Seismometer – The basic Principles
x0
xr
x
x
x0
xm
um
u
ug
u
xr
x0
Seismology and the Earth’s Deep Interior
ground displacement
displacement of seismometer mass
mass equilibrium position
Seismometry
Seismometer – The basic Principles
The motion of the seismometer mass as a
function of the ground displacement is given
through a differential equation resulting
from the equilibrium of forces (in rest):
x0
xr
x
Fspring + Ffriction + Fgravity = 0
for example
Fsprin=-k x, k spring constant
.
Ffriction=-D x, D friction coefficient
..
Fgravity=-mu, m seismometer mass
Seismology and the Earth’s Deep Interior
ug
Seismometry
Seismometer – The basic Principles
using the notation introduced the equation of
motion for the mass is
&x&r (t ) + 2εx&r (t ) + ϖ 02 xr (t ) = −u&&g (t )
D
ε=
= hϖ 0 ,
2m
x0
x
k
2
ϖ0 =
m
From this we learn that:
- for slow movements the acceleration and
velocity becomes negligible, the
seismometer records ground acceleration
xr
ug
- for fast movements the acceleration of the
mass dominates and the seismometer
records ground displacement
Seismology and the Earth’s Deep Interior
Seismometry
Seismometer – examples
x
x0
x
r
u
g
Seismology and the Earth’s Deep Interior
Seismometry
Seismometer – examples
x
x0
x
r
u
g
Seismology and the Earth’s Deep Interior
Seismometry
Seismometer – examples
x
x0
x
r
u
g
Seismology and the Earth’s Deep Interior
Seismometry
Seismometer – examples
x
x0
x
r
u
g
Seismology and the Earth’s Deep Interior
Seismometry
Seismometer – examples
x
x0
x
r
u
g
Seismology and the Earth’s Deep Interior
Seismometry
Seismometer – examples
x
x0
x
r
u
g
Seismology and the Earth’s Deep Interior
Seismometry
Seismometer – Questions
1. How can we determine the damping
properties from the observed behavior of the
seismometer?
x
x0
x
r
u
g
2. How does the seismometer amplify the
ground motion? Is this amplification
frequency dependent?
We need to answer these question in order to
determine what we really want to know:
The ground motion.
Seismology and the Earth’s Deep Interior
Seismometry
Seismometer – Release Test
1.
How can we determine the damping
properties from the observed behavior of
the seismometer?
x
x0
x
r
u
g
&x&r (t ) + hϖ 0 x& r (t ) + ϖ 02 xr (t ) = 0
xr (0) = x0 ,
x& r (0) = 0
we release the seismometer mass from a given initial
position and let is swing. The behavior depends on the
relation between the frequency of the spring and the
damping parameter. If the seismometers oscillates, we
can determine the damping coefficient h.
Seismology and the Earth’s Deep Interior
Seismometry
Seismometer – Release Test
F0= 1Hz , h= 0
F0= 1Hz , h= 0.2
1
x
x0
1
Dis plac em ent
x
0.5
0.5
0
0
u
g
-0.5
-0.5
-1
-1
0
1
2
3
4
5
0
1
F0= 1Hz , h= 0.7
Dis plac em ent
r
2
3
4
5
4
5
F0= 1Hz , h= 2.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
0
1
2
3
Tim e (s )
Seismology and the Earth’s Deep Interior
4
5
0
1
2
3
Tim e (s )
Seismometry
Seismometer – Release Test
The damping coefficients
can be determined from
the amplitudes of
consecutive extrema ak
and ak+1
We need the logarithmic
decrement Λ
ak
ak+1
x
x0
x
r
u
g
⎛ ak ⎞
⎟⎟
Λ = 2 ln⎜⎜
⎝ ak +1 ⎠
The damping constant h can then be determined through:
h=
Seismology and the Earth’s Deep Interior
Λ
4π 2 + Λ2
Seismometry
Seismometer – Frequency
x
x0
ak
T
x
r
u
g
ak+1
The period T with which the seismometer mass
oscillates depends on h and (for h<1) is always
larger than the period of the spring T0:
Seismology and the Earth’s Deep Interior
T=
T0
1 − h2
Seismometry
Seismometer – Response Function
2. How does the seismometer amplify the ground
motion? Is this amplification frequency dependent?
