Wong, How-Kin

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A NOTE ON THE APPLICATION OF IMAGE INTERFERENCE TO BOTTOM CONTOURINGby How-Kin W o n g  W oods Hole Oceanographic Institution W oods Hole, MassachusettsINTRODUCTIONImage interference in the sea is a well known phenomenon produced by phase interference between the acoustic waves transmitted to the sea bottom via a direct path and those via a path which includes one surface reflections. The effect has been extensively studied by a number o f in­vestigators [1-8] and is also discussed in several general works on under­water acoustics [9-11]. Various approximation methods have been applied to the analysis o f this phenomenon. The simplest is the isovelocity approximation used by the Research Analysis Group o f the U. S. National Research Council [1 ] and by C h e s t e r m a n  et al. [8 ], More refined theories include those o f Y o u n g  [2 ], S a n d e r s  and S t e w a r t  [3 ], and P e d e r s e n  [6 ]. Y o u n g  noted that it is the difference in propagation time along the two possible ray paths that determines the relative phases o f the interfering waves. He thus justified the replacement o f the curved rays propagating with a depth-dependent velocity by straight ones, and developed an equation for the transmission anomaly in which linear velocity gradients were taken into account. St e w a r t  made calculations using the actual curved ray paths for the case o f an almost isothermal sea, and in the analysis o f P e d e r s e n  the additional effect o f refraction on the pressure amplitudes o f the interfering waves was also taken into consideration.Since image interference patterns o f underwater sound are very sensitive to water depth and other geometric parameters, they offer a variety o f potential applications. By measuring the fringe position, C h r i s t e n s e n  [12] was able to deduce the physical size and location o f the scatterers responsible fo r  the interference. H a i n e s  [13 ], in a review  article, first pointed out the possibility o f making use o f this phenomenon to derive bottom contours from  the facsim ile records o f a side-scan sonar. However, it was C h e s t e r m a n  et al. [8 ] who made the first quantitative studies on bottom contouring by this method by plotting the reciprocal range for successive fringe maxima (or m inima) against the fringe order. From  the slope o f the resulting graphs, information on exact bottom depthsand hence on bottom contours was deduced. W here the bottom slope is constantly varying, the interference condition itself was used. The greatest advantage o f this method is that by making a single traverse, continuous coverage of a sea-bottom strip a few  hundred yards wide is achieved.The purpose o f this note is to demonstrate that in the application of underwater image interference to bottom contouring, the assumption o f isovelocity propagation (by C h e s t e r m a n  et al. [8 ], for example) is definite­ly incompatible with the accuracy desired. Temperature and pressure refraction must be taken into account, and the effect o f other factors (which w ill be mentioned later) should also be considered.Let us now re-examine the basis o f the analysis o f C h e s t e r m a n  et al.[8 ], Using the notations o f figure 1, the path difference between the interfering rays at B isi.e.,I f  we putA =  (SC +  CB) -  SB =  IB -  SBA =  [(D +  t )2 + a:2 ]1/2 — [(L) -  I ) 2 + x 2] 1'2 (1)A =  m X/2and recall that there is a phase reversal on reflection at the surface because o f the much lower acoustic impedance o f the reflecting medium [14], then we see that interference is constructive when m  assumes an odd, integral value and is destructive when m  is even, integral.From  equation (1), we haveD2 x2/fnA\2 / m\\2(— ) ‘ - ( t )=  I . (2)Thus, points for which the ray paths differ by a constant lie on a hyperbola w ith the sound source at one focus. It is customary to expand equation (2) and neglect terms o f order (X/t), 0./D) and higher. This gives the linear relation o f_  4tDwhich is an adequate approximation since it only introduces errors that are considerably less than 1 % for ranges greater than (say) 7 times the water depth.A second approximation used is the identification o f the horizontal and slant ranges (see [8 ] ) .  Thus, with the last equation, this gives the fam iliar relation ofi» 4tl)K = —  (3)m A vF ig . 