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.  Your comments, please.


    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. 
 Your comments, please.





    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@.......