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Correct unit of curves and scalling

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Thomas Dehaeze 2019-05-03 11:32:54 +02:00
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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>
<!-- 2019-05-03 ven. 08:40 --> <!-- 2019-05-03 ven. 11:32 -->
<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>Huddle Test of the L22 Geophones</title> <title>Huddle Test of the L22 Geophones</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="#org6392291">1. Experimental Setup</a></li> <li><a href="#org446511b">1. Experimental Setup</a></li>
<li><a href="#orgb55c46d">2. Signal Processing</a> <li><a href="#org0689ed7">2. Signal Processing</a>
<ul> <ul>
<li><a href="#org016f692">2.1. Load data</a></li> <li><a href="#org34a2d4c">2.1. Load data</a></li>
<li><a href="#orgc616389">2.2. Time Domain Data</a></li> <li><a href="#org28e8648">2.2. Time Domain Data</a></li>
<li><a href="#org8794282">2.3. Computation of the ASD of the measured voltage</a></li> <li><a href="#org910f3e4">2.3. Computation of the ASD of the measured voltage</a></li>
<li><a href="#org09670e4">2.4. Scaling to take into account the sensibility of the geophone and the voltage amplifier</a></li> <li><a href="#org4e984e1">2.4. Scaling to take into account the sensibility of the geophone and the voltage amplifier</a></li>
<li><a href="#org4e5224a">2.5. Computation of the ASD of the velocity</a></li> <li><a href="#org607752b">2.5. Computation of the ASD of the velocity</a></li>
<li><a href="#orge366f60">2.6. Transfer function between the two geophones</a></li> <li><a href="#org0a07c74">2.6. Transfer function between the two geophones</a></li>
<li><a href="#org4cacf62">2.7. Estimation of the sensor noise</a></li> <li><a href="#orgdc03acb">2.7. Estimation of the sensor noise</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org559b5b9">3. Compare axis</a> <li><a href="#org58e6c46">3. Compare axis</a>
<ul> <ul>
<li><a href="#org24b196e">3.1. Load data</a></li> <li><a href="#org8f74945">3.1. Load data</a></li>
<li><a href="#org00c635c">3.2. Compare PSD</a></li> <li><a href="#orgf13f88e">3.2. Compare PSD</a></li>
<li><a href="#org16927e1">3.3. Compare TF</a></li> <li><a href="#orgf389e18">3.3. Compare TF</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgf4800f0">4. Appendix</a> <li><a href="#org1ca0f74">4. Appendix</a>
<ul> <ul>
<li><a href="#org556257c">4.1. Computation of coherence from PSD and CSD</a></li> <li><a href="#org9faefb2">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-org6392291" class="outline-2"> <div id="outline-container-org446511b" class="outline-2">
<h2 id="org6392291"><span class="section-number-2">1</span> Experimental Setup</h2> <h2 id="org446511b"><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="orgddb9341" class="figure"> <div id="orgeab8098" 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="org8077b6d" class="figure"> <div id="orgdadbe53" 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>
@ -334,8 +334,8 @@ The voltage amplifiers include a low pass filter with a cut-off frequency at 1kH
</div> </div>
</div> </div>
<div id="outline-container-orgb55c46d" class="outline-2"> <div id="outline-container-org0689ed7" class="outline-2">
<h2 id="orgb55c46d"><span class="section-number-2">2</span> Signal Processing</h2> <h2 id="org0689ed7"><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> <p>
The Matlab computing file for this part is accessible <a href="signal_processing.m">here</a>. The Matlab computing file for this part is accessible <a href="signal_processing.m">here</a>.
@ -343,8 +343,8 @@ The <code>mat</code> file containing the measurement data is accessible <a href=
</p> </p>
</div> </div>
<div id="outline-container-org016f692" class="outline-3"> <div id="outline-container-org34a2d4c" class="outline-3">
<h3 id="org016f692"><span class="section-number-3">2.1</span> Load data</h3> <h3 id="org34a2d4c"><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.
