diff --git a/.gitignore b/.gitignore index 8df43c8..f09ce98 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,6 @@ +# Emacs +auto/ + # Simulink Real Time *bio.m *pt.m diff --git a/huddle-test-geophones/figs/comp_coherence_formula.png b/huddle-test-geophones/figs/comp_coherence_formula.png new file mode 100644 index 0000000..6e96b60 Binary files /dev/null and b/huddle-test-geophones/figs/comp_coherence_formula.png differ diff --git a/huddle-test-geophones/figs/compare_axis_psd.png b/huddle-test-geophones/figs/compare_axis_psd.png new file mode 100644 index 0000000..94c00a5 Binary files /dev/null and b/huddle-test-geophones/figs/compare_axis_psd.png differ diff --git a/huddle-test-geophones/figs/compare_tf_axis.png b/huddle-test-geophones/figs/compare_tf_axis.png new file mode 100644 index 0000000..921874c Binary files /dev/null and b/huddle-test-geophones/figs/compare_tf_axis.png differ diff --git a/huddle-test-geophones/figs/huddle-test.png b/huddle-test-geophones/figs/huddle-test.png index df2a0bd..bf72425 100644 Binary files a/huddle-test-geophones/figs/huddle-test.png and b/huddle-test-geophones/figs/huddle-test.png differ diff --git a/huddle-test-geophones/figs/huddle_test_results.png b/huddle-test-geophones/figs/huddle_test_results.png index 62d1797..7c6e819 100644 Binary files a/huddle-test-geophones/figs/huddle_test_results.png and b/huddle-test-geophones/figs/huddle_test_results.png differ diff --git a/huddle-test-geophones/figs/intrumental_noise_V.png b/huddle-test-geophones/figs/intrumental_noise_V.png new file mode 100644 index 0000000..14ba1a8 Binary files /dev/null and b/huddle-test-geophones/figs/intrumental_noise_V.png differ diff --git a/huddle-test-geophones/figs/intrumental_noise_velocity.png b/huddle-test-geophones/figs/intrumental_noise_velocity.png new file mode 100644 index 0000000..c8f2afc Binary files /dev/null and b/huddle-test-geophones/figs/intrumental_noise_velocity.png differ diff --git a/huddle-test-geophones/figs/psd_velocity.png b/huddle-test-geophones/figs/psd_velocity.png index 40716d8..2451e28 100644 Binary files a/huddle-test-geophones/figs/psd_velocity.png and b/huddle-test-geophones/figs/psd_velocity.png differ diff --git a/huddle-test-geophones/index.html b/huddle-test-geophones/index.html index 7d6f41b..311c4a6 100644 --- a/huddle-test-geophones/index.html +++ b/huddle-test-geophones/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + SpeedGoat @@ -246,6 +246,28 @@ for the JavaScript code in this tag. } /*]]>*///--> + +
@@ -254,22 +276,36 @@ for the JavaScript code in this tag.

Table of Contents

-
-

1 Setup

+
+

1 Setup

Two L22 geophones are used. @@ -283,14 +319,14 @@ The voltage amplifiers include a low pass filter with a cut-off frequency at 1kH

-
+

setup.jpg

Figure 1: Setup

-
+

geophones.jpg

Figure 2: Geophones

@@ -298,13 +334,16 @@ The voltage amplifiers include a low pass filter with a cut-off frequency at 1kH
-
-

2 Signal Processing

+
+

2 Signal Processing

-
-

2.1 Load data

+
+

2.1 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);
@@ -313,8 +352,8 @@ dt = t(
-
2.2 Time Domain Data

+

+
2.2 Time Domain Data

 

 

 
figure;
@@ -329,7 +368,7 @@ xlim(
+

 
data_time_domain.png
 

 
Figure 3: Time domain Data

@@ -349,7 +388,7 @@ xlim(
+

 
data_time_domain_zoom.png
 

 
Figure 4: Time domain Data - Zoom

@@ -357,23 +396,34 @@ xlim(
-
2.3 Compute PSD

+

+
2.3 Computation of the ASD of the measured voltage

 

