Abstract:
A target analysis method that includes the steps of: illuminating a target with acoustic waves; positioning a device at multiple acoustic vector sensing positions about the target in a scattered acoustic field of reflected waves to simultaneously measure acoustic pressure and particle velocity at each vector sensing position; converting using a Hilbert transform the measured acoustic pressures and particle velocities into a complex signal having active real and reactive imaginary vector component; computing respective active and reactive acoustic intensities at each vector components; and mapping field structure nulls being zero crossings of the active and reactive intensities to a bitmap representation of decomposed scattered target acoustic intensities of the target.

Description:
STATEMENT OF GOVERNMENT INTEREST 
     The invention described herein may be manufactured and used by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor. 
    
    
     CROSS REFERENCE TO OTHER PATENT APPLICATIONS 
     None. 
     BACKGROUND OF THE INVENTION 
     (1) Field of the Invention 
     The present invention is directed generally toward a system and method of use for analyzing reflected acoustic fields directed towards a resolution target underwater or in the atmosphere and more specifically toward a system and method of use for fully decomposing a reflected acoustic field into acoustic intensity vector components. 
     (2) Description of the Prior Art 
     Sonar systems are well known in the art for tracking and identifying submerged objects or objects in the atmosphere, for mine hunting, for precision underwater mapping and multistatic applications where there is more than one source and/or a receiver. 
     One sonar system example is a ship that can tow an array of sound-receiving hydrophones arranged in a passive towed array. The passive towed array, in conjunction with sound receiving and signal processing electronics, can detect sound in the water that may indicate the presence of an underwater target. In other arrangements, the ship can tow both the passive towed array and a towed acoustic projector, which together form a bi-static active sonar system. With this arrangement, the towed acoustic projector emits sound pulses. 
     Each sound pulse travels through the water, striking an object or target in the water, which in turn produces echoes. The echoes are received by the towed array of receiving hydrophones. Therefore, an echo indicates the presence of an underwater object and the direction from which the echo came; subsequently, indicating the direction of the underwater object. 
     In conventional bi-static active sonar systems, the towed acoustic projector is often deployed and towed separately from the towed array sound receiving hydrophones. A conventional towed acoustic projector typically includes a sound source mounted within a large rigid tow body. The conventional towed acoustic projector is large and heavy. 
     The towed acoustic projector is typically used to detect objects in deep water and at long ranges. Therefore, the acoustic projector is capable of generating sound having a high pressure level in order to enable the system to receive echoes from and to detect objects in the deep water at long ranges. 
     The towed array of receiving hydrophones are often deployed and recovered through a hull penetrator below the ship water line. In contrast, in part due to size and weight, the towed acoustic projector is deployed and recovered over the gunwale of the ship with winch and boom equipment. 
     U.S. Pat. No. 5,438,552 discloses a sonar system for identifying objects including a technique for providing a two-dimensional array of pixels, each one of the pixels representing the intensity of a signal at a predetermined range position and a predetermined cross-range position from a reference position and quantizing the intensity of each one of the pixels into one of a plurality of levels. The technique further includes comparing a distribution of the levels of pixels over a range scan at a cross-range position with the distribution of levels of pixels over a range scan at a different cross-range position to identify the existence of a foreign object such as a mine. 
     SUMMARY OF THE INVENTION 
     Accordingly it is a primary object and general purpose of the present invention to provide a sonar system and method capable of more precise target analysis through the use of vector components of sonar waves. 
     The target analysis method of the present invention comprises the steps of: illuminating a target with acoustic waves; positioning a device at multiple acoustic vector sensing positions about the target in a scattered acoustic field of reflected waves in order to simultaneously measure acoustic pressure and particle velocity at each vector sensing position; converting by using a Hilbert transform the measured acoustic pressures and particle velocities into a complex signal having active real and active imaginary vector components; computing active and reactive acoustic intensities at each vector component; and mapping field structure nulls being zero crossings of the active and reactive intensities to a bitmap representation of decomposed scattered target acoustic intensities of the target. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A more complete understanding of the invention and many attendant advantages thereto will be readily appreciated as the same becomes better understood by reference to the following detailed description when considered with the accompanying drawings wherein: 
         FIG. 1  is a schematic illustrating a sonar target analysis system according to the principles of the present invention; 
         FIG. 2  is a block diagram of steps used in the sonar target analysis according to the principles of the present invention; 
         FIG. 3  is a representative bitmap reflection of step  240  of  FIG. 2 ; 
         FIG. 4  is a results chart using a spherical target with an eighteen inch radius; and 
         FIG. 5  is a results chart using a spherical target with a twelve inch radius. 
     
