Patent Publication Number: US-10317543-B2

Title: Estimation of a far field signature in a second direction from a far field signature in a first direction

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to U.S. Provisional Application 61/979,176, filed Apr. 14, 2014, which is incorporated by reference. 
    
    
     BACKGROUND 
     In the past few decades, the petroleum industry has invested heavily in the development of marine seismic survey techniques that yield knowledge of subterranean formations beneath a body of water in order to find and extract valuable mineral resources, such as oil. High-resolution seismic images of a subterranean formation are helpful for quantitative seismic interpretation and improved reservoir monitoring. For a typical marine seismic survey, a marine seismic survey vessel tows one or more seismic sources below the surface of the water and over a subterranean formation to be surveyed for mineral deposits. Seismic receivers may be located on or near the seafloor, on one or more streamers towed by the source vessel, or on one or more streamers towed by another vessel. The source vessel typically contains marine seismic survey equipment, such as navigation control, seismic source control, seismic receiver control, and recording equipment. 
     The seismic source control may cause the one or more seismic sources, which can be air guns, marine vibrators, among other sources described herein, to produce acoustic signals at selected times. Each acoustic signal is essentially a sound wave that travels through the water and into subterranean formations. At each interface between different types of rock or other subterranean material, a portion of the sound wave may be refracted, a portion of the sound wave may be transmitted, and another portion may be reflected back toward the body of water to propagate toward the surface. The seismic receivers thereby measure a wavefield that was ultimately initiated by the actuation of the seismic source. Planning and executing a marine seismic survey and processing the acquired data require an accurate model of the output wavefield of the seismic sources used in the marine seismic survey. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates marine seismic surveying in which acoustic signals are emitted by a seismic source for recording by seismic receivers and subsequent processing and analysis to characterize the structures and distributions of features and materials underlying a solid surface of the earth. 
         FIG. 2  illustrates a two-dimensional schematic representation of a source array and a far field measurement point. 
         FIG. 3  illustrates a two-dimensional schematic representation of a plane wave emitted at a given angle relative to a line between a center of the source array and the far field measurement point. 
         FIG. 4  illustrates a two-dimensional schematic representation of simulated far field measurement points at angular positions relative to the line between the center of the source array and the far field measurement point. 
         FIG. 5  illustrates a variation in amplitude of a transfer function that transforms a vertical far field signature measurement to a 30 degree offset angle far field signature measurement as determined across a frequency spectrum. 
         FIG. 6  illustrates a variation in phase of a transfer function that transforms the vertical far field signature measurement to a 30 degree offset angle far field signature measurement as determined across the frequency spectrum. 
         FIG. 7  illustrates a variation in amplitude of a transfer function that transforms the vertical far field signature measurement to a 60 degree offset angle far field signature measurement as determined across the frequency spectrum. 
         FIG. 8  illustrates a variation in phase of the transfer function that transforms the vertical far field signature measurement to a 60 degree offset angle far field signature measurements determined across the frequency spectrum. 
         FIG. 9  illustrates a method flow diagram for estimation of a far field signature in a second direction from a far field signature in a first direction. 
         FIG. 10  illustrates a diagram of a system for estimation of a far field signature in a second direction from a far field signature in a first direction. 
         FIG. 11  illustrates a diagram of a machine for estimation of a far field signature in a second direction from a far field signature in a first direction. 
     
    
    
