the fault plane and that the relative displacement at a given point has the form of a ramp time function of finite duration, T, the total radiated P and S wave energies and the total energy spectral densities are evaluated in closed form in terms of the fault plane dimensions, final fault displacement, the time constant T, and the fault propagation velocity. Using fault parameters derived principally from the work of Ben-Menahem and Toksöz on the Kamchatka earthquake of November 4, 1952, the calculated total energy appears to be somewhat low and the calculated energy spectrum appears to be deficient at short periods. It is suggested that these discrepancies are due to over-simplification of the assumed model, and that they may be corrected by (1) assuming a somewhat roughened ramp for the fault displacement time function to correspond to a stick-slip type of motion, and (2) assuming that the short period components of the fault displacement wave are coherent only over distances considerably smaller than the total fault length.

scholarly journals Time-domain determination of earthquake fault parameters from short-period P-waves

1976 ◽  
Vol 66 (5) ◽  
pp. 1459-1484
Author(s):  
Paul G. Somerville ◽  
Ralph A. Wiggins ◽  
Robert M. Ellis
Keyword(s):  
Time Domain ◽  
Time Function ◽  
Wave Radiation ◽  
Model Parameters ◽  
Half Cycle ◽  
Rupture Velocity ◽  
Fault Surface ◽  
Fault Parameters ◽  
Short Period

abstract Source parameters of two shallow earthquakes have been determined by the time-domain analysis of short-period teleseismic recordings. For each event, the effect of the receiver crust was deconvolved from a set of globally distributed recordings using the homomorphic method. The resulting seismograms were compared with the form of the elastic-wave radiation computed from Savage's model of radially spreading rupture on a plane elliptical fault surface. This time-domain approach has permitted the determination of several kinematic parameters pertaining to the dynamics of rupture that are not ordinarily evaluated from spectral analysis. These parameters are rupture velocity, the direction of farthest rupture propagation, and the duration of a ramp dislocation time function which was prescribed to be the same everywhere on the fault surface. The application of a general linear inverse scheme has shown that the model parameters (notably rupture velocity and dimension) are only weakly coupled. Inversion is also used to determine the range of acceptable parameter values and indicates the importance of array recordings in constraining the models. A consistent discrepancy between the observed and model seismograms during the first half-cycle of motion is attributed to the incorrect prescription of the dislocation time function. It is suggested that a space-dependent function determined theoretically by Kostrov in 1964 would tend to remove this discrepancy.

Total energy and energy spectral density of elastic wave radiation from propagating faults. Part II. A statistical source model

1966 ◽  
Vol 56 (1) ◽  
pp. 125-140
Author(s):  
N. A. Haskell
Keyword(s):  
Spectral Density ◽  
Total Energy ◽  
Scaling Laws ◽  
Relative Displacement ◽  
Source Model ◽  
Wave Radiation ◽  
Statistical Parameters ◽  
Energy Spectral Density ◽  
Statistical Sense ◽  
Two Parameters

abstract Previously derived expressions for the total energy and energy spectral density of elastic waves radiated by a propagating fault are rewritten in terms of a spacio-temporal autocorrelation of the acceleration of relative displacement over the fault plane. This is interpreted in a statistical sense as the average autocorrelation over an ensemble of earthquakes. An explicit form of autocorrelation function is assumed, depending upon two parameters, a correlation length, and a correlation time, and the total energy and energy spectral density are derived in terms of these parameters. By using scaling laws due to Båth and Duda for earthquake volume and radiation efficiency as functions of magnitude, the statistical parameters may also be related to magnitude.

Frequency Response Analysis of Piecewise Nonlinear Vibration Isolator

2005 ◽  
Author(s):  
Patrick Stahl ◽  
G. Nakhaie Jazar
Keyword(s):  
Low Frequency ◽  
High Amplitude ◽  
Relative Displacement ◽  
Response Analysis ◽  
Frequency Response Analysis ◽  
State Behavior ◽  
Vibration Isolators ◽  
Short Period ◽  
Secondary Suspension ◽  
Low Amplitude

