## PSN-L Email List Message

Subject: Digital Filter for Seismography
From: Bobhelenmcclure@.......
Date: Mon, 22 Apr 2002 14:06:23 EDT

```Hi everyone,

Here is something for you to ponder over.  Since I am new to the field, what

HOW TO DIGITALLY EXTEND THE LONG PERIOD RESPONSE
OF A SEISMOMETER
R. E. McClure
Pseudoscientist Emeritus

From the electrical equivalent circuit diagram of an input series capacitor
feeding an inductance and a resistance in parallel, the velocity response of
a seismometer to ground velocity input is given by:

G = 1/(1 - (f0/f)^2 - j*(f0/f)/Q),

where f0 is the natural frequency and Q is inversely proportional to the
damping of the seismometer pendulum.  A Q of 0.5 is  the critically damped
condition.

To achieve a flat filtered response, the compensating filter must have a
gain of 1/G, i.e.:

Gain = 1 - (f0/f)^2 - j*(f0/f)/Q .

The digital implementation of such a filter is accomplished by double
summation (integration) of the signal, DataIn:

sum1 = sum1 + DataIn
sum2 = sum2 + sum1
DataOut = DataIn + (sum1 * sigma0 * deltaT) + (sum2 * (Omega0 * deltaT)^2)
...next data sample, etc. ...

where

fs = samples per second ,
deltaT = 1 / fs ,
Omega0 = 2* PI * f
sigma0 = (Omega0 * deltaT) / Q .

For practical purposes, this filter is useless.  The output very quickly
becomes large without limit if there is any dc bias at all in the input data.

The next necessary step is to precede the filter with a long time constant
dc bias blocking filter.  This helps, but is still not sufficient to make a
stable filter.

The final step is to close the loop on the double integration with feedback
to the signal input from both the first and second integration outputs.  The
coefficients for the feedback make the loop behave like a very long period,
damped, pendulum.  The resulting output of the filter is equivalant to that
obtained from a very long period sensor.

This filter is built into DrumPlot.exe.  The user inputs the period and
damping of the sensor, and the desired filtered period.  It works very well.
One cannot expect to get more than a ten times improvement in long period
response.  You will find that the output does not truly reproduce just ground
motion.  There will the artifacts also appearing, such as that resulting from
amplifier bias fluctuation, ambient temperature changes, atmospheric pressure
changes, etc.  You may also discover that wind has a large effect.  I live
about 1000 feet from a commuter railroad line.  I pick up the vibration of
passing trains, filter or no, but with the filter operating at long period, I
also detect the slow earth deformation induced by the weight of the passing
locomotive!

The verbatim listing for the filter in the DrumPlot program is:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
'DRUMPLOT DC-BLOCKING FILTER:
samplebare = sample
BiasRegister = BiasRegister + samplebare / Tc0
Deltabiasregister = BiasRegister / Tc0
sampleblock = samplebare - BiasRegister
BiasRegister = BiasRegister - Deltabiasregister

'DRUMPLOT EXTENDED-PERIOD FILTER:
sum1 = sampleblock + sum1 - sum1 * SigmaF - sum2 * Omega2F
sum2 = sum2 + sum1
sample = sampleblock + sum1 * SigmaP + sum2 * Omega2P
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The terms Omega2P, SigmaF,  Omega2P, and SigmaP are:

Omega2P = (OmegaPendulum * sampleperiod)^2
SigmaP  = OmegaPendulum / QPendulum

Omega2F = (OmegaFilter * sampleperiod)^2
SigmaF  = OmegaFilter / QFilter
(Set QFilter equal to 0.5)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Another nice benefit of this filter, regardless of whether a long period
response is desired or not, is that the output of the seismometer, if its
natural period and damping are accurately known, can be converted into one
based on a standard model.  There will then be a common ground on which to
compare waveforms obtained from sensors of different period and damping.

Robert E McClure
90 Maple Avenue
Locust Valley, NY 11560
bobhelenmcclure@.......
Hi everyone,

Here is something for you to ponder over.  Since I am new to the field, what is disclosed here may be old hat to you.

HOW TO DIGITALLY EXTEND THE LONG PERIOD RESPONSE
OF A SEISMOMETER
R. E. McClure
Pseudoscientist Emeritus

From the electrical equivalent circuit diagram of an input series capacitor feeding an inductance and a resistance i
n parallel, the velocity response of a seismometer to ground velocity input is given by:

G = 1/(1 - (f0/f)^2 - j*(f0/f)/Q),

where f0 is the natural frequency and Q is inversely proportional to the damping of the seismometer pendulum.  A Q of
0.5 is  the critically damped condition.

To achieve a flat filtered response, the compensating filter must have a gain of 1/G, i.e.:

Gain = 1 - (f0/f)^2 - j*(f0/f)/Q .

The digital implementation of such a filter is accomplished by double summation (integration) of the signal, DataIn:

sum1 = sum1 + DataIn
sum2 = sum2 + sum1
DataOut = DataIn + (sum1 * sigma0 * deltaT) + (sum2 * (Omega0 * deltaT)^2)
...next data sample, etc. ...

where

fs = samples per second ,
deltaT = 1 / fs ,
Omega0 = 2* PI * f
sigma0 = (Omega0 * deltaT) / Q .

For practical purposes, this filter is useless.  The output very quickly becomes large without limit if there i
s any dc bias at all in the input data.

The next necessary step is to precede the filter with a long time constant dc bias blocking filter.  This helps
, but is still not sufficient to make a stable filter.

The final step is to close the loop on the double integration with feedback to the signal input from both the first
and second integration outputs.  The coefficients for the feedback make the loop behave like a very long period, damped, p
endulum.  The resulting output of the filter is equivalant to that obtained from a very long period sensor.

This filter is built into DrumPlot.exe.  The user inputs the period and damping of the sensor, and the desired
filtered period.  It works very well.  One cannot expect to get more than a ten times improvement in long period resp
onse.  You will find that the output does not truly reproduce just ground motion.  There will the artifacts also appe
aring, such as that resulting from amplifier bias fluctuation, ambient temperature changes, atmospheric pressure changes, etc.
You may also discover that wind has a large effect.  I live about 1000 feet from a commuter railroad line.  I p
ick up the vibration of passing trains, filter or no, but with the filter operating at long period, I also detect the slow eart
h deformation induced by the weight of the passing locomotive!

The verbatim listing for the filter in the DrumPlot program is:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
'DRUMPLOT DC-BLOCKING FILTER:
samplebare = sample
BiasRegister = BiasRegister + samplebare / Tc0
Deltabiasregister = BiasRegister / Tc0
sampleblock = samplebare - BiasRegister
BiasRegister = BiasRegister - Deltabiasregister

'DRUMPLOT EXTENDED-PERIOD FILTER:
sum1 = sampleblock + sum1 - sum1 * SigmaF - sum2 * Omega2F
sum2 = sum2 + sum1
sample = sampleblock + sum1 * SigmaP + sum2 * Omega2P
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The terms Omega2P, SigmaF,  Omega2P, and SigmaP are:

Omega2P = (OmegaPendulum * sampleperiod)^2
SigmaP  = OmegaPendulum / QPendulum

Omega2F = (OmegaFilter * sampleperiod)^2
SigmaF  = OmegaFilter / QFilter
(Set QFilter equal to 0.5)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Another nice benefit of this filter, regardless of whether a long period response is desired or not, is that the out
put of the seismometer, if its natural period and damping are accurately known, can be converted into one based on a standard m
odel.  There will then be a common ground on which to compare waveforms obtained from sensors of different period and damp
ing.

Robert E McClure
90 Maple Avenue
Locust Valley, NY 11560
bobhelenmcclure@.......
```