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.

3-D true‐amplitude finite‐offset migration

Geophysics ◽  
1993 ◽  
Vol 58 (8) ◽  
pp. 1112-1126 ◽  
Author(s):  
Jorg Schleicher ◽  
Martin Tygel ◽  
Peter Hubral
Keyword(s):  
Weight Function ◽  
A Priori ◽  
Three Dimensional ◽  
Reflection Coefficients ◽  
Ray Theory ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Prestack Migration ◽  
The One ◽  
Paraxial Ray

Compressional primary nonzero offset reflections can be imaged into three‐dimensional (3-D) time or depth‐migrated reflections so that the migrated wavefield amplitudes are a measure of angle‐dependent reflection coefficients. Various migration/inversion algorithms involving weighted diffraction stacks recently proposed are based on Born or Kirchhoff approximations. Here a 3-D Kirchhoff‐type prestack migration approach is proposed where the primary reflections of the wavefields to be imaged are a priori described by the zero‐order ray approximation. As a result, the principal issue in the attempt to recover angle‐dependent reflection coefficients becomes the removal of the geometrical spreading factor of the primary reflections. The weight function that achieves this aim is independent of the unknown reflector and correctly accounts for the recovery of the source pulse in the migrated image irrespective of the source‐receiver configurations employed and the caustics occurring in the wavefield. Our weight function, which is computed using paraxial ray theory, is compared with the one of the inversion integral based on the Beylkin determinant. It differs by a factor that can be easily explained.

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.

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.

DIVERGENCE EFFECTS IN A LAYERED EARTH

Geophysics ◽  
1973 ◽  
Vol 38 (3) ◽  
pp. 481-488 ◽  
Author(s):  
P. Newman
Keyword(s):  
Normal Incidence ◽  
Wave Theory ◽  
Diagnostic Value ◽  
Earth Model ◽  
Correction Factors ◽  
Amplitude Correction ◽  
Multiple Attenuation ◽  
Seismic Recording ◽  
Zero Offset ◽  
Special Case

Of the various factors which influence reflection amplitudes in a seismic recording, divergence effects are possibly of least direct interest to the interpreter. Nevertheless, proper compensation for these effects is mandatory if reflection amplitudes are to be of diagnostic value. For an earth model consisting of horizontal, isotropic layers, and assuming a point source, we apply ray theory to determine an expression for amplitude correction factors in terms of initial incidence, source‐receiver offset, and reflector depth. The special case of zero offset yields an expression in terms of two‐way traveltime, velocity in the initial layer, and the time‐weighted rms velocity which characterizes reflections. For this model it follows that information which is needed for divergence compensation in the region of normal incidence is available from the customary analysis of normal moveout (NMO). It is hardly surprising that NMO and divergence effects are intimately related when one considers the expanding wavefront situation which is responsible for both phenomena. However, it is evident that an amplitude correction which is appropriate for the primary reflection sequence cannot in general be appropriate for the multiples. At short offset distances the disparity in displayed amplitude varies as the square of the ratio of primary to multiple rms velocities, and favors the multiples. These observations are relevant to a number of concepts which are founded upon plane‐wave theory, notably multiple attenuation processes and record synthesis inclusive of multiples.

Elastic least-squares reverse time migration using the energy norm

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. S237-S248 ◽  
Author(s):  
Daniel Rocha ◽  
Paul Sava
Keyword(s):  
Least Squares ◽  
Normal Incidence ◽  
Wave Mode ◽  
Energy Norm ◽  
Earth Model ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Mode Decomposition ◽  
High Convergence Rate

Incorporating anisotropy and elasticity into least-squares migration is an important step toward recovering accurate amplitudes in seismic imaging. An efficient way to extract reflectivity information from anisotropic elastic wavefields exploits properties of the energy norm. We derive linearized modeling and migration operators based on the energy norm to perform anisotropic least-squares reverse time migration (LSRTM) describing subsurface reflectivity and correctly predicting observed data without costly decomposition of wave modes. Imaging operators based on the energy norm have no polarity reversal at normal incidence and remove backscattering artifacts caused by sharp interfaces in the earth model, thus accelerating convergence and generating images of higher quality when compared with images produced by conventional methods. With synthetic and field data experiments, we find that our elastic LSRTM method generates high-quality images that predict the data for arbitrary anisotropy, without the complexity of wave-mode decomposition and with a high convergence rate.

