Geoscience Reference
In-Depth Information
Since the tangential displacements and stresses on the boundary are zero for a
vertically travelling P-wave, no SV-wave can be produced. Equations (4.56)-
(4.59) reduce to
A
1
+
A
2
=
A
0
(4.60)
Z
1
A
1
−
Z
2
A
2
=−
Z
1
A
0
(4.61)
A
0
are the reflection and transmission coef-
ficients. Equations (4.60) and (4.61) are solved for the amplitude ratios:
The amplitude ratios
A
1
/
A
0
and
A
2
/
A
1
A
0
=
Z
2
−
Z
1
Z
2
+
Z
1
ρ
2
α
2
−
ρ
1
α
1
ρ
2
α
2
+
ρ
1
α
1
=
(4.62)
A
2
A
0
=
2
Z
1
Z
2
+
Z
1
ρ
1
α
1
ρ
2
α
2
+
ρ
1
α
1
2
=
(4.63)
If Fig. 4.35 is studied again in the light of the foregoing discussion of reflection
and transmission coefficients, it should be clear that, although the first arrivals
should be the easiest phases to pick accurately, from the critical distance onwards
the largest-amplitude events are the reflections. Thus, in attempts to determine a
crustal structure, the reflections from major discontinuities in the crust are very
important. Normally, the main reflection seen on refraction profiles is P
m
P, the
P-wave reflection from the Moho (the crust-mantle boundary). At the critical
distance for P
n
(the mantle head wave), the amplitude of P
m
Pislarge, which
often helps to constrain the mantle velocity since P
n
does not usually become
a first arrival on continental lines until perhaps 200 km, at which distance from
the source low amplitudes relative to background noise may make accurate pick-
ing of arrival times difficult. At long range, the reflection P
m
Pisasymptotic
to the head wave from the layer above the mantle (Fig. 4.32(b)), so the large
amplitude of P
m
Patlong ranges is very useful in determining the lower-crustal
velocity.
Amplitude-distance relations
In addition to reflection and transmission coefficients, it is useful to under-
stand how the amplitude of waves decreases with increasing distance from the
source. We have already seen in Section 4.1.3 that the amplitude of body waves
varies as
x
−
1
with distance and that the amplitude of surface waves varies only
as
x
−
1/2
.
Foraninterface between two uniform elastic media, the amplitude of a head
wave decreases rapidly with increasing distance as
L
−
3
/
2
x
−
1
/
2
where
L
=
x
−
x
c
,
