2.5-D true‐amplitude Kirchhoff migration to zero offset in laterally inhomogeneous media

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 557-573 ◽  
Author(s):  
Martin Tygel ◽  
Jörg Schleicher ◽  
Peter Hubral ◽  
Lúcio T. Santos
Keyword(s):  
Seismic Source ◽  
Velocity Model ◽  
Dynamic Effects ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Seismic Section ◽  
Kirchhoff Migration ◽  
Reflection Angle ◽  
Earth Models ◽  
Zero Offset

The proposed new Kirchhoff‐type true‐amplitude migration to zero offset (MZO) for 2.5-D common‐offset reflections in 2-D laterally inhomogeneous layered isotropic earth models does not depend on the reflector curvature. It provides a transformation of a common‐offset seismic section to a simulated zero‐offset section in which both the kinematic and main dynamic effects are accounted for correctly. The process transforms primary common‐offset reflections from arbitrary curved interfaces into their corresponding zero‐offset reflections automatically replacing the geometrical‐spreading factor. In analogy to a weighted Kirchhoff migration scheme, the stacking curve and weight function can be computed by dynamic ray tracing in the macro‐velocity model that is supposed to be available. In addition, we show that an MZO stretches the seismic source pulse by the cosine of the reflection angle of the original offset reflections. The proposed approach quantitatively extends the previous MZO or dip moveout (DMO) schemes to the 2.5-D situation.

Computing true amplitude reflections in a laterally inhomogeneous earth

Geophysics ◽  
1983 ◽  
Vol 48 (8) ◽  
pp. 1051-1062 ◽  
Author(s):  
Peter Hubral
Keyword(s):  
Three Dimensional ◽  
Physical Experiment ◽  
Reflection Coefficients ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Seismic Section ◽  
Earth Models ◽  
Reflecting Boundaries ◽  
The Common ◽  
Zero Offset

Recently Bortfeld (1982) gave a cursory nonmathematical introduction to a procedure for computing the geometrical spreading factor of a primary zero‐offset reflection from the common datum point traveltime measurements of the event. To underline the significance and consequences of this method, a derivation and discussion of geometrical spreading factors is now given for two‐ and three‐dimensional earth models with curved reflecting boundaries. The spreading factors can be used easily to transform primary reflections in a zero‐offset seismic section to true amplitude reflections. These permit an estimation of interface reflection coefficients, either directly or in connection with a true amplitude migration. A seismic section with true amplitude reflections can be described by one physical experiment: the tuned reflector model. Hence the application of the wave equation (in connection with a migration after stack) is justified on such a seismic section. Also the geometrical spreading factors that are derived can be looked upon as a generalization of a well‐known formula (Newman, 1973), which is commonly used in true amplitude processing and trace inversion in the presence of a vertically inhomogeneous earth.

Controlling amplitudes in 2.5D common‐shot migration to zero offset

Geophysics ◽  
2004 ◽  
Vol 69 (5) ◽  
pp. 1299-1310 ◽  
Author(s):  
Jörg Schleicher ◽  
Claudio Bagaini
Keyword(s):  
Wave Propagation ◽  
Weight Function ◽  
Seismic Data ◽  
Constant Velocity ◽  
A Priori ◽  
Weight Functions ◽  
A Priori Knowledge ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Zero Offset

Configuration transform operations such as dip moveout, migration to zero offset, and shot and offset continuation use seismic data recorded with a certain measurement configuration to simulate data as if recorded with other configurations. Common‐shot migration to zero offset (CS‐MZO), analyzed in this paper, transforms a common‐shot section into a zero‐offset section. It can be realized as a Kirchhoff‐type stacking operation for 3D wave propagation in a 2D laterally inhomogeneous medium. By application of suitable weight functions, amplitudes of the data are either preserved or transformed by replacing the geometrical‐spreading factor of the input reflections by the correct one of the output zero‐offset reflections. The necessary weight function can be computed via 2D dynamic ray tracing in a given macrovelocity model without any a priori knowledge regarding the dip or curvature of the reflectors. We derive the general expression of the weight function in the general 2.5D situation and specify its form for the particular case of constant velocity. A numerical example validates this expression and highlights the differences between amplitude preserving and true‐amplitude CS‐MZO.

