Interval velocities
 | |
| Series | Geophysical References Series |
|---|---|
| Title | Problems in Exploration Seismology and their Solutions |
| Author | Lloyd P. Geldart and Robert E. Sheriff |
| Chapter | 5 |
| Pages | 141 - 180 |
| DOI | http://dx.doi.org/10.1190/1.9781560801733 |
| ISBN | ISBN 9781560801153 |
| Store | SEG Online Store |
Problem 5.15
An $ X^{2}-T^{2} $ survey gives the stacking velocity results in Table 5.15a. Calculate the interval velocities.
| 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 | 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 \ (\hbox {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/":): t_{i} \ (\hbox {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/":): V_{s} \ (\hbox {km/s}) |
|---|---|---|---|
| 1 | 1.20 | 1.100 | 2.18 |
| 2 | 2.50 | 1.786 | 2.80 |
| 3 | 3.10 | 1.935 | 3.20 |
| 4 | 4.10 | 2.250 | 3.64 |
Background
In equation (4.13a), the sum 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/":): \mathop{\sum}\nolimits_{i=1}^{n} \Delta t_{i} is the traveltime down to the base 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/":): n^{th} layer, 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/":): t_{n} . Therefore equation (4.13a) can be written
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_{\rm rms}^{2} t_{n} =\mathop{\sum}\limits_{i=1}^{n} V_{i}^{2} \Delta t_{i}. \end{align} ()
If we write 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_{L} 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/":): t_{L} for the rms velocity and traveltime down to the base 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/":): n^{th} layer, 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_{U} 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/":): t_{U} for the same quantities down to the top of the layer, we can get an equation for the interval velocity 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_{n} in 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/":): n^{th} bed by subtracting expressions for 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_{L} 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_{U} obtained from equation (5.15a). The result 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} V_{n}^{2} =\left(V_{L}^{2} t_{L} -V_{U}^{2} t_{U} \right)/\Delta t_{n}. \end{align} ()
Solution
We assume horizontal velocity layering so that 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/":): V_{s} in Table 5.15a are approximate 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/":): V_{\rm rms} (see Table 5.12b) and therefore we can use equation (5.15b) to find interval velocities. We take 2.18 km/s as the interval velocity from the surface to the first reflector. We now calculate the interval velocities 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_{2} 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/":): V_{4} :
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}^{2} =\left(V_{L}^{2} t_{L} -V_{U}^{2} t_{U} \right)/\Delta t_{2} =\left(2.80^{2} \times 1.786-2.18^{2} \times 1.100\right)/0.686,\\ V_{2}^{2} = 3.57\ {\rm km/s},\\ V_{3}^{2} = \left(3.20^{2} \times 1.935-2.80^{2} \times 1.786\right)/0.149,\; V_{3} = 6.25\ {\rm km/s},\\ V_{4}^{2} = \left(3.64^{2} \times 2.250-3.20^{2} \times 1.935\right)/0.315\; ,\; V_{4} =5.63\ {\rm km/s}. \end{align}
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| Geometry of seismic waves | Characteristics of seismic events |
Also in this chapter
- Maximum porosity versus depth
- Relation between lithology and seismic velocities
- Porosities, velocities, and densities of rocks
- Velocities in limestone and sandstone
- Dependence of velocity-depth curves on geology
- Effect of burial history on velocity
- Determining lithology from well-velocity surveys
- Reflectivity versus water saturation
- Effect of overpressure
- Effects of weathered layer (LVL) and permafrost
- Horizontal component of head waves
- Stacking velocity versus rms and average velocities
- Quick-look velocity analysis and effects of errors
- Well-velocity survey
- Interval velocities
- Finding velocity
- Effect of timing errors on stacking velocity, depth, and dip
- Estimating lithology from stacking velocity
- Velocity versus depth from sonobuoy data
- Influence of direction on velocity analyses
- Effect of time picks, NMO stretch, and datum choice on stacking velocity
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