Three‐dimensional time‐slice migration

Geophysics ◽  
1990 ◽  
Vol 55 (1) ◽  
pp. 10-19 ◽  
Author(s):  
Martin Karrenbach
Keyword(s):  
Constant Velocity ◽  
Fourier Transforms ◽  
Three Dimensional ◽  
Time Slice ◽  
Migration Process ◽  
Point Scatterer ◽  
Partial Energy ◽  
Data Volume ◽  
Summation Operator ◽  
Zero Offset

Three‐dimensional migration of zero‐offset data using a velocity varying with depth can be performed in one pass using Fourier transforms of time slices. The migration process is carried out entirely in the two‐dimensional spatial Fourier domain. The algorithm consecutively filters and adds time slices of the 3-D data volume in a way that is equivalent to summing energy over the diffraction surface of a point scatterer. The partial energy being distributed along a circle in a time slice is properly added in each summation step. Time‐slice migration is based on an integral solution of the acoustic wave equation known as the “Kirchhoff integral.” The wavelet shape in a 3-D data volume is preserved throughout the entire migration process. The frequency characteristics are maintained by summing weighted differences between time slices instead of summing the time slices themselves. Automatic weighting is achieved by time slicing at equal increments of diffraction radius. Tapering the summation operator reduces effects introduced by limiting the summation window. Time‐slice migration preserves the frequency content of a 3-D data volume during summation in a natural way. Since the migration scheme assumes a constant velocity within the entire time slice, it is a local process in time which migrates a 3-D data volume with a constant velocity or with a velocity which varies with depth. The migration algorithm is applied to numerical and physical model data. This method is especially suitable for a migration of a targeted subset of the 3-D data volume.

3-D time‐variant dip moveout by the f-k method

Geophysics ◽  
1993 ◽  
Vol 58 (7) ◽  
pp. 1030-1041 ◽  
Author(s):  
Hans A. Meinardus ◽  
Karl L. Schleicher
Keyword(s):  
Impulse Response ◽  
Constant Velocity ◽  
Decomposition Method ◽  
Three Dimensional ◽  
Velocity Variation ◽  
Velocity Function ◽  
Discrete Values ◽  
Zero Offset ◽  
Event Times ◽  
Arbitrary Velocity

The standard seismic imaging sequence consists of normal moveout (NMO), dip moveout (DMO), stack, and zero‐offset migration. Conventional NMO and DMO processes remove much of the effect of offset from prestack data, but the constant velocity assumption in most DMO algorithms can compromise the ultimate results. Time‐variant DMO avoids the constant velocity assumption to create better stacks, especially for steeply dipping events. Time‐variant DMO can be implemented as a 3-D, f-k domain process using the dip decomposition method. Prestack data are moved out with a set of NMO velocities corresponding to discrete values of in‐line and crossline dips. The dip‐dependent NMO velocity is computed to remove the trace offset and azimuth dependence of event times for an arbitrary velocity function of depth. After stacking the moved out CMP gathers, a three‐dimensional (3-D) dip filter is applied to select the particular in‐line and crossline dip. The final zero‐offset image is obtained by summing all the dip‐filtered sections. This process generates a saddle‐shaped 3-D impulse response for a constant velocity gradient. The impulse response is more complicated for a general depth‐variable velocity function, where the response exhibits secondary branches, or triplications, at steeper dips. These complicated impulse responses, including amplitude and phase effects, are implicitly produced by the f-k process. The dip‐decomposition method of 3-D time‐variant DMO is an efficient and accurate process to correct for the effect of offset in the presence of an arbitrary velocity variation with depth. The impulse response of this process implicitly contains complex features like a 3-D saddle shape, triplications, amplitude, and phase. Field data from the Gulf of Mexico shows significant improvement on a steep salt flank event.

