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 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@.......