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

#+TITLE:SpeedGoat
:DRAWER:
#+STARTUP: overview

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#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results output
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs
:END:

* 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.

#+name: fig:figure_name
#+caption: Setup
#+attr_html: :width 500px
[[file:./figs/setup.jpg]]

#+name: fig:figure_name
#+caption: Geophones
#+attr_html: :width 500px
[[file:./figs/geophones.jpg]]

* Signal Processing
** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

** Load data
#+begin_src matlab :results none
  load('mat/data_001.mat', 't', 'x1', 'x2');
  dt = t(2) - t(1);
#+end_src

** Time Domain Data
#+begin_src matlab :results none
  figure;
  hold on;
  plot(t, x1);
  plot(t, x2);
  hold off;
  xlabel('Time [s]');
  ylabel('Voltage [V]');
  xlim([t(1), t(end)]);
#+end_src

#+NAME: fig:data_time_domain
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/data_time_domain.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:data_time_domain
#+CAPTION: Time domain Data
#+RESULTS: fig:data_time_domain
[[file:figs/data_time_domain.png]]


#+begin_src matlab :results none
  figure;
  hold on;
  plot(t, x1);
  plot(t, x2);
  hold off;
  xlabel('Time [s]');
  ylabel('Voltage [V]');
  xlim([0 1]);
#+end_src

#+NAME: fig:data_time_domain_zoom
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/data_time_domain_zoom.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:data_time_domain_zoom
#+CAPTION: Time domain Data - Zoom
#+RESULTS: fig:data_time_domain_zoom
[[file:figs/data_time_domain_zoom.png]]

** Compute PSD
We first define the parameters for the frequency domain analysis.
#+begin_src matlab :results none
  win = hanning(ceil(length(x1)/100));
  Fs = 1/dt;
#+end_src

#+begin_src matlab :results none
  [pxx1, f] = pwelch(x1, win, [], [], Fs);
  [pxx2, ~] = pwelch(x2, win, [], [], Fs);
#+end_src

** Take into account sensibility of Geophone
The Geophone used are L22.
#+begin_src matlab :results none
  S0 = 88; % Sensitivity [V/(m/s)]
  f0 = 2; % Cut-off frequnecy [Hz]
  S = (s/2/pi/f0)/(1+s/2/pi/f0);
#+end_src

#+begin_src matlab :results none
  figure;
  bodeFig({S});
  ylabel('Amplitude [V/(m/s)]')
#+end_src

#+NAME: fig:geophone_sensibility
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/geophone_sensibility.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:geophone_sensibility
#+CAPTION: Sensibility of the Geophone
#+RESULTS: fig:geophone_sensibility
[[file:figs/geophone_sensibility.png]]


We take into account the gain of the electronics.
The cut-off frequency is set at 1kHz.

- [ ] Check what is the order of the filter
- [ ] Maybe I should not use this filter as there is no high frequencies anyway?

#+begin_src matlab :results none
  G0 = 60; % [dB]

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

#+begin_src matlab :results none
  figure;
  hold on;
  plot(f, sqrt(pxx1)./squeeze(abs(freqresp(G, f, 'Hz')))./squeeze(abs(freqresp(S, f1, 'Hz'))));
  plot(f, sqrt(pxx2)./squeeze(abs(freqresp(G, f, 'Hz')))./squeeze(abs(freqresp(S, f2, 'Hz'))));
  hold off;
  set(gca, 'xscale', 'log');
  set(gca, 'yscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('PSD [m/s/sqrt(Hz)]')
  xlim([2, 500]);
#+end_src

#+NAME: fig:psd_velocity
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/psd_velocity.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:psd_velocity
#+CAPTION: Spectral density of the velocity
#+RESULTS: fig:psd_velocity
[[file:figs/psd_velocity.png]]

** Transfer function between the two geophones
#+begin_src matlab :results none
  [T12, ~] = tfestimate(x1, x2, win, [], [], Fs);
#+end_src

#+begin_src matlab :results none
  figure;
  ax1 = subplot(2, 1, 1);
  plot(f, abs(T12));
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('Magnitude');

  ax2 = subplot(2, 1, 2);
  plot(f, mod(180+180/pi*phase(T12), 360)-180);
  set(gca, 'xscale', 'log');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  xlabel('Frequency [Hz]'); ylabel('Phase');

  linkaxes([ax1,ax2],'x');
  xlim([1, 500]);
#+end_src

#+NAME: fig:tf_geophones
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/tf_geophones.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:tf_geophones
#+CAPTION: Estimated transfer function between the two geophones
#+RESULTS: fig:tf_geophones
[[file:figs/tf_geophones.png]]

