WO2017024112A1

WO2017024112A1 - Seismic wavefield deghosting

Info

Publication number: WO2017024112A1
Application number: PCT/US2016/045494
Authority: WO
Inventors: Yousif Izzeldin Kamil Amin; Maximilian SCHUBERTH; Philippe Caprioli; Massimiliano Vassallo
Original assignee: Schlumberger Technology Corporation; Schlumberger Canada Limited; Services Petroliers Schlumberger; Geoquest Systems B.V.
Priority date: 2015-08-06
Filing date: 2016-08-04
Publication date: 2017-02-09

Abstract

A method can include receiving data that include a signal and a ghost signal that is characterized at least in part by a ghost delay; and, for a deghosting operator parameterized with respect to a ghost delay parameter, determining a ghost delay parameter value that minimizes a p-norm of a result generated by applying the deghosting operator to at least a portion of the data.

Description

SEISMIC WAVEFIELD DEGHOSTING

RELATED APPLICATIONS

[0001] This application claims priority to and the benefit of a U.S. Provisional Application having Serial No. 62/201 ,921 , filed 6 August 2015, which is incorporated by reference herein.

BACKGROUND

[0002] Reflection seismology finds use in geophysics, for example, to estimate properties of subsurface formations. As an example, reflection seismology may provide seismic data representing waves of elastic energy (e.g., as transmitted by P- waves and S-waves, in a frequency range of approximately 1 Hz to approximately 100 Hz). Seismic data may be processed and interpreted, for example, to understand better composition, fluid content, extent and geometry of subsurface rocks.

SUMMARY

[0003] In accordance with some embodiments, a method is performed that includes: receiving data that include a signal and a ghost signal that is characterized at least in part by a ghost delay; and, for a deghosting operator parameterized with respect to a ghost delay parameter, determining a ghost delay parameter value that minimizes a p-norm of a result generated by applying the deghosting operator to at least a portion of the data.

[0004] In accordance with some embodiments, a system is provided that includes a processor; memory accessible by the processor; processor-executable instructions stored in the memory to instruct the system to: receive data that include a signal and a ghost signal that is characterized at least in part by a ghost delay; and, for a deghosting operator parameterized with respect to a ghost delay parameter, determine a ghost delay parameter value that minimizes a p-norm of a result generated by application of the deghosting operator to at least a portion of the data. [0005] In accordance with some embodiments, one or more computer- readable storage media include computer-executable instructions to instruct a system to: receive data that include a signal and a ghost signal that is characterized at least in part by a ghost delay; and, for a deghosting operator parameterized with respect to a ghost delay parameter, determine a ghost delay parameter value that minimizes a p-norm of a result generated by application of the deghosting operator to at least a portion of the data.

[0006] This summary is provided to introduce a selection of concepts that are further described below in the detailed description. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in limiting the scope of the claimed subject matter.

BRIEF DESCRIPTION OF THE DRAWINGS

[0007] Features and advantages of the described implementations can be more readily understood by reference to the following description taken in conjunction with the accompanying drawings.

[0008] Fig. 1 illustrates an example of a geologic environment and an example of a technique;

[0009] Fig. 2 illustrates examples of multiple reflections and examples of techniques;

[0010] Fig. 3 illustrates

[0011] Fig. 4 illustrates

[0012] Fig. 5 illustrates

[0013] Fig. 6 illustrates

[0014] Fig. 7 illustrates

[0015] Fig. 8 illustrates

[0016] Fig. 9 illustrates

delays;

[0017] Fig. 10 illustrates example plots of spectra and function values;

[0018] Fig. 1 1 illustrates example plots of waveforms and function values;

[0019] Fig. 12 illustrates example plots;

[0020] Fig. 13 illustrates example plots associated with an example of a shot gather; [0021] Fig. 14 illustrates

ghosting;

[0022] Fig. 15 illustrates

[0023] Fig. 16 illustrates

[0024] Fig. 17 illustrates

[0025] Fig. 18 illustrates

[0026] Fig. 19 illustrates

[0027] Fig. 20 illustrates

[0028] Fig. 21 illustrates

system.

DETAI LED DESCRIPTION

[0029] The following description includes the best mode presently

contemplated for practicing the described implementations. This description is not to be taken in a limiting sense, but rather is made merely for the purpose of describing the general principles of the implementations. The scope of the described implementations should be ascertained with reference to the issued claims.

[0030] As mentioned, reflection seismology finds use in geophysics, for example, to estimate properties of subsurface formations. As an example, reflection seismology may provide seismic data representing waves of elastic energy (e.g. , as transmitted by P-waves and S-waves, in a frequency range of approximately 1 Hz to approximately 100 Hz or optionally less that 1 Hz and/or optionally more than 100 Hz). Seismic data may be processed and interpreted, for example, to understand better composition, fluid content, extent and geometry of subsurface rocks.

[0031] Fig. 1 shows an example of a geologic environment 100 (e.g. , an environment that includes a sedimentary basin, a reservoir 101 , a fault 103, one or more fractures 109, etc.) and an example of an acquisition technique 140 to acquire seismic data (see, e.g. , data 160). As an example, a system may process data acquired by the technique 140, for example, to allow for direct or indirect

management of sensing, drilling, injecting, extracting, etc., with respect to the geologic environment 100. In turn, further information about the geologic

environment 100 may become available as feedback (e.g. , optionally as input to the system). As an example, an operation may pertain to a reservoir that exists in the geologic environment 100 such as, for example, the reservoir 101 . As an example, a technique may provide information (e.g., as an output) that may specifies one or more location coordinate of a feature in a geologic environment, one or more characteristics of a feature in a geologic environment, etc.

[0032] As an example, the geologic environment 100 may be referred to as or include one or more formations. As an example, a formation may be a unit of lithostratigraphy, for example, a body of rock that is sufficiently distinctive and continuous that it can be mapped. As an example, in stratigraphy, a formation may be a body of strata of predominantly one type or combination of types, for example, where multiple formations form groups, and subdivisions of formations are members.

[0033] As an example, a sedimentary basin may be a depression in the crust of the Earth, for example, formed by plate tectonic activity in which sediments accumulate. Over a period of geologic time, continued deposition may cause further depression or subsidence. With respect to a petroleum systems analysis, if rich hydrocarbon source rocks occur in combination with appropriate depth and duration of burial, hydrocarbon generation may possibly occur within a basin. Exploration plays and prospects may be developed in basins or regions in which a complete petroleum system has some likelihood of existing. The geologic environment 100 of Fig. 1 may include one or more plays, prospects, etc.

[0034] As an example, a system may be implemented to process seismic data, optionally in combination with other data. Processing of data may include generating one or more seismic attributes, rendering information to a display or displays, etc. A process or workflow may include interpretation, which may be performed by an operator that examines renderings of information and that identifies structure or other features within such renderings. Interpretation may be or include analyses of data with a goal to generate one or more models and/or predictions (e.g., about properties and/or structures of a subsurface region).

[0035] As an example, a system may include features of a commercially available framework such as the PETREL® seismic to simulation software framework (Schlumberger Limited, Houston, Texas). The PETREL® framework provides components that allow for optimization of exploration and development operations. The PETREL® framework includes seismic to simulation software components that can output information for use in increasing reservoir performance, for example, by improving asset team productivity. Through use of such a framework, various professionals (e.g. , geophysicists, geologists, and reservoir engineers) can develop collaborative workflows and integrate operations to streamline processes. Such a framework may be considered an application and may be considered a data-driven application (e.g. , where data is input for purposes of simulating a geologic environment, decision making, operational control, etc.).

[0036] As an example, a system may include add-ons or plug-ins that operate according to specifications of a framework environment. For example, a

commercially available framework environment marketed as the OCEAN®

framework environment (Schlumberger Limited, Houston, Texas) allows for integration of add-ons (or plug-ins) into a PETREL® framework workflow. The OCEAN® framework environment leverages .NET® tools (Microsoft Corporation, Redmond, Washington) and offers stable, user-friendly interfaces for efficient development. In an example embodiment, various components (e.g. , modules, blocks, etc.) may be implemented as add-ons (or plug-ins) that conform to and operate according to specifications of a framework environment (e.g. , according to application programming interface (API) specifications, etc.).

[0037] As an example, seismic data may be processed using a framework such as the OMEGA® framework (Schlumberger Limited, Houston, TX). The OMEGA® framework provides features that can be implemented for processing of seismic data, for example, through prestack seismic interpretation and seismic inversion. A framework may be scalable such that it enables processing and imaging on a single workstation, on a massive compute cluster, etc. As an example, one or more techniques, technologies, etc. described herein may optionally be implemented in conjunction with a framework such as, for example, the OMEGA® framework.

[0038] A framework for processing data may include features for 2D line and 3D seismic surveys. Modules for processing seismic data may include features for prestack seismic interpretation (PSI), optionally pluggable into a framework such as the OCEAN® framework. A workflow may be specified to include processing via one or more frameworks, plug-ins, add-ons, etc. A workflow may include quantitative interpretation, which may include performing pre- and poststack seismic data conditioning, inversion (e.g., seismic to properties and properties to synthetic seismic), wedge modeling for thin-bed analysis, amplitude versus offset (AVO) and amplitude versus angle (AVA) analysis, reconnaissance, etc. As an example, a workflow may aim to output rock properties based at least in part on processing of seismic data. As an example, various types of data may be processed to provide one or more models (e.g. , earth models). For example, consider processing of one or more of seismic data, well data, electromagnetic and magnetic telluric data, reservoir data, etc.

[0039] In the example of Fig. 1 , the geologic environment 100 includes an offshore portion and an on-shore portion. As an example, a geologic environment may be or include one or more of an offshore geologic environment, a seabed geologic environment, an ocean bed geologic environment, etc.

[0040] As an example, the geologic environment 100 may be outfitted with any of a variety of sensors, detectors, actuators, etc. For example, equipment 102 may include communication circuitry to receive and to transmit information with respect to one or more networks 105. Such information may include information associated with downhole equipment 104, which may be equipment to acquire information, to assist with resource recovery, etc. Other equipment 106 may be located remote from a well site and include sensing, detecting, emitting or other circuitry. Such equipment may include storage and communication circuitry to store and to communicate data, instructions, etc. As an example, one or more satellites may be provided for purposes of communications, data acquisition, etc. For example, Fig. 1 shows a satellite in communication with the network 105 that may be configured for communications, noting that the satellite may additionally or alternatively include circuitry for imagery (e.g., spatial, spectral, temporal,

radiometric, etc.).

[0041] Fig. 1 also shows the geologic environment 100 as optionally including equipment 107 and 108 associated with a well that includes a substantially horizontal portion that may intersect with one or more of the one or more fractures 109. For example, consider a well in a shale formation that may include natural fractures, artificial fractures (e.g. , hydraulic fractures) or a combination of natural and artificial fractures. As an example, a well may be drilled for a reservoir that is laterally extensive. In such an example, lateral variations in properties, stresses, etc. may exist where an assessment of such variations may assist with planning, operations, etc. to develop the reservoir (e.g., via fracturing, injecting, extracting, etc.). As an example, the equipment 107 and/or 108 may include components, a system, systems, etc. for fracturing, seismic sensing, analysis of seismic data, assessment of one or more fractures, etc.

