seismology part vii: characteristics of surface wave propagation

Seismology

Part VII:

Characteristics of Surface Wave Propagation

Propagation on a Spherical Earth

Waves follow great circle paths

They emanate from a source, propagate to an antipode, and then back to the source.

Any station not at an antipode will record a wave traveling along the minor arc (first) and then the major (or longer) arc (second).

Nomenclature: Rayleigh waves are "R", Love waves are sometimes called "G" (after Beno Gutenberg). Subscripts denote the arc, with odd number for repeated minors, even for repeated majors. For example, R1 is the first minor, R3 is a minor plus 2, R2 is the first major, R4 is the major plus 2. Overtones are generally labeled "X" with the same subscript convention.

Seismograms of the 1989 Loma Prieta (central California) earthquake in record section form showing the long period Rayleigh wave (largest amplitudes) labeled R1, R2, R3.  The seismograms are vertical component records and have been filtered to include only periods longer than 125 seconds.  Seismograph stations that recorded the seismograms, in order of distance, are:  AMNO, COL, KIP, HRV, SJG, PPT, RPN, AFI, CAY, MDJ, HIA, TOL, BDF, WMQ, TAM, KMY, CAN, TWO, HYB, BCAO, NWAO, SLR, RER.  Locations of most of these stations are shown on the maps in Figures 2 and 3.  (Modified from Lay and Wallace, 1995).

Schematic diagram illustrating a cross section through the Earth and the locations of seismograph stations.  Distance from the earthquake epicenter to a station can be measured in kilometersalong the surface or by the geocentric angle.

Group vs Phase velocities.

We construct our theory based on monochromatic waves (i.e., one frequency) but the reality is that a source will emit a continuous band of frequencies. If all the frequencies travel at the same speed, then there is no distortion of the waveform. But if the medium is dispersive, then we need to be mindful of what exactly it is we are looking at in a seismogram.

We never record a monochromatic signal. If the medium is very dispersive, then if we wait long enough we might get close to seeing what a single frequency is doing, but in general we looking at the behavior of a BAND of frequencies that are interfering with each other.

The classic way to discuss this phenomenon is to observe what happens with a signal composed of 2 similar frequencies.

u cos( 't k 'x) cos(" t k"x)

let '" (so "'), and k = k' + k = k" - k and and k << k. Substitution gives:

u cos(( )t (k k)x) cos(( )t (kk)x)

u cos((t kx) (t kx)) cos((t kx)(t kx))

cos(x y) cos(x y) 2cos(x)cos(y)

u 2cos(t kx)cos(t kx)

Recalling that

and identifying x = t - kx and y = t - kx, we have

which is a "beat" wave, consisting of a wave moving at the mean phase velocity (c = /k) multiplied by an "envelope" moving at the "group" velocity U = /k.

Example of dk = 1, dw = 0, so group velocity is 0

Example of dk = 1, dw = 2, so group velocity is 2

If we have a continuous band of frequencies ( and k -> 0), then

U k

(kc)

kc k

c

kc

2

c

(2 /)c c

For non-dispersive media, c/k = 0 and U = c.

In general in the Earth, c increases with wavelength (decreases with frequency), so typically U < c.

Typically, we measure similar looking arrivals at stations at increasing distance, estimate the frequency from 1/T, and then get U from (travel time)/(distance).

This will be more accurate if we pass the wave train through a series of narrow band filters first.

Measuring group and phase velocities

Group Velocities:

If we make measurements in the time domain, we are always measuring group velocities.

Phase VelocitiesPhase velocities can be deduced from group velocities using the relations given above.

We can also measure them directly by computing FFTs at a number of stations and comparing the change in phase associated with a given frequency.

The relationship between a time series and its Fourier transform can be written as:

u(x,t) 1

ˆ u (x,

0

)cos(t

c( )xo( ))

The argument to the cosine represents the total phase shift as a function of time and space; o is the initial phase shift at the source for a given frequency. Note that we can introduce an arbitrary shift of 2n to the phase - this represents a fundamental ambiguity in determining phase velocities from phase shifts.

Let's suppose we have two seismograms, one recorded at position x1 at time t1, and the other at (x2, t2). The difference in observed phases will be:

1( ) 2( ) ( t1 t2 )

c( )(x1 x2 ) 2M

c( ) (x1 x2 )

(t1 t2 ) 2M (1( ) 2( ))

from which we get the phase velocities:

c( ) x1 x2

t1 t2 TM (T / 2 )(1( ) 2( ))

c( ) x1 x2

t1 t2 T M (1/ 2 )(1( ) 2( ))

Thus, knowing x and t, we can determine c. Once we know c, we can get U (if we want) from the relations above.

We choose M to give the most realistic result for c (closest to expected result).

Note that c usually changes rapidly for high f and slowly for low f (reflecting larger changes in wavespeed with depth near the surface), which means that while it increases monotonically it will have an inflection point. The group velocity U will thus often have a minimum, meaning that significant band of frequencies will all arrive (a) late and (b) at the same time. This late arriving phase is called the Airy phase and has a period on the order of 20s for continental and 10-15 s for oceanic paths.

For this reason, the Ms magnitude scale is typically based on the amplitude of 20 s surface waves.