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Journal articles on the topic 'Extrapolators'

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Published: 10 December 2022

Last updated: 29 January 2023

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1

Hale, Dave. "Stable explicit depth extrapolation of seismic wavefields." GEOPHYSICS 56, no. 11 (November 1991): 1770–77. http://dx.doi.org/10.1190/1.1442989.

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Stability has traditionally been one of the most compelling advantages of implicit methods for seismic wavefield extrapolation. The common 45-degree, finite‐difference migration algorithm, for example, is based on an implicit wavefield extrapolation that is guaranteed to be stable. Specifically, wavefield energy will not grow exponentially with depth as the wavefield is extrapolated downwards into the subsurface. Explicit methods, in contrast, tend to be unstable. Without special care in their implementation, explicit extrapolation methods cause wavefield energy to grow exponentially with depth, contrary to physical expectations. The Taylor series method may be used to design finite‐length, explicit, extrapolation filters. In the usual Taylor series method, N coefficients of a finite‐length filter are chosen to match N terms in a truncated Taylor series approximation of the desired filter’s Fourier transform. Unfortunately, this method yields unstable extrapolation filters. However, a simple modification of the Taylor series method yields extrapolators that are stable. The accuracy of stable explicit extrapolators is determined by their length—longer extrapolators yield accurate extrapolation for a wider range of propagation angles than do shorter filters. Because a very long extrapolator is required to extrapolate waves propagating at angles approaching 90 degrees, stable explicit extrapolators may be less efficient than implicit extrapolators for high propagation angles. For more modest propagation angles of 50 degrees of less, stable explicit extrapolators are likely to be more efficient than current implicit extrapolators. Furthermore, unlike implicit extrapolators, stable explicit extrapolators naturally attenuate waves propagating at high angles for which the extrapolators are inaccurate.

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2

Mittet, Rune, Roger Sollie, and Ketil Hokstad. "Prestack depth migration with compensation for absorption and dispersion." GEOPHYSICS 60, no. 5 (September 1995): 1485–94. http://dx.doi.org/10.1190/1.1443882.

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In prestack depth migration using explicit extrapolators, the attenuation and dispersion of the seismic wave has been neglected so far. We present a method for accommodating absorption and dispersion effects in depth migration schemes. Extrapolation operators that compensate for absorption and dispersion are designed using an optimization algorithm. The design criterion is that the wavenumber response of the operator should equal the true extrapolator. Both phase velocity and absorption macro models are used in the wavefield extrapolation. In a model with medium to high absorption, the images obtained are superior to those obtained using extrapolators without compensation for absorption.

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3

Musaev, Alexander A., and Andrey V. Gaikov. "MULTIAGENT SYSTEM OF FORECASTING OF CHAOTIC PROCESSES ON THE BASIS OF STATISTICAL EXTRAPOLATORS." Bulletin of the Saint Petersburg State Institute of Technology (Technical University) 56 (2021): 65–71. http://dx.doi.org/10.36807/1998-9849-2020-56-82-65-71.

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The problem of the of a non-stationary system state predicting is considered. The decision based on the joint processing of the results obtained by a group of independent statistical extrapolators. In the terminology of multiagent systems, each extrapolator is an intelligent agent. The quality of the agent solutions is evaluated on retrospective data and is used as weight characteristic in the problem of a terminal solution estimation. The specificity of non-stationary processes with a chaotic system component leads to the empiricca version of the forecast generation algorithm

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4

Du, Xiang, Paul J. Fowler, and Robin P. Fletcher. "Recursive integral time-extrapolation methods for waves: A comparative review." GEOPHYSICS 79, no. 1 (January 1, 2014): T9—T26. http://dx.doi.org/10.1190/geo2013-0115.1.

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We compared several families of algorithms for recursive integral time-extrapolation (RITE) algorithms for waves in isotropic and anisotropic media. These methods allow simulating accurate wave extrapolation with little numerical dispersion even when using larger time steps than are usually possible for conventional finite-difference methods. These various RITE algorithms all share the use of mixed space/wavenumber-domain operators derived from Fourier integral solutions of single-mode wave equations. We evaluated a taxonomy for RITE methods based on how they approximated the influence of medium heterogeneity. One family of methods uses mixed-domain series expansions to provide accurate approximations to heterogeneous extrapolators even for large time steps. We compared several methods for deriving coefficients for such series approximations. Another family of methods uses interpolation between different homogeneous extrapolations to approximate heterogeneous time extrapolation. Such methods can be based on interpolating either the extrapolators themselves or interpolating between reference wavefields extrapolated using different homogeneous parameters. Interpolation methods work well for smooth media, but can suffer from oscillatory artifacts at large velocity discontinuities unless the time step is small. We tested numerical examples of the various families of RITE algorithms to determine their relative strengths and limitations.

