`
`
`Ex. PGS 1070
`
`
`EX. PGS 1070
`
`
`
`
`
`

`

`GEOPHYSICS, VOL. 54, NO.3 (MARCH 1989); P. 302-308, 5 FIGS.
`
`The shape of a marine streamer in a cross current
`
`P. M. Krail* and H. Bryskt
`
`ABSTRACT
`
`A serious problem in marine data collection (particu(cid:173)
`larly 3-D) is the necessity to assign to each hydrophone
`a precise location on the surface of the earth. The data
`available for this purpose can come from the vessel's
`navigation system, a radio beacon (at the tail buoy), an
`acoustic transponder (at the head of the streamer and
`occasionally elsewhere), and magnetic compasses dis(cid:173)
`tributed over the length of the cable, The shape of the
`streamer is commonly reconstructed by fitting a curve's
`tangents to the compass readings. The shape deduced in
`this fashion is highly sensitive to compass errors.
`As an alternative to the ad hoc numerical fit, we have
`derived the shape of the streamer in a cross current
`from physical principles. The mechanical equilibrium of
`tension and drag forces leads to differential equations
`which we integrate analytically to obtain a formal solu(cid:173)
`tion for the tension. A further analytic integration yields
`an equation (containing two integration constants) that
`relates arc length along the cable to the tangent angle.
`This equation can be matched to compass readings (for
`stability and minimum error) by a least-squares esti(cid:173)
`mation of the integration constants. A satisfactory fit is
`obtained with a representative data set Parametric
`equations are also obtained for x and y along the cable
`(with the x-axis in the fluid flow direction), as functions
`of the tangent angle; they contain the same integration
`constants (evaluated by the previous procedure). Elimi(cid:173)
`nation of the angle between the two parametric equa(cid:173)
`tions yields the equation of the cable shape in Cartesian
`coordinates.
`
`INTRODUCTION
`
`In the presence of cross currents, the towed hydrophone
`cable (streamer) does not follow in a straight line behind the
`seismic vessel. A crude description of the streamer deviation is
`given by the angle between the line of the ship's course and
`the line from the ship to the tail buoy; this is called the feather
`
`angle. Actually, the currents cause the streamer to drift out of
`the line of travel of the vessel in a curved shape. This curve
`cannot be adequately described by a single angular measure(cid:173)
`ment
`Determining the streamer shape is necessary in 3-D marine
`data collection to precisely locate the hydrophones. Current
`practice employs magnetic compasses along the length of the
`cable to measure tangents to the streamer. These tangent
`measurements are the basis for a variety of curve-fitting algo(cid:173)
`rithms. In one method of curve fitting, the tangent values are
`used to determine the coefficients of an uth-order polynomial,
`which is then said to be fit to the cable shape. Several variants
`of this method, including cubic spline fits, are used. A poly(cid:173)
`nomial curve fit
`is an ad hoc procedure that requires no
`knowledge of the phenomenology (and is therefore attractive
`in the absence of such knowledge). Conversely, a polynomial
`curve may deviate substantially from the functional form of
`the actual solution to the physics problem.
`In this study, a physical model of the hydrodynamic forces
`acting on the cable will be used to derive equations whose
`solution describes the cable shape. Knowing the functional
`form of the cable curve allows us to estimate those parame(cid:173)
`ters, in addition to compass measurements, that are most im(cid:173)
`portant for accurate cable shape determination and provides a
`more realistic approach to determining the shape of
`the
`streamer from compass measurements taken along its length.
`
`THEORY
`
`Flexible cable
`
`the cable as a deformable body in equilibrium,
`We treat
`with its matter uniformly distributed along a geometrical
`curve. A deformable body is in equilibrium if and only if the
`force sum and moment sum of the external forces acting on
`every portion of the body vanish. If a cable is divided into two
`parts by an ideal section, the cable is said to be flexible if the
`action of either part on the other may be represented by a
`single force-not a force and a couple. When the. cable is cut
`at any point P, if the force F denotes the action of the part to
`the right on that to the left, the reaction of the part to the left
`on that to the right is - F. The absolute numerical value F of
`
`Manuscript received by the Editor November 16, 1987; revised manuscript received August 12, 1988.