    
    
     DESCRIPTION OF THE INVENTION 
     A sonar tracking system  10  is shown in  FIG. 1  to include a sonar transmission or sending system  20  which transmits a signal  34  of infinite-extent sonar waveform planes  22  toward a target  26  (a rigid sphere with a radius  33 ). From the θ=−τγ direction. A radius  32  (in meters) is a distance measured from the center of the target  26  and θ is the conical angle relative to the normal to the direction of plane wave propagation. The signal  34  illuminates the target sphere  26  and is reflected from the sphere to a back-scattered region  28  and a forward-scattered region  30 . 
     A plurality of acoustic vector sonar sensors or receivers  36  at different vector sensing positions receive a signal  42  from the reflection action. The sensors  36  then simultaneously measure acoustic pressure and particle velocity at each vector sensing position. The measured acoustic pressures and particle velocities are sent via sensing lines  38  to a computer  40  for signal analysis. 
     The steps of the method are shown in the block diagram of  FIG. 2  where the target  26  is illuminated by acoustic wave forms in step  200 . The acoustic vector sonar sensors  36  are positioned and measure the acoustic pressures and particle velocities at each vector sensing position in step  210 . The measured acoustic pressures and particle velocities are converted into a complex signal in step  220 . The active and reactive acoustic intensities are computed at each vector component in step  230 . A bitmap representation of decomposed scattered target acoustic intensities is created to include mapping field structure nulls in step  240 . 
     A representative bitmap structure of nulls is depicted in  FIG. 3 . Null mapping is the process of analyzing null structures in a scattered intensity field by a set of logical operations on sparse matrices constructed from separated real and imaginary components of pressure and particle velocity fields. 
     The details below describe the structure of a scattered acoustic field by using a separation of the complex intensity field into active and reactive components. Utilizing the dimensionless constant relating an incident (or illuminating) wave to a radius of a sphere 2Πa/λ (or ka); a particular scattered region of interest is in the region of interest is in the resonance range (ka˜3) where the scattered diameter is approximately equal to 1λ and creeping waves are diffracted around the scatterer and combine with pressure scattered by the illuminated surface. 
     The phase differences caused by the acoustic path lengths of the diffracted waves cause interference patterns that vary with frequency and scattered characteristics which include geometry and material properties. Through a power mapping of the real (active) and imaginary (reactive) complex acoustic intensity; the effects of the illuminated target characteristics on the total acoustic energy fields are characterized. 
     Of further interest is the understanding of how the scattered vector field characterization extends and transitions into the far-field. In the preferred embodiment, fully-developed scattered intensity fields from simple rigid spheres are examined. Numerical and measured results have been studied and modeling will extend to elastic and fluid-filled boundary conditions. 
     The following embodiment is a simple scattering case for the rigid target sphere  26  of a radius  32  as shown in  FIG. 1 . However, the method is not limited to a rigid spherical shape target. This description includes derivations for fluid-filled thin wall spheres and evacuated spherical shells. 
     Cylinder mapping is another example. The target only need be on the size order proportional to approximately one wavelength of the illumination frequency such as in the resonance region, where the acoustic wavelength of the illumination field and scatterer size (2*pi/lambda)*(radius of target) is on the order of two or three. 
     The target sphere  26  illuminated by infinite-extent plan waves  22  from the −π direction. Equation (1) represents the incident pressure 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       i 
                     
                     ⁡ 
                     
                       ( 
                       
                         R 
                         , 
                         θ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       P 
                       i 
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           ⅈ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           kr 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     Relating the scattering problem to that of a spherical radiator, the Junger reference expressed the incident pressure field of Equation (1) as a series of Legendre functions as Equation (2), 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       i 
                     
                     ⁡ 
                     
                       ( 
                       
                         R 
                         , 
                         θ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       P 
                       i 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                             + 
                             1 
                           
