Feasibility of mapping a horizon using head waves

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Problem 11.15

Construct the expected time-distance curve for the Java Sea velocity-depth relation shown in Figure 11.15a. Is it feasible to map the top of the relatively flat 4.25 km/s limestone at a depth of about 0.9 km by using head waves? What problems are likely to be encountered?


Figure 11.15a.  Java Sea velocity-depth relation.
Table 11.15a. Java Sea layering.
Depth range (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/":): {\displaystyle V_t} (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/":): {\displaystyle \Delta t_T} (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/":): {\displaystyle t_i = \sum \Delta t_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/":): {\displaystyle V_{t}^{2}\Delta t_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/":): {\displaystyle \sum V_t^2\Delta_{t_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/":): {\displaystyle V_{\rm rms}} (km/s)
0.00–0.03 1.53 0.039 0.039 0.091 0.091
0.03–0.16 1.9 0.137 0.176 0.495 0.586
0.16–0.28 1.97 0.122 0.298 0.472 1.059
0.28–0.50 2.25 0.196 0.494 0.992 2.051
0.50–0.70 2.15 0.186 0.68 0.86 2.911
0.70–0.90 2.67 0.15 0.83 1.069 3.98 2.19
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/":): {\displaystyle {\rm L}_{s}\to} 0.90–0.97 4.25 0.033 0.863 0.596 4.576 2.30
0.97–1.10 5.27 0.049 0.912 1.361 5.937

Solution

The time-depth data in Figure 11.15a are listed in the first two columns of Table 11.15a. We calculated the data in columns 3 (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/":): {\displaystyle t_{i}} two-way traveltime through the layer) to 6 to determine reflection arrival times (column 4) 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/":): {\displaystyle V_{\rm rms}} [using equation (4.13a)]. We take 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/":): {\displaystyle V_{1} \approx 1.90} km/s (see Figure 11.15a) to plot the direct wave.

We must take into account other events that might interfere, primarily the reflection and head wave from the 5.27 km/s layer. To plot the refraction curves, we need their slopes, one point on each curve, and the critical distances—where a head wave is tangent to the reflection (see Figure 4.18a). We also calculate the intercept times as a check.

The slope of the limestone refractor (assumed to be flat) is 1/4.25 s/km; taking 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/":): {\displaystyle z = 0.90} 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/":): {\displaystyle V_{1} = 2.19, \theta _{c} =\sin ^{-1} (2.19/4.25)=31^{\circ}} . From Figure 4.18a the critical distance 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/":): {\displaystyle x'=2z\tan \theta _{c} =2\times 0.90\times \tan 31^{\circ} =1.1} km; at this 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/":): {\displaystyle t'=2z/V_{1} \cos \theta _{c} = 2\times 0.90/2.19\times \cos 31^{\circ} =0.96} s. The intercept time given by equation (4.18a) 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/":): {\displaystyle t_{i} = 2z\cos \theta _{c} /V_{1} =0.70} s. Thus the head-wave curve passes through the 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/":): {\displaystyle \left(t,\; x\right)= \left(0.96,1.1\right)} with slope 1/4.25, is tangent to its reflection at 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/":): {\displaystyle x=1.1} km, and the intercept time 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/":): {\displaystyle t_{i} =0.70} s. The reflection arrives at 0.863 s at zero offset and also passes through (0.96,1.1). These curves are shown in Figure 11.15b.

Carrying out similar calculations for the 5.27 km/s layer and using 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/":): {\displaystyle V_{1} \approx 2.30} km/s (estimated from Figure 11.15a), we get

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/":): {\displaystyle \begin{align} {\rm slope} = 1/5.27,\quad \theta _{c} =26^{\circ},\quad x'=0.95\ {\rm km},\quad t'=0.94\ {\rm s},\quad t_{i} =0.86\ {\rm s}. \end{align} }

The reflection arrives at zero offset at 0.912 s and also passes through (0.95, 0.94). These curves are also plotted in Figure 11.15b.

The 4.25 km/s head wave is always a second arrival. It also follows very closely the reflection from the 5.27 km/s layer. It will almost certainly not be observed as a distinctly separate arrival because later cycles of earlier events will mask it.

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