#+begin_src matlab :results none
  [coh12, ~] = mscohere(x1, x2, win, [], [], Fs);
#+end_src

#+begin_src matlab :results none
  figure;
  plot(f, coh12);
  set(gca, 'xscale', 'log');
  xlabel('Frequency [Hz]'); ylabel('Coherence');
  ylim([0,1]); xlim([1, 500]);
#+end_src

#+NAME: fig:coh_geophones
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/coh_geophones.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:coh_geophones
#+CAPTION: Cohererence between the signals of the two geophones
#+RESULTS: fig:coh_geophones
[[file:figs/coh_geophones.png]]

** Huddle Test
#+NAME: fig:huddle_test
#+HEADER: :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/MEGA/These/LaTeX/}{config.tex}")
#+HEADER: :imagemagick t :fit yes :iminoptions -scale 100% -density 150 :imoutoptions -quality 100
#+HEADER: :results raw replace :buffer no :eval no-export :exports both :mkdirp yes
#+HEADER: :output-dir figs
#+begin_src latex :file huddle-test.pdf :post pdf2svg(file=*this*, ext="png") :exports results
  \begin{tikzpicture}
    \coordinate[] (U) at (0, 0) {};
    \node[block, above right=0.5 and 2 of U] (S1) {$S_1$};
    \node[block, below right=0.5 and 2 of U] (S2) {$S_2$};
    \node[addb={+}{}{}{}{}, right=0.5 of S1] (add1) {};
    \node[addb={+}{}{}{}{}, right=0.5 of S2] (add2) {};

    \draw[] (U) node[above right]{$U$} -- ++(1, 0) node[]{$\bullet$};
    \draw[->] ($(U)+(1, 0)$)  |- (S1.west);
    \draw[->] ($(U)+(1, 0)$)  |- (S2.west);

    \draw[->] (S1.east)  -- (add1.west);
    \draw[->] (S2.east)  -- (add2.west);

    \draw[->] (add1.east)  -- ++(1, 0) node[above]{$X_1$};
    \draw[->] (add2.east)  -- ++(1, 0) node[above]{$X_2$};

    \draw[<-] (add1.north)  -- ++(0, 0.8)node[right]{$N_1$};
    \draw[<-] (add2.north)  -- ++(0, 0.8)node[right]{$N_2$};
  \end{tikzpicture}
#+end_src

#+NAME: fig:huddle_test
#+CAPTION: Huddle test block diagram
#+RESULTS: fig:huddle_test
[[file:figs/huddle-test.png]]

We are measuring $X_1$ and $X_2$.
The goal is to determine $N$.

\begin{align*}
  X_1(\omega) &= S_1(\omega) U(\omega) + N_1(\omega)\\
  X_2(\omega) &= S_2(\omega) U(\omega) + N_2(\omega)
\end{align*}

Then
\[ X_2(\omega) = \frac{S_2(\omega)}{S_1(\omega)} X_1(\omega) + N_2(\omega) - \frac{S_2(\omega)}{S_1(\omega)}N_1(\omega) \]

We suppose $N_1 = N_2 = N$
\[ N_2(\omega) - \frac{S_2(\omega)}{S_1(\omega)}N_1(\omega) = \left( 1 - \frac{S_2(\omega)}{S_1(\omega)}\right) N(\omega) \]
and
\[ N(\omega) = \frac{S_2(\omega)}{S_1(\omega)} X_2(\omega) - N_2(\omega) - \frac{S_2(\omega)}{S_1(\omega)}N_1(\omega) \]



#+begin_src matlab :results none
  S = abs(T12.*pxx1);

  N = pxx2 - (T12.^2).*pxx1;
  N = abs(N)/2;
#+end_src

#+begin_src matlab :results none
  figure;
  hold on;
  plot(f, pxx1, '-');
  plot(f, pxx2, '-');
  plot(f, N, 'k:', 'linewidth', 1);
  hold off;
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  xlim([1, 500]);
  legend('$\Phi_{ss} (f)$','$\Phi_{nn} (f)$')
#+end_src

#+NAME: fig:huddle_test_results
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/huddle_test_results.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:huddle_test_results
#+CAPTION: Results of the Huddle test
#+RESULTS: fig:huddle_test_results
[[file:figs/huddle_test_results.png]]