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<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgc7e4a9f">1. Experimental Setup</a></li>
<li><a href="#org6abc0c5">2. Signal Processing</a>
<li><a href="#orgb47b0b2">1. Experimental Setup</a></li>
<li><a href="#org0b1e99e">2. Signal Processing</a>
<ul>
<li><a href="#orgda0769c">2.1. Load data</a></li>
<li><a href="#orgf362871">2.2. Time Domain Data</a></li>
<li><a href="#org3a306be">2.3. Computation of the ASD of the measured voltage</a></li>
<li><a href="#org0a45f4a">2.4. Scaling to take into account the sensibility of the geophone and the voltage amplifier</a></li>
<li><a href="#org63db1c4">2.5. Computation of the ASD of the velocity</a></li>
<li><a href="#org6df28e4">2.6. Transfer function between the two geophones</a></li>
<li><a href="#org2a1c732">2.7. Estimation of the sensor noise</a></li>
<li><a href="#org35507da">2.1. Load data</a></li>
<li><a href="#org69bec1f">2.2. Time Domain Data</a></li>
<li><a href="#orgf0a72da">2.3. Computation of the ASD of the measured voltage</a></li>
<li><a href="#org7acabeb">2.4. Scaling to take into account the sensibility of the geophone and the voltage amplifier</a></li>
<li><a href="#org4b3b8a7">2.5. Computation of the ASD of the velocity</a></li>
<li><a href="#orgaaca76b">2.6. Transfer function between the two geophones</a></li>
<li><a href="#org5c32c39">2.7. Estimation of the sensor noise</a></li>
</ul>
</li>
<li><a href="#orga1d7838">3. Compare axis</a>
<li><a href="#org52fed77">3. Compare axis</a>
<ul>
<li><a href="#org1153270">3.1. Load data</a></li>
<li><a href="#orgd3a0160">3.2. Compare PSD</a></li>
<li><a href="#orgcd6137b">3.3. Compare TF</a></li>
<li><a href="#org7f52c4b">3.1. Load data</a></li>
<li><a href="#orgb105293">3.2. Compare PSD</a></li>
<li><a href="#orgbce3b98">3.3. Compare TF</a></li>
</ul>
</li>
<li><a href="#org693291d">4. Appendix</a>
<li><a href="#orgd32606c">4. Appendix</a>
<ul>
<li><a href="#org58d8ebb">4.1. Computation of coherence from PSD and CSD</a></li>
<li><a href="#orge652a09">4.1. Computation of coherence from PSD and CSD</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-orgc7e4a9f" class="outline-2">
<h2 id="orgc7e4a9f"><span class="section-number-2">1</span> Experimental Setup</h2>
<div id="outline-container-orgb47b0b2" class="outline-2">
<h2 id="orgb47b0b2"><span class="section-number-2">1</span> Experimental Setup</h2>
<div class="outline-text-2" id="text-1">
<p>
Two L22 geophones are used.
@@ -327,14 +77,14 @@ The voltage amplifiers includes:
</ul>
<div id="orga476294" class="figure">
<div id="orgfc98f42" class="figure">
<p><img src="./img/setup.jpg" alt="setup.jpg" width="500px" />
</p>
<p><span class="figure-number">Figure 1: </span>Setup</p>
</div>
<div id="org2fb54a0" class="figure">
<div id="org01be4e4" class="figure">
<p><img src="./img/geophones.jpg" alt="geophones.jpg" width="500px" />
</p>
<p><span class="figure-number">Figure 2: </span>Geophones</p>
@@ -342,13 +92,13 @@ The voltage amplifiers includes:
</div>
</div>
<div id="outline-container-org6abc0c5" class="outline-2">
<h2 id="org6abc0c5"><span class="section-number-2">2</span> Signal Processing</h2>
<div id="outline-container-org0b1e99e" class="outline-2">
<h2 id="org0b1e99e"><span class="section-number-2">2</span> Signal Processing</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="orged15cd2"></a>
<a id="org28ac54a"></a>
</p>
<div class="note">
<div class="note" id="orgdb9b9cc">
<p>
All the files (data and Matlab scripts) are accessible <a href="data/huddle_test_signal_processing.zip">here</a>.