1. —  Schematic representation of the geometry fo r image interference S =  source o f acoustic energy submerged at depth t I =  image sourceR =  slant range from  S to point B on sea bottom  x =  horizontal range from  S to B A =  point on sea bottom, slope o f AB  is aIn figure 2, the range difference is plotted against water depth for various slant ranges. It can be seen that this approximation is quite inadequate for seabed contouring applications where measurements to the order o f a yard are involved.THE ANALYSISThe first step in the accurate analysis o f side-scan interferograms is the determination of the fringe order m. This can be done by tracing the fringe to a flat seabed or to a place where a precise depth measurement has been made, and then proceeding w ith equation (3). W ith  m  thus determined, the position of interference fringes can be used to deduce theWATER DEPTH (D ) IN FEETF ig . 2. —  V a ria t io n  o f  the d ifference betw een  horizon tal and  slant rangesw ith  w a te r depth.R  =  slant range  X  — horizon ta l rangewater depth at specific points. Alternately, if a seabed o f constant slope a is assumed, we have (figure 1)D =  D0 -  a (x  -  x 0) (4)Let us for the moment consider isovelocity propagation only and identify R with x.From (3), we havem =  i £ 2  -  ( î l ' j  Dq ~  a (g  ~ r 0)XR V X / R/ 4 1  ̂ D0 -  a (R  -  x 0) 4 1 _  , , 1n r ) ----------it---------------x +orm = i i / (Do + a , o ) i - l i t e  (5)where f  =  frequency o f the projected underwater sound and c =  its velocity o f propagation.Thus, a plot o f /71 versus 1/R is a straight line, from  the slope and intercept o f which the depth at any range may be calculated.I f  we now relax equating R and x, the m  versus 1/R plots w ill no longer be straight lines. They w ill become curves with a slope given by :dm , 4 » / r D a2 ( D - Q ( t - D 0 - a * n) - 2 a * n ( D - 2 D n) 1 d (l/ R ) c L Do +  “ * o +  a2 (D — t) +  (D — D0) — a x 0 J (6)The first two terms o f (6) are identical with the corresponding ones o f (5), but the last is a correction term arising from the differences between R and x. The importance o f this term depends on the values o f range and bottom slope in question, but its order o f magnitude may well be comparable to the term just preceding it.Let us next relax the isovelocity propagation condition and take, as a better approximation, Y o u n g ’s theory o f acoustic refraction. According to this theory, the interference pattern is given by [3 ] :R-instead of equation (3). Here Ac =  c 0 — Cj- = -  f *  l iy  Cn t  “'o V (y)i  _____i _  ^cx D — t  vP dy(.y)where g is the depth measured from  the sea surface downwards and v(y) gives the vertical variation of the sound velocity function.Hencec0 =  harmonic mean velocity between the surface and the trans­ducer depthct =  harmonic mean velocity between the layers at the transducer depth and at the bottomc ~  c0 ~  ClThis depth— range relationship together w ith the isovelocity approxima­tion of equation (3) is plotted in figure 3 for the typical case considered by C h e s t e r m a n  et al. [8 ], The dotted lines correspond to isovelocity conditions, and the solid lines are for upward refraction due to pressure alone as predicted by Y o u n g ’s theory.It can be seen (figure 3) that at a range o f 2 400 feet, the difference in deduced depth between the two sets o f curves can be as large as 15 feet. Also, at a water depth o f 50 feet (say), pressure refraction alone increases the range o f the interference minimum o f order 6 by 1 000 feet and that o f fringe order 12 by some 100 feet. Thus, for the application o f image interference to bottom contouring, the isovelocity theory is clearly in­adequate.É-ltùES[jhzsaHcuu3QRANGE (R ) IN FEETF ig . 3. —  P o s it io n s  o f  in terference  m in im a  fo r  a tran sd u cer dep th  o f  14.5 feet.The dotted lines are fo r  isovelocity conditions, and the solid lines fo r  upward  refraction due to pressure alone as predicted by  Y o u n g ’ s theory.Frequency / =  48 kcpsAverage sound velocity c =  4 850 feet per second Uniform  velocity gradient =  0.0182 (foot per sec) per foot.fr o m  equations (7) and (4 ), i f  we again identify R with x, we can deduce that :dm = ^ r Dd (1/R) c L 0+  a x 0 +x 2 Ac + a x 3 Acc (t +  D) c (t +  D) ] (8)Hence refraction introduces two additional correction terms to the slope o f the m  versus 1/R curves. For a positive velocity gradient, these curves are bent upwards away from  the isovelocity straight line, and for a negative velocity gradient, they are bent downwards.OTHER COMPLICATING FACTORSThere are other complicating factors. Heaving, yaw ing and pitching o f the transducer ( i f  placed within a towed body which is not stabilized) and the action o f waves would constantly alter the phase difference betweenDEPTH IN METRESTEMPERA TU RE IN DEGREES C.0 5 10 15 200L1 0 0_L_ 1200SCALE IN YARDSF ig . 