     DETAILED DESCRIPTION 
     The present disclosure is related to estimation of a far field signature of a seismic source in a second direction from a far field signature of the seismic source in a first direction. A source of an acoustic signal can emit a wavefield and, as described herein, can be termed a “seismic source”. Such seismic sources can include one or more seismic source elements, such as air guns, water guns, explosive devices, and/or vibratory devices, among others. In theory, the wavefield emitted by a point source can be described by a one-dimensional “signature” because a point source emits the same wavefield in all directions. A “notional source element signature” is a three-dimensional wavefield emitted by one source element, often approximated as a point source. The superposition of notional source element signatures from all of the source elements in a source array results in the “source signature.” Thus, a source signature is a net three-dimensional pressure variation in a body of water as a function of time caused by a transient perturbation of pressure by the acoustic signal, an “impulse” or wavefield, from the actual submerged source. 
     As used herein, an “estimation of a far field signature” is a representation of what a measured far field signature would be in a direction and location where it is not known or measured, the estimation being based at least in part on a known or measured far field signature in at least one other direction and known relative positions of a number of source elements that in combination form a seismic source. As used herein, a “seismic source” can represent a single source element or a plurality of source elements arranged at known positions relative to each other in a source array. As used herein, a “source element” represents one of the sound-emitting devices (e.g., air guns, water guns, explosive devices, and/or vibratory devices) composing the seismic source. If not otherwise stated, the terms “source” and “source array” represent the same entity and refer to the cluster of source elements whose combined output of acoustic signals composes a total wavefield emitted from the corresponding source or source array. 
     As used herein, the measured far field signature or estimated far field signature are acoustic signals that arrive directly from a source to a far field measurement point or would arrive directly from the source to a simulated far field measurement point. As such, the signals arrive at the far field measurement point or would arrive at the simulated far field measurement point without reflecting off of a free surface, a solid surface, and/or a subsurface associated with a fluid volume of water. 
     One characteristic of a seismic source is its far field signature. A signature of a seismic source refers to a shape of the signal transmitted by the seismic source as recorded by a seismic receiver. The signature of the seismic source varies with direction and with distance from the seismic source. Along a given direction, this signature varies with increasing distance from the seismic source, until at some given distance the shape of the signature achieves a relatively stable shape. At greater distances than this given distance, the signature remains relatively unchanged. The region where the signature shape does not change substantially with distance in all directions is known as the far field region and the seismic signature measured or estimated within that region is known as the far field signature of the seismic source. Often the far field region will be greater than 100 meters from the source, and 200 meters from the source will be in the far field region in most instances. The far field signature of a source array having more than one source element separated in space may vary with direction. For instance, in comparison to a first far field signature measured vertically (e.g., in-line with gravitational pull) under the geometric center of the source array, a second far field signature can vary notably when measured at a 30 degree angle (30°) offset from vertical under the center of the seismic source. The center of the seismic source can be calculated as a geometric center reference point determined from the outermost edges of the most distal source elements in one, two, or three dimensions, depending on the configuration of the source array. 
     Estimating a far field signature from a seismic source at a location where there may not be a seismic receiver can be beneficial, as described in more detail below. According to some embodiments of the present disclosure, for a seismic source including a number of source elements, an impulse response can be determined in a first direction and a second direction. A transfer function that transforms a far field signature of the seismic source in a first direction to a far field signature of the seismic source in a second direction can be determined based on the corresponding impulse responses in the first direction and the second direction. An estimated far field signature for the seismic source in the second direction then can be determined based on the transfer function. 
     It is to be understood that the present disclosure is not limited to particular devices or methods, which may, of course, vary. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to be limiting. As used herein, the singular forms “a”, “an”, and “the” include singular and plural referents, unless the context clearly dictates otherwise, as do “a number of”, “at least one”, and “one or more”. Furthermore, the words “can” and “may” are used throughout this application in a permissive sense (i.e., having the potential to, being able to), not in a mandatory sense (i.e., must). The term “include,” and derivations thereof, mean “including, but not limited to.” The term “coupled” means directly or indirectly connected. 
     The figures herein follow a numbering convention in which the first digit or digits correspond to the drawing figure number and the remaining digits identify an element or component in the drawing. Similar elements or components between different figures may be identified by the use of similar digits. For example,  108  may reference element “ 08 ” in  FIG. 1 , and a similar element may be referenced as  208  in  FIG. 2 . As will be appreciated, elements shown in the various embodiments herein can be added, exchanged, and/or eliminated so as to provide a number of additional embodiments of the present disclosure. In addition, as will be appreciated, the proportion and the relative scale of the elements provided in the figures are intended to illustrate certain embodiments of the present invention, and should not be taken in a limiting sense. 
     This disclosure is related generally to the field of marine geophysical surveying. For example, this disclosure may have applications in marine seismic surveying, in which one or more towed sources are used to generate wavefields, where seismic receivers (e.g., towed and/or on or near an ocean bottom) can receive direct seismic energy generated by the seismic sources and/or as affected by interaction with subsurface formations. In the present disclosure, such a seismic receiver can be at a far field measurement point to directly receive, detect, and/or measure the seismic energy of the wavefield generated by the seismic source. 
       FIG. 1  illustrates marine seismic surveying in which acoustic signals are emitted by a seismic source for recording by seismic receivers and subsequent processing and analysis to aid in characterizing the structures and distributions of features and materials underlying a solid surface of the earth.  FIG. 1  shows a domain volume  100  of the earth&#39;s surface.  FIG. 1  includes a Cartesian coordinate system  101  used to specify coordinate locations within the domain volume  100  with respect to three orthogonal, spatial coordinate axes x, y and z. The x coordinate uniquely specifies the position of a point in a direction substantially parallel to a front of the domain volume  100  and substantially parallel to a free surface  102  of the domain volume  100 . The y coordinate, although not shown due to the two-dimensional representation in  FIG. 1 , uniquely specifies the position of a point in a direction perpendicular to the x axis and substantially parallel to the free surface  102 . The z coordinate uniquely specifies the position of a point perpendicular to an xy plane of the domain volume  100 . 
     The domain volume  100  includes a solid volume  106  of sediment and rock below a solid surface  104  of the earth that, in turn, underlies a fluid volume  103  of water having the free surface  102 , for instance, within an ocean, an inlet or bay, or a large freshwater lake. The domain volume  100  shown in  FIG. 1  represents an example experimental domain for a class of observational and analytical techniques and systems referred to as marine seismic surveying. 
       FIG. 1  shows subsurface features of a subterranean formation in the lower portion of the domain volume  100 . While the fluid volume  103  can be explored, analyzed, and characterized with relative precision using many different types of methods and probes, including remote-sensing submersibles, sonar, and other such devices and methods, the solid volume  106  underlying the fluid volume  103  is comparatively more difficult to probe and/or characterize. Compared to the overlying fluid volume  103 , the solid volume  106  is significantly more heterogeneous and anisotropic, and includes many different types of features and materials of interest to marine surveying seismologists. For example, as shown in  FIG. 1 , the solid volume  106  may include a first sediment layer  107 , a fractured and uplifted first rock layer  108 , and an underlying second rock layer  110  below the first rock layer  108 . In certain cases, the second rock layer  110  may be porous and contain a significant concentration of liquid hydrocarbon that is less dense than the second-rock-layer material and that, therefore, rises upward within the second rock layer  110 . In the case shown in  FIG. 1 , the first rock layer  108  is not porous and, therefore, forms a lid that prevents further upward migration of the liquid hydrocarbon, which therefore pools in a hydrocarbon-saturated layer  112  below the first rock layer  108 . One goal of marine seismic surveying is to identify likely locations of hydrocarbon-saturated porous strata within volumes of the earth&#39;s crust underlying the solid surface of the earth. 
       FIG. 1  shows an example of a marine seismic survey vessel  114  equipped to carry out a series of marine seismic data acquisitions. In particular, the vessel  114  can tow one or more seismic source  116 , each of which can include a number of source elements, for example, as shown at  118 - 1  and  118 - 2 , a number of meters below the free surface  102 . In some situations, a seismic source can be towed across an approximately constant-depth plane below the free surface, although the depth can be varied as desired. The vessel  114  tows the seismic source  116  that produces pressure impulses at spatial and temporal intervals as the vessel  114  towing the seismic source  116  moves across the free surface  102 . The seismic source  116  can include cables containing power and data-transmission lines to which the source elements  118 - 1 ,  118 - 2  are connected at regular or varied intervals. The seismic source  116  and the vessel  114  can include sophisticated control electronics and/or data-processing facilities. 
     In various embodiments, a plurality of source elements and/or seismic sources can be configured to form a one-dimensional, two-dimensional, or three-dimensional array, which can be considered as a single seismic source. In some embodiments, a plurality of source elements  118 - 1 ,  118 - 2  can be defined as a source array by being arranged in a one-dimensional, two-dimensional, or three-dimensional configuration. 
     In  FIG. 1 , the source elements  118 - 1 ,  118 - 2  are shown to lie below the free surface  102 , with the source element positions correlated with overlying surface positions, such as a surface position  117  correlated with the position of source element  118 - 2 . In various embodiments, sources may be otherwise disposed in the fluid volume  103 . 
     Distal to the seismic source  116 ,  FIG. 1  shows a streamer  113  below the surface of the free surface  102  that includes a number of seismic receivers for detection of the portion of the sound wave emitted by a seismic source that is reflected back toward the surface. The seismic receivers in the streamer  113  thereby aid in characterizing the structures and distributions of features and materials underlying a solid surface of the earth. 
       FIG. 1  shows a two-dimensional representation of an expanding, spherical acoustic wavefront, represented by semicircles of increasing radius centered at the source element  118 - 1 , such as semicircle  119 , following a seismic impulse emitted by the source element  118 - 1 . The wavefronts are, in effect, shown in a vertical plane cross section in  FIG. 1 . The outward and downward expanding acoustic wavefield shown in  FIG. 1  can eventually reach the solid surface  104 , at which point the outward and downward expanding acoustic waves can partially reflect from the solid surface  104  and partially refract downward into the solid volume  106 , becoming elastic waves within the solid volume. In other words, in the fluid volume  103 , the waves are compressional pressure waves, or P-waves, the propagation of which can be modeled by an acoustic-wave equation. In the solid volume  106 , the waves can include both P-waves and transverse waves, or S-waves, the propagation of which can be modeled by an elastic-wave equation. Within the solid volume  106 , at each interface between different types of materials, at discontinuities in density, and/or in one or more of various other physical characteristics or parameters, downward propagating waves can be partially reflected and partially refracted, as at solid surface  104 . As a result, each point of a solid surface and within the underlying solid volume  106  can become a potential secondary source from which acoustic and elastic waves, respectively, may emanate upward toward seismic receivers in the streamer  113  in response to the wavefield emitted by the source element  118 - 1  and downward-propagating elastic waves resulting from the transmitted impulse. 
     In addition, as described further herein, a far field signature contributed to by each of the output impulses from the source elements  118 - 1 ,  118 - 2  of the seismic source  116  can be directly measured by a seismic receiver  123  positioned at a far field measurement point that records pressure variation over a period of time. As described herein, the position of the seismic receiver  123  is not limited to being directly under a source element and/or a center of a seismic source. That is, the seismic receiver  123  can be positioned at various offset angles relative to the source element and/or a center of a seismic source. 
       FIG. 2  illustrates a two-dimensional schematic representation of a source array and a far field measurement point. The two-dimensional representation shows a schematic representation of a number of source elements  218 . The number of source elements  218 , as described herein, is not limited to a particular number, as indicated by the source elements being numbered as  218 - 1 ,  2 ,  3 , . . . , n−1, and n. As described herein, the source elements  218  can be arranged in a source array  216 . As shown in  FIG. 2  in a one-dimensional arrangement, the source array  216  can have a source array center  221 . The source array center  221  can have its position determined as the geometrical center of the source array  216 . For example, the source array center  221  of the one-dimensional source array  216  shown in  FIG. 2  can be determined as the position of a midpoint between an edge of source element  218 - 1  that is farthest to the left and an edge of source element  218 - n  that is farthest to the right of the source array center  221 . As will be appreciated, geometric centers of two-dimensional and three-dimensional arrays can be similarly determined. The different sizes of the circles representing the source elements  218  in the source array  216  are proportional to different relative amplitudes of output impulses, as described further herein, produced by each source element, although some of the plurality of the source elements may produce the same or nearly the same amplitude of output impulse. In the illustrated example, source element  218 - 1  and source element  218 - n  have circles—representing amplitude—of similar size. 
     As shown in the two-dimensional representation in  FIG. 2 , the far field signature contributed to by each of the output impulses from the source elements  218  can be directly measured at a far field measurement point  225  by a seismic receiver, for instance, as shown by seismic receiver  123  in  FIG. 1 , that records pressure variation over a period of time. As shown, the position of each source element  218  in the source array  216  results in a variable corresponding distance  223 - r   1 , r 2 , r 3 , . . . , r n-1 , r n  from the source element  218 - 1 ,  2 ,  3 , . . . , n−1, n to the far field measurement point  225 . As shown in  FIG. 2 , each of corresponding distances  223  can be longer than a distance  224  from the source array center  221  to the far field measurement point  225  when the far field measurement point  225  is positioned directly under the source array center  221 . Hence, a wavefield emitted at the same time or nearly the same time by each of the source elements  218  would take varying lengths of time relative to each other to reach the far field measurement point  225 . Likewise, the travel time of wavefields emitted by each of the source elements  218  would differs relative to a hypothetical wavefield emitted from the source array center  221 , given a known acoustic wave velocity in the water between the source array  216  and the far field measurement point  225 . 
     Although in the description provided herein the measured far field signature is determined in the vertical direction, the vertical far field signature being commonly determined in practice, the measured far field signature can be measured from a far field measurement point in any other known direction, for instance from hypothetical far field measurement point  226 , and the calculations presented herein are equally valid. In various implementations, the far field signature measurement can be a direct recording of the source signature at some distance from the source, extracted from multi-channel seismic recordings, calculated and/or inferred from indirect near field measurements, or a combination of these implementations, among others. 
     The measured far field signature can be expressed as a linear superposition of so-called notional source element signatures from the n source elements in the source array  216 , as shown in  FIG. 2 . As such, the measured far field signature can be determined as follows in equation 1: 
                     FFS   measured     =       1     4   ⁢   π       ⁢       ∑     j   =   1     n     ⁢       1     r   j       ·       s   j     ⁡     (     t   -       r   j     v       )                     (   1   )               
Where s j  is the notional source element signature in time t from source element number j in the source array  216 , r j  is the distance from source element j to the far field measurement point  225  and v is the acoustic wave velocity in the water between the source array  216  and the far field measurement point  225 .
 
