Volumetric wavefield recording and wave equation inversion for near‐surface material properties

Geophysics ◽  
2002 ◽  
Vol 67 (5) ◽  
pp. 1602-1611 ◽  
Author(s):  
Andrew Curtis ◽  
Johan O. A. Robertsson
Keyword(s):  
Wave Equation ◽  
Material Properties ◽  
Surface Condition ◽  
Elastic Wave Equation ◽  
Wave Type ◽  
Near Surface ◽  
The Earth ◽  
Spatial Derivatives ◽  
Seismic Wavefield ◽  
Derivatives Of

“Volumetric recording” of the seismic wavefield implies that the local receiver group or array approximately encloses a volume of the earth. We show how volumetric recording can be used to measure several spatial derivatives of the wavefield. By making use of the full elastic wave equation, the free surface condition on elastic wavefields, and derivative centering techniques analagous to Lax‐Wendroff corrections used in synthetic finite‐difference modeling, these derivative estimates can be inverted for P‐ and S‐velocities in the near surface directly beneath the receiver group. The quantities estimated are the effective velocities of the P‐ and S‐components experienced by the wavefield at any point in time. Hence, the velocity estimates may vary with both wave type and wavelength. The estimates may be useful to aid statics estimation and are exactly the effective velocities required for separation of the wavefield into P‐ and S‐, and up‐ and down‐going components.

scholarly journals Traveltime calculations from frequency‐domain downward‐continuation algorithms

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1380-1388 ◽  
Author(s):  
Changsoo Shin ◽  
Seungwon Ko ◽  
Wonsik Kim ◽  
Dong‐Joo Min ◽  
Dongwoo Yang ◽  
...  
Keyword(s):  
Wave Equation ◽  
Frequency Domain ◽  
Downward Continuation ◽  
Calculation Algorithm ◽  
Near Surface ◽  
Absolute Value ◽  
Head Waves ◽  
First Arrival ◽  
The One ◽  
Derivatives Of

We present a new, fast 3D traveltime calculation algorithm that employs existing frequency‐domain wave‐equation downward‐continuation software. By modifying such software to solve for a few complex (rather than real) frequencies, we are able to calculate not only the first arrival and the approximately most energetic traveltimes at each depth point but also their corresponding amplitudes. We compute traveltimes by either taking the logarithm of displacements obtained by the one‐way wave equation at a frequency or calculating derivatives of displacements numerically. Amplitudes are estimated from absolute value of the displacement at a frequency. By using the one‐way downgoing wave equation, we also circumvent generating traveltimes corresponding to near‐surface upcoming head waves not often needed in migration. We compare the traveltimes computed by our algorithm with those obtained by picking the most energetic arrivals from finite‐difference solutions of the one‐way wave equation, and show that our traveltime calculation method yields traveltimes comparable to solutions of the one‐way wave equation. We illustrate the accuracy of our traveltime algorithm by migrating the 2D IFP Marmousi and the 3D SEG/EAGE salt models.

On the mechanism of earthquakes

1964 ◽  
Vol 54 (5A) ◽  
pp. 1283-1289
Author(s):  
M. J. Randall
Keyword(s):  
Wave Equation ◽  
General Solution ◽  
Elastic Wave ◽  
Sudden Change ◽  
Elastic Equilibrium ◽  
Equilibrium Calculation ◽  
Elastic Wave Equation ◽  
The Earth ◽  
New Form ◽  
Elastostatic Problem

Abstract An earthquake may be regarded as resulting from a sudden change in the condition of elastic equilibrium in the Earth. A new form of the general solution of the elastic wave equation relates seismic radiation to displacement from equilibrium. Calculation of the radiation pattern for a proposed mechanism is thus reduced to an elastostatic problem.

A generalization of the Fourier pseudospectral method

Geophysics ◽  
2010 ◽  
Vol 75 (6) ◽  
pp. A53-A56 ◽  
Author(s):  
José M. Carcione
Keyword(s):  
Wave Equation ◽  
Fractional Laplacian ◽  
Differential Operators ◽  
Density Wave ◽  
Pseudospectral Method ◽  
Uniform Density ◽  
Fourier Pseudospectral Method ◽  
Spatial Derivatives ◽  
Fourier Pseudospectral ◽  
Derivatives Of

The Fourier pseudospectral (PS) method is generalized to the case of derivatives of nonnatural order (fractional derivatives) and irrational powers of the differential operators. The generalization is straightforward because the calculation of the spatial derivatives with the fast Fourier transform is performed in the wavenumber domain, where the operator is an irrational power of the wavenumber. Modeling constant-[Formula: see text] propagation with this approach is highly efficient because it does not require memory variables or additional spatial derivatives. The classical acoustic wave equation is modified by including those with a space fractional Laplacian, which describes wave propagation with attenuation and velocity dispersion. In particular, the example considers three versions of the uniform-density wave equation, based on fractional powers of the Laplacian and fractional spatial derivatives.

