Streamer feathering due to cross-currents

From SEG Wiki
Jump to navigation Jump to search
ADVERTISEMENT

Problem 7.10a

A 96-channel streamer with 25-m groups has the hydrophones spaced uniformly throughout its length. The lead-in and compliant sections together are 200 m in length and the tail section and buoy connection are 150 m long. Assume a source 100 m behind the ship, ship’s speed of 5.8 knots, and a 1.9 knot current perpendicular to the direction of travel.

What are the perpendicular and inline components of the distance to the farthest active group center with respect to the traverse direction?

Figure 7.10a  Marine streamer.

Background

Marine seismic work usually employs hydrophones to receive the seismic signals. These consist of piezoelectric crystals which generate a voltage difference between opposite faces when subjected to a pressure. The hydrophones are enclosed in a neutrally buoyant tube called a streamer (Figure 7.10a) which is towed by the ship. A number of hydrophones, spread over as much as 25 m, are connected together to form each group. Lead-in and compliant sections are inserted between the ship and the near end of the streamer; these increase the distance between the ship and the cable and reduce the effects of sudden changes which might jerk or break the streamer. A location sensor and radar target are usually placed on a tail buoy so that the end of the streamer can be located and the amount of feathering (sideways drift of the streamer) determined.

Common-midpoint (CMP) recording is discussed in problem 5.12.

Solution

We assume that the source is in the line between the ship and the streamer. We take the origin at the source and the $ x $-axis along the line of traverse. With 200 m lead-in and compliant sections and active streamer length of $ 96\times 25=2400 $ m, the farthest group center is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(200+2400-12\right)=2590 m from the ship (assuming distances are to the group centers). The last group has a velocity of 5.8 knots along the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x -axis and 1.9 knots along the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y -axis, hence the streamer will be at the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \tan^{-1} \left(1.9/5.8\right)=18^{\circ} (see Figure 7.10b). The coordinates of this group center with respect to the source are $ x=2490\cos 18^{\circ }=2370 $ m, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y=2490\sin 18^{\circ} =770 m.

Problem 7.10b

If the average velocity to a reflector 2.00 km below the source is 3.00 km/s and if the reflector dips Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 20^{\circ} perpendicular to the traverse direction, (i) by how much will the arrival time be changed for the far trace, and (ii) if this should be attributed to a change in velocity rather than cross-dip, what velocity would it imply?

Figure 7.10b  Drift of streamer in crosscurrent.

Solution

(i) We shall use a derivation similar to that in problem 4.2g except that the problem is now three dimensional, so we take the receiver coordinates as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(x, y, 0\right) . Because the dip is along the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y -axis, the coordinates of the image point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): I are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (0,2h\sin \xi , 2h\cos \xi) , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h being the slant depth (2.00 km). We now have


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} (Vt)^2 = x^{2} +(y\pm 2h\sin \xi )^{2} +(2h\cos\xi )^{2}\\ =x^{2} +y^{2} +4h^{2} \pm 4hy\sin \xi, \end{align} (7.10a)

the plus sign for the last term being used when the dip and feathering are in the same direction.

When the streamer is tracking behind the boat, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y=0 and equation (7.10a) gives


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t=\left(1/3.00\right)(2.49^{2} +4^{2} )^{1/2} =22.1^{1/2} /3.00=1.571\ \mathrm{s}. \end{align} (7.10b)

When the feathering is in the downdip direction, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=2.37 km, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y=0.77 km, and


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t=\left(1/3.00\right)(2.37^{2} +0.77^{2} +4^{2} +2\times 4\times 0.77\times \sin 20^{\circ})^{1/2}\\ =24.32^{1/2} /3.00=1.644\ \mathrm{s}. \end{align} (7.10c)

When the streamer is drifting updip, the sign of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sin \xi -term is reversed, so


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t=\left(1/3.00\right)(2.37^{2} +0.77^{2} +4^{2} -2\times 4\times 0.77\times \sin 20^{\circ})^{1/2}\\ =20.103^{1/2} /3.00=1.495\ \mathrm{s}. \end{align} (7.10d)

(ii) At the source we observe the traveltime


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{0} = 2h/V=4.00/3.00=1.333\ \mathrm{s}. \end{align} (7.10e)

If we attribute the difference in traveltimes because of dip to a velocity change rather than to dip, we can determine the velocity as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} (Vt)^2 =x^{2} +y^{2} +4h^{2} =x^{2} +y^{2} +(Vt_{0} )^{2}, \end{align}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \text {so} \qquad V^{2} \left(t^{2} -t_{0}^{2} \right)=x^{2} +y^{2} =\left(2.37^{2} +0.77^{2} \right)=6.21\ \mathrm{km}^{2}, \end{align}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \text {and} \qquad V=2.49/(t^{2} -1.33^{2} )^{1/2}\ \mathrm{km/s}. \end{align}

When the streamer is feathering in the down-dip direction,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V=2.49/(1.64^{2} -1.33^{2} )^{1/2} =2.49/0.96=2.57\ \mathrm{km/s}. \end{align}

For updip feathering,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V=2.49/(1.45^{2} -1.33^{2} )^{1/2} =2.49/0.694=3.56\ \mathrm{km/s}. \end{align}

Compared with the actual value of 3.00 km/s, these results are in error by 14% and 19% for the downdip and updip cases, respectively.

