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.

High-accuracy wavefront tracing traveltime calculation

Geophysics ◽  
1993 ◽  
Vol 58 (2) ◽  
pp. 284-292 ◽  
Author(s):  
Robert. L. Coultrip
Keyword(s):  
Three Dimensional ◽  
Computation Time ◽  
High Accuracy ◽  
Minimum Time ◽  
Model Complexity ◽  
Earth Model ◽  
Arrival Times ◽  
Multiple Paths ◽  
First Arrival ◽  
Tracing Algorithm

Conventional ray-tracing algorithms for first-arrival calculation suffer from drawbacks such as (1) no guarantee of finding the globally minimum traveltime path when multiple paths exist, (2) shadow zones, and (3) trouble finding minimum traveltime paths containing refraction and/or diffraction energy. Algorithms that trace wavefronts circumvent these problems. The new wavefront-tracing algorithm presented here is based on an earth model consisting of uniform-velocity triangular cells with nodes placed at vertices and along cell edges. Nodes are places where traces of first arrival wavefronts (propagation directions and arrival times) are stored. The algorithm works by propagating wavefronts (sampled at the nodes) away from the source throughout the entire model. Wavefronts are propagated locally as diffraction, direct arrival, or critically refracted energy that implicitly describe minimum time paths. Once the first arrival wavefront is sampled throughout the model, traveltimes and raypaths from the source to receivers are easily calculated. This algorithm computes the globally minimum time paths from the source to all points in the model regardless of model complexity and the number of locally minimum traveltime paths. Traveltime calculations are highly accurate and computation time is O(n log2 n) for n nodes. Use of triangular cells allows for cell boundaries that follow, say, fault planes and dipping beds, without resorting to stair-step approximations inherent with rectangular cells. This method can be extended to three dimensional (3-D) problems.

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.

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.

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.

scholarly journals A Low-Complexity Region-Based Authentication Algorithm for 3D Polygonal Models

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Yuan-Yu Tsai ◽  
Tsung-Chieh Cheng ◽  
Yao-Hsien Huang
Keyword(s):  
Three Dimensional ◽  
Low Complexity ◽  
Secret Key ◽  
Authentication Code ◽  
Geometrical Properties ◽  
Property Evaluation ◽  
Adaptive Embedding ◽  
Local Properties ◽  
Embedding Rate ◽  
Polygonal Models

This study proposes a low-complexity region-based authentication algorithm for three-dimensional (3D) polygonal models, based on local geometrical property evaluation. A vertex traversal scheme with a secret key is adopted to classify each vertex into one of two categories: embeddable vertices and reference vertices. An embeddable vertex is one with an authentication code embedded. The algorithm then uses reference vertices to calculate local geometrical properties for the corresponding embeddable vertices. For each embeddable vertex, we feed the number of reference vertices and local properties into a hash function to generate the authentication code. The embeddable vertex is then embedded with the authentication code, which is based on a simple message-digit substitution scheme. The proposed algorithm is of low complexity and distortion-controllable and possesses a higher and more adaptive embedding capacity and a higher embedding rate than most existing region-based authentication algorithms for 3D polygonal models. The experimental results demonstrate the feasibility of the proposed algorithm.

Distribution of arrival times in cosmic ray showers

1978 ◽  
Vol 10 (4) ◽  
pp. 730-735
Author(s):  
H. S. Green
Keyword(s):  
Cosmic Radiation ◽  
Three Dimensional ◽  
Stochastic Theory ◽  
Cosmic Ray ◽  
Time Of Arrival ◽  
Air Showers ◽  
Actual Distribution ◽  
Arrival Times ◽  
Primary Particles ◽  
Theoretical Analyses

The theoretical analyses of the extensive air showers developing from the cosmic radiation has its origins in the work of Carlson and Oppenheimer (1937) and Bhabha and Heitler (1937), at a time when it was thought that such showers were initiated by electrons. The realization that protons and other nuclei were the primary particles led to a reformulation of the theory by Heitler and Janossy (1949), Messel and Green (1952) and others, in which the production of energetic pions and the three-dimensional development of air showers were accounted for. But as the soft (electromagnetic) component of the cosmic radiation is the most prominent feature of air showers at sea level, there has been a sustained interest in the theory of this component. Most of the more recent work, such as that by Butcher and Messel (1960) and Thielheim and Zöllner (1972) has relied on computer simulation; but this method has disadvantages in terms of accuracy and presentation of results, especially where a simultaneous analysis of the development of air showers in terms of several physical variables is required. This is so for instance when the time of arrival is one of the variables. Moyal (1956) played an important part in the analytical formulation of a stochastic theory of cosmic ray showers, with time as an explicit variable, and it is essentially this approach which will be adopted in the following. The actual distribution of arrival times is cosmic ray showers, for which results are obtained, is of current experimental interest (McDonald, Clay and Prescott (1977)).

