Patent Publication Number: US-2016231445-A1

Title: Systems and methods for far field signature reconstruction using data from near field, mid field, and surface field sensors

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application claims the benefit of priority under 35 U.S.C. §119 from U.S. Provisional Patent Application Ser. No. 61/979,280, filed on Apr. 14, 2014 and U.S. Provisional Patent Application Ser. No. 61/882,641, filed on Sep. 26, 2013, which are incorporated by reference in their entirety for all purposes. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates generally to seismic exploration and, more particularly, to systems and methods for far field signature reconstruction using data from near field, mid field, and surface field sensors. 
     BACKGROUND 
     In recent years, offshore drilling has become an increasingly important method of locating and retrieving oil and gas. But because drilling offshore involves high costs and high risks, marine seismic surveys are used to produce an image of subsurface geological structures. While the image may not directly show the location of oil or gas, those trained in the field can use such images to more accurately predict the location of oil and gas and thus reduce the chance of drilling a non-productive well. 
     Marine seismic surveys are usually accomplished by marine survey ships towing a signal source and seismic sensors. Each seismic sensor, or “sensor,” may be a hydrophone, which detects variations in pressure below the ocean surface. The sensors are contained within or attached to a cable that is towed behind the moving ship. The cables are often multiple kilometers in length and each has many sensors. The towing process is referred to as “streaming” the cable, and the cables themselves are referred to as “streamer cables.” In most surveys, the marine survey ship tows multiple streamer cables. The ship may tow streamer cables at the same depth or at different depths. One or multiple depth positioning devices can act to hold portions of the streamer cable below the ocean surface at a desired depth. Streamer cables may be positioned at a constant depth below the ocean surface, or they may have a variable depth profile, according to the design of the survey. 
     Marine survey ships also tow a signal source. The signal source generates a seismic signal, which is a series of seismic waves that travel in various directions including toward the ocean floor. The seismic waves penetrate the ocean floor and are at least partially reflected by interfaces between subsurface layers having different seismic wave propagation speeds. Sensors detect and receive these reflected waves. Sensors transform the seismic waves into seismic traces suitable for analysis. Sensors are in communication with a computer or recording system, which records the seismic traces from each sensor. A seismic trace thus represents the seismic waves received at a sensor from a source. 
     Each seismic trace typically contains contributions corresponding to multiple reflected waves that travel different paths from the signal source to the seismic sensor. For example, a given sensor may detect waves reflected from an interface at a shallow depth below the surface at one time, and detect waves reflected from an interface at a deeper depth at a later time. The arrival times of the waves travelling along each path may be affected by a variety of factors including the composition of the subsurface layers along each path, the depths and thicknesses of the layers along each path, the angle of the incoming wave to the interfaces between layers, and other factors. As a result, correlation of the seismic traces with the seismic signal generated by the signal source provides data on the location, orientation, structure, nature, and composition of the geological structures beneath the ocean floor. 
     A common type of signal source in a marine environment is a marine airgun. A marine airgun releases a quantity of high-pressure compressed air into the water when fired. The released air creates an air bubble with an internal pressure greatly exceeding the hydrostatic pressure of the surrounding water. As a result, the air bubble expands. Eventually, the internal pressure of the air bubble becomes lower than the surrounding pressure and the air bubble begins to contract until the internal pressure again becomes higher than the surrounding pressure. Thus, the air bubble oscillates between expansion and contraction below the ocean surface. This oscillation may continue for many cycles and produces the seismic waves that constitute a seismic signal. However, the initial pressure increase due to the released air, or the “primary pulse,” is typically of significantly higher amplitude than the amplitude of the secondary pulses that follow. 
     Single airguns do not produce sufficient energy to penetrate to the desired depths under the ocean floor. Therefore, a signal source typically includes an array of multiple individual airguns configured to fire at approximately the same time. When the array is fired, the primary pulses from the airguns generally reinforce one another and produce a higher-amplitude primary pulse, while the secondary pulses tend to cancel one another out through destructive interference. 
     Use of an array of airguns complicates the process of determining the resulting seismic signal generated by the entire array. However, at a sufficiently large distance from the signal source, the seismic waves generated by each of the individual airguns effectively merge to produce a stable wave shape whose amplitude decreases linearly with distance. The region in which the wave shape becomes stable is referred to as the “far field,” and can begin at a distance on the order of three hundred meters from the array. Subsurface geological features are typically located in the far field, so it is the shape of the seismic signal in the far field—also known as the “far field signature”—that can be important for later analysis of the seismic traces recorded during the survey. 
     In operation, it may not be feasible to directly measure the far field signature by placing sensors in the far field. However, at locations closer than the far field, the distance from the location to each airgun in the array can vary. This variability causes the wave shape to fluctuate with the distance from the array. As a result, sensors placed close to the array can measure a signal different from the far field signature. 
     Some seismic survey systems approximate the far field signature using sensors in the “mid field”—a region extending from approximately twenty meters to approximately three hundred meters away from the signal source. In these surveys, the mid field signature is treated as if it were identical to the far field signature. However, because of the differences described above between the mid field signature and the far field signature, utilizing the seismic traces produced by mid field sensors produces inferior data regarding subsurface geological structures than would be obtained were the actual far field signature used. 
     Some seismic survey systems attempt to approximate the far field signature using sensors in the “near field”—a region extending from approximately one meter to approximately twenty meters from individual airguns in the signal source. Sensors generally cannot be placed closer than one meter to an airgun because the sensor can become engulfed in the air bubble produced by the airgun, preventing the sensor from detecting at least some parts of the seismic waves. In these surveys, the pattern of seismic waves generated by each airgun in the signal source is determined and the far field signature is modeled. Approximating the far field signature using near field sensors also produces inferior data regarding subsurface geological structures than would be obtained using the actual far field signature. 
     Accordingly, methods and systems capable of utilizing measurements from sensors located in different fields to improve the far field signature are needed. 
     SUMMARY 
     In accordance with one or more embodiments of the present disclosure, a method for reconstructing the far field signature of a signal source includes receiving a first trace from a first sensor. The method also includes receiving a second trace from a second sensor located in a different field from the first sensor. The method additionally includes receiving position information for the first sensor and for the second sensor. The method further includes computing a plurality of notional signatures based on the first trace, the second trace, and the position information. The method also includes determining a far field signature by combining the plurality of notional signatures. 
     In accordance with another embodiment of the present disclosure, a seismic survey system includes a source array comprising a plurality of seismic energy sources. The system also includes a first sensor configured to transform seismic waves into a first trace, and a second sensor (located in a different field from the first sensor) configured to transform seismic waves into a second trace. The system further includes a computing system communicatively coupled to the source array, the first sensor, and the second sensor. The computing system includes a processor, a memory communicatively coupled to the processor, and instructions stored in the memory. Those instructions when executed by the processor, cause the processor to receive the first trace from the first sensor; receive the second trace from the second sensor; receive position information for the first sensor and for the second sensor; compute a plurality of notional signatures based on the first trace, the second trace, and the position information; and determine a far field signature by combining the plurality of notional signatures. 
