Holcomb Coherence Analysis of Two Side-By-Side Yuma Seismometers
Given two side-by-side seismometers, with equal input power X, and noise power N1 and N2, it has been shown(1) that the magnitude squared coherence function, gamma2, can be expressed by each system's output auto-spectral density and cross-spectral density functions P11, P22 and P12. This is a real valued function of frequency with values between 0 and 1, where 1 indicates perfect coherence.
In solving the system equations, the transfer functions of each seismometer drop out of the equation and we are left with the power signal-to-noise ratios, X/N1 and X/N2 as a function of gamma2, in the form of translated equilateral hyperbolas.
The asymptotes of the translated hyperbolas are the minimum value of SNRs that both seismometers can have. This method does not provide the power spectral densities of N1 or N2, that newer methods do(2), but does give some quantitative measure of seismometer performance. The table below provides SNRs for high values of coherence calculated from the asymptotes, X/N1 = X/N2 = (gamma2)/(1-gamma2).
---------------Coherence and SNR values---------------
The following SigView(3) coherence plots of two side-by-side Yuma seismometers shows data for five 24 hour periods. The sample rate is 200 SPS and the raw data is pre-filtered with a bandpass between 0.002 Hz to 0.07 Hz. An aggressive Hann weighted moving average function of 750 points is used to smooth out the multiple plots.
The seemingly low values of coherence and the variability is interesting, as the two seismometers are very closely matched. Here are some items which will be investigated.
Reducing the moving average Blackman smoothing to 50 points resulted in a noisier plot but with slightly higher coherence values. Here is a 24 hour plot from 1/5/14, 200 SPS resampled to 50 SPS, with several peaks in the 0.998 range. Again the data is pre-filtered with a bandpass between 0.002 Hz to 0.07 Hz.
Here is a coherence plot with raw unfiltered data. The data is from 12/06/13, 24 hours, 200 SPS resampled to 50 SPS, Blackman weighted moving average function of 100 points, with several peaks of 0.99999 to 0.999994. The high peak at <0.001 Hz is due to slow temperature variations at the instrument location.
Coherence of each Yuma output with Atmospheric Pressure
The following plots show the Coherence between each Yuma vertical output and atmospheric pressure for the same time periods. Only raw data is used; there is no filtering and the data is resampled to 100SPS. Also shown are the corresponding phase spectrums for each seismometer and pressure variations.The data clearly shows a strong influence of atmospheric pressure on the seismometer output at about 0.005 Hz to 0.01 Hz. Brett discusses several possible reasons for this:
At this time it is unclear to me which item is the probable cause.
Plots for a 6 hour period on 03/06/14. The left is the Coherence data and right is the Phase data
Plots for a 12 hour period on 03/09/14. The left is the Coherence data and right is the Phase data
(1) Holcomb, L.G., "A Direct Method for Calculating Instrument Noise Levels in Side-by-Side Seismometer Evaluations", USGS, Open-File Report 89-214, 1989.
(2) Reinoud Sleeman, Arie van Wettum, and Jeannot Trampert, "Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors", Bulletin of the Seismological Society of America, Vol. 96, No. 1, pp. 258-271, February 2006.(3) SigView V2.6.1, SignalLab, Goran Obradovic, 1995, FFTW-GNU open source FFT libraries.
(4) Rademacher, H., Guralp, C., see reference here