Fourier prestack migration by equivalent wavenumber

Geophysics ◽  
1999 ◽  
Vol 64 (1) ◽  
pp. 197-207 ◽  
Author(s):  
Gary F. Margrave ◽  
John C. Bancroft ◽  
Hugh D. Geiger
Keyword(s):  
Fourier Transform ◽  
Inverse Fourier Transform ◽  
Square Root ◽  
Constant Frequency ◽  
Prestack Migration ◽  
Change Of Variables ◽  
Time Migration ◽  
Data Volume ◽  
And Migration ◽  
Amplitude Preservation

Fourier prestack migration is reformulated through a change of variables, from offset wavenumber to a new equivalent wavenumber, which makes the migration phase shift independent of horizontal wavenumber. After the change of variables, the inverse Fourier transform over horizontal wavenumber can be performed to create unmigrated, but fully horizontally positioned, gathers at each output location. A complete prestack migration then results by imaging each gather independently with a poststack migration algorithm. This equivalent wavenumber migration (EWM) is the Fourier analog of the space‐time domain method of equivalent offset migration (EOM). The latter is a Kirchhoff time‐migration technique which forms common scatterpoint (CSP) gathers for each migrated trace and then images those gathers with a Kirchhoff summation. These CSP gathers are formed by trace mappings at constant time, and migration velocity analysis is easily done after the gathers are formed. Both EWM and EOM are motivated by the algebraic combination of a double square root equation into a single square root. This result defines equivalent wavenumber or offset. EWM is shown to be an exact reformulation of prestack f-k migration. The EWM theory provides explicit Fourier integrals for the formation of CSP gathers from the prestack data volume and the imaging of those gathers to form the final prestack migrated result. The CSP gathers are given by a Fourier mapping, at constant frequency, of the unmigrated spectrum followed by an inverse Fourier transform. The mapping requires angle‐dependent weighting factors for full amplitude preservation. The imaging expression (for each CSP gather) is formally identical to poststack migration with the result retained only at zero equivalent offset. Through a numerical simulation, the impulse responses of EOM and EWM are shown to be kinematically identical. Amplitude scale factors, which are exact in the constant velocity EWM theory, are implemented approximately in variable velocity EOM.

A REGULARIZED FOURIER TRANSFORM APPROACH FOR VALUING OPTIONS UNDER STOCHASTIC DIVIDEND YIELDS

2010 ◽  
Vol 13 (02) ◽  
pp. 211-240 ◽  
Author(s):  
BAYE M. DIA
Keyword(s):  
Fourier Transform ◽  
Option Pricing ◽  
Inverse Fourier Transform ◽  
Option Price ◽  
Square Root ◽  
Option Prices ◽  
European Options ◽  
Dividend Yield ◽  
New Approach ◽  
Main Effects

This paper studies the option pricing problem in a class of models in which dividend yields follow a time-homogeneous diffusion. Within this framework, we develop a new approach for valuing options based on the use of a regularized Fourier transform. We derive a pricing formula for European options which gives the option price in the form of an inverse Fourier transform and propose two methods for numerically implementing this formula. As an application of this pricing approach, we introduce the Ornstein-Uhlenbeck and the square-root dividend yield models in which we explicitly solve the pricing problem for European options. Finally we highlight the main effects of a stochastic dividend yield on option prices.

Efficient one‐pass 3-D time migration

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1833-1845
Author(s):  
Matthew A. Brzostowski ◽  
Fred F. C. Snyder ◽  
Patrick J. Smith
Keyword(s):  
Fourier Transform ◽  
Wave Equation ◽  
Downward Continuation ◽  
Time Dependent ◽  
Single Pass ◽  
Time Migration ◽  
Data Volume ◽  
Lateral Variations ◽  
D Wave

An efficient one‐pass 3-D time migration algorithm is introduced as an alternative to Ristow’s splitting approach. This algorithm extends Black and Leong’s [Formula: see text] approach with a time‐dependent Stolt stretch operation called dilation. Migration using [Formula: see text] dilation consists of a single pass over the 3-D data volume after [Formula: see text] slices are formed with each [Formula: see text] slice downward continued independently. A number of downward continuation algorithms based upon the 3-D wave equation may be used. Dilation accommodates any lateral variations in velocity before the 3-D data volume is decomposed into [Formula: see text] slices via a Fourier transform. An inverse dilation operation is performed after the downward‐continuation operation and after the data volume have been inverse Fourier transformed subsequently along the [Formula: see text] direction. Migration using the [Formula: see text] approach yields a one‐pass 3-D time migration algorithm that is practical and efficient where the medium velocity is smoothly varying.