</p>
@@ -356,38 +106,38 @@ All the files (data and Matlab scripts) are accessible <a href="data/huddle_test
</div>
</div>
<div id="outline-container-orgda0769c" class="outline-3">
<h3 id="orgda0769c"><span class="section-number-3">2.1</span> Load data</h3>
<div id="outline-container-org35507da" class="outline-3">
<h3 id="org35507da"><span class="section-number-3">2.1</span> Load data</h3>
<div class="outline-text-3" id="text-2-1">
<p>
We load the data of the z axis of two geophones.
</p>
<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>;
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>;
<pre class="src src-matlab">load(<span class="org-string">'mat/data_001.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'x1'</span>, <span class="org-string">'x2'</span>);
dt = t(2) <span class="org-type">-</span> t(1);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgf362871" class="outline-3">
<h3 id="orgf362871"><span class="section-number-3">2.2</span> Time Domain Data</h3>
<div id="outline-container-org69bec1f" class="outline-3">
<h3 id="org69bec1f"><span class="section-number-3">2.2</span> Time Domain Data</h3>
<div class="outline-text-3" id="text-2-2">
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>t, x1<span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>t, x2<span class="org-rainbow-delimiters-depth-1">)</span>;
plot(t, x1);
plot(t, x2);
hold off;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Time </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">s</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
ylabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Voltage </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">V</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>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>, t<span class="org-rainbow-delimiters-depth-3">(</span>end<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel(<span class="org-string">'Time [s]'</span>);
ylabel(<span class="org-string">'Voltage [V]'</span>);
xlim([t(1), t(end)]);
</pre>
</div>
<div id="orgb109da2" class="figure">
<div id="orgc871843" class="figure">
<p><img src="figs/data_time_domain.png" alt="data_time_domain.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Time domain Data</p>
@@ -397,17 +147,17 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>t, x1<span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>t, x2<span class="org-rainbow-delimiters-depth-1">)</span>;
plot(t, x1);
plot(t, x2);
hold off;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Time </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">s</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
ylabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Voltage </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">V</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel(<span class="org-string">'Time [s]'</span>);
ylabel(<span class="org-string">'Voltage [V]'</span>);
xlim([0 1]);
</pre>
</div>
<div id="org09931c1" class="figure">
<div id="org30ca9f2" class="figure">
<p><img src="figs/data_time_domain_zoom.png" alt="data_time_domain_zoom.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Time domain Data - Zoom</p>
@@ -415,16 +165,16 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
</div>
<div id="outline-container-org3a306be" class="outline-3">
<h3 id="org3a306be"><span class="section-number-3">2.3</span> Computation of the ASD of the measured voltage</h3>
<div id="outline-container-orgf0a72da" class="outline-3">
<h3 id="orgf0a72da"><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">
<p>
We first define the parameters for the frequency domain analysis.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Fs = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>dt; <span class="org-comment">% [Hz]</span>
<pre class="src src-matlab">Fs = 1<span class="org-type">/</span>dt; <span class="org-comment">% [Hz]</span>
win = hanning<span class="org-rainbow-delimiters-depth-1">(</span>ceil<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">10</span><span class="org-type">*</span>Fs<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
win = hanning(ceil(10<span class="org-type">*</span>Fs));
</pre>
</div>
@@ -432,30 +182,30 @@ win = hanning<span class="org-rainbow-delimiters-depth-1">(</span>ceil<span clas
Then we compute the Power Spectral Density using <code>pwelch</code> function.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>pxx1, f<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>x1, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-rainbow-delimiters-depth-1">[</span>pxx2, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>x2, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
<pre class="src src-matlab">[pxx1, f] = pwelch(x1, win, [], [], Fs);
[pxx2, <span class="org-type">~</span>] = pwelch(x2, win, [], [], Fs);
</pre>
</div>
<p>
And we plot the result on figure <a href="#org7cf943c">5</a>.