@ -358,8 +358,8 @@ dt = t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-high
</div> </div>
</div> </div>
<div id="outline-container-orgc616389" class="outline-3"> <div id="outline-container-org28e8648" class="outline-3">
<h3 id="orgc616389"><span class="section-number-3">2.2</span> Time Domain Data</h3> <h3 id="org28e8648"><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>;
@ -374,7 +374,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
<div id="orgeb104f7" class="figure"> <div id="orgbc90092" 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>
@ -394,7 +394,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
<div id="orgd0dc39d" class="figure"> <div id="orgbb22470" 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>
@ -402,78 +402,105 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
</div> </div>
<div id="outline-container-org8794282" class="outline-3"> <div id="outline-container-org910f3e4" class="outline-3">
<h3 id="org8794282"><span class="section-number-3">2.3</span> Computation of the ASD of the measured voltage</h3> <h3 id="org910f3e4"><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.
</p> </p>
<div class="org-src-container"> <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; <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>
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<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>;
</pre> </pre>
</div> </div>
<p>
Then we compute the Power Spectral Density using <code>pwelch</code> function.
</p>
<div class="org-src-container"> <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>; <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>; <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> </pre>
</div> </div>
<p>
And we plot the result on figure <a href="#orgbf77081">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>;
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>;
</pre>
</div>
<div id="orgbf77081" 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>
</div>
</div> </div>
</div> </div>
<div id="outline-container-org09670e4" class="outline-3"> <div id="outline-container-org4e984e1" class="outline-3">
<h3 id="org09670e4"><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="org4e984e1"><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 is shown on figure <a href="#org0a867d9">6</a>.
Their sensibility are shown on figure <a href="#orgdbecac7">5</a>.
</p> </p>
<div class="org-src-container"> <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> <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> f0 = <span class="org-highlight-numbers-number">2</span>; <span class="org-comment">% Cut-off frequnecy [Hz]</span>
S = <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><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>;
</pre> </pre>
</div> </div>
<div id="orgdbecac7" class="figure"> <div id="org0a867d9" 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 6: </span>Sensibility of the Geophone</p>
</div> </div>
<p> <p>
We also take into account the gain of the electronics which is here set to be \(60dB\). 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.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">G0 = <span class="org-highlight-numbers-number">60</span>; <span class="org-comment">% [dB]</span> <pre class="src src-matlab">G0_db = <span class="org-highlight-numbers-number">60</span>; <span class="org-comment">% [dB]</span>
G = <span class="org-highlight-numbers-number">10</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-1">(</span>G0<span class="org-type">/</span><span class="org-highlight-numbers-number">20</span><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><span class="org-highlight-numbers-number">1000</span><span class="org-rainbow-delimiters-depth-1">)</span>; G0 = <span class="org-highlight-numbers-number">10</span><span class="org-type">^</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">60</span><span class="org-type">/</span>G0_db<span class="org-rainbow-delimiters-depth-1">)</span>; <span class="org-comment">% [abs]</span>
</pre> </pre>
</div> </div>
<p> <p>
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 divide the ASD measured (in \(\text{V}/\sqrt{\text{Hz}}\)) by the gain 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 \(m/s/\sqrt{Hz}\). We further divide the result by the sensibility of the Geophone to obtain the ASD of the velocity in \(m/s/\sqrt{Hz}\).
</p> </p>
<div class="org-src-container"> <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>G<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 = <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> </pre>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org4e5224a" class="outline-3"> <div id="outline-container-org607752b" class="outline-3">
<h3 id="org4e5224a"><span class="section-number-3">2.5</span> Computation of the ASD of the velocity</h3> <h3 id="org607752b"><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="#orgf6e310e">6</a>. The ASD of the measured velocity is shown on figure <a href="#orgd9a4009">7</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
@ -484,53 +511,53 @@ plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org
hold off; 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">'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>; <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">'PSD </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">m/s/sqrt</span><span class="org-string"><span class="org-rainbow-delimiters-depth-3">(</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-3">)</span></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> 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>; 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>;
</pre> </pre>
</div> </div>
<div id="orgf6e310e" class="figure"> <div id="orgd9a4009" 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 7: </span>Amplitude Spectral Density of the Velocity</p>
</div> </div>
<p> <p>
We also plot the ASD in displacement (figure <a href="#org0ea2712">7</a>); We also plot the ASD in displacement (figure <a href="#orgc0b2ca5">8</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>;
hold on; hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, <span class="org-rainbow-delimiters-depth-2">(</span>pxx1<span class="org-type">.*</span>scaling<span class="org-type">./</span>f<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.^</span><span class="org-highlight-numbers-number">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>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>pxx2<span class="org-type">.*</span>scaling<span class="org-type">./</span>f<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.^</span><span class="org-highlight-numbers-number">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>;
hold off; 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>; <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">'PSD </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">m/s/sqrt</span><span class="org-string"><span class="org-rainbow-delimiters-depth-3">(</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-3">)</span></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> 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>; 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>;
</pre> </pre>
</div> </div>
<div id="org0ea2712" class="figure"> <div id="orgc0b2ca5" class="figure">
<p><img src="figs/asd_displacement.png" alt="asd_displacement.png" /> <p><img src="figs/asd_displacement.png" alt="asd_displacement.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Amplitude Spectral Density of the displacement as measured by the geophones</p> <p><span class="figure-number">Figure 8: </span>Amplitude Spectral Density of the Displacement</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orge366f60" class="outline-3"> <div id="outline-container-org0a07c74" class="outline-3">
<h3 id="orge366f60"><span class="section-number-3">2.6</span> Transfer function between the two geophones</h3> <h3 id="org0a07c74"><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="#orgba6f718">8</a>. The result is shown on figure <a href="#org7d8ea2b">9</a>.
