Patent Publication Number: US-10775522-B2

Title: Systems and methods for attenuating noise in seismic data and reconstructing wavefields based on the seismic data

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims priority to and the benefit of U.S. Provisional Application No. 62/350,310, entitled “A METHOD OF ACQUIRING SEISMIC DATA,” filed Jun. 15, 2016, which is hereby incorporated by reference in its entirety for all purposes. This application also claims priority to and the benefit of U.S. Provisional Application No. 62/350,371, entitled “A METHOD OF ACQUIRING AND PROCESSING SEISMIC DATA,” filed Jun. 15, 2016, which is hereby incorporated by reference in its entirety for all purposes. In addition, this application claims priority to and the benefit of U.S. Provisional Application No. 62/350,349, entitled “A METHOD OF RECONSTRUCTING A DATASET,” filed Jun. 15, 2016, which is also hereby incorporated by reference in its entirety for all purposes. 
     This application is related to International Publication No. WO 2016/179060, International Publication No. WO 2016/179206, and International Publication No. WO 2015/168130, each of which is hereby incorporated by reference in its entirety for all purposes. This application is also related to U.S. Provisional Patent Application No. 62/261,934, which is hereby incorporated by reference in its entirety for all purposes. 
    
    
     BACKGROUND 
     The present disclosure relates generally to systems and methods for processing seismic data acquired by one or more seismic receivers to identify locations of hydrocarbon formations or deposits within subterranean regions of the earth. Seismic exploration involves surveying subterranean geological formations for hydrocarbon deposits. A survey typically involves deploying seismic source(s) and seismic receivers at predetermined locations at or near the surface of the Earth. The sources generate seismic waves, which propagate into the subterranean geological formations creating pressure changes and vibrations along the way. Changes in elastic properties of a geological formation scatter the seismic waves, changing the direction of propagation and other properties of the seismic waves. Part of the energy emitted by the sources is reflected back from the geological formations toward the surface and reaches the seismic receivers. Some seismic receivers are sensitive to pressure changes (e.g. hydrophones), others to particle motion (e.g. geophones), and surveys may deploy only one type of receiver or both. 
     In response to the detected seismic events, the receivers generate electrical signals to produce seismic data. Analysis of the seismic data can be processed to indicate the presence or absence of probable locations of hydrocarbon deposits. Additionally, seismic sources and receivers may be used to monitor hydrocarbon production from a subterranean reservoir and/or other fluid flow within the reservoir. 
     Some surveys are known as “marine” surveys because they are conducted in marine environments, which may include saltwater environments, fresh water environments, and brackish water environments. In one type of marine survey, called a “towed-array” survey, an array of seismic receiver-containing streamers is towed behind a survey vessel which also tows one or more seismic sources. A possible alternative, or addition, to the use of towed streamers is the use of ocean bottom cables or ocean bottom nodes which contain seismic receivers. Unlike streamers, these lay on the sea bed and do not move during recording of seismic data. In such a survey the seismic sources may be towed by a vessel. There are also survey procedures in which the seismic sources are stationary (e.g. attached to a moored buoy). 
     Regardless of whether the receivers and seismic source(s) are moving or stationary, the received data can incorporate effects resulting from the methodology used to generate the seismic waves which penetrate into the subterranean and/or undersea geological formation. These source-side acquisition effects include signatures of the seismic sources, radiation patterns, residual shot noise, data irregularity, sparse data sampling, effects from the use of more than one seismic source, effects from motion of the seismic source (e.g., if it is moving while data is being acquired), the effect of the water surface above the seismic source(s), and the like. As such, the seismic data collected at the receivers (i.e. the seismic measurements made by the receivers) may be processed to remove or reduce artifacts which do not correspond to features of the geological formations that are being surveyed. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present disclosure is best understood from the following detailed description when read with reference to the accompanying drawings. Like characters represent like parts throughout the drawings. It is emphasized that, in accordance with standard practice in the industry, various features are not drawn to scale. In fact, the dimensions of various features may be arbitrarily increased or reduced for clarity of discussion. 
         FIG. 1  illustrates a schematic view of a seismic marine survey arrangement that includes a seismic vibrator array and ocean bottom receivers, in accordance with one or more embodiments of the present disclosure; 
         FIG. 2  illustrates a block diagram of a computer system that may perform one or more of the seismic data processing on seismic data acquired via the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 3  illustrates a schematic diagram of seismic survey geometry having one seismic source and multiple seismic receivers, in accordance with one or more embodiments of the present disclosure; 
         FIG. 4  illustrates example seismic wavefields of a common source gather that may be acquired via a seismic survey geometry of  FIG. 3 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 5  illustrates a diagram of a cost function that may be associated with the common source gather of  FIG. 4 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 6  illustrates a schematic example of seismic wavefields of a common-source gather that may be acquired via a seismic survey geometry of  FIG. 3  without noise components, in accordance with one or more embodiments of the present disclosure; 
         FIG. 7  illustrates a schematic diagram of seismic survey geometry having multiple seismic sources and one seismic receiver, in accordance with one or more embodiments of the present disclosure; 
         FIG. 8  illustrates example seismic wavefields of a common-receiver gather that may be acquired via a seismic survey geometry of  FIG. 7 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 9  illustrates a diagram of a cost function that may be associated with the common receiver gather of  FIG. 8 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 10  illustrates a flow chart of a method for simultaneously reconstructing source side data and removing noise from seismic data acquired via the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 11  illustrates a schematic survey geometry that may correspond to the seismic data processed according to the method of  FIG. 10 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 12  illustrates a time slice of seismic wavefields acquired via the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 13  illustrates a diagram of a unified source-receiver cost function that may be associated with the time slice of seismic wavefields of  FIG. 12 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 14  illustrates a schematic survey geometry that may correspond to interpolated seismic data determined according to the method of  FIG. 10 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 15  illustrates a time slice of seismic wavefields acquired via the seismic marine survey arrangement of  FIG. 1  and processed to remove noise according to the method of  FIG. 10 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 16  illustrates a flow chart of a method for simultaneously reconstructing source side data and removing noise from seismic data acquired via the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 17  illustrates a flow chart of a method for separating components from seismic data related to pressure and particle velocity measurements acquired via the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 18  illustrates a block diagram of a data flow chart for separating up-going and down-going seismic wavefields from seismic data related to pressure and particle velocity measurements acquired via the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 19  illustrates another flow chart of a method for separating components from seismic data related to pressure and particle velocity measurements acquired via the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 20  illustrates a block diagram of an embodiment of a seismic receiver that may be used in the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 21  illustrates a block diagram of a second embodiment of a seismic receiver that may be used in the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 22  illustrates a block diagram of a third embodiment of a seismic receiver that may be used in the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 23  illustrates a block diagram of a fourth embodiment of a seismic receiver that may be used in the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 24  illustrates a block diagram of a fifth embodiment of a seismic receiver that may be used in the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 25  illustrates a flow chart of a method for processing seismic data acquired via (for example) the seismic receivers of any of  FIGS. 20-24 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 26  illustrates a block diagram of a sixth embodiment of a seismic receiver that may be used in the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 27  illustrates a block diagram of a seventh embodiment of a seismic receiver that may be used in the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; 
         FIG. 28  illustrates a block diagram of an eighth embodiment of a seismic receiver that may be used in the seismic marine survey arrangement of  FIG. 1 , in accordance with one or more embodiments of the present disclosure; and 
         FIG. 29  illustrates a data flow diagram for processing seismic data acquired via (for example) the seismic receivers of any of  FIGS. 26-28 , in accordance with one or more embodiments of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     One or more specific embodiments will be described below. In an effort to provide a concise description of these embodiments, not all features of an actual implementation are described in the specification. It should be appreciated that in the development of any such actual implementation, as in any engineering or design project, numerous implementation-specific decisions must be made to achieve the developers&#39; specific goals, such as compliance with system-related and business-related constraints, which may vary from one implementation to another. Moreover, it should be appreciated that such a development effort might be complex and time consuming, but would nevertheless be a routine undertaking of design, fabrication, and manufacture for those of ordinary skill having the benefit of this disclosure. 
     Generally, seismic data acquired using marine seismic survey in shallow water (e.g., less than 300 m depth) differs from seismic data acquired in deeper water (e.g., greater than 300 m depth). For example, the seismic data acquired in shallower water depths may include certain properties such as a Scholte wave and other coherent noise similar to noise (e.g., ground roll) acquired during land seismic acquisition. 
     For example, a marine survey design may account for a minimum expected velocity of propagation for seismic (e.g., acoustic) wavefields to travel across seismic source and/or receiver arrays of the marine survey design. As such, in marine survey designs using ocean bottom nodes (OBN) in deeper water, a minimum velocity of the propagation of the seismic wavefields across the source array may correspond to a speed of sound in water. Using this minimum velocity of propagation, seismic data acquisition parameters (e.g., distance between seismic receivers) for the seismic receiver may be determine based on the speed of sound in water. However, in shallower water, a Scholte wave may be slower than the speed of sound in water. This slower velocity (at least at the Scholte wave frequencies) may cause seismic data processing (e.g., source side reconstruction) over large distances to be difficult and may produce seismic wavefields with significant noise components that reduce the ability of the wavefields to accurately indicate locations of hydrocarbon formations within the earth. 
     In addition to the source side sampling/reconstruction problem (e.g., removal of effects due to positioning and interference of seismic sources) described above, marine survey designs that use ocean bottom nodes arrange seismic receivers on the seabed with spacing between each seismic receiver of approximately 300 to 400 m. This amount of distance between each seismic receiver reduces the effectiveness of seismic data processing processes that use multiple seismic datasets acquired by multiple receivers (e.g., array processing). As such, the OBN seismic data acquired by multiple seismic receivers are processed individually using data acquired by each respective seismic receiver. 
     With the foregoing in mind, to perform effective array-based seismic data processing (e.g., noise attenuation) in shallower water depths, seismic receivers along an Ocean Bottom Cable (OBC) may be separated by approximately 25 to 50 m, thereby involving an increased number of seismic receivers as compared to the OBN survey mentioned above. As such, certain embodiments of the present disclosure are directed towards performing seismic data processing on seismic data acquired by seismic receivers disposed at various distances apart while attenuating noise that may be present in the seismic data due to the shallow water depths, receiver distances, and the like. 
     Indeed, in some embodiments, a computer system may receive seismic data acquired via a marine seismic survey and may process the seismic data using certain algorithms (e.g., Extended Generalized Matching Pursuit (EGMP)) that employ gradient data (e.g., pressure gradient, spatial velocity gradient) to identify wavefield components corresponding to coherent noise in the seismic data and remove the identified wavefield components from the input seismic data without performing any interpolation/reconstruction step. 
     In addition, in some embodiments, a computer system may employ a unified seismic data processing framework for simultaneous source separation and receiver reconstruction to perform source-side reconstruction while performing coherent noise attenuation using a multi-channel method. The combination of the source-side reconstruction and multichannel (i.e., with gradients) noise attenuation provide for improved seismic data processing that enables marine seismic surveys to employ seismic receivers at larger distances apart, thereby increasing the efficiency of the marine seismic survey. Additional details with regard to performing the seismic data processing in accordance with the techniques described above are provided below with reference to  FIGS. 1-19 . 
     In certain embodiments, different sensors may be incorporated into seismic receivers to acquire seismic data that may enable the unified seismic data processing framework described above to perform various types of seismic data processing. These sensors, in one example, may include a four-component sensor that measures particle velocity in three directions along with pressure and at least one other sensor that measures another property associated with the location of the seismic receiver. That is, the at least one other sensor may measure a single particle velocity component, two particle velocity components, three particle velocity components, pressure, or any other suitable property. The acquired data from these seismic receivers may enable the seismic data processing techniques described herein to be performed more effectively. Additional details with regard to the seismic receivers used to acquire seismic data to perform various types of seismic data processing techniques described herein are provided below with reference to  FIGS. 20-28 . 
     By way of introduction,  FIG. 1  illustrates a schematic diagram of an example marine seismic survey  10  arrangement that includes a marine vessel  12  that tows a seismic vibrator array  14  through a body of water  16 . The seismic vibrator array  14  may include seismic vibrators  18  that can be activated in response to activation signals produced by a controller  20 . The controller  20  may provide the activation signals via a link  22  to the seismic vibrator array  14 . In the example of  FIG. 1 , a series  24  of seismic receivers  26  (sensors) are deployed on the water bottom  30  (e.g., seafloor or seabed). The receivers  26  may be deployed in a cable (e.g., ocean bottom cable) or nodal (e.g., ocean bottom node) form. In certain embodiments, the disposition of the seismic receivers  26  may include without limitation directly or indirectly at or in proximity to the water bottom  30  (e.g., seafloor or seabed), with or without physical contact to the water bottom  30  (e.g., seafloor or seabed), coupled to the water bottom  30  (e.g., seafloor or seabed), and the like. 
