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Thomas Dehaeze 2019-04-18 17:11:25 +02:00
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81
huddle-test-geophones/compare_axis.m Normal file
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% Matlab Init :noexport:ignore:
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
%% Initialize ans with org-babel
ans = 0;
% Load data
% We first load the data for the three axis.
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
% The PSD for each axis of the two geophones are computed.
[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)));
% We compare them. The result is shown on figure [[fig:compare_axis_psd]].
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]);
% Compare TF
% The transfer functions from one geophone to the other are also computed for each axis.
% The result is shown on figure [[fig:compare_tf_axis]].
[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)));
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]);

179
huddle-test-geophones/index.html
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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2019-04-18 jeu. 17:02 --> <!-- 2019-04-18 jeu. 17:11 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" /> <meta name="viewport" content="width=device-width, initial-scale=1" />
<title>SpeedGoat</title> <title>SpeedGoat</title>
@ -276,36 +276,36 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#org15c3fe1">1. Setup</a></li> <li><a href="#orgeb3ab60">1. Experimental Setup</a></li>
<li><a href="#org10e7b57">2. Signal Processing</a> <li><a href="#org17b1fa2">2. Signal Processing</a>
<ul> <ul>
<li><a href="#orgce9374f">2.1. Load data</a></li> <li><a href="#orge77a4ee">2.1. Load data</a></li>
<li><a href="#orgf5bde3c">2.2. Time Domain Data</a></li> <li><a href="#org6b7befd">2.2. Time Domain Data</a></li>
<li><a href="#org53fa3f2">2.3. Computation of the ASD of the measured voltage</a></li> <li><a href="#org4e41681">2.3. Computation of the ASD of the measured voltage</a></li>
<li><a href="#orgdb1374d">2.4. Scaling to take into account the sensibility of the geophone and the voltage amplifier</a></li> <li><a href="#org83cd67f">2.4. Scaling to take into account the sensibility of the geophone and the voltage amplifier</a></li>
<li><a href="#org07e8527">2.5. Computation of the ASD of the velocity</a></li> <li><a href="#orge0f93a9">2.5. Computation of the ASD of the velocity</a></li>
<li><a href="#orgf3d4fe6">2.6. Transfer function between the two geophones</a></li> <li><a href="#orgd178821">2.6. Transfer function between the two geophones</a></li>
<li><a href="#org1336e7b">2.7. Estimation of the sensor noise</a></li> <li><a href="#orgaf0dac1">2.7. Estimation of the sensor noise</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgaee4ca7">3. Compare axis</a> <li><a href="#org126e178">3. Compare axis</a>
<ul> <ul>
<li><a href="#org4a91ee9">3.1. Load data</a></li> <li><a href="#org3f6cac8">3.1. Load data</a></li>
<li><a href="#org36adfc5">3.2. Compare PSD</a></li> <li><a href="#org608de0c">3.2. Compare PSD</a></li>
<li><a href="#org698e322">3.3. Compare TF</a></li> <li><a href="#org4db3872">3.3. Compare TF</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orge3ff149">4. Appendix</a> <li><a href="#org66ca70f">4. Appendix</a>
<ul> <ul>
<li><a href="#org2060421">4.1. Computation of coherence from PSD and CSD</a></li> <li><a href="#org7492edb">4.1. Computation of coherence from PSD and CSD</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org15c3fe1" class="outline-2"> <div id="outline-container-orgeb3ab60" class="outline-2">
<h2 id="org15c3fe1"><span class="section-number-2">1</span> Setup</h2> <h2 id="orgeb3ab60"><span class="section-number-2">1</span> Experimental Setup</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
Two L22 geophones are used. Two L22 geophones are used.
@ -319,14 +319,14 @@ The voltage amplifiers include a low pass filter with a cut-off frequency at 1kH
</p> </p>
<div id="org2286cc3" class="figure"> <div id="orgfe0dca3" class="figure">
<p><img src="./figs/setup.jpg" alt="setup.jpg" width="500px" /> <p><img src="./figs/setup.jpg" alt="setup.jpg" width="500px" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Setup</p> <p><span class="figure-number">Figure 1: </span>Setup</p>
</div> </div>
<div id="org9446018" class="figure"> <div id="orgbcec6a9" class="figure">
<p><img src="./figs/geophones.jpg" alt="geophones.jpg" width="500px" /> <p><img src="./figs/geophones.jpg" alt="geophones.jpg" width="500px" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Geophones</p> <p><span class="figure-number">Figure 2: </span>Geophones</p>
@ -334,16 +334,22 @@ The voltage amplifiers include a low pass filter with a cut-off frequency at 1kH
</div> </div>
</div> </div>
<div id="outline-container-org10e7b57" class="outline-2"> <div id="outline-container-org17b1fa2" class="outline-2">
<h2 id="org10e7b57"><span class="section-number-2">2</span> Signal Processing</h2> <h2 id="org17b1fa2"><span class="section-number-2">2</span> Signal Processing</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p>
The Matlab computing file for this part is accessible <a href="signal_processing.m">here</a>.