</p> </p>
<p> <p>
We also compute the coherence between the two signals (figure <a href="#org251a7af">9</a>). We also compute the coherence between the two signals (figure <a href="#org628544f">10</a>).
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
@ -539,10 +566,10 @@ We also compute the coherence between the two signals (figure <a href="#org251a7
</div> </div>
<div id="orgba6f718" class="figure"> <div id="org7d8ea2b" 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 8: </span>Estimated transfer function between the two geophones</p> <p><span class="figure-number">Figure 9: </span>Estimated transfer function between the two geophones</p>
</div> </div>
<div class="org-src-container"> <div class="org-src-container">
@ -551,16 +578,16 @@ We also compute the coherence between the two signals (figure <a href="#org251a7
</div> </div>
<div id="org251a7af" class="figure"> <div id="org628544f" 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 9: </span>Cohererence between the signals of the two geophones</p> <p><span class="figure-number">Figure 10: </span>Cohererence between the signals of the two geophones</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org4cacf62" class="outline-3"> <div id="outline-container-orgdc03acb" class="outline-3">
<h3 id="org4cacf62"><span class="section-number-3">2.7</span> Estimation of the sensor noise</h3> <h3 id="orgdc03acb"><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>.
@ -590,11 +617,11 @@ where:
</ul> </ul>
<p> <p>
The <code>mscohere</code> function is compared with this formula on Appendix (section <a href="#org1e77eec">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="#orgc9ed210">4.1</a>), it is shown that it is identical.
</p> </p>
<p> <p>
Figure <a href="#org8c23b33">10</a> illustrate a block diagram model of the system used to determine the sensor noise of the geophone. Figure <a href="#org9b31b02">11</a> illustrate a block diagram model of the system used to determine the sensor noise of the geophone.
</p> </p>
<p> <p>
@ -606,10 +633,10 @@ Each sensor has noise \(N\) and \(M\).
</p> </p>
<div id="org8c23b33" class="figure"> <div id="org9b31b02" 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 10: </span>Huddle test block diagram</p> <p><span class="figure-number">Figure 11: </span>Huddle test block diagram</p>
</div> </div>
<p> <p>
@ -621,7 +648,7 @@ We also assume that \(S_1 = S_2 = 1\).