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5

Mousa, Wail A., Mirko van der Baan, Said Boussakta, and Desmond C. McLernon. "Designing stable extrapolators for explicit depth extrapolation of 2D and 3D wavefields using projections onto convex sets." GEOPHYSICS 74, no. 2 (March 2009): S33—S45. http://dx.doi.org/10.1190/1.3077621.

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We have developed a robust algorithm for designing explicit depth extrapolation operators using the projections-onto-convex-sets (POCS) method. The operators are optimal in the sense that they satisfy all required extrapolation design characteristics. In addition, we propose a simple modification of the POCS algorithm (modified POCS, or MPOCS) that further enhances the stability of extrapolated wavefields and reduces the number of iterations required to design such operators to approximately 2% of that required for the basic POCS design algorithm. Various synthetic tests show that 25-coefficient 1D extrapolation operators, which have 13 unique coefficients, can accommodate dip angles up to 70°. We migrated the SEG/EAGE salt model data with the operators and compare our results with images obtained via extrapolators based on modified Taylor series and with wavefield extrapolation techniques such as phase shift plus interpolation (PSPI) and split-step Fourier. The MPOCS algorithm provides practically stable depth extrapolators. The resulting migrated section is comparable in quality to an expensive PSPI result and visibly outperforms the other two techniques. Strong dips and subsalt structures are imaged clearly. Finally, we extended the 1D extrapolator design algorithm, using MPOCS for 2D extrapolation, to the 2D case to perform 3D extrapolation; the result is a perfect circularly symmetric migration impulse response.

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6

Mousa, Wail A. "The design of stable, sparse wavefield extrapolators using projections onto convex sets." GEOPHYSICS 78, no. 1 (January 1, 2013): T11—T20. http://dx.doi.org/10.1190/geo2012-0130.1.

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We present the results of poststack explicit depth migration of the well-known 2D SEG/EAGE salt model zero-offset seismic data using sparse wavefield extrapolators. The extrapolators are designed to be sparse by forcing some of the very small complex-valued coefficients’ magnitude values to be zero. The proposed extrapolators design method combines the previously reported modified projections onto convex sets (MPOCS) for designing explicit depth frequency-space ([Formula: see text]) wavefield extrapolation operators with hard-thresholding of the small extrapolators coefficients’ magnitude. The real and imaginary parts of the MPOCS operators, with small magnitudes, are replaced by zeros during the MPOCS algorithm iterations. The migrated result of the SEG/EAGE salt model data, using such sparse designed operators, shows comparable migrated results using the nonsparse version of the MPOCS extrapolation operators as well as the image obtained using the well-known phase-shift plus interpolation (PSPI) migration technique. Overall, the sparse operators result in poststack imaging computational savings (in terms of used flops) of about 28% when compared to poststack imaging of the same data using the nonsparse MPOCS designed operators, and of more than 87.77% saved flops using the PSPI technique.

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7

Al-Saleh, Saleh M., Gary F. Margrave, and Sam H. Gray. "Direct downward continuation from topography using explicit wavefield extrapolation." GEOPHYSICS 74, no. 6 (November 2009): S105—S112. http://dx.doi.org/10.1190/1.3263914.

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Downward-continuation migration algorithms are powerful tools for imaging complicated subsurface structures. However, they usually assume that extrapolation proceeds from a flat surface, whereas most land surveys are acquired over irregular surfaces. Our method downward continues data directly from topography using a recursive space-frequency explicit wavefield-extrapolation method. The algorithm typically handles strong lateral velocity variations by using the velocity value at each spatial position to build the wavefield extrapolator in which the depth step usually is kept fixed. To accommodate topographic variations, we build space-frequency wavefield extrapolators with laterally variable depth steps (LVDS). At each spatial location, the difference between topography and extrapolation depth is used to determine the depth step. We use the velocity and topographic values at each spatial lateral position to build extrapolators. The LVDS approach does not add more data nor does it require preprocessing prior to extrapolation. We implemented the LVDS method and applied it to a source profile prestack migration technique. We also implemented the previously developed zero-velocity layer approach to use for comparison. For both algorithms, we modeled the acoustic source as an approximate free-space Green’s function, not as a simple extrapolated spatial impulse. Tests on a synthetic data set modeled from rough topography and comparisons with the zero-velocity layer approach confirm the method’s effectiveness in imaging shallow and deep structures beneath rugged topography.

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8

Graves, R. W., and R. W. Clayton. "Modeling acoustic waves with paraxial extrapolators." GEOPHYSICS 55, no. 3 (March 1990): 306–19. http://dx.doi.org/10.1190/1.1442838.

