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From seismic inversion to applied optics: A novel approach to design of complex systems
Abstract
Propagation of seismic waves studied in geophysical prospecting and optic wave propagation through defined media is based on the same physical-mathematical principles. It makes it possible to transfer modern procedures developed in the first field to the second one and back. This paper suggests transferring novel procedures developed in seismic prospecting to applied optics. We selected two progressive approaches for such conversion: (1) Homeomorphic Imaging and (2) Novel description of boundary conditions. The first approach is based on the employment of revealed local theoretical relationships between the geometrical characteristics of two fundamental beams and the geometrical properties of the studied media's geological layers (bodies). Geometrical characteristics of the fundamental beams are spreading functions and curvatures of the special wavefronts. The second approach – a new description of boundary conditions- permits the determination of a perfect seismic (optical) system with the necessary focusing and imaging properties, free from any aberrations. An optimal optical system is determined as an arrangement corresponding to some perfect system with admissible accuracy. Application of the developed procedures in the optical design will permit the application of a description of an optical surface using: (a) Parametric functions, (b) Differential equations, and (c) Mixed (parametric-differential). On this basis, the optical systems consisting of a minimal number of optical elements with complicated shapes might be promptly computed. Another important application field of the suggested methods is the design of optical systems with diffractive elements.
... The velocities between the optical surfaces are constant, and only several interfaces have a spherical form. An equation of a ray may be written in the following general form [36]: ...
... Therefore, two independent solutions permit finding any wavefront curvature using these two fundamental solutions [36]. ...
The concept of infinitesimal elastic deformation and the theory of elastic seismic waves was formed in the first part of the 19th century and was based mainly on the Fermat, Huygens and Snell developments in the theory of optics. At the same time, seismic wave propagation (utilized in geophysical prospecting) and optic wave propagation through defined media are based on the same physical-mathematical principles, making it possible to transfer nonconventional procedures developed in the first domain to the second one and back. In this investigation, we propose transferring advanced methodologies established in seismic prospecting to practical optics. We selected two advanced approaches with the following aims: (a) homeomorphic Imaging; (b) novel description of boundary conditions. The first approach is established with the utilization of the revealed local theoretical relationship between the geometrical features of two fundamental beams and the geometrical properties of hidden geological targets of the media under study. The employed geometrical characteristics of the fundamental beams are spreading functions and curvatures of the singular wavefronts. The second approach is based on a novel description of the boundary conditions. It enables the determination of a faultless seismic (optical) system with the preassigned focusing and imaging assets when any aberrations are absent. An optimal optical system is usually determined as some arrangement agreeing to some perfect system with acceptable correctness. Employment of the developed procedures in the optical design will permit the application of a description of the optical surface using: (1) parametric functions, (2) differential equations, and (3) mixed (parametric-differential). On this basis, optical systems with a minimal number of optical features with complicated shapes can be promptly computed. Another important application field of the suggested methods is the design of optical systems with diffractive elements.
... The seismic "central ray method" employed in the Filizchay deposit (southern slope of the Greater Caucasus) did not yield sufficiently reliable results since the area of high gradients of seismic velocities covered only the uppermost part of the geological section and information on the deeper horizons was extremely distorted (Karayev and Rabinovich 2000). Obviously, the recently developed seismic multifocusing technology for complex media (e.g., Gurevich et al. 2002;Berkovitch and Eppelbaum 2009) might be effectively applied in such areas. Undoubtedly, ore seismics must be integrated with other geophysical methods. ...