scholarly journals A transversely isotropic medium with a tilted symmetry axis normal to the reflector

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. A19-A24 ◽  
Author(s):  
Tariq Alkhalifah ◽  
Paul Sava
Keyword(s):  
Closed Form ◽  
Model Building ◽  
Incidence Angle ◽  
Transversely Isotropic ◽  
Velocity Model ◽  
Reflection Angle ◽  
Velocity Model Building ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Zero Offset

The computational tools for imaging in transversely isotropic media with tilted axes of symmetry (TTI) are complex and in most cases do not have an explicit closed-form representation. Developing such tools for a TTI medium with tilt constrained to be normal to the reflector dip (DTI) reduces their complexity and allows for closed-form representations. The homogeneous-case zero-offset migration in such a medium can be performed using an isotropic operator scaled by the velocity of the medium in the tilt direction. For the nonzero-offset case, the reflection angle is always equal to the incidence angle, and thus, the velocities for the source and receiver waves at the reflection point are equal and explicitly dependent on the reflection angle. This fact allows for the development of explicit representations for angle decomposition as well as moveout formulas for analysis of extended images obtained by wave-equation migration. Although setting the tilt normal to the reflector dip may not be valid everywhere (i.e., on salt flanks), it can be used in the process of velocity model building, in which such constrains are useful and typically are used.

Geometrical spreading corrections of offset reflections in a laterally inhomogeneous earth

Geophysics ◽  
1992 ◽  
Vol 57 (8) ◽  
pp. 1054-1063 ◽  
Author(s):  
M. Tygel ◽  
J. Schleicher ◽  
P. Hubral
Keyword(s):  
Three Dimensional ◽  
Velocity Model ◽  
Earth Model ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Depth Migration ◽  
Avo Analysis ◽  
Amplitude Versus Offset ◽  
One Step

Compressional primary seismic nonzero offset reflections are the most essential wavefield attributes used in seismic parameter estimation and imaging. We show how the determination of angle‐dependent reflection coefficients can be addressed from identifying such events for arbitrarily curved three‐dimensional (3-D) subsurface reflectors below a laterally inhomogeneous layered overburden. More explicitly, we show how the geometrical‐spreading factor along a reflected primary ray with offset can be calculated from the identified (i.e., picked) traveltimes of offset primary reflections. Seismic traces in which all primary reflections are corrected with the geometrical‐spreading factor are, as is well‐known, referred to as true‐amplitude traces. They can be constructed without any knowledge of the velocity distribution in the earth model. Apart from possibly finding a direct application in an amplitude‐versus‐offset (AVO) analysis, the theory developed here can be of use to derive true‐amplitude time‐ and depth‐migration methods for various seismic data acquisition configurations, which pursue the aim of performing the wavefield migration (based upon the use of a macro‐velocity model) and the AVO analysis in one step.

scholarly journals Through-Tubing Well Seismic: A Mast Campaign Across an Offshore Producing Field, Saudi Arabia-Kuwait Partitioned Neutral Zone

GeoArabia ◽  
1996 ◽  
Vol 1 (4) ◽  
pp. 571-582
Author(s):  
Kaoru Yamaguchi ◽  
Bernard G. Frignet
Keyword(s):  
Arabian Gulf ◽  
Seismic Source ◽  
Velocity Model ◽  
Data Availability ◽  
Planning Stage ◽  
Air Gun ◽  
Local Experience ◽  
Zero Offset ◽  
Tube Wave ◽  
Well Data