The <code>mat</code> file containing the measurement data is accessible <a href="mat/data_001.mat">here</a>.
</p>
</div> </div>
<div id="outline-container-orgce9374f" class="outline-3">
<h3 id="orgce9374f"><span class="section-number-3">2.1</span> Load data</h3> <div id="outline-container-orge77a4ee" class="outline-3">
<h3 id="orge77a4ee"><span class="section-number-3">2.1</span> Load data</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<p> <p>
We load the data of the z axis of two geophones. We load the data of the z axis of two geophones.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_001.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_001.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
dt = t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-type">-</span> t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>; dt = t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-type">-</span> t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -352,8 +358,8 @@ dt = t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-high
</div> </div>
</div> </div>
<div id="outline-container-orgf5bde3c" class="outline-3"> <div id="outline-container-org6b7befd" class="outline-3">
<h3 id="orgf5bde3c"><span class="section-number-3">2.2</span> Time Domain Data</h3> <h3 id="org6b7befd"><span class="section-number-3">2.2</span> Time Domain Data</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>; <pre class="src src-matlab"><span class="org-type">figure</span>;
@ -368,7 +374,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
<div id="org2d30e99" class="figure"> <div id="org0161433" class="figure">
<p><img src="figs/data_time_domain.png" alt="data_time_domain.png" /> <p><img src="figs/data_time_domain.png" alt="data_time_domain.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Time domain Data</p> <p><span class="figure-number">Figure 3: </span>Time domain Data</p>
@ -388,7 +394,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
<div id="org8e28356" class="figure"> <div id="orgab64b52" class="figure">
<p><img src="figs/data_time_domain_zoom.png" alt="data_time_domain_zoom.png" /> <p><img src="figs/data_time_domain_zoom.png" alt="data_time_domain_zoom.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Time domain Data - Zoom</p> <p><span class="figure-number">Figure 4: </span>Time domain Data - Zoom</p>
@ -396,8 +402,8 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
</div> </div>
<div id="outline-container-org53fa3f2" class="outline-3"> <div id="outline-container-org4e41681" class="outline-3">
<h3 id="org53fa3f2"><span class="section-number-3">2.3</span> Computation of the ASD of the measured voltage</h3> <h3 id="org4e41681"><span class="section-number-3">2.3</span> Computation of the ASD of the measured voltage</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<p> <p>
We first define the parameters for the frequency domain analysis. We first define the parameters for the frequency domain analysis.
@ -416,12 +422,12 @@ Fs = <span class="org-highlight-numbers-number">1</span><span class="org-type">/
</div> </div>
</div> </div>
<div id="outline-container-orgdb1374d" class="outline-3"> <div id="outline-container-org83cd67f" class="outline-3">
<h3 id="orgdb1374d"><span class="section-number-3">2.4</span> Scaling to take into account the sensibility of the geophone and the voltage amplifier</h3> <h3 id="org83cd67f"><span class="section-number-3">2.4</span> Scaling to take into account the sensibility of the geophone and the voltage amplifier</h3>
<div class="outline-text-3" id="text-2-4"> <div class="outline-text-3" id="text-2-4">
<p> <p>
The Geophone used are L22. The Geophone used are L22.
