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Seismology and the Earth’s Deep Interior Seismometry
Seismometer – The basic PrinciplesSeismometer – The basic Principles
u
x x0
xm
x
ug
um
x0 xr
u ground displacementxr displacement of seismometer massx0 mass equilibrium position
Seismology and the Earth’s Deep Interior Seismometry
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):
Fspring + Ffriction + Fgravity = 0
for example
Fsprin=-k x, k spring constant
Ffriction=-D x, D friction coefficient
Fgravity=-mu, m seismometer mass
Seismometer – The basic PrinciplesSeismometer – The basic Principles
ug
x
x0 xr
.
..
Seismology and the Earth’s Deep Interior Seismometry
Seismometer – The basic PrinciplesSeismometer – The basic Principles
ug
x
x0 xr
using the notation introduced the equation ofmotion for the mass is
mkh
mD
tutxtxtx grrr
===
−=++
200
20
,2
)()()(2)(
ϖϖε
ϖε &&&&&
From this we learn that:
- for slow movements the acceleration and velocity becomes negligible, the seismometer records ground acceleration
- for fast movements the acceleration of the mass dominates and the seismometer records ground displacement
Seismology and the Earth’s Deep Interior Seismometry
Seismometer – examplesSeismometer – examples
ug
x
x0 xr
Seismology and the Earth’s Deep Interior Seismometry
Seismometer – examplesSeismometer – examples
ug
x
x0 xr
Seismology and the Earth’s Deep Interior Seismometry
Seismometer – examplesSeismometer – examples
ug
x
x0 xr
Seismology and the Earth’s Deep Interior Seismometry
Seismometer – examplesSeismometer – examples
ug
x
x0 xr
Seismology and the Earth’s Deep Interior Seismometry
Seismometer – examplesSeismometer – examples
ug
x
x0 xr
Seismology and the Earth’s Deep Interior Seismometry
Seismometer – examplesSeismometer – examples
ug
x
x0 xr
Seismology and the Earth’s Deep Interior Seismometry
Seismometer – QuestionsSeismometer – Questions
ug
x
x0 xr1. How can we determine the damping
properties from the observed behavior of the seismometer?
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 TestSeismometer – Release Test
ug
x
x0 xr1. How can we determine the damping
properties from the observed behavior of the seismometer?
0)0(,)0(0)()()(
0
200
===++
rr
rrr
xxxtxtxhtx
&
&&& ϖϖ
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 TestSeismometer – Release Test
ug
x
x0 xr
0 1 2 3 4 5-1
-0.5
0
0.5
1F0=1Hz, h=0
Dis
plac
emen
t
0 1 2 3 4 5-1
-0.5
0
0.5
1F0=1Hz, h=0.2
0 1 2 3 4 5-1
-0.5
0
0.5
1F0=1Hz, h=0.7
Time (s)
Dis
plac
emen
t
0 1 2 3 4 5-1
-0.5
0
0.5
1F0=1Hz, h=2.5
Time (s)
Seismology and the Earth’s Deep Interior Seismometry
Seismometer – Release TestSeismometer – Release Test
ug
x
x0 xrThe damping coefficients
can be determined from the amplitudes of consecutive extrema akand ak+1We need the logarithmic decrement Λ
ak
ak+1
⎟⎟⎠
⎞⎜⎜⎝
⎛=Λ
+1
ln2k
k
aa
The damping constant h can then be determined through:
224 Λ+
Λ=
πh
Seismology and the Earth’s Deep Interior Seismometry
Seismometer – FrequencySeismometer – Frequency
ug
x
x0 xr
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:
20
1 hTT−
=
ak
ak+1
T
Seismology and the Earth’s Deep Interior Seismometry
Seismometer – Response FunctionSeismometer – Response Function
ug
x
x0 xr
tirrr eAtxtxhtx ϖϖϖϖ 0
2200 )()()( =++ &&&
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:
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:
20
22
2
20
2041
1
TTh
TTA
Ar
+⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
Seismology and the Earth’s Deep Interior Seismometry
Seismometer – Response FunctionSeismometer – Response Function
ug
x
x0 xr
20
22
2
20
2041
1
TTh
TTA
Ar
+⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
Seismology and the Earth’s Deep Interior Seismometry
Sampling rateSampling rate
Sampling frequency, sampling rate is the number of samplingpoints per unit distance or unit time. Examples?
Seismology and the Earth’s Deep Interior Seismometry
Data volumesData volumes
Real numbers are usually described with 4 bytes (singleprecision) 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(Relative) Dynamic range
What is the precision of the sampling of our physical signalin amplitude?
Dynamic range: the ratio between largest measurableamplitude Amax to the smallest measurable amplitude Amin.
The unit is Decibel (dB) and is defined as the ratio of twopower values (and power is proportional to amplitude square)
What is the precision of the sampling of our physical signalin amplitude?
Dynamic range: the ratio between largest measurableamplitude Amax to the smallest measurable amplitude Amin.