+

+We first define the parameters for the frequency domain analysis.
+

 

-
[pxx1, f1] = pwelch(x1, hanning(ceil(1/dt)), 0, [], 1/dt);
-[pxx2, f2] = pwelch(x2, hanning(ceil(1/dt)), 0, [], 1/dt);
+
win = hanning(ceil(length(x1)/100));
+Fs = 1/dt;
+

+

+
+

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

 

 

 

 
-

-
2.4 Take into account sensibility of Geophone

+

+
2.4 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 5.
 

+
 

 
S0 = 88; % Sensitivity [V/(m/s)]
 f0 = 2; % Cut-off frequnecy [Hz]
@@ -381,15 +431,8 @@ S = (s
 

 
-

-
figure;
-bodeFig({S});
-ylabel('Amplitude [V/(m/s)]')
-

-

 
-
-

+

 
geophone_sensibility.png
 

 
Figure 5: Sensibility of the Geophone

@@ -397,15 +440,10 @@ ylabel(

+

+
+

+
2.5 Computation of the ASD of the velocity

+

+

+The ASD of the measured velocity is shown on figure 6.
+

+
 

 
figure;
 hold on;
-plot(f1, sqrt(pxx1)./squeeze(abs(freqresp(G, f1, 'Hz')))./squeeze(abs(freqresp(S, f1, 'Hz'))));
-plot(f2, sqrt(pxx2)./squeeze(abs(freqresp(G, f2, 'Hz')))./squeeze(abs(freqresp(S, f2, 'Hz'))));
+plot(f, sqrt(pxx1)./scaling);
+plot(f, sqrt(pxx2)./scaling);
 hold off;
 set(gca, 'xscale', 'log');
 set(gca, 'yscale', 'log');
@@ -427,54 +484,308 @@ xlim(
+

 
psd_velocity.png
 

 
Figure 6: Spectral density of the velocity

 

 

 

-

-
2.5 Transfer function between the two geophones

-

+
+

+
2.6 Transfer function between the two geophones

+

+

+We here compute the transfer function from one geophone to the other.
+The result is shown on figure 7.
+

+
+

+We also compute the coherence between the two signals (figure 8).
+

+
 

-
[T12, f12] = tfestimate(x1, x2, hanning(ceil(length(x1)/100)), [], [], 1/dt);
+
[T12, ~] = tfestimate(x1, x2, win, [], [], Fs);
+

+

+
+
+

+
tf_geophones.png
+

+
Figure 7: Estimated transfer function between the two geophones

+

+
+

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

+

+
+
+

+
coh_geophones.png
+

+
Figure 8: Cohererence between the signals of the two geophones

+

+

+

+
+

+
2.7 Estimation of the sensor noise

+

+

+The technique to estimate the sensor noise is taken from 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 4.1), it is shown that it is identical.
+

+
+

+Figure 9 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.png
+

+
Figure 9: 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}
+\label{org5b4a541}
+  \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}
+\label{orga04bdf3}
+  |G_X(\omega)| = |G_N(\omega)| + |G_U(\omega)|
+\end{equation}
+
+

+From equations \eqref{org5b4a541} and \eqref{orga04bdf3}, we finally obtain
+

+

+\begin{equation}
+\label{orga57b1ae}
+  |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 V2/Hz is shown on figure 10.
+

+

+
pxxN = pxx1.*(1 - coh12);
 

 

 
 

 
figure;
-ax1 = subplot(2, 1, 1);
-plot(f12, abs(T12));
+hold on;
+plot(f, pxx1, '-');
+plot(f, pxx2, '-');
+plot(f, pxxN, 'k--');
+hold off;
 set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
-set(gca, 'XTickLabel',[]);
-ylabel('Magnitude');
-
-ax2 = subplot(2, 1, 2);
-plot(f12, 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');
+xlabel('Frequency [Hz]'); ylabel('PSD [$V^2/Hz$]');
 xlim([1, 500]);
 