Their sensibility are shown on figure <a href="#org59aaa6d">5</a>. Their sensibility are shown on figure <a href="#orgf03c4b4">5</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
@ -432,7 +438,7 @@ S = <span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type"
</div> </div>
<div id="org59aaa6d" class="figure"> <div id="orgf03c4b4" class="figure">
<p><img src="figs/geophone_sensibility.png" alt="geophone_sensibility.png" /> <p><img src="figs/geophone_sensibility.png" alt="geophone_sensibility.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Sensibility of the Geophone</p> <p><span class="figure-number">Figure 5: </span>Sensibility of the Geophone</p>
@ -463,11 +469,11 @@ We further divide the result by the sensibility of the Geophone to obtain the AS
</div> </div>
</div> </div>
<div id="outline-container-org07e8527" class="outline-3"> <div id="outline-container-orge0f93a9" class="outline-3">
<h3 id="org07e8527"><span class="section-number-3">2.5</span> Computation of the ASD of the velocity</h3> <h3 id="orge0f93a9"><span class="section-number-3">2.5</span> Computation of the ASD of the velocity</h3>
<div class="outline-text-3" id="text-2-5"> <div class="outline-text-3" id="text-2-5">
<p> <p>
The ASD of the measured velocity is shown on figure <a href="#orgd1800a0">6</a>. The ASD of the measured velocity is shown on figure <a href="#org9a56511">6</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
@ -484,7 +490,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
<div id="orgd1800a0" class="figure"> <div id="org9a56511" class="figure">
<p><img src="figs/psd_velocity.png" alt="psd_velocity.png" /> <p><img src="figs/psd_velocity.png" alt="psd_velocity.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Spectral density of the velocity</p> <p><span class="figure-number">Figure 6: </span>Spectral density of the velocity</p>
@ -492,16 +498,16 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
</div> </div>
<div id="outline-container-orgf3d4fe6" class="outline-3"> <div id="outline-container-orgd178821" class="outline-3">
<h3 id="orgf3d4fe6"><span class="section-number-3">2.6</span> Transfer function between the two geophones</h3> <h3 id="orgd178821"><span class="section-number-3">2.6</span> Transfer function between the two geophones</h3>
<div class="outline-text-3" id="text-2-6"> <div class="outline-text-3" id="text-2-6">
<p> <p>
We here compute the transfer function from one geophone to the other. We here compute the transfer function from one geophone to the other.
The result is shown on figure <a href="#org6abd5e1">7</a>. The result is shown on figure <a href="#orgb6c07f9">7</a>.
</p> </p>
<p> <p>
We also compute the coherence between the two signals (figure <a href="#org80d61f1">8</a>). We also compute the coherence between the two signals (figure <a href="#org1b45a36">8</a>).
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
@ -510,7 +516,7 @@ We also compute the coherence between the two signals (figure <a href="#org80d61
</div> </div>
<div id="org6abd5e1" class="figure"> <div id="orgb6c07f9" class="figure">
<p><img src="figs/tf_geophones.png" alt="tf_geophones.png" /> <p><img src="figs/tf_geophones.png" alt="tf_geophones.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Estimated transfer function between the two geophones</p> <p><span class="figure-number">Figure 7: </span>Estimated transfer function between the two geophones</p>
@ -522,7 +528,7 @@ We also compute the coherence between the two signals (figure <a href="#org80d61
</div> </div>
<div id="org80d61f1" class="figure"> <div id="org1b45a36" class="figure">
<p><img src="figs/coh_geophones.png" alt="coh_geophones.png" /> <p><img src="figs/coh_geophones.png" alt="coh_geophones.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Cohererence between the signals of the two geophones</p> <p><span class="figure-number">Figure 8: </span>Cohererence between the signals of the two geophones</p>
@ -530,8 +536,8 @@ We also compute the coherence between the two signals (figure <a href="#org80d61
</div> </div>
</div> </div>
<div id="outline-container-org1336e7b" class="outline-3"> <div id="outline-container-orgaf0dac1" class="outline-3">
<h3 id="org1336e7b"><span class="section-number-3">2.7</span> Estimation of the sensor noise</h3> <h3 id="orgaf0dac1"><span class="section-number-3">2.7</span> Estimation of the sensor noise</h3>
<div class="outline-text-3" id="text-2-7"> <div class="outline-text-3" id="text-2-7">
<p> <p>
The technique to estimate the sensor noise is taken from <a class='org-ref-reference' href="#barzilai98_techn_measur_noise_sensor_presen">barzilai98_techn_measur_noise_sensor_presen</a>. The technique to estimate the sensor noise is taken from <a class='org-ref-reference' href="#barzilai98_techn_measur_noise_sensor_presen">barzilai98_techn_measur_noise_sensor_presen</a>.
@ -561,11 +567,11 @@ where:
</ul> </ul>
<p> <p>
The <code>mscohere</code> function is compared with this formula on Appendix (section <a href="#org72623b2">4.1</a>), it is shown that it is identical. The <code>mscohere</code> function is compared with this formula on Appendix (section <a href="#orgd085438">4.1</a>), it is shown that it is identical.
