Some numerical solutions for Lamb's problem

1974 ◽  
Vol 64 (2) ◽  
pp. 473-491
Author(s):  
Harold M. Mooney
Keyword(s):  
Rayleigh Wave ◽  
Poisson’S Ratio ◽  
Pulse Width ◽  
Numerical Solutions ◽  
Pulse Height ◽  
P Wave ◽  
Poisson's Ratio ◽  
Wave Form ◽  
Lamb’S Problem ◽  
Wave Forms

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.

scholarly journals A single Rayleigh mode may exist with multiple values of phase-velocity at one frequency

2020 ◽  
Vol 222 (1) ◽  
pp. 582-594
Author(s):  
Thomas Forbriger ◽  
Lingli Gao ◽  
Peter Malischewsky ◽  
Matthias Ohrnberger ◽  
Yudi Pan
Keyword(s):  
Phase Velocity ◽  
Rayleigh Wave ◽  
Half Space ◽  
Wave Velocity ◽  
Poisson’S Ratio ◽  
P Wave ◽  
Poisson's Ratio ◽  
Wave Number ◽  
Homogeneous Layer ◽  
Rayleigh Mode

SUMMARY Other than commonly assumed in seismology, the phase velocity of Rayleigh waves is not necessarily a single-valued function of frequency. In fact, a single Rayleigh mode can exist with three different values of phase velocity at one frequency. We demonstrate this for the first higher mode on a realistic shallow seismic structure of a homogeneous layer of unconsolidated sediments on top of a half-space of solid rock (LOH). In the case of LOH a significant contrast to the half-space is required to produce the phenomenon. In a simpler structure of a homogeneous layer with fixed (rigid) bottom (LFB) the phenomenon exists for values of Poisson’s ratio between 0.19 and 0.5 and is most pronounced for P-wave velocity being three times S-wave velocity (Poisson’s ratio of 0.4375). A pavement-like structure (PAV) of two layers on top of a half-space produces the multivaluedness for the fundamental mode. Programs for the computation of synthetic dispersion curves are prone to trouble in such cases. Many of them use mode-follower algorithms which loose track of the dispersion curve and miss the multivalued section. We show results for well established programs. Their inability to properly handle these cases might be one reason why the phenomenon of multivaluedness went unnoticed in seismological Rayleigh wave research for so long. For the very same reason methods of dispersion analysis must fail if they imply wave number kl(ω) for the lth Rayleigh mode to be a single-valued function of frequency ω. This applies in particular to deconvolution methods like phase-matched filters. We demonstrate that a slant-stack analysis fails in the multivalued section, while a Fourier–Bessel transformation captures the complete Rayleigh-wave signal. Waves of finite bandwidth in the multivalued section propagate with positive group-velocity and negative phase-velocity. Their eigenfunctions appear conventional and contain no conspicuous feature.

Frame moduli of unconsolidated sands and sandstones

Geophysics ◽  
1994 ◽  
Vol 59 (9) ◽  
pp. 1352-1361 ◽  
Author(s):  
James W. Spencer ◽  
Michael E. Cates ◽  
Don D. Thompson
Keyword(s):  
Poisson’S Ratio ◽  
Empirical Relation ◽  
Ultrasonic Pulse Velocity ◽  
Ultrasonic Pulse ◽  
Pulse Velocity ◽  
P Wave ◽  
Poisson's Ratio ◽  
Unconsolidated Sands ◽  
Seismic Amplitude ◽  
Seismic Frequencies

