Fresnel volume migration of multicomponent data

Geophysics ◽  
2005 ◽  
Vol 70 (6) ◽  
pp. S121-S129 ◽  
Author(s):  
Stefan Lüth ◽  
Stefan Buske ◽  
Rüdiger Giese ◽  
Alexander Goertz
Keyword(s):  
Heterogeneous Media ◽  
Specular Reflection ◽  
Image Point ◽  
Polarization Angle ◽  
Depth Migration ◽  
Migration Operator ◽  
Measured Polarization ◽  
Paraxial Ray ◽  
Multicomponent Data ◽  
Homogeneous Media

If the aperture of a seismic reflection experiment is strongly limited, Kirchhoff migration suffers from strong artifacts attributable to incomplete summation. This can be overcome by restricting the migration operator to the region that physically contributes to a reflection event. Examples of such limited-aperture experiments include data acquisition in boreholes, tunnels, and mines. We present an extension to three-component (3C) Kirchhoff prestack depth migration, where the migration operator is restricted to the Fresnel volume of the specular reflected raypath. We use the measured polarization direction at a 3C receiver to determine points of specular reflection. In homogeneous media, the polarization angle of 3C data can be used directly to decide whether a certain image point belongs to the Fresnel volume of a specular reflection. In heterogeneous media, the Fresnel volume around an image point is approximated by means of paraxial ray tracing. The method is tested on a synthetic vertical seismic profiling experiment with strongly limited aperture. Migration artifacts and crosstalk effects from converted waves are strongly reduced compared with standard migration schemes. The method is successfully applied to seismic data acquired in a tunnel.

True‐amplitude weight functions in 3D limited‐aperture migration revisited

Geophysics ◽  
2004 ◽  
Vol 69 (4) ◽  
pp. 1025-1036 ◽  
Author(s):  
Jianguo Sun
Keyword(s):  
Weight Function ◽  
Order Approximation ◽  
Velocity Model ◽  
Seismic Signal ◽  
Weight Functions ◽  
Image Point ◽  
Depth Migration ◽  
Amplitude Distortion ◽  
Pulse Distortion ◽  
Paraxial Ray

The true‐amplitude weight function in 3D limited‐aperture migration is obtained by extending its formula at an actual reflection point to any arbitrary subsurface point. This implies that the recorded seismic signal is a delta impulse. When the weight function is used in depth migration, it results in an amplitude distortion depending on the vertical distance from the target reflector. This distortion exists even if the correct velocity model is used. If the image point lies at a depth shallower than the half‐offset, the distortion cannot be ignored, even for a spatial wavelet having a short length. Using paraxial ray theory, I find a formula for the true‐amplitude weight function causing no amplitude distortion, under the condition that the earth's surface is smoothly curved. However, the formula is reflector dependent. As a result, amplitude distortion, in parallel with pulse distortion, is an intrinsic effect in depth migration, and true‐amplitude migration without amplitude distortion is possible only when the position of the target reflector is known. If this is the case, true‐amplitude migration without amplitude distortion can be realized by filtering the output of a simple unweighted diffraction stack with the weight function presented here. Also, using Taylor expansions with respect to the vertical, I derive an alternative formula for the true‐amplitude weight function that causes no amplitude distortion. Starting from this formula, I show that the previously published reflector‐independent true‐amplitude weight function is a zero‐order approximation to the one given here.

Kinematic interpretation in the prestack depth‐migrated domain

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1226-1237 ◽  
Author(s):  
Irina Apostoiu‐Marin ◽  
Andreas Ehinger
Keyword(s):  
Signal To Noise Ratio ◽  
Velocity Model ◽  
Point Of View ◽  
Prestack Depth Migration ◽  
Depth Migration ◽  
Velocity Models ◽  
Time Domain Data ◽  
And Migration ◽  
Point To Point ◽  
Homogeneous Media