A two‐pass approximation to 3-D prestack migration

Geophysics ◽  
1996 ◽  
Vol 61 (2) ◽  
pp. 409-421 ◽  
Author(s):  
Anat Canning ◽  
Gerald H. F. Gardner
Keyword(s):  
Computation Time ◽  
Three Dimensions ◽  
Velocity Analysis ◽  
Depth Migration ◽  
Prestack Migration ◽  
Time Migration ◽  
Data Volume ◽  
Land Survey ◽  
Zero Offset ◽  
The One

A two‐pass approximation to 3-D Kirchhoff migration simplifies the migration procedure by reducing it to a succession of 2-D operations. This approach has proven very successful in the zero‐offset case. A two‐pass approximation to 3-D migration is described here for the prestack case. Compared to the one‐pass approach, the scheme presented here provides significant reduction in computation time and a relatively simple data manipulation scheme. The two‐pass method was designed using velocity independent prestack time migration (DMO‐PSI) applied in the crossline direction, followed by conventional prestack depth migration in the inline direction. Velocity analysis, an important part of prestack migration, is also included in the two‐pass scheme. It is carried out as a 2-D procedure after 3-D effects are removed from the data volume. The procedure presented here is a practical full volume 3-D prestack migration. One of its main benefits is a realistic and efficient iterative velocity analysis procedure in three dimensions. The algorithm was designed in the frequency domain and the computational scheme was optimized by processing individual frequency slices independently. Irregular trace distribution, a feature that characterizes most 3-D seismic surveys, is implicitly accounted for within the two‐pass algorithm. A numerical example tests the performance of the two‐pass 3-D prestack migration program in the presence of a vertical velocity gradient. A 3-D land survey from a fold and thrust belt region was used to demonstrate the algorithm in a complex geological setting. The results were compared with images from other 2-D and 3-D migration schemes and show improved resolution and higher signal content.

The equivalent offset method of prestack time migration

Geophysics ◽  
1998 ◽  
Vol 63 (6) ◽  
pp. 2042-2053 ◽  
Author(s):  
John C. Bancroft ◽  
Hugh D. Geiger ◽  
Gary F. Margrave
Keyword(s):  
Velocity Analysis ◽  
Square Root ◽  
Prestack Migration ◽  
Time Migration ◽  
Migration Velocity Analysis ◽  
Scattered Energy ◽  
Nmo Correction ◽  
Velocity Information ◽  
Prestack Time Migration ◽  
Receiver Input

A prestack time migration is presented that is simple, efficient, and provides detailed velocity information. It is based on Kirchhoff prestack time migration and can be applied to both 2-D and 3-D data. The method is divided into two steps: the first is a gathering process that forms common scatterpoint (CSP) gathers; the second is a focusing process that applies a simplified Kirchhoff migration on the CSP gathers, and consists of scaling, filtering, normal moveout (NMO) correction, and stacking. A key concept of the method is a reformulation of the double square‐root equation (of source‐scatterpoint‐receiver traveltimes) into a single square root. The single square root uses an equivalent offset that is the surface distance from the scatterpoint to a colocated source and receiver. Input samples are mapped into offset bins of a CSP gather, without time shifting, to an offset defined by the equivalent offset. The single square‐root reformulation gathers scattered energy to hyperbolic paths on the appropriate CSP gathers. A CSP gather is similar to a common midpoint (CMP) gather as both are focused by NMO and stacking. However, the CSP stack is a complete Kirchhoff prestack migrated section, whereas the CMP stack still requires poststack migration. In addition, the CSP gather has higher fold in the offset bins and a much larger offset range due to the gathering of all input traces within the migration aperture. The new method gains computational efficiency by delaying the Kirchhoff computations until after the CSP gather has been formed. The high fold and large offsets of the CSP gather enables precise focusing of the velocity semblance and accurate velocity analysis. Our algorithm is formulated in the space‐time domain, which enables prestack migration velocity analysis to be performed at selected locations and permits prestack migration of a 3-D volume into an arbitrarily located 2-D line.