The wave equation in generalized coordinates

Geophysics ◽  
1994 ◽  
Vol 59 (12) ◽  
pp. 1911-1919 ◽  
Author(s):  
José M. Carcione
Keyword(s):  
Wave Equation ◽  
Pseudospectral Method ◽  
Special Treatment ◽  
Lamb’S Problem ◽  
Generalized Coordinates ◽  
Topographic Features ◽  
Boundary Treatment ◽  
Collocation Scheme ◽  
Spatial Derivatives ◽  
Derivatives Of

This work introduces a spectral collocation scheme for the viscoelastic wave equation transformed from Cartesian to generalized coordinates. Both the spatial derivatives of field variables and the metrics of the transformation are calculated by the Chebychev pseudospectral method. The technique requires a special treatment of the boundary conditions, which is based on 1-D characteristics normal to the boundaries. The numerical solution of Lamb’s problem requires two 1-D stretching transformations for each Cartesian direction. The results show excellent agreement between the elastic numerical and analytical solutions, demonstrating the effectiveness of the differential operator and boundary treatment. Another example, requiring 1-D transformations, tests the propagation of a Rayleigh wave around a corner of the numerical mesh. Two‐dimensional transformations adapt the grid to topographic features: a syncline, and an anticlinal structure formed with fine layers.

Phase correction in separating P‐ and S‐waves in elastic data

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1515-1518 ◽  
Author(s):  
Robert Sun ◽  
Jinder Chow ◽  
Kuang‐Jung Chen
Keyword(s):  
Wave Equation ◽  
Velocity Model ◽  
Acoustic Wave Equation ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Wave Type ◽  
P Waves ◽  
S Waves ◽  
Elastic Data ◽  
Wavefield Extrapolation

Two‐dimensional elastic data containing reflected P‐waves and converted S‐wave generated by a P‐source may be separated using dilatation and rotation calculation (Sun, 1999). The algorithm is a combination of elastic full wavefield extrapolation (Sun and McMechan, 1986; Chang and McMechan, 1987, 1994) and wave‐type separation using dilatation (divergence) and rotation (curl) calculations (Dellinger and Etgen, 1990). It includes (1) downward extrapolating the (multicomponent) elastic data in an elastic velocity model using the elastic wave equation, (2) calculating the dilatation to represent pure P‐waves and calculating the rotation to represent pure S‐waves at some depth, and (3) upward extrapolating the dilatation in a P‐velocity model and upward extrapolating the rotation in an S‐velocity model, using the acoustic wave equation for each.

Elastic wave equation traveltime and waveform inversion of crosswell data

Geophysics ◽  
1997 ◽  
Vol 62 (3) ◽  
pp. 853-868 ◽  
Author(s):  
Changxi Zhou ◽  
Gerard T. Schuster ◽  
Sia Hassanzadeh ◽  
Jerry M. Harris
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Input Data ◽  
Equivalent Stress ◽  
Waveform Inversion ◽  
Low Frequency ◽  
Median Filtering ◽  
Elastic Wave Equation ◽  
The Earth ◽  
Stress Components

A method is presented for reconstructing P‐ and S‐velocity distributions from elastic traveltimes and waveforms. The input data consist of crosswell hydrophone records generated by a piezoelectric borehole source. Borehole effects are partially accounted for by using a low‐frequency Green's function to simulate the pressure generated in the fluid‐filled receiver well. The tube waves in the borehole are ignored, on the assumption that they can be removed from the field data by median filtering. In addition, the source‐radiation pattern is partially taken into account by inverting for the equivalent stress components acting on the earth at the source location. The elastic wave equation traveltime and waveform inversion (WTW) method is applied to both synthetic crosswell data and the McElroy field crosswell data. As predicted by theory, results show that elastic WTW tomograms provide a sharper interface image than delineated in the traveltime tomograms. The spatial resolution of the McElroy traveltime tomogram is about 20 m compared to about 3 m and 1.5 m, respectively, for the associated P‐ and S‐velocity WTW tomograms. From these tomograms, detailed porosity maps of the interwell geology are constructed. There is a very good correlation between the P‐velocity tomograms and the P‐velocity log profiles, and there is a good correlation between the smooth parts of the S‐velocity tomogram and the S‐velocity logs. Unfortunately, the high‐wavenumber parts of the S‐velocity tomograms do not correlate well with the high‐wavenumber parts of the S‐velocity logs. We believe this problem is partly caused by not taking into account attenuation effects in the WTW algorithm.

Vibration attenuation of a two-link flexible arm carried by a translational stage

2018 ◽  
Vol 24 (23) ◽  
pp. 5650-5664 ◽  
Author(s):  
Shang–Teh Wu ◽  
Shan-Qun Tang ◽  
Kuan–Po Huang
Keyword(s):  
Feedback Loop ◽  
Control Method ◽  
Flexible Manipulator ◽  
Closed Loop System ◽  
Flexible Arm ◽  
Vibration Attenuation ◽  
Shaft Encoder ◽  
High Frequency Vibrations ◽  
Spatial Derivatives ◽  
Derivatives Of

This paper investigates the vibration control of a two-link flexible manipulator carried by a translational stage. The first and the second links are each driven by a stage motor and a joint motor. By treating the joint motor as a virtual spring, the two-link manipulator can be regarded as an integral flexible arm driven by the stage motor. A noncollocated controller is devised based on feedback from the deflection of the virtual spring, which can be measured by a shaft encoder. Stability of the closed-loop system is analyzed by examining the spatial derivatives of the modal functions. By including a bandpass filter in the feedback loop, residual vibrations can be attenuated without exciting high-frequency vibrations. The control method is simple to implement; its effectiveness is confirmed by simulation and experimental results.

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