EAS 557
Introduction to Seismology
Robert L. Nowack
Lecture 13B
II. Reflection and Transmission of Waves at Boundaries
The two boundary conditions of continuity of particle displacement and traction at a boundary for an incident plane wave can be written
Assuming the phase terms are all equal at x3 = 0, then
which is Snell’s law or conservation of horizontal “slowness” from a horizontal interface. Then,
at x3 = 0. This results in six equations for six unknowns. The unknowns to be solved for are and . Since A1, A3, T1, T3 for the in-plane motion don’t depend on A2, T2 for the antiplane motions and visa versa, the system decouples.
1) Incident anti-plane “SH” case.
For this case, we have two equations for two unknowns, and . For this case
Now,
Then,
From the second equation,
Finally,
(See also Aki and Richards, 1980, page 144.) Thus, the reflection and transmission coefficients for the displacement amplitude are related to the “impedance differences” across the boundary.
For vertical incidence
This depends only on “impedances ” and not or alone. Thus, normal incident “seismic reflection studies” can only infer impedance as a function of depth. Multiple offset source and receiver data are required to separate and .
2) P-SV case – incident P-wave
In this case, we have at x3 = 0
This is four equations for four unknowns, where stands for the x1 component of an incident P-wave. (See Aki and Richards, 1980, page 150, for the formulas for the P-SV case.)
Since from Snell’s law , then we have a scattering diagram
where scattered SV waves are always closer to vertical then the similarly scattered P waves (since they sense lower velocities).
A P wave incident on a traction free surface can be used to model the scattering of waves at the Earth’s surface.
At a traction free surface, the tractions are zero at the surface x3=0. Thus,
These are two equations for two unknowns. Each of the different waves can be written
Upgoing P,
Downgoing P,
Downgoing SV,
The resulting equations for the scattered amplitudes ARP and ARS are given by 5.26-5.31 from Aki and Richards (1980).
An example of the reflection coefficients from a free surface for = 5 km/s and = 3 km/s are given below where the angle of incidence i for an incident P-wave and j for an incident S-wave are shown on the top. The slowness p is shown below.
(from Aki and Richards, 1980)
No P-S or S-P conversions occur at normal incidence. For the case of a welded interface, mode conversions also don’t occur at normal incidence.
For an incident SH wave on a free surface,
and
Then,
Thus, for the SH case at a free surface, the reflection coefficient is 1 for all angles of incidence. Combining the incident and reflected waves at the free surface gives rise to a doubling of the SH ground motion at the free surface.
The seismic impedance, I, is defined as a stress over a particle velocity giving the resistance of the particle motion to the applied stress. Hence the name “seismic impedance”. In the case of SH, antiplane motion
for a plane wave
where
Then,
and,
The impedance I can then be written
As shown earlier, the SH reflection coefficient is equal to the mismatch in seismic wave impedances across a boundary. At normal incidence, .
The acoustic impedance for a P wave at a fluid-fluid will be, , but for normal incidence this is . For the general P-SV elastic case, writing the reflection and transmission coefficients in terms of seismic impedances is more complicated for non-normal incidence.
It is sometimes convenient to work with energy normalized reflection and transmission coefficients. Away from an interface, the wavefront will be compressed or stretched depending on how the wavefront bends at the interface.
For an SH wave, the energy flux per unit area can be written
The incidence energy flux at the boundary for the incident wave is reduced by
For the reflected wave, the energy flux also includes the reflection coefficient R.
For the transmitted wave, the energy flux is given by
where T is the transmission coefficient. Then, for the energy flux,
,
For the P-SV case of a converted reflected wave, this is similar to the transmitted case.
Let’s look at the incident SH case on a welded interface
Since , then if , and . Thus, there will be some for which . This is called the critical angle .
Now for , then ! Thus, becomes complex.
We could avoid this whole business if we ignored the transmitted wave beyond the critical angle and just look at the “super critical reflection”. But, this also has possible problems since the reflection coefficient also includes .
It turns out that allowing the angles to become complex, results in complex reflection and transmission coefficients. This results in the concept called “inhomogeneous plane waves”. These are formally just the same as ordinary plane waves, except that they decay exponentially in one direction. As we will see, coupled P-SV inhomogeneous waves at a free surface will result in free Rayleigh waves at the boundary.
Consider the SH case of a welded interface.
The horizontal slowness is conserved by a horizontal interface. Thus,
For , there will be a critical incident angle for which . This incident angle is called the critical angle . (In P-SV problems, there will be two critical angles.). For , then . There is a way around this and still use the same reflection and transmission coefficients. The refelction coefficient is
Let , then from Snell’s law
with
For , then is an imaginary number and
where and
Thus,
where and . This can be written in terms of magnitude and phase as
where
and
The phase PH ranges from 0 to . A super critical reflection is called a total reflection since the magnitude of the reflection coefficient is 1.
The phase shift distorts the shape of the reflected waveform
(from Cerveny, V. (1987), Ray Methods for Three-Dimensional
Seismic Modeling,
Lecture Notes,
For example, a phase shift of 90o takes a symmetric pulse and changes it to an antisymmetric pulse. A phase shift of 180o multiples the pulse by -1.
For a layer over a half space “wave guide”, super-critically reflected energy constructively interferes in the layer to form a horizontally traveling wave packet called a Love wave in the SH case
Now, what happens to the
transmitted wave corresponding to the supercritically reflected wave? For this case, and . This results in a
purely imaginary term for increasing x3
This term is exponentially decreasing with depth away from the boundary. By contrast, the precritical transmission is oscillatory. Both of these cases are plane waves as long as (in each layer). The case when k3 is imaginary results in an evanescent (or inhomogeneous) plane wave which decays in one direction along the wavefront. Thus, the supercritical transmission does not transmit energy away from the boundary.
Another case of an inhomogeneous wave is the case of a free Rayleigh wave at a free surface. This is combined P and SV motion decaying away from a “free” surface. We will discuss this in greater detail when discussing surface waves.
Finally, let’s draw the “travel time curve” for a set of receivers at a vertical distance H above a welded interface (either acoustic or SH case)
From geometry, the time for the direct wave is (straight-line). Reflected time is
This is a hyperbola in (x1,t). At x1 = 0, the reflection time is
(reflection seismology plots t down in anticipation for converting t to depth). The line marked lateral head wave is an additional phase which relies on the curvature of the wavefront impinging on the interface (a plane wave incident won’t give it to us).
This lateral wave (or head wave) is a diffracted wave in the upper half space due to a curved wave front incident on the interface. It results in a constant moveout wave which travels at a velocity of the lower medium v2. It only appears for distances greater than the critical distance xcrit.
travel time model for head wave
The calculation of the amplitude of the head wave is a complicated integral over incidence angle (a little beyond this course). But, the travel time can be computed using the graphical construction above.
The figure
below shows an example of marine expanding spread seismic profile in the
(from Kennett, 2002, The Seismic Wavefield, Vol. II)
In the next several sections, we will investigate ray theoretical computations in smoothly varying heterogeneous media and layered media.