Estimating ‘noise-floor PSD’ by using two or three collocated seismometers: an alternative approach

Abstract

The revolutionary approach to determining self-noise of the test seismometer by the use of two additional seismometers was presented by Sleeman et al (Bull Seismol Soc Am 96(1):258–271, 2006). Yet nowadays there are common situations where only two seismometers are installed side by side. This article thus outlines the various procedures that can be used in such situations. As it will be shown by examples, these procedures do not provide independent information unlike those given by three-seismometer procedures, however they still provide relevant data that can be used to assess the condition of the tested seismometers. Three equations are presented, that can be used in two-seismometer approach. The Eqs. 1 and 2 are already explained in the article (Tasič and Runovc in J Seismol 16:183–194. https://doi.org/10.1007/s10950-011-9257-4, 2012) The Eq. 3, which represents the “average self-noise”, differs from the equation “average self-noise” from the previously described article. Experimentally we found that the latter equation is not consistent with the results obtained for the “average self-noise” with the Sleeman procedure, while the equation for the “average self-noise” presented in this article is consistent with it. It is consistent in the frequency interval, where PSD of the seismic signal is at least 5 dB above the seismometers self-noises and it is very suitable for situations where two seismometers of the same type are compared between each other. This paper also presents an alternative three-seismometers approach. It is derived from the aforementioned Eq. 3. For this reason, at the frequency interval, where PSD of the seismic signal is at least 5 dB above the seismometers self-noises the output from this algorithm should be in accordance with the output from algorithm of Sleeman et al (Bull Seismol Soc Am 96(1):258–271, 2006). If there are deviations in certain frequency interval, this indicates some irregularities in the test itself. Under optimal measuring conditions, the Sleeman procedure is sufficient. However, if the test conditions are suboptimal, a comparison between the two procedures, where one contains division and the other does not, may be used to estimate the frequency range where the results are not reliable. Alternative equations, presented in this paper, are useful to discover unknown errors of the test system. The applicability of the equations is given by examples.

Appendix

Equation

$$0.5 ( {\text{N}}_{\rm{qq}} + {\text{N}}_{\rm{rr}} ) = \sqrt {{\text{P}}_{\rm{qq}} {\text{P}}_{\rm{rr}} } - \sqrt {{\text{P}}_{\rm{rq}} {\text{P}}_{\rm{rq}}^{ *} } ,\quad {\text{for}}\,\,{\text{P}}_{\rm{qq}} , {\text{P}}_{\rm{rr}} > {\text{N}}_{\rm{rr}} ,{\text{N}}_{\rm{qq}} .$$

(8)

represent the “average self-noise” of systems ‘r’ and ‘q’ if seismic signal is above self-noise for at least 5 dB. Let now assume that three seismometers, ‘q’, ‘r’ and ‘k’ are on the same seismic pier. Equation 8 is rewritten as

$$0. 5 ( {\text{N}}_{\rm{qq}} + {\text{N}}_{\rm{kk}} ) = \sqrt {{\text{P}}_{\rm{qq}} {\text{P}}_{\rm{kk}} } - \sqrt {{\text{P}}_{\rm{kq}} {\text{P}}_{\rm{kq}}^{ *} } ,$$

(9)

$$0. 5 ( {\text{N}}_{\rm{rr}} + {\text{N}}_{\rm{kk}} ) = \sqrt {{\text{P}}_{\rm{rr}} {\text{P}}_{\rm{kk}} } - \sqrt {{\text{P}}_{\rm{kr}} {\text{P}}_{\rm{kr}}^{ *} } .$$

(10)