And we plot the result on figure <a href="#orgf270ed7">5</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>pxx1<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>pxx2<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot(f, sqrt(pxx1));
plot(f, sqrt(pxx2));
hold off;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'yscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'</span>ASD of the measured Voltage $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span>V<span class="org-rainbow-delimiters-depth-3">}{</span><span class="org-type">\</span>sqrt<span class="org-rainbow-delimiters-depth-4">{</span>Hz<span class="org-rainbow-delimiters-depth-4">}</span><span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>);
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'ASD of the measured Voltage $\left[\frac{V}{\sqrt{Hz}}\right]$'</span>)
xlim([0.1, 500]);
</pre>
</div>
<div id="org7cf943c" class="figure">
<div id="orgf270ed7" class="figure">
<p><img src="figs/asd_voltage.png" alt="asd_voltage.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Amplitude Spectral Density of the measured voltage</p>
@@ -463,23 +213,23 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
</div>
<div id="outline-container-org0a45f4a" class="outline-3">
<h3 id="org0a45f4a"><span class="section-number-3">2.4</span> Scaling to take into account the sensibility of the geophone and the voltage amplifier</h3>
<div id="outline-container-org7acabeb" class="outline-3">
<h3 id="org7acabeb"><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">
<p>
The Geophone used are L22. Their sensibility is shown on figure <a href="#org045b636">6</a>.
The Geophone used are L22. Their sensibility is shown on figure <a href="#orgc586e06">6</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">S0 = <span class="org-highlight-numbers-number">88</span>; <span class="org-comment">% Sensitivity [V/(m/s)]</span>
f0 = <span class="org-highlight-numbers-number">2</span>; <span class="org-comment">% Cut-off frequnecy [Hz]</span>
<pre class="src src-matlab">S0 = 88; <span class="org-comment">% Sensitivity [V/(m/s)]</span>
f0 = 2; <span class="org-comment">% Cut-off frequnecy [Hz]</span>
S = S0<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>f0<span class="org-rainbow-delimiters-depth-1">)</span><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>s<span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>f0<span class="org-rainbow-delimiters-depth-1">)</span>;
S = S0<span class="org-type">*</span>(s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>f0)<span class="org-type">/</span>(1<span class="org-type">+</span>s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>f0);
</pre>
</div>
<div id="org045b636" class="figure">
<div id="orgc586e06" class="figure">
<p><img src="figs/geophone_sensibility.png" alt="geophone_sensibility.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Sensibility of the Geophone</p>
@@ -491,9 +241,9 @@ We also take into account the gain of the electronics which is here set to be \(
</p>
<div class="org-src-container">
<pre class="src src-matlab">G0_db = <span class="org-highlight-numbers-number">60</span>; <span class="org-comment">% [dB]</span>
<pre class="src src-matlab">G0_db = 60; <span class="org-comment">% [dB]</span>
G0 = <span class="org-highlight-numbers-number">10</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-1">(</span>G0_db<span class="org-type">/</span><span class="org-highlight-numbers-number">20</span><span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% [abs]</span>
G0 = 10<span class="org-type">^</span>(G0_db<span class="org-type">/</span>20); <span class="org-comment">% [abs]</span>
</pre>
</div>
@@ -503,57 +253,57 @@ We further divide the result by the sensibility of the Geophone to obtain the AS
</p>
<div class="org-src-container">
<pre class="src src-matlab">scaling = <span class="org-highlight-numbers-number">1</span><span class="org-type">./</span>squeeze<span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>freqresp<span class="org-rainbow-delimiters-depth-3">(</span>G0<span class="org-type">*</span>S, f, <span class="org-string">'Hz'</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">scaling = 1<span class="org-type">./</span>squeeze(abs(freqresp(G0<span class="org-type">*</span>S, f, <span class="org-string">'Hz'</span>)));
</pre>
</div>
</div>
</div>
<div id="outline-container-org63db1c4" class="outline-3">
<h3 id="org63db1c4"><span class="section-number-3">2.5</span> Computation of the ASD of the velocity</h3>
<div id="outline-container-org4b3b8a7" class="outline-3">
<h3 id="org4b3b8a7"><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">
<p>
The ASD of the measured velocity is shown on figure <a href="#org1aa0246">7</a>.
