AVO versus AVA and effect of velocity gradient

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Problem 3.11a

How would you recalibrate the scale to change a plot showing amplitude variation with offset (AVO) into a plot of amplitude variation with angle (AVA)? What will be the effect if velocity increases with depth?

Solution

Assuming constant velocity above a horizontal reflector, the angle of incidence , where is the offset and the depth. We can calculate for a series of values and then stretch the scale to make it linear in . The amount of stretch increases as and increase.

If the velocity increases with depth, straight raypaths become curved as shown by the solid lines in Figure 3.11a. The effect is to increase as compared with the constant velocity case. Thus angles of incidence are often larger than assumed.

Figure 3.11a.  Curved raypaths.

Problem 3.11b

Calculate the angles of incidence on a reflector at a depth of 2.00 km at offsets of 2.00 and 3.00 km where the velocity increases linearly with depth from 2.20 km/s at the surface to 3.10 km/s at the reflector depth. Assume straightray travel at the average velocity.

Solution

The average velocity is , where is the depth and is the traveltime at the source. Since , . Then,

so


(2.1c)

For km, the angle of incidence (2.00/4.00) . Similarly, for km,

Table 3.11a Values for curved raypaths.
20 28.8 1.82
25 36.5 2.37
30 44.7 3.04

Problem 3.11c

Repeat part (b) for curved raypaths.

Solution

We are given , , (which is equivalent to ) and must find the values of for the different offsets. Solving equations to determine both and is not practical, so we assume values of , then use equation (4.17f) to obtain , and equation (4.17b) to find the offset , and then by trial we determine . The results of part (b) suggest that we start with . Then, , . From equation (4.17f) we have

The offset is

We next take , so .

Then

For , we get

We could calculate offsets for intermediate values of , e.g., for , to get more accuracy. However, instead we shall interpolate between these pairs of values. We tabulate the values in Table 3.11a.

Interpretation gives, for the angles of incidence corresponding to and 3.00 km, the values 31.3 and 44.2 respectively. Comparing these with the results in part (b) of 26.6 and 36.9. We see that the angles of incidence for the curved raypaths are 18% and 20% greater than those for the straight-line paths, the difference increasing with offset.

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