       FIG. 3  illustrates a two-dimensional schematic representation of a plane wave emitted at a given angle relative to a line between a center of the source array and the far field measurement point. As described herein, it may be desirable to estimate a far field signature signal at a location that does not correspond to a seismic receiver (e.g., at a location where there is no seismic receiver). An example of such a location where there may not be a seismic receiver (or where a seismic receiver is not intended to make far field signature signal detection) is a location that is not directly under the source array center  221  where placement of a seismic receiver, for instance as shown at  226  in  FIG. 2 , may be impractical and/or costly. As such, estimating a far field signature of a down-going wavefield, for example, as hypothetically measured at a simulated far field measurement point that is not directly under the source array, in lieu of measuring it directly, can provide valuable information regarding the characteristics of the down-going wavefield. Hence, it is useful to derive a far field signature at a different angle from the source array center  221  than the angle used for calculation of the measured far field signature. 
     Similar to  FIG. 2 ,  FIG. 3  shows a source array  316  of source elements  318 . As described with regard to  FIG. 2 , the different sizes of the circles representing the source elements  318  in the source array  316  are proportional to corresponding relative output amplitudes. A weighting factor w 1 , w 2 , w 3 , . . . , w n-1 , w n  is attributed to the output of each source element, as described further herein. Predetermined weighting factors, that is, “weights”, attributed to each source element are used in calculation of an impulse response in a given direction produced by the source array  316 , as described herein with regard to equation 2. The position of each source element  330  can be determined relative to an arbitrary origin on an x axis, for example, position x 0  at the source array center  321 , where x 0 =0, and/or a geometric center. A position  330 - x   1 , x 2 , x 3 , . . . x n-1 , x n  of each source element  318  is shown in  FIG. 3 . Although a one-dimensional source array is shown in  FIG. 3 , it will be appreciated that positions of source elements relative to positions of geometric centers of two-dimensional and three-dimensional source arrays can be similarly determined. 
     In various situations, it may be useful to determine an estimated far field signature in a second direction that differs from a first direction used for a measured far field signature. For example, as represented in two dimensions in  FIG. 3 , one can determine, as described herein, an estimated far field signature in a particular direction  331  that is offset at a particular desired angle  332 , denoted as θ i , relative to the direction of the measured far field signature  333 . 
     Multiple source elements in a source array can each produce an impulse substantially simultaneously to create multiple overlapping wavefields. However, in practical applications, overlapping multiple wavefields can be described, in a far field approximation and in a given direction, by a single plane wave  334  that runs through the geometric center of the source array, as shown in  FIG. 3 . 
     Because the measured far field signature and the estimated far field signatures have actual or simulated, respectively, far field measurement points in the far field region relative to the source array (e.g., as shown in  FIG. 4  along an arc  422  of fixed radius  424  centered at source array center  421 ), one can locally approximate an expanding wave front by a plane wave (e.g., a plane locally tangent to the wave front at the far field measurement point). When the situation is inverted by considering the emitting source array as a passive receiving array and propagating the same plane wave backwards, as is, to the x 0  position at the source array center  321 , a plane wave  334  results at the source array  316 , as shown in  FIG. 3 . 
     As such, pertaining to the two-dimensional representation shown in  FIG. 3 , direction  331  represents the direction offset angle  332  relative to a direction of the measured far field signature  333 , where both lines intersect at the source array center  321 . The plane wave  334  can be represented in two dimensions as a line perpendicular to the direction  331  representing the offset angle  332  in the plane of that line and the vertical line and passing through the source array center  321 . As a general case in three dimensions, whatever direction the line representing the offset angle is pointing relative to the measured far field signature, where both lines intersect at the source array center, the plane wave can be represented in three dimensions as a plane perpendicular to the line representing the offset angle and passing through the source array center. 
     Although the plane wave  334  is illustrated in  FIG. 3  as a static object passing through the position of the source array center  321 , the plane wave actually represents one moving plane wave that is sweeping through the source array at the given offset angle producing the time delays described herein. When the plane wave is not coming from the vertical direction, the plane wave will intersect each of the source positions  330 - x   1 , x 2 , x 3 , . . . , x n-1 , x n  at different times. The larger the offset angle  332 , the larger the time delay difference will be between when the plane wave  334  intersects each of the source positions. As described herein, the moving plane wave  334  is considered to intersect the position x 0  at the source array center  321  at a time corresponding to time zero (t=0). Hence, the plane wave  334  intersects the source positions  330 - x   n-1  and  330 - x   n  before time zero and the source positions  330 - x   1 , x 2 , and x 3  after time zero. When the plane wave is coming from the vertical direction, the plane wave will intersect each of the source positions  330 - x   1 , x 2 , x 3 , . . . x n  at the same time, which results in an impulse response, as described herein, from the vertical direction having a value of one. 
     Estimation of a far field signature in a second direction from a far field signature in a first direction, as described herein, is enabled by determination of a transfer function that can transform a measured far field signature in the first direction to the desired far field signature in the second direction. The transfer function should be independent from the measured far field signature itself to operate independently. A transfer function that operates as such is achieved through use of an impulse response calculated using input parameters derived from the actual source array configuration. The source array, which is an active system that can be, by analogy as described above, regarded as a passive linear system responding to plane waves at various incidence angles that, by analogy, are the offset angles of the wave front from the source array center. 
     Using this analogy with a passive linear system, the impulse response of the source array at a given offset angle θ is the weighted sum of the individual predetermined weights (w j ) of the output of each of the source elements (n) composing the source array, after the application of the appropriate time delays that are a function of the position of each of the source elements (n) in the source array and the offset angle θ (e.g., as measured from the vertical), as described herein, such as follows in equation 2: 
     
       
         
           
             
               
                 
                   
                     IR 
                     
                       θ 
                       ⁡ 
                       
                         ( 
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                   = 
                   
                     
                       
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                               - 
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                             ⁢ 
                             
                                 
                             
                             ⁢ 
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                             ⁢ 
                             
                               
                                 
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                                       x 
                                       0 
                                     