     The seismic receivers  26  may detect wavefields reflected from a subsurface structure  28  that is underneath an earth surface (e.g., under the water bottom  30 , seafloor, or seabed). The seismic receivers  26  may be capable of receiving reflected seismic wavefields having at least four properties (e.g., particle velocities, particle accelerations pressure, and/or other wavefield properties). In some embodiments, the seismic receivers  26  may be four-component (4C) seismic receivers that may receive seismic data having four properties in addition to at least one or more gradients of the four components. In addition, the seismic receivers  26  may measure characteristics of wave propagation such as the ellipticity of the wavefield at the seismic receiver  26 . The subsurface structure  28  can include one or multiple subsurface elements of interest  32 . Source wavefields propagated by the seismic sources  18  are propagated into the subsurface structure  28 . The subsurface structure  28  reflects a part of the source wavefields, such that the reflected wavefields are detected by the seismic receivers  26 . Measured data (e.g., seismic data) acquired by the seismic receivers  26  can be communicated to the controller  20  for storage or for processing. Although  FIG. 1  describes data being communicated via the controller  20 , it should be noted that the controller  20  may include any suitable computing device having one or more processors. 
     The seismic vibrators  18  in the seismic vibrator array  14  can be controlled to either be in-phase or out-of-phase to cause production of an omnidirectional source wavefield or a source gradient wavefield, respectively. The controller  20  may send activation signals to the seismic vibrator array  14  to cause the seismic vibrator array  14  to produce an omnidirectional source wavefield in a first shot (e.g., first activation of the seismic vibrator array  14 ) and to produce a source gradient wavefield in a second shot. The seismic vibrator array  14  may also be referred to as a marine vibrator source or a marine vibrator. In certain embodiments described herein, the seismic vibrator array  14  may emit source wavefields with alternating directivity patterns. Additional details regarding the seismic vibrators  18  discussed herein may be found in International Publication No. WO 2016/179060, which is incorporated herein by reference above. Although  FIG. 1  provides details with regard to a marine seismic survey, it should be noted that the processing techniques described below may be performed, in some embodiments, on land seismic surveys as well. 
     After the seismic data (e.g., reflected seismic wavefields) is acquired by the seismic receivers  26 , the seismic data may be stored via a storage component, database, or the like. The seismic data may then undergo seismic data processing to convert the raw seismic data into wavefield data that may be used to identify hydrocarbon formations or other geological features within the subsurface structure  28 . In some embodiments, the seismic data processing may also remove noise components and other artifacts that may be present in the seismic data. 
     With the foregoing in mind,  FIG. 2  illustrates a block diagram of a computer system  40  that may perform various types of seismic data processing of the present disclosure. It should be noted that the computer system  40  may also be part of the controller  20  shown in  FIG. 1 . In some embodiments, the computer system  40  may include a seismic vibrator control module  42 , which may be executable on one or multiple processors  44  to control seismic vibrators of the seismic vibrator array  14  via the controller  20 . The computer system  40  may also include a processing module  46 , which may also be executable on the processor(s)  44  to perform any of the tasks discussed below. In addition, the processing module  46  may perform a variety of seismic data processing techniques, such as crossline reconstruction, in-line reconstruction, up-down source side wavefield reconstruction, and/or multi-component imaging, and the like. It should be noted that the processing module  46  may be provided in a computer system that is separate from a computer system including the seismic vibrator control module  42 . The processor(s)  44  can be coupled to a network interface  48  (to allow the computer system  40  to communicate over a network) and a storage medium (or storage media)  50 , to store data and machine-executable instructions. 
     The storage medium (or storage media)  50  can be implemented as one or more non-transitory computer-readable or machine-readable storage media. The storage media can include different forms of memory including semiconductor memory devices such as dynamic or static random access memories (DRAMs or SRAMs), erasable and programmable read-only memories (EPROMs), electrically erasable and programmable read-only memories (EEPROMs) and flash memories; magnetic disks such as fixed, floppy and removable disks; other magnetic media including tape; optical media such as compact disks (CDs) or digital video disks (DVDs); or other types of storage devices. Note that the instructions discussed above can be provided on one computer-readable or machine-readable storage medium, or alternatively, can be provided on multiple computer-readable or machine-readable storage media distributed in a large system having possibly plural nodes. Such computer-readable or machine-readable storage medium or media is (are) considered to be part of an article (or article of manufacture). An article or article of manufacture can refer to any manufactured single component or multiple components. The storage medium or media can be located either in the machine running the machine-readable instructions, or located at a remote site from which machine-readable instructions can be downloaded over a network for execution. It should be noted that the components described above with regard to the computer system  40  are examples and the computer system  40  may include additional or fewer components as shown. 
     In some embodiments, the seismic data acquired via the seismic receivers  26  may be stored in a database  52  or other storage component. The database  52  may be accessible to the computer system  40  via the network interface  48  or some other communication protocol. In addition, after the computer system  40  performs the seismic data processing, the resulting seismic data (e.g., noise attenuated) may be stored in the database  52 . 
     Keeping the foregoing in mind, the embodiments described herein are related to combining the seismic data processing techniques related to seismic data acquired using seismic source arrays (e.g., seismic vibrator array  14 ) and seismic receiver arrays (e.g., series  24  of seismic receivers  26 ). Prior to discussing the combined approach for processing seismic data, it may be useful consider the case of processing seismic data received via a receiver array deployed on a seabed with a single seismic source in the water, as illustrated in  FIG. 3 . 
     Referring to  FIG. 3 , a seismic survey geometry  60  illustrates a single seismic source  62  (e.g., seismic vibrator  18 ) disposed within the water  16  and a number of seismic receivers  26  disposed on a water bottom (e.g., seafloor or seabed)  30 . In one embodiment, each seismic receiver  26  of  FIG. 3  may include at least one pressure sensor to measure a pressure value and at least one additional pressure sensor to measure a horizontal gradient of the pressure value. In certain embodiments, the seismic receivers  26  may be sampled at a spatial interval of approximately 100 m, which is beyond the Nyquist interval for some shallow water data. 
     After the seismic source  62  is activated and the seismic receivers  26  acquire the reflected seismic wavefields, the computer system  40  or other suitable processing device may collect the reflected seismic wavefields as a common-shot gather. An example common-shot gather  70  is illustrated in  FIG. 4 . Referring to  FIG. 4 , the common-shot gather  70  may include a collection of reflected seismic wavefields  72 . In addition to the reflected seismic wavefields  72 , the common-shot gather  70  may include noise components  74 , such as a Scholte wave and/or the like. 
     Based on the common-shot gather  70 , the computer system  40  may determine a cost function of a portion of the reflected seismic wavefields  72 . For example,  FIG. 5  illustrates a slowness spectrum (e.g., cost function) for a single frequency of a window  76  of the common-shot gather  70 . The slowness spectrum may be used to identify which parts of the common-shot gather  70  correspond to noise and which parts correspond to the desired signal (e.g., reflected seismic wavefields). For instance, in  FIG. 5 , p 1  and p 2  correspond to cut-off slowness values that may be used to distinguish between noise (e.g., Scholte wave) and the desired signal. In other words, coherent noise in the acquired seismic data can be characterized by a defined range of slowness (e.g., a slowness range p 1  to p 2 ), outside of which the reflected seismic wavefields may include coherent noise. 
     In certain embodiments, the slowness spectrum may be based on a planewave decomposition of the acquired seismic data and may corresponds to the cost function solved using a multichannel interpolation by matching pursuit (see, e.g., WO 2015/168130) or the like. 
     Applying the multichannel interpolation by matching pursuit method to the cost function of  FIG. 5  may include iteratively minimizing at each iteration by: (1) selecting the strongest (e.g., largest) component in the cost function (e.g., slowness transform); (2) computing the amplitude corresponding to this component; (3) forward modelling the desired output wavefield (e.g., an interpolated version of the input) using the amplitude and corresponding plane wave component; (4) using the computed amplitude to forward model the contribution of the component to the input; and (5) subtracting this contribution from the input data. In order to distinguish the correct slowness from the noise components, the computer system  40  may use the gradient of the wavefield acquired by the seismic receivers  26 , as described by International Publication No. WO 2015/168130. 
     With the foregoing in mind, when attempting to obtain a de-noised version of the acquired seismic data, the computer system  40  may not compute the desired interpolated output wavefield (e.g., step (3)—forward modelling the desired output wavefield), as provided in the multichannel interpolation by matching pursuit method. Instead, since the coherent noise in the seismic data can be characterized by a defined range of slowness (e.g., a slowness outside the range p 1  to p 2 ), the computer system  40  may modify the EGMP workflow detailed above, to model the noise contribution in the seismic data, rather than interpolating the desired output wavefield, using the chosen component. That is, if the chosen component lies within the range of slowness p 1  and p 2 , the computer system  40  may add the modeled contribution is added to a signal output. Alternatively, if the chosen component lies outside of the range of slowness p 1  and p 2 , the computer system  40  may add the modeled contribution to a noise output. As a result, the computer system  40  may generate a noise estimate and a signal estimate. An example signal estimate is sketched in the schematic of  FIG. 6 . As shown in  FIG. 6 , the noise components  74  previously included in the common-shot gather  70  of  FIG. 4  have been removed, thereby leaving the signal estimate (which can also be added back to the residual to preserve any remaining weak events, if desired). 
     In addition to a common-shot gather, the acquired seismic data may be organized according to a common-receiver gather. For example,  FIG. 7  illustrates an example survey geometry  90  that corresponds to a common-receiver gather, which includes a number of seismic sources  62  located in the water  16  and a single seismic receiver  26  located on the water bottom (e.g., sea floor or seabed)  30 . In one embodiment, the seismic sources  62  are marine vibrators that may emit seismic wavefields with alternating directivity patterns as described in International Publication No. WO 2015/143189 (hereinafter Halliday 2014), which is incorporated by reference above. The alternating directivity patterns may enable multi-channel interpolation on the source side. The multi-channel interpolation may also include joint processing to deal with other acquisition effects as described by Halliday 2014. By using the multi-channel interpolation, the computer system  40  may use seismic data acquired by the seismic receiver  26  with spacing beyond the spatial Nyquist interval (e.g., a spacing of 90 m). 
     Due to the issues that arise when performing seismic data acquisition in shallow water described above, a common-receiver gather  100 , as illustrated in  FIG. 8 , may be similar in appearance to the common-source gather  70  of  FIG. 4 . Thus, the cost function of the common-receiver gather  100 , as depicted in  FIG. 9 , is similar to the common-source gather  70 , as depicted in  FIG. 4 . In certain embodiments, the seismic data acquired by the common-receiver gather may be reconstructed using a similar scheme described above (where the cost function is iteratively minimized) to give a dense source output. It should be noted that the computer system  40  may be used to limit the output of the reconstruction to exclude unwanted events (e.g., ghost waveforms, replicas, noise) from the interpolated output. 
     Joint Source Wavefield Reconstruction and Noise Attenuation 
     With the foregoing in mind, the computer system  40  may process seismic data acquired via a common-source gather and a common-receiver gather using similar techniques (e.g., matching pursuit based processing) to remove coherent noise from aliased receiver data and to reconstruct aliased source data. To remove coherent noise, the computer system  40  may use seismic data having receiver gradients. To reconstruct the aliased source data, the computer system  40  may use seismic data that is acquired via marine vibrators that emit seismic wavefields with different directivities or directionalities. In certain embodiments, the computer system may process seismic data using a combination of both techniques. That is, the computer system  40  may process seismic data acquired with marine vibrators emitting wavefields with different directionalities recorded by, for example, pressure sensors and pressure gradient sensors, such that the seismic data can be simultaneously reconstructed on the source side and de-noised on the receiver side. The resulting seismic data would include seismic data from a dense grid of reconstructed shot points with coherent noise removed. As such, the computer system  40  may overcome the issues that slowly propagating coherent noise may produce on source-side reconstruction, while extending the range of applicability of marine vibrator acquisition. 
     By way of example, the computer system  40  may process the seismic data acquired via a common-shot gather and a common-receiver gather together in a higher-dimensional “source-receiver space.” By working in this higher-dimensional space (i.e., source-receiver space), the computer system  40  may compute a cost function (e.g., slowness spectrum) may be computed as a 2-dimensional or higher-dimensional cost function, rather than a 1-dimensional cost function. As a result, the cost function in higher-dimensions can be sparser, thereby making it easier to identify isolated events. In addition, the problems described above on both the source and receiver sides can be solved simultaneously. 
       FIG. 10  illustrates an example method  110  for simultaneously processing seismic data acquired by two or more seismic receivers  26  in response to two or more seismic sources  62  at two or more locations emitting seismic wavefields that change directivity patterns from shot to shot. The following description of the method  110  is described as being performed by the computer system  40 , but it should be noted that the method  110  may be performed by any suitable processing device and any number of receivers, sources, and locations, which numbers may be the same or different from each other. In addition, although the method  110  is described as being performed in a particular order, it should be understood that the method  110  may be performed in any suitable order. 
     Referring now to  FIG. 10 , at block  112 , the computer system  40  may receive seismic data (e.g., reflected seismic wavefields) acquired by two or more seismic receivers  26  in response to two or more seismic sources  62 , such as the seismic vibrator array  14 . The seismic vibrator array  14  may activate seismic sources  62  that emit wavefields at two or more locations.  FIG. 11  illustrates an example survey geometry  130  that includes multiple seismic sources  62  and multiple seismic receivers  26 . The survey geometry  130  of  FIG. 10  combines the survey geometries depicted in  FIGS. 3 and 7 . In one or more embodiments, the seismic sources  62  may be activated such that the directivity pattern changes from shot to shot. 
     For each source activation, the resulting seismic wavefield may be recorded by two or more seismic receivers  26 . Each of the seismic receiver  26  may record a pressure wavefield and a gradient of the pressure wavefield in at least one direction. As such, each seismic receiver  26  may include at least one pressure sensor and at least one pressure sensor to measure the gradient of a pressure wavefield in at least one direction. Although the following discussion may include pressure gradient sensors, it should be noted that the gradient on the seabed may also be estimated from other measurements, for example, by taking the finite-difference of two closely located pressure sensors. 
     After receiving the seismic data, at block  114 , the computer system  40  may convert the seismic data into a combined-source receiver domain.  FIG. 12  illustrates an example time slice  132  through the combined source-receiver space. As shown in  FIG. 12 , one axis corresponds to source position (x s ) and the other axis corresponds to the receiver position (x r ). In the combined source-receiver domain, the different arrivals may appear as three-dimensional waveforms or as two-dimensional waveforms in a single time slice. For example, coherent noise  134  of  FIG. 12  corresponds to the same noise component  74  illustrated in  FIG. 4 . However, by working in this higher-dimensional space, as mentioned above, the cost function may be computed as a two-dimensional or higher-dimensional cost function, rather than a one-dimensional cost function only. 