The ASD of the measured velocity is shown on figure <a href="#orgdacf8d0">7</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>pxx1<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.*</span>scaling<span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>pxx2<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.*</span>scaling<span class="org-rainbow-delimiters-depth-1">)</span>;
plot(f, sqrt(pxx1)<span class="org-type">.*</span>scaling);
plot(f, sqrt(pxx2)<span class="org-type">.*</span>scaling);
hold off;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'yscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'</span>ASD of the measured Velocity $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span>m<span class="org-type">/</span>s<span class="org-rainbow-delimiters-depth-3">}{</span><span class="org-type">\</span>sqrt<span class="org-rainbow-delimiters-depth-4">{</span>Hz<span class="org-rainbow-delimiters-depth-4">}</span><span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>);
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'ASD of the measured Velocity $\left[\frac{m/s}{\sqrt{Hz}}\right]$'</span>)
xlim([0.1, 500]);
</pre>
</div>
<div id="org1aa0246" class="figure">
<div id="orgdacf8d0" class="figure">
<p><img src="figs/psd_velocity.png" alt="psd_velocity.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Amplitude Spectral Density of the Velocity</p>
</div>
<p>
We also plot the ASD in displacement (figure <a href="#org2fbcc25">8</a>);
We also plot the ASD in displacement (figure <a href="#org47aa587">8</a>);
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, <span class="org-rainbow-delimiters-depth-2">(</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span>pxx1<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">.*</span>scaling<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">./</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>f<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, <span class="org-rainbow-delimiters-depth-2">(</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span>pxx2<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">.*</span>scaling<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">./</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>f<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot(f, (sqrt(pxx1)<span class="org-type">.*</span>scaling)<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>f));
plot(f, (sqrt(pxx2)<span class="org-type">.*</span>scaling)<span class="org-type">./</span>(2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>f));
hold off;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; set</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">gca, 'yscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'</span>ASD of the displacement $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span>m<span class="org-rainbow-delimiters-depth-3">}{</span><span class="org-type">\</span>sqrt<span class="org-rainbow-delimiters-depth-4">{</span>Hz<span class="org-rainbow-delimiters-depth-4">}</span><span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'ASD of the displacement $\left[\frac{m}{\sqrt{Hz}}\right]$'</span>)
xlim([0.1, 500]);
</pre>
</div>
<div id="org2fbcc25" class="figure">
<div id="org47aa587" class="figure">
<p><img src="figs/asd_displacement.png" alt="asd_displacement.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Amplitude Spectral Density of the Displacement</p>
@@ -561,37 +311,37 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
</div>
<div id="outline-container-org6df28e4" class="outline-3">
<h3 id="org6df28e4"><span class="section-number-3">2.6</span> Transfer function between the two geophones</h3>
<div id="outline-container-orgaaca76b" class="outline-3">
<h3 id="orgaaca76b"><span class="section-number-3">2.6</span> Transfer function between the two geophones</h3>
<div class="outline-text-3" id="text-2-6">
<p>
We here compute the transfer function from one geophone to the other.
The result is shown on figure <a href="#org008cd50">9</a>.
The result is shown on figure <a href="#org313c704">9</a>.
</p>
<p>
We also compute the coherence between the two signals (figure <a href="#org47fd0f6">10</a>).
We also compute the coherence between the two signals (figure <a href="#orgbed81d4">10</a>).
