What is Q?
Interpretation 1: Suppose A0 represents wave amplitudes, then
ln(A)
A  A0ebt  A0e 0 t /(2Q )
 0
ln( A)  ln( A0 )   t
2Q

intercept
slope
t

Interpretation 2:
Suppose u represents displacement, then
u(t)  A0e i(a ib)t  A0e i 0 t e 0 t /(2Q )
 =“modified” or “instantaneous freq”
a   0 11/4Q2 (real)
0
b
(imaginary)
2Q


Suppose: small attenuation, the
0 b  2Q
We can define b = *, where * ---> 0 as Q increases (imaginary
1
freq due to attenuation),
a  

* 
2Q
Relation with velocity:

Q1  2 * /

* 
Q1
c  ic*   i
 i
k
k
k
2k
Imaginary velocity
due to attenuation



c*
c*  Q  Q  2
2k
c
1
1
So Q is a quantity that defines the relationship between real and
imaginary frequency (or velocity) under the influence of attenuation.
Interpretation
 3: Q is the number of cycles the oscillations take to
decay to a certain amplitude level. n  t /T  t( /2 )
if Q  
then n  t  0/2 
n  2
 tn 
0
So amplitude at time tn (after n cycles)
A(tn )  A0e

w 0 t n / 2Q
 A0e
w 0 n 2  /( 0 2Q )
 A0e
n /Q
2
Attenuation and Physical Dispersion (continued…)
Different interpretations of Q (quality factor):
(1)As a damping term Q = 0/g
(2)As a fraction between imaginary and real frequencies (or imaginary
velocity to real velocity)
2 *
1
-1 2c *
Q 
or Q =

c
(3)As the number of cycles for a wave to decay to a certain amplitude.
If n = Q, then

A(t n )  A0e 0 t / 2Q  A0e( 0 / 2Q )(n*2 /  0 )  0.04 A0
(4)Connection with t* (for body wave).
N
dt
t i
t*  

 path Q(r) i=1 Qi
(N layers, r = location)
(5)Energy formula
1
E

Q( )
2E
1 number that describes Q
structure of several layers
3
(-E = energy loss per cycle)
Effects of Q (assume the SAME Q value)
(1) A( )  A0 ( )e
 0 t / 2Q
dependencies
 A0 ( )e
x /(2Qv)
 large distance x - - - - > more amplitude decay

large velocity v - - - -- > the less amplitude decay
large frequency  - - - - > more amplitude decay

(2) Physical dispersion: different frequency component travels at
different times, hence causing broadening of phase pulse.
Lets assume a Dirac delta function: area = 1, infinite height.
u(x,t)
u(x,t)  (t  x /c) (x = distance, c = velocity)
spectrum F( ) 
t



 (t  x /c)e


it
u(x,t)e
dt


it
dt
e
ix / c
F
Rough sketch of the real part
of spectrum

Key: The Fourier spectrum of a delta function contains infinite
number of frequencies/velocities, not a singlet nor a
constant nor zero!
Now: Lets find out what the delta function becomes IF there is
attenuation (or, finite Q value).
 x
J( )  e 2cQ
Inverse FFT to get time domain signal


x
2cQ
ix
1
it
it
c
u(x,t)   J( )F( )e d 
e
e
e
d

2 

Eventually
u(x,t)  {( x
/2cQ) /[( x /2cQ) 2  (x /c  t) 2 ]} /  5
u(x,t)  {( x /2cQ) /[( x /2cQ) 2  (x /c  t) 2 ]} / 
Noncausal
Causal
Attenuation
Attenuation
elastic
Operator
Operator
Observation: pulse is “spread out” which means dispersive (different
frequencies arrive at different times!)
Problem: envelope of the function is nonzero before t=x/c (it is like
receiving earthquake energy before the rupture, not physical!) 6
How to make this process causal?
One of the often-cited solutions: Azimi’s Attenuation Law:
 1  
c( )  c 0 1
ln  c  reference speed for frequency 
0
 Q  0  0
if Q = ,
then c = c0 (no dispersion)
if  = 0,
then c = c0 (no dispersion)

then c  c0 (means high freq arrives first)
if  = ,
Where does it come from?
Answer: Derived under the following causality condition:
u(x,t)  0 for all t < x/c(), where c( ) is the
highest (infinite) frequency that arrives first.
Physical Models of Anelasticity
In an nutshell, Earth is composed of lots of Viscoelastic (or
Standard Linear) Solids
7
Standard Linear Solid (or Viscoelastic Solid)
Spring const

Spring const
k1
m
k2

Specification: (1) consist of two springs and a dashpot
(2) viscosity of fluid inside dashpot
8

Governing Equation (stress):
 (t)  k1H(t)  k2e
t / 
where H(t)  step (or Heavyside function)
and  = /k 2 (relaxation time)
The response to a harmonic wave depends on the product of the
angular frequency and the relaxation time.


c()
1
Q
viscous
c(0)
elastic
 1
100
c()

 1

100
The left-hand figure is the absorption function. The absorption is small
at both very small and very large frequencies. It is the max at 9 = 1!

A given polycrystalline material in
the Earth is formed of many SLS
superimposed together. So the final
Frequency dependent Q is constant
for many seismic frequencies ----
1
Q
~constant

Question: Is the fact that high
frequencies are attenuated more a
contradiction to this flat Q observation??
100
0.0001
Frequency(Hz)
Frequency dependence of mantle attenuation
Answer: No. High frequencies are
attenuated more due to the following
equation that works with the same Q.
So it is really “frequency dependent
amplitude”, NOT “frequency dependent
Q”.
A( )  A0 ( )ex /(2Qv)
10
Eastern USA (EUS) and Basin-and-Range Attenuation
Mitchell 1995 (lower
attenuation occurring at
high frequencies, tells
us that frequency
independency is not
always true
Low Q in the asthenosphere,
Romanowicz, 1995
11
An example of Q extraction from differential waveforms.
Two phases of interest:
sS-S
sScS - ScS
Flanagan & Wiens (1990)
12
Key Realization: The two waveforms are similar in nature,
mainly differing by the segment in the above source (the small
depth phase segment)
sS()  S()R()A()
sS( )  frequency spec of sS
S( )  frequency spec of S
R( )  crustal operator

A( )  attenuation operator of interest
Approach: Spectral division of S from SS, then divide out the
13 accounts for the
Crustal operator (a function in freq that
of the additional propagation in a normal crust)

Spectral dividing S and R will leave the attenuation term A(w)
(slope = =t/2Q,
t
t / 2Q
A( )  e
 log( A( ) )  

t = time diff of sS-S)
2Q