 

 
 
-

-
tf_geophones.png
+

+
intrumental_noise_V.png
 

-
Figure 7: Estimated transfer function between the two geophones

+
Figure 10: Instrumental Noise and Measurement in \(V^2/Hz\)

+

+
+

+This is then further converted into velocity and compared with the ground velocity measurement. (figure 11)
+

+

+
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]);
+

+

+
+
+

+
intrumental_noise_velocity.png
+

+
Figure 11: Instrumental Noise and Measurement in \(m/s/\sqrt{Hz}\)

 

 

 

 

+
+

+
3 Compare axis

+

+

+

+
3.1 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');
+

+

+

+

+
+

+
3.2 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)));
+

+

+
+
+

+
compare_axis_psd.png
+

+
Figure 12: Compare the measure PSD of the two geophones for the three axis

+

+

+

+
+

+
3.3 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)));
+

+

+
+
+

+
compare_tf_axis.png
+

+
Figure 13: Compare the transfer function from one geophone to the other for the 3 axis

+

+

+

+

+

+
4 Appendix

+

+

+

+
4.1 Computation of coherence from PSD and CSD

+

+

+
+

+

+
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]);
+

+

+
+
+

+
comp_coherence_formula.png
+

+
Figure 14: Comparison of mscohere and manual computation

+

+

+

+

+
+

+
+
Bibliography

+
+

 

 

 
Author: Thomas Dehaeze

-
Created: 2019-04-18 jeu. 09:47

+
Created: 2019-04-18 jeu. 17:02

 
Validate

 

 
diff --git a/huddle-test-geophones/index.org b/huddle-test-geophones/index.org
index 41e3e1f..887b654 100644
--- a/huddle-test-geophones/index.org
+++ b/huddle-test-geophones/index.org
@@ -43,6 +43,7 @@ The voltage amplifiers include a low pass filter with a cut-off frequency at 1kH
 #+end_src
 
 ** Load data
+We load the data of the z axis of two geophones.
 #+begin_src matlab :results none
   load('mat/data_001.mat', 't', 'x1', 'x2');
   dt = t(2) - t(1);
@@ -94,7 +95,7 @@ The voltage amplifiers include a low pass filter with a cut-off frequency at 1kH
 #+RESULTS: fig:data_time_domain_zoom
 [[file:figs/data_time_domain_zoom.png]]
 
-** Compute PSD
+** Computation of the ASD of the measured voltage
 We first define the parameters for the frequency domain analysis.
 #+begin_src matlab :results none
   win = hanning(ceil(length(x1)/100));
@@ -106,15 +107,17 @@ We first define the parameters for the frequency domain analysis.
   [pxx2, ~] = pwelch(x2, win, [], [], Fs);
 #+end_src
 
-** Take into account sensibility of Geophone
+** 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]].
+
 #+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
+#+begin_src matlab :results none :exports none
   figure;
   bodeFig({S});
   ylabel('Amplitude [V/(m/s)]')
@@ -132,11 +135,8 @@ The Geophone used are L22.
 [[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?
+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.
 
 #+begin_src matlab :results none
   G0 = 60; % [dB]
@@ -144,11 +144,21 @@ The cut-off frequency is set at 1kHz.
   G = G0/(1+s/2/pi/1000);
 #+end_src
 
+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 $m/s/\sqrt{Hz}$.
+
+#+begin_src matlab :results none
+  scaling = 1./squeeze(abs(freqresp(G, f, 'Hz')))./squeeze(abs(freqresp(S, f, 'Hz')));
+#+end_src
+
+** Computation of the ASD of the velocity
+The ASD of the measured velocity is shown on figure [[fig:psd_velocity]].
+
 #+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'))));
+  plot(f, sqrt(pxx1)./scaling);
+  plot(f, sqrt(pxx2)./scaling);
   hold off;
   set(gca, 'xscale', 'log');
   set(gca, 'yscale', 'log');
@@ -168,11 +178,16 @@ The cut-off frequency is set at 1kHz.
 [[file:figs/psd_velocity.png]]
 
