... produced a number of widely recognized sages. To them we canapart from Pythagorascount, eg,... more ... produced a number of widely recognized sages. To them we canapart from Pythagorascount, eg, Lao Tse (c. 550 BCE, China), Buddha (c. 527 BCE, India),Zoroaster (c. 628 BCE, Iran) and Plato (427347 BCE, Greece). ...
A system of three ordinary non-linear first order differential equations is proposed for the comp... more A system of three ordinary non-linear first order differential equations is proposed for the computation of the geometrical spreading of the wave front of a seismic body wave in a three-dimensional medium. The variables of the system are the parameters which provide a second order approximation of the wave front.
An amplitude-preserving migration aims at imaging compressional primary (zero-or) non-zero-offset... more An amplitude-preserving migration aims at imaging compressional primary (zero-or) non-zero-offset reflections into 3D time or depth-migrated reflections so that the migrated wavefield amplitudes are a measure of angle-dependent reflection coefficients. The principal objective is the removal of the geometrical-spreading factor of the primary reflections. Various migration/inversion algorithms involving weighted diffraction stacks proposed recently are based on Born or Kirchhoff approximations. Here, a 3D Kirchhoff-type zero-offset migration approach, also known as a diffraction-stack migration, is implemented in the form of a time migration. The primary reflections of the wavefield to be imaged are described a priori by the zero-order ray approximation. The aim of removing the geometrical-spreading loss can, in the zero-offset case, be achieved by not applying weights to the data before stacking them. This case alone has been implemented in this work. Application of the method to 3D synthetic zero-offset data proves that an amplitude-preserving migration can be performed in this way. Various numerical aspects of the true-amplitude zero-offset migration are discussed.
ABSTRACT For a horizontally stratified (isotropic) earth, the rms-velocity of a primary reflectio... more ABSTRACT 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.
The computation of the geometrical-spreading factor and the number of caustics is often considere... more The computation of the geometrical-spreading factor and the number of caustics is often considered to be the most fundamental step in computing zero-order ray solutions for elementary-wave Green`s functions along a ray that originates at a point source and passes through a 3-D laterally inhomogeneous isotropic medium. Here, a new factorization method is described that establishes both quantities recursively along the ray segments into which the total ray can be subdivided. As a consequence of the proposed method, the point-source geometrical-spreading factor and the number of ray caustics along the total ray can be decomposed into (1) point-source spreading factors of the ray segments and (2) certain Fresnel zone contributions at the ray-segment connection points. In a so-called ``3-D simple medium,`` by which any 3-D laterally inhomogeneous medium can be approximated, the new factorization approach permits a simple computation of both quantities. It thus simplifies and provides new insights into the computation of ray-theoretical Green`s functions.
... produced a number of widely recognized sages. To them we canapart from Pythagorascount, eg,... more ... produced a number of widely recognized sages. To them we canapart from Pythagorascount, eg, Lao Tse (c. 550 BCE, China), Buddha (c. 527 BCE, India),Zoroaster (c. 628 BCE, Iran) and Plato (427347 BCE, Greece). ...
A system of three ordinary non-linear first order differential equations is proposed for the comp... more A system of three ordinary non-linear first order differential equations is proposed for the computation of the geometrical spreading of the wave front of a seismic body wave in a three-dimensional medium. The variables of the system are the parameters which provide a second order approximation of the wave front.
An amplitude-preserving migration aims at imaging compressional primary (zero-or) non-zero-offset... more An amplitude-preserving migration aims at imaging compressional primary (zero-or) non-zero-offset reflections into 3D time or depth-migrated reflections so that the migrated wavefield amplitudes are a measure of angle-dependent reflection coefficients. The principal objective is the removal of the geometrical-spreading factor of the primary reflections. Various migration/inversion algorithms involving weighted diffraction stacks proposed recently are based on Born or Kirchhoff approximations. Here, a 3D Kirchhoff-type zero-offset migration approach, also known as a diffraction-stack migration, is implemented in the form of a time migration. The primary reflections of the wavefield to be imaged are described a priori by the zero-order ray approximation. The aim of removing the geometrical-spreading loss can, in the zero-offset case, be achieved by not applying weights to the data before stacking them. This case alone has been implemented in this work. Application of the method to 3D synthetic zero-offset data proves that an amplitude-preserving migration can be performed in this way. Various numerical aspects of the true-amplitude zero-offset migration are discussed.
ABSTRACT For a horizontally stratified (isotropic) earth, the rms-velocity of a primary reflectio... more ABSTRACT 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.
The computation of the geometrical-spreading factor and the number of caustics is often considere... more The computation of the geometrical-spreading factor and the number of caustics is often considered to be the most fundamental step in computing zero-order ray solutions for elementary-wave Green`s functions along a ray that originates at a point source and passes through a 3-D laterally inhomogeneous isotropic medium. Here, a new factorization method is described that establishes both quantities recursively along the ray segments into which the total ray can be subdivided. As a consequence of the proposed method, the point-source geometrical-spreading factor and the number of ray caustics along the total ray can be decomposed into (1) point-source spreading factors of the ray segments and (2) certain Fresnel zone contributions at the ray-segment connection points. In a so-called ``3-D simple medium,`` by which any 3-D laterally inhomogeneous medium can be approximated, the new factorization approach permits a simple computation of both quantities. It thus simplifies and provides new insights into the computation of ray-theoretical Green`s functions.
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