[0042] As an example, a system may be used to perform one or more workflows. A workflow may be a process that includes a number of worksteps. A workstep may operate on data, for example, to create new data, to update existing data, etc. As an example, a system may operate on one or more inputs and create one or more results, for example, based on one or more algorithms. As an example, a system may include a workflow editor for creation, editing, executing, etc. of a workflow. In such an example, the workflow editor may provide for selection of one or more pre-defined worksteps, one or more customized worksteps, etc. As an example, a workflow may be a workflow implementable in the PETREL® software, for example, that operates on seismic data, seismic attribute(s), etc. As an example, a workflow may be a process implementable in the OCEAN® framework. As an example, a workflow may include one or more worksteps that access a module such as a plug-in (e.g. , external executable code, etc.). As an example, a workflow may include rendering information to a display (e.g., a display device). As an example, a workflow may include receiving instructions to interact with rendered information, for example, to process information and optionally render processed information. As an example, a workflow may include transmitting information that may control, adjust, initiate, etc. one or more operations of equipment associated with a geologic environment (e.g., in the environment, above the environment, etc.).

[0043] In Fig. 1 , the technique 140 may be implemented with respect to a geologic environment 141. As shown, an energy source (e.g., a transmitter) 142 may emit energy where the energy travels as waves that interact with the geologic environment 141 . As an example, the geologic environment 141 may include a bore 143 where one or more sensors (e.g., receivers) 144 may be positioned in the bore 143. As an example, energy emitted by the energy source 142 may interact with a layer (e.g., a structure, an interface, etc.) 145 in the geologic environment 141 such that a portion of the energy is reflected, which may then be sensed by one or more of the sensors 144. Such energy may be reflected as an upgoing primary wave (e.g., or "primary" or "singly" reflected wave). As an example, a portion of emitted energy may be reflected by more than one structure in the geologic environment and referred to as a multiple reflected wave (e.g. , or "multiple"). For example, the geologic environment 141 is shown as including a layer 147 that resides below a surface layer 149. Given such an environment and arrangement of the source 142 and the one or more sensors 144, energy may be sensed as being associated with particular types of waves.

[0044] As an example, a "multiple" may refer to multiply reflected seismic energy or, for example, an event in seismic data that has incurred more than one reflection in its travel path. As an example, depending on a time delay from a primary event with which a multiple may be associated, a multiple may be

characterized as a short-path or a peg-leg, for example, which may imply that a multiple may interfere with a primary reflection, or long-path, for example, where a multiple may appear as a separate event. As an example, seismic data may include evidence of an interbed multiple from bed interfaces, evidence of a multiple from a water interface (e.g., an interface of a base of water and rock or sediment beneath it) or evidence of a multiple from an air-water interface, etc.

[0045] As shown in Fig. 1 , the acquired data 160 can include data associated with downgoing direct arrival waves, reflected upgoing primary waves, downgoing multiple reflected waves and reflected upgoing multiple reflected waves. The acquired data 160 is also shown along a time axis and a depth axis. As indicated, in a manner dependent at least in part on characteristics of media in the geologic environment 141 , waves travel at velocities over distances such that relationships may exist between time and space. Thus, time information, as associated with sensed energy, may allow for understanding spatial relations of layers, interfaces, structures, etc. in a geologic environment.

[0046] Fig. 1 also shows various types of waves as including P, SV an SH waves. As an example, a P-wave may be an elastic body wave or sound wave in which particles oscillate in the direction the wave propagates. As an example, P- waves incident on an interface (e.g. , at other than normal incidence, etc.) may produce reflected and transmitted S-waves (e.g. , "converted" waves). As an example, an S-wave or shear wave may be an elastic body wave, for example, in which particles oscillate perpendicular to the direction in which the wave propagates. S-waves may be generated by a seismic energy sources (e.g., other than an air gun). As an example, S-waves may be converted to P-waves. S-waves tend to travel more slowly than P-waves and do not travel through fluids that do not support shear. In general, recording of S-waves involves use of one or more receivers operatively coupled to earth (e.g., capable of receiving shear forces with respect to time). As an example, interpretation of S-waves may allow for determination of rock properties such as fracture density and orientation, Poisson's ratio and rock type, for example, by crossplotting P-wave and S-wave velocities, and/or by other techniques.

[0047] As an example of parameters that may characterize anisotropy of media (e.g. , seismic anisotropy), consider the Thomsen parameters ε, δ and γ. The Thomsen parameter δ describes depth mismatch between logs (e.g., actual depth) and seismic depth. As to the Thomsen parameter ε, it describes a difference between vertical and horizontal compressional waves (e.g., P or P-wave or quasi compressional wave qP or qP-wave). As to the Thomsen parameter γ, it describes a difference between horizontally polarized and vertically polarized shear waves (e.g. , horizontal shear wave SH or SH-wave and vertical shear wave SV or SV-wave or quasi vertical shear wave qSV or qSV-wave). Thus, the Thomsen parameters ε and γ may be estimated from wave data while estimation of the Thomsen parameter δ may involve access to additional information.

[0048] As an example, seismic data may be acquired for a region in the form of traces. In the example of Fig. 1 , the technique 140 may include the source 142 for emitting energy where portions of such energy (e.g. , directly and/or reflected) may be received via the one or more sensors 144. As an example, energy received may be discretized by an analog-to-digital converter that operates at a sampling rate. For example, acquisition equipment may convert energy signals sensed by a sensor to digital samples at a rate of one sample per approximately 4 ms. Given a speed of sound in a medium or media, a sample rate may be converted to an approximate distance. For example, the speed of sound in rock may be of the order of around 5 km per second. Thus, a sample time spacing of approximately 4 ms would correspond to a sample "depth" spacing of about 10 meters (e.g. , assuming a path length from source to boundary and boundary to sensor). As an example, a trace may be about 4 seconds in duration; thus, for a sampling rate of one sample at about 4 ms intervals, such a trace would include about 1000 samples where latter acquired samples correspond to deeper reflection boundaries. If the 4 second trace duration of the foregoing example is divided by two (e.g. , to account for reflection), for a vertically aligned source and sensor, the deepest boundary depth may be estimated to be about 10 km (e.g., assuming a speed of sound of about 5 km per second).

[0049] Fig. 2 shows an example of a geologic environment 201 that includes a seabed 203 and a sea surface 205. As shown, equipment 210 such as a ship may tow an energy source 220 and a string of sensors 230 at a depth below the sea surface 205. In such an example, the energy source 220 may emit energy at a time TO, a portion of that energy may be reflected from the seabed 203 at a time T1 and a portion of that reflected energy may be received at the string of sensors 230 at a time T2.

[0050] As mentioned with respect to the technique 140 of Fig. 1 , a wave may be a primary or a wave may be a multiple. As shown in an enlarged view of the geologic environment 201 , the sea surface 205 may act to reflect waves such that sensors 232 of the string of sensors 230 may sense multiples as well as primaries. In particular, the sensors 232 may sense so-called sea surface multiples, which may be multiples from primaries or multiples of multiples (e.g. , due to sub-seabed reflections, etc.).

[0051] As an example, each of the sensors 232 may sense energy of an upgoing wave at a time T2 where the upgoing wave reflects off the sea surface 205 at a time T3 and where the sensors may sense energy of a downgoing multiple reflected wave at a time T4 (see also the data 160 of Fig. 1 and data 240 of Fig. 2). In such an example, sensing of the downgoing multiple reflected wave may be considered noise that interferes with sensing of one or more upgoing waves. As an example, an approach that includes summing data acquired by a geophone and data acquired by a hydrophone may help to diminish noise associated with downgoing multiple reflected waves. Such an approach may be employed, for example, where sensors may be located proximate to a surface such as the sea surface 205 (e.g. , arrival times T2 and T4 may be relatively close). As an example, the sea surface 205 or a water surface may be an interface between two media. For example, consider an air and water interface. As an example, due to differing media properties, sound waves may travel at about 1 ,500 m/s in water and at about 340 m/s in air. As an example, at an air and water interface, energy may be transmitted and reflected. [0052] As an example, each of the sensors 232 may include at least one geophone 234 and a hydrophone 236. As an example, a geophone may be a sensor configured for seismic acquisition, whether onshore and/or offshore, that can detect velocity produced by seismic waves and that can, for example, transform motion into electrical impulses. As an example, a geophone may be configured to detect motion in a single direction. As an example, a geophone may be configured to detect motion in a vertical direction. As an example, three mutually orthogonal geophones may be used in combination to collect so-called 3C seismic data. As an example, a hydrophone may be a sensor configured for use in detecting seismic energy in the form of pressure changes under water during marine seismic acquisition. As an example, hydrophones may be positioned along a string or strings to form a streamer or streamers that may be towed by a seismic vessel (e.g., or deployed in a bore). Thus, in the example of Fig. 2, the at least one geophone 234 can provide for motion detection and the hydrophone 236 can provide for pressure detection. As an example, the data 240 (e.g., analog and/or digital) may be transmitted via equipment, for example, for processing, etc.

[0053] As an example, a method may include analysis of hydrophone response and vertical geophone response, which may help to improve a PZ summation, for example, by reducing receiver ghost and/or free surface-multiple noise contamination (see, e.g. , PZSUM algorithm, discussed further below). As an example, a ghost may be defined as a reflection of a wavefield as reflected from a water surface (e.g. , water and air interface) that is located above a receiver, a source, etc. (e.g. , a receiver ghost, a source ghost, etc.). As an example, a receiver may experience a delay between an upgoing wavefield and its downgoing ghost, which may depend on depth of the receiver.

[0054] As an example, a surface marine cable may be or include a buoyant assembly of electrical wires that connect sensors and that can relay seismic data to the recording seismic vessel. As an example, a multi-streamer vessel may tow more than one streamer cable to increase the amount of data acquired in one pass. As an example, a marine seismic vessel may be about 75 m long and travel about 5 knots, for example, while towing arrays of air guns and streamers containing sensors, which may be located, for example, about a few meters below the surface of the water. A so-called tail buoy may assist crew in location an end of a streamer. As an example, an air gun may be activated periodically, such as about each 25 m (e.g. , about at 10 second intervals) where the resulting sound wave travels into the Earth, which may be reflected back by one or more rock layers to sensors on a streamer, which may then be relayed as signals (e.g. , data, information, etc.) to equipment on the tow vessel.

[0055] In the example of Fig. 2, the equipment 210 may include a system such as the system 250. As shown in Fig. 2, the system 250 includes one or more information storage devices 252, one or more computers 254, one or more network interfaces 260 and one or more modules 270 (e.g. , processor-executable

instructions, etc.). As to the one or more computers 254, each computer may include one or more processors (e.g., or processing cores) 256 and memory 258 for storing instructions (e.g., modules), for example, executable by at least one of the one or more processors. As an example, a computer may include one or more network interfaces (e.g., wired or wireless), one or more graphics cards, a display interface (e.g., wired or wireless), etc. As an example, a system may include one or more display devices (e.g., optionally as part of a computing device, etc.).

[0056] As an example, pressure data may be represented as "P" and velocity data may be represented as "Z"; noting, however, that the vertical component of a measured particle velocity vector may be denoted "V" and that "Z" may refer to a scaled, measured particle velocity. For example, in various equations presented herein, "V" represents a measured velocity and "Z" represents a scaling thereof.

[0057] As an example, a hydrophone may sense pressure information (e.g., P data) and a geophone may sense velocity information (e.g., V and/or Z data). As an example, hydrophone may output signals, optionally as digital data, for example, for receipt by a system. As an example, a geophone may output signals, optionally as digital data, for example, for receipt by a system. As an example, the system 250 may receive P and V/Z data via one or more of the one or more network interfaces 260 and process such data, for example, via execution of instructions stored in the memory 258 by the processor 256. As an example, the system 250 may store raw and/or processed data in one or more of the one or more information storage devices 252.

