Refractions and Head Waves

Part 5, Part 5: Beyond Convolution

Learning objectives

  • Explain a head wave as energy refracted along a faster interface
  • Read the direct, reflected, and refracted arrivals on a travel-time diagram
  • Locate the crossover where the refraction overtakes the direct wave
  • Recover velocities and depth from first-arrival slopes and intercept

The Wave That Runs Along the Interface

When a wave meets a faster layer, part of it reflects, but at one special angle, the critical angle thetac=arcsin(v1/v2)\theta_c = \arcsin(v_1/v_2)c=arcsin(v_1/v2), something new happens. The refracted ray bends to run along the interface, travelling at the fast velocity v2v_2 and continuously leaking energy back up to the surface. That up-leaking energy is the head wave, or refraction.

Because the head wave travels part of its path at the fast speed, at large offset it overtakes the direct wave and becomes the first arrival. The travel-time diagram shows the whole picture: the direct wave is a line through the origin at slope 1/v_11/v_1, the reflection is a hyperbola, and the head wave is a flatter line at slope 1/v21/v_2 that begins at the critical distance.

Refractions and head wavescrossovertimeoffsetDirect, reflection, and head wave. Beyond the crossover the fast refraction arrives first.

Reading the Earth from First Breaks

The point where the head-wave line crosses the direct line is the crossover. Beyond it, the refraction arrives first, and that first-break information is remarkably powerful. The slope of the head-wave line is 1/v_21/v_2, so it hands you the deep velocity directly. Its intercept time, where the line projects back to zero offset, encodes the interface depth. Pick the first-arrival slopes and intercepts and you recover both velocities and the depth without interpreting a single reflection. That is refraction seismology, the method that mapped basement and weathering layers for a century and still provides near-surface velocities and statics today.

None of it exists in a convolutional model, because a head wave requires energy to travel along an interface, a lateral, propagating effect with no place in a trace-by-trace convolution. It is one more member of the family this part has been reintroducing: the events that appear the moment you solve the actual wave equation.

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