Non-smooth piecewise functional isolators are smart passive vibration isolators that can provide effective isolation for high frequency/low amplitude excitation by introducing a soft primary suspension, and by preventing a high relative displacement in low frequency/high amplitude excitation by introducing a relatively damped secondary suspension. In this investigation a linear secondary suspension is attached to a nonlinear primary suspension. The primary is assumed to be nonlinear to model the inherent nonlinearities involved in real suspensions. However, the secondary suspension comes into action only during a short period of time, and in mall domain around resonance. Therefore, a linear assumption for the secondary suspension is reasonable. The dynamic behavior of the system subject to a harmonic base excitation has been analyzed utilizing the analytic results derived by applying the averaging method. The analytic results match very well in the transition between the two suspensions. A sensitivity analysis has shown the effect of varying dynamic parameters in the steady state behavior of the system.

Elastic displacements in the near-field of a propagating fault

1969 ◽  
Vol 59 (2) ◽  
pp. 865-908
Author(s):  
N. A. Haskell
Keyword(s):  
Fault Plane ◽  
Near Field ◽  
Strong Motion ◽  
Dislocation Motion ◽  
Displacement Discontinuity ◽  
Wave Form ◽  
Strong Motion Station ◽  
Ramp Function ◽  
Wave Forms ◽  
Parkfield Earthquake

abstract Displacement, particle velocity, and acceleration wave forms in the near field of a propagating fault have been computed by numerical integration of the Green's function integrals for an infinite medium. The displacement discontinuity (dislocation) on the fault plane is assumed to have the form of a unilaterally propagating finite ramp function in time. The calculated wave forms in the vicinity of the fault plane are quite similar to those observed at the strong motion station nearest the fault plane at the Parkfield earthquake. The comparison suggests that the propagating ramp time function is roughly representative of the main features of the dislocation motion on the fault plane, but that the actual motion has somewhat more high frequency complexity. Calculated amplitudes indicate that the average final dislocation on the fault at the Parkfield earthquake was more than an order of magnitude greater than the offsets observed on the visible surface trace. Computer generated wave form plots are presented for a variety of locations with respect to the fault plane and for two different assumptions on the relation between fault length and ramp function duration.

Source inversion of the 1988 Upland, California, earthquake: Determination of a fault plane for a small event

1990 ◽  
Vol 80 (3) ◽  
pp. 507-518 ◽  
Author(s):  
Jim Mori ◽  
Stephen Hartzell
Keyword(s):  
Green Function ◽  
Fault Plane ◽  
Least Squares Method ◽  
Source Area ◽  
Rupture Velocity ◽  
P Waves ◽  
Waveform Data ◽  
Empirical Green Function ◽  
Finite Fault ◽  
Short Period

Abstract We examined short-period P waves to investigate if waveform data could be used to determine which of two nodal planes was the actual fault plane for a small (ML 4.6) earthquake near Upland, California. We removed path and site complications by choosing a small aftershock (ML 2.7) as an empirical Green function. The main shock P waves were deconvolved by using the empirical Green function to produce simple far-field displacement pulses. We used a least-squares method to invert these pulses for the slip distribution on a finite fault. Both nodal planes (strike 125°, dip 85° and strike 221°, dip 40°) of the first-motion focal mechanism were tested at various rupture velocities. The southwest trending fault plane consistently gave better fitting solutions than the southeast-trending plane. We determined a moment of 4.2 × 1022 dyne-cm. The rupture velocity, and thus the source area could not be well resolved, but if we assume a reasonable rupture velocity of 0.87 times the shear wave velocity, we obtain a source area of 0.97 km2 and a stress drop of 38 bars. Choice of a southwest-trending fault plane is consistent with the trend of the nearby portion of the Transverse Ranges frontal fault zone and indicates left-lateral motion. This method provides a way to determine the fault plane for small earthquakes that have no surface rupture and no obvious trend in aftershock locations.