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>T12, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = tfestimate<span class="org-rainbow-delimiters-depth-1">(</span>x1, x2, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
<pre class="src src-matlab">[T12, <span class="org-type">~</span>] = tfestimate(x1, x2, win, [], [], Fs);
</pre>
</div>
<div id="org008cd50" class="figure">
<div id="org313c704" class="figure">
<p><img src="figs/tf_geophones.png" alt="tf_geophones.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Estimated transfer function between the two geophones</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>coh12, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = mscohere<span class="org-rainbow-delimiters-depth-1">(</span>x1, x2, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
<pre class="src src-matlab">[coh12, <span class="org-type">~</span>] = mscohere(x1, x2, win, [], [], Fs);
</pre>
</div>
<div id="org47fd0f6" class="figure">
<div id="orgbed81d4" class="figure">
<p><img src="figs/coh_geophones.png" alt="coh_geophones.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Cohererence between the signals of the two geophones</p>
@@ -599,11 +349,11 @@ We also compute the coherence between the two signals (figure <a href="#org47fd0
</div>
</div>
<div id="outline-container-org2a1c732" class="outline-3">
<h3 id="org2a1c732"><span class="section-number-3">2.7</span> Estimation of the sensor noise</h3>
<div id="outline-container-org5c32c39" class="outline-3">
<h3 id="org5c32c39"><span class="section-number-3">2.7</span> Estimation of the sensor noise</h3>
<div class="outline-text-3" id="text-2-7">
<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 href="#citeproc_bib_item_1">Barzilai, VanZandt, and Kenny 1998</a>).
</p>
<p>
@@ -630,11 +380,11 @@ where:
</ul>
<p>
The <code>mscohere</code> function is compared with this formula on Appendix (section <a href="#orgd79f363">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="#org06003b1">4.1</a>), it is shown that it is identical.
</p>
<p>
Figure <a href="#org7d058c1">11</a> illustrate a block diagram model of the system used to determine the sensor noise of the geophone.
Figure <a href="#orgae83a33">11</a> illustrate a block diagram model of the system used to determine the sensor noise of the geophone.
</p>
<p>
@@ -646,7 +396,7 @@ Each sensor has noise \(N\) and \(M\).
</p>
<div id="org7d058c1" class="figure">
<div id="orgae83a33" class="figure">
<p><img src="figs/huddle-test.png" alt="huddle-test.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Huddle test block diagram</p>
@@ -661,7 +411,7 @@ We also assume that \(S_1 = S_2 = 1\).
We then obtain:
</p>
\begin{equation}
\label{org68c59ea}
\label{orgbacdde9}
\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}
@@ -669,67 +419,67 @@ We then obtain:
Since the input signal \(U\) and the instrumental noise \(N\) are incoherent:
</p>
\begin{equation}
\label{org6525ac9}
\label{org21035bd}
|G_X(\omega)| = |G_N(\omega)| + |G_U(\omega)|
\end{equation}
<p>
From equations \eqref{org68c59ea} and \eqref{org6525ac9}, we finally obtain
From equations \eqref{orgbacdde9} and \eqref{org21035bd}, we finally obtain
</p>
<div class="important">
<div class="important" id="org594bc6f">
\begin{equation}
\label{orgcac9782}
\label{orga8ac5d9}
|G_N(\omega)| = |G_X(\omega)| \left( 1 - \sqrt{\gamma_{XY}^2(\omega)} \right)
\end{equation}
</div>
<p>
The instrumental noise is computed below. The result in V<sup>2</sup>/Hz is shown on figure <a href="#org0f85822">12</a>.
The instrumental noise is computed below. The result in V^2/Hz is shown on figure <a href="#org7943278">12</a>.
