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Nelson-Nordgren, Astatic Spring, Broadband Force Balance, Vertical (FBV) Seismometers

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---------------

Gamma**2 SNR 10*log(SNR)
0.900000 9.000000 9.5424252
0.910000 10.111111 10.047989
0.920000 11.500000 10.606978
0.930000 13.285714 11.233849
0.940000 15.666667 11.949766
0.950000 19.000000 12.787536
0.960000 24.000000 13.802112
0.970000 32.333333 15.096505
0.980000 49.000000 16.901961
0.990000 99.000000 19.956352
0.991000 110.111111 20.418311
0.992000 124.000000 20.934217
0.993000 141.857143 21.518512
0.994000 165.666667 22.192351
0.995000 199.000000 22.988531
0.996000 249.000000 23.961993
0.997000 332.333333 25.215739
0.998000 499.000000 26.981005
0.999000 999.000000 29.995655
0.999100 1110.111111 30.453664
0.999200 1249.000000 30.965624
0.999300 1427.571429 31.545978
0.999400 1665.666667 32.215881
0.999500 1999.000000 33.008128
0.999600 2499.000000 33.977663
0.999700 3332.333333 35.227484
0.999800 4999.000000 36.988831
0.999900 9999.000000 39.999566
0.999990 99999.000000 49.999957
0.999999 999998.999971 59.999996


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:
  1. "concrete basement floor being distorted by pressure-difference"
  2. "slow leak in case"
  3. "bottom of the case may distort with pressure"
  4. "variation in local g due to atmosphere pressure acting on the ground

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

03-06-14 03-06-14

Plots for a 12 hour period on 03/09/14. The left is the Coherence data and right is the Phase data

03-09-14 03-09-14

(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