Seismology · Interactive Lecture

The Haskell Source Model

Extended fault radiation, rupture propagation & directivity — graduate level

01 — Geometry

Finite Fault & Rupture Propagation

The Haskell model represents a finite rectangular fault of length L and width W. Slip D propagates unilaterally at constant rupture velocity Vr, producing a moving dislocation on the fault plane.

Controls
Fault length L 30 km
Fault width W 15 km
Rupture velocity Vr/Vs 0.80
Rise time τ 1.5 s
Mean slip D 2.0 m
M₀ = μ · L · W · D = —
Fault plane view (map)
The rupture front (orange line) sweeps the fault at Vr. Behind it, each point slips for duration τ (rise time) then locks. The apparent duration seen by a distant observer depends on azimuth — this is directivity.
02 — Source Time Function

Haskell STF: Boxcar → Trapezoid

Each point on the fault slips at rate ṡ(t) = D/τ · [H(t) − H(t−τ)] — a boxcar of duration τ (the rise time). The far-field displacement is proportional to the moment rate M̊₀(t), which is the spatial integral of ṡ over the fault as the rupture front sweeps at Vr. This integral produces a trapezoid — the Haskell STF — whose flat top has duration Tr(θ) = (L/Vr)(1 − Vr/c · cosθ) and total duration Tr + τ. No further differentiation is needed: M̊₀(t) is already the far-field waveform shape.

ṡ(t) = D/τ · [H(t) − H(t−τ)] → M̊₀(t) ∝ ∫₀ᴸ ṡ(t − x/Vᵣ) dx = trapezoid(T_r, τ)
Observer azimuth (directivity)
Azimuth θ from strike 0°
Vr/Vs 0.80
Rise time τ 1.5 s
L/Vs (rupture duration) 5.0 s
T_app = —
f_c = —
M̊₀(t) — far-field displacement (trapezoid)
ṡ(t) — point slip rate (boxcar, τ)
03 — Directivity

Rupture Directivity & Apparent Duration

Observers in the rupture propagation direction (θ = 0°) see a compressed, high-amplitude pulse. Observers in the anti-rupture direction (θ = 180°) see a long, low-amplitude pulse. The apparent duration is:

T(θ) = (L/Vᵣ) · (1 − Vᵣ/c · cosθ) + τ
Directivity rose diagram
Parameters
Vr/Vs 0.80
L/Vs 5.0 s
Rise time τ 1.5 s
Waveforms at 8 azimuths
04 — Spectrum

Far-field Displacement Spectrum

The Fourier transform of the trapezoidal STF gives the ω² (Brune) spectral shape. Two corner frequencies emerge from the finite fault: fc1 from rise time τ and fc2 from the rupture duration L/Vr. Directivity shifts fc2 with azimuth.

|Ũ(f)| ∝ M₀ · sinc(πf·τ) · sinc(πf·T_r(θ)) → flat plateau, then ω⁻², then ω⁻⁴
Spectral parameters
Vr/Vs 0.80
L/Vs 5.0 s
Rise time τ 1.5 s
Azimuth θ 0°
f_c1 (rise time) = —
f_c2 (rupture dur.) = —
At θ=0° (in-rupture direction) f_c2 is pushed to higher frequencies — apparent higher frequency content. At θ=180° f_c2 collapses to lower frequencies.
Total spectrum
Rise-time term sinc(πfτ)
Rupture-dur. term sinc(πfTr)
05 — Summary

Key Scaling Relations

The Haskell model links observable seismogram properties to physical fault parameters through well-defined scaling relations. Fault dimensions follow the empirical regressions of Wells & Coppersmith (1994, BSSA): log L = −2.44 + 0.59·Mw and log W = −1.01 + 0.32·Mw (all faults). Drag the moment magnitude to see how all quantities scale.

Moment magnitude Mw 7.0