The unit is Decibel (dB) and is defined as the ratio of twopower values (and power is proportional to amplitude square)
In terms of amplitudes
Dynamic range = 20 log10(Amax/Amin) dB
Example: with 1024 units of amplitude (Amin=1, Amax=1024)
20 log10(1024/1) dB 60 dB
In terms of amplitudes
Dynamic range = 20 log10(Amax/Amin) dB
Example: with 1024 units of amplitude (Amin=1, Amax=1024)
20 log10(1024/1) dB 60 dB
Seismology and the Earth’s Deep Interior Seismometry
Nyquist Frequency(Wavenumber, Interval)
Nyquist Frequency(Wavenumber, Interval)
The frequency half of the sampling rate dt is called theNyquist frequency fN=1/(2dt). The distortion of a physicalsignal higher than the Nyquist frequency is called aliasing.
The frequency of thephysical signal is > fNis sampled with (+) leading to theerroneous blueoscillation.
What happens in space?How can we avoidaliasing?
Seismology and the Earth’s Deep Interior Seismometry
A cattle gridA cattle grid
Seismology and the Earth’s Deep Interior Seismometry
Signal and NoiseSignal and Noise
Almost all signals contain noise. The signal-to-noise ratio isan important concept to consider in all geophysicalexperiments. Can you give examples of noise in the variousmethods?
Almost all signals contain noise. The signal-to-noise ratio isan important concept to consider in all geophysicalexperiments. Can you give examples of noise in the variousmethods?
Seismology and the Earth’s Deep Interior Seismometry
Discrete ConvolutionDiscrete Convolution
Convolution is the mathematical description of the changeof waveform shape after passage through a filter(system).
There is a special mathematical symbol for convolution (*):
Here the impulse response function g is convolved withthe input signal f. g is also named the „Green‘s function“
Convolution is the mathematical description of the changeof waveform shape after passage through a filter(system).
There is a special mathematical symbol for convolution (*):
Here the impulse response function g is convolved withthe input signal f. g is also named the „Green‘s function“
)()()( tftgty ∗=
nmk
fgym
iikik
+=
=∑=
−
,,2,1,00
K
migi ,....,2,1,0=
njf j ,....,2,1,0=
Seismology and the Earth’s Deep Interior Seismometry
Convolution Example(Matlab)
Convolution Example(Matlab)
>> x
x =
0 0 1 0
>> y
y =
1 2 1
>> conv(x,y)
ans =
0 0 1 2 1 0
>> x
x =
0 0 1 0
>> y
y =
1 2 1
>> conv(x,y)
ans =
0 0 1 2 1 0
Impulse responseImpulse response
System inputSystem input
System outputSystem output
Seismology and the Earth’s Deep Interior Seismometry
Convolution Example (pictorial)Convolution Example (pictorial)
x y„Faltung“
0 1 0 0
1 2 1
0 1 0 01 2 1
0 1 0 0
1 2 1
0 1 0 0
1 2 1
0 1 0 01 2 1
0 1 0 0
1 2 1
0
0
1
2
1
0
y x*y
Seismology and the Earth’s Deep Interior Seismometry
DeconvolutionDeconvolution
Deconvolution is the inverse operation to convolution.
When is deconvolution useful?
Deconvolution is the inverse operation to convolution.
When is deconvolution useful?
Seismology and the Earth’s Deep Interior Seismometry
Digital FilteringDigital Filtering
Often a recorded signal contains a lot of information thatwe are not interested in (noise). To get rid of thisnoise we can apply a filter in the frequency domain.
The most important filters are:
• High pass: cuts out low frequencies
• Low pass: cuts out high frequencies
• Band pass: cuts out both high and low frequencies and leaves a band of frequencies
• Band reject: cuts out certain frequency band and leaves all other frequencies
Often a recorded signal contains a lot of information thatwe are not interested in (noise). To get rid of thisnoise we can apply a filter in the frequency domain.
The most important filters are:
• High pass: cuts out low frequencies
• Low pass: cuts out high frequencies
• Band pass: cuts out both high and low frequencies and leaves a band of frequencies
• Band reject: cuts out certain frequency band and leaves all other frequencies
Seismology and the Earth’s Deep Interior Seismometry
Digital FilteringDigital Filtering
Seismology and the Earth’s Deep Interior Seismometry
Low-pass filteringLow-pass filtering
Seismology and the Earth’s Deep Interior Seismometry
Lowpass filteringLowpass filtering
Seismology and the Earth’s Deep Interior Seismometry
High-pass filterHigh-pass filter
Seismology and the Earth’s Deep Interior Seismometry
Band-pass filterBand-pass filter
Seismology and the Earth’s Deep Interior Seismometry
Seismic NoiseSeismic 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 FiltersInstrument Filters
Seismology and the Earth’s Deep Interior Seismometry
Time Scales in SeismologyTime Scales in Seismology