`*Texaco, E&P Technology Division, P.O. Box 770070, Houston, TX 77215-0070.
`:::CGG American Services Inc., 2500 Wilcrest, Ste. 200, Houston, TX 77042.
`(C' 1989 Society of Exploration Geophysicists. All rights reserved.
`
`302
`
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`
`

`

`Shape of a Marine Streamer in a Cross Current
`
`303
`
`these forces is called the tension of the cable at P. A marine
`streamer is a flexible cable, since it is easily wound on aim
`reel. The cable is taken to be inextensible with negligible flex(cid:173)
`ural rigidity. When the vessel is traveling with constant veloci(cid:173)
`ty in a constant-velocity cross current, we argue that
`the
`streamer is in equilibrium. The tension due to the boat pull is
`balanced by the drag forces.
`Oscillatory instability caused by transverse or vertical vi(cid:173)
`brations has been dealt with in several studies to determine
`under what conditions a towed, thin, flexible cylinder is stable.
`Paidoussis (1966) has shown that, for sufficiently large flow
`velocities, the towed cylinder may be subject to buckling and
`oscillatory instabilities in its first and higher flexural modes.
`For flow velocity below the critical value, the system possesses
`a natural frequency at which any free oscillation is eventually
`damped out by the fluid viscosity. Pao and Tran (1973) have
`shown that if the tail section is a free end (no tail-buoy drag), a
`phenomenon of whipping can occur during a severe reso(cid:173)
`nance. Marschall (1987) believes that, because of their great
`length and attached tail-drag sections, seismic streamers are
`nearly always stable. In this study we shall consider that all
`forces are constant for a time sufficient to damp all oscil(cid:173)
`lations, and that,
`therefore, an equilibrium condition exists.
`However, any sudden change in the forces acting on the cable
`this equilibrium. For example, a change in the
`could upset
`ship's speed or direction would change the towing tension;
`and fluctuations in the sea state or the cross currents would
`mean changes in the hydrodynamic drag forces. Under such
`conditions, a cable shape derived assuming equilibrium is in(cid:173)
`valid.
`Restricting the cable's physical description to tension and
`drag may not suffice near the ends of the cable, particularly in
`the proximate wake of the vessel. The cable is enormously
`longer than the vessel, however, so that
`the disturbed end
`zones impact on only a small fraction of the streamer geome(cid:173)
`try. We shall return to this point later.
`Drag forces are due to the resistance of the cable to the
`water flowing past it; these are distributed forces. Consider a
`cable acted on by a distributed force Q per unit length. In
`general Q will vary from point to point. Let us measure the
`arc s along the cable (Figure 1). A small element PP' of the
`
`FIG. 1. The distributed drag force Q acts on a length of cable
`~s and the tension F is tangential.
`
`o
`
`cable of length ~s is acted on by the tension forces, F at P and
`F' at P', and the resultant of the distributed load Q~s at some
`point P" between P and P'. For this element to be in equilibri(cid:173)
`um, the force sum and moment sum of these forces must
`vanish: hence,
`
`F' - F + Q~s = 0
`
`and
`
`r' x F' - r x F + r" x Q~s = 0,
`
`where r, r', and r" are the position vectors of P, P', and P"
`r x Fare
`relative to any origin O. Now F' - F and r' x F' -
`the increments of F and r x F, respectively, in passing from P
`to P'. The above equations may therefore be written
`
`and
`
`~F
`-+Q=O
`~s
`
`~(r x F) + r"xQ = O.
`
`~s
`
`In the limit as ~s approaches zero,
`
`dF
`-+Q=O
`ds
`
`and
`
`(1)
`
`d(r x F)
`---'- + r x Q = O.
`ds
`
`Expanding the derivative of the cross product and using equa(cid:173)
`tion (1), we obtain
`
`(dF)
`dr
`- x F + r x - + Q = u x F = 0,
`ds
`ds
`
`where u is the unit tangent vector at P. In other words, the
`tensile stress at any point of the cable is tangential
`to the
`cable.
