Depth and dip calculations using velocity functions

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Problem 4.15a

Repeat the calculations of problem 4.14 assuming horizontal velocity layering and dip moveout of 104 ms/km, and find the dips.

Background

While velocity generally follows the layering, especially in structurally deformed areas, isovelocity surfaces may not parallel interfaces. Where the section has not been uplifted significantly, isovelocity surfaces are apt to be nearly horizontal in spite of structural relief.

Solution

The depths will be the same as those calculated in problem 4.14 except that $ z $ is now slant depth. Vertical depths are $ z_{v}=z\cos \xi $, where the dip $ \xi $ is given by $ \sin ^{-1}(V\Delta t_{d}/2\Delta x)=\sin ^{-1}(0.052\ V) $, $ V $ usually being either $ {\bar {V}} $ or $ V_{\rm {rms}} $. Using the values of $ t_{0} $, $ {\bar {V}} $ 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/":): V_{\rm rms} from Table 4.14a, we obtain the results in Table 4.15a.

Table 4.15a. Calculated depths and dips.
Time Velocity Slant Dip Vert. depth
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/":): t_0 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 depth 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 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/":): z \cos \xi
i) Assuming average velocities:
1.00 s 2.00 km/s 1000 m 5.97Failed 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/":): ^{\circ} 990 m
2.00 2.50 2500 7.47 2480
2.10 2.67 2800 7.98 2770
3.10 3.10 4800 9.28 4740
ii) Assuming rms velocities:
1.00 s 2.99 km/s 1000 m 5.97$ ^{\circ } $ 990 m
2.00 2.55 2550 7.62 2530
2.10 2.81 2950 8.40 2920
3.10 3.24 5020 9.70 4950
iii) Assuming best-fit depth function:
1.00 s 2.02 km/s 1010 m 6.03Failed 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/":): ^{\circ} 1000 m
2.00 2.54 2540 7.59 2520
2.10 2.59 2720 7.74 2700
3.10 3.12 4840 9.34 4780
iv) Assuming best-fit traveltime function:
1.00 s 2.03 km/s 1020 m 6.00Failed 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/":): ^{\circ} 1010 m
2.00 2.62 2620 7.83 2800
2.10 2.68 2810 8.01 2780
3.10 3.27 5270 9.79 5000

Problem 4.15b

Trace rays assuming (i) the velocity layering given in Figure 4.13a, and (ii) that the velocity is constant at the values 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/":): \bar{V} 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/":): V_{\rm rms} listed in Figure 4.13b. Find the arrival times and reflecting points of reflections at each of the interfaces.

Solution

i) We first calculate the angle of approach 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/":): \alpha [using equation (4.2d)] and then use Snell’s law to find the other angles:

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} \alpha = \theta_1 &= \sin^{-1}\left(2.00\times \frac{0.104}{2}\right) = 6.0^{\circ}, \\ \sin\theta_2 &= (V_2/V_1)\sin \theta_1, \\ \theta_2 &= \sin^{-1}[(3.00/2.00) \sin 6.0^{\circ}] = 9.0^{\circ},\\ \theta_3 &= \sin^{-1}[(6.00/2.00) \sin 6.0^{\circ}] = 18.3^{\circ},\\ \theta_4 &= \sin^{-1}[(4.00/2.00) \sin 6.0^{\circ}] = 12.1^{\circ},\\ t_A &= 2z/(V_1 \cos \theta_1) = 2.000/(2.00 \cos 6.0^{\circ}) \\ &= 1.006\ \mathrm{s}, \\ t_B &= 1.006 + 2 \times 1.500/3.00 \cos 9.0^{\circ} \\ &= 2.018\ \mathrm{s}, \\ t_C &= 2.018 + 2 \times 0.300/6.00 \cos 18.3^{\circ} \\ &= 2.123\ \mathrm{s}, \\ t_D &= 2.123 + 2 \times 2.000/4.00 \cos 12.1^{\circ} \\ &= 3.146\ \mathrm{s}. \end{align}

Next we find $ x $-coordinates of intersections of rays and interfaces:

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} x_A &= z\sin 6.0^{\circ} = 0.105\ \mathrm{km}, \\ x_B &= 0.105 + 0.156 = 0.261\ \mathrm{km}, \\ x_C &= 0.261 + 0.031 = 0.292\ \mathrm{km}, \\ x_D &= 0.292 + 0.209 = 0.501\ \mathrm{km}. \end{align}

Figure 4.15a.  Raypath.

ii) Assuming 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/":): \bar{V} = 0.288 z + 1.77 (see Figure 4.13b) and the given depths,