ABSTRACT A multi-well VSP Through-Tubing campaign was conducted in order to survey 2 neighboring mature fields in the northern Arabian Gulf. Nine key wells were selected by the reservoir engineers and geophysicists from a total of more than 200 wells. The selection criteria included: (1) spatial distribution; (2) previous well data availability; and (3) completion type. At the planning stage it was estimated that 3 days (excluding night work for safety) would be required per well (one day to rig-up the mast, one day to acquire the seismic data and one day to rig-down). Local experience indicated that a single air gun would be an effective seismic source. A slimhole, monocable, single-axis geophone sonde was selected as the downhole seismic tool. The zero-offset VSP configuration, with 80 foot level-spacing from total depth to surface was adopted. A supply boat was dedicated for the duration of the campaign. The actual operation was completed in 24 days, 3 days earlier than planned. An average of 90% of the VSP levels were found suitable for first break detection, which provided accurate Time-Depth curves for all 9 wells. Geophone coupling quality is dependent on the Tubing-Casing contact. The tube wave is developed at all 9 wells; but, overall 80% of the levels were selected for VSP processing. In 5 wells, where sonic logs had been acquired, synthetic seismograms were generated which confirmed the validity of VSP reflections. The data is now integrated in a 3-D velocity model.

Three‐dimensional true‐amplitude zero‐offset migration

Geophysics ◽  
1991 ◽  
Vol 56 (1) ◽  
pp. 18-26 ◽  
Author(s):  
Peter Hubral ◽  
Martin Tygel ◽  
Holger Zien
Keyword(s):  
Time Derivative ◽  
Three Dimensional ◽  
Normal Incidence ◽  
Earth Model ◽  
Ray Theory ◽  
Transmission Losses ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Time Migration ◽  
Zero Offset

The primary zero‐offset reflection of a point source from a smooth reflector within a laterally inhomogeneous velocity earth model is (within the framework of ray theory) defined by parameters pertaining to the normal‐incidence ray. The geometrical‐spreading factor—usually computed along the ray by dynamic‐ray tracing in a forward‐modeling approach—can, in this case, be recovered from traveltime measurements at the surface. As a consequence, zero‐offset reflections can be time migrated such that the geometrical‐spreading factor for the normal‐incidence ray is removed. This leads to a so‐called “true‐amplitude time migration.” In this work, true‐amplitude time‐migrated reflections are obtained by nothing more than a simple diffraction stack essentially followed by a time derivative of the diffraction‐stack traces. For small transmission losses of primary zero‐offset reflections through intermediate‐layer boundaries, the true‐amplitude time‐migrated reflection provides a direct measure of the reflection coefficient at the reflecting lower end of the normal‐incidence ray. The time‐migrated field can be easily transformed into a depth‐migrated field with the help of image rays.

Application of RTM 3D angle gathers to wide-azimuth data subsalt imaging

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. WB175-WB182 ◽  
Author(s):  
Yan Huang ◽  
Bing Bai ◽  
Haiyong Quan ◽  
Tony Huang ◽  
Sheng Xu ◽  
...  
Keyword(s):  
Model Updating ◽  
Velocity Model ◽  
Azimuth Angle ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Data Set ◽  
Additional Information ◽  
Time Migration ◽  
Reflection Angle ◽  
Subsalt Imaging

The availability of wide-azimuth data and the use of reverse time migration (RTM) have dramatically increased the capabilities of imaging complex subsalt geology. With these improvements, the current obstacle for creating accurate subsalt images now lies in the velocity model. One of the challenges is to generate common image gathers that take full advantage of the additional information provided by wide-azimuth data and the additional accuracy provided by RTM for velocity model updating. A solution is to generate 3D angle domain common image gathers from RTM, which are indexed by subsurface reflection angle and subsurface azimuth angle. We apply these 3D angle gathers to subsalt tomography with the result that there were improvements in velocity updating with a wide-azimuth data set in the Gulf of Mexico.

