transfer functions of geophones and accelerometers and their effects on frequency content and...

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Transfer functions of geophones andTransfer functions of geophones and

accelerometers and their effects onaccelerometers and their effects onfrequency content and waveletsfrequency content and wavelets

Michael S. Hons

andRobert R. Stewart



OutlineOutline

• Intro to transfer functions• Deriving transfer functions

• Implications in the derivation• Examples

• Conclusions



Transfer FunctionsTransfer Functions

• A is input• B is output

• H is transfer function

H A

B =



Deriving transfer functionsDeriving transfer functions

• u is grounddisplacement

• x is proof mass

displacementrelative to the case

• n is the net motion,used earlier in thederivation

x

u

n



Deriving Transfer FunctionsDeriving Transfer Functions

• Must represent output divided by input• Seismic sensors are “single degree of freedom”

systems, or damped simple harmonic oscillators

2

22

002

2

2 t

u

xt

x

t

x

∂

∂−=+

∂

∂+

∂

∂ω λω




• The transducer – Detects the displacement of the proof mass relative to

the case (x)

• x is the input

– Outputs an electrical signal

• Accelerometer: capacitor responds to proof massdisplacement

• Geophone: magnetic induction responds to proof mass

velocity




• At low frequencies, then proof mass displacementis directly proportional to acceleration

• When ω<<ω0

then2

2

t

u

x ∂

∂∝

-2 3 8 13 18




• At frequencies near resonance, the proof massdisplacement is proportional to velocity

• When ω≅ ω0

thent

u x

∂

∂∝

-2 3 8 13 18 23 28 33 38




• At high frequencies, the proof mass displacementis directly proportional to ground displacement

• When ω>>ω0

then u x∝

-5 15 35 55 75 95



Deriving Transfer FunctionsDeriving Transfer Functions• Input:

– Proof mass displacement relative to case, so A=x

• x is proportional to some aspect of the ground motion, eitherdisplacement, velocity or acceleration

– Thus if ω<<ω0,

or if ω≅ ω0,

or ifω

>>ω

0

2

2

t

u A

∂

∂∝

t u A∂∂∝

u A∝




• Output: – Related to some aspect of the motion of the proof

mass relative to the case (either displacement or

velocity), depending on the transducer used

– For a geophone:

– For an accelerometer:

t

x B

∂

∂∝

x B∝




• Transform to frequency domain, rearrangeaccording to velocity (geophone) output and

assorted inputs yields:

02

00

2

2

2,

2ω ω

ω ω λω ω

ω <<

++−

−=

∂

∂

∂

∂

j

j

t

U

t

X

02

00

2

3

,2 ω ω

ω ω λω ω

ω

>>++−

=∂

∂

j

j

U t

X

2

00

2

2

,

2

ωω ω ω λω ω

ω ≅

++−=

∂

∂

∂

∂

jt

U t

X




• Arranging for displacement (accelerometer)output and various inputs yields:

2

00

2,

2ωω

ω ω λω ω

ω ≅

++−

−=

∂

∂ j

j

t

U

X

02

00

2

2

,

2

ω ω

ω ω λω ω

ω >>

++−

=

jU

X

02

00

2

2

2,

2

1ω ω

ω ω λω ω <<

++−

−=

∂

∂ j

t

U

X



ImplicationsImplications

• In both cases, raw output is a double timederivative of ground displacement

• Geophone equation retains ω in the numerator,

MEMS accelerometer equation does not• Equations as solutions = no frequency limits

• Equations as transfer functions = frequency limits• Geophone equation not a transfer function at low

frequencies



ExamplesExamples

• Geophone response curves

10 Hz, 0.7 damping

10 Hz, 0.1 damping

4 Hz, 0.7 damping



ExamplesExamples

• Accelerometer response curves

1000 Hz, 0.7 damping 1000 Hz, 0.1 damping



ExamplesExamples

Input ground displacement

Bandpass 1-8-60-70 Hz

Input ground velocity

Input ground acceleration



ExamplesExamples

Transducer input Transducer output



ConclusionsConclusions

• Equations governing proof mass motion in termsof ground motion become transfer functions when

they represent transducer output/input

• Raw output from geophone and accelerometer is

expected to be similar

• Geophone equation is not a valid transfer functionfor very low or very high frequencies



AcknowledgementsAcknowledgements• Thanks to Glenn Hauer of ARAM for helpful

comments, and all the CREWES sponsors fortheir support

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