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Modeling by paraxial extrapolators is applicable to wave‐propagation problems in which most of the energy is traveling within a restricted angular cone about a principal axis of the problem. Using this technique, frequency‐domain finite‐difference solutions accurate for propagation angles out to 60° are readily generated for both two‐dimensional (2-D) and three‐dimensional (3-D) models. Solutions for 3-D problems are computed by applying the 2-D paraxial operators twice, once along the x‐axis and once along the y‐axis, at each extrapolation step. The azimuthal anisotropy inherent to this splitting technique is essentially eliminated by adding a phase‐correction operator to the extrapolation system. For heterogeneous models, scattering effects are incorporated by determining transmission and reflection coefficients at structural boundaries within the media. The direct forward‐scattered waves are modeled with a single pass of the extrapolation operator in the paraxial direction for each frequency. The first‐order backscattered energy is then modeled by extrapolation (in the opposite direction) of the reflected field determined on the first pass. Higher order scattering can be included by sweeping through the model with more passes. The chief advantages of the paraxial approach are (1) active storage is reduced by one dimension compared to solutions which must track both forward‐scattered and backscattered waves simultaneously; thus, realistic 3-D problems can fit on today’s computers, (2) the decomposition in frequency allows the technique to be implemented on highly parallel machines, (3) attenuation can be modeled as an arbitrary function of frequency, and (4) only a small number of frequencies are needed to produce movie‐like time slices.

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9

van Breda, Leo, and Peter O. Passenier. "Effect of Path Prediction on Navigational Performance." Journal of Navigation 51, no. 2 (May 1998): 216–28. http://dx.doi.org/10.1017/s0373463397007662.

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A simulator experiment was conducted to determine the potential benefits of path prediction on the navigational performance of channel-bound vessels. Channel pilots had to sail an approach channel under critical conditions in a deep-draught vessel. For the navigation task, basic radar information was used, supplemented by three different path predictors. Predictor (a) was based on an accurate fast-time hydrodynamic model of the vessel and showed the exact future path of the vessel. Both other path predictors were less accurate, relatively simple extrapolators; predictor (b) was based on a speed and rate of turn extrapolator and showed a curved representation of the future path; predictor (c) was based on a linear speed and course extrapolator and showed the ground velocity vector. Navigational performance was determined in terms of deviation from the planned route. The results indicate that the relatively simple extrapolator (b) supported the navigational task as effectively as the highly accurate path predictor (a). In comparison with the linear extrapolator (c), the navigational accuracy increased by a factor of two. It is concluded that support in anticipating the vessel's rate of turn is essential for accurate navigation. Implications of the use of path prediction for ship control are discussed.

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10

Fadeeva, E. P. "Optimal linear extrapolators of some stationary processes." Journal of Soviet Mathematics 43, no. 1 (October 1988): 2240–48. http://dx.doi.org/10.1007/bf01095929.

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11

Angus, D. A. "True amplitude corrections for a narrow-angle one-way elastic wave equation." GEOPHYSICS 72, no. 2 (March 2007): T19—T26. http://dx.doi.org/10.1190/1.2430694.

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Wavefield extrapolators using one-way wave equations are computationally efficient methods for accurate traveltime modeling in laterally heterogeneous media, and have been used extensively in many seismic forward modeling and migration problems. However, most leading-order, one-way wave equations do not simulate waveform amplitudes accurately and this is primarily because energy flux is not accounted for correctly. I review the derivation of a leading-order, narrow-angle, one-way elastic wave equation for 3D media. I derive correction terms that enable energy-flux normalization and introduce a new higher-order, narrow-angle, one-way elastic wave extrapolator. By implementing these correction terms, the new true amplitude wave extrapolator allows accurate amplitude estimates in the presence of strong gradients. I present numerical examples for 1D velocity transition models to show that (1) the leading-order, narrow-angle propagator accurately models traveltimes, but overestimates transmitted- or primary-wave amplitudes and (2) the new amplitude corrected narrow-angle propagator accurately models both the traveltimes and amplitudes of all forward-traveling waves.

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12

Shragge, Jeff, and Guojian Shan. "Prestack wave-equation depth migration in elliptical coordinates." GEOPHYSICS 73, no. 5 (September 2008): S169—S175. http://dx.doi.org/10.1190/1.2956349.

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We extend Riemannian wavefield extrapolation (RWE) to prestack migration using 2D elliptical-coordinate systems. The corresponding 2D elliptical extrapolation wavenumber introduces only an isotropic slowness model stretch to the single-square-root operator. This enables the use of existing Cartesian finite-difference extrapolators for propagating wavefields on elliptical meshes. A poststack migration example illustrates advantages of elliptical coordinates for imaging turning waves. A 2D imaging test using a velocity-benchmark data set demonstrates that the RWE prestack migration algorithm generates high-quality prestack migration images that are more accurate than those generated by Cartesian operators of the equivalent accuracy. Even in situations in which RWE geometries are used, a high-order implementation of the one-way extrapolator operator is required for accurate propagation and imaging. Elliptical-cylindrical and oblate-spheroidal geometries are potential extensions of the analytical approach to 3D RWE-coordinate systems.

