Methods and systems for reconstruction of low frequency particle velocity wavefields and deghosting of seismic streamer data.
Abstract
Se describen métodos y sistemas de cómputo para eliminar el fantasma de los datos de cable marino sísmico marino; en particular, una embarcación de exploración sismológica arrastra una cantidad de cables marinos que forman una superficie de adquisición de datos localizada debajo de una superficie libre; los métodos eliminan el fantasma computacionalmente o remueven sustancialmente señales fantasma de los dados sísmicos grabados por receptores del cable marino; los métodos de eliminación de fantasma incluyen compensación de baja frecuencia para recuperar información del campo de onda de velocidad vertical que se pierde normalmente debido a una baja proporción de señal a ruido sobre un rango de baja frecuencia independiente de las condiciones de superficie libre o de la forma de la superficie de adquisición de datos.

Term
6.8 yearsleft in the term
Expires 10 July 2033.
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12 claims: 3 independent, 9 dependent
- 1NOVEDAD DE LA INVENCIÓN IMPI INSTITUTO MEXICANO DE LA PROPIEDAD industrial REIVINDICACIONES 5 1Un método para eliminar el fantasma de los datos sísmicos marinos que se deben llevar a cabo mediante un sistema de cómputo que incluye uno o más procesadores y uno o más dispositivos para almacenamiento de datos, el método comprende:calcular los campos de onda de velocidad vertical relacionados con receptores sísmicos marinos con base 10 en la reflectividad de un tiempo variante, para un perfil de superficie libre arbitrariamente irregular sobre los receptores sísmicos marinos y para frecuencias debajo de una frecuencia de umbral que usa campos de onda de presión medidos en los receptores;y calcular campos de onda sin fantasma del receptor con base en los campos de onda de presión medidos, campos de 15 onda de velocidad vertical medidos en receptores sobre la frecuencia de umbral, y los campos de onda de velocidad vertical calculados.
- 22,- El método de conformidad con la reivindicación 1, caracterizado además porque calcular campos de onda de velocidad vertical comprende:descomponer los campos de onda medidos en campos de onda 20 ascendentes y descendentes;calcular una imagen de perfil de superficial libre de la superficie libre como una función de los campos de onda ascendentes y descendentes;calcular una extensión del perfil de superficie libre del perfil de superficie libre asumiendo una superficie libre congelada y al usar un espectro de irregularidad espacial de la superficie libre;y caféWí^^&leq Aí»-· L> °' campo de onda de superficie libre para el perfil do ouueiíiuu JjLHé V leí extensión de perfil de superficie libre.
- 3- El método de conformidad con la reivindicación 2, 5 caracterizado además porque también comprende:calcular un gradiente del campo de onda en ubicaciones de coordenada del receptor debajo de la frecuencia de umbral al usar la reflectividad de campo de onda de superficie libre;y calcular el campo de onda de velocidad vertical debajo de la frecuencia de umbral como una función del gradiente del campo de onda de presión en 10 las ubicación de coordenada de receptor.
- 4- El método de conformidad con la reivindicación 2, caracterizado además porque descomponer los campos de onda medidos en campos de onda ascendentes y descendentes también comprende descomponer los campos de onda de presión medidos en campos de onda de 15 presión ascendentes y descendentes y descomponer los campos de onda de velocidad vertical medidos en campos de onda de velocidad vertical ascendentes y descendentes.
- 5- El método de conformidad con la reivindicación 1, caracterizado además porque calcular los campos de onda sin fantasma del 20 receptor también comprende calcular campos de onda de presión con señal fantasma removida como una función de los campos de onda de velocidad vertical medidos y calcular los campos de onda de velocidad vertical debajo de la frecuencia de umbral y calcular campos de onda de velocidad vertical sin ΙΜΡΪ fantasma del receptor como una función de los campos sin fantasma.
- 66, - El método de conformidad con la reivindicación 1, caracterizado además porque los receptores sísmicos forman una superficie 5 de adquisición de datos arbitrariamente formados.
- 77, - El método de conformidad con la reivindicación 1, caracterizado además porque la frecuencia de umbral es una frecuencia más alta que un rango de frecuencia sobre la que los campos de onda de velocidad vertical medidos tienen una baja proporción de señal a ruido. 10 8..- Un sistema de cómputo para eliminación del fantasma de los datos sísmicos marinos durante un tiempo variante, para una superficie libre arbitrariamente irregular, el sistema de cómputo comprende:uno o más procesadores;y uno o más dispositivos para almacenamiento de datos;en donde el uno o más procesadores están configurados para: recuperar de uno 15 o más dispositivos para almacenamiento de datos los datos de campo de onda de presión medidos en receptores sísmicos marinos sobre un rango de frecuencia y campos de onda de velocidad vertical medidos en los receptores sobre una frecuencia de umbral del rango de frecuencia;calcular campos de onda de velocidad vertical relacionados con receptores sísmicos marinos para 20 frecuencias debajo de la frecuencia de umbral al usar los campos de onda de presión medidos;y calcular campos de onda sin fantasma del receptor sobre un rango de frecuencia con base en campos de onda de presión medidos, los campos de onda de velocidad vertical medidos sobre la frecuencia de umbral, IMPI y los campos de onda de velocidad vertical calculadod! WT ^’I?^%7¿ 9. - El sistema de cómputo de conformidad con la reivindicación
- 88, caracterizado además porque calcular campos de onda de velocidad vertical para frecuencias debajo de la frecuencia de umbral también 5 comprende:descomponer los campos de onda medidos en campos de onda ascendentes y descendentes;calcular una imagen de perfil de superficial libre de la superficie libre como una función de los campos de onda ascendentes y descendentes;calcular una extensión del perfil de superficie libre del perfil de superficie libre asumiendo una superficie libre congelada y al usar un espectro 10 de irregularidad espacial de la superficie libre;y calcular la reflectlvidad de campo de onda de superficie libre para el perfil de superficie libre y la extensión de perfil de superficie libre. 10. - El sistema de cómputo de conformidad con la reivindicación
- 99, caracterizado además porque también comprende:calcular un gradiente del 15 campo de onda en ubicaciones de coordenada del receptor debajo de la frecuencia de umbral al usar la reflectividad de campo de onda de superficie libre;y calcular el campo de onda de velocidad vertical relacionado con los receptores para frecuencias debajo de la frecuencia de umbral como una función del gradiente del campo de onda de presión en las ubicaciones de 20 coordenada de receptor. 11. - El sistema de cómputo de conformidad con la reivindicación 9, caracterizado además porque descomponer los campos de onda medidos en campos de onda ascendentes y descendentes también comprende IMPI wrnrro msxjcan# descomponer los campos de onda de presión medidos en camp A * S9WD< presión ascendentes y descendentes y descomponer los^wnpee-ete-eeda-da. velocidad vertical medidos en campos de onda de velocidad vertical ascendentes y descendentes. 5 12,- El sistema de cómputo de conformidad con la reivindicación 8, caracterizado además porque calcular los campos de onda sin fantasma de receptor también comprende calcular campos de onda de presión con señal fantasma removida como una función de los campos de onda de velocidad vertical medidos y calcular los campos de onda de velocidad vertical debajo
- 1010 de la frecuencia de umbral y calcular campos de onda de velocidad vertical sin fantasma del receptor como una función de los campos de onda de presión sin fantasma.
- 1113 - El sistema de cómputo de conformidad con la reivindicación 8, caracterizado además porque los receptores sísmicos marinos forman una 15 superficie de adquisición de datos arbitrariamente formados.
- 1214 - El sistema de cómputo de conformidad con la reivindicación 8, caracterizado además porque la frecuencia de umbral es una frecuencia más alta que un rango de frecuencia sobre la que los campos de onda de velocidad vertical medidos tienen una baja proporción de señal a ruido.
Independent claims12
387 paragraphs in 83 sections, as filed
(54) Title: METHODS AND SYSTEMS FOR THE RECONSTRUCTION OF LOW FREQUENCY PARTICLE SPEED WAVE FIELDS AND REMOVAL OF SEA-MARINE CABLE DATA GHOSTS.
(54) Title: METHODS AND SYSTEMS FOR RECONSTRUCTION OF LOW FREQUENCY PARTIOLE VELOCITY WAVEFIELDS AND DEGHOSTING OF SEISMIC STREAMER DATA.
(57) Summary
Methods and computation systems are described to eliminate the phantom of marine seismic cable data; in particular, a seismic survey vessel drags a number of marine cables that form a data acquisition surface located below a free surface; the methods eliminate phantom computationally or substantially remove phantom signals from seismic data recorded by receivers on the marine cable; Phantom removal methods include low-frequency compensation to retrieve vertical velocity wavefield information that is normally lost due to a low signal-to-noise ratio over a low-frequency range independent of free surface or surface conditions. shape of the data acquisition surface.
(57) Abstract
Computational methods and systems for deghosting marine seismic streamer data are described. In particular, an exploration-seismology vessel tows a number of streamers that form a data acquisition surface located beneath a free surface. The methods computationally deghost or substantially remove receiver ghost signáis from seismic data recorded by steamer receivers. The deghosting methods inelude low frequeney compensation to recover vertical velocity wavefield Information that is typically lost due to a low signal-to-noise ratio over a low frequeney range independent of the free surface conditions or the shape of the data acquisition surface.
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Mexican Institute of Industrial Property
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PATENT TITLE NO. 340795
Owner (s): PGS GEOPHYSICAL AS
Address: Strandveien 4, N-1366, Lysaker, NORWAY
Name: METHODS AND SYSTEMS FOR THE RECONSTRUCTION OF LOW FREQUENCY PARTICLE SPEED WAVE FIELDS AND REMOVAL OF SEASMIC MARINE CABLE DATA GHOSTS.
Classification: lnt.CI.8: G01V1 / 36; G01V3 / 38
Inventor (s): WALTER SOLLNER; OKWUDILI ORJI; STIAN HEGNA
REQUEST
<td>Number:</td><td>Presentation date:</td><td>Hour:</td>
<td>MX / a / 2013/008073</td><td>July 10,2013</td><td> 14:52</td>
<td></td><td>PRIORITY</td><td></td>
<td>Country:</td><td>Date:</td><td>Number:</td>
<td>US</td><td>July 10, 2012</td><td> 13/545,609</td>
Validity: Twenty years
Expiration Date: July 10, 2033
The reference patent is granted based on articles 1, 2 fraction V, 6 fraction III, and 58 of the Industrial Property Law.
