Purdue University

EAS 557

Introduction to Seismology

Robert L. Nowack

Lecture 15C

Analysis of Surface Wave Phase and Group Velocities

 

 

            In the last lectures, we investigated some of the properties of Love (SH) surface waves and Rayleigh (P-SV) surface waves.  In particular, in the presence of layering, the phase velocity of Love and Rayleigh waves are frequency dependent .  In addition, we may have a number of higher modes  for a given frequency, but here, for simplicity, we will just look at the properties of the fundamental mode.

 

            The surface wave motion observed at a given distance from an earthquake or explosion is the Fourier integral of all frequency components, thus summing up the contribution of each sinusoidal wave.  Depending on the details of the faulting mechanism, each frequency component leaves the source with a specified amplitude and phase.  Propagation through the Earth then modifies the amplitude and shifts the phase.

 

            We record the time history of the surface wave motion at a particular station and want to use it to learn about the source and/or the elastic properties of the intervening path.  We could just Fourier transform the whole wave train to obtain  and .  But, basic analysis of the visual record can also be done since a lot of the surface waves look more or less sinusoidal over certain time ranges.

 

            A surface wave in a layered medium then has

 

1)   Dispersion – the wave train is more drawn out in time with increasing distance.

 

2)   The wave train has an approximate sinusoidal character at large distances with a slowly varying period.

 

 

 

 

(from Lay and Wallace, 1995)

 

The phase velocity is the velocity of any particular point or “phase” of the sinusoidal wave train (say a peak or trough).

 

            Ex)  Consider an infinite sine wave at two locations x1 and x2.

 

 

 

 

The phase peak (1) at distance x1 is delayed in time an amount (t2t1) in traveling a distance (x2x1).  However, unless we are able to follow the wave continuously between x1 and x2, we couldn’t differentiate between point (2) at x2 with any other phase peak at points (3), (4), (5), (6), etc. separated by an integral number of wave periods .  For a wave with period T0 and frequency , the phase velocity is given by

 

 

The phase difference between x1 and x2 is then uncertain by  for a sine wave.

 

            In practice, we have a surface wave u(x,t) perhaps observed at two stations and we wish to determine  between the stations from these records.  If we Fourier analysis the records, we can determine the Fourier phase as a function of frequency at each station.  The phase difference between the two stations is then

 

 

where the  needs to be included to signify the phase ambiguity.

 

Since the x dependence of each sinusoidal component is eikx, the phase shift in propagation from x1 to x2 is

 

 

So,

 

 

So,  can be determined experimentally from two stations by Fourier analysis and phase unwrapping to account for the phase ambiguity.

 

             can also be determined using just one station if the source mechanism and instrument phase response are known.  Then,

 

where

 

 = focal phase (initial phase at source) dependent on depth of source, orientation

 

 = phase shift due to propagation from the source

 

 = instrument phase response

 

 = phase shift from origin time to the start of the seismic record

 

n = integer constant to be determined by a phase unwrapping procedure

 

Then,

 

 

where n is chosen to give a continuous  curve with .  Thus, we can measure  in the frequency domain from both two stations or one station if the focal phase of the source is known.

 

 

 

Wave Packets and Group Velocity

 

            For one mode, the time domain response can be constructed by

 

 

or by a discrete summation

 

 

 

 

 

where .  For small , then  and

 

 

then,

 

 

where

 

 

where .  Then, evaluating the integral

 

 

Now, since

 

 

then

 

 

            Now,  and .  With , then

 

 

Thus, a given phase moves at  and the sinc function envelope, , moves at a velocity .  If we plot the term in brackets as well as the sine wave in front, then

 

 

 

Combining the two terms to get Ii and plotting at two distances x1 and x2, then

 

 

 

 

So, the total motion is the sum over frequency intervals of these individual wave packets.  Note that we can obtain  from  since

 

   and   

 

then

 

 

We can write this as

 

 

or

 

Then

 

For a nondispersive medium,  and  in the direction of propagation.

 

            Another simple way to derive the concepts of phase and group velocities is to consider a sum of two waves with similar frequencies and wavenumbers

 

 

where

 

 

 

 

 

Then adding the two cosines results in

 

 

 

 

 

(from Stein and Wysession, 2003)

 

where the carrier wave may move at a different speed than the envelope and also may even go in an opposite direction.

 

            One final relation.  Often one is given an implicit dispersion relation .  Now, a small change in  and k will give

 

 

The dispersion relation for a single mode is then .  If we want  to also be along the dispersion relation, then .  Thus,

 

 

Note that one must be careful using the chain rule with partial derivatives!

 

            A technique called moving window analysis (Dziewonski et al., 1969) has been used where surface waves are not well-dispersed, such as short continental paths or for long period waves across oceans.  The seismic signal is first filtered about some narrow band frequency centered on .  The group arrival time at frequency  is then defined as the arrival time of the maximum amplitude and the phase velocity is determined from the phase at this time.  This gives an estimate of the group velocity with frequency.  However, the reason why phase velocities are more useful than group velocities is that they are more directly related to the material parameters of the layered Earth.

 

            The top figure below shows the phase velocity c and group velocity cg = U for a crustal model with a 40 km thick crust with  = 3.9 km/sec.  The middle figure below shows successive peaks and troughs of an observed transverse component surface wave.  The lower figure shows the predicted group velocity curve for the reference model.  The group velocity curve that fits the data has a lower group velocity curve with a 40 km thick crust with  = 3.6 km/sec.

 

 

 

 

(from Stein and Wysession, 2003)

 

            The figure below shows Rayleigh wave phase velocities for paths through the oceanic lithosphere.  The path from the earthquake to the station TUC travels through younger lithosphere along the ridge crust and has slower phase velocities than the path to station ARE.

 

 

 

 

(from Stein and Wysession, 2003)

 

            The figure below shows a comparison between an observed seismogram and a calculated seismogram computed by summation of the contribution of the different surface wave modes.  The individual mode contributions are also shown.  Except for the initial body waves, the later parts of the wave train have been well modeled by just the surface components.

 

 

 

 

(from Kennett, 2002)