 ** 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]]).
+
 #+begin_src matlab :results none
   [T12, ~] = tfestimate(x1, x2, win, [], [], Fs);
 #+end_src
 
-#+begin_src matlab :results none
+#+begin_src matlab :results none :exports none
   figure;
   ax1 = subplot(2, 1, 1);
   plot(f, abs(T12));
@@ -206,7 +221,7 @@ The cut-off frequency is set at 1kHz.
   [coh12, ~] = mscohere(x1, x2, win, [], [], Fs);
 #+end_src
 
-#+begin_src matlab :results none
+#+begin_src matlab :results none :exports none
   figure;
   plot(f, coh12);
   set(gca, 'xscale', 'log');
@@ -225,63 +240,62 @@ The cut-off frequency is set at 1kHz.
 #+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) {};
+** Estimation of the sensor noise
+The technique to estimate the sensor noise is taken from cite:barzilai98_techn_measur_noise_sensor_presen.
 
-    \draw[] (U) node[above right]{$U$} -- ++(1, 0) node[]{$\bullet$};
-    \draw[->] ($(U)+(1, 0)$)  |- (S1.west);
-    \draw[->] ($(U)+(1, 0)$)  |- (S2.west);
+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 $Y$.
 
-    \draw[->] (S1.east)  -- (add1.west);
-    \draw[->] (S2.east)  -- (add2.west);
+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
 
-    \draw[->] (add1.east)  -- ++(1, 0) node[above]{$X_1$};
-    \draw[->] (add2.east)  -- ++(1, 0) node[above]{$X_2$};
+The =mscohere= function is compared with this formula on Appendix (section [[sec:coherence]]), it is shown that it is identical.
 
-    \draw[<-] (add1.north)  -- ++(0, 0.8)node[right]{$N_1$};
-    \draw[<-] (add2.north)  -- ++(0, 0.8)node[right]{$N_2$};
-  \end{tikzpicture}
-#+end_src
+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 $M$.
 
 #+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 $N$.
+We here assume that each sensor has the same magnitude of instrumental noise ($N = M$).
+We also assume that $H_1 = H_2 = 1$.
 
-\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
-\[ N(\omega) = \frac{S_2(\omega)}{S_1(\omega)} X_2(\omega) - N_2(\omega) - \frac{S_2(\omega)}{S_1(\omega)}N_1(\omega) \]
+We then obtain:
+#+NAME: eq:coh_bis
+\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:
+#+NAME: eq:incoherent_noise
+\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_important
+#+NAME: eq:noise_psd
+\begin{equation}
+  |G_N(\omega)| = |G_X(\omega)| \left( 1 - \sqrt{\gamma_{XY}^2(\omega)} \right)
+\end{equation}
+#+end_important
 
+The instrumental noise is computed below. The result in V^2/Hz is shown on figure [[fig:intrumental_noise_V]].
 #+begin_src matlab :results none
-  S = abs(T12.*pxx1);
-
-  N = pxx2 - (T12.^2).*pxx1;
-  N = abs(N)/2;
+  pxxN = pxx1.*(1 - coh12);
 #+end_src
 
 #+begin_src matlab :results none
@@ -289,20 +303,184 @@ and
   hold on;
   plot(f, pxx1, '-');
   plot(f, pxx2, '-');
-  plot(f, N, 'k:', 'linewidth', 1);
+  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]);
-  legend('$\Phi_{ss} (f)$','$\Phi_{nn} (f)$')
 #+end_src
 
-#+NAME: fig:huddle_test_results
+#+NAME: fig:intrumental_noise_V
 #+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")
+#+begin_src matlab :var filepath="figs/intrumental_noise_V.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
   <>
 #+end_src
 
