Reversed time migration in spatial frequency domain

Geophysics ◽  
1983 ◽  
Vol 48 (5) ◽  
pp. 627-635 ◽  
Author(s):  
Dan Loewenthal ◽  
Irshad R. Mufti
Keyword(s):  
Finite Difference ◽  
Acoustic Waves ◽  
Seismic Waves ◽  
Finite Difference Methods ◽  
Computation Time ◽  
Real Data ◽  
Coordinate Frame ◽  
Seismic Section ◽  
Time Migration ◽  
Similar Scheme

During the past decade, finite‐difference methods have become important tools for direct modeling of seismic data as well as for certain interpretation processes. One of the earliest applications of these methods to seismics is the pioneering contribution of Alterman who, in a series of papers (Alterman and Karal, 1968; Alterman and Aboudi, 1968; Alterman and Rotenberg, 1969; Alterman and Loewenthal, 1972) demonstrated the usefulness of such numerical computations for the propagation of seismic waves in elastic media. A clear exposition of these techniques, as well as a comparison of results obtained from them with the corresponding analytical solutions, can be found in Alterman and Karal (1968). This subject was further developed and extended to more complicated models by Boore (1970), Ottaviani (1971), and Kelly et al (1976). Claerbout introduced a somewhat different finite‐difference approach (Claerbout, 1970; Claerbout and Johnson, 1971) for modeling the acoustic waves which often dominate the reflection seismogram. In his approach, the original wave equation, which governs the propagation of the acoustic waves, is modified in such a way so as to allow the propagation of either only upcoming or only downgoing waves. By moving the coordinate frame with the downgoing waves, Claerbout showed that one could greatly reduce computation time. Using the same concepts, he showed (Claerbout and Doherty, 1972) how to use a similar scheme for migrating a seismic section by downward continuation of the upcoming waves. This migration method is an interesting extension of the ideas of Hagedoorn (1954) and was found to be extremely useful with real data (Larner and Hatton, 1976; Loewenthal et al, 1976).

Implicit interpolation in reverse‐time migration

Geophysics ◽  
1997 ◽  
Vol 62 (3) ◽  
pp. 906-917 ◽  
Author(s):  
Jinming Zhu ◽  
Larry R. Lines
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Real Data ◽  
Seismic Sources ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Difference Wave ◽  
Time Migration ◽  
Seismic Acquisition ◽  
Recorded Data

Reverse‐time migration applies finite‐difference wave equation solutions by using unaliased time‐reversed recorded traces as seismic sources. Recorded data can be sparsely or irregularly sampled relative to a finely spaced finite‐difference mesh because of the nature of seismic acquisition. Fortunately, reliable interpolation of missing traces is implicitly included in the reverse‐time wave equation computations. This implicit interpolation is essentially based on the ability of the wavefield to “heal itself” during propagation. Both synthetic and real data examples demonstrate that reverse‐time migration can often be performed effectively without the need for explicit interpolation of missing traces.

Mimetic seismic wave modeling including topography on deformed staggered grids

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. T125-T141 ◽  
Author(s):  
Josep de la Puente ◽  
Miguel Ferrer ◽  
Mauricio Hanzich ◽  
José E. Castillo ◽  
José M. Cela
Keyword(s):  
Finite Difference ◽  
Seismic Wave ◽  
Seismic Waves ◽  
Numerical Solutions ◽  
Surface Condition ◽  
Finite Difference Methods ◽  
Staggered Grid ◽  
Grid Deformation ◽  
Large Dispersion ◽  
Difference Methods

Finite-difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography, due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography. As a consequence, alternatives to finite-difference methods have been proposed to circumvent these issues. We present a new numerical scheme for 3D elastic wave propagation in the presence of strong topography. This finite-difference scheme is based upon a staggered grid of the Lebedev type, or fully staggered grid (FSG). It uses a grid deformation strategy to make a regular Cartesian grid conform to a topographic surface. In addition, the scheme uses a mimetic approach to accurately solve the free-surface condition and hence allows for a less restrictive grid spacing criterion in the computations. The scheme can use high-order operators for the spatial derivatives and obtain low-dispersion results with as few as six points per minimum wavelength. A series of tests in 2D and 3D scenarios, in which our results are compared to analytical and numerical solutions obtained with other numerical approaches, validate the accuracy of our scheme. The resulting FSG mimetic scheme allows for accurate and efficient seismic wave modelling in the presence of very rough topographies with the advantage of using a structured staggered grid.