Prestack depth migration can be used in the velocity model estimation process if one succeeds in interpreting depth events obtained with erroneous velocity models. The interpretational difficulty arises from the fact that migration with erroneous velocity does not yield the geologically correct reflector geometries and that individual migrated images suffer from poor signal‐to‐noise ratio. Moreover, migrated events may be of considerable complexity and thus hard to identify. In this paper, we examine the influence of wrong velocity models on the output of prestack depth migration in the case of straight reflector and point diffractor data in homogeneous media. To avoid obscuring migration results by artifacts (“smiles”), we use a geometrical technique for modeling and migration yielding a point‐to‐point map from time‐domain data to depth‐domain data. We discover that strong deformation of migrated events may occur even in situations of simple structures and small velocity errors. From a kinematical point of view, we compare the results of common‐shot and common‐offset migration. and we find that common‐offset migration with erroneous velocity models yields less severe image distortion than common‐shot migration. However, for any kind of migration, it is important to use the entire cube of migrated data to consistently interpret in the prestack depth‐migrated domain.

Leading-order seismic imaging using curvelets

Geophysics ◽  
2007 ◽  
Vol 72 (6) ◽  
pp. S231-S248 ◽  
Author(s):  
Huub Douma ◽  
Maarten V. de Hoop
Keyword(s):  
Seismic Data ◽  
Order Approximation ◽  
Angular Frequency ◽  
Leading Order ◽  
Numerical Examples ◽  
Depth Migration ◽  
Map Migration ◽  
Kirchhoff Depth Migration ◽  
Homogeneous Media ◽  
Spatial Support

Curvelets are plausible candidates for simultaneous compression of seismic data, their images, and the imaging operator itself. We show that with curvelets, the leading-order approximation (in angular frequency, horizontal wavenumber, and migrated location) to common-offset (CO) Kirchhoff depth migration becomes a simple transformation of coordinates of curvelets in the data, combined with amplitude scaling. This transformation is calculated using map migration, which employs the local slopes from the curvelet decomposition of the data. Because the data can be compressed using curvelets, the transformation needs to be calculated for relatively few curvelets only. Numerical examples for homogeneous media show that using the leading-order approximation only provides a good approximation to CO migration for moderate propagation times. As the traveltime increases and rays diverge beyond the spatial support of a curvelet; however, the leading-order approximation is no longer accurate enough. This shows the need for correction beyond leading order, even for homogeneous media.

Offset‐domain pseudoscreen prestack depth migration

Geophysics ◽  
2002 ◽  
Vol 67 (6) ◽  
pp. 1895-1902 ◽  
Author(s):  
Shengwen Jin ◽  
Charles C. Mosher ◽  
Ru‐Shan Wu
Keyword(s):  
Heterogeneous Media ◽  
Fourier Transforms ◽  
Data Migration ◽  
Computationally Efficient ◽  
Lateral Velocity ◽  
Prestack Depth Migration ◽  
Depth Migration ◽  
Narrow Angle ◽  
Domain Phase ◽  
Wavefield Extrapolation

The double square root equation for laterally varying media in midpoint‐offset coordinates provides a convenient framework for developing efficient 3‐D prestack wave‐equation depth migrations with screen propagators. Offset‐domain pseudoscreen prestack depth migration downward continues the source and receiver wavefields simultaneously in midpoint‐offset coordinates. Wavefield extrapolation is performed with a wavenumber‐domain phase shift in a constant background medium followed by a phase correction in the space domain that accommodates smooth lateral velocity variations. An extra wide‐angle compensation term is also applied to enhance steep dips in the presence of strong velocity contrasts. The algorithm is implemented using fast Fourier transforms and tri‐diagonal matrix solvers, resulting in a computationally efficient implementation. Combined with the common‐azimuth approximation, 3‐D pseudoscreen migration provides a fast wavefield extrapolation for 3‐D marine streamer data. Migration of the 2‐D Marmousi model shows that offset domain pseudoscreen migration provides a significant improvement over first‐arrival Kirchhoff migration for steeply dipping events in strong contrast heterogeneous media. For the 3‐D SEG‐EAGE C3 Narrow Angle synthetic dataset, image quality from offset‐domain pseudoscreen migration is comparable to shot‐record finite‐difference migration results, but with computation times more than 100 times faster for full aperture imaging of the same data volume.