[0058] Fig. 3 shows an example of a side view of a marine-based survey 360 of a subterranean subsurface 362. The subsurface 362 includes a seafloor surface 364. Seismic sources 366 may include marine sources such as vibroseis or air guns, which may propagate seismic waves 368 (e.g. , energy signals) into the Earth over an extended period of time or at a nearly instantaneous energy provided by impulsive sources. The seismic waves may be propagated by marine sources as a frequency sweep signal. For example, marine sources of the vibroseis type may initially emit a seismic wave at a low frequency (e.g., about 5 Hz) and increase the seismic wave to a higher frequency (e.g. , about 80 Hz to about 90Hz or more) over time.

[0059] The component(s) of the seismic waves 368 may be reflected and converted by the seafloor surface 364 (e.g., as a reflector), and seismic wave reflections 370 may be received by a plurality of seismic receivers 372. As an example, seismic waves may penetrate the subsurface 362 below the seafloor surface 364 and be reflected by one or more reflectors therein and received by one or more of the plurality of seismic receivers 372. As shown in the example of Fig. 3, the seismic receivers 372 may be disposed on a plurality of streamers (e.g. , a streamer array 374). The seismic receivers 372 may generate electrical signals representative of the received seismic wave reflections 370. The electrical signals may be embedded with information regarding the subsurface 362 and captured as a record of seismic data.

[0060] In one implementation, each streamer may include streamer steering devices such as a bird, a deflector, a tail buoy and the like. One or more streamer steering devices may be used to control streamer position.

[0061] In one implementation, the seismic wave reflections 370 may travel upward and reach the water/air interface at the water surface 376, a portion of reflections 370 may then reflect downward again (e.g., sea-surface ghost waves 378) and be received by the plurality of seismic receivers 372. As an example, the sea-surface ghost waves 378 may be referred to as surface multiples. In such an example, the point on the water surface 376 at which the wave is reflected downward may be referred to as a downward reflection point.

[0062] Electrical signals generated by one or more of the receivers 372 may be transmitted to a vessel 380 via transmission cables, wireless communication or the like. The vessel 380 may then transmit the electrical signals to a data processing center. Alternatively, the vessel 380 may include an onboard computing system capable of processing the electrical signals (e.g. , representing seismic data). As an example, surveys may be of formations deep beneath the surface. The formations may include multiple reflectors, some of which may include dipping events, and may generate multiple reflections (including wave conversion) for receipt by the seismic receivers 372. As an example, seismic data may be processed to generate a seismic image of the subsurface.

[0063] As an example, a marine seismic acquisition system may tow streamers in the streamer array 374 at an approximate even depth (e.g., about 5 m to about 10 m). However, the marine based survey 360 may tow each streamer in streamer array 374 at different depths such that seismic data may be acquired and processed in a manner that avoids the effects of destructive interference due to sea- surface ghost waves. For instance, the marine-based survey 360 of Fig. 3 illustrates eight streamers towed by the vessel 380 at eight different depths. The depth of each streamer may be controlled and maintained using the birds disposed on each streamer.

[0064] Fig. 4 shows an example of a system 420 in which one or more vessels 422 may be employed to enable seismic profiling, e.g., three-dimensional vertical seismic profiling (VSP) or rig/offset vertical seismic profiling (VSP). In the example of Fig. 4, the system 420 is illustrated as including a rig 450, the vessel 422, and one or more acoustic receivers 428 (e.g., a receiver array). As an example, a vessel may include a source 424 (e.g., or source array) and/or the rig 450 may include a source 424 (e.g. , or source array).

[0065] As an example, the vessel 422 may travel a path or paths where locations may be recorded through the use of navigation system signals 436. As an example, such signals may be associated with a satellite-based system that includes one or more satellites 452 and 438. As an example, the satellite 438 may be part of a global positioning system (GPS), which may be implemented to record position, speed, direction, and other parameters of the vessel 422. As an example, one or more satellites, communication equipment, etc. may be configured to provide for VSAT communications, VHF communications, UHF communications, etc.

[0066] In the example of Fig. 4, the acoustic receivers 428 may be part of a data acquisition system 426, for example, that may be deployed in borehole 430 via one or more of a variety of delivery systems, such as wireline delivery systems, slickline delivery systems, and other suitable delivery systems. As an example, the acoustic receivers 428 may be communicatively coupled with processing equipment 458, which may be positioned at a downhole location. By way of example, processing equipment 458 may include a telemetry system for transmitting data from acoustic receivers 428 to additional processing equipment 462 located at the surface, e.g. , on the rig 450 and/or vessels 422. As an example, information acquired may optionally be transmitted (see, e.g. , signals 459).

[0067] Depending on the specifics of a given data communication system, examples of surface processing equipment 462 may include a radio repeater 460 and/or one or more of a variety of other and/or additional signal transfer components and signal processing components. The radio repeater 460 along with other components of processing equipment 462 may be used to communicate signals, e.g. , UHF and/or VHF signals, between vessels (e.g. , the vessel 422 and one or more other vessels) and the rig 450, for example, to enable further communication with downhole data acquisition system 426.

[0068] As an example, the acoustic receivers 428 may be coupled to the surface processing equipment 462 via one or more wire connections; noting that additionally or alternatively wireless and/or optical connections may be employed.

[0069] As an example, the surface processing equipment 462 may include a synchronization unit, for example, to assist with coordination of emissions from one or more sources (e.g. , optionally dithered (delayed) source arrays). As an example, coordination may extend to one or more receivers (e.g. , consider the acoustic receivers 428 located in borehole 430). As an example, a synchronization unit may use coordinated universal time, optionally employed in cooperation with a global positioning system (e.g., to obtain UTC data from GPS receivers of a GPS system).

[0070] Fig. 4 illustrates examples of equipment for performing seismic profiling that can employ simultaneous or near-simultaneous acquisition of seismic data. By way of example, the seismic profiling may include three-dimensional vertical seismic profiling (VSP) but other applications may utilize rig/offset vertical seismic profiling or seismic profiling employing walkaway lines. As an example, an offset source may be provided by the source 424 located on the rig 450, on the vessel 422, and/or on another vessel or structure (e.g. , stationary and/or movable from one location to another location). [0071] As an example, a system may employ one or more of various arrangements of a source or sources on a vessel(s) and/or a rig(s). As shown in the example of Fig. 4, the acoustic receivers 428 of downhole acquisition system 426 are configured to receive the source signals, at least some of which are reflected off a reflection boundary 464 located beneath a sea bottom 436. The acoustic receivers 428 may generate data streams that are relayed uphole to a suitable processing system, e.g. , the processing system 462.

[0072] While the acoustic receivers 428 may generate data streams, a navigation system may determine a real-time speed, position, and direction of the vessel 422 and also estimate initial shot times accomplished via signal generators 454 of the appropriate source 424 (e.g., or source array). A source controller may be part of the surface processing equipment 462 (e.g., located on the rig 450, on the vessel 422, or at other suitable location) and may be configured with circuitry that can control firing of acoustic source generated signals so that the timing of an additional shot time (e.g., optionally a shot time via a slave vessel) may be based on an initial shot time (e.g., a shot time via a master vessel) plus a dither value.

[0073] As an example, a synchronization unit of, for example, the surface processing equipment 462, may coordinate firing of dithered acoustic signals with recording of acoustic signals by the downhole acquisition system 426. A processor system may be configured to separate a data stream of the initial shot and a data stream of the additional shot via a coherency filter. As an example, an approach may employ simultaneous acquisition and/or may not perform separation of the data streams. In such cases, the dither may be effectively zero.

[0074] After an initial shot time at T=0 (TO) is determined, subsequent firings of acoustic source arrays may be offset by a dither. The dithers may be positive or negative and sometimes created as pre-defined random delays. Use of dithers facilitates the separation of simultaneous or near-simultaneous data sets to simplify the data processing. The ability to have acoustic source arrays fire in simultaneous or near-simultaneous patterns reduces the overall amount of time used for three- dimensional vertical seismic profiling source acquisition. This, in turn, may reduce rig time. As a result, the overall cost of the seismic operation may be reduced, rendering the data intensive process much more accessible. [0075] If acoustic source arrays used in the seismic data acquisition are widely separated, the difference in move-outs across the acoustic receiver array of the wave fields generated by the acoustic sources can be sufficient to obtain a relatively clean data image via processing the data. However, even when acoustic sources are substantially co-located in time, data acquired a method involving dithering of the firing times of the individual sources may be processed to a formation image. For example, consider taking advantage of the incoherence of the data generated by one acoustic source when seen in the reference time of another acoustic source.

[0076] Also shown in Fig. 4 is an inset example of a zero-offset vertical seismic profile (VSP) scenario 490. In such an example, an acquisition geometry may be limited to an ability to position equipment that is physically coupled to the rig 450. As shown, for given the acquisition geometry, there may be no substantial offset between the source 424 and bore 430. In such an example, a zero-offset VSP may be acquired where seismic waves travel substantially vertically down to a reflector (e.g., the layer 464) and up to the receiver 428, which may be a receiver array. Where one or more vessels are employed (e.g., the vessel 422), one or more other types of surveys may be performed. As an example, a three-dimensional VSP may be performed using a vessel.

[0077] Fig. 5 shows an example of a technique 501 with respect to a geologic environment 541 , a surface 549, at least one energy source (e.g. , a transmitter) 542 that may emit energy where the energy travels as waves that interact with the geologic environment 541. As an example, the geologic environment 541 may include a bore 543 where one or more sensors (e.g., receivers) 544 may be positioned in the bore 543. As an example, energy emitted by the energy source 542 may interact with a layer (e.g. , a structure, an interface, etc.) 545 in the geologic environment 441 such that a portion of the energy is reflected, which may then be sensed by at least one of the one or more of the sensors 544.

[0078] As an example, a 3D VSP technique may be implemented with respect to an onshore and/or an offshore environment. As an example, an acquisition technique for an onshore (e.g. , land-based) survey may include positioning a source or sources along a line or lines of a grid; whereas, in an offshore implementation, source positions may be laid out in lines or in a spiral centered near a well. [0079] A 3D acquisition technique may help to illuminate one or more 3D structures (e.g., one or more features in a geologic environment). Information acquired from a 3D VSP may assist with exploration and development, pre-job modeling and planning, etc. As an example, a 3D VSP may fill in one or more regions that lack surface seismic survey information, for example, due to interfering surface infrastructure or difficult subsurface conditions, such as, for example, shallow gas, which may disrupt propagation of P-waves (e.g., seismic energy traveling through fluid may exhibit signal characteristics that differ from those of seismic energy traveling through rock).

[0080] As an example, a VSP may find use to tie time-based surface seismic images to one or more depth-based well logs. For example, in an exploration area, a nearest well may be quite distant such that a VSP is not available for calibration before drilling begins on a new well. Without accurate time-depth correlation, depth estimates derived from surface seismic images may include some uncertainties, which may, for example, add risk and cost (e.g. , as to contingency planning for drilling programs). As an example, a so-called intermediate VSP may be performed, for example, to help develop a time-depth correlation. For example, an intermediate VSP may include running a wireline VSP before reaching a total depth. Such a survey may, for example, provide for a relatively reliable time-depth conversion; however, it may also add cost and inefficiency to a drilling operation and, for example, it may come too late to forecast drilling trouble. As an example, a seismic while drilling process may be implemented, for example, to help reduce uncertainty in time-depth correlation without having to stop a drilling process. Such an approach may provide real-time seismic waveforms that can allow an operator to look ahead of a drill bit, for example, to help guide a drill string to a target total depth.