</p>
<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>(1 <span class="org-type">-</span> coh12);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, pxx1, <span class="org-string">'-'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, pxx2, <span class="org-string">'-'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, pxxN, <span class="org-string">'k--'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot(f, pxx1, <span class="org-string">'-'</span>);
plot(f, pxx2, <span class="org-string">'-'</span>);
plot(f, pxxN, <span class="org-string">'k--'</span>);
hold off;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; set</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">gca, 'yscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'</span>PSD of the measured Voltage $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span>V<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">}{</span>Hz<span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>;
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'PSD of the measured Voltage $\left[\frac{V^2}{Hz}\right]$'</span>);
xlim([0.1, 500]);
</pre>
</div>
<div id="org0f85822" class="figure">
<div id="org7943278" class="figure">
<p><img src="figs/intrumental_noise_V.png" alt="intrumental_noise_V.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Instrumental Noise and Measurement in \(V^2/Hz\)</p>
</div>
<p>
This is then further converted into velocity and compared with the ground velocity measurement. (figure <a href="#org81c8985">13</a>)
This is then further converted into velocity and compared with the ground velocity measurement. (figure <a href="#org8ecfd08">13</a>)
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>pxx1<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.*</span>scaling, <span class="org-string">'-'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>pxx2<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.*</span>scaling, <span class="org-string">'-'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>pxxN<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.*</span>scaling, <span class="org-string">'k--'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot(f, sqrt(pxx1)<span class="org-type">.*</span>scaling, <span class="org-string">'-'</span>);
plot(f, sqrt(pxx2)<span class="org-type">.*</span>scaling, <span class="org-string">'-'</span>);
plot(f, sqrt(pxxN)<span class="org-type">.*</span>scaling, <span class="org-string">'k--'</span>);
hold off;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; set</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">gca, 'yscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'</span>ASD of the Velocity $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span>m<span class="org-type">/</span>s<span class="org-rainbow-delimiters-depth-3">}{</span><span class="org-type">\</span>sqrt<span class="org-rainbow-delimiters-depth-4">{</span>Hz<span class="org-rainbow-delimiters-depth-4">}</span><span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>;
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>); <span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency [Hz]'</span>); ylabel(<span class="org-string">'ASD of the Velocity $\left[\frac{m/s}{\sqrt{Hz}}\right]$'</span>);
xlim([0.1, 500]);
</pre>
</div>
<div id="org81c8985" class="figure">
<div id="org8ecfd08" class="figure">
<p><img src="figs/intrumental_noise_velocity.png" alt="intrumental_noise_velocity.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Instrumental Noise and Measurement in \(m/s/\sqrt{Hz}\)</p>
@@ -738,14 +488,14 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
</div>
<div id="outline-container-orga1d7838" class="outline-2">
<h2 id="orga1d7838"><span class="section-number-2">3</span> Compare axis</h2>
<div id="outline-container-org52fed77" class="outline-2">
<h2 id="org52fed77"><span class="section-number-2">3</span> Compare axis</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org73446c8"></a>
<a id="org4f6c263"></a>
</p>
<div class="note">
<div class="note" id="org7d50518">
<p>
All the files (data and Matlab scripts) are accessible <a href="data/huddle_test_compare_axis.zip">here</a>.
</p>
@@ -753,44 +503,44 @@ All the files (data and Matlab scripts) are accessible <a href="data/huddle_test
</div>
</div>
<div id="outline-container-org1153270" class="outline-3">
<h3 id="org1153270"><span class="section-number-3">3.1</span> Load data</h3>
<div id="outline-container-org7f52c4b" class="outline-3">
<h3 id="org7f52c4b"><span class="section-number-3">3.1</span> Load data</h3>
<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">
<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>;
north = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_003.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<pre class="src src-matlab">z = load(<span class="org-string">'mat/data_001.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'x1'</span>, <span class="org-string">'x2'</span>);
east = load(<span class="org-string">'mat/data_002.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'x1'</span>, <span class="org-string">'x2'</span>);
