Site investigation · Geophysics & monitoring
Seismic refraction: two layers, forward and reverse shots calculator
Least-squares lines for the direct and head-wave arrivals of a forward and a reverse shot give V1, the apparent up- and down-dip velocities, the true V2, the dip and the perpendicular depth below each shot.
Standard: Intercept-time method with dip (Redpath 1973; Lankston 1990)
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Inputs in the worked example
The live calculator starts with these values; change any of them and every result, check and diagram updates instantly.
| Input | Value | Unit |
|---|---|---|
| Spread length (shot to shot) | 80 | m |
| Forward: last direct-wave point at offset | 16 | m |
| Reverse: last direct-wave point at offset | 24 | m |
| Forward shot A: offset (m), time (ms) | 4, 6.52 8, 13.41 12, 20.03 16, 26.55 20, 30.08 24, 31.95 28, 34.15 32, 36.37 36, 38.23 40, 40.60 44, 42.60 48, 44.63 52, 46.98 56, 48.83 60, 51.09 64, 53.25 68, 55.14 72, 57.53 76, 59.46 80, 61.56 | |
| Reverse shot B: offset from B (m), time (ms) | 4, 6.64 8, 13.34 12, 20.02 16, 26.63 20, 33.37 24, 39.98 28, 45.85 32, 47.11 36, 48.24 40, 49.53 44, 50.72 48, 51.89 52, 53.19 56, 54.31 60, 55.57 64, 56.81 68, 57.93 72, 59.25 76, 60.39 80, 61.60 |
Worked example — results
| Result | Value | Unit |
|---|---|---|
| V1 forward | 599.6 | m/s |
| V1 reverse | 599.8 | m/s |
| V1 (average) | 599.7 | m/s |
| Apparent down-dip velocity | 1,896 | m/s |
| Apparent up-dip velocity | 3,301 | m/s |
| Intercept time at A | 19.41 | ms |
| Intercept time at B | 37.38 | ms |
| asin(V1/Vd) | 0.322 | rad |
| asin(V1/Vu) | 0.183 | rad |
| Critical angle θc = (ad + au)/2 | 14.45 | ° |
| Dip δ = (ad − au)/2 (down towards B) | 3.99 | ° |
| V2 = V1 / sin θc | 2,403 | m/s |
| Depth below B | 11.58 | m |
| Crossover (flat-layer estimate) | 15.51 | m |
| Depth below A (perpendicular) V1·tA/(2cosθc) | 6.01 | m |
- ✓ V2 > V1 (no velocity inversion)
- ✓ Both shots give similar V1 (within 10 %)
Method, formulas and assumptions
- V1 forward = 1000/SLOPE([f.td],[f.xd]) = 599.61 m/s
- V1 reverse = 1000/SLOPE([r.td],[r.xd]) = 599.83 m/s
- V1 (average) = (599.61+599.83)/2 = 599.72 m/s
- Apparent down-dip velocity = 1000/SLOPE([f.th],[f.xh]) = 1896 m/s
- Apparent up-dip velocity = 1000/SLOPE([r.th],[r.xh]) = 3301.1 m/s
- Intercept time at A = INTERCEPT([f.th],[f.xh]) = 19.407 ms
- Intercept time at B = INTERCEPT([r.th],[r.xh]) = 37.384 ms
- asin(V1/Vd) = ASIN(599.72/1896) = 0.32183 rad
- asin(V1/Vu) = ASIN(599.72/3301.1) = 0.18269 rad
- Critical angle θc = (ad + au)/2 = DEGREES((0.32183+0.18269)/2) = 14.453 °
- Dip δ = (ad − au)/2 (down towards B) = DEGREES((0.32183-0.18269)/2) = 3.9861 °
- V2 = V1 / sin θc = 599.72/SIN(RADIANS(14.453)) = 2402.8 m/s
- Depth below B = 599.72*37.384/1000/(2*COS(RADIANS(14.453))) = 11.576 m
- Crossover (flat-layer estimate) = 2*599.72*19.407/1000/(2*COS(RADIANS(14.453)))*SQRT((2402.8+599.72)/(2402.8-599.72)) = 15.51 m
- Depth below A (perpendicular) V1·tA/(2cosθc) = 599.72*19.407/1000/(2*COS(RADIANS(14.453))) = 6.0097 m
- Rippability is commonly judged from V2 (e.g. ripper charts: below about 2000 m/s rippable for large dozers); confirm with boreholes.
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For preliminary design and education. Check every result against the current standard and your own engineering judgement before use.