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13

Wu, Bangyu, Ru-Shan Wu, and Jinghuai Gao. "Preliminary Investigation of Wavefield Depth Extrapolation by Two-Way Wave Equations." International Journal of Geophysics 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/968090.

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Most of the wavefield downward continuation migration approaches are relying on one-way wave equations, which move the seismic energy always in one direction along depth. The one-way downward continuation migrations only use the primaries for imaging and do not treat secondary reflections recorded on the surface correctly. In this paper, we investigate wavefield depth extrapolators based on the full acoustic wave equations, which can propagate wave components to opposite directions. Several two-way wavefield downward continuation propagators are numerically tested in this study. Recursively implementing of the depth extrapolator makes it necessary and important to eliminate the unstable wave modes, that is, evanescent waves. For the laterally varying velocity media, distinction between the propagating and evanescent wave mode is less clear. We demonstrate that the spatially localized two-way beamlet propagator is an effective way to remove the evanescent waves while maintain the propagating mode in laterally inhomogeneous media.

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14

Salam, Syed Abdul, and Wail A. Mousa. "The design of L1-norm visco-acoustic wavefield extrapolators." Journal of Applied Geophysics 151 (April 2018): 157–65. http://dx.doi.org/10.1016/j.jappgeo.2018.01.018.

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15

Roe, Francis J. C. "Should We Listen to Granny or to the Risk Extrapolators." Indoor and Built Environment 1, no. 1 (1992): 41–43. http://dx.doi.org/10.1159/000463407.

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16

Roe, Francis J. C. "Should We Listen to Granny or to the Risk Extrapolators." Indoor Environment 1, no. 1 (January 1992): 41–43. http://dx.doi.org/10.1177/1420326x9200100107.

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17

Matveev, Al A., and A. A. Matveev. "Second-order extrapolators in information measuring and signal processing systems." Measurement Techniques 34, no. 2 (February 1991): 111–18. http://dx.doi.org/10.1007/bf00990810.

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18

Abdul Salam, Syed, and Wail A. Mousa. "The design of visco-acoustic weightedL1-error frequency-space wavefield extrapolators." Geophysical Prospecting 66, no. 2 (April 28, 2017): 300–310. http://dx.doi.org/10.1111/1365-2478.12509.

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19

Thomson, C. J. "Accuracy and efficiency considerations for wide-angle wavefield extrapolators and scattering operators." Geophysical Journal International 163, no. 1 (October 2005): 308–23. http://dx.doi.org/10.1111/j.1365-246x.2005.02750.x.

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20

Al-Battal, Abdullah F., and Wail A. Mousa. "The Design of 2-D Explicit Depth Extrapolators Using the Cauchy Norm." IEEE Transactions on Geoscience and Remote Sensing 55, no. 5 (May 2017): 3029–36. http://dx.doi.org/10.1109/tgrs.2017.2659663.

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21

Alroy, John. "On four measures of taxonomic richness." Paleobiology 46, no. 2 (March 16, 2020): 158–75. http://dx.doi.org/10.1017/pab.2019.40.

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AbstractThe choice of measures used to estimate the richness of species, genera, or higher taxa is a crucial matter in paleobiology and ecology. This paper evaluates four methods called shareholder quorum subsampling, true richness estimated using a Poisson sampling model (TRiPS), squares, and the corrected first-order jackknife (cJ1). Quorum subsampling interpolates to produce a relative richness estimate, while the other three extrapolate to the size of the overall species pool. Here I use routine ecological data to show that squares and cJ1 pass several basic validation tests, but TRiPS does not. First, TRiPS estimates are insensitive to the shape of abundance distributions, being entirely predicted by total counts of species and of individuals regardless of the details. Furthermore, TRiPS tends not to extrapolate at all when sampling is moderate or intense. Second, all three extrapolators yield lower values when they work with small uniform subsamples of large raw inventories. The third test is a split-analyze-and-sum analysis: each inventory is divided between the most common and least common halves of the abundance distribution, the methods are applied to the half-inventories, and the estimates are summed. Squares and cJ1 perform well here, but TRiPS does not extrapolate as long as the full inventories are reasonably well-sampled. It is otherwise not particularly accurate. The extrapolators are largely insensitive to the influence of abundance distribution evenness, as quantified using Pielou's J and a new index called the ratio of means. Quorum subsampling generally performs well, but it stumbles on the split-analyze-and-sum test and is confounded somewhat by evenness.

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22

Sava, Paul, and Sergey Fomel. "Riemannian wavefield extrapolation." GEOPHYSICS 70, no. 3 (May 2005): T45—T56. http://dx.doi.org/10.1190/1.1925748.