north = load(<span class="org-string">'mat/data_003.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'x1'</span>, <span class="org-string">'x2'</span>);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgd3a0160" class="outline-3">
<h3 id="orgd3a0160"><span class="section-number-3">3.2</span> Compare PSD</h3>
<div id="outline-container-orgb105293" class="outline-3">
<h3 id="orgb105293"><span class="section-number-3">3.2</span> Compare PSD</h3>
<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">
<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>;
<pre class="src src-matlab">[pz1, fz] = pwelch(z.x1, hanning(ceil(length(z.x1)<span class="org-type">/</span>100)), [], [], 1<span class="org-type">/</span>(z.t(2)<span class="org-type">-</span>z.t(1)));
[pz2, <span class="org-type">~</span>] = pwelch(z.x2, hanning(ceil(length(z.x2)<span class="org-type">/</span>100)), [], [], 1<span class="org-type">/</span>(z.t(2)<span class="org-type">-</span>z.t(1)));
<span class="org-rainbow-delimiters-depth-1">[</span>pe1, fe<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>east.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>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>pe2, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>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.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>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>;
[pe1, fe] = pwelch(east.x1, hanning(ceil(length(east.x1)<span class="org-type">/</span>100)), [], [], 1<span class="org-type">/</span>(east.t(2)<span class="org-type">-</span>east.t(1)));
[pe2, <span class="org-type">~</span>] = pwelch(east.x2, hanning(ceil(length(east.x2)<span class="org-type">/</span>100)), [], [], 1<span class="org-type">/</span>(east.t(2)<span class="org-type">-</span>east.t(1)));
<span class="org-rainbow-delimiters-depth-1">[</span>pn1, fn<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>north.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>north.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>north.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>north.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>pn2, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>north.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>north.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>north.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>north.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>;
[pn1, fn] = pwelch(north.x1, hanning(ceil(length(north.x1)<span class="org-type">/</span>100)), [], [], 1<span class="org-type">/</span>(north.t(2)<span class="org-type">-</span>north.t(1)));
[pn2, <span class="org-type">~</span>] = pwelch(north.x2, hanning(ceil(length(north.x2)<span class="org-type">/</span>100)), [], [], 1<span class="org-type">/</span>(north.t(2)<span class="org-type">-</span>north.t(1)));
</pre>
</div>
<p>
We compare them. The result is shown on figure <a href="#org16d65ef">14</a>.
We compare them. The result is shown on figure <a href="#orgf6d7705">14</a>.
</p>
<div id="org16d65ef" class="figure">
<div id="orgf6d7705" class="figure">
<p><img src="figs/compare_axis_psd.png" alt="compare_axis_psd.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Compare the measure PSD of the two geophones for the three axis</p>
@@ -798,23 +548,23 @@ We compare them. The result is shown on figure <a href="#org16d65ef">14</a>.
</div>
</div>
<div id="outline-container-orgcd6137b" class="outline-3">
<h3 id="orgcd6137b"><span class="section-number-3">3.3</span> Compare TF</h3>
<div id="outline-container-orgbce3b98" class="outline-3">
<h3 id="orgbce3b98"><span class="section-number-3">3.3</span> Compare TF</h3>
<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="#orgdcda32f">15</a>.
The result is shown on figure <a href="#org1936d29">15</a>.
</p>
<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>;
<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>Tn, fn<span class="org-rainbow-delimiters-depth-1">]</span> = tfestimate<span class="org-rainbow-delimiters-depth-1">(</span>north.x1, north.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>north.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>north.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>north.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">[Tz, fz] = tfestimate(z.x1, z.x2, hanning(ceil(length(z.x1)<span class="org-type">/</span>100)), [], [], 1<span class="org-type">/</span>(z.t(2)<span class="org-type">-</span>z.t(1)));
[Te, fe] = tfestimate(east.x1, east.x2, hanning(ceil(length(east.x1)<span class="org-type">/</span>100)), [], [], 1<span class="org-type">/</span>(east.t(2)<span class="org-type">-</span>east.t(1)));
[Tn, fn] = tfestimate(north.x1, north.x2, hanning(ceil(length(north.x1)<span class="org-type">/</span>100)), [], [], 1<span class="org-type">/</span>(north.t(2)<span class="org-type">-</span>north.t(1)));
</pre>
</div>
<div id="orgdcda32f" class="figure">
<div id="org1936d29" class="figure">
<p><img src="figs/compare_tf_axis.png" alt="compare_tf_axis.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Compare the transfer function from one geophone to the other for the 3 axis</p>
@@ -823,46 +573,46 @@ The result is shown on figure <a href="#orgdcda32f">15</a>.
