Seismogram example Recording system Basic theory
Poles and zeros, SEED headers
How to read and use seismometer response parameters
Thomas Forbriger
Karlsruhe Institute of Technology (KIT)
Black Forest Observatory (BFO)
September 2011
Thomas Forbriger
Poles and zeros, SEED headers
Seismogram example Recording system Basic theory
North Sumatra earthquake, 12/2004, M > 9
Raw data recorded at BFO
Thomas Forbriger
Poles and zeros, SEED headers
Seismogram example Recording system Basic theory
North Sumatra earthquake, 12/2004, M > 9
Deconvolution of seismic record to ground acceleration
Thomas Forbriger
Poles and zeros, SEED headers
Seismogram example Recording system Basic theory
North Sumatra earthquake, 12/2004, M > 9
Deconvolution of seismic record to ground displacement
Thomas Forbriger
Poles and zeros, SEED headers
Seismogram example Recording system Basic theory
North Sumatra earthquake, 12/2004, M > 9
The seismometer acts as a waveform filter
Thomas Forbriger
Poles and zeros, SEED headers
Seismogram example Recording system Basic theory
Recording system
The seismic recording system consists of a sequence of signal
transformers.
from SEED manual V2.4 page 139
A proper physical model for the system and its parameters is required
for a successful signal deconvolution. With properly designed digital
systems all components can be ignored, except the seismometer and
amplifiers.
Thomas Forbriger
Poles and zeros, SEED headers
Seismogram example Recording system Basic theory
Recording system
Beware!
Although there are numerous tools available to work on seismograph
response parameters, there is (as always) no alternative to a sound
understanding of the underlying models and properties. There are
known cases of wrongly specified instrument responses in data of high
reputation and of well established software tools introducing erroneous
signal delays.
Thomas Forbriger
Poles and zeros, SEED headers
Seismogram example Recording system Basic theory
Basic elements of seismometer response theory
Differential equation
Electrodynamic seismometers (geophones) and most broad-band
feedback seismometers are described by a simple differential
equation:
Ü (t ) + 2hω0 U̇ (t ) + ω20 U (t ) = K v̈ (t )
U: output voltage (Volts)
v : ground velocity (m/s)
K : gain (Vs/m)
ω0 : angular eigenfrequency (rad/s)
h: damping as a fraction of critical damping
(dimensionless)
Thomas Forbriger
Poles and zeros, SEED headers
Seismogram example Recording system Basic theory
Basic elements of seismometer response theory
Transfer function
H (s ) =
H (s):
s:
s1,2 :
Ũ:
ṽ :
K:
ω0 :
h:
Ũ (s)
ṽ (s)
=K
s2
s2 + 2hω0 s + ω20
=K
s2
(s − s1 ) (s − s2 )
transfer function (Vs/m)
parameter of Laplace transformation (rad/s)
poles of transfer function (rad/s)
Laplace transform of output voltage
Laplace transform of ground velocity
gain (Vs/m)
angular eigenfrequency (rad/s)
damping as a fraction of critical damping
(dimensionless)
Thomas Forbriger
Poles and zeros, SEED headers
Seismogram example Recording system Basic theory
Basic elements of seismometer response theory
Poles of the transfer function
Poles and zeros must be either real values or appear as complex
conjugate pairs of second order systems.
s1,2 = ω0 (−h ± i
s:
s1,2 :
ω0 :
T0 :
h:
p
1 − h2 ) =
2π
T0
(−h ± i
p
1 − h2 )
parameter of Laplace transformation (rad/s)
poles of transfer function (rad/s)
angular eigenfrequency (rad/s)
eigenperiod (s)
damping as a fraction of critical damping
(dimensionless)
Thomas Forbriger
Poles and zeros, SEED headers
Seismogram example Recording system Basic theory
Basic elements of seismometer response theory
Descriptive functions
Amplitude response or ”gain” function:
f2
M (f ) = |H (i2πf )| = K f − s1
2πi
f2
f − 2sπ2i
= K p
(f 2 − f 2 )2 + (2hff0 )2
Phase response:
Φ(f ) = arctan
ImH (i2πf )
ReH (i2πf )
Phase delay:
∆tph (f ) = −
Φ(f )
2πf
Group delay:
∆tgr (f ) = −
1
d 2π df 0 f 0 =f
Φ(f 0 )
Thomas Forbriger
Poles and zeros, SEED headers
0
Seismogram example Recording system Basic theory
Basic elements of seismometer response theory
Descriptive parameters
Any second order system can be described by two parameters which
can easily be derived from the location of poles.
1. Eigenperiod (defines pass-band):
T0 =
2π
|s1,2 |
2. Damping as a fraction of critcal damping:
h=−
T0
2π
Re s1,2
Thomas Forbriger
Poles and zeros, SEED headers