Problem 7.10c

Assume that the amount of sideways drift of the streamer is ascertained by radar sighting on the tail buoy with an accuracy of only Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \pm 3^{\circ} ; (i) how much change will this produce in locating the far group? (ii) How much change in arrival time will be associated with this uncertainty?

Solution

i) The radar sighting gives the angle of feathering Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta . The center of the last group is 2590 from the ship, so taking an origin at the ship, the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x' and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y' coordinates of the group are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x'=2590 \cos \theta , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y'=2590\sin \theta ; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta =18^{\circ} . The errors in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x' and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y' due to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta \theta =3^{\theta} =0.052 radians are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \Delta x'=\Delta x=-2590\sin \theta \Delta \theta =-\left(2590\times \sin 18^{\circ} \right)\times 0.052=-41\ \mathrm{m},\\ \Delta y'=\Delta y=2590\cos \theta \Delta \theta =\pm 128\ \mathrm{m}. \end{align}

ii) When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta \theta is positive, i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta +\Delta \theta =21^{\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta x=-42 m, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta y=+128 m. When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta \theta is negative, the sign of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta y is reversed. We use equation (7.10a) to find the traveltimes, that is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} (Vt)^2 = x^{2} +y^{2} +4h^{2} \pm 4hy\sin \xi , \end{align}

where $ x=2.49 $ km, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y=0.77 km, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h=2.00 km, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V=3.00 km/s, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi =21^{\circ} , and we use the plus sign for the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi -term when the cable is drifting downdip, the minus sign when the drift is updip. We have four cases to consider: the drift is down- or updip, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta \theta is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): +3^{\circ} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -3^{\circ} .

Drift downdip

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \Delta \theta =+3^{\circ} : t=0.333[(2.49-0.04)^{2} +(0.77+0.13)^{2} \\ \qquad\qquad+16+8\times \left(0.77+0.13\right)\sin 21^{\circ})]^{1/2} =1.676\ \mathrm{s}.\\ \Delta \theta =-3^{\circ}, t=0.333[2.45^{2} +(0.77-0.13)^{2} +16\\ \qquad\qquad+8\times 0.64\sin 15^{\circ}]^{1/2} =1.639\ \mathrm{s}. \end{align}

Drift updip

$ {\begin{aligned}\Delta \theta =+3^{\circ },t=0.333(2.45^{2}+0.64^{2}+16-8\times 0.64\sin 20^{\circ })^{1/2}=1.515\ \mathrm {s} ,\\\Delta \theta =-3^{\circ },t=0.333(2.45^{2}+0.90^{2}+16-8\times 0.90\sin 20^{\circ })^{1/2}=1.504\ \mathrm {s} .\end{aligned}} $

Problem 7.10d

The midpoint traces that are to be combined in a CMP stack will be distributed over what distance?

Solution

We determine the shifts from the line of traverse of groups #96 and #1, subtract these shifts, then divide by 2 to get the shifts of the midpoint. Thus, the shift relative to the ship of group center #96 is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \Delta x=2590\left(\cos 18^{\circ} -1\right)=-129\ \mathrm{m},\qquad \Delta y=2590\sin 18^{\circ} =800\ \mathrm{m}. \end{align}

For group #1,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \Delta x=212\left(\cos 18^{\circ} -1\right)=-10\ \mathrm{m},\qquad \Delta y=212\sin 18^{\circ} = 66\ \mathrm{m}. \end{align}

The midpoints are shifted by half of these amounts, so the spread of the midpoints is: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta x=\left(-128+10\right)/2=-59 m, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta y=\left(800-66\right)/2=367 m.

Continue reading

Previous section Next section
Effect of coil inductance on geophone equation Filtering effect of geophones and amplifiers
Previous chapter Next chapter
Characteristics of seismic events Reflection field methods

Table of Contents (book)

Also in this chapter

External links

find literature about
Streamer feathering due to cross-currents
Retrieved from "http://wiki.seg.org/w/index.php?title=Streamer_feathering_due_to_cross-currents&oldid=131487"