Blondeau weathering corrections
Series | Geophysical References Series |
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Title | Problems in Exploration Seismology and their Solutions |
Author | Lloyd P. Geldart and Robert E. Sheriff |
Chapter | 8 |
Pages | 253 - 294 |
DOI | http://dx.doi.org/10.1190/1.9781560801733 |
ISBN | ISBN 9781560801153 |
Store | SEG Online Store |
Problem
The Blondeau method of making weathering corrections (Musgrave and Bratton, 1967, 231−246) is useful in areas where, because of appreciable compaction within the low-velocity layer, the velocity is given approximately by the equation
( )
and being constants,
The Blondeau method starts with a curve of first breaks versus offset [Figure 8.21a(ii)] plotted on log-log graph paper; the curve is approximately a straight line with slope (see Musgrave and Bratton, 1967, 244).
To remove the effect of a surface layer of thickness , we find , a tabulated function of . Then . Next we use the plot to find the corresponding . Finally, , the vertical traveltime to the depth , is given by .
Verify the Blondeau procedure by deriving these relations:
- , where is the angle of incidence measured with respect to the vertical at depth ;
- , where function of , hence also of ;
- , where ; [Note that can be obtained from the table for by writing , finding for , and multiplying the value by
- , the horizontal component of the apparent velocity at any point of the trajectory, is ;
- .
Solution
i) Note that the quantities and are the offset and traveltime for the point of emergence. For intermediate points we write except for the deepest point, where we have , , .
Solving equation (8.21a) for gives
( )
From Snell’s law we have
( )
Thus, using equations (8.21b,c) we get
( )
ii) From equations (4.17d), we get
where we have differentiated equation (8.21d) to replace with . If we integrate from 0 to and multiply by 2, the result is
( )
iii) From Figure 4.17a we have , so
where equation (8.21d) was used to replace with . Using equation (8.21c) to eliminate , we get
Integration from 0 to gives , so
Since from equations (8.21a) and from equation (8.21e), this result can be written
( )
iv) Solving equation (8.21f) for gives
( )
Equation (8.21f) can be used to eliminate , so equation (8.21g) becomes
( )
The angle between the wavefront and a horizontal line equals the angle between a ray and the vertical, both angles being the angle of incidence (see Figure 4.2c), so
[compare with equation (4.2d)]. Therefore,
using Snell’s law. Thus the apparent velocity ; since this is a constant, it holds at every point of the trajectory, including the point of emergence. Therefore, from equation (8.21h),
( )
v) We have time to travel vertically from the surface to the depth , Thus,
where we have used equation (8.21a) in the first and last steps. From equations (8.21e) and (8.21i) we have and , so
( )
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Also in this chapter
- Effect of too many groups connected to the cable
- Reflection-point smear for dipping reflectors
- Stacking charts
- Attenuation of air waves
- Maximum array length for given apparent velocity
- Response of a linear array
- Directivities of linear arrays and linear sources
- Tapered arrays
- Directivity of marine arrays
- Response of a triangular array
- Noise tests
- Selecting optimum field methods
- Optimizing field layouts
- Determining vibroseis parameters
- Selecting survey parameters
- Effect of signal/noise ratio on event picking
- Interpreting uphole surveys
- Weathering and elevation (near-surface) corrections
- Determining static corrections from first breaks
- Determining reflector location
- Blondeau weathering corrections