Head wave

In refraction seismics, the head wave is that wave which strikes a boundary surface between one seismic medium and another medium at a higher seismic speed at the critical angle and is refracted by the perpendicular at a right angle . It runs along the interface and continuously radiates back wave energy at the critical angle. It is also known as the (critically) refracted wave or - after its discoverer Ludger Mintrop - as the Mintrop wave .

Physical background

Seismic waves propagate in rock layers according to Huygens' principle and are subject to the effects of reflection and refraction known from optics . Refraction is also known as refraction and means the change in direction of the propagating wave fronts ( beam path ) due to the change in the propagation speed of the wave. Here, Snellius' law of refraction applies to the angles and speeds :

${\displaystyle {\frac {\sin(i_{0})}{v_{0}}}={\frac {\sin(i_{1})}{v_{1}}}}$    or:     ${\displaystyle {\frac {\sin(i_{0})}{\sin(i_{1})}}={\frac {v_{0}}{v_{1}}}}$

In seismics , the velocities of the upper (v ₀ ) and the underlying (v ₁ ) layer are determined by the rock material, so that at a given angle of incidence (i ₀ ), the angle of refraction (i ₁ ) is also predetermined. In the event that v _{1 is} greater than v ₀ , i ₁ is therefore also greater than i ₀ . Thus, the angle of incidence can be so large that the expression takes exactly the value 1. This angle of incidence is called the critical angle (i _c ). In this case the angle of refraction is exactly 90 °, which corresponds to a spread along the layer boundary. In the case of larger angles of incidence, the refraction is not physically realized; total reflection then occurs . ${\displaystyle \sin(i_{1})}$

The head wave in seismics

Schematic representation of the beam paths of direct and head wave and the corresponding time-of- flight diagram .

The seismic head wave is characterized by the fact that it propagates parallel to the layer boundary at the speed of the layer below (v ₁ ) and constantly radiates back wave energy into the upper layer at the critical angle. According to Snellius' law of refraction , the head wave is given by the equation

${\displaystyle \sin(i_{c})={\frac {v_{0}}{v_{1}}}}$

realized. Nevertheless, their occurrence is not trivial, since the radiated energy was theoretically only fed in by an infinitely thin bundle of rays at the angle i _c . Qualitatively, however, this effect can be reproduced if one looks at the wavefront in the elastic half-space that runs along the layer boundary, which in turn generates secondary waves according to Huygens' principle .

In the schematic illustration, the head wave is shown in green. The beam paths are shown below, above the corresponding time- of- flight curves of the direct wave and the head wave. In the example shown, the transit time curve is a straight line because it moves at a constant speed along a level layer boundary. The seismic speed can therefore be derived directly from the slope, as . ${\displaystyle {\frac {1}{v_{1}}}}$

As the head wave only exists after reaching the critical angle, it can only be measured on the surface after a certain distance. This so-called critical distance (x _c ) (or the critical point ) depends on the thickness (z ₀ ) of the upper layer:

${\displaystyle x_{c}=2z_{0}\tan(i_{c})}$ .

By lengthening the runtime curve back, the theoretical intersection point with the y-axis, the so-called intercept time (t _i ), is obtained, which is mathematically derived

${\displaystyle t_{i}={\frac {2z_{0}\cos(i_{c})}{v_{0}}}}$ results.

From this it follows as a straight line equation for the head wave:

${\displaystyle t(x)=2z_{0}{\frac {\cos(i_{c})}{v_{0}}}+{\frac {x}{v_{1}}}}$   .

Individual evidence

1. ^ ^{A b} Hans Berckhemer : Fundamentals of Geophysics , Scientific Book Society 2002, ISBN 978-3534136964
2. ↑ R. Kirsch & W. Rabbel : Seismic methods in environmental geophysics in: Martin Beblo (Hrsg.): Umweltgeophysik , Ernst & Sohn Verlag f. Architecture and technical sciences, Berlin 1997, ISBN 3433015414
3. ↑ ^{a b} WM Telford, LP Geldart & RE Sheriff: Applied Geophysics , Cambridge University Press, 1990, ISBN 978-0521339384 (Engl.)