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Riemannian spaces are described by nonorthogonal curvilinear coordinates. We generalize one-way wavefield extrapolation to semiorthogonal Riemannian coordinate systems that include, but are not limited to, ray coordinate systems. We obtain a one-way wavefield extrapolation method that can be used for waves propagating in arbitrary directions, in contrast to downward continuation, which is used for waves propagating mainly in the vertical direction. Ray coordinate systems can be initiated in many different ways; for example, from point sources or from plane waves incident at various angles. Since wavefield propagation happens mostly along the extrapolation direction, we can use inexpensive finite-difference or mixed-domain extrapolators to achieve high angle accuracy. The main applications of our method include imaging of steeply dipping or overturning reflections.

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23

Ahmed, Duraid F., and Ali H. Khalaf. "Development of Artificial Neural Network Model of Crude Oil Distillation Column." Tikrit Journal of Engineering Sciences 22, no. 1 (April 1, 2015): 24–37. http://dx.doi.org/10.25130/tjes.22.1.03.

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Artificial neural network in MATLAB simulator is used to model Baiji crude oil distillation unit based on data generated from aspen-HYSYS simulator. Thirteen inputs, six outputs and over 1487 data set are used to model the actual unit. Nonlinear autoregressive network with exogenous inputs (NARX) and back propagation algorithm are used for training. Seventy percent of data are used for training the network while the remaining thirty percent are used for testing and validating the network to determine its prediction accuracy. One hidden layer and 34 hidden neurons are used for the proposed network with MSE of 0.25 is obtained. The number of neuron are selected based on less MSE for the network. The model founded to predict the optimal operating conditions for different objective functions within the training limit since ANN models are poor extrapolators. They are usually only reliable within the range of data that they had been trained for.

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Mousa, Wail A., Said Boussakta, Desmond C. McLernon, and Mirko Van der Baan. "Implementation of 2D explicit depth extrapolation FIR digital filters for 3D seismic volumes using singular value decomposition." GEOPHYSICS 75, no. 1 (January 2010): V1—V12. http://dx.doi.org/10.1190/1.3294424.

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We propose a new scheme for implementing predesigned 2D complex-valued wavefield extrapolation finite impulse response (FIR) digital filters, which are used for extrapolating 3D seismic wavefields. The implementation is based on singular value decomposition (SVD) of quadrantally symmetric 2D FIR filters (extrapolators). To simplify the SVD computations for such a filter impulse response structure, we apply a special matrix transformation on the extrapolation FIR filter impulse responses where we guarantee the retention of their wavenumber phase response. Unlike the existing 2D FIR filter implementation methods that are used for this geophysical application such as the McClellan transformation or its improved version, this implementation via SVD results in perfect circularly symmetrical magnitude and phase wavenumber responses. In this paper, we also demonstrate that the SVD method can save (depending on the filter size) more than 23% of the number of multiplications per output sample and approximately 62% of the number of additions per output sample when compared to direct implementation with quadrantal symmetry via true 2D convolution. Finally, an application to extrapolation of a seismic impulse is shown to prove our theoretical conclusions.

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25

van Manen, Dirk-Jan, Xun Li, Marlies Vasmel, Filippo Broggini, and Johan Robertsson. "Exact extrapolation and immersive modelling with finite-difference injection." Geophysical Journal International 223, no. 1 (June 26, 2020): 584–98. http://dx.doi.org/10.1093/gji/ggaa317.

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SUMMARY In numerical modelling of wave propagation, the finite-difference (FD) injection method enables the re-introduction of simulated wavefields in model subdomains with machine precision, enabling the efficient calculation of waveforms after localized model alterations. By rewriting the FD-injection method in terms of sets of equivalent sources, we show how the same principles can be applied to achieve on-the-fly wavefield extrapolation using Kirchhoff–Helmholtz (KH)-like integrals. The resulting extrapolation methods are numerically exact when used in conjunction with FD-computed Green’s functions. Since FD injection only relies on the linearity of the wave equation and compactness of FD stencils in space, the methods can be applied to both staggered and non-staggered discretizations with arbitrary-order spatial operators. Examples for both types of discretizations show how these extrapolators can be used to truncate models with exact absorbing or immersive boundary conditions. Such immersive modelling involves the evaluation of KH-type extrapolation and representation integrals in the same simulation, which include the long-range interactions missing from conventional FD injection.

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26

Rutyna, Bruno Benato, Marcio Augusto Villela Pinto, Reverton Luis Antunes Neundorf, Marcio Alexandro Maciel Anunciação, and Marcio André Martins. "Measuring the effectiveness of extrapolation techniques associated with the multigrid method applied to the Navier-Stokes equations." Acta Scientiarum. Technology 44 (March 11, 2022): e57398. http://dx.doi.org/10.4025/actascitechnol.v44i1.57398.