scholarly journals Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-24 ◽  
Author(s):  
David W. Pravica ◽  
Njinasoa Randriampiry ◽  
Michael J. Spurr
Keyword(s):  
Fourier Transform ◽  
Theta Function ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Inverse Fourier Transform ◽  
Recursion Relations ◽  
Jacobi Theta Function ◽  
Convergence Results ◽  
The Family ◽  
Reciprocal Symmetry

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

Effect of the Condition Number of the Inverse Fourier Transform Matrix on the Solution Behavior of the Time Spectral Equation System

2020 ◽  
Author(s):  
Sen Zhang ◽  
Dingxi Wang ◽  
Yi Li ◽  
Hangkong Wu ◽  
Xiuquan Huang
Keyword(s):  
Fourier Transform ◽  
Stability Analysis ◽  
Condition Number ◽  
Inverse Fourier Transform ◽  
Real Life ◽  
Equation System ◽  
Time Instant ◽  
Transform Matrix ◽  
Spectral Equation ◽  
The Impact

Abstract The time spectral method is a very popular reduced order frequency method for analyzing unsteady flow due to its advantage of being easily extended from an existing steady flow solver. Condition number of the inverse Fourier transform matrix used in the method can affect the solution convergence and stability of the time spectral equation system. This paper aims at evaluating the effect of the condition number of the inverse Fourier transform matrix on the solution stability and convergence of the time spectral method from two aspects. The first aspect is to assess the impact of condition number using a matrix stability analysis based upon the time spectral form of the scalar advection equation. The relationship between the maximum allowable Courant number and the condition number will be derived. Different time instant groups which lead to the same condition number are also considered. Three numerical discretization schemes are provided for the stability analysis. The second aspect is to assess the impact of condition number for real life applications. Two case studies will be provided: one is a flutter case, NASA rotor 67, and the other is a blade row interaction case, NASA stage 35. A series of numerical analyses will be performed for each case using different time instant groups corresponding to different condition numbers. The conclusion drawn from the two real life case studies will corroborate the relationship derived from the matrix stability analysis.

scholarly journals The Bernstein Projector Determined by a Weak Associate Class of Good Cosets

2021 ◽  
Author(s):  
Yeansu Kim ◽  
Loren Spice ◽  
Sandeep Varma
Keyword(s):  
Fourier Transform ◽  
Characteristic Function ◽  
Lie Algebra ◽  
Equivalence Relation ◽  
Characteristic Zero ◽  
Closed Subset ◽  
Inverse Fourier Transform ◽  
Reductive Group ◽  
Associate Class ◽  
Weak Associate

Abstract Let ${\text G}$ be a reductive group over a $p$-adic field $F$ of characteristic zero, with $p \gg 0$, and let $G={\text G}(F)$. In [ 15], J.-L. Kim studied an equivalence relation called weak associativity on the set of unrefined minimal $K$-types for ${\text G}$ in the sense of A. Moy and G. Prasad. Following [ 15], we attach to the set $\overline{\mathfrak{s}}$ of good $K$-types in a weak associate class of positive-depth unrefined minimal $K$-types a ${G}$-invariant open and closed subset $\mathfrak{g}_{\overline{\mathfrak{s}}}$ of the Lie algebra $\mathfrak{g} = {\operatorname{Lie}}({\text G})(F)$, and a subset $\tilde{{G}}_{\overline{\mathfrak{s}}}$ of the admissible dual $\tilde{{G}}$ of ${G}$ consisting of those representations containing an unrefined minimal $K$-type that belongs to $\overline{\mathfrak{s}}$. Then $\tilde{{G}}_{\overline{\mathfrak{s}}}$ is the union of finitely many Bernstein components of ${G}$, so that we can consider the Bernstein projector $E_{\overline{\mathfrak{s}}}$ that it determines. We show that $E_{\overline{\mathfrak{s}}}$ vanishes outside the Moy–Prasad ${G}$-domain ${G}_r \subset{G}$, and reformulate a result of Kim as saying that the restriction of $E_{\overline{\mathfrak{s}}}$ to ${G}_r\,$, pushed forward via the logarithm to the Moy–Prasad ${G}$-domain $\mathfrak{g}_r \subset \mathfrak{g}$, agrees on $\mathfrak{g}_r$ with the inverse Fourier transform of the characteristic function of $\mathfrak{g}_{\overline{\mathfrak{s}}}$. This is a variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan, and Y. Varshavsky in [8] for the depth-$r$ Bernstein projector.