A three‐dimensional perspective on two‐dimensional dip moveout

Geophysics ◽  
1988 ◽  
Vol 53 (5) ◽  
pp. 604-610 ◽  
Author(s):  
David Forel ◽  
Gerald H. F. Gardner
Keyword(s):  
Line Segment ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Prestack Migration ◽  
Standard Normal ◽  
Ellipsoidal Surface ◽  
Zero Offset

Prestack migration in a constant‐velocity medium spreads an impulse on any trace over an ellipsoidal surface with foci at the source and receiver positions for that trace. The same ellipsoid can be obtained by migrating a family of zero‐offset traces placed along the line segment from the source to the receiver. The spheres generated by migrating the zero‐offset impulses are arranged to be tangent to the ellipsoid. The resulting nonstandard moveout equation is equivalent to two consecutive moveouts, the first requiring no knowledge of velocity and the second being standard normal moveout (NMO). The first of these is referred to as dip moveout (DMO). Because this DMO-NMO algorithm converts any trace to an equivalent set of zero‐offset traces, it can be applied to any ensemble of traces no matter what the variations in azimuth and offset may be. In particular, this three‐dimensional perspective on DMO can be used with multifold inline data. Then it becomes clear that velocity‐independent DMO operates on radial‐trace profiles and not on constant‐offset profiles. Inline data over a three‐dimensional subsurface will be properly stacked by using DMO followed by NMO.

True‐amplitude imaging and dip moveout

Geophysics ◽  
1993 ◽  
Vol 58 (1) ◽  
pp. 47-66 ◽  
Author(s):  
James L. Black ◽  
Karl L. Schleicher ◽  
Lin Zhang
Keyword(s):  
Wave Equation ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Acoustic Wave Equation ◽  
Simple Modification ◽  
Amplitude Image ◽  
Imaging Sequence ◽  
Zero Offset ◽  
Spherical Divergence ◽  
Zero Phase

True‐amplitude seismic imaging produces a three dimensional (3-D) migrated section in which the peak amplitude of each migrated event is proportional to the reflectivity. For a constant‐velocity medium, the standard imaging sequence consisting of spherical‐divergence correction, normal moveout (NMO), dip moveout (DMO), and zero‐offset migration produces a true‐amplitude image if the DMO step is done correctly. There are two equivalent ways to derive the correct amplitude‐preserving DMO. The first is to improve upon Hale’s derivation of F-K DMO by taking the reflection‐point smear properly into account. This yields a new Jacobian that simply replaces the Jacobian in Hale’s method. The second way is to calibrate the filter that appears in integral DMO so as to preserve the amplitude of an arbitrary 3-D dipping reflector. This latter method is based upon the 3-D acoustic wave equation with constant velocity. The resulting filter amounts to a simple modification of existing integral algorithms. The new F-K and integral DMO algorithms resulting from these two approaches turn out to be equivalent, producing identical outputs when implemented in nonaliased fashion. As dip increases, their output become progressively larger than the outputs of either Hale’s F-K method or the integral method generally associated with Deregowski and Rocca. This trend can be observed both on model data and field data. There are two additional results of this analysis, both following from the wave‐equation calibration on an arbitrary 3-D dipping reflector. The first is a proof that the entire imaging sequence (not just the DMO part) is true‐amplitude when the DMO is done correctly. The second result is a handy formula showing exactly how the zero‐phase wavelet on the final migrated image is a stretched version of the zero‐phase deconvolved source wavelet. This result quantitatively expresses the loss of vertical resolution due to dip and offset.

Three‐dimensional modeling and migration of seismic data using Fourier transforms

Geophysics ◽  
1988 ◽  
Vol 53 (9) ◽  
pp. 1194-1201 ◽  
Author(s):  
Jing Wen ◽  
George A. McMechan ◽  
Michael W. Booth
Keyword(s):  
Wave Equation ◽  
Seismic Data ◽  
Fourier Transforms ◽  
Three Dimensional ◽  
Scale Model ◽  
Model Data ◽  
Three Dimensional Modeling ◽  
Dimensional Modeling ◽  
Zero Offset ◽  
And Migration

