Quadratic wavefront and traveltime approximations in inhomogeneous layered media with curved interfaces

Geophysics ◽  
1982 ◽  
Vol 47 (7) ◽  
pp. 1012-1021 ◽  
Author(s):  
Bjørn Ursin
Keyword(s):  
Taylor Series ◽  
Layered Medium ◽  
Source Region ◽  
Three Dimensional ◽  
Homogeneous Medium ◽  
Quadratic Approximation ◽  
Reference Source ◽  
Layered Model ◽  
Zero Offset ◽  
Receiver Point

A quadratic approximation for the square of the traveltime from a source region to a receiver region is given for a three‐dimensional (3-D) medium consisting of inhomogeneous layers with curved interfaces. The square of the traveltime, being a function of source and receiver coordinates, is developed in a Taylor series around a reference source and receiver point. The relationships of the traveltime parameters to the ray parameters and the wavefront curvature matrices are shown. Using midpoint, half‐offset coordinates gives a simplified traveltime function compared to using source‐receiver coordinates only in the case that the reference source point and the reference receiver point coincide (zero‐offset approximation). For a medium consisting of homogeneous layers with plane dipping interfaces, the traveltime approximation is further simplified. The derived traveltime approximation is shown to be exact for a reflection from a plane dipping interface in a homogeneous medium. Explicit expressions for the traveltime parameters in terms of the layer parameters are derived for a horizontally layered medium. The traveltime errors of two different approximations are compared for a given layered model in a numerical example.

Investigations of the Adhesive Contact Behavior of Elastic Layered Media With Surface Roughness

2019 ◽  
Vol 141 (4) ◽  
Author(s):  
Yuyan Zhang ◽  
Lina Si ◽  
Xiaoqing Zhang ◽  
Juan Li ◽  
Wanjun Wang
Keyword(s):  
Surface Roughness ◽  
Layered Medium ◽  
Three Dimensional ◽  
Homogeneous Medium ◽  
Surface Force ◽  
Adhesive Contact ◽  
Diamond Like Carbon ◽  
Contact Behavior ◽  
Average Value ◽  
Homogeneous Half Space

A deterministic adhesive model for the contact between an elastic layered medium with surface roughness and a smooth elastic microsphere was developed on the basis of the Lennard–Jones surface force law. Through numerical simulations, the adhesive contact behavior of the layered medium with the measured three-dimensional (3D) surface topography was comparatively analyzed with that of the homogeneous medium. Furthermore, the contact characteristics of the layered medium with pre-assigned roughness parameters were investigated with the aid of a computer-generated technique for simulating surface roughness. Results showed that the pull-off force for the contact problem involving rough surfaces was influenced by the contact location, and the average value for the contact between an alumina (SiO2) microsphere and a diamond-like carbon/silicon (DLC/Si)-layered medium was smaller than that for the contact between a SiO2 microsphere and a Si homogeneous half-space. In addition, the effect of the diamond-like carbon (DLC) layer on reducing adhesion was smaller than that of the surface roughness. Finally, the average pull-off force for a DLC/Si-layered medium with computer-generated surface roughness rapidly decreased; however, it eventually became almost unchangeable with the increase in the root-mean-square (RMS) deviation.

Traveltime and wavefront curvature calculations in three‐dimensional inhomogeneous layered media with curved interfaces

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1466-1494 ◽  
Author(s):  
H. Gjøystdal ◽  
J. E. Reinhardsen ◽  
B. Ursin
Keyword(s):  
Taylor Series ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Normal Incidence ◽  
Reference Source ◽  
Second Derivative ◽  
Second Derivatives ◽  
Derivative Matrix ◽  
Three Dimensional Models ◽  
Derivatives Of