In accordance with article 23 of the Industrial Property Law, this patent has a non-renewable term of twenty years, counted from the filing date of the application and will be subject to the payment of the fee to keep the rights in force.
Whoever signs this title> does so based on the provisions of articles 84 sections III and 7 ° bis 2 of the Industrial Property Law (Official Gazette of the Federation (D.OF.) 06/27/1991, amended on 08/02/1984, 10/25/1986, 12/26/1887, 05/17/1999, 01/26/2004, 06/16/2005, 05/25/2006, 05/06/2008, 01/06/2010, 06/18/2010, 06/28/20'0 01/27/2012 and 04/08/2012), articles 1, 3, section V, subsection a), 4 and 12th fractions l and lll of the Regulations of the Mexican Institute of Industrial Property (DO.F 14/12/1988, amended on 07/01/2002, 07/07/2004 # 07/28/2004 and 09/07/2007 ); articles 1 ° 3 °. 4 '5 ”fracclór V .naso a), 18“ factions I and lll and 30 of the Organic Eatatuto of the Mexican Institute of Industrial Property (DOF 2 * · 12''9Β · ί o, i «ms a' 3/10 / 2002,28 / 07/7004, 08/04/2004 and 13ft) 9/2007); 1st, 3rd and 5th subsection a) of the Agreement that delegates powers to the Deputy Directors General, Cocrdinsrior, Divisional Directors #: Holders of the Regional Offices, Sabd rector® Divisionales, Coordinators ÓepeOMlnlplií / ÚR * subordinates of the Maxiumude Property Institute Industnal. (DOF 15 / 13PI999, rearmed on 02/04/2000, 07/29/2004, 08/04/2004 and 09/13/2007). R
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METHODS AND SYSTEMS FOR RECONSTRUCTION OF FIELDS
LOW FREQUENCY PARTICLE SPEED WAVE AND
REMOVAL OF DATA GHOSTS FROM MARINE CABLES
SEISMIC
BACKGROUND OF THE INVENTION
In the last decades, the oil industry has invested heavily in the development of marine seismic study techniques that generate knowledge of underground formations below a body of water in order to find and extract valuable mineral resources, such as oil. High resolution seismic images of an underground formation are essential for quantitative seismic interpretation and improved monitoring of deposits. For a typical marine seismic survey, an exploratory seismology vessel tows one or more seismic sources and one or more marine cables below the water surface and over an underground formation to be investigated for mineral deposits. The vessel contains seismic acquisition equipment, such as navigation control, seismic source control, seismic receiver control, and recording equipment. Seismic source control causes one or more seismic sources, which are normally air guns, to produce acoustic pulses at selected times. Each impulse is a sound wave that travels through the water and into the underground formation. At each interface
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INDUSTKIAE Among the different types of rocks, a portion of the sound wave is refracted, a portion of the sound wave is transmitted, and another portion is reflected back to the body of water to propagate to the surface. The marine cables dragged behind the boat are elongated cable-like structures. Each marine cable includes a number of seismic receivers or sensors that detect the pressure and / or changes in particle motion in the water created by the reflected sound waves back into the water from the underground formation.
Sound waves that propagate upward from the underground formation are referred to as ascending field waves that are detected by the receivers and converted into seismic signals that are recorded by the recording equipment and processed to produce seismic images that characterize the geological structure and properties of the underground formation to be investigated. However, seismic signals can also include a source ghost produced by sound waves that are first reflected from the sea surface before the waves move on the sub-surface to produce stray wave fields detected by the receivers. The ghosts of the source are lagged in time relative to the sound waves that travel directly from the source to the underground formation. As a result, source ghosts can amplify some frequencies and attenuate other frequencies and typically manifest as spectral notches in the seismic waveform.
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recorded, making it difficult to obtain accurate high-resolution seismic images of the underground formation. In addition to source ghosts, the seismic signal may also include receiver ghosts produced by scattered sound waves that are first reflected from the sea surface before reaching the receivers. Receiver phantoms can also amplify some frequencies and attenuate other frequencies and typically manifest as receiver phantom notches. As a result, those working in the oil industry continue to search for systems and methods to eliminate the effects of phantom reflections, or removal of phantoms from seismic signals.
BRIEF DESCRIPTION OF THE DRAWINGS
Figure 1 shows a domain volume of the earth's surface.
Figure 2 shows the characteristics of the subsurface of an underground formation in the lower portion of the domain volume shown in Figure 1.
Figures 3A-3C show a seismic exploration method whereby digitally encoded data is instrumentally acquired for subsequent seismic exploration processing and analysis in order to characterize the structures and distributions of underlying features and materials. to
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solid surface of the earth.
Figures 4A-4C show the processed waveforms generated from hydrophone and geophone outputs.
Figure 5 shows an elevation view of a seismic survey vessel towing a source and marine cable located below a free surface.
Figure 6 shows a flowchart related to a method for a source that removes phantoms of seismic signal data.
Figures 7A-7B introduce coordinates and terminology related to computation processes for ghost removal from the source.
Figure 8 shows an elevation view of a seismic survey vessel and a marine cable and includes an enlarged view of a receiver located along the marine cable.
Figures 9A-9C each show an exemplary graph representing one aspect of the wave field decomposition.
Figures 10A-10C each show an exemplary graph representing an aspect for calculating an image-processed free surface profile.
Figure 11 shows a flow control diagram of a computational method to calculate a free surface profile.
Figure 12A shows a graph of a hypothetical free surface and an image-processed free surface profile that is
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approaches the free surface.
Figure 12B shows a graph of a hypothetical free surface and an extended image-processed free surface profile approaching the free surface.
Figure 13 shows parameters of an equation representing the reflectivity of the free surface wave field.
Figures 14A-14B show a two-dimensional view of an exemplary model space related to a volume of fluid.
Figure 15 shows a segment of a marine cable located below a free surface imaged from a volume of fluid.
Figure 16 shows an example of a generalized computing system that executes an efficient method for removing ghosts from a scattered wave field.
Figures 17A-17E show graphs of a marine cable and free surface.
DETAILED DESCRIPTION OF THE INVENTION
Methods and computation systems are described for the receiver to remove the phantoms from the marine seismic cable data. In particular, a seismic survey vessel tows a number of marine cables that form a data acquisition surface
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located below an air / fluid surface referred to as a free surface. ”Marine cables include receivers that measure pressure and wave fields of particle motion, which are digitally encoded and stored. The methods computationally remove phantoms or substantially remove receiver phantom signals from seismic data recorded by receivers independent of free surface conditions or the shape of the data acquisition surface. In other words, the methods described below computationally remove the receiver ghost signals from seismic data without assuming free surface constraints or assuming constraints on the shape of the data acquisition surface, such as assuming a frozen flat free surface (i.e. stationary) and assume a frozen flat and horizontal data acquisition surface. Phantom removal methods include low frequency compensation to recover vertical velocity wavefield information that is normally lost due to a low signal to noise ratio on a low frequency scale.
The following discussion includes two subdivisions: Subdivision (I) provides a general idea of seismological exploration; and in subdivision (II) a discussion of computational processing methods for the receiver to remove phantoms from seismic signal data as an example of computational processing methods and systems to which this description is directed. The reading of the
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<sub>7</sub> IMPI 'Μ »? ΠΤϋΤΟ MEXICAN
1 «INDUSTRIAL PROPERTY first subdivision may be omitted by those already familiar with marine seismological exploration.
I. A general idea of marine seismological exploration
Figure 1 shows a domain volume of the earth's surface. Domain volume 102 comprises a solid volume of sediments and rocks 104 below the solid surface 106 of the earth which, in turn, is below a volume of water fluid 108 within an ocean, inlet or bay, or a large freshwater lake. The domain volume shown in Figure 1 represents an example of an experimental domain for a class of analytical and observational seismic exploration techniques and systems called marine seismological exploration.
Figure 2 shows the characteristics of the subsurface of an underground formation in the lower portion of the domain volume shown in Figure 1. As shown in Figure 2, for seismological exploration purposes, the volume of fluid 108 is a relatively uninteresting, generally homogeneous volume covering the solid volume 104 of interest. However, although fluid volume 108 can be explored, analyzed, and characterized with relative precision using many different types of methods and probes, including remote sensing submersibles, sonar, and other such devices and methods, the volume of the cortex solid 104 below the volume of wwnrrTa mrxscakd ΰί the rK.¡nWAi>
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fluent, comparatively more difficult to test and characterize. In contrast to the underlying fluid volume108, the solid volume 104 is significantly heterogeneous and anisotropic and includes many types of features and materials of interest to exploration seismologists. For example, as shown in Figure 2, the solid volume 104 may include a first layer of sediment 202, a first layer of fractured and elevated rock 204, and a second layer of underlying rock 206 below the first layer of rock. . In certain cases, the second rock layer 206 may be porous and contain a significant concentration of liquid hydrocarbon 208 that is less dense than the material in the second rock layer and therefore rises within the second rock layer 206. In the case shown in Figure 2, the first rock layer 204 is nonporous, and therefore forms a cap that prevents further upward migration of the liquid hydrocarbon, which consequently accumulates in a saturated layer of hydrocarbon 208 below first layer of rock 204. An objective of seismological exploration is to identify the locations of hydrocarbon-saturated porous strata within the volumes of the earth's crust below the earth's solid surface.
Figures 3A-3C show a seismological exploration method whereby digitally encoded data is acquired instrumentally for subsequent seismological exploration processing and analysis in order to characterize the structures and distributions of characteristics and materials of a training
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Mexican instituvtq THE underground property. Figure 3A shows an example of a "seismic survey vessel 302, equipped to perform a continuous series of seismic survey experiments and data collections. In particular, vessel 302 tows one or more marine cables 304-305, through an approximately constant depth plane, generally located a number of meters below fluid surface 306. Marine 304-305 cables are long cables containing power and data transmission lines, to whose receivers, also referred to as sensors, are connected at regular intervals. In a type of seismic survey, each receiver, such as the receiver represented by the shaded disk 308 in Figure 3A, comprises a pair of seismic receivers including a geophone, which detects vertical displacement within the fluid medium over time at detect speeds of movement or accelerations of the particle, and a hydrophone that detects the variations in pressure that occur over time. Marine cables 304305 and vessel 302 include sophisticated electronic sensing systems and data processing facilities that allow receiver readings to be correlated with absolute positions on the free surface and absolute three-dimensional positions relative to a three-dimensional coordinate system. arbitrary. In Figure 3A, receivers along marine cables are shown below free surface 306, with receiver positions correlated with underlying surface positions, such as a surface position
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310 correlated to the position of receiver 308. Vessel 302 also tows one or more acoustic wave sources 312 that produce pressure pulses at spatial and temporal intervals as vessel 302 and towed marine cables 304-305 move toward forward through free surface 306. Sources 312 can also be towed by other vessels or otherwise arranged in fluid volume 108.
