SME
Geophysics
Seismology
Use this for “Direct waves cross the ground at 1,500 m/s and head waves from a faster layer at 4,500 m/s; the two lines cross 60 m out. How deep is the layer?”
Diagram
sin
ic
=
v₁
v₂
=
1,500
4,500,
so
ic
=
19.5°.
h
=
(
xc
2
)√(
v₂ − v₁
v₂ + v₁
)
=
21.21
m.
Past xc the head wave arrives first; it reaches the surface only from 2h tan ic = 15 m.
This is an example. Type your own values to start a new problem.
h (m)
=
xc (m)
2
×
v₂ (m/s)
−
v₁ (m/s)
v₂ (m/s)
+
v₁ (m/s)
ic
Critical angle
calculated · °
tᵢ
Intercept time
calculated
Formulas
sin(ic°)
=
v₁
v₂
sin(19.4712°)
=
1,500
4,500
h
=
(
xc
2
)
×
√(
v₂
−
v₁
v₂
+
v₁
)
21.2132
=
(
60
2
)
×
√(
4,500 − 1,500
4,500 + 1,500
)
tᵢ
=
2
×
h
×
cos(ic°)
v₁
×
1,000
26.6667
=
2
×
21.2132
×
cos(19.4712°)
1,500
×
1,000
Assumptions
•
The layers are flat, and the lower one is faster: otherwise no head wave comes back up.
•
The head wave runs along the top of layer 2 at v₂ after meeting it at the critical angle.
•
Beyond the crossover distance the head wave arrives first; nearer the source the direct wave does.
Step-by-step
WE KNOW
v₁ = 1,500 m/s, v₂ = 4,500 m/s, xc = 60 m
FIND
critical angle (ic), depth to layer 2 (h), intercept time (tᵢ)
Step 1 · Find critical angle (ic)
Use sin(ic°) = v₁ ÷ v₂
At the critical angle the ray bends to run along the top of the faster layer.
ic
=
sin⁻¹(
v₁
v₂
)
ic
=
sin⁻¹(
1,500
4,500
)
ic = 19.4712°
Step 2 · Find depth to layer 2 (h)
Use h = (xc ÷ 2) × √((v₂ − v₁) ÷ (v₂ + v₁))
Half the crossover distance, times the root of the speeds’ difference over their sum.
h
=
(
60
2
)
×
√(
4,500 − 1,500
4,500 + 1,500
)
h
=
60
2
×
√(
3,000
6,000
)
h
=
60
2
×
√
0.5
h
=
60
2
×
0.7071
h
=
30
×
0.7071
h
=
21.2132
m
Step 3 · Find intercept time (tᵢ)
Use tᵢ = 2 × h × cos(ic°) ÷ v₁ × 1,000
The head wave’s extra time down and back up through layer 1, in milliseconds.
tᵢ
=
2
×
21.2132
×
cos(19.4712°)
1,500
×
1,000
tᵢ
=
2
×
21.2132
×
0.9428
1,500
×
1,000
tᵢ
=
42.4264
×
0.9428
1,500
×
1,000
tᵢ
=
40
1,500
×
1,000
tᵢ = 0.02667 × 1,000
tᵢ = 26.6667 ms
Check
Put the numbers back in. Both sides should match.
sin(19.4712°)
=
1,500
4,500
✓
21.2132
=
(
60
2
)
×
√(
4,500 − 1,500
4,500 + 1,500
)
✓
26.6667
=
2
×
21.2132
×
cos(19.4712°)
1,500
×
1,000
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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