[0081] Fig. 6 shows an example scenario 601 where drilling equipment 603 operates a drill bit 604 operatively coupled to an equipment string that includes one or more sensors (e.g. , one or more receivers) 644. In the scenario 601 , the drill bit 604 is advanced in a geologic environment 641 that includes stratified layers disposed below a sea bed surface where the layers include a layer 645. As shown in the example of Fig. 6, at a water surface 649 of the geologic environment 641 , seismic equipment 605 includes a seismic energy source 642 that can emit seismic energy into the geologic environment 641 . [0082] As an example, the seismic equipment 605 may be moveable, duplicated, etc., for example, to emit seismic energy from various positions, which may be positions about a region of the geologic environment 641 that includes the drill bit 604. As an example, the scenario 601 may be a VSP scenario, for example, where the equipment 603, 644, 605 and 642 can perform a seismic survey (e.g. , a VSP while drilling survey).

[0083] As an example, a survey may take place during one or more so-called "quiet" periods during which drilling is paused. As an example, data acquired via a survey may be analyzed where results from an analysis or analyses may be used, at least in part, to direct further drilling, make assessments as to a drilled portion of a geologic environment, etc. As an example, a method may optionally include processing in near real-time, which may, for example, be instructive for seismic while drilling, etc.

[0084] In Fig. 6, the method 650 includes an acquisition block 654 for acquiring data, an analysis block 658 for analyzing at least a portion of the data and an adjustment block 662 for adjusting one or more field operations, for example, based at least in part on an output from analyzing data.

[0085] The method 650 may be associated with various computer-readable media (CRM) blocks or modules 653, 657 and 663. Such blocks or modules may include instructions suitable for execution by one or more processors (or processor cores) to instruct a computing device or system to perform one or more actions. As an example, a single medium may be configured with instructions to allow for, at least in part, performance of various actions of the method 650. As an example, a computer-readable medium (CRM) may be a computer-readable storage medium (e.g., a non-transitory medium).

[0086] As noted above, in marine seismic acquisitions, sensors may record desired upgoing wavefield energy reflected from one or more geological formations and reflections from the sea surface (e.g., an air and water interface), which may be referred to as downgoing wavefield energy (e.g. , or seismic ghosts).

[0087] As an example, a ghost may cause one or more notches in a frequency spectrum, for example, at one or more frequencies that may be described as a function of receiver depth below an air and water interface and angle of incidence of a wavefield at a receiver. In practice, receiver depths may be chosen so that these frequency notches are beyond the range of frequencies desired for the seismic data. Depths of less than about 10 meters may be used, making the first occurrence of a notch in the spectrum of the pressure wavefield above 75 Hz in standard scenarios. However, deploying seismic streamers deeper than 10 m may allow for recording more energy at low frequency. Yet, such an approach may produce ghost notches at lower frequencies than about 75 Hz. In the foregoing example, 75 Hz is provided as an example frequency for purposes of explaining various types of phenomena.

[0088] As an example, to address the presence of ghosts or ghosting, a method can include deghosting (e.g., to reduce effects caused by the presence of ghosts, etc.). As an example, deghosting may optionally be applied in an adaptive manner (e.g., adaptive deghosting). As an example, a method can include estimating a "true" ghost delay. In such an example, an estimated ghost delay may be a ghost delay parameter value that can be applied to deghost data that includes a signal and a ghost signal (e.g. , an upgoing signal and a downgoing ghost signal). As an example, a method may be applied to a single measured signal that includes a ghost signal and/or to multi-measurement signal acquisitions.

[0089] As explained, a ghost reflection can be a delayed replica of a seismic wavefield that is reflected at the sea surface. A sea surface, as an air and water interface, may be characterized by an impedance mismatch that is sufficient large to cause the sea surface to act substantially as a mirror for an upgoing wavefield.

When a ghost reflection (e.g., downgoing wavefield) reaches a sensor or sensors, it can interfere with a subsursface reflected wavefield (e.g., upgoing wavefield) and, for example, generate one or more notches in a spectrum of the measured wavefield (e.g., sensed wavefield), which, in turn, can reduce usable bandwidth of a measured wavefield signal and hence, resolution of the seismic data.

[0090] A ghosting effect may be modelled mathematically as a filter, which may be referred to as, for example, a ghost operator. As an example, a ghost operator may be parameterized by a delay parameter that characterizes a delay between upgoing and downgoing waves. Such a delay may be referred to as a ghost delay and may be valued with respect to time (e.g. , consider time in

milliseconds, etc.). As an example, a method may aim to "remove" a ghost(s) at a receiver(s) in an effort to restore the effective bandwidth of the signal and, in turn, to increase resolution of a processed seismic image. To reduce the presence of a ghost, a ghost operator can be estimated with a desired amount of accuracy (e.g. , which may depend on one or more factors). Ghost operator estimation can depend on accuracy of estimation of a ghost delay (e.g., a ghost delay parameter value).

[0091] Where depth of a streamer is known and data are well sampled at the surface, a ghost delay may be calculated. However, such conditions are seldom met in practice (e.g., during acquisition at sea using subsurface sensors). For example, data may be sampled in a crossline direction with some amount of uncertainty compared to data sampled in an inline direction. Further, rough seas and inaccurate depth measurements can result in uncertainty in receiver depth (e.g., sensor depth) and hence give rise to uncertainty in a calculated ghost delay value. To address such issues, an algorithm may make relatively unrealistic assumptions about the acquired data and/or use additional measurements (e.g., by using dual streamers, particle motion related measurements when these measurements are available at a sensor cable).

[0092] As an example, a method can be data adaptive. As an example, a method can include deghosting based on p-norm minimization, optionally with one or more constraints to estimate a ghost operator. Such a method may be a data adaptive deghosting method that includes determining a ghost delay parameter value, which may be performed, for example, in a time domain and/or in another domain (e.g. , consider a frequency domain).

[0093] As an example, a method can include ghost operator parameter estimation and deghosting data using one or more types of deterministic

deconvolution. Such a method may be implemented for single sensor

measurements and/or for multisensory measurements. As an example, a method may provide a relatively robust shield against noise, for example, where parameter estimation may be performed using a filtered version of measured data. As an example, a method may be flexible in that it can be applied in one or more domains (e.g., time, frequency, etc.). As an example, a method may optionally include performing quality control, for example, where output is generated via an operator multiplying at least a portion of received data.

[0094] Fig. 7 shows an example of a method 750 that includes a reception block 754 and a determination block 758. As shown, the reception block 754 can include receiving data that include a signal and a ghost signal that is characterized at least in part by a ghost delay and the determination block 758 can include, for a deghosting operator parameterized with respect to a ghost delay parameter, determining a ghost delay parameter value that minimizes a p-norm of a result generated by applying the deghosting operator to the data. In such an example, the deghosting operator can be or include an inverse deghosting filter.

[0095] As shown in the example of Fig. 7, a method can include performance block 730 for performing data acquisition (e.g. , a seismic survey, etc.) and a method can include generating at least one image based at least in part on a result (e.g., a result of the determination block 758 of the method 750). In such an example, at least one image may be rendered to a display. As an example, the method 750 of Fig. 7 may be part of a workflow that may include, for example, one or more actions of a method such as the method 650 of Fig. 6 (e.g., or one or more other methods, etc.).

[0096] The method 750 may be associated with various computer-readable media (CRM) blocks or modules 755 and 759. Such blocks or modules may include instructions suitable for execution by one or more processors (or processor cores) to instruct a computing device or system to perform one or more actions. As an example, a single medium may be configured with instructions to allow for, at least in part, performance of various actions of the method 750. As an example, a computer-readable medium (CRM) may be a computer-readable storage medium (e.g., a non-transitory medium, one that is not a carrier wave). As an example, CRM blocks or modules 731 and/or 771 may be included (e.g. , as media or a medium).

[0097] In Fig. 7, the example method 750 may be a workflow, part of a workflow, etc. As an example, the method 750 may be performed on a computing system (see, e.g. , the system 250 of Fig. 2). As an example, the method 750, or a portion thereof, may be applied to deghosting an input dataset that corresponds to a multi-dimensional region of interest (e.g. , of a geologic environment). As an example, the method 750 may be used in conjunction with one or more other techniques for processing collected data, modelling, etc.

[0098] As an example, equipment such as, for example, acquisition equipment (see, e.g., Fig. 2), may include circuitry that can implement at least a portion of a method. For example, a sensor unit may include circuitry that can process signals acquired (e.g., measured) via one or more sensors, optionally during a survey. In such an example, consider an approach that can apply deghosting to data, for example, prior to transmission (e.g. , transmission to a vessel, to a satellite, etc.).

[0099] In the example of Fig. 7, the method 750 may include calculating p- norm values for a plurality of candidate results for corresponding candidate ghost delay parameter values (e.g., optionally as part of the determination block 758). As an example, for a p-norm, a value of "p" may be, for example, greater than or equal zero and less than infinity (e.g., 0 < p <∞).

[00100] As an example, an L^p space may be defined to be a function space where, for example, such a space is a natural generalization of a p-norm for a finite- dimensional vector space (e.g., consider Lebesgue spaces).

[00101 ] As an example, by letting p > 1 be a real number, consider the following relationship:

[00102] In such an example, for p = 1 , a so-called taxicab norm may result and, for p = 2, a Euclidean norm may result. In such an example, as p approaches infinity (∞), the p-norm approaches the infinity norm or maximum norm. As an example, the p-norm can be related to the Holder mean (e.g. , power mean).

[00103] As an example, a method may implement a p-norm where p = 1 . As an example, a p-norm may be implemented with p = 2 (e.g., a Euclidean norm).

[00104] As an example, a method can include determining a ghost delay parameter value that minimizes a norm where the ghost delay parameter value, when utilized in an operator (e.g., a deghosting operator) applied to data, minimizes a ringing artefact in a result. For example, a ringing artefact may stem from a ghost delay parameter value of an applied deghosting operator where the ghost delay parameter value does not, with sufficient accuracy, represent a ghost delay between a signal and a ghost signal. As an example, some amount of ringing artefact may be tolerable, for example, within a tolerance limit.

[00105] As an example, a method can include determining a ghost delay parameter value for a trace at least in part by applying a constraint using determined delays extrapolated from adjacent traces in different domains. As an example, a method may implement one or more constraints. As an example, a constraint may pertain to a bandwidth, a shape, a delay, etc. As an example, a method can include determining a ghost delay parameter value at least in part by applying a constraint to a resultant wavelet shape of one or more adjacent traces in one or more different domains.

[00106] As an example, a method may include generating, as a result, substantially deghosted data representative of a signal (e.g., with minimal influence from a ghost signal).

[00107] As an example, a method can include filtering data, for example, filtering prior to determining a ghost delay parameter value.

[00108] Fig. 8 shows an example of a process 800 with respect to data, including a plot of input data 810, a plot of deterministic results 830 and a plot of adaptive results 850.

[00109] Specifically, in Fig. 8, the plot 810 shows values of an autocorrelation function of the data where the y axis is time in milliseconds (ms) and where on the x axis different traces may be discerned (e.g., as in a stack of seismic traces). For individual traces, at approximately 0 ms, a zero lag autocorrelation is visible and, at about 20 ms, the effect of a ghost can be discerned (see, e.g., peak extending to the left in the data 810). Superimposed on the plot (see red curve at the word stack along the x axis) is an average of the traces in the x axis direction. The plots 830 and 850 represent results from the application of two different deghosting methods. In the plot 850, an example method such as the method 750 of Fig. 7 has been applied to generate the results. In such an example, in comparison to the plots 810 and 830, there is a substantial visual absence of the ghost at 20 ms as well as a substantial visual absence of ringing (see, e.g., Fig. 9 as to ringing).