                           ) 
                         
                         ⁢ 
                         
                           i 
                           n 
                         
                         ⁢ 
                         
                           
                             P 
                             n 
                           
                           ⁡ 
                           
                             ( 
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             j 
                             n 
                           
                           ⁡ 
                           
                             ( 
                             
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               R 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Where P n (cos θ) and j n (kR) are respectively the Legendre polynomial and spherical Bessel function of the first kind. F M, Junger, D. Feit, “Sound, Structures and Their Interactions”, copyright 1993 by Acoustic Society of America, Chapter 10. 
     The general form of the scattered pressure field from a rigid sphere (∞ denotes the rigid boundary condition of infinite acoustic impedance) is then Equation (3): 
     
       
         
           
             
               
                 
                   
                     
                       
                         p 
                         
                           s 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ∞ 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           R 
                           , 
                           θ 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         - 
                         
                           P 
                           i 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           n 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                               + 
                               1 
                             
                             ) 
                           
                           ⁢ 
                           
                             i 
                             n 
                           
                           ⁢ 
                           
                             
                               P 
                               n 
                             
                             ⁡ 
                             
                               ( 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             b 
                             n 
                           
                           ⁢ 
                           
                             
                               h 
                               n 
                             
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 R 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
                       b 
                       n 
                     
                     = 
                     
                       
                         
                           j 
                           n 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             a 
                           
                           ) 
                         
                       
                       
                         
                           h 
                           n 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             a 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     The symbols h′ n  and j′ n  are the Hankel and Bessel function of the first kind and their derivatives, respectively. 
     The vector field describing the complex scattered acoustic intensity is given by Equation (4): 
     
       
         
           
             
               
                 
                   
                     J 
                     = 
                     
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         p 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           u 
                           * 
                         
                       
                       = 
                       
                         I 
                         + 
                         
                           i 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Q 
                             4 
                           
                         
                       
                     
                   
                   , 
                   
                     p 
                     = 
                     
                       
                         p 
                         i 
                       
                       + 
                       
                         p 
                         
                           s 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ∞ 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     The symbols p and u* are respectively the complex acoustic scalar pressure and conjugated particle velocity. I is the scattered active Intensity. Q is the scattered reactive intensity. 
     The particle velocity field is related to the gradient of the scalar pressure field by the momentum equation, which in axis-symmetric spherical coordinates becomes Equation (5): 
     
       
         
           
             
               
                 
                   
                     ∇ 
                     
                       p 
                       ⁡ 
                       
                         ( 
                         
                           R 
                           , 
                           θ 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     i 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ck 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         u 
                         ⁡ 
                         
                           ( 
                           
                             R 
                             , 
                             θ 
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     The scattered velocity field u is found in Equation (6) by combining Equations (2), (3), and (5): 
     
       
         
           
             
               
                 
                   
                     u 
                     ⁡ 
                     
                       ( 
                       
                         R 
                         , 
                         θ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         P 
                         i 
                       
                       
                         p 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         c 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         n 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                             + 
                             1 
                           
                           ) 
                         
                         ⁢ 
                         
                           i 
                           
                             n 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           { 
                           
                             
                               
                                 
                                   
                                     P 
                                     n 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       θ 
                                     
                                     ) 
                                   
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       
                                         j 
                                         n 
                                         ′ 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           k 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           R 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         b 
                                         n 
                                       
                                       ⁢ 
                                       
                                         
                                           h 
                                           n 
                                           ′ 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             k 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             R 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                               ⁢ 
                               
                                 r 
                                 ^ 
                               
                             
                             - 
                             
                               
                                 
                                   P 
                                   n 
                                   ′ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                   
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     R 
                                   
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       
                                         j 
                                         n 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           k 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           R 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         b 
                                         n 
                                       
                                       ⁢ 
                                       
                                         
                                           h 
                                           n 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             k 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             R 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           } 
                         
                         ⁢ 
                         
                           θ 
                           ^ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     For analysis, it is desirable to maintain the separable spatial vector and complex components of the time: averaged scattered acoustic intensity field as shown in Equation (7) where J (R,θ) is the complex scattered acoustic intensity, I (R,θ) is the real part of J, and Q (R,θ) is the imaginary part of J. 
     