</p> </p>
<p> <p>
Figure <a href="#org88b4597">9</a> illustrate a block diagram model of the system used to determine the sensor noise of the geophone. Figure <a href="#orga4af110">9</a> illustrate a block diagram model of the system used to determine the sensor noise of the geophone.
</p> </p>
<p> <p>
@ -577,7 +583,7 @@ Each sensor has noise \(N\) and \(M\).
</p> </p>
<div id="org88b4597" class="figure"> <div id="orga4af110" class="figure">
<p><img src="figs/huddle-test.png" alt="huddle-test.png" /> <p><img src="figs/huddle-test.png" alt="huddle-test.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Huddle test block diagram</p> <p><span class="figure-number">Figure 9: </span>Huddle test block diagram</p>
@ -592,7 +598,7 @@ We also assume that \(H_1 = H_2 = 1\).
We then obtain: We then obtain:
</p> </p>
\begin{equation} \begin{equation}
\label{org5b4a541} \label{org65b3ddf}
\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} \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} \end{equation}
@ -600,23 +606,23 @@ We then obtain:
Since the input signal \(U\) and the instrumental noise \(N\) are incoherent: Since the input signal \(U\) and the instrumental noise \(N\) are incoherent:
</p> </p>
\begin{equation} \begin{equation}
\label{orga04bdf3} \label{org14038cd}
|G_X(\omega)| = |G_N(\omega)| + |G_U(\omega)| |G_X(\omega)| = |G_N(\omega)| + |G_U(\omega)|
\end{equation} \end{equation}
<p> <p>
From equations \eqref{org5b4a541} and \eqref{orga04bdf3}, we finally obtain From equations \eqref{org65b3ddf} and \eqref{org14038cd}, we finally obtain
</p> </p>
<div class="important"> <div class="important">
\begin{equation} \begin{equation}
\label{orga57b1ae} \label{org84c455c}
|G_N(\omega)| = |G_X(\omega)| \left( 1 - \sqrt{\gamma_{XY}^2(\omega)} \right) |G_N(\omega)| = |G_X(\omega)| \left( 1 - \sqrt{\gamma_{XY}^2(\omega)} \right)
\end{equation} \end{equation}
</div> </div>
<p> <p>
The instrumental noise is computed below. The result in V<sup>2</sup>/Hz is shown on figure <a href="#orgd3329eb">10</a>. The instrumental noise is computed below. The result in V<sup>2</sup>/Hz is shown on figure <a href="#orgd0b903d">10</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">pxxN = pxx1<span class="org-type">.*</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">-</span> coh12<span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">pxxN = pxx1<span class="org-type">.*</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">-</span> coh12<span class="org-rainbow-delimiters-depth-1">)</span>;
@ -637,14 +643,14 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
<div id="orgd3329eb" class="figure"> <div id="orgd0b903d" class="figure">
<p><img src="figs/intrumental_noise_V.png" alt="intrumental_noise_V.png" /> <p><img src="figs/intrumental_noise_V.png" alt="intrumental_noise_V.png" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>Instrumental Noise and Measurement in \(V^2/Hz\)</p> <p><span class="figure-number">Figure 10: </span>Instrumental Noise and Measurement in \(V^2/Hz\)</p>
</div> </div>
<p> <p>
This is then further converted into velocity and compared with the ground velocity measurement. (figure <a href="#org6fa2e55">11</a>) This is then further converted into velocity and compared with the ground velocity measurement. (figure <a href="#org0542ab9">11</a>)
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>; <pre class="src src-matlab"><span class="org-type">figure</span>;
@ -660,7 +666,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
<div id="org6fa2e55" class="figure"> <div id="org0542ab9" class="figure">
<p><img src="figs/intrumental_noise_velocity.png" alt="intrumental_noise_velocity.png" /> <p><img src="figs/intrumental_noise_velocity.png" alt="intrumental_noise_velocity.png" />
</p> </p>
<p><span class="figure-number">Figure 11: </span>Instrumental Noise and Measurement in \(m/s/\sqrt{Hz}\)</p> <p><span class="figure-number">Figure 11: </span>Instrumental Noise and Measurement in \(m/s/\sqrt{Hz}\)</p>
@ -669,13 +675,26 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
</div> </div>
<div id="outline-container-orgaee4ca7" class="outline-2"> <div id="outline-container-org126e178" class="outline-2">
<h2 id="orgaee4ca7"><span class="section-number-2">3</span> Compare axis</h2> <h2 id="org126e178"><span class="section-number-2">3</span> Compare axis</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p>
The Matlab computing file for this part is accessible <a href="compare_axis.m">here</a>.