Aftershocks of the February 21, 1973 Point Mugu, California earthquake

1976 ◽  
Vol 66 (6) ◽  
pp. 1931-1952
Author(s):  
Donald J. Stierman ◽  
William L. Ellsworth
Keyword(s):  
Fault Zone ◽  
Main Shock ◽  
Fault Plane ◽  
Body Wave ◽  
P Wave ◽  
Fault System ◽  
Wave Radiation ◽  
Fault Plane Solutions ◽  
Complex Deformation ◽  
Transverse Ranges

abstract The ML 6.0 Point Mugu, California earthquake of February 21, 1973 and its aftershocks occurred within the complex fault system that bounds the southern front of the Transverse Ranges province of southern California. P-wave fault plane solutions for 51 events include reverse, strike slip and normal faulting mechanisms, indicating complex deformation within the 10-km broad fault zone. Hypocenters of 141 aftershocks fail to delineate any single fault plane clearly associated with the main shock rupture. Most aftershocks cluster in a region 5 km in diameter centered 5 km from the main shock hypocenter and well beyond the extent of fault rupture estimated from analysis of body-wave radiation. Strain release within the imbricate fault zone was controlled by slip on preexisting planes of weakness under the influence of a NE-SW compressive stress.

Use of reciprocity theorem for obtaining Rayleigh wave radiation patterns

1966 ◽  
Vol 56 (4) ◽  
pp. 925-936 ◽  
Author(s):  
I. N. Gupta
Keyword(s):  
Rayleigh Wave ◽  
Rayleigh Waves ◽  
Three Dimensional ◽  
Reciprocity Theorem ◽  
Point Sources ◽  
Dimensional Case ◽  
Wave Radiation ◽  
Radiation Patterns ◽  
Homogeneous Half Space ◽  
Double Couple

abstract The reciprocity theorem is used to obtain Rayleigh wave radiation patterns from sources on the surface of or within an elastic semi-infinite medium. Nine elementary line sources first considered are: horizontal and vertical forces, horizontal and vertical double forces without moment, horizontal and vertical single couples, center of dilatation (two dimensional case), center of rotation, and double couple without moment. The results are extended to the three dimensional case of similar point sources in a homogeneous half space. Haskell's results for the radiation patterns of Rayleigh waves from a fault of arbitrary dip and direction of motion are reproduced in a much simpler manner. Numerical results on the effect of the depth of these sources on the Rayleigh wave amplitudes are shown for a solid having Poisson's ratio of 0.25.

Pipeline Design Method Against Large Displacement of Strike-Slip Fault

2016 ◽  
Author(s):  
Keita Oda ◽  
Takahiro Ishihara ◽  
Masakatsu Miyajima
Keyword(s):  
Elastic Limit ◽  
Design Method ◽  
Fem Analysis ◽  
Relative Displacement ◽  
Strike Slip ◽  
Strike Slip Fault ◽  
Large Displacement ◽  
Pipeline System ◽  
Tensile Direction ◽  
Fault Displacement

This study proposes a method for designing a water pipeline system against fault displacement by incorporating earthquake resistant ductile iron pipes (ERDIPs). An ERDIP pipeline is capable of absorb the large ground displacements that occur during severe earthquakes by movement of its joint (expansion, contraction and deflection) and the use of the joint locking system. Existing ERDIP pipelines have been exposed to several severe earthquakes such as the 1995 Kobe Earthquake and the 2011 Great East Japan Earthquake, and there has been no documentation of their failure in the last 40 years. In the case of a pipeline that crosses a fault, there is the possibility of the occurrence of a local relative displacement between the pipeline and the ground. It is known that an ERDIP pipeline withstands a fault of axial compression direction by past our study. Hence, this present study was targeted at developing a method for designing an ERDIP pipeline that is capable of withstanding a strike-slip fault of axial tensile direction for a pipeline. This was done by FEM analysis wherein the ERDIPs and spring elements were used to model the soil and ERDIP joints. An ERDIP pipeline can accommodate a fault displacement of about 2 m by joint expansion/contraction and deflection, while maintaining the stress in the pipeline within the elastic limit. However, additional countermeasure is required when the fault displacement exceeds 2 m because such could stress the pipeline beyond the elastic limit. The use of large displacement absorption unit is an effective countermeasure for displacements exceeding 2 m. The expansion/contraction capacity of a unit is 10 times that of an ERDIP joint and it is able to absorb a locally-concentrated axial displacement of the pipeline. It was confirmed in the present study that an ERDIP pipeline with large displacement absorption unit, referred to as a large displacement absorption system, could accommodate fault displacement in excess of 2 m within the elastic stress range of the pipeline.

