Feasibility of mapping a horizon using head waves
 | |
| Series | Geophysical References Series |
|---|---|
| Title | Problems in Exploration Seismology and their Solutions |
| Author | Lloyd P. Geldart and Robert E. Sheriff |
| Chapter | 11 |
| Pages | 415 - 468 |
| DOI | http://dx.doi.org/10.1190/1.9781560801733 |
| ISBN | ISBN 9781560801153 |
| Store | SEG Online Store |
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?
| Depth range (km) | $ 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/":): \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/":): 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/":): 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/":): \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/":): 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/":): {\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/":): 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/":): 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/":): 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/":): 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/":): V_{1} = 2.19, \theta _{c} =\sin ^{-1} (2.19/4.25)=31^{\circ} . From Figure 4.18a the critical distance $ 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/":): 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/":): 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/":): \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/":): 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/":): 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/":): 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/":): \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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| Comparison of refraction interpretation methods | Refraction blind spot |
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| Geologic interpretation of reflection data | 3D methods |
Also in this chapter
- Salt lead time as a function of depth
- Effect of assumptions on refraction interpretation
- Effect of a hidden layer
- Proof of the ABC refraction equation
- Adachi’s method
- Refraction interpretation by stripping
- Proof of a generalized reciprocal method relation
- Delay time
- Barry’s delay-time refraction interpretation method
- Parallelism of half-intercept and delay-time curves
- Wyrobek’s refraction interpretation method
- Properties of a coincident-time curve
- Interpretation by the plus-minus method
- Comparison of refraction interpretation methods
- Feasibility of mapping a horizon using head waves
- Refraction blind spot
- Interpreting marine refraction data
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