     In accordance with another embodiment of the present disclosure, a non-transitory computer-readable medium includes instructions that, when executed by a processor, cause the processor to receive a first trace from a first sensor; receive a second trace from a second sensor; receive position information for the first sensor and for the second sensor; compute a plurality of notional signatures based on the first trace, the second trace, and the position information; and determine a far field signature by combining the plurality of notional signatures. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a more complete understanding of the present disclosure and its features and advantages, reference is now made to the following description, taken in conjunction with the accompanying drawings, which may include drawings that are not to scale and wherein like reference numbers indicate like features, in which: 
         FIG. 1  illustrates a side view of an example marine seismic survey system used for reconstructing the far field signature of a signal source in accordance with some embodiments of the present disclosure; 
         FIG. 2  illustrates a flow chart of an example method for reconstructing the far field signature of a signal source using data from near field and mid field sensors in accordance with some embodiments of the present disclosure; 
         FIG. 3  illustrates a flow chart of an example method for computing notional signatures using data from near field and mid field sensors in accordance with some embodiments of the present disclosure; 
         FIG. 4  illustrates a flow chart of an example method for computing notional signatures using a surface reflection coefficient in accordance with some embodiments of the present disclosure; 
         FIG. 5  illustrates a flow chart of an example method for computing notional signatures by combining preliminary notional signatures in accordance with some embodiments of the present disclosure; 
         FIG. 6  illustrates a flow chart of an example method for computing notional signatures by iteratively adjusting the position information and a surface reflection coefficient in accordance with some embodiments of the present disclosure; and 
         FIG. 7  illustrates a flow chart of an example method for computing notional signatures using additional data from surface field sensors in accordance with some embodiments of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     The present disclosure is directed to methods and systems of reconstructing the far field signature of a seismic energy source by combining data from near field and mid field sensors. The system includes at least one sensor positioned in the “near field,” which extends from approximately one meter to twenty meters away from the signal source towed behind the survey ship, and at least one sensor positioned in the “mid field,” which extends from approximately twenty meters to three hundred meters from the signal source. The system may include multiple sensors in each field, and may also include sensors in the “surface field” located within approximately one meter of the ocean surface. Using seismic traces from sensors located in both the near field and the mid field in the same computation, a more accurate reconstruction of the far field signature of the source may be obtained than is available using seismic traces from one field alone. Subsurface geological features are typically located in the far field, so it is the far field signature that can be important for later analysis of the seismic traces recorded during the survey. 
       FIG. 1  illustrates a side view of an example marine seismic survey system  100  used for reconstructing the far field signature of signal source in accordance with some embodiments of the present disclosure. Seismic survey system  100  includes signal source  130  and marine survey ship  110 . Ship  110  may be traveling in direction “T” shown by directional arrow  114 . Signal source  130  includes an array of seismic energy sources  160   a ,  160   c , and  160   e  (collectively “seismic energy sources  160 ”) towed behind ship  110 . Seismic energy sources  160  may also be referred to as “seismic sources,” “energy sources,” or “sources.” Seismic survey system  100  may also include near field sensors  170   a - 170   h  (collectively referred to as “near field sensors  170 ”), mid field sensors  180   a - 180   c  (collectively referred to as “mid field sensors  180 ”), and surface field sensor  190 . Near field sensors  170 , mid field sensors  180 , and surface field sensor  190  may be towed behind ship  110  and positioned relative to signal source  130 . 
     In some embodiments, near field sensors  170  are positioned within the near field—between approximately one meter and twenty meters from one of seismic energy sources  160 . For example, near field sensors  170   a  and  170   b  may be positioned approximately two meters above and two meters below seismic energy source  160   a , respectively. As another example, near field sensors  170   c  and  170   d  may be positioned approximately two meters above and two meters below seismic energy source  160   b , respectively, and near field sensors  170   e  and  170   f  may be positioned approximately two meters above and two meters below seismic energy source  160   e , respectively. In some embodiments, other near field sensors  170  are positioned at other distances or in other directions. For example, near field sensor  170   g  may be positioned above and behind seismic energy source  160   c  at a distance of approximately five meters, or near field sensor  170   h  may be positioned below and behind seismic energy source  160   e  at a distance of approximately ten meters. Although a specific number of near field sensors  170  is depicted, any appropriate number of near field sensors  170  may be used. Furthermore, near field sensors  170  may be positioned at any appropriate set of locations within the near field, with any appropriate combination of in-line offset (along the direction T), cross-line offset (perpendicular to direction T with respect to the sea surface plane), and depth offset from seismic energy sources  160 . In some embodiments, near field sensors  170  can be coupled to seismic energy sources  160  or other near field sensors  170  via cables  172 . For example, near field sensor  170   a  may be coupled to seismic energy source  160   a  via cable  172 . In some embodiments, one or more near field sensors  170  may be coupled to a streamer line (not shown), or maintained in a selected position or location using any other suitable positioning system. 
     In some embodiments, mid field sensors  180  are positioned within the mid field—between approximately twenty meters and three hundred meters from seismic energy sources  160 . For example, mid field sensor  180   a  may be positioned approximately one hundred and fifty meters below seismic energy source  160   a  and mid field sensor  180   b  may be positioned approximately fifty meters below mid field sensor  180   a . A particular mid field sensor  180  may be positioned approximately directly below a particular seismic energy source  160  or may be offset in any direction from directly below a particular seismic energy source  160 . In some embodiments, other mid field sensors  180  are positioned at other distances or in other directions. For example, mid field sensor  180   c  may be positioned approximately one hundred meters behind mid field sensor  180   a , at a distance of approximately one hundred eighty meters from seismic energy source  160   a . Although a specific number of mid field sensors  180  is depicted, any appropriate number of mid field sensors  180  may be used. Furthermore, mid field sensors  180  may be positioned at any appropriate set of locations within the mid field, with any appropriate combination of in-line offset, cross-line offset, and depth offset from seismic energy sources  160 . In some embodiments, mid field sensors  180  may be attached to a particular streamer line  182  or maintained in a selected position or location using any other suitable positioning system. 
     In some embodiments, surface field sensor  190  may be positioned within one meter of ocean surface  112 . Although only one surface field sensor  190  is depicted, any appropriate number of surface field sensors  190  may be used. Furthermore, surface field sensor  190  may be positioned at any appropriate locations within the surface field, with any appropriate combination of in-line offset, cross-line offset, and depth offset from seismic energy sources  160 . In some embodiments, surface field sensor  190  is coupled to streamer line  192  or maintained in a selected position or location using any other suitable positioning system. 
     A sensor may be described as “located in a different field” from another sensor based on the positions of the two sensors. For example, near field sensor  170   a  is located in a different field from mid field sensor  180   a  because near field sensor is positioned in the near field while mid field sensor  180   a  is positioned in the mid field. By contrast, near field sensor  170   a  is located in the same field as near field sensor  170   b  because both are positioned in the near field. 
     In some embodiments, near field sensors  170 , mid field sensors  180 , and surface field sensor  190  may be hydrophones, which detect pressure fluctuations in the surrounding water, or any other suitable sensor. Each near field sensor  170 , mid field sensor  180 , and surface field sensor  190  detects seismic waves received from seismic energy sources  160  and transforms the seismic waves into a seismic trace. A seismic trace may be digital sample data, an analog electrical signal, or any other appropriate representation of the seismic waves detected by the sensor. Seismic traces may be referenced based on the location of the sensor that detected the waves represented in the trace. For example, a “near field trace” may be a seismic trace from a particular near field sensor  170 , a “mid field trace” may be a seismic trace from a particular mid field sensor  180 , and a “surface field trace” may be a seismic trace from a particular surface field sensor  190 . A seismic trace from an unspecified sensor may be referred to as a “trace.” 