     With this in mind, at block  116 , the computer system  40  may transform the seismic data in the combined source-receiver domain into a cost function (e.g., slowness transform). By way of example,  FIG. 13  illustrates a cost function  136  that is generated by transforming the seismic data in the combined source-receiver domain into a unified source-receiver cost function according to a slowness transform. The cost function  136  includes axes that correspond to the receiver slowness (e.g., p r ) and the source slowness (e.g., p s ). 
     At block  118 , the computer system  40  may iteratively identify the strongest (e.g., having highest peak, amplitude, or the like) component(s) in the cost function. As discussed above, since the cost function in higher-dimensions is more sparse as compared to the cost function in lower dimensions, the computer system  40  may more easily identify isolated events that correspond to coherent noise more easily. That is, the isolated events that do not correspond to the reflected seismic wavefields may be more identifiable as being disparate from the reflected seismic wavefields. 
     After identifying the strongest component(s) of the cost function, at block  119 , the computer system  40  may classify each identified component as a signal component of the seismic data or as a noise component of the seismic data. The computer system  40  may classify the identified component based on whether the associated slowness of the strongest component lies within the expected slowness range for signal or not. 
     If the identified component is noise, the computer system  40  may remove the noise component from input data and add the noise component to a noise output. Conversely, if the identified component is part of the signal, the computer system  40  may remove the signal component from input data and add the signal component to a signal output. That is, at block  120 , the computer system  40  may remove the identified component from the seismic data received at block  112 . In some embodiments, the computer system  40 , at each iteration performed during the block  118 , may model the desired output (e.g., signal component or noise component) based on the strongest component. The computer system  40  may then subtract the identified noise component from the input seismic data. As a result, the source sampling of the modeled signal component corresponds to a dense grid of seismic sources  62  recorded by a single seismic receiver  26 , as illustrated in  FIG. 14 . 
     In embodiments, the computer system  40  may reconstruct the seismic data based on source points that correspond to a dense grid, as illustrated in  FIG. 14 , using a matching pursuit method or the like. The matching pursuit method may involve transforming data into a cost function (transform domain) using a chosen basis function (e.g., linear basis functions), iteratively selecting the strongest (e.g., largest) component in the transform domain, computing an amplitude corresponding to that component, computing and updating the contribution to the output for that component, and subtracting the component from the input data. When the strongest (e.g., largest) component has a slowness that is inside the region defined as containing a signal, the computer system  40  may classify the component as part of the desired signal and add the component to a desired output signal model. Conversely, when the strongest (e.g., largest) component has a slowness that is outside the region defined as containing a signal, the computer system  40  may classify the component as noise and add the component to a desired output noise model. 
     After identifying the noise components, the computer system  40  may remove the noise components from the input seismic data, and the resulting seismic data may not include the coherent noise  134  (e.g., Scholte wave) previously included in the input seismic data. For example,  FIG. 15  illustrates a time slice  138  similar to the time slice  132  of  FIG. 12  with the coherent noise  134  removed in light of the processing described with respect to the method  110 . The resulting seismic data is achieved by removing the slowly propagating coherent noise that occurs during source-side reconstruction, as discussed above. That is, in the combined source-receiver domain, the removal of coherent noise can also be applied to the seismic sources. It should be noted that although the coherent noise removed from the seismic data is described as being related to a slowness property, in other embodiments, the computer system  40  may choose any desired slowness limitation in the combined cost function. 
     In certain embodiments (e.g., see  FIG. 10 , block  119 ), when iteratively selecting the strongest (e.g., largest) component in the transform domain (e.g., slowness transform), the computer system  40  may determine whether a slowness of the identified component lies within the region specified as belonging to a desired signal, compute a contribution of the desired signal for the desired densely sampled sources and adding to the output (e.g., adding the solution for the current iteration to the solutions from the previous iteration), compute a contribution of the desired signal to the input data, and subtract the contribution of the desired signal from the input seismic data. Alternatively, if the slowness lies outside of the region specified as belonging to a desired signal, the computer system  40  may compute a contribution of the noise component to the input seismic data, subtract the noise component from the input seismic data, and generate a “noise output” signal based on the contribution of the noise component. It should be noted that input data for the above process corresponds the raw seismic data or a selected window of raw seismic data, and contribution of the desired signal refers to the “matched” portion of the input data at the current iteration. 
     By using the combined source-receiver domain, the computer system  40  may produce the same type of output as the two separate source and receiver methods described above, but in a single processing step. In this way, the embodiments described herein may prevent errors from one step from affecting the other, improving the quality of results, as the data become sparser in higher dimensions (e.g., wavefield components are more easily distinguished), and allowing slowness limitations to be applied simultaneously on both the source and receiver side. In addition, seismic receivers  26  in OBC and OBN acquisitions may include more space or distance therebetween, thereby allowing seismic surveys to be conducted using fewer sensing and computing resources. 
     Keeping the foregoing in mind,  FIG. 16  illustrates another method  150  for simultaneously reconstructing source side data and removing noise from seismic data acquired via the seismic marine survey arrangement of  FIG. 1 . The following description of the method  150  will be discussed as being performed by the computer system  40 , but it should be understood that (like the other methods of the present disclosure) the method  150  may be performed by any suitable processing device. In addition, although the method  150  is described as being performed in a particular order, it should be noted that (like the other methods of the present disclosure) the method  150  may be performed in any suitable order. 
     Referring now to  FIG. 16 , at block  152 , the computer system  40  may receive seismic data, as discussed above with reference to block  112  of  FIG. 10 . At block  154 , the computer system may select a window or portion of the received seismic data. The portion of seismic data may include two or more time samples from the reflected seismic wavefields from at least two of the activated seismic sources  62  and at least two of the seismic receivers  26 . 
     At block  156 , the computer system  40  may generate one or more basis functions suitable to describe the pressure measurements and pressure gradient measurements provided in the portion of the seismic data for each seismic source  62  that contributed to the data provided in the window of seismic data selected at block  154 . The basis functions may include or be any suitable basis function that characterizes noise components, signal components, and the like in a variety of domains (e.g., slowness). For example, the basis functions may be in the time (e.g., tau-p) domain, may correspond to parabolic or hyperbolic events, may correspond to curvelets, wavelets, or any other function that may describe and/or decompose seismic data. 
     After generating the basis functions, the computer system  40  may determine a weight for each basis function to model a portion of the seismic data that corresponds to the window of seismic data selected at block  154 . In embodiments, the weight for each basis function may correspond to the amplitude of the identified component (e.g., peak/strongest). 
     At block  160 , the computer system  40  may identify criteria to classify the generated basis functions as signal components or noise components. The criteria may include a determination as to whether the identified slowness, as described above, is within an expected range of the signal. The expected range of the signal may correspond to user defined range, ranges that typically include signals, and the like. 
     After the basis functions are classified, at block  162 , the computer system  40  may model a signal that corresponds to the desired seismic data output as a sum of the weighted basis functions classified as signal components. In the same manner, at block  164 , the computer system  40  may model the noise component as a sum of weighted basis functions classified as noise components. 
     After modelling the signal component and the noise component of the received seismic data, at block  166 , the computer system  40  may interpolate the signal components for additional pseudo seismic sources at additional pseudo source locations (e.g., source reconstruction). That is, the computer system  40  may model signal components using additional basis functions that correspond to source locations where there was not a physical source. 
     At block  168 , the computer system  40  may determine whether another window of seismic data is available. Alternatively, the computer system  40  may determine whether additional analyses on other portions of the seismic data are desired to be performed. If another window of seismic data is available for analysis, the computer system  40  may return to block  156  and perform the method  150  for the next window of seismic data. If, however, another window of seismic data is not available for analysis, the computer system  40  may proceed to block  170  and display the modelled signal component, the modelled noise component, the interpolated seismic data, and the like. 
     In addition, the data generated by the method  150  may be stored in the database  52  or another suitable storage device, such that the data may be analyzed to identify locations of hydrocarbon deposits and the like. For example, at block  172 , the computer system  40  may perform subsequent processing on the data generated by the method  150  to identify the hydrocarbon locations or other features of interest. The identified locations may then be explored and operations to extract the deposits may occur. 
     By employing the presently disclosed embodiments, the computer system  40  may more efficiently process seismic data acquired via ocean bottom node or cable systems, including for 4-dimensional or time-lapse applications. As described above, the input data for the processing operations described above may include raw acquired seismic data acquired with a marine vibrator source (e.g., where the marine vibrator emits alternating directivity patterns) and sea-bed sensors (e.g., 4C, 4C with additional gradients of one or more of the four components). The output may include source data on a dense grid with source-related acquisition effects removed, and the receiver data on the same grid as the input data was acquired, but with coherent noise removed. As a result, coherent noise attenuation can be carried out within the same process as the source-side reconstruction. In this way, the applicability of the marine vibrator ocean bottom node acquisition may be extendable to shallower water without the need to perform additional processing either before or after the source reconstruction step. 
     Moreover, the embodiments described above allow for the sampling of the seabed receivers to increase in distance (e.g., from 25 m to 100 m) without compromising the ability to attenuate coherent noise. Thus, the seismic data processing involves less time to record the source data, while using fewer sensors to cover the same receiver area. 
     Wavefield Separation During Source-Side Reconstruction 
     In addition to simultaneously reconstructing seismic data on the source side and de-noising the seismic data on the receiver side as described above, the computer system  40  may also process the acquired seismic data to separate components of the seismic data. For example, up-going wavefields and down-going wavefields may be separated from the acquired seismic data, pressure wavefields and shear wavefields may be separated from the acquired seismic data, multiple free up-going wavefields may be separated from the acquired seismic data, and the like. 
     Before discussing the various embodiments for processing the seismic data to separate these components, it may be useful to note that in ocean bottom node (OBN) seismic surveys, due to the distance between each deployed node (e.g., seismic receiver  26 ), the reflected seismic data acquired at each node may be processed individually. That is, common-source gather-type processing may not be used. Rather, seismic sources  62  may be distributed on a dense grid (e.g., either in acquisition, or through interpolation/reconstruction of sparser acquisition grids), such that the data acquired by each node in the OBN survey may be processed accurately. Using some assumptions, the presence of the source grid may enable the data to be processed using 3-dimensional data processing algorithms. For example, using this dense grid of shots, the computer system  40  may analyze the acquired seismic data using wavefield separation and de-multiple processes, prior to imaging. 
     To process the seismic data acquired at each individual node, the computer system  40  may receive seismic data from seismic sensors  24  that may include 4-component (4C) sensors. The 4C sensors may measure pressure and three components of particle velocity (e.g., x-direction velocity, y-direction velocity, z-direction velocity). The properties (e.g., recording components) measured by the seismic receivers  26  may allow some local data processing steps to be applied, despite that the seismic data is acquired from a single location. For example, the weighted combination of pressure and vertical particle velocity may allow the measured pressure wavefield to be separated into the up-going and down-going wavefields. In addition, the weighted combination of all four recording components can also allow the data to be separated into pressure (P) wave and shear (S) wave arrivals. 
     With the foregoing in mind, the presently disclosed embodiments may enable the computer system  40  to process the seismic data acquired by the OBN-type seismic survey by incorporating the receiver-side processing steps during the part of the processing flow that solves the problems on the source-side. This is possible by recognizing that the assumptions involved (e.g., the sea-bed is flat) in performing up-down separation and de-multiple processes by up-down deconvolution may involve decomposing the acquired seismic wavefield into slowness or wavenumber wavefield components on the source-side. These wavenumber wavefield components correspond to the source wavefields used to perform source-side reconstruction. That is, instead of performing seismic data processing using multi-channel reconstruction, which employs multiple channels on the source-side to assist in the reconstruction of source points (or conversely on the receiver-side to assist in the reconstruction of receiver points), the presently disclosed embodiments may separate components of acquired seismic data using data acquired by at least one seismic receiver  26 . Additional details with regard to separating components of acquired seismic data while attenuating noise components due to the seismic sources  62  will be discussed below with reference to  FIGS. 17-19 . 
     By way of example,  FIG. 17  illustrates a flow chart of a method  190  for determining up-going and down-going seismic wavefields in the seismic data acquired by one or more seismic receivers  26 . Although the method  190  is described below in a particular order, it should be noted that the method  190  may be performed in any suitable order. In addition, the method  190  will be described as being performed by the computer system  40 , but any suitable computing device may perform the following method. 
     At block  192 , the computer system  40  may receive seismic data acquired by one or more seismic receivers  26  in response to seismic sources  62  emitting wavefields from two or more source locations. In some embodiments, the directivity pattern of the emitted wavefields may change from shot to shot. In ocean bottom node acquisition, multiple wavefield components may be acquired via the seismic data and the seismic receivers  26 . The wavefield components may include pressure and three-components of particle velocity. By way of example, for each source activation, the one or more seismic receivers  26  may record the reflected seismic wavefield, which may include the pressure wavefield and the vertical particle velocity wavefield. 
     In the frequency-wavenumber domain, the pressure, P(ω, k), and vertical particle velocity, V Z (ω, k), recorded in the water layer can be related by: 
                       V   z     ⁡     (     ω   ,   k     )       =         k   x     ωρ     ⁢       P   ⁡     (     ω   ,   k     )       .               (   1   )               
where k is the horizontal wavenumber vector (k=k x  for 2D acquisition, and k=[k x , k y ] for 3D acquisition), ω is the angular frequency, ρ is the water density, and k is the vertical wavenumber wavefield component. The directionality of the vertical particle velocity allows the two recording components to be combined to separate the data into up- and down-going wavefields:
 