The specific nature of mining geophysics, models of media, and the details and the sequence of the interpretation process, including petrophysical studies, have been amply covered in the books (Ismailzadeh et al., 1983a, b; Khesin et al., 1983, 1996; Gadjiev et al., 1984; Karayev and Rabinovich, 2000; Eppelbaum and Khesin, 2012; Eppelbaum et al., 2014) and numerous articles. A separate chapter (Khesin et al., 1988) in (Borisovich and Eppelbaum, 1988) deals with optimizing mining geophysical method interpretation for detailed prospecting under mountainous conditions. These works form the basis for the Azerbaijanian mining geophysics presented in this chapter. The most specific conditions for geophysical mining studies are described below. Solid heterogeneous associations usually outcrop at the Earth’s surface in these regions. These associations are multiply folded, with intensive rupture tectonics (including thrust tectonics). Rocks of various origins and compositions with a broad range of physical properties rapidly change in vertical and lateral directions. The factors mentioned above predetermine the complexity of the images of geophysical fields. The dissected relief, complex geology, difficulties in transporting, and observation all affect the investigation procedure. Due to the curvature of the earth-to-air interface and the rugged observation surface, the relief exerts a pronounced effect on the geophysical observations. They restrict efficiency because they require more intricate and cumbersome equipment and survey systems. At the same time, it would be a mistake to view mountainous conditions solely as an obstacle. Deep erosional truncation and lack or low loose deposit thickness encourage visual geological methods. These make it possible to obtain extensive geological evidence on the nature of anomaly sources, correlate them with geophysical data, and study the physical properties of rocks and ores in natural and artificial exposures. Rugged relief can be used to calculate the actual physical parameters of a section exposed to erosion based on measurements of the corresponding fields on an uneven surface. It also facilitates the geological application of topography data. Outcroppings of mountainous regions promote the integrated application of conventional and nonconventional methodologies.
Outcroppings of mountainous regions promote the integrated application of conventional geophysical methods not only with visual and geological methods but also with geophysical, geochemical, petrophysical, and physical–chemical investigations (including areal study), which can dramatically lessen the ambiguity of geophysical interpretation. According to Solovov (1985), open areas where ore-bearing rocks outcrop onto the surface or are covered with eluvial and diluvial products of their weathering can be divided into two groups. The first has a severe topography and is considered unfavorable for geophysical prospecting. This opinion, as regards typical orogens (mountainous folded areas) of the open (one stage) type, can be easily explained since complex geophysical equipment is challenging to transport and use in highly inaccessible mountainous regions, and anomalies caused by the relief are pronounced and difficult to take into account. However, mountainous regions objectively call for a systematic application of geophysical methods of investigation. Moreover, only these methods can ensure a sequential, deep, and sufficiently rapid study of the endogenic mineralization distribution and its relations to the geological structure. Prospecting for large hidden deposits is conducted with these issues in mind. Many valuable deposits of different types are located in mountainous regions above 2000 m, which comprise about 12 % of the total continental area, and in areas over 1000 m, which account for as much as 30% of the total land. Most part of this area consists of mountainous structures formed or rejuvenated during the Alpine epoch (Khain, 1984).
... This is due to a much higher fold in the MF gathers than in the CMP gathers. 1mcr Interestingly, that MF technology has multi-dimension orientation and may be successfully applied in other physical domains: first of all in optics (Berkovitch and Eppelbaum, 2000) and GPR processing. Indeed, the classic optics and GPR theory are based on the same physical lows. ...
A multifocusing seismic processing (MFSP) is based on the homeomorphic imaging theory and consists of stacking seismic data with arbitrary source-receiver distribution according to a new local moveout correction. Three-parameter travel time representation allows a precise approximation for the CMP travel time curves and the entire multi-coverage travel time surfaces. The MFSP observed data are stacked along an entire travel time surface, whereas only the CMP uses only hyperbolas. The MFSP does not require knowledge of the subsurface model and produces an accurate zero-offset section, even in the cases of complicated geological structure and low relation signal/noise. The optimal sets of parameters obtained in the MFSP (emergence angle of normal rays and two fundamental solutions of differential equations of wavefront) play an essential role in analyzing wavefield distribution. These parameters can be used as initial ones for inverse problem solutions and the construction of the velocity model. Thus, the modified MFSP version may be applied for ground-penetrating radar (GPR) data processing. We illustrate the application of the MFSP technique to the GPR profile aimed to map the stratigraphy at the quarry “Nesher” in central Israel. The results indicate that the presented method may be effectively used with conventional GPR procedures.
... The seismic "central ray method" employed in the Filizchay deposit (southern slope of the Greater Caucasus) did not yield sufficiently reliable results since the area of high gradients of seismic velocities covered only the uppermost part of the geological section and information on the deeper horizons was extremely distorted (Karayev and Rabinovich 2000). Obviously, the recently developed seismic multifocusing technology for complex media (e.g., Gurevich et al. 2002;Berkovitch and Eppelbaum 2009) might be effectively applied in such areas. Undoubtedly, ore seismics must be integrated with other geophysical methods. ...