In this study, we investigate the elastic moduli of the empty grain framework (the “frame” moduli) in unconsolidated sands and consolidated sandstones. The work was done to improve the interpretation of seismic amplitude anomalies and amplitude variations with offset (AVO) associated with hydrocarbon reservoirs. We developed a laboratory apparatus to measure the frame Poisson’s ratio and Young’s modulus of unconsolidated sands at seismic frequencies (0.2 to 155 Hz) in samples approximately 11 cm long. We used ultrasonic pulse velocity measurements to measure the frame moduli of consolidated sandstones. We found that the correlation coefficient between the frame Poisson’s ratio [Formula: see text] and the mineral Poisson’s ratio [Formula: see text] is 0.84 in consolidated sandstones and only 0.28 in unconsolidated sands. The range of [Formula: see text] values in unconsolidated sands is 0.115 to 0.237 (mean = 0.187, standard deviation = 0.030), and [Formula: see text] cannot be estimated without core or log analyses. Frame moduli analyses of core samples can be used to calibrate the interpretation of seismic amplitude anomalies and AVO effects. For use in areas without core or log analyses, we developed an empirical relation that can be used to estimate [Formula: see text] in unconsolidated sands and sandstones from [Formula: see text] and the frame P‐wave modulus.

scholarly journals Fluid prediction of a deep carbonate reservoir using walkaround 3D-3C vertical seismic profiling data

2019 ◽  
Author(s):  
Haohao Zhang ◽  
Jun Lu ◽  
Benchi Chen ◽  
Xuejun Ma ◽  
Zhidong Cai
Keyword(s):  
Poisson’S Ratio ◽  
Wave Impedance ◽  
Carbonate Reservoir ◽  
P Wave ◽  
Poisson's Ratio ◽  
Inversion Method ◽  
Seismic Profiling ◽  
Tahe Oilfield ◽  
Vertical Seismic Profiling ◽  
Time Migration

Abstract The considerable depth and complicated structure of the Tahe Oilfield in the Tuofutai area of China make it very difficult to delineate its Ordovician carbonate fracture-cavity reservoir. The resolution of conventional ground seismic data is inadequate to satisfy current exploitation requirements. To further improve the understanding of the carbonate fracture-cavity reservoir of the Tahe Oilfield and to provide predictions of reservoir properties that are more accurate, a walkaround 3D-3C vertical seismic profiling (VSP) survey was conducted. First, we preprocessed raw VSP data and developed a method of joint PP- and PSV-wave prestack time migration. In contrast to ground seismic imaging profiles, VSP imaging profiles have a higher resolution and wider spectrum range that provide more detailed strata information. Then, using the joint PP- and PSV-wave prestack inversion method, we obtained the PP- and PSV-wave impedance and Poisson's ratio parameters of the Ordovician carbonate reservoir. Compared with the P-wave impedance of the ground seismic inversion, we found the VSP inversion results had higher accuracy, which enabled clearer identification of the internal characteristics of the carbonate reservoir. Finally, coupled with the Poisson's ratio attribute, we predicted the distribution of favorable reservoirs and interwell connectivity. The prediction results were verified using both logging and production data. The findings of this study demonstrate the applicability of the proposed technical method for the exploration of deep carbonate fracture-cavity reservoirs.

Elastic impedance inversion of multichannel seismic data from unconsolidated sediments containing gas hydrate and free gas

Geophysics ◽  
2004 ◽  
Vol 69 (1) ◽  
pp. 164-179 ◽  
Author(s):  
Shaoming Lu ◽  
George A. McMechan
Keyword(s):  
Elastic Properties ◽  
Gas Hydrate ◽  
Poisson’S Ratio ◽  
P Wave ◽  
Poisson's Ratio ◽  
Unconsolidated Sediments ◽  
Free Gas ◽  
Impedance Inversion ◽  
Elastic Impedance ◽  
New Algorithms