                       P   up     ⁡     (     ω   ,   k     )       =       1   2     ⁡     [       P   ⁡     (     ω   ,   k     )       +       a   ⁡     (   ω   )       ⁢     ωρ     k   z       ⁢       V   z     ⁡     (     ω   ,   k     )           ]               (   2   )                   P   down     ⁡     (     ω   ,   k     )       =       1   2     ⁡     [       P   ⁡     (     ω   ,   k     )       -       a   ⁡     (   ω   )       ⁢     ωρ     k   z       ⁢       V   z     ⁡     (     ω   ,   k     )           ]               (   3   )               
This separation may be performed in the frequency-wavenumber or frequency-slowness domain as the scaling factor
 
             ωρ     k   z           
is dependent on wavenumber and frequency (or slowness). The factor α(ω) is a calibration factor that accounts for differences in the response of the pressure and particle velocity sensors. In some embodiments, the calibration factor may be determined from the acquired seismic data by, in one example, identifying part of the seismic data where only upgoing waves will be present (e.g., between the first arrival at the seabed and the first sea-surface reflection). In this case, the down-going wavefields should be exactly zero, so equations (2) and (3) can be solved for the calibration factor. In the embodiments described herein, however, the separation of up-going waves and down-going waves may be determined while performing the source-side reconstruction data processing operations.
 