       
         
           
             
               
                 
                   
                     J 
                     ⁡ 
                     
                       ( 
                       
                         R 
                         , 
                         θ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       I 
                       ⁡ 
                       
                         ( 
                         
                           R 
                           , 
                           θ 
                         
                         ) 
                       
                     
                     + 
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Q 
                         ⁡ 
                         
                           ( 
                           
                             R 
                             , 
                             θ 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     I 
                     ⁡ 
                     
                       ( 
                       
                         R 
                         , 
                         θ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       R 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       e 
                       ⁢ 
                       
                         { 
                         p 
                         } 
                       
                       ⁢ 
                       R 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       e 
                       ⁢ 
                       
                         { 
                         
                           u 
                           * 
                         
                         } 
                       
                     
                     - 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       I 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       m 
                       ⁢ 
                       
                         { 
                         p 
                         } 
                       
                       ⁢ 
                       I 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       m 
                       ⁢ 
                       
                         { 
                         
                           u 
                           * 
                         
                         } 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     Q 
                     ⁡ 
                     
                       ( 
                       
                         R 
                         , 
                         θ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       Im 
                       ⁢ 
                       
                         { 
                         p 
                         } 
                       
                       ⁢ 
                       Re 
                       ⁢ 
                       
                         { 
                         
                           u 
                           * 
                         
                         } 
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       Re 
                       ⁢ 
                       
                         { 
                         p 
                         } 
                       
                       ⁢ 
                       Im 
                       ⁢ 
                       
                         { 
                         
                           u 
                           * 
                         
                         } 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Empirical data was obtained from Equation (7) with an eighteen inch (0.4572 m) diameter rigid spherical scatterer  200  (radius=0.2294 m), 716.4 Hz incident plane wave  204  from the −π direction, the complex acoustic scalar pressure ρ=1.21 kg/m 3 , c=343 m/s in air, which represents ka=3.0, where c is the sound speed of acoustic propagation in air. The total power in the scattered instantaneous acoustic intensity field (normalized by ρc) can be obtained as well as the spatial and complex decomposition of the scattered intensity field components. 
     An experiment exemplifies the viability of extracting field structures that can be seen in the spatial and complex separated components of the scattered acoustic intensity field from direct measurements. 
     In order to create a scattered field where ka=3, spheres milled from oak with diameters of twelve inches (30.48 cm) and eighteen inches (45.72 cm) were illuminated by a source at 3.6 m of 1000 Hz and 716.4 Hz, respectively. Measurements were collected using an acoustic vector sensor probe with radial velocity recorded on the axial velocity sensor, angular velocity component on the “y” velocity sensor, and scalar pressure on the microphone (See  FIG. 4 ). 
     In order to ensure that the spherical coordinate velocity components could be measured using orthogonal sensors, the probes were aligned to the equator of the spheres while maintaining the probe axis normal to the surface of the scatterer. A self-leveling laser guide was used to align the geometry. Angles referenced to the maximum response angle (MRA) were hand-measured using distances to a stationary target offset in the test cell. 
     The raw sensor data was collected through a signal conditioner with gain set to “high” and corrections turned “off”. At each position, 20,000 samples at a rate of 5120 Hz (approximately four seconds) were acquired (known herein as a data record). Samples were taken primarily along an arc in the forward scattered region. Data was also collected in the forward scattered region for both spheres. 
     The data samples (time series) from the sensors were processed by first applying phase and sensitivity calibrations, and then filtered to remove 60 Hz noise components. The real signals x(t) were then combined with the Hilbert transform to create the analytic signal 
                       x   ~     ⁡     (   t   )       =         x   ⁡     (   t   )       +     i   ⁢       h   ^     ⁡     (     x   ⁡     (   t   )       )           =       Re   ⁢     {     x   ⁡     (   t   )       }       +       i   ⁢   I     ⁢           ⁢   m   ⁢     {       x   ~     ⁡     (   t   )       }                   (   8   )               
where h(x(t)) is the Hilbert transform of x(t). The analytic signals of pressure and velocity were then used to compute the four components of the scattered intensity field in spherical coordinates. For each acquired data record (n) of 20,000 samples, at a position R n θ n  the spatial and complex separated components of the time-averaged scattered acoustic intensity field are computed in Equation (9).
 