Figure 3B shows a spherical acoustic expanding wavefront, represented by semicircles of increasing radius centered on acoustic source 312, such as semicircle 316 followed by an acoustic pulse emitted by acoustic source 312. Wave fronts, in In effect, they are shown in the cross section of the vertical plane in Figure 3B. As shown in Figure 3C, the outward and downward expanding sound wave field, shown in Figure 3B, finally reaches solid surface 106, at which point the outward and downward expanding sound waves are reflected partially from the solid surface and are partially refracted down into the solid volume, becoming elastic waves within the solid volume. In other words, in the fluid volume, the waves are compression pressure waves, or P waves, the propagation of which can be shaped by the acoustic wave equation whereas, in a solid volume, the waves include both the P waves. as to transverse waves, or S waves, whose propagation can be modeled by the elastic wave equation. Within said solid volume, at each interface
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between different types of materials or in discontinuities in intensity or in one or more other diverse physical characteristics or parameters, the waves that propagate downward are partially reflected and partially refracted, on the solid surface 106. As a result, each point on the solid surface and within the underlying solid volume 104 becomes a potential secondary point source from which the acoustic and elastic waves, respectively, can emanate upward toward the receptors in response to the pressure impulse. emitted by acoustic source 312 and the downward propagating elastic waves generated from the pressure pulse.
As shown in Figure 3C, secondary waves of significant amplitude are generally emitted from points at or near solid surface 106, such as point 320, and from points at or very near a discontinuity in solid volume 104, such as Points 322 and 324. Tertiary waves can be emitted from the free surface 306 towards the solid surface 106 in response to the secondary waves emitted by the solid surface and the subsurface characteristics.
Figure 3C also shows the fact that secondary waves are generally emitted at different times within a time interval following the initial pressure pulse. A point on solid surface 106, such as point 320, receives a pressure disturbance corresponding to the initial pressure pulse faster than a point within solid volume 104, such as points 322 and 324. Similarly,
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a point on the solid surface directly underlying the sound source receives the pressure boost faster than a point that resides more distant on the solid surface. Accordingly, the times at which secondary and higher order waves are emitted from various points within the solid volume are related to the distance, in three-dimensional space, of the points from the acoustic source.
Acoustic and elastic waves, however, travel at different speeds within different materials, as well as within the same material under different pressures. Consequently, the displacement times of the initial pressure pulse and the secondary waves emitted in response to the initial pressure pulse are complex functions of the distances from the acoustic source, as well as the materials and physical characteristics of the materials through which, the acoustic wave corresponds to the displacement of the initial pressure pulse.
Additionally, as shown in Figure 3C, for the secondary wave emitted from point 322, the shapes of the expanding wavefronts can be altered as the cross interfaces of wavefronts and as the speed of sound varies in the half traversed by the wave. Overlapping waves emitted from within domain volume 102 in response to the initial pressure pulse is a generally very complicated wave field that includes information on the shapes, sizes, and material characteristics of domain volume 102 that includes information on the shapes, sizes and locations of
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Interest for the exploration of seismologists.
The complicated wave field that occurs in response to the Initial Pressure Boost is sampled, over time, by receivers positioned along the marine cables towed by a seismic survey vessel. Figures 4A-4B show the processed waveforms generated by a hydrophone and geophone, respectively. As shown in Figure 4A, the waveform recorded by the hydrophone represents the pressure at moments after the Initial Pressure Boost, with the amplitude of the waveform at a point in time related to the pressure at the hydrophone at the point in time. Similarly, as shown in Figure 4B, the geophone provides an Indication of the movement, velocity, or acceleration of the fluid particle, in a vertical direction, with respect to time. The pressure and particle motion signals represented by the waveforms in Figures 4A and 4B can be used to generate pressure wave fields and vertical velocity wave fields which, in turn, can be used for processing seismic, as attenuation of multiple marine seismic data. However, since recorded particle motion signals are often contaminated with low-frequency noise due to vibrations from towed marine cables, the signal-to-noise ratio on a low-frequency scale is poor for the combined signals. Figure 4C shows a graph of relative amplitude against a scale of acoustic frequencies of
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the exemplary particle motion and pressure signals depicted in Figures 4A and 4B. Solid curve 401 and dotted curve 402 represent the recorded signals of pressure and movement, respectively. Figure 4C reveals a portion 403 of particle motion signals with relative amplitudes greater than zero, corresponding to a low signal-to-noise ratio on a low-frequency scale from 0Hz to an acoustic threshold frequency, fth ·, which in the example in figure 4C is about 20Hz. As a result, the particle motion signal below the threshold frequency fth · cannot normally be used to perform seismic processing. On the other hand, the pressure wave field below the threshold frequency f * can be used to recreate some information lost by the particle motion sensor when the spectrum of the pressure signal has a signal to noise ratio. satisfactory and when the depth of the particle motion and pressure sensors is known. As a result, the depth of the marine cable is selected so that the frequency of the first spectral notch in the pressure sensor signal caused by surface reflections is greater than the threshold frequency fth ·.
The free surface acts as an almost perfect acoustic reflector, causing phantom effects on the recorded seismic data. At the source location of each pulse, a time-delayed reflection from the free surface, called a source phantom, routes the seismic wave field that travels directly from the source.
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Wrí'STRtAL source in the subsurface of the earth. As a result, both low-frequency and high-frequency information are penalized, and the earth's subsurface cannot be imaged accurately at all seismic frequencies. Also at each receiver location along a marine cable, a time-delayed reflection from the free surface, called a receiver phantom, interferes in a continuous and undesirable manner with the seismic wave field scattered directly from the subsurface. from the ground to the receiver. Figure 5 shows an elevation view of a seismological survey vessel 502 towing a source 504 and a marine cable 506 located below a free surface 508. Figure 5 includes four beam paths representing four ways in which the Wave fields related to the same pulse can be shifted from source 504 to a receiver 510 located along marine cable 506. Dashed line 514 represents a primary ray or direct path where an impulse output from source 504 is displaced directly on subsurface 512 and related stray wave fields are displaced directly from subsurface 512 to receiver 510. Solid line 516 represents a ray path that includes a phantom from source 571 produced by sound waves that are first reflected from free surface 508 before entering subsurface 512 to produce scattered wave fields moving directly from subsurface 512 to receiver 510. Long dashed line 518 represents a ray path that includes a phantom of the
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receiver 519 produced by sound waves moving directly from source 504 to subsurface 512 to produce scattered wave fields that are reflected from free surface 508 before being detected by receiver 510. Dotted dashed line 520 represents a trajectory of the lightning including a phantom from source 521 and a phantom from receiver
522.
The scattered wave fields related to the primary ray paths are ideally the desired seismic wave fields to be detected by the receivers. However, in practice source ghosts and receiver ghosts are also detected, and since ghosts are time-delayed, ghosts can constructively and destructively intervene with the recorded waveforms of the seismic wave field. As a result, ghosts can lead to imprecise seismic images of the underground formation located below the fluid volume. The computational methods and systems described below are directed to the receiver that removes the phantom images from the seismic signal data for an arbitrarily irregular free surface, variable with time and without restrictions on the shape of the data acquisition surface defined by marine cables. The methods are also not limited to using vertical velocity and pressure wave fields above the threshold frequency Ak. In other words, the receiver that removes the phantom images from the seismic signal data can be achieved for the vertical velocity and pressure wave fields over the entire
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frequency scale.
II. Methods for removing phantoms from the receiver as an example of computational processing methods and systems 5 Figure 6 shows a control flow diagram 600 related to a computational method for the receiver that removes phantom images from data from seismic signal with low frequency compensation. Flowchart 600 summarizes a general computation process for the receiver that removes phantom images from seismic data.
Each block of diagram 600 is described below in a separate subdivision.
Estimate the depth of the marine cable
In block 601 of Figure 6, the depth of the marine cable is estimated. Figures 7A-7B introduce coordinates and terminology related to computational processes for ghost removal from the receiver. Figure 7A shows a top or xy plane view of an exemplary seismic survey vessel 702 towing a source 704 and four separate marine cables 706-709, located below a free surface. Each marine cable is attached at one end to vessel 702 and at the opposite end to a buoy, such as buoy 710 attached to marine cable 708. Marine cables 706-709 ideally form a flat horizontal acquisition surface located below the surface
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free. However, in practice, the acquisition surface can be smoothly variable in the z direction due to active marine currents and weather conditions. In other words, towed marine cables can also undulate as a result of dynamic fluid conditions. FIG. 7B shows an elevation or xz plane view of marine cable 708 located below a free surface 712. Figure 7B represents a snapshot, at an instant in time, of the undulating free surface 712 and corresponding smooth waveform on marine cable 708. Figure 7A includes an xy plane 714 and Figure 7B includes an xz plane thereof Cartesian coordinate system used to specify the locations of coordinates within the fluid volume with respect to the three spatial, orthogonal coordinate axes labeled x, y, and z. The x coordinate only specifies the position of a point in a direction parallel to the length of the marine cables, and the y coordinate only specifies the position of a point in a direction perpendicular to the x axis and substantially parallel to the free surface 712, and the z coordinate only specifies the position of a point perpendicular to the xy plane. As shown in Figure 7B, marine cable 708 is at a depth, z<sub>r</sub>, which can be estimated at various locations along the marine cables from hydrostatic pressure measurements made by depth controllers (not shown) attached to the marine cables. Depth controllers are typically placed at intervals of approximately 300 meters along each marine cable. The estimated depths of the
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Marine cable is then used to calculate an interpolated two-dimensional shape of the marine cable that approximates the waveform of a real marine cable at an instant in time. Furthermore, the estimated depths of the marine cable can be used to calculate a three-dimensional interpolated surface approximation of the acquisition surface. Depth z<sub>r</sub> and the elevation of the free surface profile are estimated with respect to the geoid, which is represented in Figure 7B by the dotted line 718. The geoid is the hypothetical surface of the earth that worldwide coincides with the mean sea level and it is used to define the elevation of zero (i.e. z = 0). Shaded disks, like shaded disk 720, represent separate receivers at regular intervals, Δχ. The coordinates of receiver 720 are given by, where the depth z<sub>r</sub> it can be an interpolated value.