Kirchhoff modeling, inversion for reflectivity, and subsurface illumination

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1195-1209 ◽  
Author(s):  
Bertrand Duquet ◽  
Kurt J. Marfurt ◽  
Joe A. Dellinger
Keyword(s):  
Least Squares ◽  
A Priori ◽  
Computational Cost ◽  
Image Point ◽  
Surface Sampling ◽  
Constrained Least Squares ◽  
Amplitude Variation With Offset ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Least Squares Migration

Because of its computational efficiency, prestack Kirchhoff depth migration is currently one of the most popular algorithms used in 2-D and 3-D subsurface depth imaging. Nevertheless, Kirchhoff algorithms in their typical implementation produce less than ideal results in complex terranes where multipathing from the surface to a given image point may occur, and beneath fast carbonates, salt, or volcanics through which ray‐theoretical energy cannot penetrate to illuminate underlying slower‐velocity sediments. To evaluate the likely effectiveness of a proposed seismic‐acquisition program, we could perform a forward‐modeling study, but this can be expensive. We show how Kirchhoff modeling can be defined as the mathematical transpose of Kirchhoff migration. The resulting Kirchhoff modeling algorithm has the same low computational cost as Kirchhoff migration and, unlike expensive full acoustic or elastic wave‐equation methods, only models the events that Kirchhoff migration can image. Kirchhoff modeling is also a necessary element of constrained least‐squares Kirchhoff migration. We show how including a simple a priori constraint during the inversion (that adjacent common‐offset images should be similar) can greatly improve the resulting image by partially compensating for irregularities in surface sampling (including missing data), as well as for irregularities in ray coverage due to strong lateral variations in velocity and our failure to account for multipathing. By allowing unstacked common‐offset gathers to become interpretable, the additional cost of constrained least‐squares migration may be justifiable for velocity analysis and amplitude‐variation‐with‐offset studies. One useful by‐product of least‐squares migration is an image of the subsurface illumination for each offset. If the data are sufficiently well sampled (so that including the constraint term is not necessary), the illumination can instead be calculated directly and used to balance the result of conventional migration, obtaining most of the advantages of least‐squares migration for only about twice the cost of conventional migration.

On Fermat’s principle and Snell’s law in lossy anisotropic media

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. T107-T116 ◽  
Author(s):  
José M. Carcione ◽  
Bjorn Ursin
Keyword(s):  
Electromagnetic Waves ◽  
Heterogeneous Media ◽  
Transversely Isotropic ◽  
Slowness Vector ◽  
First Case ◽  
Fermat’S Principle ◽  
Snell’S Law ◽  
Snell's Law ◽  
Fermat's Principle ◽  
Homogeneous Media

Fermat’s principle of least action is one of the methods used to trace rays in inhomogeneous media. Its form is the same in anisotropic elastic and anelastic media, with the difference that the velocity depends on frequency in the latter case. Moreover, the ray, envelope, and energy velocities replace the group velocity because this concept has no physical meaning in anelastic media. We have first considered a lossy (anelastic) anisotropic medium and established the equivalence between Fermat’s principle and Snell’s law in homogeneous media. Then, we found that the different ray velocities defined in the literature were the same for stationary rays in homogeneous media, with phase and inhomogeneity angles satisfying the principle and the law. We considered an example of a transversely isotropic medium with a vertical symmetry axis and wavelike and diffusionlike properties. In the first case, the differences were negligible, which was the case of real rocks having a quality factor greater than five. Strictly, ray tracing should be based on the so-called stationary complex slowness vector to obtain correct results, although the use of homogeneous viscoelastic waves (zero inhomogeneity angle) is acceptable as an approximation for earth materials. However, from a rigorous point of view, the three velocities introduced in the literature to define the rays present discrepancies in heterogeneous media, although the differences are too small to be measured in earth materials. The findings are also valid for electromagnetic waves by virtue of the acoustic-electromagnetic analogy.