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In this work, we applied different extrapolation techniques in association with the multigrid method to discover which one is the most effective in reducing the iteration error and the processing time (CPU time), as well as in improving the convergence factors. The mathematical model studied refers to the two-dimensional laminar flow of an isothermal time-dependent incompressible fluid modeled by the Navier-Stokes equations, with , solved iteratively with the projection method and the Finite Volume Method. The extrapolation methods used were: Aitken, Empiric, Mitin, scalar Epsilon, scalar Rho, topological Epsilon, and topological Rho. A two-step application was performed: first, extrapolators methods were applied individually after the use of the multigrid method. Then, the best-performing extrapolation techniques were used in the second step, where they were applied between the cycles of the multigrid method. The methods that presented the best convergence properties in the first stage were topological and scalar Epsilon. In the second stage, both methods maintained their performance, however, the topological Epsilon method presented more significant convergence rates than the scalar Epsilon. The other parameters analyzed were: the storage memory peak, the dimensionless norm of the residual based on the initial estimate, and the error norms of iteration. Thus, it was possible to state which extrapolation technique performed best and to compare it with the multigrid method with no extrapolation, which in this study was the topological Epsilon method.

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27

Graves, Robert W., and Robert W. Clayton. "Modeling path effects in three-dimensional basin structures." Bulletin of the Seismological Society of America 82, no. 1 (February 1, 1992): 81–103. http://dx.doi.org/10.1785/bssa0820010081.

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Abstract Path effects for seismic wave propagation within three-dimensional (3-D) basin structures are analyzed using a reciprocal source experiment. In this experiment, a numerical simulation is performed in which a point source is excited at a given location and then the wave field is propagated and recorded throughout a 3-D grid of points. Using the principle of reciprocity, source and receiver locations are reversed. This allows the modeling of path effects into a particular observation site for all possible source locations using only one simulation. The numerical technique is based on the use of paraxial extrapolators and currently tracks only acoustic waves. However, the method is capable of handling arbitrary media variations; thus, effects due to focusing, diffraction, and the generation of multiple reflections and refractions are modeled quite well. The application of this technique to model path effects for local earthquakes recorded at stations in the Los Angeles area of southern California indicates the strong influence of the 3-D crustal basins of this region on the propagation of seismic energy. The modeling results show that the Los Angeles, San Fernando, and San Gabriel basins create strong patterns of focusing and defocusing for paths into these stations from various source locations. These simulations correlate well with earthquake data recorded at both stations. By comparing these calculations with earthquake data, we can begin to evaluate the importance of these basin effects on observed patterns of strong ground motions.

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Arntsen, B., A. Kritski, B. Ursin, and L. Amundsen. "Shot-profile true amplitude crosscorrelation imaging condition." GEOPHYSICS 78, no. 4 (July 1, 2013): S221—S231. http://dx.doi.org/10.1190/geo2012-0211.1.

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The U/D imaging condition for shot profile migration can be used to estimate the angle dependent reflection coefficient, but is difficult to implement numerically because of the spectral division involved. Most techniques for stabilizing the division require a damping factor which might be difficult to estimate and which also introduces bias into the final result. A stable result can be achieved by approximating the imaging condition with a crosscorrelation of the up- and downgoing wavefields at zero time lag, but this will lead to incorrect amplitude-versus-angle (AVA) behavior of the estimated reflection coefficient. We use a simple model for wave propagation of primary reflections in the wavenumber frequency domain and invert the model with respect to the reflection coefficient. By using the properties of wavefield extrapolators it can then be shown that the reflection coefficients can be estimated by crosscorrelation of the upgoing wavefield and a downgoing wavefield where the initial wavefield is the inverse of the wavefield generated by a point source. The new imaging condition gives the correct AVA behavior for horizontal reflectors. For dipping reflectors it is shown that a postmigration correction factor can be used to recover the correct angle behavior of the reflection coefficient. The new imaging condition is numerically stable, does not involve damping factors, is simple to implement numerically, and is a simple modification of the classical crosscorrelation imaging condition. Numerical examples confirm the correct AVA behavior of the new imaging condition.

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29

Vasconcelos, Ivan. "Source-receiver, reverse-time imaging of dual-source, vector-acoustic seismic data." GEOPHYSICS 78, no. 2 (March 1, 2013): WA123—WA145. http://dx.doi.org/10.1190/geo2012-0300.1.