To answer this question we excite our seismometer
with a monofrequent signal and record the response of
the seismometer:
x
x0
x
r
u
g
&x&r (t ) + hϖ 0 x&r (t ) + ϖ 02 xr (t ) = ϖ 2 A0 e iϖt
the amplitude response Ar of the seismometer depends
on the frequency of the seismometer w0, the
frequency of the excitation w and the damping
constant h:
Ar
1
=
2
A0
2
2
⎞
⎛T
T
⎜⎜ 2 − 1⎟⎟ + 4h 2 2
T0
⎠
⎝ T0
Seismology and the Earth’s Deep Interior
Seismometry
Seismometer – Response Function
Ar
1
=
2
A0
2
2
⎞
⎛T
T
⎜⎜ 2 − 1⎟⎟ + 4h 2 2
T0
⎠
⎝ T0
Seismology and the Earth’s Deep Interior
x
x0
x
r
u
g
Seismometry
Sampling rate
Sampling frequency, sampling rate is the number of sampling
points per unit distance or unit time. Examples?
Seismology and the Earth’s Deep Interior
Seismometry
Data volumes
Real numbers are usually described with 4 bytes (single
precision) or 8 bytes (double precision). One byte consists of
8 bits. That means we can describe a number with 32 (64)
bits. We need one switch (bit) for the sign (+/-)
-> 32 bits -> 231 = 2.147483648000000e+009 (Matlab output)
-> 64 bits -> 263 = 9.223372036854776e+018 (Matlab output)
(amount of different numbers we can describe)
How much data do we collect in a typical seismic experiment?
Relevant parameters:
Sampling rate 1000 Hz, 3 components
Seismogram length 5 seconds
200 Seismometers, receivers, 50 profiles
50 different source locations
Single precision accuracy
How much (T/G/M/k-)bytes to we end up with? What about compression?
Seismology and the Earth’s Deep Interior
Seismometry
(Relative) Dynamic range
What
Whatisisthe
theprecision
precisionof
ofthe
thesampling
samplingof
ofour
ourphysical
physicalsignal
signal
ininamplitude?
amplitude?
Dynamic
Dynamicrange:
range:the
theratio
ratiobetween
betweenlargest
largestmeasurable
measurable
amplitude
to the smallest measurable amplitude Amin. .
amplitudeAAmax
max to the smallest measurable amplitude A
min
The
Theunit
unitisisDecibel
Decibel(dB)
(dB)and
andisisdefined
definedas
asthe
theratio
ratioof
oftwo
two
power
powervalues
values(and
(andpower
powerisisproportional
proportionalto
toamplitude
amplitudesquare)
square)
In
Interms
termsof
ofamplitudes
amplitudes
Dynamic
/A )) dB
Dynamicrange
range==20
20log
log1010(A
(Amax
dB
max/Amin
min
Example:
=1, Amax=1024)
Example:with
with1024
1024units
unitsof
ofamplitude
amplitude(A
(Amin
min=1, A
max=1024)
20
20log
log1010(1024/1)
(1024/1) dB
dB¡
¡60
60dB
dB
Seismology and the Earth’s Deep Interior
Seismometry
Nyquist Frequency
(Wavenumber, Interval)
The frequency half of the sampling rate dt is called the
Nyquist frequency fN=1/(2dt). The distortion of a physical
signal higher than the Nyquist frequency is called aliasing.
The frequency of the
physical signal is > fN
is sampled with (+)
leading to the
erroneous blue
oscillation.
What happens in space?
How can we avoid
aliasing?
Seismology and the Earth’s Deep Interior
Seismometry
A cattle grid
Seismology and the Earth’s Deep Interior
Seismometry
Signal and Noise
Almost
Almostall
allsignals
signalscontain
containnoise.
noise.The
Thesignal-to-noise
signal-to-noiseratio
ratioisis
an
animportant
importantconcept
conceptto
toconsider
considerininall
allgeophysical
geophysical
experiments.
experiments.Can
Canyou
yougive
giveexamples
examplesof
ofnoise
noiseininthe
thevarious
various
methods?
methods?
