Suppressing the unwanted reflections of the full wave equation

Geophysics ◽  
1987 ◽  
Vol 52 (7) ◽  
pp. 1007-1012 ◽  
Author(s):  
Dan Loewenthal ◽  
Paul L. Stoffa ◽  
Eduardo L. Faria
Keyword(s):  
Wave Equation ◽  
Phase Shift ◽  
Wave Fields ◽  
Shift Method ◽  
Depth Direction ◽  
Full Wave ◽  
And Migration ◽  
The One ◽  
Phase Shift Method ◽  
Natural Way

In many instances in exploration geophysics we are interested in the so‐called one‐way wave equation. This equation allows the wave fields to propagate in the positive depth direction, but not in the reverse (−Z) direction. Some modeling and migration methods, such as the f-k method (Stolt, 1978) and the phase‐shift method (Gazdag, 1978), produce in a natural way the one‐way wave equation. The main advantage of the one‐way wave equation is that it does not give rise to multiples or interlayer reverberations and enables the observation of primary events only.

Migration of seismic data by phase shift plus interpolation

Geophysics ◽  
1984 ◽  
Vol 49 (2) ◽  
pp. 124-131 ◽  
Author(s):  
Jeno Gazdag ◽  
Piero Sguazzero
Keyword(s):  
Phase Shift ◽  
Wave Field ◽  
Seismic Data ◽  
Three Dimensional ◽  
Reference Wave ◽  
Second Step ◽  
Lateral Velocity ◽  
Wave Fields ◽  
Shift Method ◽  
Phase Shift Method

Under the horizontally layered velocity assumption, migration is defined by a set of independent ordinary differential equations in the wavenumber‐frequency domain. The wave components are extrapolated downward by rotating their phases. This paper shows that one can generalize the concepts of the phase‐shift method to media having lateral velocity variations. The wave extrapolation procedure consists of two steps. In the first step, the wave field is extrapolated by the phase‐shift method using ℓ laterally uniform velocity fields. The intermediate result is ℓ reference wave fields. In the second step, the actual wave field is computed by interpolation from the reference wave fields. The phase shift plus interpolation (PSPI) method is unconditionally stable and lends itself conveniently to migration of three‐dimensional data. The performance of the methods is demonstrated on synthetic examples. The PSPI migration results are then compared with those obtained from a finite‐difference method.

Wave equation migration with the phase‐shift method

Geophysics ◽  
1978 ◽  
Vol 43 (7) ◽  
pp. 1342-1351 ◽  
Author(s):  
Jenö Gazdag
Keyword(s):  
Wave Equation ◽  
Phase Shift ◽  
Finite Difference Methods ◽  
Migration Process ◽  
Asymptotic Equation ◽  
Shift Method ◽  
Seismic Records ◽  
Zero Offset ◽  
Degree Equation ◽  
Phase Shift Method

Accurate methods for the solution of the migration of zero‐offset seismic records have been developed. The numerical operations are defined in the frequency domain. The source and recorder positions are lowered by means of a phase shift, or a rotation of the phase angle of the Fourier coefficients. For applications with laterally invariant velocities, the equations governing the migration process are solved very accurately by the phase‐shift method. The partial differential equations considered include the 15 degree equation, as well as higher order approximations to the exact migration process. The most accurate migration is accomplished by using the asymptotic equation, whose dispersion relation is the same as that of the full wave equation for downward propagating waves. These equations, however, do not account for the reflection and transmission effects, multiples, or evanescent waves. For comparable accuracy, the present approach to migration is expected to be computationally more efficient than finite‐difference methods in general.

A HIGHLY ACCURATE ULTRASONIC MEASUREMENT SYSTEM FOR TREMOR USING BINARY AMPLITUDE-SHIFT-KEYING AND PHASE-SHIFT METHOD

2003 ◽  
Vol 15 (02) ◽  
pp. 61-67 ◽  
Author(s):  
MENG-HSIANG YANG ◽  
K. N. HUANG ◽  
C. F. HUANG ◽  
S. S. HUANG ◽  
M. S. YOUNG
Keyword(s):  
Phase Shift ◽  
Measurement System ◽  
Low Cost ◽  
Phase Shifts ◽  
Ultrasonic Waves ◽  
Shift Method ◽  
Narrow Bandwidth ◽  
Shift Keying ◽  
Modulation Signal ◽  
Phase Shift Method

A highly accurate Binary Amplitude-Shift-Keyed (BASK) ultrasonic tremor measurement system for use in isothermal air is developed. In this paper, we present a simple but efficient algorithm based upon phase shifts generated by three ultrasonic waves of different frequencies. By the proposed method, we can conduct larger range measurement than the phase-shift method and also get higher accuracy compared with the time-of-flight (TOF) method. Our microcomputer-based system includes two important parts. One of which is BASK modulation signal generator. The other is a phase meter designed to record and compute the phase shifts of the three different frequencies and the result motion is then sent to either an LCD for display or a PC for calibration. Experiments are done in the laboratory using BASK modulation for the frequencies of 200 Hz and 1 kHz with a 40 kHz carrier. The measurement accuracy of this measurement system in the reported experiments is within +/- 0.98 mm. The main advantages of this ultrasonic tremor measurement system are high resolution, narrow bandwidth requirement, low cost, and easy to be implemented.

A digital phase shift method for phase compensation of electronic transformer

2017 ◽  
Vol 40 (13) ◽  
pp. 3690-3695 ◽  
Author(s):  
Wei Wei ◽  
Han-miao Cheng ◽  
Fan Li ◽  
Deng-ping Tang ◽  
Shui-bin Xia
Keyword(s):  
Phase Shift ◽  
Phase Error ◽  
Recursion Formula ◽  
Rogowski Coil ◽  
Shift Method ◽  
Electronic Transformer ◽  
Digital Phase ◽  
Fitting Algorithm ◽  
Fixed Phase ◽  
Phase Shift Method

When sampling analog signal, the electronic transformer generally produces a fixed phase error that will compromise the measurement accuracy and require a phase shift method for correction. In this paper, we propose a digital phase shift method based on least squares fitting algorithm and derive the recursion formula of digital phase shift. The simulation has also been done to analysis its performance. The result shows that the method has high phase shift resolution and precision. By applying the method to an electronic transformer based on Rogowski coil, we have experimentally verified the feasibility and validity of the method.

Sign in / Sign up

Export Citation Format

Share Document