The <code>mat</code> files containing the measurement data are accessible with the following links:
</p>
<ul class="org-ul">
<li>z axis: <a href="mat/data_001.mat">here</a>.</li>
<li>east axis: <a href="mat/data_002.mat">here</a>.</li>
<li>north axis: <a href="mat/data_003.mat">here</a>.</li>
</ul>
</div> </div>
<div id="outline-container-org4a91ee9" class="outline-3">
<h3 id="org4a91ee9"><span class="section-number-3">3.1</span> Load data</h3> <div id="outline-container-org3f6cac8" class="outline-3">
<h3 id="org3f6cac8"><span class="section-number-3">3.1</span> Load data</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<p>
We first load the data for the three axis.
</p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">z = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_001.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">z = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_001.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
east = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_002.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>; east = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_002.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -685,9 +704,12 @@ north = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="or
</div> </div>
</div> </div>
<div id="outline-container-org36adfc5" class="outline-3"> <div id="outline-container-org608de0c" class="outline-3">
<h3 id="org36adfc5"><span class="section-number-3">3.2</span> Compare PSD</h3> <h3 id="org608de0c"><span class="section-number-3">3.2</span> Compare PSD</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<p>
The PSD for each axis of the two geophones are computed.
</p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>pz1, fz<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>z.x1, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>z.x1<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>pz1, fz<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>z.x1, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>z.x1<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-rainbow-delimiters-depth-1">[</span>pz2, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>z.x2, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>z.x2<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-rainbow-delimiters-depth-1">[</span>pz2, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>z.x2, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>z.x2<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -700,8 +722,11 @@ north = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="or
</pre> </pre>
</div> </div>
<p>
We compare them. The result is shown on figure <a href="#orge0ebe78">12</a>.
</p>
<div id="orgc2c3d69" class="figure"> <div id="orge0ebe78" class="figure">
<p><img src="figs/compare_axis_psd.png" alt="compare_axis_psd.png" /> <p><img src="figs/compare_axis_psd.png" alt="compare_axis_psd.png" />
</p> </p>
<p><span class="figure-number">Figure 12: </span>Compare the measure PSD of the two geophones for the three axis</p> <p><span class="figure-number">Figure 12: </span>Compare the measure PSD of the two geophones for the three axis</p>
@ -709,9 +734,14 @@ north = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="or
</div> </div>
</div> </div>
<div id="outline-container-org698e322" class="outline-3"> <div id="outline-container-org4db3872" class="outline-3">
<h3 id="org698e322"><span class="section-number-3">3.3</span> Compare TF</h3> <h3 id="org4db3872"><span class="section-number-3">3.3</span> Compare TF</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<p>
The transfer functions from one geophone to the other are also computed for each axis.
The result is shown on figure <a href="#org2a4c622">13</a>.
