Optimization of the parameters in complex Padé Fourier finite-difference migration

Geophysics ◽  
2017 ◽  
Vol 82 (3) ◽  
pp. S259-S269 ◽  
Author(s):  
Marco Salcedo ◽  
Amélia Novais ◽  
Jörg Schleicher ◽  
Jessé C. Costa
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Rotation Angle ◽  
Computational Cost ◽  
Wide Angle ◽  
Model Data ◽  
Data Set ◽  
Dependent Parameter ◽  
The One

Complex Padé Fourier finite-difference migration is a stable one-way wave-equation technique that allows for better treatment of evanescent modes than its real counterpart, in this way producing fewer artifacts. As for real Fourier finite-difference (FFD) migration, its parameters can be optimized to improve the imaging of steeply dipping reflectors. The dip limitation of the FFD operator depends on the variation of the velocity field. We have developed a wide-angle approximation for the one-way continuation operator by means of optimization of the Padé coefficients and the most important velocity-dependent parameter. We have evaluated the achieved quality of the approximate dispersion relation in dependence on the chosen function of the ratio between the model and reference velocities under consideration of the number of terms in the Padé approximation and the branch-cut rotation angle. The optimized parameters are chosen based on the migration results and the computational cost. We found that by using the optimized parameters, a one-term expansion achieves the highest dip angles. The implementations were validated on the Marmousi data set and SEG/EAGE salt model data.

Wide-angle FD and FFD migration using complex Padé approximations

Geophysics ◽  
2007 ◽  
Vol 72 (6) ◽  
pp. S215-S220 ◽  
Author(s):  
Daniela Amazonas ◽  
Jessé C. Costa ◽  
Jörg Schleicher ◽  
Reynam Pestana
Keyword(s):  
Finite Difference ◽  
Special Treatment ◽  
Wide Angle ◽  
Model Data ◽  
Padé Approximations ◽  
Data Set ◽  
Domain Boundaries ◽  
Pade Approximations ◽  
Impulsive Response ◽  
The One

Seismic migration by downward continuation using the one-way wave-equation approximations has two shortcomings: imaging steep-dip reflectors and handling evanescent waves. Complex Padé approximations allow a better treatment of evanescent modes, stabilizing finite-difference migration without requiring special treatment for the migration domain boundaries. Imaging of steep-dip reflectors can be improved using several terms in the Padé expansion. We discuss the implementation and evaluation of wide-angle complex Padé approximations for finite-difference and Fourier finite-difference migration methods. The dispersion relation and the impulsive response of the migration operator provide criteria to select the number of terms and coefficients in the Padé expansion. This ensures stability for a prescribed maximum propagation direction. The implementations are validated on the Marmousi model data set and SEG/EAGE salt model data.

Optimum split-step Fourier 3D depth migration: Developments and practical aspects

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. S167-S175 ◽  
Author(s):  
Jianfeng Zhang ◽  
Linong Liu
Keyword(s):  
Finite Difference ◽  
Heterogeneous Media ◽  
Fourier Transforms ◽  
Computational Cost ◽  
Fourier Method ◽  
Approximate Algorithm ◽  
Wide Angle ◽  
Data Set ◽  
Depth Migration ◽  
Varying Coefficients

We present an efficient scheme for depth extrapolation of wide-angle 3D wavefields in laterally heterogeneous media. The scheme improves the so-called optimum split-step Fourier method by introducing a frequency-independent cascaded operator with spatially varying coefficients. The developments improve the approximation of the optimum split-step Fourier cascaded operator to the exact phase-shift operator of a varying velocity in the presence of strong lateral velocity variations, and they naturally lead to frequency-dependent varying-step depth extrapolations that reduce computational cost significantly. The resulting scheme can be implemented alternatively in spatial and wavenumber domains using fast Fourier transforms (FFTs). The accuracy of the first-order approximate algorithm is similar to that of the second-order optimum split-step Fourier method in modeling wide-angle propagation through strong, laterally varying media. Similar to the optimum split-step Fourier method, the scheme is superior to methods such as the generalized screen and Fourier finite difference. We demonstrate the scheme’s accuracy by comparing it with 3D two-way finite-difference modeling. Comparisons with the 3D prestack Kirchhoff depth migration of a real 3D data set demonstrate the practical application of the proposed method.

Application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide-angle wave equation

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. T313-T321 ◽  
Author(s):  
Hanming Chen ◽  
Hui Zhou ◽  
Yanqi Li
Keyword(s):  
Wave Equation ◽  
Computational Cost ◽  
Grazing Incidence ◽  
Equation System ◽  
Perfectly Matched Layer ◽  
Staggered Grid ◽  
Complex Frequency ◽  
Wide Angle ◽  
First Order ◽  
General Complex