</div>
</div>
<div id="outline-container-org693291d" class="outline-2">
<h2 id="org693291d"><span class="section-number-2">4</span> Appendix</h2>
<div id="outline-container-orgd32606c" class="outline-2">
<h2 id="orgd32606c"><span class="section-number-2">4</span> Appendix</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-org58d8ebb" class="outline-3">
<h3 id="org58d8ebb"><span class="section-number-3">4.1</span> Computation of coherence from PSD and CSD</h3>
<div id="outline-container-orge652a09" class="outline-3">
<h3 id="orge652a09"><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">
<p>
<a id="orgd79f363"></a>
<a id="org06003b1"></a>
</p>
<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>;
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>;
Fs = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>dt;
win = hanning<span class="org-rainbow-delimiters-depth-1">(</span>ceil<span class="org-rainbow-delimiters-depth-2">(</span>length<span class="org-rainbow-delimiters-depth-3">(</span>x1<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<pre class="src src-matlab">load(<span class="org-string">'mat/data_001.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'x1'</span>, <span class="org-string">'x2'</span>);
dt = t(2) <span class="org-type">-</span> t(1);
Fs = 1<span class="org-type">/</span>dt;
win = hanning(ceil(length(x1)<span class="org-type">/</span>100));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">pxy = cpsd<span class="org-rainbow-delimiters-depth-1">(</span>x1, x2, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
pxx = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>x1, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
pyy = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>x2, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
coh = mscohere<span class="org-rainbow-delimiters-depth-1">(</span>x1, x2, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
<pre class="src src-matlab">pxy = cpsd(x1, x2, win, [], [], Fs);
pxx = pwelch(x1, win, [], [], Fs);
pyy = pwelch(x2, win, [], [], Fs);
coh = mscohere(x1, x2, win, [], [], Fs);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, abs<span class="org-rainbow-delimiters-depth-2">(</span>pxy<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">./</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>pxx<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">./</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>pyy<span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-string">'-'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, coh, <span class="org-string">'--'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot(f, abs(pxy)<span class="org-type">.^</span>2<span class="org-type">./</span>abs(pxx)<span class="org-type">./</span>abs(pyy), <span class="org-string">'-'</span>);
plot(f, coh, <span class="org-string">'--'</span>);
hold off;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'Coherence'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'xscale'</span>, <span class="org-string">'log'</span>);
xlabel(<span class="org-string">'Frequency'</span>); ylabel(<span class="org-string">'Coherence'</span>);
xlim([1, 500]);
</pre>
</div>
<div id="orgc65c7db" class="figure">
<div id="org3ae3bcc" class="figure">
<p><img src="figs/comp_coherence_formula.png" alt="comp_coherence_formula.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Comparison of <code>mscohere</code> and manual computation</p>
@@ -873,15 +623,16 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
<p>
<h1 class='org-ref-bib-h1'>Bibliography</h1>
<ul class='org-ref-bib'><li><a id="barzilai98_techn_measur_noise_sensor_presen">[barzilai98_techn_measur_noise_sensor_presen]</a> <a name="barzilai98_techn_measur_noise_sensor_presen"></a>Aaron Barzilai, Tom VanZandt & Tom Kenny, Technique for Measurement of the Noise of a Sensor in the Presence of Large Background Signals, <i>Review of Scientific Instruments</i>, <b>69(7)</b>, 2767-2772 (1998). <a href="https://doi.org/10.1063/1.1149013">link</a>. <a href="http://dx.doi.org/10.1063/1.1149013">doi</a>.</li>
</ul>
</p>
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
<div class="csl-bib-body">
<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Barzilai, Aaron, Tom VanZandt, and Tom Kenny. 1998. “Technique for Measurement of the Noise of a Sensor in the Presence of Large Background Signals.” <i>Review of Scientific Instruments</i> 69 (7):2767–72. <a href="https://doi.org/10.1063/1.1149013">https://doi.org/10.1063/1.1149013</a>.</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2019-05-15 mer. 10:08</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
<p class="date">Created: 2020-11-12 jeu. 10:28</p>
</div>
</body>
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