     After receiving the seismic data, at block  194 , the computer system  40  may select a window or portion of the seismic data. As mentioned above with regard to  FIG. 16 , the window of the seismic data may include two or more time samples from the seismic data associated with at least two of the activated sources as received by one of the seismic receivers  26 . 
     After receiving the selection of the window of seismic data, the computer system may, at block  196 , generate one or more basis functions to describe the pressure and particle velocity measurements for each seismic source  62  contributing to the data present in the window of seismic data. In certain embodiments, the basis function may be defined as follows:
 
 b =exp( ik   n   ·x )  (4)
 
where x is a vector describing spatial coordinates of the seismic sources  24  within the current analysis window, and k n  is the wavenumber vector corresponding to the chosen wavefield component at iteration n.
 
     At block  198 , the computer system  40  may determine a weight for each basis function to model the window of seismic data, as described above with respect to block  158 . 
     After determining the weights for the basis functions, at block  200 , the computer system  40  may apply the weights to the basis functions associated with the pressure measurements and the particle velocity measurements. For example, if A P  and A Z  correspond to the determined weights (e.g., output amplitudes) determined for the pressure and particle velocity data, respectively, the weights may be applied to the respective basis functions at the nth iteration as:
 
 P   n (ω, k )= A   P  exp( ik   n   ,x )  (5)
 
 V   zn (ω, k )= A   Z  exp( ik   n   ,x )  (6)
 
     At block  202 , the computer system  40  may combine the weighted basis functions using the physical relationship between the pressure measurements and the particle velocity measurements. For example, the weighted pressure measurement and particle velocity measurement described above in equations (5) and (6) may be combined with equations (2) and (3) to determine the up-going and down-going wavefields corresponding to the wavefield component at the nth iteration according to: 
     
       
         
           
             
               
                 
                   
                     
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     At block  204 , the computer system  40  may determine the separated up-going and down-going seismic wavefields from the seismic data according to equations (7) and (8). At block  206 , the computer system  40  may interpolate seismic data for additional pseudo seismic sources based on the determined up-going and down-going wavefields. That is, the output dataset may be modelled using additional basis functions corresponding to source locations where there was not a physical seismic source  62 . 
     After interpolating the seismic data, at block  208 , the computer system  40  may determine whether another window of seismic data is available or desired to be processed. If another window of seismic data is available, the computer system  40  may return to block  196  and perform the method  190  again with respect to the new window of seismic data. Alternatively, if another window of seismic data is not available, the computer system  40  may proceed to block  210  and display the separated components of the seismic data and/or the interpolated data. In addition, the computer system may store the data determined via the method  190  in the database  52  or any other suitable storage component. The computer system  40  may then perform subsequent analysis of the separated components of the seismic data and/or the interpolated data to identify locations of hydrocarbon deposits or the like. 
     By performing the processing operation of the method  190 , the computer system  40  may determine the up-going and down-going wavefields as part of the source reconstruction algorithm, thereby forgoing i.e. avoiding the need for a separate processing step. That is, since the seismic data is decomposed into plane-wave (or wavenumber) wavefield components during the source-side reconstruction, the wavefield separation results are achieved with minimal additional computational cost. A block diagram of the inputs and outputs of the method  190  is illustrated in  FIG. 18  for reference. 
     In the foregoing discussion of the method  190 , the seismic data may include a combination of pressure and vertical particle velocity measurements, but it should be noted that it is also possible to use other recording components to do different types of wavefield processing/separation. That is, the presently disclosed embodiments with regard to the method  190 , as well as the other systems and methods of the present disclosure, should not be limited to just these two recording components. 
     With the foregoing in mind,  FIG. 19  illustrates another method  220  for determining up-going and down-going seismic wavefields in the seismic data acquired by one or more seismic receivers  26 . Although the method  220  is described below in a particular order, it should be noted that the method  220  may be performed in any suitable order. In addition, the method  220  will be described as being performed by the computer system  40 , but any suitable computing device may perform the following method. 
     Referring now to  FIG. 19 , the blocks  222  and  224  may correspond to the blocks  192  and  196 , respectively, of  FIG. 17 . At block  226 , the computer system  40  may determine an amplitude for the strongest (e.g., largest) wavefield component in the basis functions for the pressure measurement and the particle velocity measurement. That is, in some embodiments, for a given frequency, the computer system  40  may use a specified basis function library to identify the strongest (e.g., largest) wavefield component in the data across both the pressure and particle velocity measurements. The computer system  40  may then compute the corresponding amplitude of the strongest (e.g., largest) wavefield component for both measurements. 
     Generally, the source-side reconstruction operation may include determining a wavefield made up of a sum of weighted plane-wave basis functions, according to the matching pursuit method or the like. During this operation, the computer system  40  may identify the strongest wavenumber (or slowness) wavefield component at each frequency, and the amplitude of that wavefield component may be determined based on a description of the seismic data that may include without limitation the source ghost, source motion, source signature effects, and the like. The computer system  40  may then identify the basis function corresponding to the strongest wavenumber, scale the identified basis function by the determined amplitude, and add the scaled basis function to the output wavefield. 
     If there will be two datasets that the seismic data is matched with—the pressure dataset and the particle velocity dataset—in some embodiments, the computer system  40  may determine an amplitude of a wavefield component for each strongest wavefield component in both the pressure and the particle velocity datasets. 
     Referring back to  FIG. 19 , at block  228 , the computer system  40  may determine whether the acquired seismic data falls between the direct wave and the first sea surface multiple. If the seismic data does fall between the direct wave and the first sea surface multiple, the computer system  40  may proceed to block  232  and assume that the down-going wavefield in the seismic data is zero and compute the calibration factor α(ω) accordingly. 
     If, however, the seismic data is after the first sea surface multiple, the computer system  40  may proceed to block  230  and compute the average of the calibration factors α(ω). In certain embodiments, calibration factors can be computed for each seismic receiver  26  in turn, or computed as a sum of the contributions from neighboring seismic receivers  26 . After determining the calibration factor α(ω), the computer system  40  may proceed to block  234  and determine whether the identified wavefield component exists on the pressure measurement and the particle velocity measurement based on the calibration factor α(ω). That is, the computer system  40  may use the determined calibration factor α(ω) and the corresponding determined amplitudes to determine whether the wavefield component exists on both basis functions that describe the pressure and particle velocity measurements using equation (1) mentioned above. 
     If the computer system  40  determines that the wavefield component exists on both measurements, the computer system  40  may proceed to block  236 . At block  236 , the computer system  40  may use the determined amplitudes to compute the equivalent up-going and down-going wavefields of the seismic data. The computer system  40  may then, at block  238 , remove the identified wavefield component from the seismic data received at block  222 . 
     If, however, the computer system  40  determines that the wavefield component does not exist on both measurements, the computer system  40  may proceed to block  242 . At block  242 , the computer system  40  may remove the identified wavefield component from the seismic data received at block  222 . 
     In certain embodiments, the method  220  may be iteratively repeated for the next strongest (e.g., largest) wavefield component of the basis functions that describe the pressure and velocity measurements of the seismic data. 
     In addition to the foregoing discussions related to processing seismic data, while separating the data into up-going and down-going pressure fields, the computer system  40  may also compute a horizontal gradient of the pressure by multiplying the reconstructed basis function by the optimum wavenumber. That is, using the relationships described in equations (9) and (10) below, the computer system  40  may determine the P-wave contributions on the horizontal and vertical components (v x   P , v x   P ) within the same workflow described above. 
                       v   x   P     ⁡     (     ω   ,     p   x       )       =         a   ⁡     (   ω   )       ⁢       -   k     ωρ     ⁢     P   ⁡     (     ω   ,     p   x       )         +         2   ⁢     c   s   2     ⁢     k   2         ω   2       ⁢       v   x     ⁡     (     ω   ,     p   x       )                   (   9   )                   v   z   P     ⁡     (     ω   ,     p   x       )       =       (     1   -       2   ⁢     c   s   2     ⁢     k   2         ω   2         )     ⁢       v   z     ⁡     (     ω   ,     p   x       )                 (   10   )               
where c S  is the shear wave velocity at the seabed.
 