                                   
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                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                               
                                 v 
                                 ⁢ 
                                 
                                     
                                 
                               
                             
                           
                         
                       
                     
                     
                       
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     Further, as presented in equation 2, co is the angular frequency (e.g., 2πf, where f is the frequency of the impulse in cycles per second or hertz), x 0  is the position of the geometrical center of the source array relative to an arbitrary origin of the x-axis, x j  is the position of source element number j in the source array relative to the same arbitrary origin of the x-axis, v is the acoustic wave velocity in water between the source array and the far field measurement point, the appropriate time delays being a function of the difference between x j  and x 0  divided by v and taking into account the offset angle θ. 
     A weighted sum of the individual predetermined weighting factors w j  of each of the source elements n of the source array can be determined by utilizing the relative amplitudes of each of the output impulses produced by the source elements in the source array. For example, because the output pressure from an air gun serving as a source element is generally proportional to the cube root of a volume of the gun chamber of each air gun, the weighting factors w j  in equation 2 can be determined as follows in equation 3:
 
 w   j = 3 √{square root over (volume SOURCE     i   )}  (3)
 
For example, as shown in  FIGS. 2-3 , where the different sizes of the circles represent the source elements  218  in the source array  216  as being proportional to different relative amplitudes of an output impulse.
 
     In general, with a substantially equal operating pressure, an air gun with a relatively larger chamber volume can produce relatively larger peak amplitudes that contribute to a relatively higher weighting factor than produced by an air gun with a relatively smaller chamber volume. The air guns of a source array can be selected with different chamber volumes and/or arranged in a particular manner in order to generate a resulting far field seismic wave with, for example, a short and narrow signature in the vertical-downward direction and with a spectrum that is smooth and broad over a frequency band of interest. 
     In various embodiments, each of the plurality of source elements can be selected from a group of source elements that can include air guns, water guns, explosives, and/or vibratory devices, among others. In various embodiments, the selected source elements can all be of the same type or can be a mixture of different types as long as the following conditions are satisfied: the measured far field signature of the plurality of the source elements is a sum of notional source element signatures for each of the plurality of source elements, a position of each source element is known relative to the geometric center of the plurality of the source elements, and a relative output amplitude for a wavefield created in water is predetermined for each of the plurality of the source elements. 
     For the purpose of clarity, the sources in equation 2 and  FIGS. 2-3  are distributed along a one-dimensional axis. In practice, source elements in a source array can be distributed in two dimensions or in all three dimensions. Accordingly, equation 2 can be expressed as a function in two or three dimensions. 
     Continuing the analogy with a passive linear system, impulse response equation 2 is also known as a wave number response for the source array that describes the source array directivity, or response with offset angle θ, which is independent from the incident waveform. Staying with the one-dimensional expression, equation 2 can be rewritten as a wave number response, such as follows in equation 4: 
                     I   ⁢           ⁢     R     θ   ⁡     (     K   x     )           =         ∑     j   =   1     n     ⁢       w   j     ⁢     e         -   i     ⁢           ⁢       K   x     ⁡     (       x   j     -     x   0       )         ⁢                       ∑     j   =   1     n     ⁢     w   j                 (   4   )               
where
 
               K   x     =     ω   ⁢       sin   ⁢           ⁢   θ     v             
is termed a horizontal wave number.
 
     A transfer function that transforms a vertical far field signature (or a far field signature in any other measured direction) to any other desired direction can be derived, as described herein, by a ratio between the source array impulse response in the desired direction and the impulse response measured in the vertical, or other, direction, such as follows in equation 5: 
     
       
         
           
             
               
                 
                   
                     TF 
                     
                       θ 
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                       IR 
                       
                         
                           θ 
                           0 
                         
                         ⁡ 
                         
                           ( 
                           ω 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     The transfer function shown in equation 5 can be applied as a filter to a single measured far field signature to estimate far field signatures that would have been measured and/or extracted at the desired angles, such as follows in equation 6:
 
FFS θ =FFS measured   *tf   θ   (6)
 
where tf θ  is now in the time domain (e.g., tf θ (t)=FFT −1 (TF θ(ω) ), where FFT −1  denotes an inverse Fourier transform) and * denotes convolution.
 
       FIG. 4  illustrates a two-dimensional schematic representation of simulated far field measurement points at angular positions relative to the line between the center of the source array and the far field measurement point. As described herein, estimated far field signatures can be determined for a number of simulated far field measurement points. 
     Among various embodiments consistent with the present disclosure,  FIG. 4  shows a source array  416  of source elements  418  distributed in one dimension relative to a source array center  421 . The measured far field signature has an actual far field measurement point  425  and the estimated far field signatures have simulated far field measurement points  437 ,  439 , each at one position far from the source array  416  along an arc  422  of fixed radius  424  centered at source array center  421 . For example, as represented in two dimensions in  FIG. 3 , one can determine, as described herein, an estimated far field signature in a particular direction  331  that is offset at a particular desired angle  332 , denoted as θ i , relative to the direction of the measured far field signature  333 . The direction of the measured far field signature  333 , as shown in  FIG. 3 , can be a line corresponding to the fixed radius  424  between the source array center  421  at x 0  and the actual far field measurement point  425 , as shown in  FIG. 4 . Directions for estimated far field signatures in a number of particular directions  331  that are offset at particular desired angles  332 , denoted as θ i , as shown in  FIG. 3 , can be, for example, lines  435 - 1 ,  435 - 2  corresponding to the fixed radius  424  between the source array center  421  at x 0  and the positions of the simulated far field measurement points  437 ,  439 , as shown in  FIG. 4 . 
     Determination of at least one measured far field signature and a plurality of estimated far field signatures can provide a foundation for a system of equations, as described herein, for determination of notional source element signatures of each individual source element in a source array. For example, estimated far field signatures can be calculated in m−1 different directions, where m is greater than or equal to the number of source elements n in the source array for which the notional source element signatures are desired. Repeating the calculations in equations 2 to 6 for a number m different directions θ i  (where i=1, m−1) and including the measured far field signature yields a total of m far field signatures that can be used to build the system of equations. 
     The estimated far field signatures and the measured far field signature can be input together into equation (1), which is reproduced below for the sake of clarity, to provide a system of linear equations, such as follows in equation 7: 
                             FFS     θ   ⁢           ⁢   i       =       ⁢       1     4   ⁢   π       ⁢       ∑     j   =   1     n     ⁢       1     r   ij       ·       s   j     ⁡     (     t   -       r   ij     v       )               ,     i   =   1     ,     m   ;     m   ≥   n                     =       ⁢       1     4   ⁢   π       ⁢       ∑     j   =   1     n     ⁢         1     r   ij       ·       s   j     ⁡     (   t   )         ⋆     δ   ⁡     (     t   -       r   ij     v       )               ,     i   =   1     ,     m   ;     m   ≥   n                     (   7   )               
where r ij  is the distance from a source element&#39;s position j to a position of the i th  simulated far field measurement point, * denotes convolution, and δ is the Dirac delta function:
 
     
       
         
           
             
               δ 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         1 
                         , 
                       
                     
                     
                       
                         t 
                         = 
                         0 
                       
                     
                   
                   
                     
                       
                         0 
                         , 
                       
                     
                     
                       
                         t 
                         ≠ 
                         0 
                       
                     
                   
                 
                 , 
               
             
           
         
       