Measurement of velocity and pressure wave fields
Returning to Figure 6, at block 602, the pressure and velocity wave fields are measured at the receivers. Figure 8 also shows an elevation view of vessel 702 and marine cable 708 and includes an enlarged view 802 of a receiver 804 located along marine cable 708. The enlarged view 802 reveals that receivers are truly dual sensors that they include a pressure sensor 806, such as a hydrophone, and a motion sensor 808, such as a geophone. Each pressure sensor measures a pressure wave field denoted by p (* r.yr,<sup>z</sup>r.<sup>c</sup>) and
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each of the motion sensors measures a velocity wave field denoted by v ^ (x<sub>r</sub>,and<sub>r</sub>, z<sub>r</sub>, t), where x<sub>r</sub> = yy represent the spatial coordinate x- yy of the sensor, and t represents time. Index m is a positive integer used to identify the receiver and is also referred to as a "channel" index. The acquisition surface, as shown in Figure 7A, has 4 marine cables 706-709 and each marine cable includes 14 receivers. Receivers can be indexed from 0 to 13. For example, a receiver 810 on marine cable 708 located closest to vessel 702 may be assigned a channel index m = 0, and a receiver 812 located along the same marine cable 708 farthest from vessel 702 may be assigned a channel index m = 13, with dual sensors between them indexed consecutively from 1 to 12. The motion sensors can be mounted inside a gimbal to orient the motion sensors to detect movement of particles in a normal direction to the marine cable as represented by the normal unit vector 814 in the enlarged view 802. Thus, the Motion sensors measure a velocity wave field directed normal to the smooth varied marine cable
708, where the subscript vector * · represents a normal unit vector pointing in the normal direction 814 to marine wire 608 in the xz plane. However for particle motion measured below the threshold frequency, the signal-to-noise ratio is normally below as previously described with reference to Figure 4C. As a result,
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vertical velocities below the threshold frequency-¿A ... narmatenente-Bo · can be determined exactly. Methods include low frequency compensation described below to calculate vertical velocity wave fields at marine cable receivers below threshold frequency
Wave field decomposition
Returning to figure 6, in block 603 the decomposition of the wave field is performed in the pressure and velocity wave fields for frequencies greater than the threshold frequency ^<sup>th</sup> described above with reference to Figure 4C. In the following description the spatial component is ignored in order to simplify the description: It should be noted that in practice the spatial component y is included. In other words, in the approach below, the three spatial coordinates of the measured pressure wave field<sup>z</sup>r> * 0 are reduced to two spatial coordinates in the representation of the pressure wave field,, and the three spatial coordinates of the measured velocity wave field Vü (x<sub>r</sub>,and<sub>r</sub>, z<sub>rt</sub>t) are reduced to two spatial coordinates in the representation of the velocity wave field, vx (x<sub>r</sub>, z<sub>r</sub>, t). Reducing the two spatial coordinates provides a clear understanding, while preserving the main characteristics of the method.
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The velocity and pressure wave fields can be broken down into vertical velocity and ascending and descending pressure components. In Figure 8, directional arrow 816 represents the direction of a rising wave field detected by a receiver 818 and dashed line directional arrow 820 represents the direction of a falling wave field reflected from free surface 712 and is detected by receiver 818. In other words, the pressure wave field Pfo.W) is composed of a rising pressure component and a falling pressure component, and the velocity wave field * »·» *) is made up of a rising vertical velocity component and a descending vertical velocity component. The descending pressure wave field and the descending vertical velocity wave field are phantom wave fields of the receiver, and the ascending pressure wave field and the ascending vertical velocity wave field are wave fields removed from phantoms of the receiver. receiver.
Figures 9A-9C each show an exemplary graph representing one aspect of the wave field decomposition. In Figures 9A-9C, vertical axis 902 is a z coordinate axis representing depth or spatial dimension z, and in Figures 9A-9B, horizontal axis 904 is an x coordinate axis representing spatial dimension x. The spatial coordinate "0" z corresponds to a tangent plane a * =<sup>0</sup> for the gimbal 718 described above with reference to Figure 7B. Figure 9A shows an exemplary graph of an interpolated marine cable 906, based
<img file="MX340795B_D0023.tif" />
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The pressure wave field <sup>Zr</sup>-<sup>m</sup>'and the velocity wave field v<sub>n</sub>(max, z<sub>rim</sub>, t) <sub>associate</sub>¡<sub>ac</sub>|<sub>you with ca</sub>j<sub>to pray</sub>pt<sub>or I know</sub> transform from the domain,. Ρ (πχΔχ.ζ<sub>Γ</sub>, ^ ω) from space-time to a pressure wave field and a velocity wave field * ·. «*) <sub>Jan</sub>Space-frequency domain, where <sup>ω</sup> ”Is the angular frequency for the acoustic frequencies f detected by the dual sensors. For example, the pressure wave field and velocity wave field related to a receiver can be transformed using Fourier transforms:
F
ΡζζηΔχ, z<sub>r</sub>_<sub>m</sub>, <sup>=</sup> I ρ {τηΔχ, ζ<sub>ΓιΤη</sub>, ί) β dt (1) ^ (τηΔχ, ^, ω) = vg (m & x, z<sub>rim</sub>, t) e ~ ^<sup>t</sup> dt (2) to where j is the imaginary unit. In practice, the transformation can be performed using a discrete Fast Fourier Transform (FFT) for computational efficiency. It should be noted that the letters
<img file="MX340795B_D0024.tif" />
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After the velocity and pressure wave fields associated with each receiver have been transformed from the spacetime domain to the space-frequency domain, the pressure wave fields and velocity wave fields combine to produce a pressure component ascending on the geoid (i.e. z = 0) in the wave-frequency number domain. Pressure sensors alone cannot distinguish between the opposite polarity of the scattered seismic wave field upward from the underground formation and the time-delayed seismic wave field reflected downward from the sea surface (i.e. receiver ghost) , but the information obtained from the particle motion sensors can be used to distinguish the polarity of the seismic wave field. A mathematical expression that relates the velocity and pressure wave fields to an upward pressure component of the pressure wave field in the geoid is given by:
<sup>20</sup> P “P (fc<sub>x</sub>, z = 0, w)
Ml = Z2 ^ "Σ ί /<sup>ω</sup>Ρ<sup>ν</sup>'Γ (<sup>7ηΔ; κ</sup>-ΖΓ.π, .ω) β ^<sup>πιΔ</sup>*<sup>7/£</sup>^·’<sup>η</sup>) ' <sup>z</sup>mM) + P (znAx, z<sub>nm</sub>, w) eÚ<sup>kj:</sup>’<sup>neither</sup>*’’/<sup>fc</sup>^ - ') (/ k<sub>x</sub>n<sub>x</sub> (3)
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where k<sub>z</sub> is the vertical sling number in the z direction provided by:
with c the speed of sound in the fluid;
k<sub>x</sub> is the horizontal sling number in the spatial direction xm is the dual sensor or channel index;
M is the total number of dual sensors located along the marine cable, p is the density of the fluid;
* r, m <sub>is</sub> |<sub>to</sub> Interpolated depth of the marine cable in the dual sensor m °;
n<sub>x</sub> is the x component of the normal vector;
n<sub>z</sub> is the z component of the normal vector <sup>n</sup>»; and V / (m Δχ, Ζπη, ω) is the velocity sling field for angular frequencies greater than a limiting angular frequency (i.e.<sup>= 2π</sup>&*>'
Advantageously, the downward pressure component in the wave-frequency number domain az = 0 is calculated in a similar way by:
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P<sup>down</sup>(fc<sub>x</sub>, z = 0, *>) _
Ml = 2j ^ - (<sup>m</sup> Δ *> <sup>z</sup>rm> ü)) eO ^<sup>m</sup>^ J ^ .rn) (4) + P (max, z<sub>r</sub>,<sub>m</sub>, (f)) e ^<sup>kxmÍix + íkzgr</sup>^ (jk<sub>x</sub>n<sub>x</sub> + jk<sub>z</sub>n<sub>z</sub>)}
Note that the wave field of the upward pressure, <sup>pup</sup>, and the downward pressure wave field, P<sup>down</sup>t are calculated from p
pressure wave field,, and velocity wave field,
<img file="MX340795B_D0027.tif" />
Figure 9C shows an exemplary graph of the wave-frequency number domain. The horizontal axis 910 represents the axis of the coordinate k<sub>x</sub> of the wave number. Note that the axis ω of the angular frequency is perpendicular to the k-plane<sub>x</sub>z, which is not shown in figure 9C. Shaded disk 912 represents a point in the wave-frequency number domain that is located along the coordinate axis k<sub>x</sub> 910. Point 912 in the wave-frequency number domain has associated upward and downward pressure components <sup>ρνρ</sup>&<sub>χ</sub>, ζ - ο, ώ) gq4 and
P<sup>down</sup>(^,<sub>2</sub> = 0, <and)<sub>916</sub>
After the up and down pressure wave field components in the geoid in the wave-frequency number domain have been determined, the pressure wave field components are changed to an observation level at a depth , z<sup>oüs</sup>, between the geoid and the marine cable. In Figure 9C, the dashed line 918 represents the k axis<sub>x</sub> at an observation level that is located between the geoid and the marine cable
<img file="MX340795B_D0028.tif" />
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918 lies between the geoid and the depth of the 920 marine cable. The upward pressure component at a point ^ χ, ζ<sup>065</sup>) 921 along the observation level z °<sup>bs</sup> 918 is calculated from P<sup>up</sup>(kx><sup>z =</sup> θ »<sup>ω</sup>) by:
P'VQt<sub>x</sub>, z<sup>obs</sup>, or>) = P “P (k<sub>x</sub>, z = 0. ώ) β ^ (5)
Similarly, the downward pressure component at the observation level z<sup>obs</sup> 918 is calculated by:
Ρ<sup>Λον, η</sup>0ίχ, ζ ^, ω) = P<sup>down</sup>Qcx, z = Q ^ e-<sup>Ik</sup>^<sup>obs</sup> (<sup>θ</sup>)
The vertical velocity component up at the point (t, <sub>7</sub>obs \ '921 is calculated from the upward pressure component P<sup>up</sup>(k<sub>x</sub>, z<sup>obs</sup> , ω) by:
V ^ ik<sub>x</sub>, z<sup>0bs</sup>, O) = --ΤρΦ (Λ<sub>χ</sub>, χ <* »,<sub>ω</sub>) P <ú (7)
The downward vertical velocity component at point {k<sub>x</sub>, z<sup>obs</sup>} g2i is calculated from the downward pressure component; P<sup>down</sup>(fcx, z<sup>obs</sup>, a>) by and dcwn <sub>(A z</sub>«L ~ <sub>ω</sub>) <sub>=</sub> ¿Ote, (Q) ρω
In other embodiments, the calculation of the free surface profile can also be accomplished using the vertical velocity components up and down.