     Seismic energy sources  160  can be any suitable seismic energy source. For example, seismic energy sources  160  may be marine airguns, which generate a high-pressure oscillating air bubble. In some embodiments, seismic energy source  160   a  and seismic energy source  160   c  are at approximately the same depth below ocean surface  112 , for example approximately ten meters. Seismic energy sources  160   a  and  160   c  may be positioned at a predetermined horizontal distance from one another, for example approximately three meters. In some embodiments, seismic energy source  160   e  is at a different depth from seismic energy sources  160   a  and  160   c . Although only three seismic energy sources  160  are depicted, any appropriate number of seismic energy sources  160  may be used. Furthermore, seismic energy sources  160  may be arranged in any appropriate geometry and positioned at any appropriate depth, according to the design of the seismic survey. For example, seismic energy sources  160  may be coupled to streamer line  162  or maintained in a selected position or location using any other suitable positioning system. 
     In some embodiments, the positions of seismic energy sources  160 , near field sensors  170 , mid field sensors  180 , and surface field sensor  190  are monitored using one or more position-measurement mechanisms. For example, seismic survey system  100  may include an ultra-short baseline (USBL) transceiver  195  attached to ship  110 , which measures an angle and distance to each element of signal source  130  using acoustic pulses. Seismic survey system  100  may also include depth sensors, GPS sensors, visible light or infrared transceivers, or any other mechanisms suitable for measuring the positions of the elements of signal source  130 . 
     In some embodiments, seismic survey system  100  also includes computing system  198 . Near field sensors  170 , mid field sensors  180 , and surface field sensor  190  transmit seismic traces to computing system  198 . Computing system  198  receives position data reflecting the positions of the elements of seismic survey system  100 . In some embodiments, computing system  198  analyzes the seismic traces and position data to reconstruct a far field signature according to the methods discussed below. In some embodiments, computing system  198  stores the seismic traces and position data for later analysis, either by computing system  198  or by some other computing system. In some embodiments, computing system  198  or some other computing system utilizes the reconstructed far field signature to analyze other seismic traces created during the seismic survey. In some embodiments, computing system  198  or some other computing system utilizes the reconstructed far field signature to generate an image of subsurface geological structures. 
     Computing system  198  may include any instrumentality or aggregation of instrumentalities operable to compute, classify, process, transmit, receive, store, display, record, or utilize any form of information, intelligence, or data. For example, computing system  198  may include one or more personal computers, storage devices, servers, or any other suitable device and may vary in size, shape, performance, functionality, and price. Computing system  198  may include random access memory (RAM), one or more processing resources, such as a central processing unit (CPU) or hardware or software control logic, or other types of volatile or non-volatile memory. Additional components of computing system  198  may include one or more disk drives, one or more network ports for communicating with external devices, one or more input/output (I/O) devices, such as a keyboard, a mouse, or a video display. Computing system  198  may be configured to permit communication over any type of network, such as a wireless network, a local area network (LAN), or a wide area network (WAN) such as the Internet. Furthermore, computing system  198  may be located in any suitable enclosure, and may be located on ship  110 , on another ship, on shore, or in any other suitable location. 
       FIG. 2  illustrates a flow chart of an example method  200  for reconstructing the far field signature of a signal source using data from near field and mid field sensors in accordance with some embodiments of the present disclosure. For illustrative purposes, method  200  is described with respect to signal source  130  in seismic survey system  100 , discussed with respect to  FIG. 1 ; however, method  200  may be used to reconstruct the far field signature of any appropriate signal source. The steps of method  200  can be performed by a user, electronic or optical circuits, various computer programs, models, or any combination thereof, configured to process seismic traces. The programs and models may include instructions stored on a non-transitory computer-readable medium and operable to perform, when executed, one or more of the steps described below. The computer-readable media can include any system, apparatus, or device configured to store and retrieve programs or instructions such as a hard disk drive, a compact disc, flash memory, or any other suitable device. The programs and models may be configured to direct a processor or other suitable unit to retrieve and execute the instructions from the computer readable media. Collectively, the user, circuits, or computer programs and models used to process seismic traces may be referred to as a “processing tool.” For example, the processing tool may be computing system  198  located on ship  110 , discussed with reference to  FIG. 1 . In some embodiments, the processing tool is located elsewhere, and receives information stored during the seismic survey. For example, computing system  198  may record seismic traces and position information and deliver them to an on-shore computing system for processing at a later time. 
     Method  200  begins at step  210 , where the processing tool receives a near field trace from a near field sensor. For example, the processing tool may receive a near field trace from near field sensor  170   a , discussed with reference to  FIG. 1 . In some embodiments, the processing tool receives multiple near field traces, each from a separate near field sensor. For example, the processing tool may receive separate near field traces from near field sensors  170   a - 170   h . In some embodiments, the processing tool receives the near field traces indirectly from the near field sensors. For example, computing system  198  may store near field traces from near field sensors  170   a - 170   h  and deliver them to the processing tool at a later time. 
     In step  220 , the processing tool receives a mid field trace from a mid field sensor. For example, the processing tool may receive a mid field trace from mid field sensor  180   a , discussed with reference to  FIG. 1 . In some embodiments, the processing tool receives multiple mid field traces, each from a separate mid field sensor. For example, the processing tool may receive separate mid field traces from mid field sensors  180   a - 180   c . In some embodiments, the processing tool receives the mid field traces indirectly from the mid field sensors. For example, computing system  198  may store mid field traces from mid field sensors  180   a - 180   c  and deliver them to the processing tool at a later time. 
     In step  230 , the processing tool receives a surface field trace from a surface field sensor. For example, the processing tool may receive a surface field trace from surface field sensor  190 , discussed with reference to  FIG. 1 . In some embodiments, the processing tool receives multiple surface field traces, each from a separate surface field sensor. In some embodiments, the processing tool receives the surface field traces indirectly from the surface field sensors. For example, computing system  198  may store surface field traces from surface field sensor  190  and deliver them to the processing tool at a later time. 
     In step  240 , the processing tool receives position information regarding the sensors and the seismic energy sources. For example, the processing tool may receive position information regarding near field sensors  170   a - 170   h , mid field sensors  180   a - 180   c , surface field sensor  190 , and seismic energy sources  160   a - 160   c , discussed with respect to  FIG. 1 . In some embodiments, the processing tool receives the position information indirectly. For example, computing system  198  may store the position information and deliver it to the processing tool at a later time. The position information may be generated using ultra-short baseline (USBL) systems, global positioning systems (GPS), depth sensors, visible or infrared signaling systems, or any other suitable position-measuring mechanism. 
     In step  250 , the processing tool computes a set of “notional signatures” based at least on the near field traces and the mid field traces. In some embodiments, the notional signatures are also based on the surface field trace, the position information, or other suitable data. A notional signature is a mathematical model of a hypothetical seismic energy source, for example an oscillating air bubble expelled by a marine airgun, that does not interact with the seismic waves generated by other seismic energy sources. When an array of airguns is fired, the seismic waves created by each air bubble change the hydrostatic pressure of the water surrounding every other bubble, thus altering their oscillation patterns in a non-linear fashion. Rather than modelling the actual oscillations of the physical bubbles, the processing tool computes a model of a “notional” non-interacting bubble corresponding to each bubble created by the signal source. For example, in some embodiments the nth notional signature P n (t, r) is modeled as having spherical symmetry according to the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       n 
                     