First sum Eqs. 8 and 9, and from that we subtract the Eq. 10. New estimator for self-noise of seismometer ‘q’ is expressed as

$${\text{N}}_{\rm{qq}} = \sqrt {{\text{P}}_{\rm{qq}} {\text{P}}_{\rm{rr}} } - \sqrt {{\text{P}}_{\rm{rq}} {\text{P}}_{\rm{rq}}^{ *} } + \sqrt {{\text{P}}_{\rm{qq}} {\text{P}}_{\rm{kk}} } - \sqrt {{\text{P}}_{\rm{kq}} {\text{P}}_{\rm{kq}}^{ *} } - (\sqrt {{\text{P}}_{\rm{rr}} {\text{P}}_{\rm{kk}} } - \sqrt {{\text{P}}_{\rm{kr}} {\text{P}}_{\rm{kr}}^{ *} } ),\,{\text{P}}_{\rm{qq}} , {\text{P}}_{\rm{rr}} , {\text{P}}_{\rm{kk}} > {\text{N}}_{\rm{rr}} ,{\text{N}}_{\rm{qq}} ,{\text{N}}_{\rm{kk}} .$$

(11)

The equation is valid if the same condition as in Eq. 8 is fulfilled.

The similar is valid for Eq. 12. The equation

$$(0.5{\text{N}}_{\rm{rr}} + 0.5{\text{N}}_{\rm{qq}} )_{q} {\text{ = P}}_{\rm{qq}} - \frac{{{\text{P}}_{\rm{rq}} {\text{P}}_{\rm{rq}}^{*} }}{{{\text{P}}_{\rm{rr}} - (\sqrt {{\text{P}}_{\rm{qq}} {\text{P}}_{\rm{rr}} } - \sqrt {{\text{P}}_{\rm{rq}} {\text{P}}_{\rm{rq}}^{ *} } )}},\quad {\text{for}}\,\,{\text{P}}_{\rm{qq}} , {\text{P}}_{\rm{rr}} > |({\text{N}}_{\rm{rr}} - {\text{N}}_{\rm{qq}} )|.$$

(12)

produce equal result as the Eq. 8 if seismic signal is above self-noise for at least 5 dB. This can be shown by writing the right side of Eq. 12 as follows

$${\text{X = P}}_{\rm{qq}} - \frac{{{\text{P}}_{\rm{qr}} {\text{P}}_{\rm{qr}}^{*} }}{{{\text{P}}_{\rm{rr}} - (0.5({\text{N}}_{\rm{rr}} + {\text{N}}_{\rm{qq}} ) )}}.$$

(13)

From this equation it follows

$${\text{X}}(0.5({\text{N}}_{\rm{rr}} + {\text{N}}_{\rm{qq}} )- {\text{P}}_{\rm{rr}} )){\text{ = P}}_{\rm{qr}} {\text{P}}_{\rm{qr}}^{*} + {\text{P}}_{\rm{qq}} (0.5({\text{N}}_{\rm{rr}} + {\text{N}}_{\rm{qq}} )- {\text{P}}_{\rm{rr}} ).$$

(14)

By using relations ${\text{P}}_{\rm{qq}} {\text{ = P}}_{\rm{xx}} {\text{ + N}}_{\rm{qq}}$, ${\text{P}}_{\rm{rr}} {\text{ = P}}_{\rm{xx}} {\text{ + N}}_{\rm{rr}}$ and ${\text{P}}_{\rm{qr}} {\text{ = P}}_{\rm{xx}}$, Eq. 14 is rewritten as

$${\text{X}}(0.5({\text{N}}_{\rm{qq}} - {\text{N}}_{\rm{rr}} ) - {\text{XP}}_{\rm{xx}} {\text{ = N}}_{\rm{qq}} 0.5({\text{N}}_{\rm{qq}} - {\text{N}}_{\rm{rr}} )- {\text{P}}_{\rm{xx}} 0.5({\text{N}}_{\rm{qq}} + {\text{N}}_{\rm{rr}} ).$$

(15)

Under the assumption, that expression $( {\text{N}}_{\rm{rr}} - {\text{N}}_{\rm{qq}} )$ is negligible with respect to the seismic signal, then ${\text{X}} = 0.5 ( {\text{N}}_{\rm{rr}} + {\text{N}}_{\rm{qq}} )$.

As long as seismic signal is above the self-noise of at least 5 db, outputs from Eqs. 8 and 12 are comparable. However, when this condition is not fulfilled the outputs from Eqs. 8 and 12 differ. When only two seismometers of equal type are on test bed, Eq. 12 can be used, to define the frequency interval, where Eq. 8 is valid.