     
     For example,  FIG. 4  shows two simulated far field signature directions  435 - 1 ,  435 - 2 , where the first far field signature direction  435 - 1  is associated with a position of a first simulated far field measurement point  437  and the second far field signature direction  435 - 2  is associated with a position of a second simulated far field measurement point  439 . For example, the number of source elements  418 , as indicated by  418 - 1 ,  2 ,  3 , . . . n−1, and n, are shown with their respective distances, as denoted by r ij , from the position of the first simulated far field measurement point  437  and the position of the second simulated far field measurement point  439 . That is, the r ij  distances  436  from the source elements  418  to the position of the first simulated far field measurement point  437  are denoted as r 1,1 , r 1,2 , r 1,3 , r 1,n-1 , and r 1,n  and the r ij  distances  438  from the source elements  418  to the position of the second simulated far field measurement point  439  are denoted as r 2,1 , r 2,2 , r 2,3 , r 2,n-1 , and r 2,n . 
     In various embodiments, equation 7 can be converted into Fourier domain notation, such as follows in equation 8: 
                         FFS   i     ⁡     (   ω   )       =       1     4   ⁢   π       ⁢       ∑     j   =   1     n     ⁢       1     r   ij       ⁢     e       -   i     ⁢           ⁢   ω   ⁢       r   ij     v         ⁢       s   j     ⁡     (   ω   )               ,     
     ⁢     i   =   1     ,     m   ;     m   ≥   n               (   8   )               
In various embodiments, equation 7 can be converted into matrix notation, such as follows in equation 9:
 
FFS= G·s   (9)
 
where FFS=[FFS 1 (ω), FFS 2 (ω), FFS 3 (ω), . . . , FFS m (ω)] T  is a vector containing the m calculated and measured individual far field signatures at different angles, with T denoting transposition, s=[s 1 (ω), s 2  (ω), s 3  (ω), . . . , s n (ω)] T  is a vector containing the n unknown notional source element signatures from the n individual source elements in the source array, and G, as shown in Table 1, is a matrix of homogeneous 3D Green&#39;s functions for acoustic wave propagation from source element position j to simulated far field measurement point i. Equation 8 and equation 9 are essentially the same equation written in two different ways. They are both expressed in the Fourier (ω) domain. Equation 8 is actually a set of equations, which is represented by a more convenient matrix form in equation 9. For clarity, an expanded form of the Green&#39;s functions matrix (G) is shown in Table 1.
 
     
       
         
           
               
             
               
                 TABLE 1 
               
               
                   
               
             
            
               
                 
                   
                     
                       
                         G 
                         = 
                         
                           
                             1 
                             
                               4 
                               ⁢ 
                               π 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         11 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
                                           i 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                           
                                             r 
                                             11 
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         12 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
                                           i 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                           
                                             r 
                                             12 
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         
                                           1 
                                           , 
                                           
                                             n 
                                             - 
                                             1 
                                           
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
                                           i 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               1 
                                               , 
                                               
                                                 n 
                                                 - 
                                                 1 
                                               
                                             
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         
                                           1 
                                           , 
                                           n 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
                                           i 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               1 
                                               , 
                                               n 
                                             
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         21 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
                                           i 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                           
                                             r 
                                             21 
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         22 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
                                           i 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                           
                                             r 
                                             22 
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         
                                           2 
                                           , 
                                           
                                             n 
                                             - 
                                             1 
                                           
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
                                           i 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               2 
                                               , 
                                               
                                                 n 
                                                 - 
                                                 1 
                                               
                                             
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         
                                           2 
                                           , 
                                           n 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
                                           i 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               2 
                                               , 
                                               n 
                                             
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   ⋮ 
                                 
                                 
                                   ⋮ 
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   ⋮ 
                                 
                                 
                                   ⋮ 
                                 
                               
                               
                                 
                                   ⋮ 
                                 
                                 
                                   ⋮ 
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   ⋮ 
                                 
                                 
                                   ⋮ 
                                 
                               
                               
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         
                                           
                                             m 
                                             - 
                                             1 
                                           
                                           , 
                                           1 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
                                           i 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               
                                                 m 
                                                 - 
                                                 1 
                                               
                                               , 
                                               1 
                                             
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         
                                           
                                             m 
                                             - 
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                                           , 
                                           2 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
                                           i 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
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                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               
                                                 m 
                                                 - 
                                                 1 
                                               
                                               , 
                                               2 
                                             
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         
                                           
                                             m 
                                             - 
                                             1 
                                           
                                           , 
                                           n 
                                         
                                       
                                     
                                     ⁢ 
                                     
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                                         ⁢ 
                                         
                                             
                                         
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                                         ⁢ 
                                         
                                           
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                                                 - 
                                                 1 
                                               
                                               , 
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                                           v 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         
                                           m 
                                           , 
                                           1 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
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                                         ⁢ 
                                         
                                             
                                         
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                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               m 
                                               , 
                                               1 
                                             
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         
                                           m 
                                           , 
                                           2 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
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                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
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                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               m 
                                               , 
                                               2 
                                             
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         
                                           m 
                                           , 
                                           
                                             n 
                                             - 
                                             1 
                                           
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
                                           i 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                           
                                             r 
                                             
                                               m 
                                               , 
                                               
                                                 n 
                                                 - 
                                                 1 
                                               
                                             
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       
                                         r 
                                         mn 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         
                                           - 
                                           i 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                           
                                             r 
                                             mn 
                                           
                                           v 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                   
               
            
           
         
       
     
     Green&#39;s functions can be used in marine seismic surveying to calculate, in a given medium, an impulse response at some known seismic receiver location from a wavefield generated at some known source position. As used herein, the Green&#39;s functions may be calculated in a homogeneous medium because the propagation from the source to the seismic receiver occurs in a relatively homogeneous medium with a constant and known acoustic velocity. The above linear system of equations in the matrix notation G can be inverted to provide the notional source element signatures for each source element in the source array, such as follows in equation 10:
 
 s=G   −1 ·FFS  (10)
 