                      
                     
                       ( 
                       
                         t 
                         , 
                         r 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         r 
                         n 
                       
                     
                      
                     
                       
                         p 
                         n 
                       
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             
                               r 
                               n 
                             
                             c 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where
         t=time;   r n =distance to notional bubble number n; and   p n (t)=“driving pressure” of notional bubble number n (equal to the difference between the modeled internal pressure of notional bubble number n at time t and the hydrostatic pressure outside notional bubble n at time t); and   c=speed of sound in water.
 
The modeled oscillations of each notional bubble incorporate the non-linear interactions between the physical bubbles. For example, the driving pressure of each notional bubble may include terms that represent the effect of seismic waves generated by each other notional bubble. This model is referred to as a “notional signature.” There are many possible methods by which the traces and the position information may be used to compute a set of notional signatures. A number of example methods for step  250  are discussed in  FIGS. 3-7  below. However, any method which utilizes at least one near field and at least one mid field trace is intended to be encompassed within the present disclosure.
       

     In step  260  the processing tool determines a far field signature by combining the notional signatures. In some embodiments, the far field signature is a linear superposition of the notional signatures. Because the notional bubbles represented by the notional signatures do not interact, they may be linearly superimposed to produce a model of the far field signature. For example, in some embodiments the far field signature F(t, loc) is modeled as a sum of notional signatures each having spherical symmetry according to the following equation: 
         F ( t,loc )=Σ N=1   N ( P   n ( t,r   n,loc )+ k*P   n ( t,rg   n,loc )  (2)
 
     where
         t=time;   loc=location at which F is being calculated;   N=number of notional signatures;   r n,loc =distance from location loc to notional bubble number n;   rg n,loc =distance from location loc to the “ghost” of notional bubble number n (caused by reflection from the sea surface); and   k=the surface reflection coefficient.       