Gaussian beam migration

Geophysics ◽  
1990 ◽  
Vol 55 (11) ◽  
pp. 1416-1428 ◽  
Author(s):  
N. Ross Hill
Keyword(s):  
Gaussian Beam ◽  
Wave Field ◽  
Seismic Source ◽  
Velocity Model ◽  
Downward Continuation ◽  
Gaussian Beams ◽  
Seismic Velocities ◽  
Migration Method ◽  
Gaussian Beam Migration ◽  
Lateral Variations

Just as synthetic seismic data can be created by expressing the wave field radiating from a seismic source as a set of Gaussian beams, recorded data can be downward continued by expressing the recorded wave field as a set of Gaussian beams emerging at the earth’s surface. In both cases, the Gaussian beam description of the seismic‐wave propagation can be advantageous when there are lateral variations in the seismic velocities. Gaussian‐beam downward continuation enables wave‐equation calculation of seismic propagation, while it retains the interpretive raypath description of this propagation. This paper describes a zero‐offset depth migration method that employs Gaussian beam downward continuation of the recorded wave field. The Gaussian‐beam migration method has advantages for imaging complex structures. Like finite‐difference migration, it is especially compatible with lateral variations in velocity, but Gaussian beam migration can image steeply dipping reflectors and will not produce unwanted reflections from structure in the velocity model. Unlike other raypath methods, Gaussian beam migration has guaranteed regular behavior at caustics and shadows. In addition, the method determines the beam spacing that ensures efficient, accurate calculations. The images produced by Gaussian beam migration are usually stable with respect to changes in beam parameters.

Finite‐difference calculation of direct‐arrival traveltimes using the eikonal equation

Geophysics ◽  
2002 ◽  
Vol 67 (4) ◽  
pp. 1270-1274 ◽  
Author(s):  
Le‐Wei Mo ◽  
Jerry M. Harris
Keyword(s):  
Finite Difference ◽  
Ray Tracing ◽  
Finite Differences ◽  
Eikonal Equation ◽  
Velocity Model ◽  
Uniform Grid ◽  
Square Grid ◽  
Kirchhoff Migration ◽  
Difference Calculation

Traveltimes of direct arrivals are obtained by solving the eikonal equation using finite differences. A uniform square grid represents both the velocity model and the traveltime table. Wavefront discontinuities across a velocity interface at postcritical incidence and some insights in direct‐arrival ray tracing are incorporated into the traveltime computation so that the procedure is stable at precritical, critical, and postcritical incidence angles. The traveltimes can be used in Kirchhoff migration, tomography, and NMO corrections that require traveltimes of direct arrivals on a uniform grid.

Anisotropic geometrical-spreading correction for wide-azimuth P-wave reflections

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. D161-D170 ◽  
Author(s):  
Xiaoxia Xu ◽  
Ilya Tsvankin
Keyword(s):  
Transversely Isotropic ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Fractured Reservoir ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Velocity Anisotropy ◽  
Reflection Data ◽  
Fitting Procedure ◽  
Avo Attributes

Compensation for geometrical spreading along a raypath is one of the key steps in AVO (amplitude-variation-with-offset) analysis, in particular, for wide-azimuth surveys. Here, we propose an efficient methodology to correct long-spread, wide-azimuth reflection data for geometrical spreading in stratified azimuthally anisotropic media. The P-wave geometrical-spreading factor is expressed through the reflection traveltime described by a nonhyperbolic moveout equation that has the same form as in VTI (transversely isotropic with a vertical symmetry axis) media. The adapted VTI equation is parameterized by the normal-moveout (NMO) ellipse and the azimuthally varying anellipticity parameter [Formula: see text]. To estimate the moveout parameters, we apply a 3D nonhyperbolic semblance algorithm of Vasconcelos and Tsvankin that operates simultaneously with traces at all offsets andazimuths. The estimated moveout parameters are used as the input in our geometrical-spreading computation. Numerical tests for models composed of orthorhombic layers with strong, depth-varying velocity anisotropy confirm the high accuracy of our travetime-fitting procedure and, therefore, of the geometrical-spreading correction. Because our algorithm is based entirely on the kinematics of reflection arrivals, it can be incorporated readily into the processing flow of azimuthal AVO analysis. In combination with the nonhyperbolic moveout inversion, we apply our method to wide-azimuth P-wave data collected at the Weyburn field in Canada. The geometrical-spreading factor for the reflection from the top of the fractured reservoir is clearly influenced by azimuthal anisotropy in the overburden, which should cause distortions in the azimuthal AVO attributes. This case study confirms that the azimuthal variation of the geometrical-spreading factor often is comparable to or exceeds that of the reflection coefficient.

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