</p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>Tz, fz<span class="org-rainbow-delimiters-depth-1">]</span> = tfestimate<span class="org-rainbow-delimiters-depth-1">(</span>z.x1, z.x2, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>z.x1<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>Tz, fz<span class="org-rainbow-delimiters-depth-1">]</span> = tfestimate<span class="org-rainbow-delimiters-depth-1">(</span>z.x1, z.x2, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>z.x1<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-rainbow-delimiters-depth-1">[</span>Te, fe<span class="org-rainbow-delimiters-depth-1">]</span> = tfestimate<span class="org-rainbow-delimiters-depth-1">(</span>east.x1, east.x2, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>east.x1<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>east.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>east.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-rainbow-delimiters-depth-1">[</span>Te, fe<span class="org-rainbow-delimiters-depth-1">]</span> = tfestimate<span class="org-rainbow-delimiters-depth-1">(</span>east.x1, east.x2, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>east.x1<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>east.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>east.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -720,7 +750,7 @@ north = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="or
</div> </div>
<div id="orgf7b4d80" class="figure"> <div id="org2a4c622" class="figure">
<p><img src="figs/compare_tf_axis.png" alt="compare_tf_axis.png" /> <p><img src="figs/compare_tf_axis.png" alt="compare_tf_axis.png" />
</p> </p>
<p><span class="figure-number">Figure 13: </span>Compare the transfer function from one geophone to the other for the 3 axis</p> <p><span class="figure-number">Figure 13: </span>Compare the transfer function from one geophone to the other for the 3 axis</p>
@ -728,15 +758,16 @@ north = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="or
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orge3ff149" class="outline-2">
<h2 id="orge3ff149"><span class="section-number-2">4</span> Appendix</h2> <div id="outline-container-org66ca70f" class="outline-2">
<h2 id="org66ca70f"><span class="section-number-2">4</span> Appendix</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
</div> </div>
<div id="outline-container-org2060421" class="outline-3"> <div id="outline-container-org7492edb" class="outline-3">
<h3 id="org2060421"><span class="section-number-3">4.1</span> Computation of coherence from PSD and CSD</h3> <h3 id="org7492edb"><span class="section-number-3">4.1</span> Computation of coherence from PSD and CSD</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
<p> <p>
<a id="org72623b2"></a> <a id="orgd085438"></a>
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_001.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>; <pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_001.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
@ -767,7 +798,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
<div id="org3184d2c" class="figure"> <div id="org0bcdbf7" class="figure">
<p><img src="figs/comp_coherence_formula.png" alt="comp_coherence_formula.png" /> <p><img src="figs/comp_coherence_formula.png" alt="comp_coherence_formula.png" />
</p> </p>
<p><span class="figure-number">Figure 14: </span>Comparison of <code>mscohere</code> and manual computation</p> <p><span class="figure-number">Figure 14: </span>Comparison of <code>mscohere</code> and manual computation</p>
@ -785,7 +816,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p> <p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-04-18 jeu. 17:02</p> <p class="date">Created: 2019-04-18 jeu. 17:11</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p> <p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div> </div>
</body> </body>

29
huddle-test-geophones/index.org
View File

@ -18,7 +18,7 @@
#+PROPERTY: header-args:matlab+ :output-dir figs #+PROPERTY: header-args:matlab+ :output-dir figs
:END: :END:
* Setup * Experimental Setup
Two L22 geophones are used. Two L22 geophones are used.
They are placed on the ID31 granite. They are placed on the ID31 granite.
They are leveled. They are leveled.
@ -37,6 +37,13 @@ The voltage amplifiers include a low pass filter with a cut-off frequency at 1kH
[[file:./figs/geophones.jpg]] [[file:./figs/geophones.jpg]]
* Signal Processing * Signal Processing
:PROPERTIES:
:header-args:matlab+: :tangle signal_processing.m
:header-args:matlab+: :comments org :mkdirp yes
:END:
The Matlab computing file for this part is accessible [[file:signal_processing.m][here]].
The =mat= file containing the measurement data is accessible [[file:mat/data_001.mat][here]].
** Matlab Init :noexport:ignore: ** Matlab Init :noexport:ignore:
#+begin_src matlab :exports none :results silent :noweb yes #+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>> <<matlab-init>>
@ -44,6 +51,7 @@ The voltage amplifiers include a low pass filter with a cut-off frequency at 1kH
** Load data ** Load data
We load the data of the z axis of two geophones. We load the data of the z axis of two geophones.
#+begin_src matlab :results none #+begin_src matlab :results none
load('mat/data_001.mat', 't', 'x1', 'x2'); load('mat/data_001.mat', 't', 'x1', 'x2');
dt = t(2) - t(1); dt = t(2) - t(1);
@ -346,12 +354,23 @@ This is then further converted into velocity and compared with the ground veloci
[[file:figs/intrumental_noise_velocity.png]] [[file:figs/intrumental_noise_velocity.png]]
* Compare axis * Compare axis
:PROPERTIES:
:header-args:matlab+: :tangle compare_axis.m
:header-args:matlab+: :comments org :mkdirp yes
:END:
The Matlab computing file for this part is accessible [[file:compare_axis.m][here]].
The =mat= files containing the measurement data are accessible with the following links:
- z axis: [[file:mat/data_001.mat][here]].
- east axis: [[file:mat/data_002.mat][here]].
- north axis: [[file:mat/data_003.mat][here]].
** Matlab Init :noexport:ignore: ** Matlab Init :noexport:ignore:
#+begin_src matlab :exports none :results silent :noweb yes #+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>> <<matlab-init>>
#+end_src #+end_src
** Load data ** Load data
We first load the data for the three axis.