Programs for 3-D modeling and migration, using 3-D Fourier transforms to solve the scalar wave equation in frequency‐wavenumber space, are developed, implemented, tested, and applied to synthetic and scale‐model data. With microtasking to fully use four CRAY processors in parallel, we can solve a complete [Formula: see text] modeling problem in about 2.5 minutes (elapsed time); of this time, the two 3-D Fourier transforms take 1 minute and the wave‐equation calculations take 1.5 minutes. The corresponding migration also takes 2.5 minutes. Thus, even iterative 3-D processing is now feasible. The two main assumptions in our algorithm are that the earth has a constant velocity and that the data are zero‐offset (or stacked). Tests with model data verify that the algorithm produces the correct results when these assumptions are satisfied. Tests with scale‐model data show that approximate images may still be obtained when the assumptions are not strictly met; but the images contain a variety of distortions, primarily related to undermigration and overmigration, so caution is required in interpretation.

scholarly journals A critical look at the prediction of the temperature field around a laser-induced melt pool on metallic substrates

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Yi Shu ◽  
Daniel Galles ◽  
Ottman A. Tertuliano ◽  
Brandon A. McWilliams ◽  
Nancy Yang ◽  
...  
Keyword(s):  
Laser Beam ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Laser Beams ◽  
Poor Control ◽  
Metallic Substrates ◽  
Melt Pool ◽  
Mechanical Models ◽  
Transport Effects ◽  
Thermal Histories

AbstractThe study of microstructure evolution in additive manufacturing of metals would be aided by knowing the thermal history. Since temperature measurements beneath the surface are difficult, estimates are obtained from computational thermo-mechanical models calibrated against traces left in the sample revealed after etching, such as the trace of the melt pool boundary. Here we examine the question of how reliable thermal histories computed from a model that reproduces the melt pool trace are. To this end, we perform experiments in which one of two different laser beams moves with constant velocity and power over a substrate of 17-4PH SS or Ti-6Al-4V, with low enough power to avoid generating a keyhole. We find that thermal histories appear to be reliably computed provided that (a) the power density distribution of the laser beam over the substrate is well characterized, and (b) convective heat transport effects are accounted for. Poor control of the laser beam leads to potentially multiple three-dimensional melt pool shapes compatible with the melt pool trace, and therefore to multiple potential thermal histories. Ignoring convective effects leads to results that are inconsistent with experiments, even for the mild melt pools here.

Identification of Static Unbalance Wheel of Passenger Car Carried out on a Road

2014 ◽  
Vol 214 ◽  
pp. 48-57 ◽  
Author(s):  
Krzysztof Prażnowski ◽  
Sebastian Brol ◽  
Andrzej Augustynowicz
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Constant Velocity ◽  
Fourier Transforms ◽  
Static Condition ◽  
Rotational Frequency ◽  
Acceleration Sensor ◽  
Short Time Fourier Transform ◽  
Road Roughness ◽  
Short Time

This paper presents a method of identification of non-homogeneity or static unbalance of the structure of a car wheel based on a simple road test. In particular a method the detection of single wheel unbalance is proposed which applies an acceleration sensor fixed on windscreen. It measures accelerations cause by wheel unbalance among other parameters. The location of the sensor is convenient for handling an autonomous device used for diagnostic purposes. Unfortunately, its mounting point is located away from wheels. Moreover, the unbalance forces created by wheels spin are dumped by suspension elements as well as the chassis itself. It indicates that unbalance acceleration will be weak in comparison to other signals coming from engine vibrations, road roughness and environmental effects. Therefore, the static unbalance detection in the standard way is considered problematic and difficult. The goal of the undertaken research is to select appropriate transformations and procedures in order to determine wheel unbalance in these conditions. In this investigation regular and short time Fourier transform were used as well as wavelet transform. It was found that the use of Fourier transforms is appropriate for static condition (constant velocity) but the results proves that the wavelet transform is more suitable for diagnostic purposes because of its ability of producing clearer output even if car is in the state of acceleration or deceleration. Moreover it was proved that in the acceleration spectrum of acceleration measured on the windscreen a significant peak can be found when car runs with an unbalanced wheel. Moreover its frequency depends on wheel rotational frequency. For that reason the diagnostic of single wheel unbalance can be made by applying this method.