The seismic rays and wavefront curvatures are determined by solving a system of nonlinear ordinary differential equations. For media with constant velocity and for media with constant velocity gradient, simplified solutions exist. In a general inhomogeneous medium these equations must be solved by numerical approximations. The integration of the ray‐tracing and wavefront curvature equations is then performed by a modified divided difference form of the Adams PECE (Predict‐Evaluate‐Correct‐Evaluate) formulas and local extrapolation. The interfaces between the layers are represented by bicubic splines. The changes in ray direction and wavefront curvature at the interfaces are computed using standard formulas. For three‐dimensional media, two quadratic traveltime approximations have been proposed. Both are based on a Taylor series expansion with reference to a ray from a reference source point to a reference receiver point. The first approximation corresponds to expanding the square of the traveltime in a Taylor series and taking the square root of the result. The second approximation corresponds to expanding the traveltime in a Taylor series. The two traveltime approximations may be expressed in source‐receiver coordinates or in midpoint‐half‐offset coordinates. Simplified expressions are obtained when the reference source and receiver coincide, giving zero‐offset approximations, for which the reference ray is a normal‐incidence ray. A new method is proposed for computing the second derivatives of the normal‐incidence traveltime with respect to the source‐receiver midpoint coordinates. By considering a beam of normal‐incidence rays it is shown that the second‐derivative matrix may be found by computing the wavefront curvature along a reference normal‐incidence ray starting at the reflection point with the wavefront curvature equal to the curvature of the reflecting interface. From this second‐derivative matrix the normal moveout velocity can be computed for any seismic line through the reference source‐receiver midpoint. It is also shown how a reverse wavefront curvature calculation may be used, in a time‐to‐depth migration scheme, to compute the curvature of the reflecting interface from the estimated second derivatives of the normal‐incidence traveltime. Numerical results for different three‐dimensional models indicate that the first traveltime approximation, based on an expansion of the square of the traveltime, is the most accurate for shallow reflectors and for simple models. For deeper reflectors the two approximations give comparable results, and for models with complicated velocity variations the second approximation may be slightly better than the first one, depending on the particular model chosen. A simplified traveltime approximation may be used in a three‐dimensional seismic velocity analysis. Instead of estimating the stacking velocity one must estimate three elements in a [Formula: see text] symmetric matrix. The accuracy and range of validity of the simplified traveltime approximation are investigated for different three‐dimensional models.

Synthetic reflection seismograms in three dimensions by a locked‐mode approximation

Geophysics ◽  
1989 ◽  
Vol 54 (3) ◽  
pp. 350-358 ◽  
Author(s):  
G. Nolet ◽  
R. Sleeman ◽  
V. Nijhof ◽  
B. L. N. Kennett
Keyword(s):  
Finite Difference ◽  
Born Approximation ◽  
Large Scale ◽  
Layered Medium ◽  
Three Dimensional ◽  
Simple Algorithm ◽  
Realistic Model ◽  
Three Dimensions ◽  
Acoustic Response ◽  
Many Sources

We present a simple algorithm for computing the acoustic response of a layered structure containing three‐dimensional (3-D) irregularities, using a locked‐mode approach and the Born approximation. The effects of anelasticity are incorporated by use of Rayleigh’s principle. The method is particularly attractive at somewhat larger offsets, but computations for near‐source offsets are stable as well, due to the introduction of anelastic damping. Calculations can be done on small minicomputers. The algorithm developed in this paper can be used to calculate the response of complicated models in three dimensions. It is more efficient than any other method whenever many sources are involved. The results are useful for modeling, as well as for generating test signals for data processing with realistic, model‐induced “noise.” Also, this approach provides an alternative to 2-D finite‐difference calculations that is efficient enough for application to large‐scale inverse problems. The method is illustrated by application to a simple 3-D structure in a layered medium.

Elastic solution of a polyhedral particle with a polynomial eigenstrain and particle discretization

2021 ◽  
pp. 1-35
Author(s):  
Chunlin Wu ◽  
Liangliang Zhang ◽  
Huiming Yin
Keyword(s):  
Taylor Series ◽  
Three Dimensional ◽  
Stress Singularity ◽  
Elastic Field ◽  
Analytical Form ◽  
Tetrahedral Elements ◽  
Equivalent Inclusion Method ◽  
Equivalent Inclusion ◽  
Eshelby’S Tensor ◽  
Domain Discretization

Abstract The paper extends the recent work (JAM, 88, 061002, 2021) of the Eshelby's tensors for polynomial eigenstrains from a two dimensional (2D) to three dimensional (3D) domain, which provides the solution to the elastic field with continuously distributed eigenstrain on a polyhedral inclusion approximated by the Taylor series of polynomials. Similarly, the polynomial eigenstrain is expanded at the centroid of the polyhedral inclusion with uniform, linear and quadratic order terms, which provides tailorable accuracy of the elastic solutions of polyhedral inhomogeneity by using Eshelby's equivalent inclusion method. However, for both 2D and 3D cases, the stress distribution in the inhomogeneity exhibits a certain discrepancy from the finite element results at the neighborhood of the vertices due to the singularity of Eshelby's tensors, which makes it inaccurate to use the Taylor series of polynomials at the centroid to catch the eigenstrain at the vertices. This paper formulates the domain discretization with tetrahedral elements to accurately solve for eigenstrain distribution and predict the stress field. With the eigenstrain determined at each node, the elastic field can be predicted with the closed-form domain integral of Green's function. The parametric analysis shows the performance difference between the polynomial eigenstrain by the Taylor expansion at the centroid and the 𝐶0 continuous eigenstrain by particle discretization. Because the stress singularity is evaluated by the analytical form of the Eshelby's tensor, the elastic analysis is robust, stable and efficient.