     The notional source element signatures derived as such can be used with linear superposition equation 1 to calculate an estimated far field signature at any desired simulated far field measurement position and/or at any desired offset angle. Although equations 1-6 can be used to directly calculate an estimated far field signature at any desired simulated far field measurement position and/or at any desired offset angle without having to invert for the notional source element signatures, it may be more efficient and/or practical to do so by first determining at least some of the notional source element signatures, as described herein. 
     In various embodiments, two wave forms can be defined as a function of time. For example, one of the wave forms can be termed “wave form 1”, which can be a measured vertical far field signature and the other wave form can be termed “wave form 2”, which can be a measured far field signature determined at an offset angle relative to the vertical. In that case, another function of time can be calculated, which can be termed “wave form 3”, which when subject to mathematical convolution with “wave form 1” can result in “wave form 2”. Formally, this convolution can written as “wave form 1”*“wave form 3”=“wave form 2”, where the * sign indicates the convolution. As described herein, “wave form 3” represents the transfer function in the time domain. 
     Convolution is a mathematical operation that can involve many multiplications and additions, which can be complex and/or time consuming. As an alternative, the convolution operation can be performed in the Fourier frequency domain, where the operation can be simpler and/or less time consuming than convolution in the time domain. As such, transformation of the wave forms from the time domain to the frequency domain results in: “Fourier(wave form 1)”×“Fourier(wave form 3)”=“Fourier(wave form 2)”, where × indicates the multiplication. As such, “Fourier(wave form 3)” now represents the transfer function in the frequency domain. 
     The real coefficients of a time function may transform into complex numbers in the Fourier frequency domain with one complex number for every frequency. Continuous signal functions of time can be decomposed into a sum of sinusoids with different periods that have certain amplitudes and time delays. 
     The Fourier transform is one such decomposition into frequencies. For a real function, Fourier coefficients are complex numbers as a function of the frequency. For example, frequency f i  can have a Fourier coefficient given by the complex number x+iy, where x is the real part and y is the imaginary part. The amplitude and the phase at frequency f i  are provided by √{square root over (x 2 +y 2 )} and tan −1  y/x, respectively. Accordingly, the amplitude and phase spectra of a signal are the above quantities as a function of frequency and graphs of these parameters can be formed to provide amplitude and phase spectra versus frequency plots. Such amplitude and phase spectra can provide a description of the corresponding time signal in the frequency domain. 
       FIG. 5  illustrates a variation in amplitude of a transfer function that transforms a vertical far field signature measurement to a 30 degree offset angle far field signature measurement as determined across a frequency spectrum. The plots  540  of a variation in amplitude  544  of the transfer function from vertical to 30° across the frequency spectrum  545  shown in  FIG. 5  represent a plot  543  of transfer functions determined from a measured far field signature in the vertical direction, below the geometric center of the source array, to a measured far field signature at a 30° offset angle from the vertical and a plot  549  of a variation in amplitude  544  of a theoretical transfer function from vertical to 30° across the frequency spectrum  545 . The quantities being compared in plots  540  are transfer functions from one measurement angle to another angle. 
     The theoretical transfer function can be calculated without measurement of any far field signatures by just using the source array geometry, as described herein. A theoretical transfer function curve is a plot of equation 5, as presented above. The theoretical transfer function curves  550 ,  659 ,  766 , and  873  shown in plots  549 ,  658 ,  765 , and  872  in  FIGS. 5-8  plot the amplitude and phase spectra as a function of frequency f (amplitude spectra in plots  549  and  765  and phase spectra in plots  658  and  872 ). The theoretical transfer function can be compared to the transfer function calculated as “Fourier(wave form 3)”, which is calculated from two measured far field signatures that have been modeled, as in plot  543 , using the same source array used to calculate the theoretical transfer function. 
     As shown on the vertical axis of plots  543  and  549 , amplitude in the context of a transfer function has no real units. The scale is relative amplitude. The decibel (dB) scale comes from plotting 20·log(amplitude) instead of just the amplitude. The dB scale is a logarithmic scale used in physics to express a ratio between two values of a physical quantity. The ratio being expressed on the vertical axis of plot  543  is: Fourier(wave form 3) representing the transfer function=Fourier(wave form 2)/Fourier(wave form 1). The ratio being expressed on the vertical axis of plot  549  is the corresponding theoretical ratio represented in equation 5. As shown on the horizontal axis of plots  543  and  549 , frequency spectrum  545  is the inverse, or dual, of time and has units of hertz (Hz), which indicates cycles/second. 
     The modeled and theoretical transfer function plots shown in  FIGS. 5-8  are derived from known standard source arrays. The modeled transfer function plots shown at  543 ,  653 ,  761 , and  868  are derived from calculation of far field signatures produced by two different air gun source arrays that are formed from one or more strings (i.e., one-dimensional source arrays) with a number of air guns on each string. A first source array having a single string with 10 individual air guns, shown at  541  in  FIG. 5 , is denoted as “Source Array 1”. A second source array having two strings with 20 individual air guns, shown at  542  in  FIG. 5 , is denoted as “Source Array 2”. As such, Source Array 2 has a total gun volume that is twice the total gun volume of Source Array 1 and each air gun in the two source arrays has a substantially equal output impulse amplitude. 
     In some embodiments, for each source array, a calculated Wiener filter can be used to convert a vertical far field signature into an off-vertical far field signature. As such, the calculated Wiener filter can be a reference transfer function used for plots  543 ,  653 ,  761 , and  868 . In some embodiments, the theoretical wave-number responses can be calculated with the source array geometries presented above. In some embodiments, the theoretical wave-number responses can be compared with the corresponding Wiener filter responses used as the reference transfer function. 
     In plot  543  of  FIG. 5 , the quantities being compared are amplitudes  544  of transfer functions from a measured vertical far field signature relative to a measured far field signature at a 30° offset angle, as determined across the frequency spectrum  545 , resulting from source arrays  541 ,  542  of different geometries. As shown in plot  543 , the plots for the two different source array geometries are largely indistinguishable at this scale because of overlap. However, for the sake of clarity, a zoom  546  is provided to show at  547  that the relative amplitude produced by the source array  541  having the single string is slightly higher than the amplitude produced by the source array  542  having two strings, as shown at  548 . Moreover, the shapes of the amplitude curves derived from measured far field signatures shown in plot  543  are notably similar to the shape of the theoretical transfer function curve  550  in the theoretical plot  549  based on the theoretical wave number response. 
       FIG. 6  illustrates a variation in phase of a transfer function that transforms the vertical far field signature measurement to a 30 degree offset angle far field signature measurement as determined across the frequency spectrum. The plots  651  of a variation in phase angle  654  of the transfer function from vertical to 30° across the frequency spectrum  645  shown in  FIG. 6  represent a plot  653  of transfer functions determined from a measured far field signature vertically, below the geometric center of the source array, to a measured far field signature at a 30° offset angle from the vertical and a plot  658  of a variation in phase angle  654  of a theoretical transfer function from vertical to 30° across the frequency spectrum  645 . The quantities being compared in plots  651  are transfer functions from one measurement angle to another angle. 
     The theoretical transfer function can be calculated without measurement of any far field signatures by just using the source array geometry, as described herein. The theoretical transfer function can be compared to the transfer function calculated as “PHASE(Fourier(wave form 3))”, which is calculated from two measured far field signatures that have been modeled, as in plot  653 , using the same source array used to calculate the theoretical transfer function. 
     The scale of the vertical axis of plots  653  and  568  expresses the phase angle  654  of the transfer functions described with regard to  FIG. 5  in degrees. As shown on the horizontal axis of plots  654  and  658 , frequency  645  is the inverse, or dual, of time and has units of Hz. 
     In plot  653  of  FIG. 6 , the quantities being compared are phase angles  654  of transfer functions from a measured vertical far field signature relative to a measured far field signature at a 30° offset angle, as determined across the frequency spectrum  645 , resulting from source arrays  641 ,  642  of different geometries, such as the geometries of the source arrays previously described with regard to  FIG. 5 . As shown in plot  653 , the plots for the two different source array geometries are largely indistinguishable at this scale because of overlap. However, for the sake of clarity, a zoom  655  is provided to show at  657  that the relative phase angle produced by the source array  641  having the single string is slightly lower than the phase angle produced by the source array  642  having two strings, as shown at  656 . Moreover, the shapes of the phase angle curves derived from measured far field signatures shown in plot  653  are notably similar to the shape of the phase angle curve  659  in the theoretical plot  658  based on the theoretical wave number response. 
       FIG. 7  illustrates a variation in amplitude of a transfer function that transforms the vertical far field signature measurement to a 60 degree offset angle far field signature measurement as determined across the frequency spectrum. The plots  760  of a variation in amplitude  744  of the transfer function from vertical to 60° across the frequency spectrum  745  shown in  FIG. 7  represent a plot  761  of transfer functions determined from a measured far field signature in the vertical direction, below the geometric center of the source array, to a measured far field signature at a 60° offset angle from the vertical and a plot  765  of a variation in amplitude  744  of a theoretical transfer function from vertical to 60° across the frequency spectrum  745 . The quantities being compared in plots  760  are transfer functions from one measurement angle to another angle. The theoretical transfer function can be compared to the transfer function calculated as “Fourier(wave form 3)”, which is calculated from two measured (e.g., known) far field signatures that have been modeled, as in plot  761 , using the same source array used to calculate the theoretical transfer function. 