[00110] In Fig. 8, the results of the plot 830 correspond to a deterministic deghosting approach where the ghost delay used is a nominal one (calculated according to strong assumptions as explained above). The ghost delay is not sufficiently accurate as is evident via visible presence of ringing artefacts and the substantial residual ghost left in the data (see, e.g., time of about 20 ms).

[00111 ] As to the results of the plot 850, where the ghost delay is estimated from the data itself (e.g., without adopting one or more unrealistic assumptions as in the approach for the results of the plot 830), there is a substantial absence of ringing artefact and a substantial reduction of the ghost at 20 ms. [00112] Below, various example equations are presented along with various example methods, systems, etc. , some of which are shown and/or explained with respect to synthetic data; noting that data from sensors may be received and processed, for example, as illustrated in the plots 810, 830 and 850 of Fig. 8.

[00113] In marine seismic acquisitions, sensors can record the desired upgoing wavefield reflected from one or more geological formation features and reflections from the sea surface, which are known as the downgoing wavefield or the seismic ghost (e.g. , ghost signal).

[00114] Assuming that a direct arrival has been removed from the measured data (e.g., noting that a source may be shooting at a depth greater than the measuring cable depth), measured pressure data can be written as the combination of upgoing and downgoing wavefields as well as measured noise:

P_n = U + D + ri_j,. (l)

[00115] In the foregoing expression, U represents the upgoing wavefield; D represents the downgoing wavefield; and n_p is the pressure noise. The combination U + D result in a constructive and destructive interference in different frequencies along the signal spectrum. This can create nulls or notches in the recorded spectrum reducing the effective bandwidth of the recorded seismic wavefield.

[00116] In the frequency domain, D can be written as a function of U by using the wavefield extrapolation operator Ψ and the reflection coefficient ε at the water- air interface as follows:

D{f) = εΨί/( ) = e-i^2,If^TU(f). (2)

[00117] In the foregoing equation, j is the imaginary unit, / represents frequency and τ is the time delay that the upgoing wave will take to travel to the sea surface and reflect back to the recording seismic array. For a flat sea surface, the reflection coefficient can be approximated as ε « 1. Note that in the special case of vertical incidence angle, the delay τ = where z is the cable depth and c is the acoustic speed of seismic wave in water. Also, the foregoing expression may be written in the frequency wavenumber (FK) domain as: D(f, k) = £e-'^2zk*U(f, k), (3)

where k_z represents the vertical wavenumber and given by:

where 0 the incidence angle of the wavefield at the receiver, kx and k_y denote the inline and cross line wavenumbers, respectively. By substituting the expression for D given by (2) or (3) into (1 ), it is possible to obtain the following expression for the total pressure:

P_n = (l £e-'^2zk*)U + Πρ = G_pU + rip, (5) where G_P may be referred to as the pressure ghost operator and the dependency on frequency parameter / is omitted for convenience. As can be seen from this expression, the ghost notches are function of the frequency, the depth of the streamer and the incidence angle.

[00118] As an example, the ghost operator can also be written as a function of the ghost delay t_A as follows:

P_n = (l _£e-'²"f^t )u + Πρ = G_pU + rip, (6) where: ί_Δ =— cos Θ (7) [00119] As an example, for a deghosting problem, an objective can be to estimate U (e.g. , to generate the upgoing wavefield with minimal influence of ghosting). To achieve such a result, a ghost operator can be calculated.

[00120] As an example, it is possible to assume that the data are sufficiently well sampled at the surface and that the depth and the reflection coefficient are accurately known such that the ghost operator can be calculated explicitly by taking the Fourier transform in the inline and crossline direction, for example, to obtain the vertical wavenumber as in Eq. (4). After calculating the ghost operator, deterministic deghosting techniques can be applied to obtain the upgoing wavefield. One or more different deterministic deghosting processes may be applied to obtain the upgoing wavefield (e.g., consider inverse and recursive filtering).

[00121 ] As an example, a method can include inverse filtering. For example, consider a division in frequency domain by the ghost operator as follows: ϋ = f ^e (8)

[00122] However, inverse filtering can be problematic as an approach for deghosting because the denominator in Eq. (8) goes to zero at the notch frequencies (e.g., division by zero). Accordingly, for such frequencies, U can be undefined (i.e., division by zero "error"). As an example, an approach may aim to avoid division by zero, for example, by somehow modifying the denominator. For example, a crude stabilization approach can include multiplying the complex conjugated ghost operator and adding a very small number to its power in the denominator. With such an approach, the phase adjustment may be preserved by the complex conjugate in the numerator (subtraction of the phases) and the amplitudes can be stabilized by γ.

(9) where( ) is a complex conjugate operator.

[00123] Such an approach can regularize a deconvolution problem and as a result deghosting may be achieved with an inverse deghosting filter operating on a frequency-by-frequency basis. While a recursive approach may yield such results, it can tend to be computationally more expensive. Both, recursive and inverse filtering, are deterministic approaches to deghosting that apply an assumed to be known ghost model that can be parameterized by the delay and the reflection coefficient.

[00124] As mentioned, data may be subject to sampling deficiencies, for example, consider inline and crossline data where crossline data may include some uncertainty that exceeds that of inline data. As an example, k_y may be somewhat ambiguous, for example, due to one or more aliasing effects. In such an example, a deterministic deghosting algorithm may treat the ghost operator as being known by making assumptions about the direction of propagation. For example, it can be assumed that waves propagate in a vertical plane between source and receiver. Such an assumption tends to deviate from reality because, in practice, the earth can include heterogeneity in multiple dimensions (e.g. , 3D). Further, even where the wavefield is well sampled in 3D, rough seas and uncertainty in receiver depth can also invalidate deterministic ghost models based on Eqs. (5) and (6). In addition, for various approaches, the propagation velocity in water c is assumed to be known. In reality, however, this can be an invalid assumption. For example, thermoclines, which can be sharp variations in water temperature (therefore also perturbations in density and consequently velocity), can introduce an error in c. Deterministic deghosting techniques that make strong and sometimes unrealistic assumptions regarding such parameters, while used to invert the ghost effects, tend to generate suboptimal results where assumptions can introduce errors in the deghosting process and may result in visible artefacts that can affect the quality of the data as processed and used for one or more purposes.

[00125] Various methods can include making additional measurements and using these measurements to reduce ghost effects, restore the bandwidth of the signal and increase the resolution of data, for example, either by using dual streamers or by additionally making use of the particle motion related measurements when these measurements are available at a cable; however, in single sensor acquisition systems, such additional measurements at a cable may not be available, thus, for example, an adaptive approach to estimate a ghost operator may be applied. [00126] As an example, a deghosting process can be adapted to uncertainties in ghost operator parameters, for example, optionally without use of additional measurements (e.g., whether available or not). In such an example, the

parameterized ghost model in Eqs. (5) and (6) may be used and a process that includes searching for the parameters that can optimize the resultant upgoing wave.

[00127] As an example, a metric can be employed that is based at least in part on the p-norm (e.g.,, ^p) of a hypothetical resultant upgoing wave (e.g. , a candidate resultant upgoing wave). Such an approach can be sensitive to ghost delay.

Parameterizing the ghost operator in terms of ghost delay time can introduce additional nonlinearity, however, it can constrain the parameter space considerably, which may, for example, lead to a robust solution. As an example, one or more constraints can be added that can result in additional robustness.

Example Single Sensor Adaptive Deghosting

[00128] As an example, a wavefield that can yield recorded data when ghosted with a delay may be considered to be a candidate upgoing result. In other words, the application of one or more deterministic deghosting operators parametrized by delay, may result in one or more candidate upgoing wavefields. Such an approach may be performed relatively rapidly, for example, via an inverse deghosting filter operating on a frequency-by-frequency basis.

[00129] As an example, consider the following application of an inverse deghosting filter: t/(t .) = ^Gp{t*l Ρ_η, ϊ_Δ Ε [0, ί_Δ I (10) where t_Amax is the maximum limit for searching defined by maximum depth and the minimum incidence and water velocity: (11)

[00130] As an example, one or more recursive techniques may be applied to yield such results. As an example, a candidate upgoing wavefield ϋ(ΐ_Δ), corresponding to a ghost delay ^~t_&, may act to reduce a substantial amount of data misfit as it may describe the input data. In such an example, the more closely it describes the input data, the greater the amount of reduction in data misfit.

[00131 ] As an example, a method may aim to generate information that can be used to identify an optimal upgoing wavefield among candidate wavefields (e.g. , to identify the "true" upgoing wavefield).

[00132] Fig. 9 shows an example of a process 900 with respect to a plot of synthetic recorded data 910, plots of data with a delay (Delay A) that results in substantial ringing artefacts 930 and plots of data with a different delay (Delay B) that results in relatively "clean" signals for lacking ringing artefacts 950 (e.g. , as in a synthetic data example, the delay may be known with accuracy, for example, identical to that used to generate the synthetic data). Fig. 9 also shows an example of a Ricker wavelet 912, which may be considered for use in generating synthetic data. A Ricker wavelet may be a zero-phase wavelet, the second derivative of a Gaussian function or the third derivative of a normal-probability density function. As an example, a Ricker wavelet may be used as a zero-phase embedded wavelet in modeling and synthetic seismogram manufacture.

[00133] Fig. 9 illustrates an approach that includes applying an inverse filter with two delays (e.g. , accurate and inaccurate) and adding its delayed and polarity- reversed counterpart back (e.g. , ghosting the result). As shown in Fig. 9, it can be readily seen that there exists an upgoing wavefield for a t_& that can be consistent with the recorded data.

[00134] As shown in Fig. 9, if an inaccurate delay is used for deghosting, in general, periodic artefacts appear in deterministic deghosting. These artefacts are noted as ringing, which may be viewed as dispersion of energy across time in the first two example plots 930 of Fig. 9. The frequency of these periodic artefacts can correspond to notch frequencies of the inaccurate applied ghost delay time, because the high power of its inverse ghost operator at these frequencies boosts the local signal amplitudes. In the example plots 950, the absence of ringing when deghosting using an accurate ghost delay can be utilized to identify the "true" upgoing wavefield. In other words, as demonstrated in the examples of Fig. 9, a method can include assuming that the true up-going wavefield, which does not contain any ringing, would be of lower energy (e.g. , or other p norm) than cases where artefacts are present. For example, consider the following formulation: nun||fi(t₄)||_p (₁₂) where u(t_A) is the candidate upgoing wavefield in the τ p_x or t % domains.

[00135] As an example, after specifying a set of candidate upgoing wavefields obtained by deghosting data using a range of plausible delays, a method may include performing a linesearch over a range of delays that minimizes the i^v norm of the candidate upgoing wavefields, while deghosting the recorded data, may be used to estimate the upgoing wavefield. In such an example, a single ghost delay may be an inherent assumption, which may be satisfied, for example, by using appropriately sized space time windows.

[00136] Regarding the domain in which such a minimization may be carried out, consider one or more of the following:

Since events are better separated and the p_x parameterization ties more easily to the ray-based ghost model in Eq. (1 ), the p_x domain may be selected rather than the % domain. As an example, in cases where the ghost model changes substantially in space, carrying out the minimization in the % domain may be performed.