3-D interpretation of reflected arrival times by finite‐difference techniques

Geophysics ◽  
1994 ◽  
Vol 59 (5) ◽  
pp. 844-849 ◽  
Author(s):  
M. Ali Riahi ◽  
Christopher Juhlin
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Real Data ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Computationally Efficient ◽  
Reflected Waves ◽  
Arrival Times ◽  
First Arrival ◽  
Difference Methods

Finite‐difference methods have generally been used to solve dynamic wave propagation problems over the last 25 years (Alterman and Karal, 1968; Boore, 1972; Kelly et al., 1976; and Levander, 1988). Recently, finite‐difference methods have been applied to the eikonal equation to calculate the kinematic solution to the wave equation (Vidale, 1988 and 1990; Podvin and Lecomte, 1991; Van Trier and Symes, 1991; Qin et al., 1992). The calculation of the first‐arrival times using this method has proven to be considerably faster than using classical ray tracing, and problems such as shadow zones, multipathing, and barrier penetration are easily handled. Podvin and Lecomte (1991) and Matsuoka and Ezaka (1992) extended and expanded upon Vidale’s (1988) algorithm to calculate traveltimes for reflected waves in two dimensions. Based on finite‐difference calculations for first‐arrival times, Hole et al. (1992) devised a scheme for inverting synthetic and real data to estimate the depth to refractors in the crust in three dimensions. The method of Hole et al. (1992) for inversion is computationally efficient since it avoids the matrix inversion of many of the published schemes for refraction and reflection traveltime data (Gjøystdal and Ursin, 1981).

Cascadic Newton’s method for the elliptic Monge–Ampère equation

2020 ◽  
Vol 18 (03) ◽  
pp. 2050018
Author(s):  
Qin Li ◽  
Zhiyong Liu
Keyword(s):  
Finite Difference ◽  
Newton’S Method ◽  
Newton's Method ◽  
Finite Difference Methods ◽  
Computation Time ◽  
Type Interpolation ◽  
Local Type ◽  
Cascadic Multigrid ◽  
Monge Ampere ◽  
Prolongation Operator

In this paper, a cascadic Newton’s method is designed to solve the Monge–Ampère equation. In the process of implementing the cascadic multigrid, we use the Full-Local type interpolation as prolongation operator and Newton iteration as smoother. In order to obtain Full-Local type interpolation, we provide several finite difference stencils. Especially, the skewed finite difference methods are first applied by us for the elliptic Monge–Ampère equation. Based on Full-Local interpolation techniques and cascade principle, the new algorithm can save a large amount of computation time. Some numerical experiments are provided to confirm the efficiency of our proposed method.

scholarly journals Staggered-Grid Finite Difference Method with Variable-Order Accuracy for Porous Media

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Jinghuai Gao ◽  
Yijie Zhang
Keyword(s):  
Porous Media ◽  
Finite Difference ◽  
Finite Difference Methods ◽  
Computation Time ◽  
High Order ◽  
Staggered Grid ◽  
Difference Operators ◽  
Variable Order ◽  
Low Velocity ◽  
Order Accuracy

The numerical modeling of wave field in porous media generally requires more computation time than that of acoustic or elastic media. Usually used finite difference methods adopt finite difference operators with fixed-order accuracy to calculate space derivatives for a heterogeneous medium. A finite difference scheme with variable-order accuracy for acoustic wave equation has been proposed to reduce the computation time. In this paper, we develop this scheme for wave equations in porous media based on dispersion relation with high-order staggered-grid finite difference (SFD) method. High-order finite difference operators are adopted for low-velocity regions, and low-order finite difference operators are adopted for high-velocity regions. Dispersion analysis and modeling results demonstrate that the proposed SFD method can decrease computational costs without reducing accuracy.

Finite difference automatically generated non-uniform grids in the numerical solution of the Reynolds' compressible one-dimensional squeeze-film equation

2003 ◽  
Vol 217 (3) ◽  
pp. 243-249 ◽  
Author(s):  
E. A. M. Almas ◽  
F. A. P. De Silva
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Computation Time ◽  
Squeeze Film ◽  
Uniform Grid ◽  
Efficient Computation ◽  
Uniform Grids ◽  
Uniform Model ◽  
Squeeze Number ◽  
Solution Accuracy

With the Reynolds equation, for compressible squeeze-film thrust bearings, the use of a finite difference discretization with a grid of nodes having a non-uniform spacing can result in a more efficient computation of the solution. Comparative tests of two variable spatial grid models against a uniform model were conducted with the classical finite difference methods for chosen combinations of squeeze number and excursion ratio values. For problems with a high value of σ, one of the non-uniform grid models has shown some advantages over the uniform model, requiring a smaller number of nodes and less computation time with the same solution accuracy.