The specific nature of mining geophysics, models of media, the details and the sequence of the interpretation process, including petrophysical studies, have been amply covered in books (Khesin 1969, 1976; Khesin et al. 1983, 1996) and Eppelbaum’s (1989) Ph.D. dissertation. A separate chapter (Khesin et al. 1988) in (Borisovich and Eppelbaum 1988) deals with the optimization of interpretation for detailed prospecting under mountainous conditions. These works form the basis for the Caucasian mining geophysics presented in this and previous chapters.
... This is due to a much higher fold in the MF gathers than in the CMP gathers. 1mcr Interestingly, that MF technology has multi-dimension orientation and may be successfully applied in other physical domains: first of all in optics (Berkovitch and Eppelbaum, 2000) and GPR processing. Indeed, the classic optics and GPR theory are based on the same physical lows. ...
A multifocusing seismic processing (MFSP) is based on the homeomorphic imaging theory and consists of stacking seismic data with arbitrary source-receiver distribution according to a new local moveout correction. In MFSP, observed data are stacked along a full travel time surface, whereas only hyperbolas are used in the CMP. Three-parameter travel time representation allows obtaining a precise approximation for the CMP travel time curves and the complete multi-coverage travel time surfaces. MFSP does not require knowledge of the subsurface model and produces an accurate zero offset section, even in the cases of complicated geological structure and low relation signal/noise. The optimal sets of parameters obtained in MFSP (emergence angle of normal rays and two fundamental solutions of differential equations of wavefront) play an essential role in analyzing wavefield distribution. These parameters can be used as initial ones for inverse problem solutions and the construction of velocity models. MFSP, with some modification, may be applied for the ground penetrating radar data processing. We illustrate the application of the MFSP technique to the GPR profile aimed to map stratigraphy at quarry “Nesher” in Central Israel. The obtained results show that the presented method may be effectively used together with conventional procedures. Introduction The conventional approach to seismic data processing is based on considering specific types of physical-geological models and determining the time correction procedures. The latter is used in media of arbitrary structures, which, in turn, gives rise to uncertainty as regards their applicability.
... Three-parameter travel time representation permits to obtain a precise approximation for the full multi-coverage travel time surfaces. Interestingly, that MF technology has multi-dimension orientation and may be successfully applied in other physical domains (Berkovitch and Eppelbaum, 2009). The proposed approach was tested in a well-studied geological section of the quarry "Nesher" in Central Israel ( Fig. 1) near town of Ramla. ...
In Israel, a giant number of archaeological objects of various ages, origins, and sizes occur. Different kinds of noise complicate geophysical methods' employment at archaeological sites. In many cases, geodynamical active, multi-layered, and geologically variable surrounding media damages ancient objects and disturbs their physical properties. This calls for applying different geophysical methods armed with modern interpretation technology. The primary focus is on the geophysical methods most frequently applied in Israeli archaeological sites: GPR and high-precise magnetic survey. Other methods (paleomagnetic, resistivity, near-surface seismics, piezoelectric, etc.) are briefly described and reviewed. The number of employed geophysical methodologies is constantly increasing, and now Israeli territory may be considered a peculiar polygon for various geophysical methods testing. Several examples illustrate the practical application of geophysical methods over some typical archaeological remains. The geophysical investigations at archaeological sites in Israel could be tentatively divided into three stages: (1) past (1990), (2) present (1990–2009), and (3) future (2010). The past stage, with several archaeoseismic reviews and minimal application of geophysical methods, was replaced by the present stage with the violent employment of numerous geophysical techniques. It is supposed that the extensive development of multidiscipline physical-archaeological databases will characterize the future stage, employment of all possible indicators for 4-D monitoring and ancient site reconstruction, as well as the application of combined geophysical multilevel surveys using remotely operated vehicles at low altitudes.