In summary, the decomposition of the wave field is the
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process to transform the pressure wave field <sup>i</sup>’(’’’<sup><Kv Trn</sup>'and θ (velocity wavefield * «(” »4 *» * »·.« · 0 measured on the corresponding dual sensors of each receiver in the space-time domain, as shown in Figure 9A, in a downward pressure component P ^ mQc<sub>xl</sub>z<sup>ob5</sup>, ω) Q22 <sub>a</sub> upward pressure component <sup>pup</sup>(.<sup>k</sup>x '<sup>zObs</sup>> ^ ydownfj ^ <sub>z</sub>obs <sub>ω</sub>)
923, a component of vertical speed down <sup>1</sup> * '' 924 and (fc,, ω) an upward vertical velocity component <sup>2 xr</sup> '925 in the wave-frequency number domain, as shown in Figure 9C. The pup pdown V<sub>z</sub> yyún ™ <sub>they are ca</sub>|<sub>cu</sub>|<sub>ac</sub>|<sub>I will</sub> from the pressure wave field P and the speed wave field above the threshold frequency described above with reference to Figure 4C.
Note that the three-dimensional space version of the up and down pressure components and the vertical up and down speed components can be obtained by replacing the vertical slingshot number. <sup>k</sup>* by:
(-) where k<sub>and</sub> is the horizontal wave number in the spatial direction y-.
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Calculation of a free surface profile
Returning to Figure 6, after completion of the wave field decomposition in block 603, an image free surface profile is calculated in block 604. Figures 10A-10C show three different graphs in a process to calculate a profile free surface with image of a true free surface. An image free surface profile is calculated by first computing the upward and downward pressure components, or the vertical upward and downward speed components, for each point in a series of depth levels extending upward from the observation level to beyond the geoid or <sup>2 = 0</sup> axis 910. Figure 10A shows an example of a series of points represented by a column of open circles 1002 extending from the point (k<sub>x</sub>, z °<sup>bs</sup>) 921 at observation level z<sup>obs</sup> 918 to beyond <sup>z</sup> θ axis 910. Each open circle in series 1002 represents a 15 point with & coordinates<sup>χ, ζ</sup>\ where the coordinate k<sub>x</sub>- is the same for each point and the z-coordinates are gradually increased in a process called extrapolation. The upward pressure wave field is calculated at each point in the 1002 series by:
<sup>20</sup> P<sup>or</sup>Pfc, z, ü)) = P "<sup>p</sup>(fc<sub>I</sub>.z<sup>l ,,</sup>’<sup>s</sup>, ft)) e ^ (<sup>2<</sup>’<sup>!</sup>'<sup>i</sup>-<sup>z</sup>) (9a) where z is a coordinate value in the series of points
1002. The downward pressure wave field is calculated at each point in
<img file="MX340795B_D0031.tif" />
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P<sup>d (Wn</sup>(fcx, z, <d) = P<sup>down</sup>(kx, z °<sup>bs</sup>, ω) β-<sup>;</sup>^°<sup>βί</sup>-^ (<sub>9b</sub>)
In another embodiment, the upward vertical velocity component at each point in series 1002 is calculated by:
Υ ^ Ο ^, ζ, a) = (10a)
The downward vertical velocity component at each point in the 1002 series is calculated by:
and/<sup>own</sup>(k<sub>x</sub>, z, w) = (10b)
After the extrapolated up and down pressure components, or extrapolated up and down vertical velocity components, have been calculated in the wave-frequency number domain, an inverse Fourier transform is used to transform the upward and downward pressure components extrapolated and / or upward and downward vertical velocity components extrapolated in the space-frequency domain. Figure 10B shows a series of points 1010 in the space-frequency domain that was described above with reference to Figures 9A-9B. A transform is used
Inverse Fourier 2θ to transform the pressure and vertical speed components associated with the points in the series 1002 in Figure 10A to obtain corresponding pressure and vertical speed components of the points 1010 in Figure 10B. Pressure wave fields
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Transformed to the space-frequency domain are provided by:
P<sup>up</sup>(xr, z, ¿Il = ^ f dkxP<sup>up</sup>(kx, z, ü) ') e ^<sup>x</sup>> · (11a) pdawn (<sub>Xri z ω</sub>) <sub>=</sub> 1 í dk<sub>x</sub>P<sup>aown</sup>(.k<sub>x</sub>, z, u) e '<sup>kxX</sup>r (11b)
The vertical velocity wave fields transformed into the space-frequency domain are provided by:
ν ^ χ ,, ζ, ω) = dk ^ Oc ,,, ^^ (12a) ν, ^ χ ,, ζ, ω) = (<sub>12b</sub>)
For example, a Fourier transform can be used to transform the pressure and vertical velocity components upward and <sup>15</sup> down 1005-1008 associated with point 1004, shown in Figure 10A, on a downward pressure component <sup>pimm</sup>(<sup>x</sup>r · ^ 1013, a component of upward pressure Ρ<sup>ηρ</sup>(* Γ><sup>ζ</sup>· Ω) 1014, a component of V2<sup>down</sup>(xrrz ^) vertical speed down
1015 and a component of <sup>20</sup> vertical speed up <sup>Ρ</sup>(<sup>χ</sup>Γ »^ ω) <sub>1016 at a point</sub> corresponding (* rz) 1018 in the 1010 series shown in Figure 10B. In practice, the transformations represented by equations 11a-11b and 12a-12b can be performed using an inverse fast discrete Fourier transform
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An imaging condition is used to calculate an I & r image value.<sup>2</sup>) at each point in the series of points in the space-frequency domain. For example in FIG. 10C, an imaging condition is applied to each point in the series of points 1010 for ¡(<sub>x</sub> ζΊ get an image value, like an image value ' <sup>n</sup> 'associated with point 1018. The imaging condition may be a cross-correlation of the upward and downward vertical velocity or pressure components extrapolated in the space-frequency domain. In one embodiment, the imaging condition representing a free surface image value for a selected receptor position x and the extrapolation depth z is calculated by applying the following cross-correlation equation:
l (Xf, z) as D (x<sub>r</sub>, (ü) u (x<sub>r</sub>. ζ, ω) (13) one where the upper bar designates the complex conjugation. In modality,<sub>re</sub>p<sub>re</sub>sits and U (Xr »z, a) represents
P <sup>ρ</sup>(Χτ, ζ, ω) In another modality, ΰ (χ<sub>Γ</sub>, ζ, ω) <sub>represent</sub>t<sub>to</sub>
U (x<sub>r</sub>, z, (ú) <sub>re</sub>p<sub>resen</sub>ta ν ^ ζχτ, ζ, ω). In other modalities, the imaging condition can be a normalized cross correlation provided by:
/ (x<sub>r</sub>, z) =
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Σ<sub>ω</sub> υ (χ<sub>Γ</sub>> <sup>ζ</sup>· Ω) Ρ (χ<sub>Γ</sub>, z, ω) Σω U (x<sub>r</sub>, z, ω) U (x<sub>r</sub>, z, ω)
The value of the coordinate z or the extrapolation depth associated with the maximum image value <sup>Imax</sup>^<sup>Xrr</sup> for a given channel position 5 x corresponds to a free surface elevation z at receiver position x. In the example of Figure 10C, the points on the series 1010 associated with the receiver 908 are represented with open circles on a dashed line except for the point 1022 that corresponds to a point q<sub>EU</sub> produces a maximum cross correlation ^ οχ (.Χγ><sup>ζ</sup>ο \ Like <sup>10</sup> () result, the point <sub>is a</sub> image point of a free surface profile. The modalities are not limited to the conditions of formation of
Images that were described before. Other types of imaging conditions can also be used. Other Image points represented by open circles, such as open circles 1024, are calculated in the same way by applying the same Imaging condition to each point at a series of points, as described above. The Interpolation over the Image point collection forms an approximate time-stationary Image of the free surface profile above the marine cable 908 that corresponds to a selected time window of the input data. For example, in Figure 10C a curve 1026 is a free surface profile with Image of a hypothetical free surface profile represented by dot curve 1028 in a time window
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selected. A free surface profile with Image above of a marine cable can be interpolated using Linear Interpolation, Lagrange Interpolation, multi-line interpolation, or other suitable Interpolation technique. In other modalities, Image points associated with two or more marine cables can be used to calculate an approximate three-dimensional free surface above the marine cables using multi-dimensional interpolation techniques, such as Barnes interpolation, Bezler's interpolation, blcublca interpolation, and Linear interpolation. The points along the free surface profile of Image 1026 are represented by (* / (*)]><sub>i in</sub> where <sup>10</sup> x represents a horizontal coordinate along the x-axis 904 and / (*) is the Interpolated function value representing the height of the free surface. A varied time free surface estimate is obtained by continually moving the entered data time window and combining the calculated frozen free surface profiles for all 1 ζ time windows from a Start record time to an end record time.
Figure 11 shows a control flow diagram of a computational method for calculating the free surface profile of block 604. In block 1101, pressures and vertical speeds up and down for points in the domain are calculated. of wavefrequency number, as described above with reference to FIG. 10A. In block 1102, the pressures and vertical velocities associated with the points are transformed from the wave-frequency number domain to the space-frequency domain. In block 1103, a condition is applied
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imaging as described above with reference to Figure 10C. At block 1104, it is calculated at a point associated with the maximum imaging condition for a given receiver channel, as described above with reference to FIG. 10C. At block 1105, the image points associated with a marine cable are used to calculate a freeze frame profile of the free surface.