New Formulation of a General Three-Dimensional Cascade Theory

1954 ◽  
Vol 96 (6) ◽  
pp. 1651-1654 ◽  
Author(s):  
B. A. Chartres ◽  
H. Messel
Keyword(s):  
Three Dimensional ◽  
Cascade Theory ◽  
New Formulation

scholarly journals P-wave crustal tomography of Greece with use of an accurate two-point ray tracer

1997 ◽  
Vol 40 (1) ◽  
Author(s):  
G. Drakatos ◽  
G. Karantonis ◽  
G. N. Stavrakakis
Keyword(s):  
Large Scale ◽  
Velocity Structure ◽  
Three Dimensional ◽  
Seismic Energy ◽  
Aegean Sea ◽  
P Wave ◽  
Low Velocity ◽  
Aegean Area ◽  
Arrival Times ◽  
Horizontal Variations

The three-dimensional velocity structure of the crust in the Aegean sea and the surrounding regions (34.0º-42.OºN, 19.0ºE-29.0ºE) is investigated by inversion of about 10000 residuals of arrival times of P-wave from local events. The resulting velocity structure shows strong horizontal variations due to the complicated crustal structure and the variations of crustal thickness. The northern part of the region generally shows high velocities. In the inner part of the volcanic arc (Southern Aegean area), relatively low velocities are observed, suggesting a large-scale absorption of seismic energy as confirmed by the low seismicity of the region. A low velocity zone was observed along the subduction zone of the region, up to a depth of 4 km. The existence of such a zone could be due to granitic or other intrusions in the crust during the uplift of the region during Alpidic orogenesis.

A Large Measuring Range Profilometer for Three-Dimensional Surface Topography Measurement

2007 ◽  
Vol 364-366 ◽  
pp. 750-755 ◽  
Author(s):  
Xu Dong Yang ◽  
Jia Chun Li ◽  
Tie Bang Xie
Keyword(s):  
Surface Topography ◽  
Three Dimensional ◽  
Displacement Sensor ◽  
Resistance Force ◽  
Sampling Point ◽  
Measuring Range ◽  
3D Surface Topography ◽  
Control Circuits ◽  
Industrial Control Computer ◽  
Zero Offset

A novel profilometer for three-dimensional (3D) surface topography measurement is presented. The profilometer has large measuring range, high precision and small measuring touch force. It is composed of a two-dimensional (2D) displacement sensor, a 3D platform based on vertical scanning, measuring and control circuits and an industrial control computer. When a workpiece is measured, the vertical undulation of the profile at a sampling point leads to a zero offset of the 2D displacement sensor. According to the zero offset, a piezoelectric actuator and a servo motor drive the vertical scanning platform to move vertically to ensure that the lever returns to its balance position. So the non-linear error caused by the rotation of the lever is very small even if the measuring range is large. When the stylus barges up against a steep wall, the horizontal resistance force results in another zero offset of the 2D displacement sensor. If the zero offset exceeds a quota, the vertical scanning platform descends to make the stylus climb the steep wall successfully. According to the theoretical and experimental analysis, the profilometer can measure roughness, profile of sphere, step, groove and other 3D surfaces with curvature precisely.

Large‐scale three‐dimensional seismic models and their interpretive significance

Geophysics ◽  
1990 ◽  
Vol 55 (9) ◽  
pp. 1166-1182 ◽  
Author(s):  
Irshad R. Mufti
Keyword(s):  
Finite Difference ◽  
Wave Field ◽  
Large Scale ◽  
Three Dimensional ◽  
Synthetic Data ◽  
Real Data ◽  
The Real ◽  
Need To Evaluate ◽  
Zero Offset ◽  
Seismic Models

Finite‐difference seismic models are commonly set up in 2-D space. Such models must be excited by a line source which leads to different amplitudes than those in the real data commonly generated from a point source. Moreover, there is no provision for any out‐of‐plane events. These problems can be eliminated by using 3-D finite‐difference models. The fundamental strategy in designing efficient 3-D models is to minimize computational work without sacrificing accuracy. This was accomplished by using a (4,2) differencing operator which ensures the accuracy of much larger operators but requires many fewer numerical operations as well as significantly reduced manipulation of data in the computer memory. Such a choice also simplifies the problem of evaluating the wave field near the subsurface boundaries of the model where large operators cannot be used. We also exploited the fact that, unlike the real data, the synthetic data are free from ambient noise; consequently, one can retain sufficient resolution in the results by optimizing the frequency content of the source signal. Further computational efficiency was achieved by using the concept of the exploding reflector which yields zero‐offset seismic sections without the need to evaluate the wave field for individual shot locations. These considerations opened up the possibility of carrying out a complete synthetic 3-D survey on a supercomputer to investigate the seismic response of a large‐scale structure located in Oklahoma. The analysis of results done on a geophysical workstation provides new insight regarding the role of interference and diffraction in the interpretation of seismic data.

Sign in / Sign up

Export Citation Format

Share Document