A better strategy for interpolating gravity and magnetic data

2020 ◽  
Author(s):  
Santiago Rubén Soler ◽  
Leonardo Uieda
Keyword(s):  
Observation Point ◽  
Point Sources ◽  
Source Distribution ◽  
Magnetic Data ◽  
Relative Depth ◽  
Regular Grid ◽  
Gravity And Magnetic ◽  
Layer Technique ◽  
Gravity And Magnetic Data ◽  
Equivalent Layer

<p>We present a new strategy for gravity and magnetic data interpolation and processing. Our method is based on the equivalent layer technique (EQL) and produces more accurate interpolations when compared with similar EQL methods. It also reduces the computation time and memory requirements, both of which have been severe limiting factors.</p><p>The equivalent layer technique (also known as equivalent source, radial basis functions, or Green’s functions interpolation) is used to predict the value of gravity and magnetic fields (or transformations thereof) at any point based on the data gathered on some observation points. It consists in estimating a source distribution that produces the same field as the one measured and using this estimate to predict new values. It generally outperforms other general-purpose 2D interpolators, like the minimum curvature or bi-harmonic splines, because it takes into account the height of measurements and the fact that these fields are harmonic functions. Nevertheless, defining a layout for the source distribution used by the EQL is not trivial and plays an important role in the quality of the predictions.</p><p>The most widely used source distributions are: (a) a regular grid of point sources and (b) one point source beneath each observation point. We propose a new source distribution: (c) divide the area into blocks, calculate the average location of observation points inside each block, and place one point source beneath each average location. This produces a smaller number of point sources in comparison with the other source distributions, effectively reducing the computational load. Traditionally, the source points are located: (i) all at the same depth or (ii) each source point at a constant relative depth beneath its corresponding observation point. Besides these two, we also considered (iii) a variable relative depth for each source point proportional to the median distance to its nearest neighbours. The combination of source distributions and depth configurations leads to seven different source layouts (the regular grid is only compatible with the constant depth configuration).</p><p>We have scored the performance of each configuration by interpolating synthetic ground and airborne gravity data, and comparing the interpolation against the true values of the model. The block-averaged source layout (c) with variable relative depth (iii) produces more accurate interpolation results (R² of 0.97 versus R² of 0.63 for the traditional grid layout) in less time than the alternatives (from 2 to 10 times faster on our test cases). These results are consistent between ground and airborne survey layouts. Our conclusions can be extrapolated to other applications of equivalent layers, such as upward continuation, reduction-to-the-pole, and derivative calculation. What is more, we expect that these optimizations can benefit similar spatial prediction problems beyond gravity and magnetic data.</p><p>The source code developed for this study is based on the EQL implementation available in Harmonica (fatiando.org/harmonica), an open-source Python library for modelling and processing gravity and magnetic data.</p>

Simultaneous Noise and Performance Measurements for High-Speed Jet Noise Reduction Technologies

2009 ◽  
Author(s):  
Dean Long ◽  
Steven Martens
Keyword(s):  
Noise Reduction ◽  
High Speed ◽  
Near Field ◽  
Jet Noise ◽  
Source Distribution ◽  
Test Facility ◽  
Wave Radiation ◽  
Far Field ◽  
Shock Cell ◽  
Field Noise

Model scale tests are conducted to assess the Noise/Performance trade for high speed jet noise reduction technologies. It is demonstrated that measuring the near field acoustic signature with a microphone array can be used to assess the far field noise using a procedure known as acoustic holography. The near field noise measurement is mathematically propagated producing an estimate of the noise level at the new location. Outward propagation produces an estimate of the far field noise. Propagation toward the jet axis produces the source distribution. Tests are conducted on convergent/divergent nozzles with three different area ratios, and several different chevron geometries. Noise is characterized by two independent processes: Shock cell noise radiating in the forward quadrant is produced when the nozzle is operated at non-ideally expanded conditions. Mach wave radiation propagates into the aft quadrant when the exhaust temperature is elevated. These results show good agreement with actual far field measurements from tests in the GE Cell 41 Acoustic Test Facility. Simultaneous performance measurement shows the change in thrust coefficient for different test conditions and configurations. Chevrons attached to the nozzle exit can reduce the noise by several dB at the expense of a minimal thrust loss.

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