Seismic Refraction Simulator: Ray Tracing & Travel-Time Analysis

simulator intermediate ~10 min
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X_co = 122 m — refracted wave arrives first beyond this distance

With V₁=1.5 km/s, V₂=4.0 km/s, and h=50 m, the critical angle is 22° and the crossover distance is 122 m. Beyond this offset, the head wave traveling along the faster layer arrives before the direct wave.

Formula

θ_c = arcsin(V₁ / V₂)
t(x) = x/V₁ (direct) or 2h×cos(θ_c)/V₁ + x/V₂ (refracted)
X_co = 2h × √((V₂ + V₁) / (V₂ - V₁))

Seeing Below the Surface

Seismic refraction is one of the oldest and most reliable geophysical methods for mapping subsurface structure. A seismic source (hammer blow, explosive charge, or vibrator) generates waves that travel through the ground. When these waves encounter an interface where the velocity increases, they refract according to Snell's law. At the critical angle, the refracted wave travels horizontally along the interface and continuously re-radiates energy upward — the 'head wave' — which is recorded by a line of surface geophones.

Travel-Time Curves

Plotting arrival time versus distance reveals two distinct branches: the direct wave (a straight line through the origin with slope 1/V₁) and the refracted arrival (a straight line with slope 1/V₂ and a positive time intercept). The slopes directly give the layer velocities, while the time intercept yields the layer depth through t_i = 2h×cos(θ_c)/V₁. The crossover distance where the two lines intersect marks the offset beyond which the head wave arrives first.

Critical Angle and Head Waves

The critical angle θ_c = arcsin(V₁/V₂) is the geometric key to refraction. A ray hitting the interface at exactly this angle refracts to travel horizontally in the faster medium. As it propagates along the interface at velocity V₂, it continuously generates secondary waves that return to the surface at the critical angle. This head wave has a conical wavefront and always arrives with apparent velocity V₂ along the surface, regardless of the angle.

Field Applications

Refraction surveys are standard practice in engineering geology (mapping bedrock depth for foundations), hydrogeology (finding water table depth), and crustal seismology (determining Moho depth). Forward and reverse shooting — placing sources at both ends of the geophone spread — resolves dipping interfaces. Modern tomographic refraction methods relax the flat-layer assumption and can image complex velocity gradients and lateral heterogeneity.

FAQ

What is seismic refraction?

Seismic refraction is a geophysical method that uses the travel times of waves refracted at subsurface interfaces to determine layer velocities and depths. When a seismic wave hits an interface at the critical angle, it travels along the interface at the higher velocity and re-radiates energy back to the surface as a 'head wave'.

What is the critical angle?

The critical angle θ_c = arcsin(V₁/V₂) is the incidence angle at which the refracted ray travels horizontally along the interface. Below this angle, energy is both refracted and reflected. At the critical angle, total internal reflection begins, but critically refracted energy propagates along the boundary.

What is the crossover distance?

The crossover distance X_co is where the refracted head wave arrives at the same time as the direct wave. Beyond X_co, the head wave arrives first because it traveled through the faster layer. The crossover distance depends on layer thickness, and the velocity contrast between layers.

What are the limitations of seismic refraction?

Refraction cannot detect low-velocity layers (velocity inversions) because no critical angle exists when the lower layer is slower. It assumes flat, laterally homogeneous layers. Dipping interfaces cause different travel times in forward and reverse shots, requiring reversed profiles for correct interpretation.

Sources

Embed

<iframe src="https://homo-deus.com/lab/seismology/seismic-refraction/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub