Measuring Noise; Low Noise Model

Transcription

1 Noise in data Specifying noise Measuring noise How to quantify seismic noise Karlsruhe Institute of Technology (KIT) Black Forest Observatory (BFO) September 2011

2 Noise in data Specifying noise Measuring noise Seismogramm example Sources of noise Data example: Where s the quake? Raw data recorded at BFO

3 Noise in data Specifying noise Measuring noise Seismogramm example Sources of noise Data example: Where s the quake? Quake signal rides on ocean microseisms

4 Noise in data Specifying noise Measuring noise Seismogramm example Sources of noise Data example: Where s the quake? High-pass filter removes noise

5 Noise in data Specifying noise Measuring noise Seismogramm example Sources of noise Data example: Where s the quake? High-pass filter removes noise

6 Noise in data Specifying noise Measuring noise Seismogramm example Sources of noise Sources of noise Ambient vibrations due to natural sources (like ocean microseisms, wind, etc) Man-made vibrations (like industry, traffic, etc) Unwanted components of gravitation (like Newtonian attraction of atmosphere, gravity on horizontal seismometers due to loading or strain tilt coupling, etc) Unwanted sensitivity to ambient conditions (like temperature, air density, magnetic field, etc) Deterioration of seismograph components (like corrosion, leakage currents, electronic interference, etc) Intrinsic self-noise of the seismograph (like Brownian noise, quantization noise, etc)

7 Noise in data Specifying noise Measuring noise Properties Theory Numerics Amplitudes NLNM How to specify noise Stochastic noise The word noise frequently is used to summarize all unwanted components in a seismic record. This comprises drifts, spikes, and stationary stochastic signals. In the following we will focus on the latter. Properties: Signal appears not deterministic. Signal phase varies randomly with time. Only stochastic properties can be specified. If the stochastic properties remain constant with time the signal is termed stationary.

8 Noise in data Specifying noise Measuring noise Properties Theory Numerics Amplitudes NLNM How to specify noise Noise amplitude A mean amplitude can be defined. The proper value for a signal of N samples s l is the rms (root mean square) amplitude N 1 s rms = N l=1 s 2 l. The variance (mean square srms 2 ) is a measure of signal power.

9 Noise in data Specifying noise Measuring noise Properties Theory Numerics Amplitudes NLNM How to specify noise Spectral properties A Fourier amplitude spectrum does not exist, since the signal has infinite energy. To study spectral properties: Apply a bandpass and specify the signal power or rms-amplitude within the frequency band Power spectral density is signal power bandwidth Integral of signal power over all frequencies gives total signal power Do not integrate rms-amplitude!

10 Noise in data Specifying noise Measuring noise Properties Theory Numerics Amplitudes NLNM How to specify noise Benefit from Parceval s theorem Energy of transient signal (not stationary noise!): E = + s(t) 2 dt = + s(ω) 2 dω 2π s(ω) is the Fourier transform of s(t). s(ω) 2 can be understood as the energy density spectrum of the signal. We will calculate the power spectral density similarly.

11 Noise in data Specifying noise Measuring noise Properties Theory Numerics Amplitudes NLNM How to specify noise Proper definition of power spectral density Normalized autocorrelation function: 1 P(τ) = lim T 2T +T T Power spectral denisty: P(ω) = + P(τ)e iωτ dτ s(t)s(t + τ) dt One-sided power spectral density: PSD(f) = 2 P(2πf)

12 Noise in data Specifying noise Measuring noise Properties Theory Numerics Amplitudes NLNM How to specify noise Practical calculation 1. Take a sequence of duration T = N t from samples s l of the seismic record with sampling interval t 2. Apply a smooth and properly normalized time domain taper to reduce spectral leakage 3. Calculate the Fourier series s k = N l=1 (l 1)(k 1) 2iπ s l e N t 4. The one-sided power spectral density at frequency f k = k/t is PSD k = 2 s k 2 T 5. Either apply smoothing to PSD k or average over many realizations of PSD k