The elastic properties of hydrated sediments are not well‐known, which leads to inaccuracy in the evaluation of the amount of gas hydrate worldwide. Elastic impedance inversion is useful in estimating the elastic properties of sediments containing gas hydrate, or free gas trapped beneath the gas hydrate, from angle‐dependent P‐wave reflections. We reprocess the multichannel U.S. Geological Survey seismic line BT‐1 from the Blake Ridge off the east coast of North America to obtain migrated common‐angle aperture data sets, which are then inverted for elastic impedance. Two new algorithms to estimate P‐impedance and S‐impedance from the elastic impedance are developed and evaluated using well‐log data from Ocean Drilling Program (ODP) Leg 164; these new algorithms are stable, even in the presence of modest noise in the data. The Vs/Vp ratio, Poisson's ratio, and Lamé parameter terms λρ and λ/μ are estimated from the P‐impedance and S‐impedance. The hydrated sediments have high elastic impedance, high P‐impedance, high S‐impedance, high λρ, slightly higher Vs/Vp ratio, slightly lower Poisson's ratio, and slightly lower λ/μ values compared to those of the surrounding unhydrated sediments. The sediments containing free gas have low elastic impedance, low P‐impedance, nonanomalous background S‐impedance, high Vs/Vp ratio, low Poisson's ratio, low λρ, and low λ/μ values. We conclude that some parameters such as Vs/Vp ratio, Poisson's ratio, and λ/μ, although they help identify the free‐gas charged layers, cannot differentiate between the hydrated sediments and nonhydrated sediments when gas hydrate concentration is low, and cannot differentiate between the hydrated sediments and free‐gas charged sediments when the gas hydrate concentration is high. Three distinct layers of gas hydrate are interpreted as being caused by gas hydrates with gas of different molecular weights, with correspondingly different stability zones in depth. Free gas appears to be present below the two deeper gas‐hydrate layers, but not below the shallowest one because the lack of a trapping structure. The gas hydrate has an average concentration of ∼3–5.5% by volume, and is highest (9%) at the base of the lower gas hydrate stability zone. The free‐gas concentration ranges from 1 to 8% by volume, and is most developed beneath the local topographic high of the ocean bottom.

Seismic wave velocity investigation at The Geysers‐Clear Lake geothermal field, California

Geophysics ◽  
1982 ◽  
Vol 47 (5) ◽  
pp. 819-824 ◽  
Author(s):  
Harsh K. Gupta ◽  
Ronald W. Ward ◽  
Tzeu‐Lie Lin
Keyword(s):  
Wave Velocity ◽  
Poisson’S Ratio ◽  
Poisson Ratio ◽  
Volcanic Field ◽  
P Wave ◽  
Poisson's Ratio ◽  
Clear Lake ◽  
Seismic Wave Velocity ◽  
S Waves ◽  
The Geysers

Analysis of P‐ and S‐waves from shallow microearthquakes in the vicinity of The Geysers geothermal area, California, recorded by a dense, telemetered seismic array operated by the U.S. Geological Survey (USGS) shows that these phases are easily recognized and traced on record sections to distances of 80 km. Regional average velocities for the upper crust are estimated to be [Formula: see text] and [Formula: see text] for P‐ and S‐waves, respectively. Poisson’s ratio is estimated at 23 locations using Wadati diagrams and is found to vary from 0.13 to 0.32. In general, the Poisson’s ratio is found to be lower at the locations close to the steam production zones at The Geysers and Clear Lake volcanic field to the northeast. The low Poisson ratio corresponds to a decrease in P‐wave velocity in areas of high heat flow. The decrease may be caused by fracturing of the rock and saturation with gas or steam.

Elastic impedance parameterization and inversion with Young’s modulus and Poisson’s ratio

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. N35-N42 ◽  
Author(s):  
Zhaoyun Zong ◽  
Xingyao Yin ◽  
Guochen Wu
Keyword(s):  
Parameter Estimation ◽  
Young’S Modulus ◽  
Seismic Data ◽  
Poisson’S Ratio ◽  
Young's Modulus ◽  
Incident Angle ◽  
P Wave ◽  
Poisson's Ratio ◽  
Data Set ◽  
Elastic Impedance