Application of Backus average to simulate acoustic logs in thinly bedded formations

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. D93-D104
Author(s):  
Elsa Maalouf ◽  
Carlos Torres-Verdín ◽  
Jingxuan Li
Keyword(s):  
Finite Difference ◽  
Numerical Simulations ◽  
Finite Difference Methods ◽  
Homogeneous Medium ◽  
Computation Time ◽  
Transversely Isotropic ◽  
Simulation Method ◽  
Thin Layers ◽  
Homogeneous Layer ◽  
Spatial Averaging

Slowness logs acquired in layered formations are not only affected by spatial averaging associated with the borehole acoustic tool. Layers with thicknesses smaller than the acoustic wavelength can cause measurable effects on the associated wave propagation phenomena. While spatial averaging functions can be used to model tool averaging effects, computer-intensive numerical methods such as finite differences must be used to simulate slowness logs across formations with thin layers. We adopted Backus averaging as a faster alternative to model borehole slownesses when layer thicknesses are smaller than the acoustic wavelength (i.e., in the long-wavelength limit). Using synthetic models and numerical simulations via finite-element and finite-difference methods, we have determined that borehole slownesses of a stack of horizontal layers first approach the average slowness of the individual layers. However, as the layer thickness decreases, sonic slownesses approach the slowness of a homogeneous medium with elastic properties obtained from the Backus average. Therefore, to model acoustic logs acquired in layered formations, we first approximate thin layers as a single homogeneous layer with stiffness coefficients calculated using the Backus average. Next, we apply a spatial averaging function to reproduce the spatial averaging effect inherent to borehole acoustic tools. Results indicate that the latter method is accurate and efficient for fast modeling borehole slownesses of formations with thin layers that are isotropic and intrinsically vertical transversely isotropic. The fast simulation method decreases computation time by at least a factor of 10 and yields slowness logs with a relative error below 2% compared with finite-difference numerical simulations. We also determine that the moving Backus average that is typically applied to upscale acoustic logs for seismic applications is not accurate to model borehole acoustic logs acquired across thinly layered formations.

Full elastic finite-difference modeling and interpretation of karst system in a subsalt carbonate reservoir

2014 ◽  
Vol 2 (1) ◽  
pp. T49-T56 ◽  
Author(s):  
Xin Zhan ◽  
Xinding Fang ◽  
Reza Daneshvar ◽  
Enru Liu ◽  
Christopher E. Harris
Keyword(s):  
Finite Difference ◽  
Seismic Data ◽  
Primary Objective ◽  
Carbonate Reservoir ◽  
P Wave ◽  
Seismic Section ◽  
Uncertainty Range ◽  
Time Migration ◽  
The Impact ◽  
Seismic Images

We evaluated an advanced forward-modeling-based reservoir characterization technique that uses full elastic finite-difference simulation to investigate the limits of karst identification in stacked seismic data. Identification of karsts is important for field development in carbonated reservoirs because paleokarst features can result in a loss of circulation and/or sometimes lost drill bits. Our primary objective was to verify whether we can detect and interpret the location and size of karsts from seismic data, especially given a complex overburden. We constructed an elastic reservoir model consisting of compressional velocity ([Formula: see text]), shear velocity ([Formula: see text]), and density for the study area using interpreted horizons and well log information. Karsts with varying widths, thicknesses, dip angles, and porosities were inserted to generate multiple versions of the model. We also evaluated the imaging impact of overlying faults and salt on karst detection. Full elastic simulation was performed on the various reservoir models using a realistic acquisition geometry to generate gathers, which were then prestack time migrated to quantify the impact of different karst properties on the seismic images and study the effect of reservoir property changes on the seismic response. Finally, a wave-equation target-oriented analysis was presented to improve the understanding of subsalt amplitude and illumination. From the finite-difference modeling and analysis that we performed, we obtained an uncertainty range on karst property estimation from seismic images and gained insights into future survey design for subsalt interpretation and amplitude analysis. For our specific model, we found the limit of karst identification from seismic data is a 30-m-wide horizontal karst or a 500 m karst dipping at 60°. Also, the karst image width reflected its true width only when the actual karst width was larger than the P-wave wavelength (240 m in this case). With a dipping overburden above the reservoir, apparent positions of karsts were shifted in the updip direction by prestack time migration up to 50 m from their true position. This lateral uncertainty should be kept in mind in well planning to avoid karst features interpreted from a time-migrated seismic section.

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