     Modifications, additions, or omissions may be made to method  200  without departing from the scope of the present disclosure. For example, the steps may be performed in a different order than that described and some steps may be performed at the same time. For example, the processing tool may receive traces and position information in any order or simultaneously. Further, more steps may be added or steps may be removed without departing from the scope of the disclosure. Additionally, each individual step may include additional steps without departing from the scope of the present disclosure. In particular, step  250 —computing notional signatures—may be implemented in a variety of ways, as discussed previously. 
       FIG. 3  illustrates a flow chart of an example method  300  for computing notional signatures using data from near field and mid field sensors in accordance with some embodiments of the present disclosure. For illustrative purposes, method  300  is described with respect to signal source  130  in seismic survey system  100 , discussed with respect to  FIG. 1 ; however, method  300  may be used to compute notional signatures for any appropriate signal source. The steps of method  300  can be performed by a user, electronic or optical circuits, various computer programs, models, or any combination thereof, configured to process seismic traces. The programs and models may include instructions stored on a non-transitory computer-readable medium and operable to perform, when executed, one or more of the steps described below. The computer-readable media can include any system, apparatus, or device configured to store and retrieve programs or instructions such as a hard disk drive, a compact disc, flash memory, or any other suitable device. The programs and models may be configured to direct a processor or other suitable unit to retrieve and execute the instructions from the computer readable media. As discussed previously, collectively, the user, circuits, or computer programs and models used to process seismic traces may be referred to as a “processing tool.” For example, the processing tool may be computing system  198  located on ship  110 , discussed with respect to  FIG. 1 . In some embodiments, the processing tool is located elsewhere, and receives information stored during the seismic survey. For example, computing system  198  may record seismic traces and position information and deliver them to an on-shore computing system for processing at a later time. Method  200 , discussed with respect to  FIG. 2  may utilize method  300  to implement step  250 ; however, the processing tool performing the steps of method  300  may be the same processing tool performing the steps of method  200 , or a different processing tool. 
     Method  300  begins at step  310 , where the processing tool determines the number of notional signatures needed to model the seismic sources. The number of notional signatures may be based on the number of seismic energy sources that generated the seismic traces recorded in steps  210 - 230  of method  200  discussed with reference to  FIG. 2 . For example, the number of notional signatures may equal or approximate the number of seismic energy sources  160   a ,  160   c , and  160   e  (or three notional signatures) in signal source  130 , discussed with reference to  FIG. 1 . In some embodiments, the number of notional signatures may account for reflection of the seismic waves off the ocean surface, known as “surface reflection.” Surface reflection can be modeled by the addition of a virtual seismic energy source for each actual seismic energy source. In this case, the number of notional signatures is twice the number of seismic energy sources  160  in signal source  130  (six notional signatures). The selection of the number of notional signatures may be based on the design of the seismic survey, the depth of the seismic energy sources  160 , the time needed for or cost associated with subsequent seismic analysis, or any other suitable parameter. 
     In step  312 , the processing tool calculates the total number of near field and mid field traces received from the near field and mid field sensors. For example, the processing tool may have received eight near field traces from near field sensors  170   a - 170   h , and three mid field traces from mid field sensors  180   a - 180   c , for a total of eleven traces. 
     In step  320 , the processing tool determines if the number of traces received is less than the determined number of notional signatures. If the number of traces is less than the number of notional signatures, method  300  returns to step  210 . For example, if only five traces were received by the processing tool and six notional signatures are needed, an error may have occurred within the seismic survey or analysis system. If the number of traces is greater than or equal to the number of notional signatures needed, method  300  proceeds to step  330 . 
     In step  330 , the processing tool determines whether the number of traces is greater than the number of notional signatures needed. If the number of traces is equal to the number of notional signatures needed, method  300  proceeds to step  340 . 
     In step  340 , the processing tool generates a set of simultaneous equations based on the near field traces, the mid field traces, and the position information. In some embodiments, the seismic waves generated by an air bubble, P(t,r), are modeled as having spherical symmetry according to the following equation: 
     
       
         
           
             
               
                 
                   
                     P 
                      
                     
                       ( 
                       
                         t 
                         , 
                         r 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       r 
                     
                      
                     
                       p 
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             r 
                             c 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where
         t=time;   r=distance to the bubble;   p(t)=“driving pressure” of the air bubble (equal to the difference between internal pressure of the air bubble at time t and the hydrostatic pressure outside the bubble at time t); and   c=speed of sound in water.       