The initial development of a medium model is the most crucial stage since the results of interpretation and the investigation, in general, depend considerably upon its quality. A preliminary model is devised to represent the geological objective when preparing a geophysical project for the given area. Otherwise, it is impossible to select the set of methods and interpretation procedures. The latter is revised before interpreting, if necessary.
Geophysical studies of the Caucasus need to cope with the mountainous environments of many of its regions and their inclined (oblique) magnetization in temperate latitudes. The uneven topography impedes geophysical surveys in mountainous areas due to poor accessibility and distorts the measurement results.
The Caucasus is one of the most active segments of the Alpine-Himalayan seismic belt (Khain 2000). The Caucasian region is characterized by intensive deformation and seismicity that accommodates the continental shortening between the Eurasian and Arabian plates, which converge at a rate of about 30 mm/year (De Mets et al. 1990; Jackson 1992). The Caucasus is considered a key area for seismic hazard assessment for the following main reasons: (1) the active tectonics and seismicity rate of the whole area, (2) the availability of abundant multi-disciplinary data and a long-established tradition of hazard assessment, (3) the unique opportunity to test different methodologies in one test area. Ancient earthquakes in the Caucasus have been reported in hundreds of publications. Nikonov (1982) described a very powerful earthquake in the Eastern Caucasus that occurred on January 14, 1668. The author noted that the quake was the strongest detected in the last five centuries, with an estimated M ~ 8. At the same time, Gasanov (2001) described a catastrophic earthquake at M ~ 9 in the Goygol Lake area (Lesser Caucasus) in 1139 (generally speaking, this earthquake triggered the formation of this famous lake). The Dagestan earthquake of 1830 with M ~ 6 was among the strongest in the Northern Caucasus (New Catalog 1977). Gravity, magnetic, electric, VLF, radon gas, temperature, and other geodynamic precursors are analyzed. Separate corresponding mathematical apparatuses are briefly considered.
This book describes the advantages and applications of potential and quasi-potential field interpretation in the complex environments of the Caucasus. The specifics of the media, the petrophysical supports, and various geophysical surveys are discussed. Extensive findings have shown that a common approach to analyzing magnetic (primary), gravitational, and thermal fields, resistivity, self-potential, the electromagnetic field of VLF transmitters, and the field of induced polarization is not only possible but fruitful. This book introduces the reader to interpretation processes' informational content and structure. Methods of reducing noise, especially in zones of rugged relief, are exemplified. The book also presents bright examples of 3D physical-geological modeling of gravity and magnetic fields in complex media. It explores different variants of integrated interpretation based on probabilistic and deterministic approaches, their reliability, and data presentation. Several ways of identifying the precursors of earthquakes from the noise background are discussed. Its detailed description of successful techniques in the rugged, varied terrains of the Caucasus has applications for geophysicists worldwide.
Thermal surveys in the Dead Sea-Arava Valley area have been conducted to investigate the sedimentary cover's present-day geothermal characteristics. The horizontal geothermal gradients have been assessed, and the heat-absorbing and heat-radiating intervals have been identified. The second temperature versus depth derivative is used as an index of the thermal state of the sedimentary cover. Average vertical gradients are low and vary between 20^o and 3O^o C/km. Horizontal geothermal gradients in the study area do not usually exceed 1^o C/km, although higher within the central Dead Sea Graben fault zone. Horizontal gradients may cause excess pressures, which influence the direction of fluid flow within the sedimentary cover. Basaltic intrusions may also cause the migration of overheated waters. Analytical methods based on those applied in magnetic prospecting are used to interpret temperature anomalies quantitatively. These methods can contribute to the location of faults and salt domes.
The book describes a developed system for processing and interpreting geophysical fields under conditions of uneven relief, variable geological medium, oblique magnetization (polarization), and an unknown level of the normal field. The factors mentioned above consist of the main differences from other books. The following procedures are developed or elaborated: methods of terrain correction and use of terrain complexity to reveal additional geological-geophysical information; methods to select 'useful' anomalies against the background noise; and advanced methods for solving inverse and direct geophysical problems. The developed system and its components have been successfully applied in complex physical-geological conditions in the Caucasus and other regions during integrated studies of deep structure, prospecting for economic minerals, and near-surface inhomogeneities.