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_A = 0.288 + 1.77 = 2.058\ \mathrm{km/s},\ \alpha = 6.1^{\circ}, \\ t_A = 2z/(V_1\cos\theta_1) = 0.977\ \mathrm{s}, \\ x_A = 2z \sin 6.1^{\circ} = 0.107\ \mathrm{km}; \\ V_B = 0.72 + 1.77 = 2.49\ \mathrm{km/s}, \alpha = 7.4^{\circ}, \\ t_B = 5.00/2.562\cos 7.4^{\circ} = 1.968\ \mathrm{s}, x_B = 0.324\ \mathrm{km}; \\ V_C = 0.806 + 1.77 = 2.576\ \mathrm{km/s}, \alpha = 7.7^{\circ}, \\ t_C = 5.600/2.660 \cos 7.7^\circ = 2.124\ \mathrm{s}, x_C = 0.365\ \mathrm{km}; \\ V_D = 1.382 + 1.77 = 2.940\ \mathrm{km/s}, \alpha = 9.4^{\circ} \\ t_D = 9.600/3.310\cos 9.4^{\circ} = 3.146\ \mathrm{s}, x_D = 0.787\ \mathrm{km}. \end{align}

Assuming 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_{\rm rms} = 0.325 z + 1.75 and the given depths,

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_A = 0.325 + 1.75 = 2.075\ \mathrm{km/s}, \alpha = 6.2^\circ \\ t_A = 2z(V_1\cos\theta_1) = 2.00/(2.075\cos 6.2^\circ) = 0.970\ \mathrm{s} \\ x_A = 2z\sin 6.2^\circ = 0.108\ \mathrm{km}; \\ V_B = 0.812 + 1.75 = 2.562\ \mathrm{km/s}, \alpha = 7.7^\circ \\ t_B = 5.000/2.562\cos 7.7^\circ = 2.018\ \mathrm{s}, x_B = 0.335\ \mathrm{km}; \\ V_C = 0.910 + 1.75 = 2.660\ \mathrm{km/s}, \alpha = 8.0^\circ, \\ t_C = 5.600/2.660\cos 8.0^\circ = 2.123\ \mathrm{s}, x_C = 0.390\ \mathrm{km}; \\ V_D = 1.560 + 1.75 = 3.310\ \mathrm{km/s}, \alpha = 9.9^\circ, \\ t_D = 9.600/3.310\cos 9.9^\circ = 3.146\ \mathrm{s}, x_D = 0.825\ \mathrm{km}. \end{align}

Assuming 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/":): \bar{V} = 0.526t + 1.49 and times in part (i):

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_A = 1.006\ \mathrm{s}, \bar{V} = 2.019\ \mathrm{km/s}, \alpha = 6.0^\circ, z_A = 2.042\ \mathrm{km}, x_A = 0.213\ \mathrm{km}, \\ t_B = 2.018\ \mathrm{s}, \bar{V} = 2.551\ \mathrm{km/s}, \alpha = 7.6^\circ, z_B = 5.194\ \mathrm{km}, x_B = 0.683\ \mathrm{km}, \\ t_C = 2.123\ \mathrm{s}, \bar{V} = 2.607\ \mathrm{km/s}, \alpha = 7.8^\circ, z_C = 5.586\ \mathrm{km}, x_C = 0.751\ \mathrm{km}, \\ t_D = 3.146\ \mathrm{s}, \bar{V} = 3.145\ \mathrm{km/s}, \alpha = 9.6^\circ, z_D = 10.029\ \mathrm{km}, x_D = 1.618\ \mathrm{km}. \end{align}

Assuming 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_{\rm rms} = 0.595\mathrm{t} + 1.43 and times in part (i):

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_A = 1.006\ \mathrm{s}, V_{\rm rms} = 2.029\ \mathrm{km/s}, \alpha = 6.1^\circ, z_A = 2.053\ \mathrm{km}, x_A = 0.215\ \mathrm{km},\\ t_B = 2.018\ \mathrm{s}, V_{\rm rms} = 2.631\ \mathrm{km/s}, \alpha = 7.9^\circ, z_B = 5.360\ \mathrm{km}, x_B = 0.726\ \mathrm{km},\\ t_C = 2.123\ \mathrm{s}, V_{\rm rms} = 2.693\ \mathrm{km/s}, \alpha = 8.1^\circ, z_C = 5.774\ \mathrm{km}, x_C = 0.800\ \mathrm{km},\\ t_D = 3.146\ \mathrm{s}, V_{\rm rms} = 3.302\ \mathrm{km/s}, \alpha = 9.9^\circ, z_D = 10.454\ \mathrm{km}, x_D = 1.784\ \mathrm{km}. \end{align}

The use of any functional form involves approximation, so it is not surprising that values of the depths and horizontal displacements depend upon the way in which they are calculated.

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