`Since the moment equation implies that F has the direction
`of u, we may write the equations of equilibrium in the form
`
`dF
`-+Q=O
`ds
`
`F= Tn,
`
`(2)
`
`and
`
`where T is the tension.
`
`Hydrodynamic drag
`
`For viscous flow past a solid body, the Reynolds number is
`defined as
`
`R = pvD/l1,
`
`the ratio of the dynamic force represented by the product of
`flow velocity v, the diameter of the body D, and the fluid
`density p to the friction force represented by the viscosity
`coefficient 11. The value of R characterizes the type of flow,
`steady or turbulent, and consequently the functional form of
`the hydrodynamic drag. For a typical seismic-vessel tow speed
`
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`
`

`

`304
`
`Krail and Brysk
`
`(2 m/s) and streamer diameter (0.05 m), the Reynolds number
`is large (lOS), which means that the pressure drag dominates
`over skin friction. The drag force in this range is given by
`Landau and Lifshitz (1959) as
`
`depends on the outer skin material and its roughness. Hoerner
`suggests the value for the coefficients to be Co = 1.1 and Cf =
`0.02 for a Reynolds number between 10' and 106
`•
`
`where C is the drag coefficient, A is the cross-sectional area of
`the body, and v is the main stream velocity.
`The value of the drag coefficient depends on the shape of
`the body and can be determined empirically. The angular de(cid:173)
`pendence of drag (in the direction of fluid flow) and of lift
`(perpendicular to fluid flow) are derived from the cross-flow
`principle. Hoerner (1965) reported the results of placing a cyl(cid:173)
`inder in a flow stream inclined to the flow direction with an
`angle <P between the flow stream and the cylinder axis. By
`measuring experimentally the drag coefficients parallel (Cd) and
`perpendicular (en) to the flow, Hoerner verified the coefficients
`to have a functional dependence of the form
`cd(<p) = Co sin" <p sgn <p + cf
`
`and
`
`Cn (<p) = Co sin? <p cos <p sgn <p.
`The coefficient Co depends on the shape of the body, and Cf
`
`Forces on the cable
`
`Suppose, as shown in Figure 2, that the ship is sailing into a
`cross current. Let VB be the velocity of the vessel and Vc the
`velocity of the cross current. The resultant velocity vector V of
`the water acting on the cable is v = Vc - VB' Let (x, y) be a
`coordinate system with its x-axis aligned with the velocity
`vector v, i.e., with the resultant fluid flow direction that the
`cable sees. We will find the cable shape in this coordinate
`system and then perform a rotation to locate the cable in an
`absolute frame of reference determined by the ship's navi(cid:173)
`gation data.
`Consider a point P(x, y) on the cable towed in the cross
`current (Figure 3). The tension T at P(x, y) on the cable due to
`the pull of the vessel is directed along the tangent to the cable
`curve with tangent angle <p. The drag forces opposing the
`tension are due to the resistance of the cable to water flowing
`past it. The water resistance has components parallel and per(cid:173)
`pendicular to the stream flow with drag coefficients Cd and cn
`per unit length. The drag force per unit length in the x direc(cid:173)
`tion is given by
`
`VESSEL '5 COURSE
`fa
`
`-y
`
`CROSS
`CURRENT
`
`\
`\
`\
`\
`\
`\
`\
`\
`\
`\
`\
`\
`\
`\
`\
`
`\\
`
`\
`\
`
`NORTH
`
`CROSS
`CURRENT
`
`-------\
`
`\
`\
`\
`\
`\
`\
`\
`\
`\
`\
`\
`\
`
`TAIL
`BUOY
`
`x
`
`Sx
`
`FIG. 2. The vessel's course is shown as it sails in a cross
`current. The rotated coordinate system (x, y) is aligned with
`the resultant velocity direction v,
`
`FIG 3. Force balance, in which the tension is balanced by the
`hydrodynamic drag forces.