Seismology and the Earth’s Deep Interior
Seismometry
Discrete Convolution
Convolution
Convolutionisisthe
themathematical
mathematicaldescription
description of
ofthe
thechange
change
of
ofwaveform
waveformshape
shapeafter
afterpassage
passagethrough
throughaafilter
filter
(system).
(system).
There
Thereisisaaspecial
specialmathematical
mathematicalsymbol
symbolfor
forconvolution
convolution(*):
(*):
y (t ) = g (t ) ∗ f (t )
Here
Herethe
theimpulse
impulseresponse
responsefunction
functionggisisconvolved
convolvedwith
with
the
input
signal
f.
g
is
also
named
the
„Green‘s
function“
the input signal f. g is also named the „Green‘s function“
y k = ∑ g i f k −i
gi
i = 0,1,2,...., m
k = 0,1,2, K , m + n
fj
j = 0,1,2,...., n
m
i =0
Seismology and the Earth’s Deep Interior
Seismometry
Convolution Example
(Matlab)
Impulse
Impulseresponse
response
>>
>>xx
xx==
00
>>
>>yy
00
00
System
Systeminput
input
yy==
11
22
>>
>>conv(x,y)
conv(x,y)
ans
ans==
00
11
00
Seismology and the Earth’s Deep Interior
11
11
22
11
00
System
Systemoutput
output
Seismometry
Convolution Example (pictorial)
„Faltung“
x
0
0
1
0
2
1
1
0
0
1
2
1
0
1
0
0
1
2
1
0
2
0
2
Seismology and the Earth’s Deep Interior
0
2
0
1
0
x*y
1
0
1
1
y
1
1
1
1
0
0
0
0
1
0
0
1
2
1
0
1
Seismometry
Deconvolution
Deconvolution
Deconvolutionisisthe
theinverse
inverseoperation
operationto
toconvolution.
convolution.
When
Whenisisdeconvolution
deconvolutionuseful?
useful?
Seismology and the Earth’s Deep Interior
Seismometry
Digital Filtering
Often
Oftenaarecorded
recordedsignal
signalcontains
containsaalot
lotof
ofinformation
informationthat
that
we
weare
arenot
notinterested
interestedinin(noise).
(noise).To
Toget
getrid
ridof
ofthis
this
noise
noisewe
wecan
canapply
applyaafilter
filterininthe
thefrequency
frequencydomain.
domain.
The
Themost
mostimportant
importantfilters
filtersare:
are:
••
High
Highpass:
pass:cuts
cutsout
outlow
lowfrequencies
frequencies
••
Low
Lowpass:
pass:cuts
cutsout
outhigh
highfrequencies
frequencies
••
Band
Bandpass:
pass:cuts
cutsout
outboth
bothhigh
highand
andlow
lowfrequencies
frequenciesand
and
leaves
leavesaaband
bandof
offrequencies
frequencies
••
Band
Bandreject:
reject:cuts
cutsout
outcertain
certainfrequency
frequencyband
bandand
and
leaves
all
other
frequencies
leaves all other frequencies
Seismology and the Earth’s Deep Interior
Seismometry
Digital Filtering
Seismology and the Earth’s Deep Interior
Seismometry
Low-pass filtering
Seismology and the Earth’s Deep Interior
Seismometry
Lowpass filtering
Seismology and the Earth’s Deep Interior
Seismometry
High-pass filter
Seismology and the Earth’s Deep Interior
Seismometry
Band-pass filter
Seismology and the Earth’s Deep Interior
Seismometry
Seismic Noise
Observed seismic noise as a function of frequency (power spectrum).
Note the peak at 0.2 Hz and decrease as a distant from coast.
Seismology and the Earth’s Deep Interior
Seismometry
Instrument Filters
Seismology and the Earth’s Deep Interior
Seismometry
Time Scales in Seismology
Seismology and the Earth’s Deep Interior
Seismometry