scholarly journals On the Inverse Fourier Transform of the Planck-Einstein law

2021 ◽  
Author(s):  
Alireza Jamali
Keyword(s):  
Fourier Transform ◽  
Inverse Fourier Transform ◽  
Maximum Frequency ◽  
Principle Of Maximum

After proposing a natural metric for the space in which particles spin which implements the principle of maximum frequency, E=hf is generalised and its inverse Fourier transform is calculated.

Imaging reflection-blind areas using transmitted PS-waves

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. S241-S250 ◽  
Author(s):  
Yi Luo ◽  
Qinglin Liu ◽  
Yuchun E. Wang ◽  
Mohammed N. AlFaraj
Keyword(s):  
Mode Conversion ◽  
Imaging Techniques ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Reflected Waves ◽  
Prestack Migration ◽  
Reflection Data ◽  
Time Migration ◽  
Transmitted Waves ◽  
Vsp Data

We illustrate the use of mode-converted transmitted (e.g., PS- or SP-) waves in vertical seismic profiling (VSP) data for imaging areas above receivers where reflected waves cannot illuminate. Three depth-domain imaging techniques — move-out correction, common-depth-point (CDP) mapping, and prestack migration — are described and used for imag-ing the transmitted waves. Moveout correction converts an offset VSP trace into a zero-offset trace. CDP mapping maps each sample on an input trace to the location where the mode conversion occurs. For complex media, prestack migration (e.g., reverse-time migration) is used. By using both synthetic and field VSP data, we demonstrate that images derived from transmissions complement those from reflections. As an important application, we show that transmitted waves can illuminate zones above highly de-viated or horizontal wells, a region not imaged by reflection data. Because all of these benefits are obtained without extra data acquisition cost, we believe transmission imag-ing techniques will become widely adopted by the oil in-dustry.

Quantitative characterization, classification and reconstruction of oocyst shapes of Eimeria species from cattle

Parasitology ◽  
1998 ◽  
Vol 116 (1) ◽  
pp. 21-28 ◽  
Author(s):  
C. SOMMER
Keyword(s):  
Fourier Transform ◽  
Quantitative Data ◽  
Inverse Fourier Transform ◽  
Morphological Variability ◽  
Eimeria Species ◽  
Equal Weight ◽  
Agglomerative Clustering ◽  
Multivariate Statistical ◽  
Mean Values ◽  
Size And Shape

This study reports on morphological variability of Eimeria species, which may be given either by drawings or as quantitative data. The drawings may be used to facilitate identification by eye of ‘unknown’ Eimeria specimens, whereas quantitative data may serve as a reference set for identification by multivariate statistical techniques. The morphology of 810 Eimeria specimens was defined in binary (b/w) digital images by pixels of their oocyst outline. A Fourier transform of pixel positions yielded size and shape features. To classify coccidia, the quantitative data were employed in an agglomerative clustering by average linkage algorithm with equal weight assigned to size and shape. An inverse Fourier transform served to reconstruct oocyst outlines, i.e. outlines of average shape and size, from mean values of features in resulting clusters. Clusters were subsequently identified based on their average morphology by comparison with drawings of species in an earlier taxonomical work. Five hundred oocyst outlines were simulated for each cluster representing a species, and shape/size variability was presented in contour diagrams. Differences in species shapes, and correspondence in length and width, were seen after reconstruction by inverse Fourier transform and comparison with earlier studies.

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