Free surface profile extension
Returning to Figure 6, at block 605, the free surface profile is extended to cover the origin location. Figure 12A shows a graph of a hypothetical free surface using the discontinuous curve 1028 and the free image surface profile 1026 that approximates the free surface 1028 above marine cable 906. Circle 1202 represents the coordinate location of a source towed by a seismic survey vessel (not shown). As shown in Figure 12A, and in Figures 5, 7B and 8 described above, source 1202 is not located on the marine cables, but is located between the vessel and the marine cables. Since the image free surface profile 1026 is calculated from subsurface reflections, the image free surface profile 1026 is limited to the scattered marine cable as indicated by broken lines 1204 and 1206. As a result, the image free surface profile 1026 does not cover the source 1202 and can only be used to remove ghosts that are reflected from the free surface directly above the
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marine cable and cannot be used to fade away l5§'Táñtasmas ^ ue that are reflected from the free surface above source 1202. For example, directional arrows 1208 and 1210 represent two sound impulse paths leaving the source 1208 and 1210. The two paths 5 result in direct phantom downward wave fields and phantom reflections from the source as described above with reference to Figure 5. The direct downward wavefield calculation including its phantom is used for the reconstruction of the low-frequency vertical velocity wavefield and the phantom removal of seismic data 10 with sources that are above the level of the marine cable. The free image surface 1026 can be used to remove phantom data from seismic data with ghosts following path 1210, but since the free image surface 1026 does not extend to cover the region above source 1202, the surface Free with image 1026 cannot be used to remove phantom data from seismic data following path 1208.
To compensate for the ghosts associated with reflections from the free surface above the source, the image free surface profile is extended by calculating a free surface profile extension of the image free surface profile that covers the region that is above the coordinate location of the source. FIG. 12B shows a graph of the hypothetical free surface 1028 and an extended image free surface profile. The free surface profile with extended image is the
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<img file="MX340795B_D0037.tif" />
free surface 1026 extended to include a free surface profile extent 1212. Free surface profile extent 1212 can be calculated based on parameters associated with the realistic physical model of a fully developed marine state that is determined at the time of measuring the seismic data. For example, a realistic physical model may be a Pierson-Moskowitz model that provides parameters such as wavelength, wavelengths, and the speed of free surface waves. The Pierson-Moskowitz model of free surface can be used to calculate the extent of free surface profile 1212 covering source 1202.
The Pierson-Moskowitz model assumes that when the wind blows stably for a long period over a large area, the waves eventually reach a state of equilibrium with the wind. This condition is known as a fully developed sea. The Pierson-Moskowitz spatial irregularity spectrum for a fully developed sea in one dimension is given by:
<img file="MX340795B_D0038.tif" />
(15) where K is the spatial wave number of the 2θ free surface wave;
OR<sub>w</sub> it is the wind speed measured at a height of approximately 19 meters;
a is 8.0x10 '<sup>3</sup>;
β is 0.74; and
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<img file="MX340795B_D0039.tif" />
is the acceleration due to gravity.
The function of the free surface height / & ') is generated at a point x as follows:
Nl
Í = Q (16) where for the index i> 0,
F (K¿) = j2irLW (Ki) '[íV (0.1) +; W (0, l)] / VZ for i * 0, N / 2 „JV (Ol) for í = 0, N / 2 (17 ) and for / <0, = W ·.
In equations (16) and (17), the spatial wave number for component / is provided by <sup>Ki</sup> -<sup>2Ki / L</sup>', where L is the length of the free surface. The random number ^ (°<sup>1</sup>) can be generated from a Gaussian distribution that has a zero average and a unit variance.
As a result, the free surface is formed by adding each wave number component that imposes random phase shifts. A time-frozen Pierson-Moskowitz free surface can be calculated from equation (16) using a fast Fourier transform for calculation efficiency.
Free surface waves are dispersive and in deep water the frequency and the wave number are related by a scattering ratio given by:
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<img file="MX340795B_D0040.tif" />
Equation (18) implies that each spatial harmonic component of the surface can move with a defined phase velocity. As a result, in general, free surface waves with longer wavelengths travel faster relative to waves that have shorter wavelengths. The combination of equations (16) and (18) gives a Pierson-Moskowitz free surface of varied time:
<img file="MX340795B_D0041.tif" />
(19) where instantaneous time is. Equation (19) characterizes an irregular free surface of one dimension moving in the positive x direction and can be used to calculate the free surface extent in earlier or later cases in time.
In general, a frozen free surface for earlier or later times can be calculated as follows.
Considering a free surface shape at an instant in time f with wave heights given by, the wave number spectrum (ft) of the free surface is calculated and an arbitrary known scattering ratio can be used to calculate the free surface frozen in an earlier or later case (<sup>t +</sup> 4t) <sub>and</sub>n the time, by:
<img file="MX340795B_D0042.tif" />
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<img file="MX340795B_D0043.tif" />
Free surface wave field reflectivity calculation
Returning to Figure 6, at block 606 the reflectivity of the free surface wave field (i.e., the response of a unit point source at the receiver position rV) can be calculated on the extended free surface using:
R & .rJ - + | / ~ r<sub>r</sub>|) H<sub>1</sub><sup>(1)</sup>(fc | plMxO <íx 'with (21)
Ipi
Ipi
where <sup>v</sup> '' is the asymptotic form of the first-order Hankel function with n = 0 and 1. The parameters of equation (21) are represented in Figure 13 as follows:
k is the wave number of the propagation wave field;
/ (O is the height of the free surface described above;
MWi is a coordinate position 1302 of a current scatter point on the surface;
<sup>r</sup>o is a vector 1304 from the origin of the current scatter point 1302;
<sup>Tr</sup> is a vector 1306 from the origin to a fixed receiver position 1308 with spatial coordinates go<sub>0</sub> - fies a vector 1310 from the point of current dispersion
1302 to receiver position 1408;
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<img file="MX340795B_D0044.tif" />
<sup>r</sup>s is a vector 1312 from origin to source position 1314;
<sup>r <</sup>* is a vector 1316 from source position 1314 to receiver 1308;
P is a vector 1318 from source 1314 to scatter point 1302; y = ñ-p = sew<sub>that</sub>| oblique factor with normal vectors ñ 1320 and 4 1322 corresponding to the normal free surface and the direction of the unit vector of the incident field at fo »/ (* <>)] 1302 and Θ is the angle between vectors # 1320 and β 1322.
The pressure wave field over the entire frequency range can be obtained from the pressure measurements on the pressure sensors. However, as already explained above with reference to Figure 4C, the recorded particle movement signal may be contaminated with low frequency noise due to vibrations from towed marine cables. As a result, particle motion signals recorded below the threshold frequency may not be useful for determining vertical velocity wave fields at receiver locations along the marine cable below the threshold frequency f<sup>ch</sup> . Instead, only the vertical velocity wavefield V ^ (r is normally reliable<sub>r></sub>6)) obtained from particle movement measurements on a Mexican ικίτιτυτο
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<img file="MX340795B_D0045.tif" />
frequency range greater than .A vertical velocity wavefield can be calculated for frequencies below the threshold col frequency '<sup>r</sup>'\ from the normal derivative of the pressure wave field, <sup>n</sup>'^ r<sup>p</sup>t in a process called "low frequency compensation." The wave field decomposition can then be performed to calculate the receiver phantom pressure and the velocity wave fields.
Pressure wave field calculation gradient
At block 607, the gradient of the pressure wave field,
Vp <sup>r</sup> , is calculated. The calculation of the gradient of the pressure wave field,
Vp <sup>r</sup> , is determined depending on whether the depth of the source is greater or less than the depth of the marine cable. Figures 14A-14B show two-dimensional xz plane views of an exemplary model space associated with a volume of fluid. In Figures 14A-14B, the space includes a representation of a 1402 marine cable, denoted by S<sub>r</sub>, at 1404 source, that produces sound impulses. Marine cable 1402 and source 1404 can be towed by an exploratory seismology vessel (not shown) within the fluid volume below the free surface that can be imaged as described above to produce a free surface profile with image represented by curve 1406 and denoted by So. The free surface with So images includes an extension of
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<img file="MX340795B_D0046.tif" />
Dt LA ΜΟΤΙΕΙΜΟ free surface profile 1408 on the source 1404. Uos'Vectores ^ omo el vector <sup>r¡</sup> 1410 represent spatial coordinates of source te - *) · the
JE vectors like vector <sup>r</sup>"1412 represent coordinates in the fluid volume, and vectors as the vector <sup>Ts</sup> 1414 represent 5 coordinates of the receiver ^<sup>r, Zr</sup>^ <sub>to</sub> |<sub>0</sub> |<sub>ar</sub>g<sub>0</sub> of the <sub>AC</sub>Marine Bie 1402. When the depth of source 1404 is less than the depth of the marine cable, as shown in Figure 14A, the expression used to calculate the pressure wave field gradient over the entire frequency range is given by:
J dS<sub>r</sub>ñ - /? (f<sub>r</sub>, r) V<sub>r</sub>P (r<sub>r</sub>, úi) = a (á)) fi (r, .f) + J dS<sub>r</sub>ñ- P (r<sub>r</sub>, w) V<sub>r</sub>R (f<sub>r</sub>, r) (22) where is the Fourier transform of the source-time function for the source at the coordinate location and <sup>R</sup>(<sup>r</sup>><sup>r</sup>J is the free surface wave field reflectivity 15 given by equation (21).
To calculate free surface wave field reflectivity with a hypothetical source in <sup>Ts</sup> below the surface of the receptor and a hypothetical position of the receptor in <sup>T</sup>s Knowledge of an extended free surface profile is required. Equation (22) is a Fredholm integral equation of the first type for the gradient of the pressure wave field v<sub>r</sub>p (r<sub>r</sub>, w) <sub>in</sub> where <sub>θ</sub>| |<sub>ac</sub>j<sub>0</sub> d<sub>erec</sub>ho of equation (22) contains only Ρ (η., ω) parameters known as the pressure wave field of the field
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<img file="MX340795B_D0047.tif" />
and the free surface wave field reflectivity The free surface wave field reflectivity with a hypothetical source under the receiver surface and a hypothetical receiver at <sup>r</sup>r is calculated from the free surface profile with images. The reflectivity gradient, 5
V<sub>r</sub>/? (r<sub>r</sub>, r) <sub>I know</sub> p<sub>EU</sub>to calculate using numerical gradient techniques. On the other hand, when the source is located at a depth below the marine cable S<sub>r</sub>, as shown in Figure 14B, the expression used to calculate the gradient of the pressure wave field <sup>v</sup>r<sup>/</sup>’(<sup>r</sup>r-<sup>w</sup>) over the entire frequency range, is provided by:
[dS<sub>r</sub>ñ-R (f<sub>r</sub>, r) V<sub>r</sub>P (f<sub>r</sub>, o)) = f dS<sub>r</sub>n- P (f<sub>r</sub>,av<sub>r</sub>R (f<sub>r</sub>, f) (23)
Js<sub>r</sub><sup>J</sup>Mr
In equation (23), the source function is not used to calculate Note q<sub>EU</sub> |<sub>ace</sub> Solutions of equations (22) and (23) become unstable when the spectrum of the pressure wave field has 15 very small values (for example, close to the receiver phantom notches). Generally these spectral notches occur for typical towing depths at frequencies<sup>ω> ω,: Ιί</sup>'. As a result the gradient of the pressure wave field can be calculated for the frequencies below the threshold frequency for an arbitrarily irregular free surface, variable with time. A derivation of equations (22) and (23) is provided in an annex below.