     The P-wave contributions result from summing the up-going and down-going wavefields on the horizontal and vertical components. To determine the calibration factor α(ω), the computer system  40  may identifying a range of times where there are only up-going wavefields in the seismic data. In some embodiments, the computer system  40  may split the source-side reconstruction operation into two parts. The first part may perform the source-side reconstruction operation in the region where there are only up-going wavefields, and the second part may include performing the source-side reconstruction operation in the region where there are both up-going and down-going wavefields. 
     In some embodiments, the computer system  40  may determine the calibration factor α(ω) for the second part from the first part. In other embodiments, for each analysis window that should have only up-going wavefields, the computer system  40  may compute and store the values for the calibration factor α(ω) at each iteration and use the average of this value across all iterations as the calibration factor α(ω) for later windows where both up- and down-going wavefields are present. As a result, the computer system may compute the calibration factor α(ω) “on the fly” during the source-side reconstruction operation. 
     In addition, prior to performing the up-down separation, in some embodiments, it may be useful to de-noise or filter certain noise-modes that may be present in the seismic data. An example of one of the noise-modes may include “shear on vertical” noise, which corresponds to leakage of shear energy onto the vertical component geophone. The computer system  40  may filter this noise node using data acquired from a horizontal component geophone as a noise reference for adaptive subtraction. In some embodiments, it may be possible to attempt to identify noise modes, such as the shear on vertical noise, that occur on one recording component (e.g., vertical particle velocity) and not on another recording component (e.g., pressure). For example, during source-side reconstruction operations, the computer system  40  may extract weights or amplitude values A P  and A Z  for a given slowness wavefield component. If the calibration factor α(ω) is known, the absolute values of A P  and A Z  should be predictable from one another, since all terms in equation (1) are known. Thus, if equation (1) does not hold (within some defined threshold), the computer system  40  may identify the identified wavefield component can be identified as noise occurring on either the pressure or particle velocity recording component. The computer system  40  may then review or analyze the two recording components independently to identify which recording component includes the noise. As such, the computer system  40  may identify receiver-related noise while performing a source-side reconstruction operation. 
     By separating the wavefields in the seismic data during source-side reconstruction, certain embodiments described herein save both used disk-space (e.g., memory, storage) and processing effort. As a result, the time related to processing seismic data and identifying the locations of hydrocarbon deposits may be reduced. In addition, the improved processing methods of the present disclosure described herein may enable certain seismic data processing operations to be performed locally (e.g., onboard the recording and/or node recovery/harvesting vessel). 
     The embodiments described herein may include identifying noise that exists on one channel that does not exist on another. For example, horizontally propagating scattered waves in the water layer may not be recorded on the vertical geophone recording component, but will be recorded on the hydrophone. Indeed, shear wave leakage on the vertical geophone may not be recorded on a hydrophone. As such, provided that the seismic receivers  26  (e.g., sensors) are correctly calibrated, the amplitude of an event on one recording component may be predicted for another, and hence it may be possible to determine if an event picked during reconstruction is an event to be reconstructed or is noise to be discarded. As such, certain embodiments of the present disclosure may avoid removing the shear on vertical noise by adaptively subtracting the horizontal recording component data from the vertical recording component data. Instead, the presently disclosed techniques may enable identifying and suppressing noise during reconstruction which may be a more effective way to this part of the data while preserving storage space and processing power. 
     Seismic Receiver Designs 
     The foregoing processing techniques have been described as being performed with seismic receivers  26  having 4C sensors that may measure pressure and three components of particle velocity (e.g., x-direction velocity, y-direction velocity, z-direction velocity). In addition to the 4C sensors, the seismic receivers  26  may include at least one additional sensor that may be used to determine a gradient measurement. Gradient measurements have been suggested for land seismic to acquire a noise reference that can be adaptively subtracted from the vertical component of the wavefield (Edme et al., 2011) and to combine the gradient with the vertical component of the wavefield to attenuate spatially aliased noise by matching pursuit (El Allouche et al., 2015). 
     In some instances, the lateral gradients may be estimated by determining a difference between the vertical component of two adjacent geophones. Alternatively, gradients may also be approximated from two horizontal geophones placed at two different depth levels (Goujon et al., 2013). 
     For the marine seismic survey  10 , the gradient of the pressure field may be measured using accelerometers. For example, pressure data and acceleration data may be combined to separate the pressure field in up-going and down-going waves using the matching pursuit approach described above. In some instances, a cluster of hydrophones and/or geophones placed in the water  16  may be used to record the pressure gradients. That is, the multicomponent sensor cluster may measure the gradients of the pressure field in three orthogonal directions by determining the difference between data acquired by two hydrophones placed at a distance from each other or by measuring the acceleration data in the water. 
     With the foregoing in mind, certain embodiments of the present disclosure may include one or more of various node designs for the seismic receiver  26  that may allow the recording of spatial gradients of the pressure along with a translational wavefield. By recording the gradients along with the translational wavefield, such embodiments enable a range of improved seismic data processing operations, such as coherent noise attenuation, wavefield reconstruction, and wavefield separation, as detailed above. 
     More specifically, aspects of the present disclosure are related to extending the seismic receivers  26  employed in an ocean-bottom node (OBN) that record four components of the reflected seismic wavefield (e.g., 3C particle velocity and pressure) at the seabed. The extension may include adding one or more hydrophones and/or geophones to measure the spatial gradients of pressure in the water  16  and the particle velocity gradients (or acceleration) at the water bottom  30  (e.g., seafloor or seabed). By adding the additional sensor(s) to measure these properties, the presently disclosed seismic receiver designs may acquire different data (e.g., particle velocity or acceleration) that may not correspond to spatial derivatives of the pressure field in the water  16  due to the seismic receiver&#39;s  26  being coupled to the water bottom  30  (e.g., seafloor or seabed). Indeed, the incorporation of the additional sensor(s) enable the computer system  40  to more effectively perform seismic data processing, such as up-down separation, P-S separation, wavefield reconstruction, and noise attenuation processing using the matching pursuit framework in a sparse receiver configuration. 
     Keeping the foregoing in mind,  FIGS. 20-24  illustrate various designs in which a seismic receiver  26  may incorporate different arrangements of sensors. Referring now to  FIG. 20 , a side view  250  of one embodiment of the seismic receiver  26  is illustrated. The seismic receiver  26  may include a 3C particle-velocity sensor  252  disposed within a housing  254  of the seismic receiver  26 . In addition, the seismic receiver  26  of  FIG. 20  may include a pressure sensor  256  disposed on each side of the seismic receiver  26 . The two pressure sensors  256  may be used to determine the pressure gradient between sides. It should be noted that each or of the sensors described as being disposed within the housing  254  may be in fluid contact with the surrounding medium (e.g., water  16 ). In some instances, some of the sensors described herein may be enclosed within the housing  254 . In certain embodiments, the node configuration described in  FIG. 20  may be coupled to the seafloor. 
     Although the seismic receiver  26  of  FIG. 20  may acquire pressure gradient data, in certain embodiments, it may be beneficial to include the additional sensors within the housing  254  of the seismic receiver  26 . With this in mind,  FIG. 21  illustrates a side view  260  of another design of the seismic receiver  26  that may be used in the marine seismic survey  10  described herein. In one embodiment, the seismic receiver  26  of  FIG. 21  may include a 4C sensor  262  that measures the 3C particle velocity and pressure. The 4C sensor  262  may be disposed on one end or corner of the seismic receiver  26 . In addition to the 4C sensor  262 , the seismic receiver  26  may include a 1C particle velocity sensor  264  disposed on an opposite end of the seismic receiver  26  with respect to the 4C sensor  262 . It should be noted that the sensors described herein are not limited to be disposed at certain ends or corners of the seismic receiver  26 . Indeed, each of the designs of the seismic receivers  26  described below include sensors that are positioned, such that the sensors of a respective seismic receiver  26  include a certain distance (e.g. less than half a meter) between another one of the sensors therein. By way of example, in  FIG. 21 , the 4C sensor  262  may be disposed a distance (e.g., less than half of a meter) away from the 1C particle velocity sensor  264 . The 1C particle velocity sensor  264  may measure the particle motion (e.g., velocity, acceleration) in one direction. This additional sensor may be a hydrophone and/or a geophone placed at a fixed distance  266  (e.g., less than half a meter) away from the 4C sensor  262 . Both sensors may thus be located on the same node of an OBN receiver. The spatial gradients may be estimated by determining a difference between the data acquired by two hydrophones or geophones. The seismic receiver designs having one spatial gradient may be deployed on a rope  268  at a sparse spacing (e.g., greater than 50 m) and may record the gradient along the receiver line. The sensors that measure particle motion as described herein may include a geophone, an accelerometer, or the like. 
     In addition to seismic receivers  26  with one spatial gradient,  FIGS. 22-24  illustrate sample seismic receiver designs with two spatial gradients. As such, additional sensors including hydrophones, geophones, and/or one to three component geophones may be enclosed within the housing  254  of the seismic receiver  26 . 
     For example,  FIG. 22  illustrates a top view  270  of a seismic receiver design that may include the 4C sensor  262  and two pressure sensors  272  disposed at two different corners of the housing  254  to provide two horizontal gradients. In the same manner,  FIG. 23  illustrates a top view  280  of a seismic receiver design that may include the 4C sensor  262  and two 1C particle velocity sensors  264  disposed at two different corners of the housing  254  to provide two horizontal gradients. In another example,  FIG. 24  illustrates a top view  290  of a seismic receiver design that may include the 4C sensor  262  and two 3C particle velocity sensors  272  disposed at two different corners of the housing  254  to provide two horizontal gradients. 
       FIGS. 22-24  thus illustrate examples of seismic receiver nodes with two spatial gradients that can be deployed as autonomous systems where the gradients are estimated in at least two orthogonal directions. It should be noted that nodes with pressure gradients may allow for the estimation of the horizontal accelerations (ax, ay) and particle velocities (v x , v y ) in the water  16  using the equation of motion provided below: 
                     a   x     ⁢          w     ⁢     =         ∂   t     ⁢     v   x       ⁢          w     ⁢       =         -   1       ρ   w       ⁢       ∂   x     ⁢   P         ,     
     ⁢       a     y   ⁢        w         =         ∂   t     ⁢     v   y       ⁢          w     ⁢       =         -   1       ρ   w       ⁢       ∂   y     ⁢   P         ,                             (   12   )               
Where P corresponds to pressure and ρ w  represents the water density.
 