A classical split perfectly matched layer (PML) method has recently been applied to the scalar arbitrarily wide-angle wave equation (AWWE) in terms of displacement. However, the classical split PML obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence. Our goal was to improve the computational efficiency of AWWE and to enhance the suppression of edge reflections by applying a convolutional PML (CPML). We reformulated the original AWWE as a first-order formulation and incorporated the CPML with a general complex frequency shifted stretching operator into the renewed formulation. A staggered-grid finite-difference (FD) method was adopted to discretize the first-order equation system. For wavefield depth continuation, the first-order AWWE with the CPML saved memory compared with the original second-order AWWE with the conventional split PML. With the help of numerical examples, we verified the correctness of the staggered-grid FD method and concluded that the CPML can efficiently absorb evanescent and propagating waves.

scholarly journals A Comparison of Splitting Techniques for 3D Complex Padé Fourier Finite Difference Migration

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Jessé C. Costa ◽  
Débora Mondini ◽  
Jörg Schleicher ◽  
Amélia Novais
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Computational Cost ◽  
Three Dimensional ◽  
Depth Level ◽  
Numerical Examples ◽  
The Cost ◽  
Comparable Quality ◽  
3D Problem ◽  
Dimensional Wave

Three-dimensional wave-equation migration techniques are still quite expensive because of the huge matrices that need to be inverted. Several techniques have been proposed to reduce this cost by splitting the full 3D problem into a sequence of 2D problems. We compare the performance of splitting techniques for stable 3D Fourier finite-difference (FFD) migration techniques in terms of image quality and computational cost. The FFD methods are complex Padé FFD and FFD plus interpolation, and the compared splitting techniques are two- and four-way splitting as well as alternating four-way splitting, that is, splitting into the coordinate directions at one depth and the diagonal directions at the next depth level. From numerical examples in homogeneous and inhomogeneous media, we conclude that, though theoretically less accurate, alternate four-way splitting yields results of comparable quality as full four-way splitting at the cost of two-way splitting.

scholarly journals A competitive comparison of multiparameter stacking operators

Geophysics ◽  
2017 ◽  
Vol 82 (4) ◽  
pp. V275-V283 ◽  
Author(s):  
Jan Walda ◽  
Benjamin Schwarz ◽  
Dirk Gajewski
Keyword(s):  
Optimization Technique ◽  
Computational Cost ◽  
Initial Guess ◽  
Time Shift ◽  
Data Set ◽  
Velocity Shift ◽  
Common Reflection Surface ◽  
The One ◽  
Better Than ◽  
Good Agreement

The classic common-midpoint (CMP) stack, which sums along offsets, suffers in challenging environments in which the acquisition is sparse. In the past, several multiparameter stacking techniques were introduced that incorporate many neighboring CMPs during summation. This increases data redundancy and reduces noise. Multiparameter methods that can be parameterized by the same wavefront attributes are multifocusing (MF), the common-reflection-surface (CRS), implicit CRS, and nonhyperbolic CRS (nCRS). The CRS-type operators use a velocity-shift mechanism to account for heterogeneity by changing the slope of the asymptote. On the other hand, MF uses a different mechanism: a shift of reference time while preserving the slope of the asymptote. We have formulated MF such that it uses the same mechanism as the CRS-type operators and compare them on a marine data set. In turn, we investigate the behavior of time-shifted versions of the CRS-type approximations. To provide a fair comparison, we use a global optimization technique, differential evolution, which allows to accurately estimate a solution without an initial guess solution. Our results indicate that the velocity-shift mechanism performs, in general, better than the one incorporating a time shift. The double-square-root operators are also less sensitive to the choice of aperture. They perform better in the case of diffractions than conventional hyperbolic CRS, and this fact is in good agreement with previous works. In our work the nCRS is of almost the same computational cost as that of conventional hyperbolic CRS, but it generally leads to a superior fit; therefore, we recommend its use in the future.

An implicit finite-difference operator for the Helmholtz equation

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. T97-T107 ◽  
Author(s):  
Chunlei Chu ◽  
Paul L. Stoffa
Keyword(s):  
Finite Difference ◽  
Helmholtz Equation ◽  
Frequency Domain ◽  
Difference Operator ◽  
Computational Cost ◽  
Finite Difference Discretization ◽  
Implicit Finite Difference ◽  
Explicit Finite Difference ◽  
The One ◽  
Difference Discretization

We have developed an implicit finite-difference operator for the Laplacian and applied it to solving the Helmholtz equation for computing the seismic responses in the frequency domain. This implicit operator can greatly improve the accuracy of the simulation results without adding significant extra computational cost, compared with the corresponding conventional explicit finite-difference scheme. We achieved this by taking advantage of the inherently implicit nature of the Helmholtz equation and merging together the two linear systems: one from the implicit finite-difference discretization of the Laplacian and the other from the discretization of the Helmholtz equation itself. The end result of this simple yet important merging manipulation is a single linear system, similar to the one resulting from the conventional explicit finite-difference discretizations, without involving any differentiation matrix inversions. We analyzed grid dispersions of the discrete Helmholtz equation to show the accuracy of this implicit finite-difference operator and used two numerical examples to demonstrate its efficiency. Our method can be extended to solve other frequency domain wave simulation problems straightforwardly.

scholarly journals GLEAM v3: satellite-based land evaporation and root-zone soil moisture