`
`CABLE
`
`x
`
`Downloaded 08/31/14 to 64.134.103.236. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`

`

`Shape of a Marine Streamer in a Cross Current
`
`305
`
`Cable tension
`
`Consider the flow pattern of the fluid around the vessel and
`visualize the contours of constant velocity (streamlines). A
`vessel sailing in still water with its own axis oriented in the
`direction of motion produces a pattern of streamlines that is
`different from that produced by a vessel sailing in still water
`with its axis oriented at an angle to the direction of motion.
`The situation is analogous to a wing: when the wing is orient(cid:173)
`ed parallel with the flow, there is no lift force on the wing
`because the streamlines are axially symmetric behind the wing.
`When the wing is inclined at an angle to the flow, the sym(cid:173)
`metric flow is disturbed and a lift force is exerted on the wing.
`When dynamic lift on an object occurs, it is always associated
`with an unsymrnetric set of streamlines relatively close togeth(cid:173)
`er on one side and relatively far apart on the other side of the
`object, which correspond to circulation of fluid around the
`object. Landau and Lifshitz (1959) showed that, because of
`unsymmetric flow, the longitudinal velocity falls off more rap(cid:173)
`idly outside the wake than inside it. Therefore, an object like
`the cable trailing behind the boat experiences a lift force
`directed toward the wake. The resultant of the lift force and
`the pull of the vessel is the tension that is exerted on the cable
`at the stern. If the direction of the tension were aligned with
`the axis of the vessel, the cable shape would be a straight line,
`with the streamlines symmetric about the vessel's axis. That
`situation only occurs when the course of the vessel is parallel
`to the current; <p is then constant at zero. This trivial special
`case is explicitly excluded hereafter.
`Dividing equation (7) by equation (8) yields an expression
`that relates the tension to the tangent angle, eliminating the
`arc length:
`
`..!. dT = -Qx cos <p + Qy sin <p
`Q; sin <p + Qy cos <p
`T d<p
`After substitution of equations (3) and (4) for Qx and Qy' this
`differential equation is separable and may be integrated,
`
`(11)
`
`.
`
`and in the y direction, by
`
`1
`1
`Qy = 2: pv DCn = 2: pv2D(co sirr' <P cos <P sgn <p).
`
`2
`
`(4)
`
`At the point PIx, y), the x and y components of the tension are
`countered by the corresponding components of the drag force
`from the point PIx, y) back to the end of the cable. Since the
`vessel is traveling with constant velocity in a straight line, a
`steady-state condition is assumed to exist. The shape that the
`cable assumes is such that the tension forces from the vessel
`and the tail buoy are balanced by the drag forces.
`Separating equation (2) into its x, y components, then ex(cid:173)
`panding the derivatives, we have
`
`and
`
`dT
`-
`ds
`
`d<p
`cos <p - T sin <p -
`ds
`
`= - Q
`x
`
`dT .
`d<p
`Sill <p + T cos <p -
`-
`ds
`ds
`
`= Q .
`
`Y
`
`Equations (5) and (6) can be manipulated to produce
`
`and
`
`(5)
`
`(6)
`
`(8)
`
`Provided only that the magnitude of <p never exceeds 90° (i.e.,
`the cable does not double back), the right-hand side of equa(cid:173)
`tion (7) is negative; hence, unsurprisingly,
`the tension de(cid:173)
`creases down the cable. Equation (8) indicates that the mag(cid:173)
`nitude of the angle <p increases down the cable; i.e., the cable
`shape curls away from the x-axis,
`The coordinates x, yare related to the tangent angle <p and
`the arc length s by the relations
`
`iT dT _ 1<1>
`
`To
`
`T
`
`-
`
`d<p,
`
`-Cf cos <p
`. A-(
`.
`<1>0 Sill 'I'C j+Co Sill <psgn <p)
`from the stern to an arbitrary point along the cable. The
`integration can be performed analytically and yields (provided
`<p is never zero)
`
`and
`
`dx
`-
`ds
`
`= cos <p
`
`dy
`-= -sin A(cid:173)
`ds
`'1"
`
`T(A- = T c j + Co sin <p sgn <p
`'1')
`A-
`.
`Sill 'I'
`
`1
`
`'
`
`(12)
`
`where T] is a constant given by
`
`sin <Po
`T - T.