Depending on whether the source is at a depth above or
<img file="MX340795B_D0048.tif" />
below the marine cable, as shown in Figures 14A and 14B respectively, equations (22) and (23) can be solved numerically for the pressure wave field gradient, <sup>V</sup>'<sup>F</sup>· At receiving locations along the marine cable using quadrature or expansion methods. For quadrature methods, the integrals are approximated by quadrature formulas and the resulting system of algebraic equations is solved. For expansion methods, the solution is approximated by expansion in terms of base functions.
Normal velocity wave field calculation
Returning to Figure 6 at block 608, the normal component of the normal velocity wavefield below the threshold frequency Λ * at each receiving location along the marine cable can be calculated by:
Vg ^ Xr, V («r.« R · «> (24) where p is the density of the fluid; y» is a normal vector for a receiver.
Figure 15 shows a segment of a marine cable 1502 <
located below a free surface processed by 1604 images in the xz plane of a fluid volume. The normal vector 1508 for marine cable 1502 on receiver 1506 is provided by:
/ dz<sub>r</sub> dx<sub>r</sub>\ n = (n<sub>x</sub>, n<sub>z</sub>) »(-Sen φ, ακφ) =
V<sub>r</sub>P
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<img file="MX340795B_D0049.tif" />
where <sup>r</sup> is the gradient of the pressure wave field at receiver 1606; and<sup>n</sup>'<sup>Vrp</sup> is the normal derivative of the pressure wave field P at receiver 1606.
At this point, the pressure wave field over the total frequency range is known from measurements of the pressure sensor, the vertical velocity wave field <sub>SW</sub>over a frequency range above the low frequency threshold <sup>ω</sup>*<sup>Λ</sup> is obtained from particle motion measurements and the vertical velocity wavefield below the threshold frequency <sup>Mtlt</sup> it is calculated from equation (24). The vertical velocity wave field<sup>v</sup>neither<sup>x</sup>r><sup>z</sup>r '<*>) over the total frequency range can be calculated by:
Vs<sup>the</sup>(x<sub>r</sub>, z<sub>r</sub>.co) <sup>for</sup> ^ 25) = (1 - νν (ω)) ν £<sup>α1</sup>(χΓ, Zr, ω) + ιν (ω) I saw<sup>h</sup>(xr, z<sub>r</sub>, ω) for ω<sub>Λ</sub> <ω <
<V-ñ βτ ·> <sup>z</sup>r> to <sup>ωι</sup> < <sup>ω</sup> where is a weight function with the property that w (<y) = o <sub>when</sub>| or <sup>ω =</sup> yw (w) = i when <sup>ω</sup> - <sup>ω</sup>ι for the smooth transition of <sup>Fs</sup> to <sup>v</sup>* <sup>1</sup> As ω increases from wefc to and *> »is less than <sup>nc</sup>¡<sup>Zr</sup> (that is to say, <sup>£ hr</sup>) to maintain a good signal to noise ratio of the hydrophone signal.
For example, the weight function <sup>W</sup>G>) can be a function of
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<img file="MX340795B_D0050.tif" />
linear weight determined by:
νν (ω) &>! - iü<sub>tfl</sub> for
In other embodiments, other types of linear weight functions can be used as well as nonlinear weight functions such as a window weight function.
Calculation of the receiver phantom-free wave field
In block 609 the recorded pressure wave field and the vertical velocity wave field determined by equation (25) are used to calculate the receiver-side non-phantom pressure wave field (i.e., the wave field pressure rise P<sup>or?</sup>) over the total frequency range in the wave number and frequency domain when using:
<sup>15</sup> P<sup>up</sup>(k<sub>x</sub>, z = Ο, ω)
Λ <sup>Μ_1</sup> = Σ \ jo> pVñ {™. δx, z<sub>r</sub>.m. ω)<sub>β</sub>υ ^<sup>χ</sup>-^·^ (26) <sup>J ζ</sup> m ^ O + P (mAx, z<sub>r> m</sub>, üj) e (/<sup>k</sup>’'<sup>max-</sup>-<sup>i</sup>'<sup>Cs</sup>'<sup>i</sup>'->’<sup>n</sup>) O'A:<sub>x</sub>n<sub>x</sub>-;TO:<sub>z</sub>n<sub>z</sub>)}
The upward pressure component at each receiver at p<sup>or</sup>P (L · z = 0 ¿i /)
2θ length of the marine cable is calculated from <sup>k</sup> *' ' through:
P<sup>up</sup>(fcx, zr, w) = P<sup>vp</sup>(kx, z = 0, ω) β “^ (27)
<img file="MX340795B_D0051.tif" />
IΑ4 ΡI
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Dt INDUSTRIAL FRCRIFTY and phantom-free vertical velocity wavefield at each receiver in the wave-frequency number domain is determined by:
= -—Ρ ^, Ζτ, ω) ρω (28)
It must be considered that the pressure wave field without z debe) phantom <sup>K x</sup>' <sup>r</sup>'and the ghostless velocity wavefield <sup>z</sup> <x> r> / <sub>are</sub> |<sub>I cam you</sub>p<sub>0S</sub> d<sub>e onc</sub>(at rising pressure and vertical speed over the full frequency range.
The phantom-free pressure wave field can be transformed from the wave-frequency number domain into the space-frequency domain to obtain:
Ml
P<sup>up</sup> (x<sub>r</sub>, z<sub>r</sub>, ηΙ-μΣ<sup>1</sup>™ (<sup>kx</sup>'<sup>Zr</sup>' ^<sup>eiXrm Δ </sup>m = 0 (29) followed by a transformation from the space-frequency domain to the space-time domain:
}<sup>or</sup>P (xr, 7.r, t) = z ~~ f dvP<sup>up</sup>(xr, z<sub>r</sub>, <ú) e<sup>i <at</sup> (30)
Similarly, the phantom-free vertical velocity wavefield can be transformed from the wave-frequency number domain to the space-frequency domain to obtain:
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Ml ^ (χ ,, ζ ,, ω) = -¿ tí'feír.ü ^^ aiSi- (U) πι = 0 followed by a transformation of the space velocity wavefield from the space-frequency domain to the domain of space time:
<<sup>32</sup>>
The inverse Fourier transformations of equations (29) (32) can be performed using inverse fast Fourier transformations for computational efficiency. In short, the phantomless wave fields,
Λ f \ M<sup>OR</sup>P rising wave fields <sup>p</sup> and * in receivers are used to calculate images of underground formations located below the fluid volume.
Figure 16 shows an illustrative example of a generalized computing system that executes an efficient method for phantom removal of the source of a scattered wave field and therefore represents a seismic analysis data processing system to which the description. The internal components for many small, medium, and large computing systems, as well as specialized processor-based storage systems, can be described with respect to this generalized architecture, although each particular system can characterize many additional, sub-components. systems, and the like, parallel systems with architectures similar to this generalized architecture. The computer system contains one or more units of
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<img file="MX340795B_D0052.tif" />
central processing (CPU) 1602-1605, one or more eféfcnümcas 1608 mé'H'TóVláh interconnected with the CPU in a common sub-system link CPU / memory 1610 or several common links, a first bridge 1612 interconnecting the common sub -1610 CPU / memory system with additional common links 1614 and 1616, or other types of high-speed interconnect media, including multiple serial high-speed interconnects. These common links or serial interconnects, in turn, connect the CPU and memory with specialized processors, such as a 1618 graphics processor and with one or more additional 1620 bridges, which are interconnected with high-speed serial connections or with various 1622-1627 controllers, such as the 1627 controller, which provide access to different types of computer readable media, such as 1628 computer readable media, electronic displays, input devices and other such components, subcomponents, and computational resources. Electronic displays, including the display screen, speakers, and other output interfaces, and input devices, such as mice, keyboards, touchscreens, and other input interfaces, together constitute input and output interfaces that allow the computer system to operate reciprocally with human users. The 1628 computer readable medium is a storage device, which includes electronic memory, optical or magnetic disk drives, USB drives, flash memory, and other such data storage devices. The 1628 computer readable media can be used to
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storing machine-readable instructions, associated with the computational font ghosting methods described above and can be used to store encoded data, during storage operations and from which encoded data can be obtained, during read operations, by computer systems, data storage systems and peripheral devices.
Annexed
Referring to Figures 17A-17E, we describe a derivation of integral equations that can be used to calculate the gradient of the pressure wave field at a receiver on a marine cable, V, P, (see also, Extraction of the normal component of the part velocity from marine pressure data, ”L. Amundsen et al., Geophysics, Vol. 60, No. 1, pp 212-222 (January-February 1995)).
Figure 17A shows a two-dimensional view of the xz plane of an example of model space 1700 related to a volume of fluid. Space 1700 includes a representation of a 1703 marine cable, designated by S<sub>r</sub>, and a source 1704 that produces sound impulses. Marine cable 1702 and source can be towed by an exploratory seismology vessel (not shown) in the fluid volume below an image-processed surface profile represented by a 1706 curve and marked with S<sub>or</sub>. The free surface processed by S images<sub>or</sub> includes an extension of free surface profile 1708 over font 1704. Vectors
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<img file="MX340795B_D0054.tif" />
just like the vector <sup>Ts</sup> 1704 represent the spatial coordinates fa,, ¾) of the source, vectors such as vector r 1710 represent the coordinates (xy) <sub>Jan</sub>| space 1700, and vectors such as vector 1712 _ represent the coordinates of the receiver (* rz<sub>r</sub>) located along Marine Cable 5 1702. The shaded ruler 1714 in Figure 17A and subsequent Figures 17B-17E represent a scattering ruler under Marine Cable 1702. The speed of sound in the ruler 1714 is determined by :
c (r) =
Cq (Al) where Co is the speed of sound in a fluid; i <or (r) <l is the refractive index over the 1714 region.
The pressure field generated by source 1704 at point lo
1710 is r<sub>Mr</sub> e) and can be transformed from the space-time domain to the space-frequency domain by using a Fourler transformation to obtain a pressure wave field P (r, r<sub>s</sub>, ω) at point 1710. In practice, the transformation from the domain of space-time to the domain of
2θ space-frequency can be achieved by using Fast Fourler Transformation using computational efficiency.