Preliminary results of numerical simulation of submarine landslide-generated waves

2021 ◽  
Author(s):  
Ramtin Sabeti ◽  
Mohammad Heidarzadeh
Keyword(s):  
Numerical Simulation ◽  
Numerical Modelling ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Source Region ◽  
Three Dimensional ◽  
Terminal Velocity ◽  
Sensitivity Analyses ◽  
Physical Experiments ◽  
Set Up

<p>Landslide-generated waves have been major threats to coastal areas and have led to destruction and casualties. Their importance is undisputed, most recently demonstrated by the 2018 Anak Krakatau tsunami, causing several hundred fatalities. The accurate prediction of the maximum initial amplitude of landslide waves (<em>η<sub>max</sub></em>) around the source region is a vital hazard indicator for coastal impact assessment. Laboratory experiments, analytical solutions and numerical modelling are three major methods to investigate the (<em>η<sub>max</sub></em>). However, the numerical modelling approach provides a more flexible and cost- and time-efficient tool. This research presents a numerical simulation of tsunamis due to rigid landslides with consideration of submerged conditions. In particular, this simulation focuses on studying the effect of landslide parameters on <em>η<sub>max</sub>.</em> Results of simulations are compared with our conducted physical experiments at the Brunel University London (UK) to validate the numerical model.</p><p>We employ the fully three-dimensional computational fluid dynamics package, FLOW-3D Hydro for modelling the landslide-generated waves. This software benefit from the Volume of Fluid Method (VOF) as the numerical technique for tracking and locating the free surface. The geometry of the simulation is set up according to the wave tank of physical experiments (i.e. 0.26 m wide, 0.50 m deep and 4.0 m). In order to calibrate the simulation model based on the laboratory measurements, the friction coefficient between solid block and incline is changed to 0.41; likewise, the terminal velocity of the landslide is set to 0.87 m/s. Good agreement between the numerical solutions and the experimental results is found. Sensitivity analyses of landslide parameters (e.g. slide volume, water depth, etc.) on <em>η<sub>max </sub></em>are performed. Dimensionless parameters are employed to study the sensitivity of the initial landslide waves to various landslide parameters.</p>

scholarly journals TSUNAMI GENERATION AND PROPAGATION

1972 ◽  
Vol 1 (13) ◽  
pp. 146
Author(s):  
Joseph L. Hammack ◽  
Frederic Raichlen
Keyword(s):  
Linear Theory ◽  
Source Region ◽  
Three Dimensional ◽  
Frequency Dispersion ◽  
Nonlinear Effects ◽  
Its Region ◽  
Experimental Results ◽  
Specific Class ◽  
Two Dimensional ◽  
Three Dimensional Models

A linear theory is presented for waves generated by an arbitrary bed deformation {in space and time) for a two-dimensional and a three -dimensional fluid domain of uniform depth. The resulting wave profile near the source is computed for both the two and three-dimensional models for a specific class of bed deformations; experimental results are presented for the two-dimensional model. The growth of nonlinear effects during wave propagation in an ocean of uniform depth and the corresponding limitations of the linear theory are investigated. A strategy is presented for determining wave behavior at large distances from the source where linear and nonlinear effects are of equal magnitude. The strategy is based on a matching technique which employs the linear theory in its region of applicability and an equation similar to that of Korteweg and deVries (KdV) in the region where nonlinearities are equal in magnitude to frequency dispersion. Comparison of the theoretical computations with the experimental results indicates that an equation of the KdV type is the proper model of wave behavior at large distances from the source region.