     To compute the driving pressure of each bubble, the processing tool may need to account for the non-linear interactions of the air bubbles discussed above in connection with step  250  in  FIG. 2 . For example, where two bubbles are involved, the seismic waves generated by the first bubble will cause changes in the hydrostatic pressure outside the second bubble, and vice-versa. Thus, the driving pressure P i  (t) of the first notional bubble may include a term corresponding to the seismic waves generated by the second bubble. Additional terms and equations can be added to model the effects of a third bubble or any additional number of bubbles. Although one potential method of generating a set of simultaneous equations is discussed in detail, any other suitable method for generating a set of simultaneous equations given at least a set of traces and position information may be used in step  340 . For example, the model of each notional air bubble may include one or more parameters that vary according to direction, rather than being spherically symmetric. The equations may also include terms that model weather conditions or other relevant factors. 
     In step  350 , the processing tool solves the set of simultaneous equations generated in step  340  to generate the notional signatures. Because the number of variables (corresponding to notional signatures) equals the number of equations (corresponding to traces), the system of simultaneous equations is solvable. For example, the system of simultaneous equations may be solved using matrix inversion in either the time or frequency domain. Method  300  returns the resulting notional signatures to step  260  of method  200  shown in  FIG. 2 . 
     If at step  330 , the number of traces is greater than the number of notional signatures needed, method  300  proceeds to step  360 , where the processing tool generates an error function based on the measurement design and perceived measurement quality. In this case, if a system of equations were constructed according to step  340 , the number of equations (corresponding to traces) would exceed the number of variables (corresponding to notional signatures), making the system of equations overdetermined and therefore, in general, unsolvable. However, an error function may be constructed which incorporates additional unknown quantities. For example, the error function may include the root mean square of the difference between an estimated signature at the location of a sensor and the seismic signature detected by that sensor. In some embodiments, the error function may be a weighted average of components corresponding to the near field traces and components corresponding to the mid field traces. Near field measurements may give improved estimates of the notional signatures in low frequencies, while mid field measurements give improved estimates of the notional signatures in high frequencies. As a result, the weighting factors may be chosen to suit the needs of the survey and later processing of the recorded traces. In some embodiments, higher weight is placed on the components corresponding to the near field traces. In some embodiments, higher weight is placed on the components corresponding to the mid field traces. In some embodiments, the weight placed on each component varies as a function of frequency. 
     In step  370 , the processing tool generates a set of simultaneous equations based on the near field traces, the mid field traces, the position information, and the error function generated in step  360 . In some embodiments, the simultaneous equations are generated using the equations described in connection with step  340 , with an additional term incorporating the error function generated in step  360 . 
     In step  380 , the processing tool minimizes the error function in the simultaneous equations generated in step  370  to generate the notional signatures. Mathematical methods for solving an overdetermined set of simultaneous equations by minimizing error functions can include algebraic functions, matrix transformations, or other suitable methods. Method  300  returns to step  260  of method  200  shown in  FIG. 2 . 
     Modifications, additions, or omissions may be made to method  300  without departing from the scope of the present disclosure. For example, the steps may be performed in a different order than that described and some steps may be performed at the same time. For example step  310  may be performed before or after step  312 . Further, more steps may be added or steps may be removed without departing from the scope of the disclosure. 
       FIG. 4  illustrates a flow chart of an example method  400  for computing notional signatures using a surface reflection coefficient in accordance with some embodiments of the present disclosure. For illustrative purposes, method  400  is described with respect to signal source  130  in seismic survey system  100 , discussed with respect to  FIG. 1 ; however, method  400  may be used to compute notional signatures for any appropriate signal source. The steps of method  400  can be performed by a user, electronic or optical circuits, various computer programs, models, or any combination thereof, configured to process seismic traces. The programs and models may include instructions stored on a non-transitory computer-readable medium and operable to perform, when executed, one or more of the steps described below. The computer-readable media can include any system, apparatus, or device configured to store and retrieve programs or instructions such as a hard disk drive, a compact disc, flash memory, or any other suitable device. The programs and models may be configured to direct a processor or other suitable unit to retrieve and execute the instructions from the computer readable media. As discussed previously, collectively, the user, circuits, or computer programs and models used to process seismic traces may be referred to as a “processing tool.” For example, the processing tool may be computing system  198  located on ship  110 , discussed with respect to  FIG. 1 . In some embodiments, the processing tool is located elsewhere, and receives information stored during the seismic survey. For example, computing system  198  may record seismic traces and position information and deliver them to an on-shore computing system for processing at a later time. Method  200 , discussed with respect to  FIG. 2  may utilize method  400  to implement step  250 ; however, the processing tool performing the steps of method  400  may be the same processing tool performing the steps of method  200 , or a different processing tool. 
     Method  400  begins at step  410 , where the processing tool estimates a surface reflection coefficient based on the mid field traces. When the seismic waves generated by a seismic energy source, for example seismic energy source  160 , discussed with respect to  FIG. 1 , encounter ocean surface  112 , the seismic waves are reflected back downward with less amplitude. The surface reflection coefficient represents the ratio between the amplitude of the original wave and the amplitude of the reflected wave. For example, mid field sensors  180  can detect an original wave, which travels on a direct path from seismic energy source  160   a , and an inverted “ghost” wave corresponding to the reflection from the surface. Because the depth of seismic energy source  160   a  is generally smaller than or comparable to the wavelength of the seismic waves, the ghost wave may overlap with and even destructively interfere with the original wave. This destructive interference may result in a “notch” in the spectrum of energy detected by mid field sensors  180 . The peak depth of the notch indicates the amount of the original wave canceled by the ghost wave, and thus indicates the surface reflection coefficient. 
     In step  420 , the processing tool generates a set of simultaneous equations based on the near field traces, the mid field traces, the position information, and the surface reflection coefficient estimated in step  410 . For example, the processing tool may model the seismic waves generated by each air bubble using equations (3) and (4), discussed above in connection with step  340 , discussed with relation to FIG.  3 . The processing tool may model surface reflection by including in each equation a term corresponding to the reflection of the wave from ocean surface  112 , discussed with relation to  FIG. 1 . 
     In step  430 , the processing tool solves the set of simultaneous equations to generate the notional signatures. For example, if the number of desired notional signatures equals the number of traces, the system may be solved using matrix inversion in either the time or frequency domain as described in step  350 , discussed with respect to  FIG. 3 . If the number of desired notional signatures is less than the number of traces, the system may be solved by minimizing an error function as described in step  380 , discussed with respect to  FIG. 3 . Method  400  returns the resulting notional signatures to step  260  of method  200  shown in  FIG. 2 . 
     Modifications, additions, or omissions may be made to method  400  without departing from the scope of the present disclosure. For example, step  420  may be performed before step  410 . Further, more steps may be added or steps may be removed without departing from the scope of the disclosure. 