13 Noise in data Specifying noise Measuring noise Properties Theory Numerics Amplitudes NLNM How to specify noise Estimating time domain amplitudes P(ω) specifies power spectral density The total power (or variance) in the frequency band from f 1 to f 2 then is f 2 P f1,f 2 = 2 P(2πf)df f 1 The rms-amplitude of the band-pass filtered time series then is s rms f1,f 2 = P f1,f 2

14 Noise in data Specifying noise Measuring noise Properties Theory Numerics Amplitudes NLNM How to specify noise Estimating time domain amplitudes To specify the spectral distribution for the time domain amplitudes, it is most appropriate to use a bandwidth proportional to a center frequency f c = f 1 f 2 relative bandwidth R BW = f = f 2 f 1 = 2n 1 = 10m 1 f c f c 2 n/2 10 m/2 bandwidth is specified as being n = log 2 f 2 f 1 octaves or m = log 10 f 2 f 1 decades

15 Noise in data Specifying noise Measuring noise Properties Theory Numerics Amplitudes NLNM How to specify noise Estimating time domain amplitudes Frequency distribution of time domain rms-amplitude: s rms (f c ) = 2 P(fc )f c R BW bandwidth R BW 1 decade 2,846 1/2 decade 1,215 1 octave 0,707 1/6 decade 0,386 1/2 octave 0,348 1/3 octave 0,232

16 Noise in data Specifying noise Measuring noise Properties Theory Numerics Amplitudes NLNM How to specify noise New Low Noise Model (NLNM, Peterson, 1993) 0 db M =4, 0.1 o L o M S=8, 30 NLNM STS1 clip level Tides 240 STS1 instr. noise Period [s] Courtesy of Erhard Wielandt Signal levels are specified in decibels relative to 1 m/s 2. Noise levels can be understood as rms-amplitudes in a bandwidth of 1/6 decade or as mean peak amplitudes in a bandwidth of 1/3 octave.

17 Noise in data Specifying noise Measuring noise Measuring instrumental self noise Huddle test configuration It is not possible to distinguish between ground noise and instrumental selfnoise in the signal recorded from a single instrument. Huddle test: Setup two or more instruments close to each other on the same pier. Assumptions: 1. Ground noise appears coherent in the output of all instruments (after correction for transfer function). 2. Consequently the incoherent part (difference signal) must be procuded by the instruments themselves. Open question: Which of both instruments contributes at which degree to the difference signal?

18 Noise in data Specifying noise Measuring noise Measuring instrumental self noise Method of Sleeman et al. (2006) In a huddle test with three seismometer (1,2,3) the contribution from these seismometers can be distinguished. The power spectral density of self noise of seismometer 1 then is ( N 1 (f) = P 1 (f) 1 C 12(f)C 31 (f) C 32 (f) ), where f is frequency, P 1 (f) is the power spectral density of the output of seismometer 1, and C kl (f) is the coherence spectrum for seismometers k and l. R. Sleeman, A. van Wettum, J. Trampert, Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismoc Sensors. Bull. Seism. Soc. Am., 96(1),

19 Noise in data Specifying noise Measuring noise Measuring instrumental self noise Example: STS-2 of GRSN station at BFO PSD in db relativ zu 10*log(m 2 /s 4 /Hz) Leistungsdichtespektrum: STS2 (GRSN) NLNM (Peterson, 1993) Eigenrauschen: STS2 (GRSN) Frequenz in Hz Courtesy of Daniel Armbruster

20 Noise in data Specifying noise Measuring noise Measuring instrumental self noise Example: STS-1 of IRIS/IDA station at BFO PSD in db relativ zu 10*log(m 2 /s 4 /Hz) Leistungsdichtespektrum: STS1 (IRIS/IDA) NLNM (Peterson, 1993) Eigenrauschen: STS1 (IRIS/IDA) Frequenz in Hz Courtesy of Daniel Armbruster

21 Noise in data Specifying noise Measuring noise Measuring instrumental self noise Example: STS-2 of GRSN prototype station at BFO PSD in db relativ zu 10*log(m 2 /s 4 /Hz) Leistungsdichtespektrum: STS2 (GRSN Nachfolge) NLNM (Peterson, 1993) Eigenrauschen: STS2 (GRSN Nachfolge) Frequenz in Hz Courtesy of Daniel Armbruster