Multifolding coverage is a basic of modern acquisition systems which record a vast size of seismic information. However existing technique of seismic data processing allows to extract only a small part of this information. Only wave fields, corresponding to the CMP and CS(Rec)P configurations are used in data processing in practice . But multifold systems also record wave fields corresponding to many other types of source receiver pair configurations.
This chapter discusses micro fresnel lenses. It reviews the principles, characteristics, and fabrication techniques of micro Fresnel lenses. If the zone pattern is precisely fabricated, lens characteristics such as the focal length are obtained as designed. It is stressed that the electron-beam writing technique is important for obtaining a precise pattern and blazing zone profiles, and also for redesigning and fabricating the lenses of the different specifications. The problem that needs to be investigated is the efficiency; in particular the optimum relief profile for maximum efficiency requires further theoretical examination. As the relief structure, replicas can be obtained by the stamping method, and therefore the lenses are suitable for mass production. Micro Fresnel lenses is used more widely in various optical systems in the future.
The seismic methods of geophysical exploration utilize the fact that elastic waves travel with different velocities in different rocks. The principle is to initiate such waves at a point and determine at a number of other points the time of arrival of the energy that is refracted or reflected by the discontinuities between different rock formations. This then enables the position of the discontinuities to be deduced.
A derivation and discussion of geometrical spreading factors are given for two- and three-dimensional earth models with curved reflecting boundaries. The spreading factors can be used easily to transform primary reflections in a zero-offset seismic section into true amplitude reflections. These permit an estimation of interface reflection coefficients, either directly or in connection with a true amplitude migration. A seismic section with true amplitude reflections can be described by one physical experiment: the tuned reflector model. 18 refs.
The common shot point method is a new zero-offset stacking and imaging algorithm for multifold-covered seismic reflection data. The basis of the CSP stacking method is a new normal moveout (NMO) time correction based on so-called topological (homeomorphic) imaging. The stacked section obtained provides an optimum zero-offset section and, in addition, new types of images are constructed. Both the CSP and CMP methods are applied to synthetic and real data. Compared with the CMP stack, the CSP section reveals a higher level of coherent reflection energy.
Principles of Optics is one of the classic science books of the
twentieth century, and probably the most influential book in optics
published in the past forty years. This edition has been thoroughly
revised and updated, with new material covering the CAT scan,
interference with broad-band light and the so-called Rayleigh-Sommerfeld
diffraction theory. This edition also details scattering from
inhomogeneous media and presents an account of the principles of
diffraction tomography to which Emil Wolf has made a basic contribution.
Several new appendices are also included. This new edition will be
invaluable to advanced undergraduates, graduate students and researchers
working in most areas of optics.
We propose a new approach for a fast calculation of the so-called multi-offset time field in a 2-D laterally heterogeneous medium. It comprises the traveltimes of all waves emerging from a set of sources, reflected from a given interface and recorded at the corresponding receivers. A common reflecting elements (CRE) system of observation is used. It is simulated by fans of rays emitted upward by fictitious sources (elements) located on the reflecting interface. The calculated traveltimes for the pairs of rays within the fans are used to construct a global time surface representing the multi-offset time field. They are projected through interpolation (for each branch of arrivals) on an organized table which is used for establishing coefficients for 2-D interpolation. The coefficients determine the desired time surface(s). Some results and formulae from the CRE stacking method are used to show how the distribution of the locations of the shot-offset pairs connected to a CRE and the corresponding traveltimes can be obtained by tracing only the central (normal incidence) rays. The accuracy of these formulae is investigated for two types of models. With the proposed approach, the problem of computing traveltimes for any desired configuration of sources and receivers requires only a minimal number of operations. The method is thus very fast, and its efficiency increases with the total number of shot gathers.
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Scientific Israel-Technological Advantages Scientific Herald " of Voronezh State University of
Journal "Scientific Israel-Technological Advantages"
" Scientific Herald " of Voronezh State University of Architecture and Civil Engineering,
Vol.11, № 2, 2009
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Scientific Herald" of Voronezh State University of
Journal "Scientific Israel-Technological Advantages"
"Scientific Herald" of Voronezh State University of Architecture and Civil Engineering,
Vol.11, № 2, 2009
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