Blondeau weathering corrections

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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


(8.21a)

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:

  1. , where is the angle of incidence measured with respect to the vertical at depth ;
  2. , where function of , hence also of ;
  3. , where ;

    [Note that can be obtained from the table for by writing , finding for , and multiplying the value by

  4. , the horizontal component of the apparent velocity at any point of the trajectory, is ;
  5. .

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


(8.21b)

From Snell’s law we have


(8.21c)

Thus, using equations (8.21b,c) we get


(8.21d)

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


(8.21e)

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


(8.21f)

iv) Solving equation (8.21f) for gives


(8.2lg)

Equation (8.21f) can be used to eliminate , so equation (8.21g) becomes


(8.21h)

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),


(8.21i)

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


(8.21j)

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