2016 ◽  
Author(s):  
Brecht Martens ◽  
Diego G. Miralles ◽  
Hans Lievens ◽  
Robin van der Schalie ◽  
Richard A. M. de Jeu ◽  
...  
Keyword(s):  
Soil Moisture ◽  
Eddy Covariance ◽  
Optical Depth ◽  
Root Zone ◽  
European Space Agency ◽  
Data Sets ◽  
Data Set ◽  
Previous Version ◽  
The One

Abstract. The Global Land Evaporation Amsterdam Model (GLEAM) is a set of algorithms dedicated to the estimation of terrestrial evaporation and root-zone soil moisture from satellite data. Ever since its development in 2011, the model has been regularly revised aiming at the optimal incorporation of new satellite-observed geophysical variables, and improving the representation of physical processes. In this study, the next version of this model (v3) is presented. Key changes relative to the previous version include: (1) a revised formulation of the evaporative stress, (2) an optimized drainage algorithm, and (3) a new soil moisture data assimilation system. GLEAM v3 is used to produce three new data sets of terrestrial evaporation and root-zone soil moisture, including a 35-year data set spanning the period 1980–2014 (v3.0a, based on satellite-observed soil moisture, vegetation optical depth and snow water equivalents, reanalysis air temperature and radiation, and a multi-source precipitation product), and two fully satellite-based data sets. The latter two share most of their forcing, except for the vegetation optical depth and soil moisture products, which are based on observations from different passive and active C- and L-band microwave sensors (European Space Agency Climate Change Initiative data sets) for the first data set (v3.0b, spanning the period 2003–2015) and observations from the Soil Moisture and Ocean Salinity satellite in the second data set (v3.0c, spanning the period 2011–2015). These three data sets are described in detail, compared against analogous data sets generated using the previous version of GLEAM (v2), and validated against measurements from 64 eddy-covariance towers and 2338 soil moisture sensors across a broad range of ecosystems. Results indicate that the quality of the v3 soil moisture is consistently better than the one from v2: average correlations against in situ surface soil moisture measurements increase from 0.61 to 0.64 in case of the v3.0a data set and the representation of soil moisture in the second layer improves as well, with correlations increasing from 0.47 to 0.53. Similar improvements are observed for the two fully satellite-based data sets. Despite regional differences, the quality of the evaporation fluxes remains overall similar as the one obtained using the previous version of GLEAM, with average correlations against eddy-covariance measurements between 0.78 and 0.80 for the three different data sets. These global data sets of terrestrial evaporation and root-zone soil moisture are now openly available at http://GLEAM.eu and may be used for large-scale hydrological applications, climate studies and research on land-atmosphere feedbacks.

Finite‐difference migration derived from the Kirchhoff‐Helmholtz integral

Geophysics ◽  
1996 ◽  
Vol 61 (5) ◽  
pp. 1394-1399 ◽  
Author(s):  
Thomas Rühl
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Continued Fraction ◽  
Velocity Fields ◽  
Finite Difference Schemes ◽  
Square Root ◽  
Continued Fraction Expansion ◽  
The One ◽  
Root Operator ◽  
Approximate Version

Finite‐difference (FD) migration is one of the most often used standard migration methods in practice. The merit of FD migration is its ability to handle arbitrary laterally and vertically varying macro velocity fields. The well‐known disadvantage is that wave propagation is only performed accurately in a more or less narrow cone around the vertical. This shortcoming originates from the fact that the exact one‐way wave equation can be implemented only approximately in finite‐difference schemes because of economical reasons. The Taylor or continued fraction expansion of the square root operator in the one‐way wave equation must be truncated resulting in an approximate version of the one‐way wave equation valid only for a restricted angle range.

An optimized wave equation for seismic modeling and reverse time migration

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA153-WCA158 ◽  
Author(s):  
Faqi Liu ◽  
Guanquan Zhang ◽  
Scott A. Morton ◽  
Jacques P. Leveille
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Computational Cost ◽  
Numerical Dispersion ◽  
New Wave ◽  
Acoustic Wave Equation ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration

The acoustic wave equation has been widely used for the modeling and reverse time migration of seismic data. Numerical implementation of this equation via finite-difference techniques has established itself as a valuable approach and has long been a favored choice in the industry. To ensure quality results, accurate approximations are required for spatial and time derivatives. Traditionally, they are achieved numerically by using either relatively very fine computation grids or very long finite-difference operators. Otherwise, the numerical error, known as numerical dispersion, is present in the data and contaminates the signals. However, either approach will result in a considerable increase in the computational cost. A simple and computationally low-cost modification to the standard acoustic wave equation is presented to suppress numerical dispersion. This dispersion attenuator is one analogy of the antialiasing operator widely applied in Kirchhoff migration. When the new wave equation is solved numerically using finite-difference schemes, numerical dispersion in the original wave equation is attenuated significantly, leading to a much more accurate finite-difference scheme with little additional computational cost. Numerical tests on both synthetic and field data sets in both two and three dimensions demonstrate that the optimized wave equation dramatically improves the image quality by successfully attenuating dispersive noise. The adaptive application of this new wave equation only increases the computational cost slightly.

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