4. —  Im age interference o f direct ray and ray  reflected at the boundary  between the isotherm al layer and the thermocline.the direct and surface-reflected rays, and the depth dependence of tidal currents would set up velocity gradients. The effects o f these factors must be evaluated, and taken into account if they are likely to be significant. More refined theories than that o f Y o u n g  are probably not necessary, except where considerably great depths or long ranges or severe refraction conditions are encountered. Experimentally, a method o f stabilizing the fringes would be essential. This can be done by placing an artificial reflector (o f dimensions say 3 feet by 20 feet) just underneath the surface waves, so that a stable reflected beam is assured.IMAGE INTERFERENCE OF OTHER ORIGINThere are other physical conditions which can give rise to interference patterns and must be recognized during analysis. F igure 4 shows a fringe system (accompanied by severe downward acoustic refraction) together with the bathythermogram taken just prior to the survey. The isothermal layer extended through the first 5 metres (5.5 yards) o f water, and is succeeded by a steep thermocline. The transducer was initially towed at 8 yards. A t the point A, it was raised to 4 yards; at B it was again sub­merged to 8 yards. The disappearance o f the interference fringes between points A  and B showed that the pattern in this case was caused by inter­ference between the direct ray and the ray reflected at the interface o f the isothermal layer and thermocline. The persistance o f the fringes indicated that abrupt changes in the thickness o f the isothermal layer were probably absent. The acoustic beam was pointing one degree up for this traverse, thus rather favoring image interference.F iner interference fringes superimposed on a pattern produced in the normal w ay have been reported to exist in a stratified ocean bottom [15]. These are form ed as a result o f phase interference between the direct ray and the ray which has penetrated the top layer o f the bottom and suffered one reflection at the interface o f the bottom strata. Short-period irregular­ities resembling interference fringes (also superimposed on the main pattern), as are seen in figures 33 and 35 o f reference [8 ], can also be due to heaving o f the transducer, which changes the path lengths o f the sound rays.ACKNOWLEDGEMENTI am indebted to Dr. W . D. C h e s t e r m a n  (Bath University o f Technology, England) for a critical review  o f the manuscript.NOTE AND REFERENCES[1 ] B e r g m a n n , P.G. and Y a s p a n , A .  (ed.) (1946) : Physics of sound in thesea, Part 1 : Transmission. NDRC Summary Technical Rept. Div. 6, Vol. 8, Washington, D.C. (Research Analysis Group), Sections 2.6.3. (p. 31 et seq.) and 5.2.1 (p. 95 et seq.).[2 ] Y o u n g , R.W . (1947) : Image interference in the presence o f refraction.J. Acoust. Soc. Am. 19, No. 1, 1-7.[3 ] Sa n d e r s , F.H. and S t e w a r t , R.W . : Image interference in calm, near-isothermal water; w ith an appendix : “ L loyd  m irror" interference in the presence o f slight uniform  refraction —  an approximate theory. Canad. J. Phys. 32, 599-619.[4 ] A h r e n s , E. (1957) : Use o f horizontal sounding for wreck detection.In t. Hydr. Rev. XX X IV , 73-81 (p. 77).[5 ] M c K e n z i e , K. V. (1961) : Bottom reverberation for 530- and 1030-cpssound in deep water. J. Acoust. Soc. Am . 33, 1498.[6 ] P e d e r s e n , M. A .  (1962) : Comparison of experimental and theoreticalimage interference in deep water acoustics. J. Acoust. Soc. Am. 34, No. 9, 1197-1203.[7 ] St u b b s , A.R. (1963) ; Identification of patterns on Asdic records.In t. Hydr. Rev. XL, No. 2, 53-68 (pp. 60-63).[8 ] Ch e s t e r m a n , W.D., St. Q u i n t o n , J.M.P., C h a n  Y .  and M a t t h e w s , H.R.(1967) : Acoustic surveys o f the sea floor near Hong Kong. In t. Hydr. Rev. XLIV, No. 1, 35-54.[9 ] H o r t o n , J.W. (1957) : Fundamentals o f Sonar. U.S. Naval Institute,Annapolis, Maryland. 387 pp. (Art. 3C-4, 113-117).[10] R i c h a r s o n , E.G. (1957) : Technical Aspects o f Sound, Vol. II. ElsevierPublishing Co. (Chapter 1, section 3).[11] A l b e r s , V.M. (1965) : Underwater Acoustics Handbook II. Pennsylva­nia State University Press, 365 pp. (Section 4.3.3m, 49-52).[12] Discussion by C h r is t e n s e n  in H e r s e y , J.B. (1954) : Acoustic Instru­mentation as a tool in oceanography. National Research Council Publ. No. 309, Washington, D.C.[13] H a i n e s , R.G. (1963) : Developments in ultrasonic instruments. In t.Hydr. Rev ., X L  No. 1, 49-57 (p. 53-54).[14] This phase reversal was inadvertently neglected in the analysis o fCh e s t e r m a n  et al. [ 8 ] .[15] B r o w n , M.V. and R i c a r d , J.H. (1965) : Interference pattern observedin reflection from the ocean bottom. J. Acoust. Soc. Am . 37, No. 6, 1033-1036.

A Note on the Application of Image Interference to Bottom Contouring

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A Note on the Application of Image Interference to Bottom Contouring

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