-#+NAME: fig:huddle_test_results
-#+CAPTION: Results of the Huddle test
-#+RESULTS: fig:huddle_test_results
-[[file:figs/huddle_test_results.png]]
+#+NAME: fig:intrumental_noise_V
+#+CAPTION: Instrumental Noise and Measurement in $V^2/Hz$
+#+RESULTS: fig:intrumental_noise_V
+[[file:figs/intrumental_noise_V.png]]
+
+This is then further converted into velocity and compared with the ground velocity measurement. (figure [[fig:intrumental_noise_velocity]])
+#+begin_src matlab :results none
+  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]);
+#+end_src
+
+#+NAME: fig:intrumental_noise_velocity
+#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
+#+begin_src matlab :var filepath="figs/intrumental_noise_velocity.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
+  <>
+#+end_src
+
+#+NAME: fig:intrumental_noise_velocity
+#+CAPTION: Instrumental Noise and Measurement in $m/s/\sqrt{Hz}$
+#+RESULTS: fig:intrumental_noise_velocity
+[[file:figs/intrumental_noise_velocity.png]]
+
+* Compare axis
+** Matlab Init                                              :noexport:ignore:
+#+begin_src matlab :exports none :results silent :noweb yes
+  <>
+#+end_src
+
+** Load data
+#+begin_src matlab :results none
+  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');
+#+end_src
+
+** Compare PSD
+#+begin_src matlab :results none
+  [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)));
+#+end_src
+
+#+begin_src matlab :results none :exports  none
+  figure;
+  hold on;
+  plot(fz, sqrt(pz1), '-',   'Color', [0 0.4470 0.7410], 'DisplayName', 'z');
+  plot(fz, sqrt(pz2), '--',  'Color', [0 0.4470 0.7410], 'HandleVisibility', 'off');
+  plot(fe, sqrt(pe1), '-',   'Color', [0.8500 0.3250 0.0980], 'DisplayName', 'east');
+  plot(fe, sqrt(pe2), '--',  'Color', [0.8500 0.3250 0.0980], 'HandleVisibility', 'off');
+  plot(fn, sqrt(pn1), '-',   'Color', [0.9290 0.6940 0.1250], 'DisplayName', 'north');
+  plot(fn, sqrt(pn2), '--',  'Color', [0.9290 0.6940 0.1250], 'HandleVisibility', 'off');
+  hold off;
+  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('PSD [m/s/sqrt(Hz)]');
+  legend('Location', 'northeast');
+  xlim([0, 500]);
+#+end_src
+
+#+NAME: fig:compare_axis_psd
+#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
+#+begin_src matlab :var filepath="figs/compare_axis_psd.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+  <>
+#+end_src
+
+#+NAME: fig:compare_axis_psd
+#+CAPTION: Compare the measure PSD of the two geophones for the three axis
+#+RESULTS: fig:compare_axis_psd
+[[file:figs/compare_axis_psd.png]]
+
+** Compare TF
+#+begin_src matlab :results none
+  [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)));
+#+end_src
+
+#+begin_src matlab :results none :exports none
+  figure;
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  plot(fz, abs(Tz), 'DisplayName', 'z');
+  plot(fe, abs(Te), 'DisplayName', 'east');
+  plot(fn, abs(Tn), 'DisplayName', 'north');
+  hold off;
+  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
+  set(gca, 'XTickLabel',[]);
+  ylabel('Magnitude');
+  legend('Location', 'southwest');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  plot(fz, mod(180+180/pi*phase(Tz), 360)-180);
+  plot(fe, mod(180+180/pi*phase(Te), 360)-180);
+  plot(fn, mod(180+180/pi*phase(Tn), 360)-180);
+  hold off;
+  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:compare_tf_axis
+#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
+#+begin_src matlab :var filepath="figs/compare_tf_axis.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
+  <>
+#+end_src
+
+#+NAME: fig:compare_tf_axis
+#+CAPTION: Compare the transfer function from one geophone to the other for the 3 axis
+#+RESULTS: fig:compare_tf_axis
+[[file:figs/compare_tf_axis.png]]
+* Appendix
+** Computation of coherence from PSD and CSD
+  <>
+#+begin_src matlab :results none
+  load('mat/data_001.mat', 't', 'x1', 'x2');
+  dt = t(2) - t(1);
+  Fs = 1/dt;
+  win = hanning(ceil(length(x1)/100));
+#+end_src
+
+#+begin_src matlab :results none
+  pxy = cpsd(x1, x2, win, [], [], Fs);
+  pxx = pwelch(x1, win, [], [], Fs);
+  pyy = pwelch(x2, win, [], [], Fs);
+  coh = mscohere(x1, x2, win, [], [], Fs);
+#+end_src
+
+#+begin_src matlab :results none
+  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]);
+#+end_src
+
+#+NAME: fig:comp_coherence_formula
+#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
+#+begin_src matlab :var filepath="figs/comp_coherence_formula.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
+  <>
+#+end_src
+
+#+NAME: fig:comp_coherence_formula
+#+CAPTION: Comparison of =mscohere= and manual computation
+#+RESULTS: fig:comp_coherence_formula
+[[file:figs/comp_coherence_formula.png]]
+
+* Bibliography                                                       :ignore:
+bibliographystyle:unsrt
+bibliography:ref.bib
diff --git a/huddle-test-geophones/readme.org b/huddle-test-geophones/readme.org
index e478308..aab2d19 100644
--- a/huddle-test-geophones/readme.org
+++ b/huddle-test-geophones/readme.org
@@ -1,8 +1,9 @@
-* DONE Register data on the computer
-  CLOSED: [2019-04-17 mer. 17:26]
+* TODO [#B] Find the documentation of the amplifier to know the order of the filters
+* TODO [#A] Shake a little bit the geophones to see if we have better measurements on X and Y axis
+* Measurements
 