Young’s modulus and Poisson’s ratio are related to quantitative reservoir properties such as porosity, rock strength, mineral and total organic carbon content, and they can be used to infer preferential drilling locations or sweet spots. Conventionally, they are computed and estimated with a rock physics law in terms of P-wave, S-wave impedances/velocities, and density which may be directly inverted with prestack seismic data. However, the density term imbedded in Young’s modulus is difficult to estimate because it is less sensitive to seismic-amplitude variations, and the indirect way can create more uncertainty for the estimation of Young’s modulus and Poisson’s ratio. This study combines the elastic impedance equation in terms of Young’s modulus and Poisson’s ratio and elastic impedance variation with incident angle inversion to produce a stable and direct way to estimate the Young’s modulus and Poisson’s ratio, with no need for density information from prestack seismic data. We initially derive a novel elastic impedance equation in terms of Young’s modulus and Poisson’s ratio. And then, to enhance the estimation stability, we develop the elastic impedance varying with incident angle inversion with damping singular value decomposition (EVA-DSVD) method to estimate the Young’s modulus and Poisson’s ratio. This method is implemented in a two-step inversion: Elastic impedance inversion and parameter estimation. The introduction of a model constraint and DSVD algorithm in parameter estimation renders the EVA-DSVD inversion more stable. Tests on synthetic data show that the Young’s modulus and Poisson’s ratio are still estimated reasonable with moderate noise. A test on a real data set shows that the estimated results are in good agreement with the results of well interpretation.

scholarly journals PPV Criterion of a Rock Slope Imbedded with a Fault subjected to Blasting P-Waves

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Shiwei Lu ◽  
Chuanbo Zhou ◽  
Zhen Zhang ◽  
Ling Ji ◽  
Nan Jiang
Keyword(s):  
Slope Stability ◽  
Young’S Modulus ◽  
Poisson’S Ratio ◽  
Rock Slope ◽  
Young's Modulus ◽  
Intact Rock ◽  
P Wave ◽  
Poisson's Ratio ◽  
Rock Slopes ◽  
Near Surface

The open-pit mining slopes continue to become higher and steeper with the continuous exploitation of near-surface mineral resources. The blasting excavation exerts a significance influence on the slope stability. In fact, intact rock slopes do not exist and slope stability is controlled by the geological defects. In this paper, the stability of a rock slope imbedded with a fault is considered. The P-wave component of blasting seismic waves is focused on and the fault is simplified as a semi-infinite crack. In background of Daye iron mine, the peak particle velocity (PPV) threshold is determined based on the linear elastic fracture dynamics. The influence of frequency, Young's modulus, and Poisson's ratio is studied to modify the PPV threshold. Results show that (1) the PPV threshold decreases with the increasing Young's modulus and Poisson's ratio, but increases with the increasing frequency; (2) the initiation angle is immune to Young's modulus and the frequency, and only depends on the Poisson's ratio; (3) the PPV criterion is finally determined as 1.47 cm/s when the frequency f ≤ 10 Hz, 1.47 cm/s–3.30 cm/s when 10 Hz < f ≤ 50 Hz and 3.37 cm/s–6.59 cm/s when f > 50 Hz, which are far less than that of intact rock slopes; (4) The north slope is quite safe if the proposed PPV threshold is not violated due to the variation range of the initiation angle θ0.

Model‐based inversion of amplitude‐variations‐with‐offset data using a genetic algorithm

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 939-954 ◽  
Author(s):  
Subhashis Mallick
Keyword(s):  
Reflection Coefficient ◽  
Acoustic Impedance ◽  
Poisson’S Ratio ◽  
Input Data ◽  
Synthetic Data ◽  
Single Layer ◽  
P Wave ◽  
Poisson's Ratio ◽  
Reasonable Accuracy ◽  
Avo Inversion