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Novel technologies in seismic data acquisition allow for recording full vector-acoustic (VA) data: pointwise recordings of pressure and its multicomponent gradient, excited by pressure only as well as dipole/gradient sources. Building on recent connections between imaging and seismic interferometry, we present a wave-equation-based, nonlinear, reverse-time imaging approach that takes full advantage of dual-source multicomponent data. The method’s formulation relies on source-receiver scattering reciprocity, thus making proper use of VA fields in the wavefield extrapolation and imaging condition steps in a self-consistent manner. The VA imaging method is capable of simultaneously focusing energy from all in- and outgoing waves: The receiver-side up- and downgoing (receiver ghosts) fields are handled by the VA receiver extrapolation, whereas source-side in- and outgoing (source ghosts) arrivals are accounted for when combining dual-source data at the imaging condition. Additionally, VA imaging handles image amplitudes better than conventional reverse-time migration because it properly handles finite-aperture directivity directly from dual-source, 4C data. For nonlinear imaging, we provide a complete source-receiver framework that relies only on surface integrals, thus being computationally applicable to practical problems. The nonlinear image can be implicitly interpreted as a superposition of several nonlinear interactions between scattering components of data with those corresponding to the extrapolators (i.e., to the model). We demonstrate various features of the method using synthetic examples with complex subsurface features. The numerical results show, e.g., that the dual-source, VA image retrieves subsurface features with “super-resolution”, i.e., with resolution higher than the limits of Born imaging, but at the cost of introducing image artifacts not present in the linear image. Although the method does not require any deghosting as a preprocessing step, it can use separated up- and downgoing fields to generate independent subsurface images.

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Greene, Jennifer C. "Evaluation Extrapolations." American Journal of Evaluation 22, no. 3 (September 2001): 397–402. http://dx.doi.org/10.1177/109821400102200317.

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Greene, J. "Evaluation extrapolations." American Journal of Evaluation 22, no. 3 (2001): 397–402. http://dx.doi.org/10.1016/s1098-2140(01)00153-9.

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Pozniak, Natalija, and Leonidas Sakalauskas. "Fractional Euclidean Distance Matrices Extrapolator for Scattered Data." Jaunųjų mokslininkų darbai 47, no. 2 (December 8, 2017): 56–61. http://dx.doi.org/10.21277/jmd.v47i2.156.

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The paper deals with application of fractional distance matrices to construct the efficient extrapolator of scattered data. The properties of fractional distance matrices are studied in order to develop the linear extrapolator. Study and comparison of developed extrapolator with Shepard extrapolator is performed by computer simulation.

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Baranov, L. A., V. G. Sidorenko, E. P. Balakina, and L. N. Loginova. "Intelligent centralized traffic management of a rapid transit system under heavy traffic." Dependability 21, no. 2 (June 2, 2021): 17–23. http://dx.doi.org/10.21683/1729-2646-2021-21-2-17-23.

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Aim. In today’s major cities, increased utilization and capacity of the rapid transit systems (metro, light rail, commuter trains with stops within the city limits) – under condi[1]tions of positive traffic safety – is achieved through smart automatic train traffic management. The aim of this paper is to choose and substantiate the design principles and architecture of such system.Methods. Using systems analysis, the design principles and architecture of the system are substantiated. Genetic algorithms allow automating train traffic planning. Methods of the optimal control theory allow managing energy-efficient train movement patterns along open lines, assigning individual station-to-station running times following the principle of mini[1]mal energy consumption, developing energy-efficient target traffic schedules. Methods of the automatic control theory are used for selecting and substantiating the train traffic algorithms at various functional levels, for constructing random disturbance extrapolators that minimize the number of train stops between stations.Results. Development and substantiation of the design principles and architecture of a centralized intelligent hierarchical system for automatic rapid transit traffic management. The distribution of functions between the hierarchy levels is described, the set of subsystems is shown that implement the purpose of management, i.e., ensuring traffic safety and comfort of passengers. The criteria are defined and substantiated of management quality under compensated and non-compensated disturbances. Traffic management and target scheduling automation algorithms are examined. The application of decision algorithms is demonstrated in the context of uncertainty, use of disturbance prediction and genetic algorithms for the purpose of train traffic planning automation. The design principles of the algorithms of traffic planning and management are shown that ensure reduced traction energy consumption. The efficiency of centralized intelligent rapid transit management system is demonstrated; the fundamental role of the system in the digitalization of the transport system is noted.Conclusion. The examined design principles and operating algorithms of a centralized intelligent rapid transit management system showed the efficiency of such systems that ensured by the following: increased capacity of the rapid transit system; improved energy efficiency of train traffic planning and management; improved train traffic safety; assurance of operational traffic management during emergencies and major traffic disruptions; improved passenger comfort.

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Swartz, Clifford E. "Extrapolations and prognostications." Physics Teacher 38, no. 1 (January 2000): 5. http://dx.doi.org/10.1119/1.880435.

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Křížek, M., and L. Somer. "Excessive extrapolations in cosmology." Gravitation and Cosmology 22, no. 3 (July 2016): 270–80. http://dx.doi.org/10.1134/s0202289316030105.

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LV, Tao. "Extrapolations for hypersingular integrals." SCIENTIA SINICA Mathematica 45, no. 8 (July 1, 2015): 1345–60. http://dx.doi.org/10.1360/n012014-00210.

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During, Élie. "Art, science : extrapolations, expérimentations." Critique 759-760, no. 8 (2010): 792. http://dx.doi.org/10.3917/criti.759.0792.

