Poles and zeros with positive normalisation factors

When this article was written, Güralp Systems published poles and zeros with negative normalisation factors by default. The requirement for positive normalisation factors mentioned in this article is now ubiquitous, so the calibration documents for new instruments now always include positive normalisation factors. The information below is retained for users with calibration documents predating this change.

The nominal phase and amplitude response of a seismometer at different frequencies is described by a set of numbers known as the "poles and zeros" of the transfer functionSeismometers operated within their limits can be modelled as linear, time-invariant (LTI) systems where the transfer function is the ratio of the Laplace transform of the output to that of the input. The transfer function is thus the ratio of two complex polynomials, the roots of which provide complete information about the response of the instrument. The poles are the roots of the denominator polynomial and the zeros are the roots of the numerator polynomial.. Each Güralp instrument model is available with a choice of nominal frequency responses and a standard set of poles and zeros is available for each such response. They are measured using an HP spectrum analyser and a curve is then mathematically fitted to these data. All poles and zeros supplied with Güralp equipment are derived from actual measured data and not generated from examination of theoretical calculations.

The system that Güralp Systems uses produces the lowest order transfer function that fits the data but this function often has a negative normalisation factor.

Whilst negative normalisation factors are mathematically correct and give accurate results, the SEED format does not support them and, furthermore, the standard rdseed conversion program does not properly handle negative normalisation factors.

By increasing the order of the transfer function, we have been able to provide an alternative fit to the data with a set of poles and zeros having a positive normalisation factor.

All instruments that have their frequency responses measured at GSL have their frequency response data held on file here.

Because the instruments are true feedback instruments, their response is entirely defined by the parameters in the feedback path. The frequency response of instruments is within 0.1% of nominal at the long period end and within 2% at the high frequency end.

For equivalent values in radians per second, please see this conversion tool.

Values

360 Second 3T & 1T (in Hz)

Pass-band: 360 seconds to 50 Hz
RESPONSE codes: 360s or CMG-3_360s_50Hz

Zeros	Poles	Normalisation Factor at 1 Hz
0	−80.0	A = 2 304 000
0	−160.0
	−180.0
	−0.001964 + 0.001964j
	−0.001964 – 0.001964j

120 Second 3T, 3ESP & 1T (in Hz)

Pass-band: 120 seconds to 50 Hertz
RESPONSE codes: 120s or CMG-3_120s_50Hz

Zeros	Poles	Normalisation Factor at 1 Hz
0	−80.0	A = 2 304 000
0	−160.0
	−180.0
	−0.00589 + 0.00589j
	−0.00589 – 0.00589j

100 Second 3T, 3ESP & 1T (in Hz)

Pass-band: 100 seconds to 50 Hertz
RESPONSE codes: 100s, CMG-3_100s_50Hz or CMG-40_100s_50Hz

Zeros	Poles	Normalisation Factor at 1 Hz
0	−80.0	A = 2 304 000
0	−160.0
	−180.0
	−0.00707 + 0.00707j
	−0.00707 – 0.00707j

60 Second 3T, 3ESP, & 40T (in Hz)

Pass-band: 60 seconds to 50 Hertz
RESPONSE codes: 60s, CMG-3_60s_50Hz or CMG-40_60s_50Hz

Zeros	Poles	Normalisation Factor at 1 Hz
0	−80.0	A = 2 304 000
0	−160.0
	−180.0
	−0.01178 + 0.01178j
	−0.01178 – 0.01178j

30 Second 3T, 3ESP & 40T (in Hz)

Pass-band: 30 seconds to 50 Hertz
RESPONSE codes: 30s, CMG-3_30s_50Hz or CMG-40_30s_50Hz

Zeros	Poles	Normalisation Factor at 1 Hz
0	−80.0	A = 2 304 000
0	−160.0
	−180.0
	−0.02356 + 0.02356j
	−0.02356 – 0.02356j

1 Second to 100 Hz, 40T (in Hz)

Pass-band: 1 Second to 100 Hz
RESPONSE codes: CMG-6_1Hz_100Hz, CMG-6_1s_100Hz, CMG-40_1Hz_100Hz or CMG-40_1s_100Hz

Zeros	Poles	Normalisation Factor at 10^* Hz
0	−75.0	A = 587 100 000
0	−350.0
	−0.707 + 0.707j
	−0.707 – 0.707j
	−62.3816 + 135.392j
	−62.3816 – 135.392j

*The normalisation factor is given at 10 Hz because the normal choice, 1 Hz, is at the edge of the pass-band.

DC to 100 Hz, Fortis / 5T / 5TD / 5TB (in Hz)

Pass-band: DC to 100 Hz
RESPONSE codes: DC100 or CMG-5_100Hz

Zeros	Poles	Normalisation Factor at 1 Hz
none	−755.9	A = 1 939 000 000
	−209.7
	−63.79 – 90.39j
	−63.79 + 90.39j

DC to 200 Hz, 5T / 5TD / 5TB (in Hz)

Pass-band: DC to 200 Hz
RESPONSE codes: DC200 or CMG-5_200Hz

Zeros	Poles	Normalisation Factor at 1 Hz
none	–174.0 – 177.1j	A = 1.634 × 10¹⁰
	–174.0 + 177.1j
	–309.7
	−855.9

DC to 200 Hz Fortis / 5TC (in Hz)

Pass-band: DC to 200 Hz
RESPONSE codes: CMG-5TC_200Hz, 5TC_200Hz or Fortis_200Hz

Zeros	Poles	Normalisation Factor at 1 Hz
none	−60.8 – 155j	A = 4.6839 × 10⁹
	−60.8 + 155j
	−123
	−1320