I cast the inversion of amplitude‐variation‐with‐offset (AVO) data into the framework of Bayesian statistics. Under such a framework, the model parameters and the physics of the forward problem are used to generate synthetic data. These synthetic data are then matched with the observed data to obtain an a‐posteriori probability density (PPD) function in the model space. The genetic algorithm (GA) uses a directed random search technique to estimate the shape of the PPD. Unlike the classical inversion methods, GA does not depend upon the choice of an initial model and is well suited for the AVO inversion. For the single‐layer AVO inversion where the amplitudes from a single reflection event are inverted, GA estimates the normal incidence reflection coefficient [Formula: see text] and the contrast of the Poisson’s ratio (Δσ) to reasonable accuracy, even when the signal‐to‐noise ratio is poor. Comparisons of single‐layer amplitude inversion using synthetic data demonstrate that GA inversion obtains more accurate results than does the least‐squares fit to the approximate reflection coefficients as is usually practiced in the industry. In the multilayer AVO waveform inversion, all or a part of the prestack data are inverted. Inversion of this type is nonunique for the estimation of the absolute values of velocities, Poisson’s ratios, and densities. However, by applying simplified approximations to the P‐wave reflection coefficient, I verify that [Formula: see text], the contrast in the acoustic impedance (ΔA), and the gradient in the reflection coefficient (G), can be estimated from such an inversion. From the GA estimated values of [Formula: see text], ΔA, and G, and from reliable estimates of velocity and Poisson’s ratio at the start time of the input data, an inverted model can be generated. I apply this procedure to marine data and demonstrate that the the synthetics computed from such an inverted model match the input data to reasonable accuracy. Comparison of the log data from a nearby well shows that the GA inversion obtains both the low‐ and the high‐frequency trends (within the bandwidth of seismic resolution) of the P‐wave acoustic impedance. In addition to P‐wave acoustic impedance, GA also obtains an estimate of the Poisson’s ratio, an extremely important parameter for the direct detection of hydrocarbons from seismic data.

Imaging the Moho and Main Himalayan Thrust beneath the Kumaon Himalaya: constraints from receiver function analysis

2020 ◽  
Vol 224 (2) ◽  
pp. 858-870
Author(s):  
Devajit Hazarika ◽  
Somak Hajra ◽  
Abhishek Kundu ◽  
Meena Bankhwal ◽  
Naresh Kumar ◽  
...  
Keyword(s):  
Poisson’S Ratio ◽  
Receiver Function ◽  
Function Analysis ◽  
Crustal Thickness ◽  
Receiver Functions ◽  
P Wave ◽  
Poisson's Ratio ◽  
Kumaon Himalaya ◽  
Receiver Function Analysis ◽  
Lower Crustal

SUMMARY We analyse P-wave receiver functions across the Kumaon Himalaya and adjoining area to constrain crustal thickness, intracrustal structures and seismic velocity characteristics to address the role of the underlying structure on seismogenesis and geodynamic evolution of the region. The three-component waveforms of teleseismic earthquakes recorded by a seismological network consisting of 18 broad-band seismological stations have been used for receiver function analysis. The common conversion point (CCP) depth migrated receiver function image and shear wave velocity models obtained through inversion show a variation of crustal thickness from ∼38 km in the Indo-Gangetic Plain to ∼42 km near the Vaikrita Thrust. A ramp (∼20°) structure on the Main Himalayan Thrust (MHT) is revealed beneath the Chiplakot Crystalline Belt (CCB) that facilitates the exhumation of the CCB. The geometry of the MHT observed from the receiver function image is consistent with the geometry revealed by a geological balanced cross-section. A cluster of seismicity at shallow to mid-crustal depths is detected near the MHT ramp. The spatial and depth distribution of seismicity pattern beneath the CCB and presence of steep dipping imbricate faults inferred from focal mechanism solutions suggest a Lesser Himalayan Duplex structure in the CCB above the MHT ramp. The study reveals a low-velocity zone (LVZ) with a high Poisson's ratio (σ ∼0.28–0.30) at lower crustal depth beneath the CCB. The high value of Poisson's ratio in the lower crust suggests the presence of fluid/partial melt. The shear heating in the ductile regime and/or decompression and cooling associated with the exhumation of the CCB plausibly created favorable conditions for partial melting in the lower crustal LVZ.

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