     With this in mind, nodes with particle velocity gradients may measure the spatial gradient in the seabed according to:
 
∂ x   v   z (ω, p )=− jωp   x   v   z ,
 
∂ y   v   z (ω, p )=− jωp   y   v   x ,
 
∂ y   v   x (ω, p )=− jωp   y   v   x ,
 
∂ x   v   y (ω, p )=− jωp   x   v   y ,  (13)
 
where p=(p x , p y , q) is the slowness vector.
 
     By measuring spatial gradients of pressure and particle velocity on a seismic receiver node coupled to the seabed, the computer system  40  may perform a variety of types of seismic data processing for a range of applications.  FIG. 25  illustrates a general process flow diagram of a method  300  for processing seismic data acquired by one or more types of seismic receivers  26  depicted in  FIGS. 20-24 . 
     Generally, the method  300  may include using the computer system  40  or other suitable device to receive seismic data acquired via one or more seismic receivers  26 , such that each seismic receiver  26  may include a 4C sensor and at least one other sensor at a horizontal gradient, as described above with reference to  FIGS. 20-24 . 
     At block  304 , the computer system  40  may process the acquired seismic data with the data acquired by each respective sensor disposed within or around the respective seismic receiver  26 . The seismic data processing may include without limitation source-side reconstruction, receiver-side reconstruction, noise attenuation, wavefield separation, interpolation, imaging operations, and/or the like. The processed seismic data may be employed to identify hydrocarbon deposits in the earth, such that hydrocarbon exploration and extraction equipment (e.g., drill, pumps) may be installed near the location of the hydrocarbon deposits to produce the hydrocarbons, or other subterranean feature(s) of interest. 
     After processing the seismic data, the computer system  40  may display the processed data or store the processed seismic data in the database  52 , a storage component, or the like. By way of example, a few of the seismic data processing operations that may be performed at block  304  are discussed below with respect to a type of seismic receiver design may be best suited to perform the respective operation. 
     Sparse Acquisition for Increased Efficiency 
     In shallow water surveys, OBNs are densely deployed in one direction. In this configuration, the nodes are attached by a rope (or deployed as part of a cabled system) and placed every 25 to 50 m. In certain embodiments, the seismic receivers of  FIG. 20, 22 , or  23  may be used to acquire spatial gradients. The data acquired by these sensors may enable the computer system  40  to reconstruct source-side seismic data while de-noising the seismic data in accordance with the following description. 
     By acquiring spatial gradients, the seismic receivers  26  may enable the spacing between the nodes of the OBN system to be increased. Using the reconstruction by matching pursuit approach mentioned above, the computer system  40  may represent the pressure and its spatial gradient as the sum of the same basis functions (e.g., an exponential basis function) and may be combined to de-alias the acquired seismic data and reconstruct it on a denser grid according to: 
     
       
         
           
             
               
                 
                   
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     When the spacing between the nodes is increased, the number of nodes per area decreases, thereby extending the spatial coverage of the receiver spread. Consequently, the number of receiver patch deployments can be reduced, which may enable a rolling receiver spread, rather than a series of patch deployments. Furthermore, since the required source areal coverage is larger than the receiver area, the number of redundant/repeated shot points may also be reduced, thereby significantly increasing the efficiency of shallow water OBN acquisition. 
     While reconstructing the pressure field, the computer system  40  may also reconstruct the horizontal component of particle velocity/acceleration in the water  16  can also be reconstructed using equation 12, such that: 
     
       
         
           
             
               
                 
                   
                     
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     This estimated horizontal component of acceleration/particle velocity in water can be combined with the horizontal component measured in the water-bottom (i.e., using the OBN) to infer the medium properties of the seabed. 
     Up and Down Separation by Matching Pursuit for Sparse Configuration 
     In certain embodiments, the seismic receivers  26  of  FIG. 20, 22 , or  23  may be used to acquire a horizontal gradient of the pressure field that can be measured by the sensors within the respective seismic receivers  26 . The data acquired by these sensors may enable the computer system  40  to estimate separate up-going and down-going components in the seismic data using the matching pursuit method in accordance with the following description. 
     As discussed above, separating up-going and down-going wavefields from the seismic data may be part of the processing of multicomponent seabed data where the up-going pressure field P −  (reflections from the subsurface) is separated from the down-going pressure field P +  (surface-related multiples) by combining pressure and vertical component measurements in the water column: 
                       P   +     =         1   2     ⁢   P     +         ρ   w       2   ⁢     q   w         ⁢     v   z           ⁢     
     ⁢       P   -     =         1   2     ⁢   P     -         ρ   w       2   ⁢     q   w         ⁢     v   z           ⁢     
     ⁢       where   ⁢           ⁢     q   w       =         1     c   w   2       -     p   x   2                   (   16   )               
is the vertical slowness, c w  and ρ w  are the wavespeed and density of the water, respectively. If the computer system  40  receives pressure gradient data from the seismic receivers  26 , the computer system  40  may reconstruct and decompose the pressure field in a sparser configuration, such that the seismic data is separated into the up- and down-going pressure wavefields as the seismic data is being interpolated. In this case, the up and down separation applied in the shot gather domain, can be formulated as multichannel input data:
 
     
       
         
           
             
               
                 
                   
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     These input channels can be used to formulate a cost function that allows the reconstruction of the pressure field in the water. The computer system  40  may decompose the seismic data while reconstructing the seismic data by formulating a cost function for the up-going pressure field that suppresses the down-going wavefield (by attributing the corresponding sign of the vertical slowness to the desired field). Alternatively, the computer system  40  may reconstruct the pressure and vertical particle velocity data separately and retrieve the amplitude from the vertical component. Depending on the sign of the amplitude, the reconstructed basis function can be identified as up-going or down-going. In both approaches, the computer system  40  may calibrate the vertical component of particle velocity. Generally, the calibration takes into account the difference in coupling as well as the fact that the particle velocity is measured at the water bottom  30  (e.g., seafloor or seabed) and not in the water  16 . 
     Coherent Noise Attenuation by Matching Pursuit for Sparse Configuration 
     In certain embodiments, the seismic receivers  26  of  FIG. 21, 23 , or  24  may be used to acquire a horizontal gradient of the vertical particle velocity that can be measured by the sensors within the respective seismic receivers  26 . The data acquired by these sensors may enable the computer system  40  to attenuate coherent noise in the seismic data using a matching pursuit method in accordance with the following description. 
     In a shallow water environment, the wavefield generated by a seismic source interacts with the water bottom  30  (e.g., seafloor or seabed) in the near-field. As a result, Scholte waves (also known as mud-roll) and guided waves may be excited. Generally, the seismic data acquired in this type of environment (e.g., the Persian Gulf and the Red Sea) share many similarities with land data. Specifically, the low-frequency slowly-propagating Scholte waves have similar properties to those of ground-rolls present in land seismic surveys. Therefore, noise attenuation methods based on gradients developed for sparse land acquisition (where conventional array-based methods fail) are also be applicable for shallow water environments. 
     For example, when performing noise attenuation by matching pursuit (El Allouche et al., 2015), the computer system  40  may separate aliased coherent noise while reconstructing the multicomponent input data. The separation is based on a wavenumber filter, where noise is expected to be dominant in a specific frequency-wavenumber (or slowness) range, which may be outside the signal cone. As such, the noise attenuation method may keep the signal in the residual signal and subtract the more energetic noise from the residual signal. This workflow, as described by El Allouche et al. (2015), is valid for the existing multicomponent OBN and OBC systems where there are no available spatial gradients. However, for the OBN with seismic receiver designs illustrated in  FIGS. 22-24 , this approach can be extended to the pressure gradient (in a similar manner as equation 14) or equivalently to the vertical particle velocity gradient 
                     (             v   z     ⁡     (     x   ,   ω     )                   ∂   x     ⁢       v   z     ⁡     (     x   ,   ω     )                   P   ⁡     (     x   ,   ω     )             )     =       (           C   ⁡     (   ω   )                   -   j     ⁢           ⁢   ω   ⁢           ⁢     p   x     ⁢     C   ⁡     (   ω   )                     ±     ρ   w         j   ⁢           ⁢   ω   ⁢           ⁢     q   w               )     ⁢     Ae     j   ⁢           ⁢   ω   ⁢           ⁢     p   x     ⁢   x                 (   18   )               
where C(ω) denotes the coupling filter accounting for differences in coupling between the hydrophone and the geophone.
 