`] - 0 Cj + Co sin <Po sgn <Po'
`
`(9)
`
`(to)
`
`Upon substituting equation (9) into equation (7) and inte(cid:173)
`grating, we find that T is a linear function of x,
`
`The cable shape
`
`1
`T = To - 2: pv2Dc
`
`fx,
`
`the stern. Since the
`where To is the tension in the cable at
`cable shape is not exactly a straight line, the tension is not a
`linear function of the arc length s along the cable. The devi(cid:173)
`ation is, however, relatively small.
`
`A-
`s '1') =
`(
`
`The tension equation, when substituted into equation (8),
`yields a differential expression that relates the arc length to the
`tangent angle. When this expression is integrated,
`- T] {cot <p - cot <Po}
`I
`- pv2D
`2
`
`.
`
`(13)
`
`Downloaded 08/31/14 to 64.134.103.236. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`

`

`306
`
`Krail and Brysk
`
`If L is the cable length and the angle at the tail buoy is <PT'
`
`Determination of the integration constants
`
`The analysis of the theoretical mechanics of the cable led to
`differential equations. The formal integration of these equa(cid:173)
`tions has produced mathematical expressions that contain in(cid:173)
`tegration constants <Po and <PT' the tangent angles at the two
`ends of the cable. One way to determine <Po and <PT is to
`measure them. However, tying the solution to the slopes at the
`ends, where the cable emerges from the water, would be inad(cid:173)
`visable because the equations were derived for the steady-state
`dynamics in the central part of the cable. The equations might
`be perturbed at the ends, especially near the vessel where the
`interaction with the wake has been neglected. The simplified
`physical description of the cable is justified because the cable
`is some 50 times longer than the vessel. The perturbed section
`is a small fraction of the cable length and does not seriously
`alter the overall shape. Nonetheless, some deviations from the
`theory can be expected near the ends, and it would not be
`desirable to damp the solution to the precise details of the
`cable shape at its terminations.
`The most commonly available data are compass measure(cid:173)
`ments at known emplacements along the cable. We shall illus(cid:173)
`trate the use of these. Additional data may come from radio
`(location of the tail buoy) and from transponders at the ends
`of the cable.
`
`Compass measurements
`
`A preliminary complication for processing the compass
`measurements is that the raw compass readings are referenced
`to a true north coordinate system, whereas the equations are
`in a frame aligned with the fluid flow. The latter direction is
`just the vector difference of the course of the vessel and the
`cross current; it may be estimated from the navigation record,
`the vessel heading, and the wind direction. Thus, the first pro(cid:173)
`cessing step is to apply a constant azimuthal shift to the com(cid:173)
`pass measurements.
`The number of measurements substantially exceeds the two
`integration constants, 4>0 and <PT' that remain to be deter(cid:173)
`mined. The redundancy is exploited, in least-squares sense, to
`minimize the impact of individual compass errors. Equation
`(16) relates (scaled) distances along the cable to tangent angles.
`These are precisely the pairs of variables that are represented
`by the compass readings, except for the shift of the reference
`azimuth. Setting <P = a + e, where e is the measured angle
`and a the (constant) coordinate rotation angle, equation (16)
`becomes
`
`S
`- = A - B cot (9 + a).
`L
`This is treated as a linear equation relating (s/L) to cot (9 + a).
`The compass readings are inserted into this expression, and a
`least-squares fit is obtained for the constants A and B. The
`constants <Po and <PT are then determined from equations (17)
`and (18).
`In Figure 4 we plot stl: against cot (a + e) to obtain a
`straight
`line. Compass measurements taken during calm
`weather in a cross current of approximately 0.8 knot are also
`plotted in Figure 4. The agreement between theory and
`measurements is satisfactory.