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Wells, Benjamin B. "Minimal Extrapolations of Filters." SIAM Journal on Mathematical Analysis 22, no. 3 (May 1991): 796–809. http://dx.doi.org/10.1137/0522050.

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Guidotti, Tee L. "Extrapolations and Public Policy." Archives of Environmental Health: An International Journal 59, no. 10 (October 2004): 527–28. http://dx.doi.org/10.1080/00039890409605169.

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Mohanty, P. M., and M. Acharya. "AN EXTRAPOLATORY QUADRATURE RULEFOR ANALYTIC FUNCTIONS." International Journal of Research -GRANTHAALAYAH 4, no. 6 (June 30, 2016): 8–11. http://dx.doi.org/10.29121/granthaalayah.v4.i6.2016.2631.

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A quadrature rule for the numerical evaluation of integrals of analytic functions along directed line segments in the complex plane has been formulated using the transformed rule based on Gauss Legendre two point quadrature formulas and an interpolatory three point rule. The degree of precision has been increased from five to seven

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GOTO, Fumiaki. "Precision of Richardson-Like Extrapolations." Transactions of the Society of Instrument and Control Engineers 24, no. 2 (1988): 207–9. http://dx.doi.org/10.9746/sicetr1965.24.207.

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Furnstahl, R. J., G. Hagen, T. Papenbrock, and K. A. Wendt. "Infrared extrapolations for atomic nuclei." Journal of Physics G: Nuclear and Particle Physics 42, no. 3 (February 5, 2015): 034032. http://dx.doi.org/10.1088/0954-3899/42/3/034032.

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Travis, Curtis C. "Interspecies extrapolations in risk analysis." Toxicology 47, no. 1-2 (December 1987): 3–13. http://dx.doi.org/10.1016/0300-483x(87)90157-0.

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Hauke, Jan, Augustyn Markiewicz, and Tomasz Szulc. "Inter- and extrapolatory properties of matrix partial orderings." Linear Algebra and its Applications 332-334 (August 2001): 437–45. http://dx.doi.org/10.1016/s0024-3795(01)00294-4.

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Hine, D. "Ovid's Dead Letters [Extrapolations Ex Ponto]." Literary Imagination 3, no. 2 (January 1, 2001): 205–10. http://dx.doi.org/10.1093/litimag/3.2.205.

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Clewell, Harvey J., and Melvin E. Andersen. "Risk Assessment Extrapolations and Physiological Modeling." Toxicology and Industrial Health 1, no. 4 (October 1985): 111–34. http://dx.doi.org/10.1177/074823378500100408.

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The process of assessing the risk associated with human exposure to environmental chemicals inevitably relies on a number of assumptions, estimates and rationalizations. One of the more challenging aspects of risk assessment involves the need to extrapolate beyond the range of conditions used in experimental animal studies to predict anticipated human risks. The most obvious extrapolation required is that from the tested animal species to humans; but others are also generally required, including extrapolating from high dose to low dose, from one route of exposure to another and from one exposure timeframe to another. Several avenues are available for attempting these extrapolations, ranging from the assumption of strict correspondence of dose to the use of statistical correlations. One promising alternative for conducting more scientifically sound extrapolations is that of using physiologically based pharmacokinetic models that contain sufficient biological detail to allow pharmacokinetic behavior to be predicted for widely different exposure scenarios. In recent years, successful physiological models have been developed for a variety of volatile and nonvolatile chemicals, and their ability to perform the extrapolations needed in risk assessment has been demonstrated. Techniques for determining the necessary biochemical parameters are readily available, and the computational requirements are now within the scope of even a personal computer. In addition to providing a sound framework for extrapolation, the predictive power of a physiologically based pharmacokinetic model makes it a useful tool for more reliable dose selection before beginning large-scale studies, as well as for the retrospective analysis of experimental results.

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Chen, Weidong. "Some Aspects of Band-Limited Extrapolations." IEEE Transactions on Signal Processing 58, no. 5 (May 2010): 2647–53. http://dx.doi.org/10.1109/tsp.2009.2036066.

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Lopez, S. "An effective parametrization for asymptotic extrapolations." Computer Methods in Applied Mechanics and Engineering 189, no. 1 (August 2000): 297–311. http://dx.doi.org/10.1016/s0045-7825(99)00297-2.

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Thomas, Anthony W., and Adam P. Szczepaniak. "Chiral extrapolations and exotic meson spectrum." Physics Letters B 526, no. 1-2 (January 2002): 72–78. http://dx.doi.org/10.1016/s0370-2693(01)01481-2.

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Dmochowski, Roger. "On Epidemiology: Extrapolations, Foibles, and Fictions." European Urology 68, no. 1 (July 2015): 113–14. http://dx.doi.org/10.1016/j.eururo.2014.12.039.

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