       FIG. 5  illustrates a flow chart of an example method  500  for computing notional signatures by combining preliminary notional signatures in accordance with some embodiments of the present disclosure. For illustrative purposes, method  500  is described with respect to signal source  130  in seismic survey system  100 , discussed with respect to  FIG. 1 ; however, method  500  may be used to compute notional signatures for any appropriate signal source. The steps of method  500  can be performed by a user, electronic or optical circuits, various computer programs, models, or any combination thereof, configured to process seismic traces. The programs and models may include instructions stored on a non-transitory computer-readable medium and operable to perform, when executed, one or more of the steps described below. The computer-readable media can include any system, apparatus, or device configured to store and retrieve programs or instructions such as a hard disk drive, a compact disc, flash memory, or any other suitable device. The programs and models may be configured to direct a processor or other suitable unit to retrieve and execute the instructions from the computer readable media. As discussed previously, collectively, the user, circuits, or computer programs and models used to process seismic traces may be referred to as a “processing tool.” For example, the processing tool may be computing system  198  located on ship  110 , discussed with respect to  FIG. 1 . In some embodiments, the processing tool is located elsewhere, and receives information stored during the seismic survey. For example, computing system  198  may record seismic traces and position information and deliver them to an on-shore computing system for processing at a later time. Method  200 , discussed with respect to  FIG. 2 , may utilize method  500  to implement step  250 ; however, the processing tool performing the steps of method  500  may be the same processing tool performing the steps of method  200 , or a different processing tool. 
     Method  500  begins at step  510 , where the processing tool generates a first set of simultaneous equations based on the near field traces and the position information for the near field sensors and the seismic energy sources. For example, the processing tool may model the seismic waves generated by each air bubble using equations (3) and (4), discussed above in connection with step  340 , discussed with relation to  FIG. 3 . 
     In step  520 , the processing tool computes a first set of preliminary notional signatures by solving the first set of simultaneous equations. Mathematical methods for solving a determined set of simultaneous equations can include algebraic functions, matrix transformations, or other suitable methods. For example, if the number of desired notional signatures equals the number of traces, the system may be solved using matrix inversion in either the time or frequency domain as described in step  350 , discussed with respect to  FIG. 3 . If the number of desired notional signatures is less than the number of traces, the system may be solved by minimizing an error function as described in step  380 , discussed with respect to  FIG. 3 . 
     In step  530 , the processing tool generates a second set of simultaneous equations based on the mid field traces and the position information for the mid field sensors and the seismic energy sources. For example, the processing tool may model the seismic waves generated by each air bubble using equations (3) and (4), discussed above in connection with step  340 , discussed with relation to  FIG. 3 . 
     In step  540 , the processing tool computes a second set of preliminary notional signatures by solving the second set of simultaneous equations. Mathematical methods for solving a determined set of simultaneous equations can include algebraic functions, matrix transformations, or other suitable methods. For example, if the number of desired notional signatures equals the number of traces, the system may be solved using matrix inversion in either the time or frequency domain as described in step  350 , discussed with respect to  FIG. 3 . If the number of desired notional signatures is less than the number of traces, the system may be solved by minimizing an error function as described in step  380 , discussed with respect to  FIG. 3 . 
     In step  550 , the processing tool combines the first and second sets of notional signatures. In some embodiments, the notional signatures are a weighted average of the first set of notional signatures and the second set of notional signatures. In general, near field measurements give more accurate estimates of the notional signatures in low frequencies, while mid field measurements give more accurate estimates of the notional signatures in high frequencies. As a result, the weighting factors may be chosen to suit the design of the survey and later processing of the recorded traces. In some embodiments, higher weight is placed on the near field traces, while in other embodiments, higher weight is placed on the mid field traces. In some embodiments, the weight placed on each set of traces varies as a function of frequency. Method  500  returns to step  260  of method  200  shown in  FIG. 2 . 
     Modifications, additions, or omissions may be made to method  500  without departing from the scope of the present disclosure. For example, the steps may be performed in a different order than that described and some steps may be performed at the same time. Further, more steps may be added or steps may be removed without departing from the scope of the disclosure. 
       FIG. 6  illustrates a flow chart of an example method  600  for computing notional signatures by iteratively adjusting the position information and a surface reflection coefficient in accordance with some embodiments of the present disclosure. For illustrative purposes, method  600  is described with respect to signal source  130  in seismic survey system  100 , discussed with respect to  FIG. 1 ; however, method  600  may be used to compute notional signatures for any appropriate signal source. The steps of method  600  can be performed by a user, electronic or optical circuits, various computer programs, models, or any combination thereof, configured to process seismic traces. The programs and models may include instructions stored on a non-transitory computer-readable medium and operable to perform, when executed, one or more of the steps described below. The computer-readable media can include any system, apparatus, or device configured to store and retrieve programs or instructions such as a hard disk drive, a compact disc, flash memory, or any other suitable device. The programs and models may be configured to direct a processor or other suitable unit to retrieve and execute the instructions from the computer readable media. As discussed previously, collectively, the user, circuits, or computer programs and models used to process seismic traces may be referred to as a “processing tool.” For example, the processing tool may be computing system  198  located on ship  110 , discussed with respect to  FIG. 1 . In some embodiments, the processing tool is located elsewhere, and receives information stored during the seismic survey. For example, computing system  198  may record seismic traces and position information and deliver them to an on-shore computing system for processing at a later time. Method  200 , discussed with respect to  FIG. 2 , may utilize method  600  to implement step  250 ; however, the processing tool performing the steps of method  600  may be the same processing tool performing the steps of method  200 , or a different processing tool. 
     Method  600  begins at step  610 , where the processing tool estimates a surface reflection coefficient based on the near field traces. Step  610  may use any of the methods for estimating a surface reflection coefficient discussed above in connection with step  410 , discussed with respect to  FIG. 4 , except that the near field traces are used in place of the mid field traces. 
     In step  620 , the processing tool adjusts the position information for seismic energy sources  160 , discussed with respect to  FIG. 1 , based on a first subset of the near field traces. For example, in some embodiments the processing tool locates the “notch” in the frequency spectrum of seismic energy source  160   a  as discussed above in connection with step  410 , discussed with respect to  FIG. 4 . The position (frequency of the peak) of the notch indicates the depth of seismic energy source  160   a.    
     In step  630 , the processing tool adjusts the source reflection coefficient and the position information for seismic energy sources  160 , discussed with respect to  FIG. 1 , based on the mid field traces. Step  630  may use any of the methods for estimating the source reflection coefficient and the depth and position of seismic energy sources  160  discussed above in connection with step  410 , discussed with respect to  FIG. 4 . In some embodiments, other methods are used. For example, the processing tool may adjust the source reflection coefficient and the position information to minimize an error function. 
     In step  640 , the processing tool determines whether the source reflection coefficient and the position information are within a threshold error. For example, the processing tool may calculate an estimate of the error in the source reflection coefficient and the position information. In some embodiments, the processing tool compares the largest individual error to a threshold error. In some embodiments, the processing tool combines the error estimates for each measurements into a single error estimate and compares the single error estimate to a threshold error. If the source reflection coefficient and position information are not within the threshold error, method  600  returns to step  630  to further adjust the source reflection coefficient and the position information. If they are within the threshold error, method  600  proceeds to step  650 . 