-* Measurements:
-
-| data_001.mat | Z axis |
-| data_002.mat | East   |
-| data_003.mat | North  |
\ No newline at end of file
+| Filename     | Description |
+|--------------+-------------|
+| data_001.mat | Z axis      |
+| data_002.mat | East        |
+| data_003.mat | North       |
diff --git a/huddle-test-geophones/ref.bib b/huddle-test-geophones/ref.bib
new file mode 100644
index 0000000..a602ec3
--- /dev/null
+++ b/huddle-test-geophones/ref.bib
@@ -0,0 +1,29 @@
+@article{barzilai98_techn_measur_noise_sensor_presen,
+  author =       {Aaron Barzilai and Tom VanZandt and Tom Kenny},
+  title =        {Technique for Measurement of the Noise of a Sensor in the
+                  Presence of Large Background Signals},
+  journal =      {Review of Scientific Instruments},
+  volume =       69,
+  number =       7,
+  pages =        {2767-2772},
+  year =         1998,
+  doi =          {10.1063/1.1149013},
+  url =          {https://doi.org/10.1063/1.1149013},
+}
+
+
+@article{kirchhoff17_huddl_test_measur_near_johns,
+  author =       {R. Kirchhoff and C. M. Mow-Lowry and V. B. Adya and G.
+                  Bergmann and S. Cooper and M. M. Hanke and P. Koch and S. M.
+                  K{\"o}hlenbeck and J. Lehmann and P. Oppermann and J.
+                  W{\"o}hler and D. S. Wu and H. L{\"u}ck and K. A. Strain},
+  title =        {Huddle Test Measurement of a Near Johnson Noise Limited
+                  Geophone},
+  journal =      {Review of Scientific Instruments},
+  volume =       88,
+  number =       11,
+  pages =        115008,
+  year =         2017,
+  doi =          {10.1063/1.5000592},
+  url =          {https://doi.org/10.1063/1.5000592},
+}
\ No newline at end of file