We then obtain: We then obtain:
</p> </p>
\begin{equation} \begin{equation}
\label{org3a8a59f} \label{orgf197c52}
\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}
@ -629,23 +656,23 @@ 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{org221f31e} \label{org845ba9b}
|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{org3a8a59f} and \eqref{org221f31e}, we finally obtain From equations \eqref{orgf197c52} and \eqref{org845ba9b}, we finally obtain
</p> </p>
<div class="important"> <div class="important">
\begin{equation} \begin{equation}
\label{org3574759} \label{org0941f4e}
|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="#orge1676ed">11</a>. The instrumental noise is computed below. The result in V<sup>2</sup>/Hz is shown on figure <a href="#orged7b0f2">12</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>;
@ -660,20 +687,20 @@ plot<span class="org-rainbow-delimiters-depth-1">(</span>f, pxx2, <span class="o
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<span class="org-rainbow-delimiters-depth-1">(</span>f, pxxN, <span class="org-string">'k--'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
hold off; 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>; <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>'Frequency <span class="org-rainbow-delimiters-depth-2">[</span>Hz<span class="org-rainbow-delimiters-depth-2">]</span>'<span class="org-rainbow-delimiters-depth-1">)</span>; ylabel<span class="org-rainbow-delimiters-depth-1">(</span>'PSD <span class="org-rainbow-delimiters-depth-2">[</span>$V<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span>Hz$<span class="org-rainbow-delimiters-depth-2">]</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>; 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>;
</pre> </pre>
</div> </div>
<div id="orge1676ed" class="figure"> <div id="orged7b0f2" 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 11: </span>Instrumental Noise and Measurement in \(V^2/Hz\)</p> <p><span class="figure-number">Figure 12: </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="#org30e84b0">12</a>) This is then further converted into velocity and compared with the ground velocity measurement. (figure <a href="#org3b9b556">13</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>;
@ -683,23 +710,23 @@ plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org
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<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>;
hold off; 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>; <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 <span class="org-rainbow-delimiters-depth-2">[</span>$m<span class="org-type">/</span>s<span class="org-type">/\</span>sqrt<span class="org-rainbow-delimiters-depth-3">{</span>Hz<span class="org-rainbow-delimiters-depth-3">}</span>$<span class="org-rainbow-delimiters-depth-2">]</span>'<span class="org-rainbow-delimiters-depth-1">)</span>; xlabel<span class="org-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>; 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>;
</pre> </pre>
</div> </div>
<div id="org30e84b0" class="figure"> <div id="org3b9b556" 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 12: </span>Instrumental Noise and Measurement in \(m/s/\sqrt{Hz}\)</p> <p><span class="figure-number">Figure 13: </span>Instrumental Noise and Measurement in \(m/s/\sqrt{Hz}\)</p>
</div> </div>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org559b5b9" class="outline-2"> <div id="outline-container-org58e6c46" class="outline-2">
<h2 id="org559b5b9"><span class="section-number-2">3</span> Compare axis</h2> <h2 id="org58e6c46"><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> <p>
The Matlab computing file for this part is accessible <a href="compare_axis.m">here</a>. The Matlab computing file for this part is accessible <a href="compare_axis.m">here</a>.
@ -712,8 +739,8 @@ The <code>mat</code> files containing the measurement data are accessible with t
</ul> </ul>
</div> </div>
<div id="outline-container-org24b196e" class="outline-3"> <div id="outline-container-org8f74945" class="outline-3">
<h3 id="org24b196e"><span class="section-number-3">3.1</span> Load data</h3> <h3 id="org8f74945"><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> <p>
We first load the data for the three axis. We first load the data for the three axis.
@ -727,8 +754,8 @@ north = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="or
</div> </div>
</div> </div>
<div id="outline-container-org00c635c" class="outline-3"> <div id="outline-container-orgf13f88e" class="outline-3">
<h3 id="org00c635c"><span class="section-number-3">3.2</span> Compare PSD</h3> <h3 id="orgf13f88e"><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> <p>
The PSD for each axis of the two geophones are computed. The PSD for each axis of the two geophones are computed.
@ -746,23 +773,23 @@ The PSD for each axis of the two geophones are computed.
</div> </div>
<p> <p>
We compare them. The result is shown on figure <a href="#org6734ed2">13</a>. We compare them. The result is shown on figure <a href="#orgdfcdc16">14</a>.
</p> </p>
<div id="org6734ed2" class="figure"> <div id="orgdfcdc16" 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 13: </span>Compare the measure PSD of the two geophones for the three axis</p> <p><span class="figure-number">Figure 14: </span>Compare the measure PSD of the two geophones for the three axis</p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org16927e1" class="outline-3"> <div id="outline-container-orgf389e18" class="outline-3">
<h3 id="org16927e1"><span class="section-number-3">3.3</span> Compare TF</h3> <h3 id="orgf389e18"><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> <p>
The transfer functions from one geophone to the other are also computed for each axis. The transfer functions from one geophone to the other are also computed for each axis.
The result is shown on figure <a href="#org8df4d36">14</a>. The result is shown on figure <a href="#orgdd8cabb">15</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
@ -773,24 +800,24 @@ The result is shown on figure <a href="#org8df4d36">14</a>.