Note that minimizing the Euclidean norm (p = 2) may be considered to be a special case of Eq. (12). By Parseval's theorem, the time-domain and frequency-domain ² norm can be equal. As a result, Eq. (12) can be written as:

■ II (M P II

1_Δ +r " II

As such, in the case of the Euclidean norm, minimization may be achieved relatively expeditiously with an inverse deghosting filter operating on a frequency-by-frequency basis, and then computing the energy of the deghosted wavefield (e.g., without going to the time domain). As an example, working in time rather than frequency may be, in some cases, advantageous, as it may allow for more adaptivity.

[00137] Upon performing a linesearch, which includes the deghosting of the recorded data for individual tested delays, an adaptive deghosting algorithm can be split up in a part for parameter estimation and for deghosting the data. In such an example, the latter may include employing one or more recursive or inverse filters.

[00138] As an example, splitting adaptive deghosting can optionally provide for one or more of:

(a) Simplifying the problem, as the data misfit naturally trends to zero and rather than searching for two variables, a linesearch may be implemented to search for a single variable (e.g. , a ghost delay parameter value, etc.).

(b) Providing a relatively robust shield against noise, as the parameter estimation may be performed without an actual measurement, for example, a filtered version of it may be used (e.g. , consider filtering data prior to applying an inverse deghosting operator). As an example, p norm minimization may be carried out with respect to one or more functions of an upgoing wavefield to noise. Moreover, the deghosting applied at the level of parameter estimation can be applied to optimize signal to noise ratio. For example, by applying a Wiener filter, minimal noise leakage to the parameter estimation problem can be achieved.

(c) Deghosting results can be quality checked (QC), as the output is an operator multiplying the data.

[00139] In various following examples, the Euclidean norm case is described, as shown, it is relatively domain independent and may be applied in one or more different domains (e.g. , t x, f χ, τ p_{X l} f p_x).

Example Analysis of Euclidean Norm Minimization

[00140] As an example, by multiplying a recorded spectrum with an inverse ghost operator, signal amplitudes in a vicinity of its notches can be boosted (e.g. , toward infinity or otherwise high values). This may be considered to be an instability issue that exists with deconvolution where a frequency domain filter has quite small values and its inverse therefore poles. As an example, a method may leverage such instability, for example, where an inaccurate ghost operator creates power around its "assumed" notches, to determine the operator with an acceptably accurate delay t_A, which can result in minimum power after filtering the ghosted data.

[00141 ] As an example, consider synthetically generated data using a Ricker wavelet (see, e.g., the Ricker wavelet 912 of Fig. 9) defined by a peak frequency of about 30 Hz. In such an example, a ghost may be simulated using a flat sea-surface reflectivity of and a receiver depth of about 18 m, about 5 m and about 25 m corresponding to ghost delay values of about 24 ms, about 6.6 ms and about 33.3 ms, respectively, given about 1500 m/s as the acoustic water velocity.

[00142] As the ghost delay values span from about 6 ms to about 33 ms, in this synthetic example, ghost operators with delay values ranging from 2 ms to 40 ms may be considered and constructed as well as applied to the data with varying stabilization γ values.

[00143] Fig. 10 shows plots 1010, 1012, 1030, 1032, 1050, 1052 where the plots 1010, 1030 and 1050 show synthetic ghosted signal spectra and where the plots 1012, 1032 and 1052 include the normalized inverse of the squared 2-norm as cost (see, e.g., Eq. (13)). In Fig. 10, the plot 1012 shows that there are two peaks in the cost function. While the right peak lies at the accurate delay, the slightly higher left peak at a lower delay of 13.3 ms corresponds to a notch frequency of about 80 Hz. In Fig. 10, the plot 1032 shows the cost function as including the peak for lower delay approximately at the same location and of similar shape, however, the even lower accurate delay (i.e., at about 6.6 ms indicated by a red arrow) does not appear to impact shape of the cost function. In Fig. 10, the plot 1052 illustrates the case for the higher delay (i.e. , of about 33.3 ms) where an additional, slightly higher peak is created at around half of the accurate ghost delay value. This peak is due to the inverse ghost operator (green line) having its poles at each second notch frequency of the accurate delay (red line), thus it does not boost those signal amplitudes as much as when not located at a notch within the data. However, the increased width of the inverse filter around the notches tends to indicate that those frequency components are increased more than in the case of applying the accurate delay. [00144] From estimating the delay by linesearch minimizing the total power of the upgoing wavefield, the following may be considered:

While the total power of wavefields with higher delays is decreased upon deghosting, the power of data containing lower delays is increased. Power minimization thus holds in the former case and there is a minimum delay below which the concept of minimizing the power may not hold and additional constraints may be implemented.

Such an approach may aim to explain an estimation limit observed in the plot 1032, at which lower delays may not be properly estimated.

An overall minimum power appears to lie at about an estimation limit, and not at about a peak of an accurate delay.

Secondary peaks occurring at higher delays may pose an additional challenge (e.g., as to picking a peak corresponding to minimal power).

[00145] If the cost function in Eq. (13) is analyzed by assuming that the upgoing wavefield has a flat band-limited spectrum with an upper bandwidth limit denoted as f_{H l} consider the following equation:

II ^ I ₂ = ¾ (l + ^o² 2e₀sinc{2f_Ht_Afi)) +

i^2k-^l [ + e₀ ² sinc(2f_Hlt_A) i (sinc (2f_H(lt_A t_A>0)))] +sinc (2f_H(lt_A + t_Aj0))

[00146] In the foregoing equation, t_Afi, ε₀ denote the "true" delay and sea- surface-reflectivity while the tested delay and reflectivity are denoted by t_A, i, respectively. As the terms include sine functions, several statements can be made. First, due to the nature of the sine functions, the expression is periodic, which can somewhat confound further analysis. Secondly, it appears that peaks occur when the argument of the sine function tends to zero, yielding their highest amplitude of one. Given the foregoing, as an example, for a given delay t_{A 0} this makes the first term a constant, while the second term is dominated by its second sine function containing the difference of the tested and "true" delay. This term becomes one when the tested delay is equal to an Zth of the "true" delay. Thus, this can generate secondary peaks at larger delays.

[00147] The shape of a function may be analyzed, for example, given a delay t ₀ and varying t .

[00148] Fig. 1 1 shows plots 1 130, 1 132, 1 150 and 1 152 that include data for the power for a finite pulse with f_H =100 Hz (plots 1 130 and 1 132) and 150 Hz (plots 1 150 and 1 152) and an arbitrary "true" delay of about 20 ms (e.g. , and an

assumption that the tested is equal to the true reflection coefficient). For

comparison, cost functions were calculated with a Ricker wavelet (blue), which is normalized so its power is equal to the power of the flat pulses.

[00149] Specifically, in Fig. 1 1 , the top plots 1 130 and 1 150 show synthetic recorded spectra with a Ricker wavelet (blue) and a flat pulse (red) while the bottom plots 1 132 and 1 152 show the corresponding cost functions evaluated

experimentally and analytically for the Ricker wavelet and flat pulse, respectively. Again, the plots 1 130 and 1 132 correspond to an example with 100 Hz as the signal bandwidth and the plots 1 150 and 1 152 correspond to an example with 150 Hz as the signal bandwidth.

[00150] As an example, an approach can include, firstly, an estimation limit beyond which minimizing the upgoing wavefield may encounter one or more issues, for example, if one or more additional constraints are not considered (e.g., some examples of which are described below). As an example of a constraint, consider an estimation limit that may be determined based at least in part on the bandwidth of a signal. Secondly, for flatband signals, the "true" upgoing wavefield is in fact of minimum power as long as its signal bandwidth includes at least one notch. Thirdly, secondary peaks can occur at integer fractions of the "true" delay. With some knowledge as to location of one or more of such peaks, one or more approaches may be constructed to reduce such secondary peaks.

Example Sea Surface Reflectivity Estimation

[00151 ] As an example, a cost function can be used to jointly optimize the ghost delay and the reflection coefficient. In such an example, a cost function may be a modified version of that presented above, for example, consider:

[00152] Fig. 12 shows a series of plots 1200 that illustrate how sensitivity to the reflection coefficient can be lower than the ghost delay. As an example, a method may include reflectivity estimation.

[00153] More specifically, Fig. 12 shows impact of varying delay and surface reflectivity on parameter estimation. The outer x-axis shows varying "true" delays, the outer y-axis varying "true" reflectivities. In the plots 1200, an inner panel corresponds to a grid search computing the Euclidean norm of the upgoing wave for the combination of the outer axes values. In the plots 1200, squares show the "true" maxima and circles show maxima of the cost function. Dotted lines correspond to a delay variation, due to a receiver depth error of about 1 m and dashed lines, the corresponding delay for about a 30 degree angle of propagation.

Some Example Constraints

[00154] As an example, due to the band-limited nature of seismic data, delay estimation by power minimization may be restricted to a bandwidth equivalent delay range. As an example, a method may include deghosting events that include one or more lower valued delays that may fall outside such a range. Such a method may, for example, utilize one or more several constraints.

Example Extrapolation:

[00155] As an example, a pseudo-adaptive approach based on events where delays are known with higher certainty may be implemented. For example, those events, generally around lower p_x values, can form a statistical basis for a deterministic ghost operator that can deghost events of higher p_x .

[00156] Fig. 13 shows a series of plots 1330, 1332, 1350 and 1352 for a synthetic example that illustrates an example process. In the example of Fig. 13, one event in a shotgather 1330 based on an approximately 18 m receiver depth that is transformed into the τ p_x domain per the plot 1332.

[00157] As shown in the plot 1350, a cost function can yield delays up to an arbitrary estimation limit. Given a plane wave in 3D, its delay can be written as a function of the receiver depth and vertical slowness p_z (e.g., the angle of propagation with respect to the vertical), where the latter is a function of p_x and p_y , see Eqs. (7) and (4):

[00158] If the previous equation is squared, it can be written as a first order polynomial where the crossline component is separated from the receiver depth:

t ² = 4z²p_x ² + 4z² (i p ) (16)

[00159] Given a number of estimated delays with different p_x values, they may be located on a line that gives access to the receiver depth and p_y by linear regression, for example, as shown in the plot 1352 of Fig. 13. In such an example, extrapolation may be performed, for example, to extend estimates.

Example Wavelet Shape:

[00160] As an example, a method may include utilizing the fact that different traces with different ghost delays (in x or p_x) can have a similar upgoing wavelet. In other words, the ghost delay that results in a maximum correlation between the resultant zero-phased wavelets may be deemed to be the likely "true" delay.

[00161 ] Fig. 14 shows a series of plots 141 0, 1430 and 1450 where the plot 1410 shows an upgoing wavelet, the plot 1430 shows a total wavelet with a ghost delay of approximately 10 ms and the plot 1450 shows a total wavelet with a ghost delay of approximately 6 ms.

[00162] Fig. 14 also shows an example of a method 1470 that includes a reception block 1474 for receiving data and a determination block 1478 for determining a ghost delay (e.g. , as a ghost delay parameter value). I n such a method, the determination block 1478 can operate based on an assumption that different traces with different ghost delays (e.g. , in % or p_x) may have relatively similar upgoing wavelets such that, for example, the determination block 1478 can include determining that the ghost delay that results in maximum correlation between the resultant zero-phased wavelets can be selected to be the ghost delay parameter value for appropriate deghosting. For example, a selected value may be at an appropriate level of accuracy for deghosting, which may depend, for example, on one or more subsequent processes (e.g. , interpretation workflow, etc.). As an example, a method may include receiving data that includes a signal and a ghost signal that is characterized at least in part by a ghost delay; and determining a ghost delay parameter value based at least in part on a maximum correlation between wavelets (e.g. , wavelets that result from application of an inverse ghost operator, etc.).