     In plot  761  of  FIG. 7 , the quantities being compared are amplitudes  744  of transfer functions from measured a vertical far field signature relative to a measured far field signature at a 60° offset angle, as determined across the frequency spectrum  745 , resulting from source arrays  741 ,  742  of different geometries, such as the geometries of the source arrays previously described with regard to  FIG. 5 . As shown in plot  761 , the plots for the two different source array geometries are largely indistinguishable at this scale because of overlap. However, for the sake of clarity, a zoom  762  is provided to show at  763  that the relative amplitude produced by the source array  741  having the single string is slightly higher than the amplitude produced by the source array  742  having two strings, as shown at  764 . Moreover, the shapes of the amplitude curves derived from measured far field signatures shown in plot  761  are notably similar to the shape of the amplitude curve  766  in the theoretical plot  765  based on the theoretical wave number response. 
       FIG. 8  illustrates a variation in phase of the transfer function that transforms the vertical far field signature measurement to a 60 degree offset angle far field signature measurement as determined across the frequency spectrum. The plots  867  of a variation in phase angle  854  of the transfer function from vertical to 60° across the frequency spectrum  845  shown in  FIG. 8  represent a plot  868  of transfer functions determined from a measured far field signature vertically, below the geometric center of the source array, to a measured far field signature at a 60° offset angle from the vertical and a plot  872  of a variation in phase angle  854  of a theoretical transfer function from vertical to 60° across the frequency spectrum  845 . The quantities being compared in plots  867  are transfer functions from one measurement angle to another angle. 
     The theoretical transfer function can be calculated without measurement of any far field signatures by just using the source array geometry, as described herein. The theoretical transfer function can be compared to the transfer function calculated as “PHASE(Fourier(wave form 3))”, which is calculated from two measured far field signatures that have been modeled, as in plot  868 , using the same source array used to calculate the theoretical transfer function. 
     In plot  868  of  FIG. 8 , the quantities being compared are phase angles  854  of transfer functions from a measured vertical far field signature relative to a measured far field signature at a 60° offset angle, as determined across the frequency spectrum  845 , resulting from source arrays  841 ,  842  of different geometries, such as the geometries of the source arrays previously described with regard to  FIG. 5 . As shown in plot  868 , the plots for the two different source array geometries are largely indistinguishable at this scale because of overlap. However, for the sake of clarity, a zoom  869  is provided to show at  870  that the relative phase angle produced by the source array  841  having the single string is slightly higher than the phase angle produced by the source array  842  having two strings, as shown at  871 . Moreover, the shapes of the phase angle curves derived from measured far field signatures shown in plot  868  are notably similar to the shape of the phase angle curve  873  in the theoretical plot  872  based on the theoretical wave number response. 
     In addition to using source modeling tools to calculate far field signatures in different directions, as used for the modeled transfer function plots shown at  543 ,  653 ,  761 , and  868 , in some embodiments, so-called near field measurements can be used, when available, to approximate these results. 
       FIG. 9  illustrates a method flow diagram for estimation of a far field signature in a second direction from a far field signature in a first direction. At block  975 , the method can include determining an impulse response in a first direction and an impulse response in a second direction of a seismic source. As described herein, methods (e.g., determining, calculating, predicting, estimating, etc.) can be performed by a machine, for example, a computing device. In various embodiments, a seismic source can include any number of source elements as long as the geometry of each source element relative to the other source elements and/or relative to the geometric center of a source array of the source elements is known. Although various embodiments that have more than one source element have been described herein, derivation of the impulse response by equation 2 is equally valid using a single source element. That is, if n=1 is used in equation 2, then x j  will be equal to x 0 , so that the exponential will equal 1 and the relative weight of the single source element w j  will also equal 1. As such, the IR θ     i     (ω)  value will be 1 at any angle relative to the single source element. 
     As described herein, the method can include determining the impulse response of a plurality of source elements positioned in a source array. In various embodiments, a source array can be a one-dimensional, two-dimensional, or three-dimensional source array of the plurality of source elements at known positions. For example, each source element can be positioned at a known distance from a geometric center of the source array. In various embodiments, as described herein, the method can include determining a relative amplitude of an impulse produced by each source element in the source array. 
     As described herein with regard to equation 2, the method can include determining the impulse response in the first direction and the impulse response in the second direction by calculating the impulse responses based at least in part on the position of each source element and the determined relative amplitude of the impulse produced by each source element. As further described herein with regard to equation 2, the method can include determining the impulse response in the first direction and the impulse response in the second direction by calculating the impulse responses based at least in part on summing a number of source element weights including a time delay for the impulse of each source element. 
     At block  976 , the method can include determining a transfer function that transforms a far field signature of the seismic source in the first direction to a far field signature of the seismic source in the second direction based on corresponding impulse responses in the first direction and the second direction. In various embodiments, as described herein, the method can include measuring the far field signature in the first direction. In various embodiments, as described herein, the method can include determining a ratio between a calculated impulse response of the seismic source in the second direction and a calculated impulse response of the seismic source in the first direction to contribute to determining the transfer function. Accordingly, at block  977 , the method can include determining an estimated far field signature for the seismic source in the second direction based on the transfer function. 
     In accordance with a number of embodiments of the present disclosure, a geophysical data product may be produced from the far field signature of a source and/or data acquired in a marine seismic survey utilizing the source. Geophysical data may include, among various embodiments, an impulse response of a seismic source in a first direction, an impulse response of the seismic in a second direction, a far field signature of the seismic source in the first direction, a far field signature of the seismic source in the second direction based on corresponding impulse responses in the first direction and the second direction, an estimated far field signature for the seismic source in the second direction based on the transfer function, and marine seismic survey data acquired using the seismic source. A geophysical data product may be produced by obtaining at least a portion of the geophysical data and processing such geophysical data to generate the geophysical data product. 
     The geophysical data product may be accessed and/or stored on a non-transitory, tangible machine-readable medium suitable for importing onshore. The geophysical data product may be produced by acquiring geophysical data, processing the geophysical data offshore and/or processing the geophysical data onshore either within the United States or in another country. If the geophysical data product is produced offshore and/or in another country, it may be imported onshore to a facility in the United States. In some instances, once onshore in the United States, further data processing and/or geophysical analysis may be performed on the geophysical data product. In some instances, geophysical analysis may be performed on the geophysical data product offshore. For example, the transfer function that transforms the far field signature of the seismic source in the first direction to the far field signature of the seismic source in the second direction can be determined from data offshore to facilitate other processing of the measured data either offshore or onshore. As another example, the estimated far field signature for the seismic source in the second direction based on the transfer function can be determined from data offshore or onshore to facilitate other processing of the measured data either offshore or onshore. 
       FIG. 10  illustrates a diagram of a system for estimation of a far field signature in a second direction from a far field signature in a first direction. The system  1078  can include a subsystem  1080 , and/or a number of engines, such as far field signature engine  1081 , impulse response engine  1082 , transfer function engine  1083 , and/or estimate engine  1084 , and can be in communication with a data store, such as memory, via a communication link. The system  1078  can include additional or fewer engines than illustrated to perform the various functions described herein. The system can represent program instructions and/or hardware of a machine (e.g., machine  1185  as referenced in  FIG. 11 , etc.). As used herein, an “engine” can include program instructions and/or hardware, but at least includes hardware. Hardware is a physical component of a machine that enables it to perform a function. Examples of hardware can include a processing resource, a memory resource, a logic gate, etc. 
     The number of engines can include a combination of hardware and program instructions that is configured to perform a number of functions described herein. The program instructions (e.g., software, firmware, etc.) can be stored in a memory resource (e.g., machine-readable medium (MRM), computer-readable medium (CRM), etc.) as well as in a hard-wired program (e.g., logic). Hard-wired program instructions (e.g., logic) can be considered as both program instructions and hardware. 
     The far field signature engine  1081  can include a combination of hardware and program instructions that is configured to determine a measured far field signature in a first direction of a seismic source. The impulse response engine  1082  can include a combination of hardware and program instructions that is configured to determine an impulse response in a first direction and an impulse response in a second direction for the impulses emitted by the seismic source. The transfer function engine  1083  can include a combination of hardware and program instructions that is configured to determine a transfer function that transforms the measured far field signature in the first direction to a far field signature in the second direction based on the impulse responses in the first direction and the second direction. Accordingly, the estimate engine  1084  can include a combination of hardware and program instructions that is configured to estimate the far field signature of the seismic source in the second direction based on the transfer function. 
     In various embodiments, as described herein, the far field signature engine  1082  can further determine a measured far field signature in a first direction representing a superposition of impulse data representing detected impulses emitted by a plurality of source elements. As described herein, the impulse response engine  1082  can further determine an impulse response in a first direction and an impulse response in a second direction for the impulses emitted by the plurality of source elements. As described herein, the transfer function engine  1083  can further determine a transfer function that transforms the measured far field signature in the first direction to a far field signature in a second direction based on the impulse responses in the first direction and the second direction. In addition, as described herein, the estimate engine  1084  can further estimate the far field signature of at least one of the plurality of source elements in the second direction based on the transfer function. 
