nass-micro-station-measurements/huddle-test-geophones/index.org

• Setup
• Signal Processing
  • Load data
  • Time Domain Data
  • Computation of the ASD of the measured voltage
  • Scaling to take into account the sensibility of the geophone and the voltage amplifier
  • Computation of the ASD of the velocity
  • Transfer function between the two geophones
  • Estimation of the sensor noise
• Compare axis
  • Load data
  • Compare PSD
  • Compare TF
• Appendix
  • Computation of coherence from PSD and CSD
• Bibliography

#+TITLE:SpeedGoat

Setup

Two L22 geophones are used. They are placed on the ID31 granite. They are leveled.

The signals are amplified using voltage amplifier with a gain of 60dB. The voltage amplifiers include a low pass filter with a cut-off frequency at 1kHz.

Setup

Geophones

Signal Processing

Load data

We load the data of the z axis of two geophones.

  load('mat/data_001.mat', 't', 'x1', 'x2');
  dt = t(2) - t(1);

Time Domain Data

  figure;
  hold on;
  plot(t, x1);
  plot(t, x2);
  hold off;
  xlabel('Time [s]');
  ylabel('Voltage [V]');
  xlim([t(1), t(end)]);

  <<plt-matlab>>

Time domain Data

  figure;
  hold on;
  plot(t, x1);
  plot(t, x2);
  hold off;
  xlabel('Time [s]');
  ylabel('Voltage [V]');
  xlim([0 1]);

  <<plt-matlab>>

Time domain Data - Zoom

Computation of the ASD of the measured voltage

We first define the parameters for the frequency domain analysis.

  win = hanning(ceil(length(x1)/100));
  Fs = 1/dt;

  [pxx1, f] = pwelch(x1, win, [], [], Fs);
  [pxx2, ~] = pwelch(x2, win, [], [], Fs);

Scaling to take into account the sensibility of the geophone and the voltage amplifier

The Geophone used are L22. Their sensibility are shown on figure fig:geophone_sensibility.

  S0 = 88; % Sensitivity [V/(m/s)]
  f0 = 2; % Cut-off frequnecy [Hz]
  S = (s/2/pi/f0)/(1+s/2/pi/f0);

  <<plt-matlab>>

Sensibility of the Geophone

We also take into account the gain of the electronics which is here set to be $60dB$. The amplifiers also include a low pass filter with a cut-off frequency set at 1kHz.

  G0 = 60; % [dB]

  G = G0/(1+s/2/pi/1000);

We divide the ASD measured (in $\text{V}/\sqrt{\text{Hz}}$) by the transfer function of the voltage amplifier to obtain the ASD of the voltage across the geophone. We further divide the result by the sensibility of the Geophone to obtain the ASD of the velocity in $m/s/\sqrt{Hz}$.

  scaling = 1./squeeze(abs(freqresp(G, f, 'Hz')))./squeeze(abs(freqresp(S, f, 'Hz')));

Computation of the ASD of the velocity

The ASD of the measured velocity is shown on figure fig:psd_velocity.

  figure;
  hold on;
  plot(f, sqrt(pxx1)./scaling);
  plot(f, sqrt(pxx2)./scaling);
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [m/s/sqrt(Hz)]')
  xlim([2, 500]);

  <<plt-matlab>>

Spectral density of the velocity

Transfer function between the two geophones

We here compute the transfer function from one geophone to the other. The result is shown on figure fig:tf_geophones.

We also compute the coherence between the two signals (figure fig:coh_geophones).

  [T12, ~] = tfestimate(x1, x2, win, [], [], Fs);

  <<plt-matlab>>

Estimated transfer function between the two geophones

  [coh12, ~] = mscohere(x1, x2, win, [], [], Fs);

  <<plt-matlab>>

Cohererence between the signals of the two geophones

Estimation of the sensor noise

The technique to estimate the sensor noise is taken from cite:barzilai98_techn_measur_noise_sensor_presen.

The coherence between signals $X$ and $Y$ is defined as follow \[ \gamma^2_{XY}(\omega) = \frac{|G_{XY}(\omega)|^2}{|G_{X}(\omega)| |G_{Y}(\omega)|} \] where $|G_X(\omega)|$ is the output Power Spectral Density (PSD) of signal $X$ and $|G_{XY}(\omega)|$ is the Cross Spectral Density (CSD) of signal $X$ and $Y$.

The PSD and CSD are defined as follow:

\begin{align} |G_X(\omega)| &= \frac{2}{n_d T} \sum^{n_d}_{n=1} \left| X_k(\omega, T) \right|^2 \\ |G_{XY}(\omega)| &= \frac{2}{n_d T} \sum^{n_d}_{n=1} [ X_k^*(\omega, T) ] [ Y_k(\omega, T) ] \end{align}

where:

• $n_d$ is the number for records averaged
• $T$ is the length of each record
• $X_k(\omega, T)$ is the finite Fourier transform of the kth record
• $X_k^*(\omega, T)$ is its complex conjugate

The mscohere function is compared with this formula on Appendix (section sec:coherence), it is shown that it is identical.