The constant-density acoustic wave equation that characterizes a pressure wave field P (r, r<sub>5</sub>, ω) caused by a
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<img file="MX340795B_D0055.tif" />
single source that generates sound impulses, such as source 1704, located at spatial coordinates r<sub>s</sub> it is determined by:
,2
V<sup>2</sup> + r<sub>s</sub>, ω) = α (ω) δ (τ - f<sub>s</sub>) (A-2)
Where V<sup>2</sup> it is the Laplaclano;
ω is the angular frequency;
5 (r - 7¿) is the three-dimensional Dirac delta function representing the sound output pulse from source 1704; and α (ω) is the Fourier transformation of the source-time function for the source at the coordinate location f<sub>s</sub>.
The acoustic wave equation (A-2) characterizes the propagation of the acoustic pressure wave field P (r, r<sub>Sf</sub> ώ) in the volume of fluid, where the sound pressure waves originate from the impulse sound source 1704. Substitute equation (A-1) for the speed of sound c (r) without the sound wave equation (A-2) reorganize, results in:
(V<sup>2</sup> + / cJ) P (r, r<sub>s</sub>, o>) - k¡a (r) P (r, r<sub>s</sub>, a>) + a (fi>) S (r (A-3)
Green's second identity is now considered from the vector calculation:
<img file="MX340795B_D0056.tif" />
dr [FV<sup>2</sup>C - CV<sup>2</sup>B] = dSn · (BVC - CV5] (A-4) where B and C are both doubly differentiable scalar fields in a volume V bound by a closed surface S with a
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<img file="MX340795B_D0057.tif" />
vector signage out n. Figure 17B shows a volume V in space 1700. Volume V lies within a closed surface S, which is the surface of the marine cable S<sub>r</sub> 1702 and a hemispherical layer Sr 1716 of radius /? = | r '|. Volume V covers the free surface processed by S images<sub>or</sub> but does not include dispersion region 1714. Figure 17B shows two examples of normal vectors out n. 1718 and 1720. An integral equation for the pressure wave field, P, and the gradient of the pressure wave field, VP, can be obtained by setting B = Py<sup>C = </sup>in Green's second identity (A-4) to result in:
df [P (f>) V '<sup>2</sup>C<sub>ko</sub>(F ', O - G<sub>kfl</sub>(f ', r) V'<sup>2</sup>P (r »] <sup>Jv</sup> (A-5) = f dSn · [P (f ', 6)) V'Gfc<sub>0</sub>(f ', r) “G<sub>ko</sub>(r ', r) V'P (r', w)] where it is restricted to points in volume V. The Green G function<sub>fco</sub>(r ', r) r'acterizes free surface reflections and is a solution of the acoustic wave equation (21) for a drive source V and is determined by:
(V<sup>z</sup> + ^) G<sub>what</sub>(r ', f) = or (f' - r) (A-6) where k<sub>Q</sub> = 6u / c<sub>0</sub>: yr 'represents a point in space 1700 and Ha the source location of the Green function.
For example, the reflectivity R (r<sup>r</sup>, r) determined by equation (17) can be used as the Green G function<sub>feQ</sub>(r ', r) in equations (A-5) and (A-6) (i.e. G<sub>k <></sub>(f ', r) = R (r', r)). Replace the
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<img file="MX340795B_D0058.tif" />
equations (A-3) and (A-5) determine:
I dr<sup>r</sup>[P (r ', r<sub>s</sub>, ct>) $ (r '- r) - G<sub>ko</sub>(f ', r) a (^) S (r' - Tg)
Jv ~ G<sub>ko</sub>(F<sup>,</sup>, r) kla (f<sup>r</sup>) P (.f<sup>,</sup>, r<sub>s</sub>^ ')] (A-7) = f d5n4P (f; f<sub>ilW</sub>) V'C<sub>fco</sub>(r ', r) -G<sub>ko</sub>(f ', r) V<sup>,</sup>P (r<sup>/</sup>) r<sub>i</sub>,<sub><and</sub>)] <sup>}</sup>s
By letting the radius R of the hemispheric layer Sr go to infinity, Sr reaches an infinite hemispherical layer, as shown in Figure 17C. The surface integral on surface S in equation (A-7) is reduced to an equation of integral on the surface of marine cable S<sub>r</sub>. Furthermore, the first term in equation (A-7) goes to zero when the source of the reflectivity represented by the Green function is outside volume V. Because the scattering region at 1714 is outside volume V , the third term in equation (A-7) also goes to zero, which dr'Gk.CrírW'-ñ) (A-8) = f dS<sub>r</sub>n- [P (r<sub>r</sub>, r<sub>s> W</sub>) V<sub>r</sub>G<sub>ft0</sub>(r<sub>r</sub>, r) -G<sub>k (</sub>,(F<sub>r</sub>, r) V<sub>r</sub>P (r<sub>n</sub>r<sub>s</sub>, w)] leaves:
—Α (ω) í Jv
Equation (A-8) represents a functional relationship between the pressure wave field P (r<sub>r</sub>»<sup>ω</sup>), which is obtained from measurements taken on pressure sensors along the marine cable S<sub>r</sub>, and the gradient of the pressure wave field ν<sub>Γ</sub>Ρ0ΐ · .ω) on the pressure sensors along the marine cable S<sub>r</sub>. This functional relationship between the pressure wave field
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and the pressure wave field gradient determined by T ecuSClÜn (A8) can be used to calculate the pressure wave field gradient V<sub>r</sub>P (r<sub>r</sub>, ai) when only the pressure wave field P (r<sub>r</sub>,(¡>) .
Equation (A-8) can also be applied to address two 5 cases where source 1704 lies above the level of marine cable 1702 or lies below the level of marine cable 1702. When source 1704 is located within volume V at a depth between the free surface and the receptor surface as shown in the exemplary representation of Figure 17D, according to the alternating property related to the delta function of
Dirac, the integral over volume V at the left end of equation (A-8) reduces to:
a ^ G<sub>k</sub>(r<sub>s</sub>, r) = - (dS<sub>r</sub>n · [P (r<sub>r</sub>, ω) V<sub>r</sub>G<sub>ko</sub>(r<sub>r</sub>, r) - G<sub>ka</sub>(r<sub>r</sub>, r) V<sub>r</sub>P (f<sub>r</sub>, ω)] (A-9)
The terms of equation (A-9) can be rearranged to give:
í dS<sub>T</sub>n * G<sub>fco</sub> (r<sub>r</sub>, r) V<sub>r</sub>P (f<sub>r</sub>, ω) <sup>Mr</sup> (A-10) = aQttyGkféü, t) + í dS<sub>T</sub>n · P (and<sub>r</sub>, (»))^<sub>r</sub>G<sub>k (l</sub>(F<sub>rt</sub>F} <sup>J</sup>s<sub>r</sub>
Equation (A-10) is a Fredholm integral equation of the first type for the pressure wave field gradient V<sub>r</sub>P (f<sub>r</sub>, ω) where the right end of equation (A-10) contains only known parameters such as the field pressure wave field P (r<sub>r</sub>, ω) and the function of Green G<sub>ko</sub>(r, r<sub>r</sub>) described above. On the other hand, when the source is
IMPI
INDUSTRIAL MEXICAN INSTITUTE OF LA MOHEDAL)
<img file="MX340795B_D0059.tif" />
located outside of volume V at a depth below marine cable S<sub>r</sub>, as shown in Figure 17E, equation (A-7) reduces to
<img file="MX340795B_D0060.tif" />
for the pressure gradient V<sub>r</sub>P (f<sub>r</sub>, a>) · In equation (A-11), the source function α (ω) is not used to determine 7<sub>r</sub>P (r<sub>r</sub>, ω).
Although the present invention has been described in terms of the particular embodiments, it is not intended that the present invention be limited to these embodiments. Modifications within the spirit of the present invention will be evident to those skilled in the art. For example, any number of different implementations of the computational processing method efficiently perform source and receiver phantom removal can be designed and developed using several different programming languages and computing platforms and by varying the different Implementation parameters, including control structures, variables, data structures, modular organization and other such parameters. Computational representations of wave fields, operators, and other computational objects can be implemented in different ways. Although the efficient wavefield extrapolation method discussed above can be performed in a single dimensional sample, phantom removal of the source and receiver can be accomplished by using multiple two-dimensional samples or experimentally obtained three-dimensional sampling for different depths, It can be carried out from greater depths or shallow depths and can be applied in many other contexts.
It is appreciated that the above description of the described modalities is provided to enable any expert in the field to make or use the present description. Various modifications to these embodiments will be apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the description. Thus, the present description is not limited to the modalities shown herein, but must be in accordance with the broadest scope consistent with the novel principles and features described herein.
Contents83
82 sheets
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12 members in 6 offices
Priority claims5
| Document | Office | Kind | Date |
|---|---|---|---|
| 13545609 | United States of America | – | |
| 201213545609 | United States of America | A | |
| 201213545609 | United States of America | A | |
| 13545609 | – | – | – |
| US201213545609 | – | – | – |
Members12
| Document | Office | Kind | |
|---|---|---|---|
| EP2685288A2 | European Patent Office (EPO) | A2 | |
| US2014016436A1 | United States of America | A1 | |
| MX2013008073A | Mexico | A | |
| AU2013206627A1 | Australia | A1 | |
| BR102013017551A2 | Brazil | A2 | |
| EP2685288A3 | European Patent Office (EPO) | A3 | |
| SG10201510275TA | Singapore | A | |
| MX340795BThis record | Mexico | B | |
| US9442209B2 | United States of America | B2 | |
| AU2013206627B2 | Australia | B2 | |
| EP2685288B1 | European Patent Office (EPO) | B1 | |
| BR102013017551B1 | Brazil | B1 |
1 legal event, as the office reported them to INPADOC
Events
| Event | Code | |
|---|---|---|
| Grant or registrationFG | FG |
Numbers
- Publication
- 340795
- Publication, DOCDB
- 340795
- Publication, EPODOC
- MX340795
- Application
- 8073
- Application, DOCDB
- 2013008073
- Application, EPODOC
- MX202013008073
Titles
- Spanish
- METODOS Y SISTEMAS PARA RECONSTRUCCION DE CAMPOS DE ONDA DE VELOCIDAD DE PARTICULA DE BAJA FRECUENCIA Y REMOCION DE FANTASMAS DE DATOS DE CABLES MARINOS SISMICOS.
Classification
- CPC, 2
- G01V1/364
- G01V2210/56
- IPC, 2
- G01V1 36
- G01V3 38