     
       
         
           
             
               
                 
                   
                     I 
                     ⁢ 
                     
                       
                         r 
                         ^ 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             R 
                             n 
                           
                           , 
                           
                             θ 
                             n 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     mean 
                     ⁢ 
                     
                        
                       
                         Re 
                         ⁢ 
                         
                           { 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               
                                 pu 
                                 green 
                                 * 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     R 
                                     n 
                                   
                                   , 
                                   
                                     θ 
                                     n 
                                   
                                 
                                 ) 
                               
                             
                           
                           } 
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     I 
                     ⁢ 
                     
                       
                         θ 
                         ^ 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             R 
                             n 
                           
                           , 
                           
                             θ 
                             n 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     mean 
                     ⁢ 
                     
                        
                       
                         Re 
                         ⁢ 
                         
                           { 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               
                                 pu 
                                 red 
                                 * 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     R 
                                     n 
                                   
                                   , 
                                   
                                     θ 
                                     n 
                                   
                                 
                                 ) 
                               
                             
                           
                           } 
                         
                       
                        
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     Q 
                     ⁢ 
                     
                       
                         r 
                         ^ 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             R 
                             n 
                           
                           , 
                           
                             θ 
                             n 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     mean 
                     ⁢ 
                     
                        
                       
                         Im 
                         ⁢ 
                         
                           { 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               
                                 pu 
                                 green 
                                 * 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     R 
                                     n 
                                   
                                   , 
                                   
                                     θ 
                                     n 
                                   
                                 
                                 ) 
                               
                             
                           
                           } 
                         
                       
                        
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     Q 
                     ⁢ 
                     
                       
                         θ 
                         ^ 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             R 
                             n 
                           
                           , 
                           
                             θ 
                             n 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     mean 
                     ⁢ 
                     
                        
                       
                         Im 
                         ⁢ 
                         
                           { 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               
                                 pu 
                                 red 
                                 * 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     R 
                                     n 
                                   
                                   , 
                                   
                                     θ 
                                     n 
                                   
                                 
                                 ) 
                               
                             
                           
                           } 
                         
                       
                        
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
       FIG. 4  illustrates the agreement of the structure in the forward-scattered region between the model and field measurements for time-average steady-state acoustic intensity for the eighteen inch sphere illuminated by 716 Hz plane waves (ka=3). The structure of most components was observed to agree with the exception of the reactive radial intensity. 
       FIG. 5  illustrates from testing of the twelve inch sphere. In both  FIGS. 4 and 5 , the horizontal axis represents the angle in degrees relative to zero which is the direction of the illumination wave in the forward region. The vertical axis represents pressure and intensity measured in decibels. 
     In both the twelve inch and eighteen inch sphere analysis, an inspection of the measured reactive radial intensity component indicates that the data matches the model more precisely at a slightly closer range than measured. Given the sensitivity to range for these particular features in the forward scattered region, the range differences noted can be accounted for by small misalignment in both altitude and attitude of the sensor. 
     The results for testing of the eighteen inch sphere in  FIG. 4  include the active radial intensity  430 , the active angular intensity  420 , the reactive radial intensity  400 , the reactive angular intensity  410 , and the pressure  440 . 
     The results for testing of the twelve inch sphere in  FIG. 5  include the active radial intensity  530 , the active angular intensity  520 , the reactive radial intensity  510 , the reactive angular intensity  500 , and the pressure  540 . 
     The analytical model and experimental data illustrates the ability to extract scattered field features from direct measurement of the time-averaged acoustic intensity field for simple rigid objects. 
     It will be understood that many additional changes in the details, materials, steps and arrangement of parts, which have been herein described and illustrated in order to explain the nature of the invention, may be made by those skilled in the art within the principle and scope of the invention as expressed in the appended claims. 
     The foregoing description of the invention has been presented for purposes of illustration and description only. It is not intended to be exhaustive nor to limit the invention to the precise form disclosed; and obviously many modifications and variations are possible in light of the above teaching. Such modifications and variations that may be apparent to a person skilled in the art are intended to be included within the scope of this invention as defined by the accompanying claims.