     By adding more channels, the computer system  40  may constrain the problem of de-aliasing. That is, because the coherent noise is expected to be present on the pressure as well as the particle velocity (noise exist in the same wavenumber range but with different phase and amplitude), the computer system  40  may attenuate the noise on all components even in the case of having one spatial gradient. This is possible because the computer system  40  can combine one multichannel cost function (e.g. pressure and pressure gradient) aimed at defining the optimum wavenumber with a single channel cost function aimed at determining the phase and amplitude for each component individually (vertical and horizontal particle velocity). The computer system  40  may identify the noise on the multichannel cost function and may subsequently subtract the noise from each component separately using the single channel cost function. 
     Coherent Noise Attenuation by Adaptive Subtraction 
     In other embodiments, the seismic receivers  26  of  FIGS. 20-24  may be used to acquire a horizontal gradient of the vertical particle velocity or the pressure that can be measured by the sensors within the respective seismic receivers  26 . The data acquired by these sensors may enable the computer system  40  to estimate seabed properties and attenuate noise in the seismic data using adaptive subtraction in accordance with the following description. 
     Using the acquired seismic data, the computer system  40  may address the coherent noise (e.g., Scholte waves and guided waves) using noise attenuation by adaptive subtraction. For instance, the computer system  40  may use the estimates of the horizontal gradients of the vertical particle velocity or pressure to provide a scaled version of the same physical quantity. In some embodiments, the scaling may correspond to the horizontal wavenumber, as described in equations 13. 
     Since the coherent noise described above is slower than reflections from the subsurface, the computer system  40  may scale the horizontal gradients of the vertical particle velocity or the pressure with greater values. As such, the gradient estimate, which may be dominated mainly by coherent noise, may be used as a noise reference that can be adaptively subtracted from the pressure or particle velocity wavefields (Edme et al., 2011). In this way, the analysis is performed trace-by-trace and therefore is independent of the spatial sampling. 
     As discussed above, in addition to the Scholte wave and guided waves, another type of coherent noise that may be present on the vertical component may be known as “shear noise on the vertical.” This noise may generally be attributed to bad coupling and/or to scattering from small heterogeneities near the seismic receiver  26  (Paffenholtz et al., 2006). By acquiring the horizontal gradients of the vertical component in two directions (e.g., OBN shown in  FIG. 23 ), the computer system  40  may estimate a noise reference of the slowly propagating shear wave and may adaptively subtract the noise reference from the vertical component. 
     Seabed Properties Estimation and P- and S-Wave Separation by Matching Pursuit 
     In other embodiments of the present disclosure, the seismic receivers  26  of  FIGS. 20, 22, and 23  may be used to acquire a horizontal gradient of the pressure field that can be measured by the sensors within the respective seismic receivers  26 . The data acquired by these sensors may enable the computer system  40  to estimate seabed properties and perform P-wave and S-wave separation in accordance with the following description. 
     With the foregoing in mind, it should be noted that elastic separation of the wavefield into pressure-waves (P-waves) and shear-waves (S-waves) may not be performed while the seismic data acquired at the seabed is being processed. As such, the computer system  40  may assume that the horizontal component contains all of the S-waves and the vertical component contains all of the P-waves. A proper elastic separation, however, may involve data regarding medium parameters at the water-seabed interface. 
     Schalkwijk et al. (2008) showed that the spatial gradient of the pressure field is related to the horizontal component of the particle velocity according to:
 
∂ x   P (ω, p   x )=− j ωρ( v   x   P −2 c   s   2   p   x   2   v   x )  (19)
 
     Hence, by acquiring the spatial gradient of the pressure field, the computer system  40  may then estimate the S-wave velocity (c s ). In embodiments, the P-wave velocity can be estimated by applying a time window on an event identified on the pressure and the vertical component data as described by Schalkwijk et al. (2003). After the seabed medium parameters are estimated, the computer system  40  may separate the up-going P- and S-wave using the matching pursuit approach. For each iteration n, the computer system  40  may thus determine: 
     
       
         
           
             
               
                 
                   
                     
                       
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                                     , 
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                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     , 
                                     p 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     P n (ω,p), v x,n (ω,p) and v z,n (ω,p) are the components of the input wavefield solved for at the n th  matching pursuit iteration, and P n   up  and S n   up  denote the result of processing these components to give the equivalent up-going P-wave and S-wave, respectively. 
     Vertical Gradient Measuring Seismic Receivers 
     In addition or alternative to the seismic receiver designs described in  FIGS. 20-24 , other types of designs may be incorporated within the seismic receiver  26 . For example,  FIGS. 26-28  illustrate various designs in which the seismic receiver  26  may include a 4C sensor and at least one additional sensor disposed to measure a vertical gradient. 
     Referring first to  FIG. 26 , a side view  310  of an example seismic receiver  26  depicts the 4C sensor  262  disposed at or near the surface of the water bottom  30  (e.g., seafloor or seabed) and a 2C sensor disposed vertically underneath the 4C sensor  262  by a distance  314 . The seismic receiver  26  of  FIG. 26  may thus measure the vertical gradients, which may be used to estimate the P-wave and S-wave velocities at the seabed. 
     That is, similar to the free-surface on land, the boundary conditions at the water bottom  30  (e.g., seafloor or seabed) may include the vanishing of the tangential components of stress resulting in the following relations between the spatial gradients of the translational wavefield:
 
∂ z   v   x =−∂ x   v   z  
 
∂ z   v   y =−∂ y   v   z   (21)
 
     Consequently, the computer system  40  may approximate the horizontal spatial gradients by taking the difference between two horizontal translational components buried at different vertical levels. 
     In addition to these applications, the vertical gradients provide the vertical slowness of the P-waves and S-waves depending on the type of the wave according to:
 
∂ z   v   x (ω, p )=− jωq   p   v   x ,
 
∂ z   v   x (ω, p )=− jωq   s   v   x   (22)
 
     These slownesses are related to the P-wave and S-wave velocities at the seabed (e.g., the water bottom  30 , seafloor, or seabed)) according to: 
     
       
         
           
             
               
                 
                   
                     q 
                     p 
                   
                   = 
                   
                     
                       
                         
                           
                             1 
                             
                               c 
                               p 
                               2 
                             
                           
                           - 
                           
                             p 
                             x 
                             2 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         q 
                         p 
                       
                     
                     = 
                     
                       
                         
                           1 
                           
                             c 
                             s 
                             2 
                           
                         
                         - 
                         
                           p 
                           x 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     As such, by measuring the vertical gradients with the seismic shown in  FIG. 26 , the computer system  40  may estimate the P-wave and S-wave velocities at the seabed. 
     Referring now to  FIG. 27 , a side view  320  of another design for the seismic receiver  26  includes a pressure sensor  272  disposed at the water bottom  30  (e.g., seafloor or seabed) and the 4C sensor  262  disposed vertically underneath the pressure sensor  272 . The pressure sensor  272  and the 4C sensor  262  may correspond, for example, to two hydrophones, such that one is in the water  16  and another one buried under the water bottom  30  (e.g., seafloor or seabed). By measuring the pressure (P w ) in the water  16  and at the seabed (P w ), the computer system  40  may relate the divergence (sensitive to P-waves) at the seabed to the pressure fields in the water  16  according to: 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       w 
                     
                     = 
                     
                       
                         
                           1 
                           K 
                         
                         ⁢ 
                         
                           ∇ 
                           
                             · 
                             
                               v 
                               w 
                             
                           
                         
                       
                       = 
                       
                         
                           ρ 
                           w 
                         
                         ⁢ 
                         
                           c 
                           w 
                           2 
                         
                         ⁢ 
                         
                           ∇ 
                           
                             · 
                             
                               v 
                               w 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       P 
                       s 
                     
                     = 
                     
                       
                         
                           ρ 
                           s 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               c 
                               p 
                               2 
                             
                             - 
                             
                               
                                 4 
                                 3 
                               
                               ⁢ 
                               
                                 c 
                                 s 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ∇ 
                         
                           · 
                           
                             v 
                             s 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     As a result, the computer system  40  may separate the P-waves and S-waves from the seismic data acquired at the seabed and may constrain the medium parameters estimation. 
     Referring now to  FIG. 28 , a side view  320  of another design for the seismic receiver  26  includes a 4C sensor  262  disposed at the water bottom  30  (e.g., seafloor or seabed) and another 4C sensor  262  disposed vertically underneath the 4C sensor  262  disposed at the water bottom  30  (e.g., seafloor or seabed). This configuration is beneficial because the decoupled geophone in water  16  may detect a signal that is proportional to the gradients of the pressure field (see equation 12). As such, the computer system  40  may perform many of the operations discussed above. That is, by recording the pressure and the three components of the particle velocity/acceleration in water  16  and at the water bottom  30  (e.g., seafloor or seabed), the computer system  40  may more effectively estimate seabed properties, the up and down wavefield separation, the P-wave and S-wave separation, and the like. 
     With the foregoing in mind,  FIG. 29  illustrates a data flow diagram  350  that represents example input data and output data of the various types of systems and methods described herein. Although the data flow diagram  350  illustrates a number of inputs and a number of outputs, it should be understood that the various embodiments of the present disclosure described above may include one or more of each input and output. In addition, the flow data diagram  350  is provided as an example diagram and the embodiments described herein should not be limited to the example data types provided in  FIG. 29 . 
     Referring now to  FIG. 29 , the computer system  40  may receive pressure data  352 , horizontal pressure gradient data  354 , vertical particle velocity  356 , or any combination of these inputs. The computer system  40  may then determine a cost function of the received data at block  358 , as described above. Based on the cost function, the computer system  40  may identify (block  360 ) optimum wavenumbers from replicas present in the received data and distinguish the signal components from the noise components in the received data, as described above. 
     At each iteration, based on the wavenumber, identified signal, and identified noise, the computer system  40  may output noise in pressure data  362 , noise in pressure gradient data  364 , and noise in particle velocity data  366 . These outputs are subtracted from the pressure data  352 , the horizontal pressure gradient data  354 , and the vertical particle velocity data, respectively. 
     In addition, based on the wavenumber, identified signal, and identified noise, the computer system  40  may output up-going pressure data  368 , down-going pressure data  370 , up-going pressure gradient data  372 , down-going pressure gradient data  374 , up-going vertical particle velocity data  376 , and down-going vertical particle velocity data  378 . These outputs are subtracted from the corresponding input data. The process described in the data flow diagram  350  may be iteratively performed until the seismic data is determined to accurately represent the desired wavefields. 
     Reference throughout this specification to “one embodiment,” “an embodiment,” “embodiments,” “some embodiments,” or similar language means that a particular feature, structure, or characteristic described in connection with the embodiment(s) may be included in at least one embodiment of the present disclosure. Appearances of these phrases throughout this specification may, but do not necessarily, all refer to the same embodiment. 
     Although the present disclosure has been described with respect to specific details, it is not intended that such details should be regarded as limitations on the scope of the invention, except to the extent that they are included in the accompanying claims.