     In various embodiments, the impulse data can be input to the far field signature engine  1081  from the plurality of source elements positioned in a source array, as described herein, where each source element can emit an impulse substantially simultaneously. In various embodiments, as described herein, each of the source elements can be positioned at a known distance from a geometric center of the source array when emitting the impulse. In various embodiments, the system  1078  can include a far field measurement engine to send the impulse data representing impulses detected by a seismic receiver at least one far field measurement point to the far field signature engine  1081 . 
     In various embodiments, the system  1078  can include an estimated notional source element signature engine to, as described with regard to  FIG. 4 , determine, based on the transfer function, a plurality of estimated far field signatures in a plurality of directions for the plurality of source elements in addition to the measured far field signature and determine an estimated notional source element signature for each of the plurality of source elements based at least in part on the plurality of estimated far field signatures. As described herein, the estimated notional source element signature engine can determine the notional source element signature for each source element from a known position of each of the plurality of source elements to each of a number of simulated far field measurement points. 
       FIG. 11  illustrates a diagram of a machine for estimation of a far field signature in a second direction from a far field signature in a first direction. The machine  1185  can utilize software, hardware, firmware, and/or logic to perform a number of functions. The machine  1185  can be a combination of hardware and program instructions configured to perform a number of functions (e.g., actions). The hardware, for example, can include a number of processing resources  1186  and a number of memory resources  1187 , such as a MRM, CRM, or other memory resources  1187 . The memory resources  1087  can be internal and/or external to the machine  1185 . For example, the machine  1185  can include internal memory resources and have access to external memory resources, among other embodiments. The program instructions (e.g., machine-readable instructions (MRI), computer-readable instructions (CRI), etc.) can include instructions stored on the MRM to implement a particular function (e.g., an action). For example, a set of MRI can be executable by one or more of the processing resources  1186 . The memory resources  1187  can be coupled to the machine  1185  in a wired and/or wireless manner. For example, the memory resources  1187  can be an internal memory, a portable memory, a portable disk, and/or a memory associated with another resource (e.g., enabling MRI to be transferred and/or executed across a network, such as the Internet). As used herein, a “module” can include program instructions and/or hardware, but at least includes program instructions. 
     Memory resources  1187  can be non-transitory and can include volatile and/or non-volatile memory. Volatile memory can include memory that depends upon power to store information, such as various types of dynamic random access memory (DRAM), among others. Non-volatile memory can include memory that does not depend upon power to store information. Examples of non-volatile memory can include solid state media such as flash memory, electrically erasable programmable read-only memory (EEPROM), phase change random access memory (PCRAM), magnetic memory, optical memory, and/or a solid state drive (SSD), etc., as well as other types of MRM. 
     The processing resources  1186  can be coupled to the memory resources  1187  via a communication path  1188 . The communication path  1188  can be local or remote to the machine  1185 . Examples of a local communication path  1188  can include an electronic bus internal to a machine, where the memory resources  1187  are in communication with the processing resources  1186  via the electronic bus. Examples of such electronic buses can include Industry Standard Architecture (ISA), Peripheral Component Interconnect (PCI), Advanced Technology Attachment (ATA), Small Computer System Interface (SCSI), Universal Serial Bus (USB), among other types of electronic buses and variants thereof. The communication path  1188  can be such that the memory resources  1187  are remote from the processing resources  1186 , such as in a network connection between the memory resources  1187  and the processing resources  1186 . That is, the communication path  1188  can be a network connection. Examples of such a network connection can include a local area network (LAN), wide area network (WAN), personal area network (PAN), and the Internet, among others. 
     As shown in  FIG. 11 , the MRI stored in the memory resources  1187  can be segmented into a number of modules  1189 ,  1190 ,  1191 ,  1192  that when executed by the processing resources  1186  can perform a number of functions. As used herein, a module includes a set of instructions included to perform a particular task or action. The number of modules  1189 ,  1190 ,  1191 ,  1192  can be sub-modules of other modules. For example, the far field signature module  1189  can be a sub-module of the impulse response module  1190  and/or the far field signature module  1189  and the impulse response module  1190  can be contained within a single module. Furthermore, the number of modules  1189 ,  1190 ,  1191 ,  1192  can include individual modules separate and distinct from one another. Examples are not limited to the specific modules  1189 ,  1190 ,  1191 ,  1192  illustrated in  FIG. 11 . 
     Each of the number of modules  1189 ,  1190 ,  1191 ,  1192  can include program instructions and/or a combination of hardware and program instructions that, when executed by a processing resource  1186 , can function as a corresponding engine as described with respect to  FIG. 10 . For example, the far field signature module  1189  can include program instructions and/or a combination of hardware and program instructions that, when executed by a processing resource  1186 , can function as the far field signature engine  1081 , the impulse response module  1190  can include program instructions and/or a combination of hardware and program instructions that, when executed by a processing resource  1186 , can function as the impulse response engine  1082 , the transfer function module  1191  can include program instructions and/or a combination of hardware and program instructions that, when executed by a processing resource  1186 , can function as the transfer function engine  1083 , and/or the estimate module  1192  can include program instructions and/or a combination of hardware and program instructions that, when executed by a processing resource  1186 , can function as the estimate engine  1084 . 
     As described with regard to  FIG. 4 , the machine  1185  can include an impulse response module  1190  that can include instructions to determine a number of impulse responses of a seismic source, where the seismic source can include a plurality of source elements. In various embodiments, as described herein, the plurality of source elements can be positioned in a source array. The machine  1185  can further include a transfer function module  1191  that can include instructions to determine a number of transfer functions that transform a far field signature of the seismic source in a particular measured direction to a far field signature of the seismic source in a plurality of other directions based on the number of impulse responses of the seismic source. In addition, the machine  1185  can further include an estimate module  1192  that can include instructions to determine a plurality of estimated far field signatures for the seismic source in the plurality of other directions based on the transfer functions and determine an estimated notional source element signature for at least one source element in the seismic source based at least in part on the plurality of estimated far field signatures. In some embodiments, the transfer function module  1191  can include instructions to determine m−1 transfer functions for m−1 different directions for which estimated far field signatures can be calculated, for example, to invert for the notional source element signatures, as described herein. 
     The machine  1185  can further include instructions to determine the estimated notional source element signature for the at least one source element in the seismic source based at least in part on the plurality of estimated far field signatures in addition to at least one measured far field signature. The measured far field signature can be obtained by the machine  1185  including the far field signature engine  1081 , which can include instructions to determine at least one measured far field signature, as described herein. The total number of estimated far field signatures in addition to the at least one measured far field signature can be at least equal to a total number of source elements in the seismic source. 
     As further described with regard to  FIG. 4 , the machine  1185  can include instructions to determine an estimated notional source element signature for each source element based at least in part on a matrix of homogeneous three-dimensional Green&#39;s functions for wave propagation from a known position of each of the plurality of source elements to each of a number of simulated far field measurement points. In addition, the machine  1185  can include instructions to determine a revised estimated far field signature for the seismic source in one of the plurality of other directions based on input of a notional source element signature for a plurality of the source elements in the source array. 
     As described herein, determination of estimated far field signatures for a source array involves a determination of the positions of the source elements in the source array geometry. As such, the source array directivity is not dependent on the actual source element signatures themselves. That is, the directivity is determined by the actual source array geometry and source element weights w j  determined from air gun volumes. In various embodiments, determination of estimated far field signatures for a source array can be extended to use a plurality of measured far field signatures instead of just one. Using more than one measured far field signature can improve the accuracy of the estimated signatures and thus improve robustness. 
     Although specific embodiments have been described above, these embodiments are not intended to limit the scope of the present disclosure, even where only a single embodiment is described with respect to a particular feature. Examples of features provided in the disclosure are intended to be illustrative rather than restrictive unless stated otherwise. The above description is intended to cover such alternatives, modifications, and equivalents as would be apparent to a person skilled in the art having the benefit of this disclosure. 
     The scope of the present disclosure includes any feature or combination of features disclosed herein (either explicitly or implicitly), or any generalization thereof, whether or not it mitigates any or all of the problems addressed herein. Various advantages of the present disclosure have been described herein, but embodiments may provide some, all, or none of such advantages, or may provide other advantages. 
     In the foregoing Detailed Description, some features are grouped together in a single embodiment for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that the disclosed embodiments of the present disclosure have to use more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive subject matter lies in less than all features of a single disclosed embodiment. Thus, the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as a separate embodiment.