Seismology Measuring the interior - The Solid Earth - page 146

where z u is the perpendicular distance from the new shotpoint to the interface.
This is called shooting up-dip .Inthis case, the apparent velocity is

α u :

α 1
sin( i c − δ )

α u = (4.45)

which is greater than the true velocity of the lower layer α 2 . When a refraction line

is reversed in this way, the true velocity of the dipping layer

α 2 can be determined

from Eqs. (4.43) and (4.45).
In one method, we rearrange Eqs. (4.43) and (4.45) and obtain

α 1
α d

sin( i c + δ ) =

and

α 1
α u

sin( i c − δ ) =

Hence

i c + δ = sin − 1 α 1
α d

(4.46)

and

i c − δ = sin − 1 α 1
α u

(4.47)

Therefore, i c and δ are easily obtained by adding and subtracting Eqs. (4.46) and

(4.47):

sin − 1 α 1
α d

sin − 1 α 1
α u

1
2

i c = + (4.48)

sin − 1 α 1
α d

− sin − 1 α 1
α u

1
2

δ = (4.49)

The velocity α 2 is known once i c has been determined since i c has been defined

by the equation sin i c = α 1 /

α 2 .
An alternative method that does not involve the use of inverse sines can be
used in situations in which the dip angle

δ is small enough for the approximations

sin δ = δ and cos δ = 1tobemade ( δ must be measured in radians, not degrees).

In this case, by expanding sin( i c + δ ) and sin( i c − δ )inEqs. (4.43) and (4.45),

we obtain

α 1
α d = sin( i c + δ )
= sin i c cos δ + cos i c sin δ

(4.50)

= sin i c + δ cos i c

and

α 1
α u = sin( i c − δ )
= sin i c cos δ − cos i c sin δ

(4.51)

= sin i c − δ cos i c