Three‐dimensional primary zero‐offset reflections

Geophysics ◽  
1993 ◽  
Vol 58 (5) ◽  
pp. 692-702 ◽  
Author(s):  
Peter Hubral ◽  
Jorg Schleicher ◽  
Martin Tygel
Keyword(s):  
Ray Tracing ◽  
Large Scale ◽  
Initial Conditions ◽  
Fresnel Zone ◽  
Three Dimensional ◽  
Normal Incidence ◽  
Careful Analysis ◽  
Point Sources ◽  
Dynamic Ray Tracing ◽  
Zero Offset

Zero‐offset reflections resulting from point sources are often computed on a large scale in three‐dimensional (3-D) laterally inhomogeneous isotropic media with the help of ray theory. The geometrical‐spreading factor and the number of caustics that determine the shape of the reflected pulse are then generally obtained by integrating the so‐called dynamic ray‐tracing system down and up to the two‐way normal incidence ray. Assuming that this ray is already known, we show that one integration of the dynamic ray‐tracing system in a downward direction with only the initial condition of a point source at the earth’s surface is in fact sufficient to obtain both results. To establish the Fresnel zone of the zero‐offset reflection upon the reflector requires the same single downward integration. By performing a second downward integration (using the initial conditions of a plane wave at the earth’s surface) the complete Fresnel volume around the two‐way normal ray can be found. This should be known to ascertain the validity of the computed zero‐offset event. A careful analysis of the problem as performed here shows that round‐trip integrations of the dynamic ray‐tracing system following the actually propagating wavefront along the two‐way normal ray need never be considered. In fact some useful quantities related to the two‐way normal ray (e.g., the normal‐moveout velocity) require only one single integration in one specific direction only. Finally, a two‐point ray tracing for normal rays can be derived from one‐way dynamic ray tracing.

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.

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.

t-x reflection curves for arbitrary three‐dimensional media in terms of geometry of a reflector and a near‐reflector wavefront

Geophysics ◽  
1986 ◽  
Vol 51 (10) ◽  
pp. 1912-1922
Author(s):  
G. Nedlin
Keyword(s):  
General Relation ◽  
Three Dimensional ◽  
Earth Model ◽  
Arrival Times ◽  
Geometrical Properties ◽  
Parametric Description ◽  
Time Offset ◽  
Zero Offset ◽  
New Formulation ◽  
Reflecting Surface

A general relation between a normal‐moveout velocity (NMOV) for t-x (time‐offset) reflection curves and the geometrical properties of a reflector and a wavefront in the vicinity of the reflector has been found. Furthermore, by considering the reflector as a set of zero‐offset reflecting points for different shot locations on the earth’s surface, a new formulation of the special “seismic” parametric description of a reflecting surface allows the arrival times to be related directly to the wavefront equation, without introducing any earth model above the reflector. The NMOV is expressed in terms of the local velocity near the reflector and the curvatures of the reflector and of the near‐reflector wavefront. New equations for geometrical migration make it possible to do direct wavefront modeling without earth modeling (above the reflector). If t-x curves are approximated by hyperbolas (i.e., terms higher than those quadratic in the offsets are neglected), all rays in a common‐midpoint (CMP) panel with a fixed midpoint have the same reflecting point, for any earth model.

Determination of Fresnel zones from traveltime measurements

Geophysics ◽  
1993 ◽  
Vol 58 (5) ◽  
pp. 703-712 ◽  
Author(s):  
Peter Hubral ◽  
Jörg Schleicher ◽  
Martin Tygel ◽  
Ch. Hanitzsch
Keyword(s):  
Fresnel Zone ◽  
Three Dimensional ◽  
Earth Model ◽  
Simple Method ◽  
Interval Velocity ◽  
Stratigraphic Analysis ◽  
Zero Offset ◽  
Fresnel Zones ◽  
Key Parameter

For a horizontally stratified (isotropic) earth, the rms‐velocity of a primary reflection is a key parameter for common‐midpoint (CMP) stacking, interval‐velocity computation (by the Dix formula) and true‐amplitude processing (geometrical‐spreading compensation). As shown here, it is also a very desirable parameter to determine the Fresnel zone on the reflector from which the primary zero‐offset reflection results. Hence, the rms‐velocity can contribute to evaluating the resolution of the primary reflection. The situation that applies to a horizontally stratified earth model can be generalized to three‐dimensional (3-D) layered laterally inhomogeneous media. The theory by which Fresnel zones for zero‐offset primary reflections can then be determined purely from a traveltime analysis—without knowing the overburden above the considered reflector—is presented. The concept of a projected Fresnel zone is introduced and a simple method of its construction for zero‐offset primary reflections is described. The projected Fresnel zone provides the image on the earth’s surface (or on the traveltime surface of primary zero‐offset reflections) of that part of the subsurface reflector (i.e., the actual Fresnel zone) that influences the considered reflection. This image is often required for a seismic stratigraphic analysis. Our main aim is therefore to show the seismic interpreter how easy it is to find the projected Fresnel zone of a zero‐offset reflection using nothing more than a standard 3-D CMP traveltime analysis.

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