`In Figure 5 the constants <Po and <PT determined from the
`
`(14)
`
`-T
`L = __1_ (cot <PT - cot <Po)'
`pv2D
`
`1-
`
`2
`
`Dividing equation (13) by equation (14) eliminates a multi(cid:173)
`plicity of constants (induding the tension on the cable at the
`stern,
`the cross-current velocity, and the drag coefficients),
`which are not an wen known, in favor of the angles at
`the
`head of the cable and at the tail buoy; i.e.,
`
`or
`
`where
`
`and
`
`s
`,
`s=-=
`L
`
`cot <Po - cot <P
`cot <Po - cot <PT
`
`,
`
`s' = A - B cot <P,
`
`A = cot <Po/{cot <Po - cot <PT}'
`
`B = l/{cot <Po - cot <PT}'
`
`(15)
`
`(16)
`
`(17)
`
`(18)
`
`Equation (16) shows a linear relation between the scaled arc
`length s' and cot <p. In a later section we plot this equation
`and compare it to actual field measurements.
`In order to obtain the cable shape in the x-y coordinate
`system, we write the derivative in equation (9) as
`
`~; =G;)/(:;) = cos <P;
`
`we differentiate equation (16) to obtain ds/d<p, then integrate
`equation (19) over <P for
`
`1)
`(1
`x
`x' = L = B sin <Po - sin <P
`
`.
`
`(19)
`
`(20)
`
`(21)
`
`A similar treatment of equation (10) yields
`
`<P
`tan -
`2
`y
`y'=-= -Bln--.
`<Po
`L
`tan(cid:173)
`2
`
`The coordinates of the tail buoy are obtained by setting <P =
`<PT in the last two equations.
`We find the equation for the cable shape in the form
`y = f(x) by eliminating the parameter <P in equations (20) and
`(21). After some trigonometric manipulations,
`
`y'=
`
`1 + cos 4>0
`-B In ------:=========:;==~~
`2x' sin <Po
`(x' sin <po)2
`+ cos,l,.-
`+
`1-
`'+'0
`
`B
`
`B
`
`x'sin <Po J 2
`
`B
`
`This is the form of the cable shape curve; note x' = 0 implies
`that y' = O. The solution process is not yet complete, however,
`since equation (22) contains two undetermined constants, <Po
`and <PT [see definition of B in equation (18)].
`
`(22)
`
`Downloaded 08/31/14 to 64.134.103.236. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`

`

`Shape of a Marine Streamer in a Cross Current
`3.3 r---------------------,
`
`0.0
`
`307
`
`-THEORY
`L'1 MEASURED
`
`-0.Q1
`
`-0.02
`
`-0.03
`
`:J
`>;
`I
`
`(f)
`
`-0.04
`
`X-c
`>.
`0w
`-J« -0.06
`
`-0.05
`
`o(
`
`f)
`
`3.2
`
`~ 3.1
`
`3.0
`
`+~
`
`I-m2.9
`oz
`~ 2.8
`oo
`
`2.7
`
`2.6
`
`2.5 :-_'--_.L_~:'-:--:'---::"--::"--::"---'----L.-'>.--'
`o
`0.1
`0.2
`0.3
`0.4
`0.5
`0.6
`0.7
`0.8
`0.9
`1.0
`
`SCALED ARC LENGTH (siLl
`
`FIG. 4. Scaled arc length along the cable versus the cotangent
`of the angle, compared with compass measurements.
`
`the cable shape
`least-squares procedure were used to plot
`curve in scaled coordinates x', y' from equation (22). The max(cid:173)
`imum deviation of the cable occurs at the tail buoy.
`
`DISCUSSION AND CONCLUSIONS
`
`In processing seismic data, it is necessary to assign spatial
`coordinates to every shot and every receiver. With marine
`acquisition, those coordinates must be deduced indirectly from
`a variety of measurements. The absolute location of the vessel
`is tracked by radio navigation (triangulation from land-based
`stations or satellites). The position of the tail buoy relative to
`the vessel is usually determined from passive radio (or radar)
`observations. Acoustic range-finding devices locate the source
`assembly and the head of the cable (especially for multistream(cid:173)
`er operation) from the vessel. Compass measurements record
`the orientation of the cable at fixed intervals along its length.
`It is not immediately obvious how best to piece together these
`diverse fragments of information, with their respective errors
`and uncertainties, into a map of the seismic geometry.