[00163] The method 1470 may be associated with various computer-readable media (CRM) blocks or modules 1475 and 1479. Such blocks or modules may include instructions suitable for execution by one or more processors (or processor cores) to instruct a computing device or system to perform one or more actions. As an example, a single medium may be configured with instructions to allow for, at least in part, performance of various actions of the method 1470. As an example, a computer-readable medium (CRM) may be a computer-readable storage medium (e.g., a non-transitory medium).

[00164] In the plot 1410, a R i eke r wavelet is defined by a peak frequency of about 30 Hz while the plots 1430 and 1450 show two traces with a simulated ghost reflection using a flat sea-surface reflectivity and a receiver depth of about 7.5 m and of about 4.5 m and corresponding ghost delay values of about 10 ms and about 6.0 ms, respectively, given about 1500 m/s as the acoustic water velocity.

[00165] As an example, a method can include searching a plurality of possible ghost delays for one or more that tend to maximize a cost function such as, for example, the following cost function:

[00166] Fig. 15 shows an example plot 1500 where maxima correspond to the delays of two traces (per cost function of Eq. (17)).

[00167] As an example, where ghost delay in M different traces in τ p_x s related (e.g. , akin to Eq. (15)), a single delay may be searched. As an example, a cost function that utilizes the correlation between candidate upgoing wavefield of different traces obtained by testing different ghost delays may be used, for example, consider the following formulation:

[00168] Such an approach may be considered a "blind" way to obtain an accurate ghost delay. Such an approach may be considered as an independent cost function for delay estimation. As an example, a method such as, for example, the method 1470 may implement such a function.

[00169] As an example, a hybrid cost function from an approach (e.g. , a blind approach) and, for example, from a p-norm minimization approach (e.g. , per Eq. (12)) may be implemented. As an example, a method may include implementing a plurality of techniques, for example, consider a "blind" technique and a p-norm technique (e.g. , or other statistical metric approach that may account for reduction in ringing, etc.).

Example Multi-measurement Adaptive Deghosting

[00170] As an example, a method may be applied to multi-measurement data. As an example, consider measured vertical particle motion to be given by the following formulation:

where G^is the vertical motion ghost operator. In such an example, the ghost operators affecting pressure and vertical velocity component have different signs because the downgoing vertical velocity component reverses its direction upon reflection.

[00171 ] As an example, one or more approaches may be applied. For example, consider one or more of two approaches that may extend an adaptive algorithm of single measurement data to multi-measurement acquisition data, where such approaches may be referred to as incoherent and coherent p-norm

minimization approaches. Note that, following the estimation of the ghost model parameters, such parameters may be used to deghost data in a deterministic fashion using one or more multi-measurement deterministic deghosting techniques, some examples of which are explained below.

Example Incoherent p-norm Minimization:

[00172] In this approach, hypothetical upgoing wavefields may be estimated using pressure data and vertical particle motion data, independently, where a cost function may be formulated, for example, as follows: + ll Ct Ij,

II (20) where the first term and the second term are the hypothetical upgoing wavefield for ghost delay t_A estimated from the pressure data and the particle motion data.

[00173] As an example, a hypothetical wavefield can be estimated akin to pressure data in Eq. (10) using a nominal depth and water velocity first and then refined for these parameters afterwards. In other words, the upgoing wavefield can be estimated in the frequency domain using an inverse filter, for example, as follows:

where t_Amax is the maximum limit for the delay search. In the cost function of Eq. (21 ), terms may be normalized, for example, by second order statistics of the noise in pressure and particle motion data, for example, to minimize the impact of noise as follows:

[00174] As an example, one or more linear combinations of the p-norm of the hypothetical wavefields from each measurement may be used.

Example Coherent p-norm Minimization:

[00175] As an example, rather than estimating the upgoing wavefield from individual measurements independently, an upgoing wavefield may be estimated from multi-measurements jointly and the parameters may estimated from the jointly estimated upgoing wavefield. For example, consider the following formulation: min II u(t„) |_P „ II (23)

[00176] As an example, a hypothetical upgoing wavefield may be estimated in such a case using one or more approaches (e.g., one or more different methods, etc.). As an example, consider deghosting algorithms for multisensory acquisitions such as the PZ sum and the optimal deghosting algorithm (ODG).

PZ sum (PZSUM)

[00177] The PZSUM algorithm is a model-independent deghosting method that estimates the upgoing wavefield as the average of the noisy P_n and Z_n

measurements:

u P, ZSUM ⁼ ;<^P«⁺SK> ⁽²⁴⁾

[00178] The PZSUM algorithm uses a small subset of propagation parameters, namely the density of the medium and the acoustic speed of sound in water to compute the obliquity factor given in Eq. (23). Such an approach can be insensitive to the ghost model (e.g., cable depth and the rough sea perturbations). However, the PZSUM algorithm ignores noise statistics on pressure and particle motion measurements. Such an approach may be unfavourable particularly at the lower end of a frequency spectrum where particle velocity measurements may be noisy.

ODG

[00179] The Optimal Deghosting (ODG) algorithm may be derived from Eqs. (5) and (19), for example, as follows:

[00180] The ODG algorithm utilizes the ghost model in addition to the noise statistics in the pressure and vertical particle velocity measurements to optimize the combination weights and minimize the amount of noise on deghosted data. The ODG achieves this by formulating the deghosting problem as a weighted least squares minimization problem. An ODG solution may be formulated as follows:

"R ¹H) ¹H"R ¹y (26)

— nn — nn where is the 2^nd order statistics of the noise vector n. When the ghost model and assumed 2^nd order statistics of the noise are accurate, the ODG solution minimizes the deghosting noise. However, being a model dependent algorithm, ODG tends to be quite sensitive to the ghost model and perturbations in the estimated depth can result in a suboptimal performance.

[00181 ] Fig. 16 shows an example of a method 1600 that includes a reception block 1610 for receiving seismic data from single sensor measurements, a estimation block 1620 for estimating ghost model parameters based at least in part on a portion of the received data, and an application block 1630 for applying determinisitic deghosting based at least in part on the estimated parameters. [00182] Fig. 17 shows an example of a method 1700 that includes a reception block 1610 for receiving seismic data from multi-sensor measurements, a estimation block 1620 for estimating ghost model parameters based at least in part on a portion of the received data, and an application block 1630 for applying multi-measurement determinisitic deghosting based at least in part on the estimated parameters.

[00183] Fig. 18 shows an example of a method 1800 that includes an estimation block 1810 for estimating hypothetical upgoing wavefields for different ghost model parameters, a calculation block 1820 for calculating the p-norm of at least a portion of the hypothetical wavefields (e.g. , candidate wavefields), an application block 1830 for applying one or more types of constraints (e.g., extrapolation, wavelet shape, etc.), and a determination block 1840 for determining ghost model parameters that minimize a p-norm of at least a portion of the hypothetical wavefields (e.g., candidate wavefields) where at least one or more of the constraints are applied.

[00184] Fig. 19 shows an example of a method 1900 that includes an estimation block 1910 for estimating hypothetical upgoing wavefields for different ghost model parameters from individual measurements independently, a calculation block 1920 for calculating the p-norm of at least a portion of the hypothetical wavefields (e.g., candidate wavefields) from individual measurements, an application block 1930 for applying one or more types of constraints (e.g., extrapolation, wavelet shape, etc.), and a determination block 1940 for determining ghost model parameters that minimize a linear combination of the p-norm of at least a portion of the hypothetical wavefields (e.g. , candidate wavefields) from multi-measurements where at least one or more of the constraints are applied.

[00185] Fig. 20 shows an example of a method 2000 that includes an estimation block 2010 for estimating hypothetical upgoing wavefields for different ghost model parameters from individual measurements (e.g., multi-measurements) jointly, a calculation block 2020 for calculating the p-norm of at least a portion of the hypothetical wavefields (e.g., candidate wavefields) from the multi-measurements, an application block 2030 for applying one or more types of constraints (e.g., extrapolation, wavelet shape, etc.), and a determination block 2040 for determining ghost model parameters that minimize the p-norm of at least a portion of the hypothetical wavefields (e.g., candidate wavefields) from multi-measurements where at least one or more of the constraints are applied.

[00186] As an example, one or more of the methods of Figs. 16 to 20 may be associated with various computer-readable media (CRM) blocks or modules. Such blocks or modules may include instructions suitable for execution by one or more processors (or processor cores) to instruct a computing device or system to perform one or more actions. As an example, a single medium may be configured with instructions to allow for, at least in part, performance of various actions of one or more of the methods. As an example, a computer-readable medium (CRM) may be a computer-readable storage medium (e.g. , a non-transitory medium).

[00187] As an example, data may be filtered data. As an example, data may be time domain data. As an example, data may be frequency-wavenumber domain data. As an example, a ghost delay may be a travel time that depends at least in part on a distance between a sensor and an air and water interface.

[00188] As an example, a method may include generating a deghosting operator and applying the deghosting operator. As an example, applying can include multiplication of the deghosting operator and at least a portion of data, for example, to deghost the portion of the data. As an example, a deghosting operator can be a function of ghost delay. As an example, a method can include receiving a parameterized ghost model and searching for parameter values that optimize a resulting upgoing wave, for example, wherein the parameterized ghost model is parameterized with respect to a ghost delay parameter. As an example,

parameterization may act to constrain a parameter space.

[00189] As an example, a method may include a function that can be minimized to determine a ghost delay in a relatively "blind" manner. For example, such an approach may examine wavelets and, for example, correlations between wavelets (e.g., to determine a ghost delay). Such a method may be adaptive in that it can be adaptive to data. As an example, a method may aim to find wavelets that appear substantially the same (e.g., as to one or more characteristics). As an example, one or more methods may be applied to data. For example, consider a hybrid method that includes applying a wavelet-based approach (e.g., wavelet correlation) and that includes applying a p-norm-based approach (e.g. , optionally minimization of energy, etc.). [00190] As an example, a method can include receiving data that include a signal and a ghost signal that is characterized at least in part by a ghost delay; and, for a deghosting operator parameterized with respect to a ghost delay parameter, determining a ghost delay parameter value that minimizes a p-norm of a result generated by applying the deghosting operator to at least a portion of the data. In such an example, the data can include representations of constructive interference and destructive interference from the signal being an upgoing wavefield signal and the ghost signal being a downgoing ghost wavefield signal reflected from a sea surface.

[00191 ] As an example, a deghosting operator may be, or include, an inverse deghosting filter.

[00192] As an example, a method can include calculating p-norm values for a plurality of candidate results for corresponding candidate ghost delay parameter values. Such a method may include selecting one of the candidate ghost delay parameter values as an accurate representation of a real ghost delay.

[00193] As an example, a p value of a p-norm may be greater than or equal to zero and less than infinity. As an example, consider a p value of a p-norm to be unity or, for example, two (e.g. , consider a Euclidean norm).

[00194] As an example, a ghost delay parameter value may be selected at least in part on a basis that it minimizes a p-norm and/or, for example, that it minimizes a ringing artefact in a result.

[00195] As an example, a ghost delay parameter value for a trace may be determined at least in part by applying a constraint using determined delays extrapolated from adjacent traces in different domains.

[00196] As an example, a ghost delay parameter value may be determined at least in part by applying a constraint to resultant wavelet shape of adjacent traces in different domains.

[00197] As an example, a result can be or include substantially deghosted data representative of a signal (e.g. , an upgoing signal).