Adaptive focusing window for seismic angle migration

Geophysics ◽  
2009 ◽  
Vol 74 (1) ◽  
pp. S1-S10 ◽  
Author(s):  
Mathias Alerini ◽  
Bjørn Ursin
Keyword(s):  
Synthetic Data ◽  
Main Idea ◽  
Three Dimensions ◽  
User Friendliness ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Migration Operator ◽  
Local Geology ◽  
Original Approach

Kirchhoff migration is based on a continuous integral ranging from minus infinity to plus infinity. The necessary discretization and truncation of this integral introduces noise in the migrated image. The attenuation of this noise has been studied by many authors who propose different strategies. The main idea is to limit the migration operator around the specular point. This means that the specular point must be known before migration and that a criterion exists to determine the size of the migration operator. We propose an original approach to estimate the size of the focusing window, knowing the geologic dip. The approach benefits from the use of prestack depth migration in angle domain, which is recognized as the most artifact-free Kirchhoff-type migration. The main advantages of the method are ease of implementation in an existing angle-migration code (two or three dimensions), user friendliness, ability to take into account multiorientation of the local geology as in faulted regions, and flexibility with respect to the quality of the estimated geologic dip field. Common-image gathers resulting from the method are free from migration noise and can be postprocessed in an easier way. We validate the approach and its possibilities on synthetic data examples with different levels of complexity.

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.

scholarly journals A Significant Detection of X-ray Polarization in Sco X-1 with PolarLight and Constraints on the Corona Geometry

2022 ◽  
Vol 924 (1) ◽  
pp. L13
Author(s):  
Xiangyun Long ◽  
Hua Feng ◽  
Hong Li ◽  
Jiahuan Zhu ◽  
Qiong Wu ◽  
...  
Keyword(s):  
Polarization Measurement ◽  
Polarization Angle ◽  
X Ray ◽  
Astrophysical Source ◽  
Accretion Rates ◽  
Measured Polarization ◽  
Low Mass ◽  
Marginal Detection ◽  
Energy Dependent ◽  
Polarization Fraction

Abstract We report the detection of X-ray polarization in the neutron-star low-mass X-ray binary Scorpius (Sco) X-1 with PolarLight. The result is energy-dependent, with a nondetection in 3–4 keV but a 4σ detection in 4–8 keV; it is also flux-dependent in the 4–8 keV band, with a nondetection when the source displays low fluxes but a 5σ detection during high fluxes, in which case we obtain a polarization fraction of 0.043 ± 0.008 and a polarization angle of 52.°6 ± 5.°4. This confirms a previous marginal detection with OSO-8 in the 1970s and marks Sco X-1 as the second astrophysical source with a significant polarization measurement in the keV band. The measured polarization angle is in line with the jet orientation of the source on the sky plane (54°), which is supposedly the symmetry axis of the system. Combining previous spectral analysis, our measurements suggest that an optically thin corona is located in the transition layer under the highest accretion rates, and disfavor the extended accretion disk corona model.

Case Study: Improving the quality of the seismic reflection image for a Fujian mineral exploration data set with offset-domain common-image gathers

Geophysics ◽  
2021 ◽  
pp. 1-45
Author(s):  
Guofeng Liu ◽  
Xiaohong Meng ◽  
Johanes Gedo Sea
Keyword(s):  
Wave Equation ◽  
Image Quality ◽  
Seismic Data ◽  
Seismic Reflection ◽  
Mineral Exploration ◽  
Velocity Model ◽  
Image Point ◽  
Single Shot ◽  
Data Set ◽  
Depth Migration

Seismic reflection is a proven and effective method commonly used during the exploration of deep mineral deposits in Fujian, China. In seismic data processing, rugged depth migration based on wave-equation migration can play a key role in handling surface fluctuations and complex underground structures. Because wave-equation migration in the shot domain cannot output offset-domain common-image gathers in a straightforward way, the use of traditional tools for updating the velocity model and improving image quality can be quite challenging. To overcome this problem, we employed the attribute migration method. This worked by sorting the migrated stack results for every single-shot gather into the offset gathers. The value of the offset that corresponded to each image point was obtained from the ratio of the original migration results to the offset-modulated shot-data migration results. A Gaussian function was proposed to map every image point to a certain range of offsets. This helped improve the signal-to-noise ratio, which was especially important in handing low quality seismic data obtained during mineral exploration. Residual velocity analysis was applied to these gathers to update the velocity model and improve image quality. The offset-domain common-image gathers were also used directly for real mineral exploration seismic data with rugged depth migration. After several iterations of migration and updating the velocity, the proposed procedure achieved an image quality better than the one obtained with the initial velocity model. The results can help with the interpretation of thrust faults and deep deposit exploration.

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