     In step  650 , the processing tool generates a set of simultaneous equations based on a second subset of the near field traces (different from the first subset) and the mid field traces, along with the position information and the surface reflection coefficient adjusted in steps  620 - 630 . Step  650  may use any method for constructing a system of equations described above in connection with step  420 , discussed with respect to  FIG. 4 . For example, the processing tool may model the seismic waves generated by each air bubble using equations (3) and (4), discussed above in connection with step  340 , discussed with relation to  FIG. 3 . The processing tool may model surface reflection by including in each equation a term corresponding to the reflection of the wave from ocean surface  112 , discussed with relation to  FIG. 1 . 
     In step  660 , the processing tool solves the set of simultaneous equations to generate the notional signatures. Mathematical methods for solving a determined set of simultaneous equations can include algebraic functions, matrix transformations, or other suitable methods. For example, if the number of desired notional signatures equals the number of traces, the system may be solved using matrix inversion in either the time or frequency domain as described in step  350 , discussed with respect to  FIG. 3 . If the number of desired notional signatures is less than the number of traces, the system may be solved by minimizing an error function as described in step  380 , discussed with respect to  FIG. 3 . Method  600  returns the resulting notional signatures to step  260  of method  200  shown in  FIG. 2 . 
     Modifications, additions, or omissions may be made to method  600  without departing from the scope of the present disclosure. For example, the steps may be performed in a different order than that described and some steps may be performed at the same time. For example, steps  620  and  630  may be performed simultaneously. Further, more steps may be added or steps may be removed without departing from the scope of the disclosure. 
       FIG. 7  illustrates a flow chart of an example method  700  for computing notional signatures using additional data from surface field hydrophones in accordance with some embodiments of the present disclosure. For illustrative purposes, method  700  is described with respect to signal source  130  in seismic survey system  100 , discussed with respect to  FIG. 1 ; however, method  700  may be used to compute notional signatures for any appropriate signal source. The steps of method  700  can be performed by a user, electronic or optical circuits, various computer programs, models, or any combination thereof, configured to process seismic traces. The programs and models may include instructions stored on a non-transitory computer-readable medium and operable to perform, when executed, one or more of the steps described below. The computer-readable media can include any system, apparatus, or device configured to store and retrieve programs or instructions such as a hard disk drive, a compact disc, flash memory, or any other suitable device. The programs and models may be configured to direct a processor or other suitable unit to retrieve and execute the instructions from the computer readable media. As discussed previously, collectively, the user, circuits, or computer programs and models used to process seismic traces may be referred to as a “processing tool.” For example, the processing tool may be computing system  198  located on ship  110 , discussed with respect to  FIG. 1 . In some embodiments, the processing tool is located elsewhere, and receives information stored during the seismic survey. For example, computing system  198  may record seismic traces and position information and deliver them to an on-shore computing system for processing at a later time. Method  200 , discussed with respect to  FIG. 2 , may utilize method  700  to implement step  250 ; however, the processing tool performing the steps of method  700  may be the same processing tool performing the steps of method  200 , or a different processing tool. 
     Method  700  begins at step  710 , where the processing tool estimates a surface reflection coefficient based on the surface field traces and the near field traces. Step  710  may use any of the methods for estimating a surface reflection coefficient discussed above in connection with step  410 , except that the surface field traces and the near field traces are used in place of the mid field traces. 
     In step  720 , the processing tool generates a set of simultaneous equations based on the near field traces, the mid field traces, the position information, and the surface reflection coefficient estimated in step  710 . Step  720  may use any method for constructing a system of equations described above in connection with step  420 . For example, the processing tool may model the seismic waves generated by each air bubble using equations (3) and (4), discussed above in connection with step  340 , discussed with relation to  FIG. 3 . The processing tool may model surface reflection by including in each equation a term corresponding to the reflection of the wave from ocean surface  112 , discussed with relation to  FIG. 1 . 
     In step  730 , the processing tool solves the set of simultaneous equations to generate the notional signatures. Mathematical methods for solving a determined set of simultaneous equations can include algebraic functions, matrix transformations, or other suitable methods. For example, if the number of desired notional signatures equals the number of traces, the system may be solved using matrix inversion in either the time or frequency domain as described in step  350 , discussed with respect to  FIG. 3 . If the number of desired notional signatures is less than the number of traces, the system may be solved by minimizing an error function as described in step  380 , discussed with respect to  FIG. 3 . Method  700  returns the resulting notional signatures to step  260  of method  200  shown in  FIG. 2 . 
     Modifications, additions, or omissions may be made to method  700  without departing from the scope of the present disclosure. For example, the steps may be performed in a different order than that described and some steps may be performed at the same time. Further, more steps may be added or steps may be removed without departing from the scope of the disclosure. 
     This disclosure encompasses all changes, substitutions, variations, alterations, and modifications to the example embodiments herein that a person having ordinary skill in the art would comprehend. Similarly, where appropriate, the appended claims encompass all changes, substitutions, variations, alterations, and modifications to the example embodiments herein that a person having ordinary skill in the art would comprehend. For example, the set of simultaneous equations constructed in step  320  with respect to  FIG. 3  may model the wave propagation pattern as a parameterized pattern rather than a spherically symmetric pattern. Moreover, reference in the appended claims to an apparatus or system or a component of an apparatus or system being adapted to, arranged to, capable of, configured to, enabled to, operable to, or operative to perform a particular function encompasses that apparatus, system, component, whether or not it or that particular function is activated, turned on, or unlocked, as long as that apparatus, system, or component is so adapted, arranged, capable, configured, enabled, operable, or operative. For example, a sensor does not have to be turned on but must be configured to receive reflected energy. 
     Any of the steps, operations, or processes described herein may be performed or implemented with one or more hardware or software modules, alone or in combination with other devices. In some embodiments, a software module is implemented with a computer program product comprising a computer-readable medium containing computer program code, which can be executed by a computer processor for performing any or all of the steps, operations, or processes described. 
     Embodiments of the disclosure may also relate to an apparatus for performing the operations herein. This apparatus may be specially constructed for the required purposes, and/or it may comprise a general-purpose computing device selectively activated or reconfigured by a computer program stored in the computer. Such a computer program may be stored in a tangible computer readable storage medium or any type of media suitable for storing electronic instructions, and coupled to a computer system bus. Furthermore, any computing systems referred to in the specification may include a single processor or may be architectures employing multiple processor designs for increased computing capability. For example, the computing system described in methods  200 ,  300 ,  400 ,  500 ,  600 , and  700  with respect to  FIGS. 2-7  may be stored in tangible computer readable storage media. 
     Although the present disclosure has been described with several embodiments, changes, variations, alterations, transformations, and modifications may be suggested to one skilled in the art, and it is intended that the present disclosure encompass such changes, variations, alterations, transformations, and modifications as fall within the scope of the appended claims. Moreover, while the present disclosure has been described with respect to various embodiments, it is fully expected that the teachings of the present disclosure may be combined in a single embodiment as appropriate. Instead, the scope of the disclosure is defined by the appended claims.