[00198] As an example, a method can include filtering data prior to determining a ghost delay parameter value. In such an example, one or more of a high-pass, low-pass and/or a band-pass filter may be applied. As an example, a filter may alter shape of one or more signals in data (e.g. , smoothing, etc.). [00199] As an example, received data can include pressure values, particle velocity values and/or pressure values and include particle velocity values. As an example, single sensor measurement data can be pressure sensor measurement data. For example, consider pressure sensor measurement data that includes upgoing and downgoing signals (e.g. , desired signals/signal information and ghost signals/ghost signal information). As an example, multi-measurement data can include pressure measurement data and particle motion measurement data.

[00200] As an example, a system can include a processor; memory accessible by the processor; processor-executable instructions stored in the memory to instruct the system to: receive data that include a signal and a ghost signal that is characterized at least in part by a ghost delay; and, for a deghosting operator parameterized with respect to a ghost delay parameter, determine a ghost delay parameter value that minimizes a p-norm of a result generated by application of the deghosting operator to at least a portion of the data. In such an example, the deghosting operator can include or be an inverse deghosting filter.

[00201 ] As an example, a system may include processor-executable instructions to instruct the system to render an image to a display based at least in part on the ghost delay parameter value. As an example, the system may include the display or displays.

[00202] As an example, one or more computer-readable storage media can include computer-executable instructions to instruct a system to: receive data that include a signal and a ghost signal that is characterized at least in part by a ghost delay; and, for a deghosting operator parameterized with respect to a ghost delay parameter, determine a ghost delay parameter value that minimizes a p-norm of a result generated by application of the deghosting operator to at least a portion of the data. In such an example, the deghosting operator can be or include an inverse deghosting filter (e.g. , or inverse deghosting filters).

[00203] As an example, a computing system may include circuitry that can render information for display via a display, a projector, etc. For example, a computing system may include one or more graphics processors (e.g., GPUs, etc.). A computing system may include a wired and/or a wireless interface for transmission of information to a device such as, for example, a display, a projector, etc. [00204] As an example, one or more functional modules may be implemented with one or more information processing apparatus such as general purpose processors or application specific chips, such as ASICs, FPGAs, PLDs, or other appropriate devices.

[00205] While certain implementations have been disclosed in the context of seismic data collection and processing, one or more of the methods, techniques, and computing systems disclosed herein may optionally be applied in another field and context, for example, where data involving structures arrayed in a multi-dimensional space and/or subsurface region of interest may be collected and processed, e.g., medical imaging techniques such as tomography, ultrasound, MRI and the like for human tissue; radar, sonar, and LIDAR imaging techniques; mining area surveying and monitoring, oceanographic surveying and monitoring, and other appropriate multi-dimensional imaging problems.

[00206] In some embodiments, the multi-dimensional region of interest is selected from the group consisting of a subterranean region, human tissue, plant tissue, animal tissue, solid volumes, substantially solid volumes, volumes of liquid, volumes of gas, volumes of plasma and volumes of space near and/or outside the atmosphere of a planet, asteroid, comet, moon or other body.

[00207] In some embodiments, the multi-dimensional region of interest includes one or more volume types selected from the group consisting of a subterranean region, human tissue, plant tissue, animal tissue, solid volumes, substantially solid volumes, volumes of liquid, volumes of air, volumes of plasma, and volumes of space near and/or or outside the atmosphere of a planet, asteroid, comet, moon, or other body.

[00208] As an example, a system may include one or more modules (e.g. , processor-executable instructions, etc.), which may be provided to analyze data, control a process, perform a task, perform a workstep, perform a workflow, etc.

[00209] Fig. 21 shows components of an example of a computing system 2100 and an example of a networked system 21 10. The system 2100 includes one or more processors 2102, memory and/or storage components 2104, one or more input and/or output devices 2106 and a bus 2108. In an example embodiment, instructions may be stored in one or more computer-readable media (e.g. , memory/storage components 2104). Such instructions may be read by one or more processors (e.g. , the processor(s) 2102) via a communication bus (e.g., the bus 2108), which may be wired or wireless. The one or more processors may execute such instructions to implement (wholly or in part) one or more attributes (e.g., as part of a method). A user may view output from and interact with a process via an I/O device (e.g. , the device 2106). In an example embodiment, a computer-readable medium may be a storage component such as a physical memory storage device, for example, a chip, a chip on a package, a memory card, etc. (e.g., a computer- readable storage medium).

[00210] In an example embodiment, components may be distributed, such as in the network system 21 10. The network system 21 10 includes components 2122-1 , 2122-2, 2122-3, . . . 2122-N. For example, the components 2122-1 may include the processor(s) 2102 while the component(s) 2122-3 may include memory accessible by the processor(s) 2102. Further, the component(s) 2102-2 may include an I/O device for display and optionally interaction with a method. The network may be or include the Internet, an intranet, a cellular network, a satellite network, etc.

[00211 ] As an example, a device may be a mobile device that includes one or more network interfaces for communication of information. For example, a mobile device may include a wireless network interface (e.g. , operable via IEEE 802.1 1 , ETSI GSM, BLUETOOTH®, satellite, etc.). As an example, a mobile device may include components such as a main processor, memory, a display, display graphics circuitry (e.g. , optionally including touch and gesture circuitry), a SIM slot, audio/video circuitry, motion processing circuitry (e.g., accelerometer, gyroscope), wireless LAN circuitry, smart card circuitry, transmitter circuitry, GPS circuitry, and a battery. As an example, a mobile device may be configured as a cell phone, a tablet, etc. As an example, a method may be implemented (e.g. , wholly or in part) using a mobile device. As an example, a system may include one or more mobile devices.

[00212] As an example, a system may be a distributed environment, for example, a so-called "cloud" environment where various devices, components, etc. interact for purposes of data storage, communications, computing, etc. As an example, a device or a system may include one or more components for

communication of information via one or more of the Internet (e.g., where

communication occurs via one or more Internet protocols), a cellular network, a satellite network, etc. As an example, a method may be implemented in a distributed environment (e.g., wholly or in part as a cloud-based service).

[00213] As an example, information may be input from a display (e.g. , consider a touchscreen), output to a display or both. As an example, information may be output to a projector, a laser device, a printer, etc. such that the information may be viewed. As an example, information may be output stereographically or

holographically. As to a printer, consider a 2D or a 3D printer. As an example, a 3D printer may include one or more substances that can be output to construct a 3D object. For example, data may be provided to a 3D printer to construct a 3D representation of a subterranean formation. As an example, layers may be constructed in 3D (e.g. , horizons, etc.), geobodies constructed in 3D, etc. As an example, holes, fractures, etc., may be constructed in 3D (e.g. , as positive structures, as negative structures, etc.).

[00214] Although only a few example embodiments have been described in detail above, those skilled in the art will readily appreciate that many modifications are possible in the example embodiments. Accordingly, all such modifications are intended to be included within the scope of this disclosure as defined in the following claims. In the claims, means-plus-function clauses are intended to cover the structures described herein as performing the recited function and not only structural equivalents, but also equivalent structures. Thus, although a nail and a screw may not be structural equivalents in that a nail employs a cylindrical surface to secure wooden parts together, whereas a screw employs a helical surface, in the

environment of fastening wooden parts, a nail and a screw may be equivalent structures. It is the express intention of the applicant not to invoke 35 U.S.C. § 1 12, paragraph 6 for any limitations of any of the claims herein, except for those in which the claim expressly uses the words "means for" together with an associated function.

Bibliography

[00215] The following documents are incorporated by reference herein:

1 . Amundsen, L. (1993). "Wavenumber-based filtering of marine point-source data."

Geophysics, 58(9), 1335-1348. 2. Beasley, C.J., R. Coates, and C. Lapilli (2013). " Wave Equation Receiver

Deghosting". 75th EAGE Conference and Exhibition incorporating SPE EUROPEC 2013.

3. Ferber, R. and C.J. Beasley (2014). " Simulating Ultra-deep-tow Marine Seismic Data for Receiver Deghosting". 76th EAGE Conference and Exhibition 2014.

4. Ozbek, A. , Vassallo, M., Ozdemir, K. , van-Manen D.J. and Eggenberger, K. [2010] Crossline wavefield reconstruction from multicomponent streamer data: Part 2 - Joint interpolation and 3-D up/down separation by generalized matching pursuit. Geophysics, 57, WB69-WB85

5. Ozdemir, A.K., and A. Ozbek, 2007, Method for optimal wave field separation: US Patent 7,676,327.

6. Ozdemir, A.K., P. Caprioli, A. Ozbek, E. Kragh, et al. (2008). Optimized deghosting of over/under towed-streamer data in the presence of noise". The Leading Edge 27 (2)

7. Posthumus, B. J., 1993. Deghosting using a twin streamer configuration.

Geophysical Prospecting, 41 , 267-286.

8. Robertsson, J.O.A., J.E. Kragh, and J.E. Martin (2001 ). "Method and system for reducing effects of sea surface ghost contamination in seismic data". Schlumberger Technology Corporation. US Patent 6,775,618.

Claims

CLAIMS What is claimed is:

1 . A method comprising:

receiving data that comprise a signal and a ghost signal that is characterized at least in part by a ghost delay; and

for a deghosting operator parameterized with respect to a ghost delay parameter, determining a ghost delay parameter value that minimizes a p-norm of a result generated by applying the deghosting operator to at least a portion of the data.

2. The method of claim 1 wherein the data comprises representations of constructive interference and destructive interference from the signal being an upgoing wavefield signal and the ghost signal being a downgoing ghost wavefield signal reflected from a sea surface.

3. The method of claim 1 wherein the deghosting operator comprises an inverse deghosting filter.

4. The method of claim 1 wherein determining comprises calculating p-norm values for a plurality of candidate results for corresponding candidate ghost delay parameter values.

5. The method of claim 1 wherein a p value of the p-norm is greater than or equal to zero and less than infinity.

6. The method of claim 1 wherein a p value of the p-norm is unity.

7. The method of claim 1 wherein a p value of the p-norm is two.

8. The method of claim 1 wherein the ghost delay parameter value that minimizes the p-norm minimizes a ringing artefact in the result.

9. The method of claim 1 wherein the ghost delay parameter value for a trace is determined at least in part by applying a constraint using determined delays extrapolated from adjacent traces in different domains.

10. The method of claim 1 wherein the ghost delay parameter value is determined at least in part by applying a constraint to resultant wavelet shape of adjacent traces in different domains.

1 1 . The method of claim 1 wherein the result comprises substantially deghosted data representative of the signal.

12. The method of claim 1 comprising filtering the data prior to the determining a ghost delay parameter value.

13. The method of claim 1 wherein the received data comprise pressure values.

14. The method of claim 1 wherein the received data comprise particle velocity values.

15. The method of claim 1 wherein the received data comprise pressure values and comprise particle velocity values.

16. A system comprising:

a processor;

memory accessible by the processor;

processor-executable instructions stored in the memory to instruct the system to:

receive data that comprise a signal and a ghost signal that is characterized at least in part by a ghost delay; and

for a deghosting operator parameterized with respect to a ghost delay parameter, determine a ghost delay parameter value that minimizes a p-norm of a result generated by application of the deghosting operator to at least a portion of the data.

17. The system of claim 16 wherein the deghosting operator comprises an inverse deghosting filter.

18. The system of claim 16 wherein the processor-executable instructions to instruct the system comprise processor-executable instructions to render an image to a display based at least in part on the ghost delay parameter value.

19. One or more computer-readable storage media comprising computer- executable instructions to instruct a system to:

20. The one or more computer-readable storage media of claim 19 wherein the deghosting operator comprises an inverse deghosting filter.