`The analysis of the receiver geometry problem is simpler
`and more orderly if it is untangled into three qualitatively
`disparate parts. The first part, the determination of the abso(cid:173)
`lute location and course of the vessel, is treated as given. The
`second part,
`the determination of the curved shape of the
`cable from the compass readings, embodies most of the con(cid:173)
`ceptual subtleties of the analysis. The third part, correlating
`range and direction measurements from radio and acoustic
`transponders to deduce the absolute locations of these devices
`and of the streamer, is discussed further only insofar as it
`impacts on the cable shape determination.
`To start with, the length of the cable is known, as are the
`emplacements on it of the hydrophones and of the compasses.
`Each compass reading provides the slope of the cable curve at
`a specified arc length on it. Drawing a curve, given its total
`length and the discrete readings of the slope, has previously
`been treated as a curve-fitting exercise. Most commonly this
`problem is solved by fitting a high-order polynomial through
`the data points, but errors in the compass readings can propa(cid:173)
`gate in the curve fit. A single bad value displaces the entire
`curve behind it laterally ("bayonet effect"). More generally,
`the fitting procedure is prone to "tail-wagging." Quality con-
`
`-0.07
`
`-0.08
`
`_009l.-~_.......l._-l.._-L_ __L._::"__.L__'--~_ __._J
`0.0
`0.1
`0.2
`0.3
`0.4
`0.5
`0.6
`0.7
`0.8
`0.9
`1.0
`
`SCALED x-AXIS (x/L)
`
`FIG. 5. Cable shape in the scaled x-y rectangular coordinate
`system for the constants of integration determined in Figure 4.
`To emphasize the curvature, different scales have been used
`for x and y.
`
`trol consists of screening out individual readings that deviate
`too much from neighboring ones and also smoothing the
`values (for instance, by using a spline).
`Instead of settling for an arbitrary curve from a numerical
`fit with a tendency to instability, we decided to derive the
`shape of the streamer in a cross current from physics. The
`mechanical equilibrium of tension and drag forces leads to
`differential equations which we solved analytically. In particu(cid:173)
`lar, we obtained an expression for the arc length along the
`cable (in units of the cable length) as a function of the tangent
`angle (with respect to the fluid flow axis). Except for a con(cid:173)
`stant shift in the reference of azimuthal axis (the course versus
`north), the arc length and tangent angle are just the variables
`in terms of which the compass measurements are recorded.
`The theoretical expression [equation (16)] contains two inte(cid:173)
`gration constants to be determined, while the compass data
`sets contain a larger number of points (16 in the field results
`we used). The obvious approach for stability and minimum
`error is a least-squares fit for the integration constants. In an
`example the fit was reasonably good (Figure 4).
`Instead of recalibrating the compass angles from true north
`to the course direction, one might be tempted to treat the
`requisite azimuthal shift as one more parameter to fit to the
`thus decoupling the compass data set from all other
`data,
`measurements. It is not particularly difficult to obtain such a
`fit, but
`there is no reason why that fit should choose the
`azimuth corresponding to the fluid flow. If it does not, equa(cid:173)
`tion (16) with a shifted angle will no longer be a solution to
`the original physical problem, but merely another ad hoc
`mathematical function.
`We have also obtained analytic formulas for the Cartesian
`coordinates of a point on the cable (with the x-axis along the
`direction of fluid flow) expressed parametrically in terms of the
`tangent angle [equations (20) and (21)], and we have com(cid:173)
`bined these to obtain the equation of the cable curve in Car(cid:173)
`tesian coordinates [y as a function of x, equation (22)]. With
`the integration constants determined from the compass read(cid:173)
`ings (as illustrated in Figure 4), the cable shape can be dis(cid:173)
`played (Figure 5). It is relatively close to a straight line (note
`that
`the x and y scales differ) but, in view of the extreme
`
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`
`

`

`308
`
`Krail and Brysk
`
`length of the cable, the discrepancy amounts to a substantial
`number of offset intervals. Neglecting the curvature of the
`cable would seriously undermine a 3-D survey.