A Large Measuring Range Profilometer for Three-Dimensional Surface Topography Measurement

2007 ◽  
Vol 364-366 ◽  
pp. 750-755 ◽  
Author(s):  
Xu Dong Yang ◽  
Jia Chun Li ◽  
Tie Bang Xie
Keyword(s):  
Surface Topography ◽  
Three Dimensional ◽  
Displacement Sensor ◽  
Resistance Force ◽  
Sampling Point ◽  
Measuring Range ◽  
3D Surface Topography ◽  
Control Circuits ◽  
Industrial Control Computer ◽  
Zero Offset

A novel profilometer for three-dimensional (3D) surface topography measurement is presented. The profilometer has large measuring range, high precision and small measuring touch force. It is composed of a two-dimensional (2D) displacement sensor, a 3D platform based on vertical scanning, measuring and control circuits and an industrial control computer. When a workpiece is measured, the vertical undulation of the profile at a sampling point leads to a zero offset of the 2D displacement sensor. According to the zero offset, a piezoelectric actuator and a servo motor drive the vertical scanning platform to move vertically to ensure that the lever returns to its balance position. So the non-linear error caused by the rotation of the lever is very small even if the measuring range is large. When the stylus barges up against a steep wall, the horizontal resistance force results in another zero offset of the 2D displacement sensor. If the zero offset exceeds a quota, the vertical scanning platform descends to make the stylus climb the steep wall successfully. According to the theoretical and experimental analysis, the profilometer can measure roughness, profile of sphere, step, groove and other 3D surfaces with curvature precisely.

Large‐scale three‐dimensional seismic models and their interpretive significance

Geophysics ◽  
1990 ◽  
Vol 55 (9) ◽  
pp. 1166-1182 ◽  
Author(s):  
Irshad R. Mufti
Keyword(s):  
Finite Difference ◽  
Wave Field ◽  
Large Scale ◽  
Three Dimensional ◽  
Synthetic Data ◽  
Real Data ◽  
The Real ◽  
Need To Evaluate ◽  
Zero Offset ◽  
Seismic Models

Finite‐difference seismic models are commonly set up in 2-D space. Such models must be excited by a line source which leads to different amplitudes than those in the real data commonly generated from a point source. Moreover, there is no provision for any out‐of‐plane events. These problems can be eliminated by using 3-D finite‐difference models. The fundamental strategy in designing efficient 3-D models is to minimize computational work without sacrificing accuracy. This was accomplished by using a (4,2) differencing operator which ensures the accuracy of much larger operators but requires many fewer numerical operations as well as significantly reduced manipulation of data in the computer memory. Such a choice also simplifies the problem of evaluating the wave field near the subsurface boundaries of the model where large operators cannot be used. We also exploited the fact that, unlike the real data, the synthetic data are free from ambient noise; consequently, one can retain sufficient resolution in the results by optimizing the frequency content of the source signal. Further computational efficiency was achieved by using the concept of the exploding reflector which yields zero‐offset seismic sections without the need to evaluate the wave field for individual shot locations. These considerations opened up the possibility of carrying out a complete synthetic 3-D survey on a supercomputer to investigate the seismic response of a large‐scale structure located in Oklahoma. The analysis of results done on a geophysical workstation provides new insight regarding the role of interference and diffraction in the interpretation of seismic data.

Three-Dimensional Finite Element Analysis of Elastic-Plastic Layered Media Under Thermomechanical Surface Loading

2002 ◽  
Vol 125 (1) ◽  
pp. 52-59 ◽  
Author(s):  
N. Ye ◽  
K. Komvopoulos
Keyword(s):  
Finite Element ◽  
Layered Medium ◽  
Numerical Solutions ◽  
Equivalent Stress ◽  
Three Dimensional ◽  
Layered Media ◽  
Thin Layers ◽  
Surface Loading ◽  
Elastic Plastic ◽  
Thermomechanical Surface

The simultaneous effects of mechanical and thermal surface loadings on the deformation of layered media were analyzed with the finite element method. A three-dimensional model of an elastic sphere sliding over an elastic-plastic layered medium was developed and validated by comparing finite element results with analytical and numerical solutions for the stresses and temperature distribution at the surface of an elastic homogeneous half-space. The evolution of deformation in the layered medium due to thermomechanical surface loading is interpreted in light of the dependence of temperature, von Mises equivalent stress, first principal stress, and equivalent plastic strain on the layer thickness, Peclet number, and sliding distance. The propensity for plastic flow and microcracking in the layered medium is discussed in terms of the thickness and thermal properties of the layer, sliding speed, medium compliance, and normal load. It is shown that frictional shear traction and thermal loading promote stress intensification and plasticity, especially in the case of relatively thin layers exhibiting low thermal conductivity.

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