Coarse-To-Fine 3D Randomized Hough Transform for Dim Target Detection

2014 ◽  
Vol 519-520 ◽  
pp. 1040-1045
Author(s):  
Ling Fan
Keyword(s):  
Target Detection ◽  
Hough Transform ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Randomized Hough Transform ◽  
Straight Line ◽  
Memory Complexity ◽  
Coarse To Fine

This paper makes some improvements on Roberts representation for straight line in space and proposes a coarse-to-fine three-dimensional (3D) Randomized Hough Transform (RHT) for the detection of dim targets. Using range, bearing and elevation information of the received echoes, 3D RHT can detect constant velocity target in space. In addition, this paper applies a coarse-to-fine strategy to the 3D RHT, which aims to solve both the computational and memory complexity problems. The validity of the coarse-to-fine 3D RHT is verified by simulations. In comparison with the 2D case, which only uses the range-bearing information, the coarse-to-fine 3D RHT has a better practical value in dim target detection.

Identification of Micro Fractures in Cretaceous Bioclastic Limestone in Iraq and Its Influence on Development of Reservoirs

2021 ◽  
Author(s):  
Genjiu Wang ◽  
Dandan Hu ◽  
Qianyao Li
Keyword(s):  
Water Movement ◽  
Rock Fracture ◽  
Water Injection ◽  
Three Dimensional ◽  
Carbonate Reservoir ◽  
Distribution Data ◽  
Fracture Development ◽  
Reservoir Conditions ◽  
Data Volume ◽  
Southern Iraq

Abstract It is generally believed that Cretaceous bioclastic limestone in Mesopotamia basin in central and southern Iraq is a typical porous reservoir with weak fracture development. Therefore, previous studies on the fracture of this kind of reservoir are rare. As a common seepage channel in carbonate rock, fracture has an important influence on single well productivity and waterflooding development of carbonate reservoir. Based on seismic, core and production data, this study analyzes the development characteristics of fractures from various aspects, and discusses the influence of fractures on water injection development of reservoirs. Through special processing of seismic data, it is found that there are a lot of micro fractures in Cretaceous bioclastic limestone reservoir. Most of these micro fractures are filled fractures without conductivity under the original reservoir conditions. However, with the further development of the reservoir, the reservoir pressure, oil-water movement, water injection and other conditions have changed, resulting in the original reservoir conditions of micro fractures with conductivity. The water cut of many production wells in the high part of reservoir rises sharply. In order to describe the three-dimensional spatial distribution of fractures, the core data is used to verify the seismic fracture distribution data volume. After the verification effect is satisfied, the three-dimensional fracture data volume is transformed into the geological model to establish the permeability field including fracture characteristics. The results of numerical simulation show that water mainly flows into the reservoir through high angle micro fractures. Fractures are identified by seismic and fracture model is established to effectively recognize the influence of micro fractures on water injection development in reservoir development process, which provides important guidance for oilfield development of Cretaceous bioclastic limestone reservoir in the central and southern Iraq fields.

scholarly journals Quantum chaos for point scatterers on flat tori

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120509 ◽  
Author(s):  
Henrik Ueberschär
Keyword(s):  
Wave Function ◽  
Strong Coupling ◽  
Quantum Chaos ◽  
Recent Progress ◽  
Three Dimensional ◽  
Point Scatterer ◽  
Open Problems ◽  
Survey Article ◽  
Delta Potential ◽  
Flat Tori

This survey article deals with a delta potential—also known as a point scatterer—on flat two- and three-dimensional tori. We introduce the main conjectures regarding the spectral and wave function statistics of this model in the so-called weak and strong coupling regimes. We report on recent progress as well as a number of open problems in this field.

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