#+begin_src matlab :results none #+begin_src matlab :results none
z = load('mat/data_001.mat', 't', 'x1', 'x2'); z = load('mat/data_001.mat', 't', 'x1', 'x2');
east = load('mat/data_002.mat', 't', 'x1', 'x2'); east = load('mat/data_002.mat', 't', 'x1', 'x2');
@ -359,6 +378,7 @@ This is then further converted into velocity and compared with the ground veloci
#+end_src #+end_src
** Compare PSD ** Compare PSD
The PSD for each axis of the two geophones are computed.
#+begin_src matlab :results none #+begin_src matlab :results none
[pz1, fz] = pwelch(z.x1, hanning(ceil(length(z.x1)/100)), [], [], 1/(z.t(2)-z.t(1))); [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))); [pz2, ~] = pwelch(z.x2, hanning(ceil(length(z.x2)/100)), [], [], 1/(z.t(2)-z.t(1)));
@ -370,7 +390,8 @@ This is then further converted into velocity and compared with the ground veloci
[pn2, ~] = pwelch(north.x2, hanning(ceil(length(north.x2)/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 #+end_src
#+begin_src matlab :results none :exports none We compare them. The result is shown on figure [[fig:compare_axis_psd]].
#+begin_src matlab :results none :exports none
figure; figure;
hold on; hold on;
plot(fz, sqrt(pz1), '-', 'Color', [0 0.4470 0.7410], 'DisplayName', 'z'); plot(fz, sqrt(pz1), '-', 'Color', [0 0.4470 0.7410], 'DisplayName', 'z');
@ -398,6 +419,9 @@ This is then further converted into velocity and compared with the ground veloci
[[file:figs/compare_axis_psd.png]] [[file:figs/compare_axis_psd.png]]
** Compare TF ** Compare TF
The transfer functions from one geophone to the other are also computed for each axis.
The result is shown on figure [[fig:compare_tf_axis]].
#+begin_src matlab :results none #+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))); [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))); [Te, fe] = tfestimate(east.x1, east.x2, hanning(ceil(length(east.x1)/100)), [], [], 1/(east.t(2)-east.t(1)));
@ -442,6 +466,7 @@ This is then further converted into velocity and compared with the ground veloci
#+CAPTION: Compare the transfer function from one geophone to the other for the 3 axis #+CAPTION: Compare the transfer function from one geophone to the other for the 3 axis
#+RESULTS: fig:compare_tf_axis #+RESULTS: fig:compare_tf_axis
[[file:figs/compare_tf_axis.png]] [[file:figs/compare_tf_axis.png]]
* Appendix * Appendix
** Computation of coherence from PSD and CSD ** Computation of coherence from PSD and CSD
<<sec:coherence>> <<sec:coherence>>

233
huddle-test-geophones/signal_processing.m Normal file
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@ -0,0 +1,233 @@
% Matlab Init :noexport:ignore:
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
%% Initialize ans with org-babel
ans = 0;
% 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)]);
% #+NAME: fig:data_time_domain
% #+CAPTION: Time domain Data
% #+RESULTS: fig:data_time_domain
% [[file:figs/data_time_domain.png]]
figure;
hold on;
plot(t, x1);
plot(t, x2);
hold off;
xlabel('Time [s]');
ylabel('Voltage [V]');
xlim([0 1]);
% 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);
figure;
bodeFig({S});
ylabel('Amplitude [V/(m/s)]')
% #+NAME: fig:geophone_sensibility
% #+CAPTION: Sensibility of the Geophone
% #+RESULTS: fig:geophone_sensibility
% [[file:figs/geophone_sensibility.png]]
% 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]);
% 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);
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]);
% #+NAME: fig:tf_geophones
% #+CAPTION: Estimated transfer function between the two geophones
% #+RESULTS: fig:tf_geophones
% [[file:figs/tf_geophones.png]]
[coh12, ~] = mscohere(x1, x2, win, [], [], Fs);
figure;
plot(f, coh12);
set(gca, 'xscale', 'log');
xlabel('Frequency [Hz]'); ylabel('Coherence');
ylim([0,1]); xlim([1, 500]);
% 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 $M$.
% #+NAME: fig:huddle_test
% #+CAPTION: Huddle test block diagram
% [[file:figs/huddle-test.png]]
% 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:
% #+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]].
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]);
% #+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]])
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]);