Cross-dip

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Problem 4.10a

Two intersecting spreads have bearings N10$ ^{\circ } $E and N140$ ^{\circ } $E. If the first spread shows an event at $ t_{0}=1.760 $ s with dip moveout of 56 ms/km down to the south and the same event on the second spread has a dip moveout of 32 ms/km down to the northwest. Assume average velocity of 3.00 km/s.

  1. Find the true dip, depth, and strike.
  2. What are the values if the dip on the second spread is southeast?

Solution

We first give numerical solutions, then graphical solutions.

i) We take the $ x $-axis in the N40$ ^{\circ } $W direction, the $ y $-axis in the N130$ ^{\circ } $W direction, and the $ y' $-axis in the N170$ ^{\circ } $W direction (see Figure 4.10a). We now use equation (4.2j) to calculate $ (\Delta t/\Delta y) $ for $ \alpha =130^{\circ } $.

$ {\begin{aligned}56=32\cos 130^{\circ }+(\Delta t_{y}/\Delta y)\cos 50^{\circ },\end{aligned}} $

Figure 4.10a.  Dips northwest and southwest.

hence, $ (\Delta t_{y}/\Delta _{y})=100 $ ms/km; total dip moveout$ {}=(32^{2}+100^{2})^{1/2}=105 $ ms/km, dip $ \xi =\sin ^{-1}(3.00\times 0.105/2)=9.1^{\circ } $,

$ {\begin{aligned}\mathrm {strike} \ \Xi &=\tan ^{-1}(32/100)\\&=17.7^{\circ }\ \mathrm {relative\ to} \ x-\mathrm {axis} \\&=\mathrm {N} (40^{\circ }-17.7^{\circ })\mathrm {W} =\mathrm {N} 22.3^{\circ }\mathrm {W} ,\end{aligned}} $

depth $ h={\frac {1}{2}}\times 3.00\times 1.76=2.64 $ km (normal to bed).

ii) We take the $ x $-axis in the S10$ ^{\circ } $W direction (see Figure 4.10b) where $ \alpha =50^{\circ } $, so

$ {\begin{aligned}32=56\cos 50^{\circ }+(\Delta t_{y}/\Delta y)\sin 40^{\circ },(\Delta t_{y}/\Delta y)=-5.2\ \mathrm {ms/km} ,\end{aligned}} $

The positive $ y $-axis is toward S80$ ^{\circ } $E, so the minus sign means that the $ y $-component of dip is in the N80$ ^{\circ } $W direction.

$ {\begin{aligned}{\hbox{Total dip moveout}}&=(56^{2}+5.2^{2})^{1/2}\\&=56.2\ \mathrm {ms/km} \end{aligned}} $

$ {\begin{aligned}{\hbox{so}}\qquad \qquad \mathrm {dip} \ \xi =\sin ^{-1}\left({\frac {1}{2}}\times 0.056\times 3.00\right)=4.8^{\circ },\end{aligned}} $

$ {\begin{aligned}{\hbox{and}}\qquad \qquad \mathrm {strkie} \ \Xi =\tan ^{-1}[56/(-5.2)]=-84.7^{\circ }.\end{aligned}} $

Figure 4.10b.  Dips southeast and southwest.

The strike is measured relative to the negative direction of the $ x $-axis as shown in Figure 4.2d where both dip components are positive; when the $ y $-component of dip is negative, as it is here, the strike line goes from $ M $ to a point on the negative $ y $-axis. Referring to Figure 4.10b, we see that the strike line is rotated counter-clockwise $ 84.7^{\circ } $ from the negative direction. Graphical solutions for (i) and (ii) are shown in Figure 4.10c.

Figure 4.10c.  Graphical solutions to cross-dip problem.

Problem 4.10b

Calculate the position of the reflecting point (migrated position) for each spread in (i) as if the cross information had not been available and each had been assumed to indicate total moveout; compare with the result of part (a). Would the errors be more or less serious if the calculations were made for the usual situation where the velocity increases with depth?

Solution

We find the coordinates of the migrated reflecting points assuming the velocity is constant at the value of the average velocity. We take the $ x $-, $ y $-, and $ z $-axes positive to the south, west, and downward, respectively, the source being at the origin.

In (i) the spread along the $ x $-axis with bearing N10$ ^{\circ } $E has dip moveout down to the south and west (Figure 4.10b), hence

$ {\begin{aligned}\xi =\mathrm {dip} =\sin ^{-1}(3.0\times 0.056/2)=4.8^{\circ }.\\h={\hbox{depth normal to bed}}\approx {\hbox{vertical depth}}={\frac {1}{2}}\times 3.00\times 1.760=2.64\ \mathrm {km} .\end{aligned}} $

Since the dip is mainly south and west, the reflection point is shifted north and east along the spread direction, the distance $ 2640\sin 4.8^{\circ }=221 $ m. Resolving this along the $ x $- and $ y $-axes, we get $ x=-221\cos 10^{\circ }=-218 $ m and $ y=-221\sin 10^{\circ }=-38 $ m. The vertical depth is $ 2640\times \cos 4.8^{\circ }=2630 $ m, giving coordinates ($ -218,-38,2630 $).

For the spread bearing S140$ ^{\circ } $E (Figure 4.10b),

$ {\begin{aligned}\xi =\sin ^{-1}(3.0\times 0.032/2)=2.8^{\circ }\ \mathrm {S} 40^{\circ }\mathrm {E} .\end{aligned}} $

The reflecting point is $ 2640\times \sin 2.8^{\circ }=129 $ m north and west of the source. Thus, $ x=-129\cos 40^{\circ }=-99 $ m, $ y=129\sin 40^{\circ }=83 $ m, $ z=2640\cos 2.8^{\circ }=2640 $ m, so the coordinates are $ (-99,83,2640) $.

Taking into account cross-dip, the total dip is $ 9.1^{\circ } $ down to the south and west. The horizontal displacement of the reflecting point is $ 2640\sin 9.1^{\circ }=418 $ m in the direction $ \mathrm {N} (22^{\circ }+90^{\circ })\mathrm {E} =\mathrm {N} 112^{\circ }\mathrm {E} (\mathrm {since} \Xi =\mathrm {N} 22^{\circ }\mathrm {W} ) $. Thus, $ x=-418\cos(180^{\circ }-112^{\circ })=-157 $ m, $ y=-418\sin(180^{\circ }-112^{\circ })=-388 $ m, $ z=2640\cos 9.1^{\circ }=2610 $ m. The coordinates are now $ (-157,-388,2610) $. The change in $ z $ is small $ (30\ \mathrm {m} \approx 1\%) $ but the $ x $- and $ y $-coordinates vary considerably, both percentagewise and in absolute values.

The errors become more serious when the velocity increases with depth because these calculations are based on the average velocity $ {\bar {V}} $ rather than the interval velocity $ V_{i} $, which is usually greater than $ {\bar {V}} $.

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