Figure fig:huddle_test illustrate a block diagram model of the system used to determine the sensor noise of the geophone.

Two geophones are mounted side by side to ensure that they are exposed by the same motion input $U$.

Each sensor has noise $N$ and $M$.

Huddle test block diagram

We here assume that each sensor has the same magnitude of instrumental noise ($N = M$). We also assume that $H_1 = H_2 = 1$.

We then obtain:

\begin{equation} \gamma_{XY}^2(\omega) = \frac{1}{1 + 2 \left( \frac{|G_N(\omega)|}{|G_U(\omega)|} \right) + \left( \frac{|G_N(\omega)|}{|G_U(\omega)|} \right)^2} \end{equation}

Since the input signal $U$ and the instrumental noise $N$ are incoherent:

\begin{equation} |G_X(\omega)| = |G_N(\omega)| + |G_U(\omega)| \end{equation}

From equations eq:coh_bis and eq:incoherent_noise, we finally obtain

\begin{equation} |G_N(\omega)| = |G_X(\omega)| \left( 1 - \sqrt{\gamma_{XY}^2(\omega)} \right) \end{equation}

The instrumental noise is computed below. The result in V^2/Hz is shown on figure fig:intrumental_noise_V.

  pxxN = pxx1.*(1 - coh12);

  figure;
  hold on;
  plot(f, pxx1, '-');
  plot(f, pxx2, '-');
  plot(f, pxxN, 'k--');
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$V^2/Hz$]');
  xlim([1, 500]);

  <<plt-matlab>>

Instrumental Noise and Measurement in $V^2/Hz$

This is then further converted into velocity and compared with the ground velocity measurement. (figure fig:intrumental_noise_velocity)

  figure;
  hold on;
  plot(f, sqrt(pxx1).*scaling, '-');
  plot(f, sqrt(pxx2).*scaling, '-');
  plot(f, sqrt(pxxN).*scaling, 'k--');
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
  xlim([1, 500]);

  <<plt-matlab>>

Instrumental Noise and Measurement in $m/s/\sqrt{Hz}$

Compare axis

Load data

  z     = load('mat/data_001.mat', 't', 'x1', 'x2');
  east  = load('mat/data_002.mat', 't', 'x1', 'x2');
  north = load('mat/data_003.mat', 't', 'x1', 'x2');

Compare PSD

  [pz1, fz] = pwelch(z.x1, hanning(ceil(length(z.x1)/100)), [], [], 1/(z.t(2)-z.t(1)));
  [pz2, ~]  = pwelch(z.x2, hanning(ceil(length(z.x2)/100)), [], [], 1/(z.t(2)-z.t(1)));

  [pe1, fe] = pwelch(east.x1, hanning(ceil(length(east.x1)/100)), [], [], 1/(east.t(2)-east.t(1)));
  [pe2, ~]  = pwelch(east.x2, hanning(ceil(length(east.x2)/100)), [], [], 1/(east.t(2)-east.t(1)));

  [pn1, fn] = pwelch(north.x1, hanning(ceil(length(north.x1)/100)), [], [], 1/(north.t(2)-north.t(1)));
  [pn2, ~]  = pwelch(north.x2, hanning(ceil(length(north.x2)/100)), [], [], 1/(north.t(2)-north.t(1)));

  <<plt-matlab>>

Compare the measure PSD of the two geophones for the three axis

Compare TF

  [Tz, fz] = tfestimate(z.x1, z.x2, hanning(ceil(length(z.x1)/100)), [], [], 1/(z.t(2)-z.t(1)));
  [Te, fe] = tfestimate(east.x1, east.x2, hanning(ceil(length(east.x1)/100)), [], [], 1/(east.t(2)-east.t(1)));
  [Tn, fn] = tfestimate(north.x1, north.x2, hanning(ceil(length(north.x1)/100)), [], [], 1/(north.t(2)-north.t(1)));

  <<plt-matlab>>

Compare the transfer function from one geophone to the other for the 3 axis

Appendix

Computation of coherence from PSD and CSD

<<sec:coherence>>

  load('mat/data_001.mat', 't', 'x1', 'x2');
  dt = t(2) - t(1);
  Fs = 1/dt;
  win = hanning(ceil(length(x1)/100));

  pxy = cpsd(x1, x2, win, [], [], Fs);
  pxx = pwelch(x1, win, [], [], Fs);
  pyy = pwelch(x2, win, [], [], Fs);
  coh = mscohere(x1, x2, win, [], [], Fs);

  figure;
  hold on;
  plot(f, abs(pxy).^2./abs(pxx)./abs(pyy), '-');
  plot(f, coh, '--');
  hold off;
  set(gca, 'xscale', 'log');
  xlabel('Frequency'); ylabel('Coherence');
  xlim([1, 500]);

  <<plt-matlab>>

Comparison of mscohere and manual computation

Bibliography   ignore

bibliographystyle:unsrt bibliography:ref.bib