`Our theoretical treatment assumes steady fluid flow. This
`means that instantaneously (in the time that one shot is re(cid:173)
`corded) the cross current does not vary substantially over the
`length of the cable. This assumption is probably not too bad,
`since acquisition will usually cease under turbulent conditions
`that would violate it. The equilibrium assumption further
`implies steady progress by the vessel and no whiplash motion
`of the cable. These constraints are most likely to be broken
`when the vessel turns around at the end of a line, if shooting is
`resumed too hastily.
`In our formal solution for the streamer shape, the integra(cid:173)
`tion constants are expressed in terms of <Po and <PT' the nom(cid:173)
`inal values of the tangent angle at the head and the tail of the
`cable [equation (15)]. This formulation has the appearance of
`a boundary-value problem, to be completed by substituting
`the values of <P at the ends of the cable. The resemblance is
`deceiving, however, since <Po and <PT are not measured quan(cid:173)
`tities. Instead, we use a quite different procedure which evalu(cid:173)
`ates the integration constants from a least-squares fit
`to
`compass measurements throughout
`the cable. Actually,
`the
`the ends. It is derived
`theoretical curve is least reliable at
`considering only tension and drag forces, an adequate model
`over most of the cable because the length of the streamer
`dwarfs the length of the vessel. The description is perturbed at
`the tail of the cable by the presence of the buoy and the
`emergence from the water. It is more seriously faulty at the
`head of the cable, where the wake behind the stern is a sub(cid:173)
`stantial disturbance which we do not know how to model
`satisfactorily.
`there
`After the shape of the cable has been determined,
`remains the task of correctly emplacing that curve on the
`surface of the earth. At the least, that requires tying it to the
`known location of one transponder, most often the tail buoy,
`perhaps the head of the cable. From the prior discussion, it is
`evident that the accuracy of positioning would be improved
`by the availability of at least one location-finding device on
`the cable away from the ends. If the cable curve could be
`anchored in its midsection, transponders on its terminations
`could serve to correct the shape at the ends. The alternative of
`
`accounting theoretically for the physics of the end effects is
`unappealing, not only because of the complexity of the physics
`but because the results would depend on details of fluid-flow
`patterns whose measurement would be difficult.
`Further uses of data redundancy for error reduction can be
`considered. If there are extra transponders, a relative weight
`should be assigned to range
`information and compass
`measurements when they are combined. In view of the fast
`repetition rate, stacking of compass readings from successive
`shots should be explored. A very limited examination of
`records
`indicated, however,
`that
`the shot-to-shot
`repro(cid:173)
`ducibility seems to be closer for each individual compass than
`the scatter of readings from one compass to the next;
`this
`leads us to suspect that the errors have less to do with random
`noise than with the imperfections of the instruments.
`A priori, we feel more comfortable with fitting the compass
`measurements statistically to an expression derived from the
`physics than with the current practice of applying a numerical
`curve-fitting procedure prone to instability. Our apparent suc(cid:173)
`cess with a small sample of results has been satisfactory. The
`ultimate judgment of the relative practicality of the two ap(cid:173)
`proaches must be determined by extensive field testing, under
`a variety of sea-state conditions and with different levels of
`error contamination of the compass readings.
`
`ACKNOWLEDGMENTS
`
`Paul Krail would like to acknowledge many stimulating
`discussions with Mike Greene and Elmer Eisner and to thank
`the EPTD Publications committee for helpful suggestions and
`Texaco USA for permission to publish. Finally, we thank Ms.
`Kathy Gough for her patience with typing equations.
`
`REFERENCES
`
`Hoerner, S. F., 1965, Fluid dynamic drag: Midland Park.
`Landau, L., and Lifshitz, E., 1959, Fluid mechanics: Addison-Wesley
`Pub!. Co.
`Marschall, R. A., 1987, Stability of flexible bodies of revolution in
`axial fluid flow: M.Sc. thesis, Univ. of Virginia.
`Paidoussis, M. P., 1966, Dynamics of flexible slender cylinders in axial
`flow; Part
`I. Theory: J. Fluid Mech., 26, 717-736.
`Pao, H. P., and Tran, Q., 1973, Response of a towed thin flexible
`cylinder in a viscous fluid: J. Acoust. Soc. Am., 53, 1441-1444.
`
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`