</div> </div>
<div id="org8df4d36" class="figure"> <div id="orgdd8cabb" 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 14: </span>Compare the transfer function from one geophone to the other for the 3 axis</p> <p><span class="figure-number">Figure 15: </span>Compare the transfer function from one geophone to the other for the 3 axis</p>
</div> </div>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orgf4800f0" class="outline-2"> <div id="outline-container-org1ca0f74" class="outline-2">
<h2 id="orgf4800f0"><span class="section-number-2">4</span> Appendix</h2> <h2 id="org1ca0f74"><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-org556257c" class="outline-3"> <div id="outline-container-org9faefb2" class="outline-3">
<h3 id="org556257c"><span class="section-number-3">4.1</span> Computation of coherence from PSD and CSD</h3> <h3 id="org9faefb2"><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="org1e77eec"></a> <a id="orgc9ed210"></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>;
@ -821,10 +848,10 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div> </div>
<div id="org190093c" class="figure"> <div id="orgea88bec" 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 15: </span>Comparison of <code>mscohere</code> and manual computation</p> <p><span class="figure-number">Figure 16: </span>Comparison of <code>mscohere</code> and manual computation</p>
</div> </div>
</div> </div>
</div> </div>
@ -839,7 +866,7 @@ 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: 2019-05-03 ven. 08:40</p> <p class="date">Created: 2019-05-03 ven. 11:32</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>

66
huddle-test-geophones/index.org
View File

@ -106,29 +106,56 @@ We load the data of the z axis of two geophones.
** Computation of the ASD of the measured voltage ** Computation of the ASD of the measured voltage
We first define the parameters for the frequency domain analysis. We first define the parameters for the frequency domain analysis.
#+begin_src matlab :results none #+begin_src matlab :results none
Fs = 1/dt; Fs = 1/dt; % [Hz]
win = hanning(ceil(10*Fs)); win = hanning(ceil(10*Fs));
#+end_src #+end_src
Then we compute the Power Spectral Density using =pwelch= function.
#+begin_src matlab :results none #+begin_src matlab :results none
[pxx1, f] = pwelch(x1, win, [], [], Fs); [pxx1, f] = pwelch(x1, win, [], [], Fs);
[pxx2, ~] = pwelch(x2, win, [], [], Fs); [pxx2, ~] = pwelch(x2, win, [], [], Fs);
#+end_src #+end_src
And we plot the result on figure [[fig:asd_voltage]].
#+begin_src matlab :results none
figure;
hold on;
plot(f, sqrt(pxx1));
plot(f, sqrt(pxx2));
hold off;
set(gca, 'xscale', 'log');
set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD of the measured Voltage $\left[\frac{V}{\sqrt{Hz}}\right]$')
xlim([0.1, 500]);
#+end_src
#+NAME: fig:asd_voltage
#+HEADER: :tangle no :exports results :results value raw replace :noweb yes
#+begin_src matlab :var filepath="figs/asd_voltage.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:asd_voltage
#+CAPTION: Amplitude Spectral Density of the measured voltage
#+RESULTS: fig:asd_voltage
[[file:figs/asd_voltage.png]]
** Scaling to take into account the sensibility of the geophone and the voltage amplifier ** Scaling to take into account the sensibility of the geophone and the voltage amplifier
The Geophone used are L22. The Geophone used are L22. Their sensibility is shown on figure [[fig:geophone_sensibility]].
Their sensibility are shown on figure [[fig:geophone_sensibility]].
#+begin_src matlab :results none #+begin_src matlab :results none
S0 = 88; % Sensitivity [V/(m/s)] S0 = 88; % Sensitivity [V/(m/s)]
f0 = 2; % Cut-off frequnecy [Hz] f0 = 2; % Cut-off frequnecy [Hz]
S = (s/2/pi/f0)/(1+s/2/pi/f0);
S = S0*(s/2/pi/f0)/(1+s/2/pi/f0);
#+end_src #+end_src
#+begin_src matlab :results none :exports none #+begin_src matlab :results none :exports none
figure; figure;
bodeFig({S}); bodeFig({S}, logspace(-1, 2, 1000));
ylabel('Amplitude [V/(m/s)]') ylabel('Amplitude $\left[\frac{V}{m/s}\right]$')
#+end_src #+end_src
#+NAME: fig:geophone_sensibility #+NAME: fig:geophone_sensibility
@ -144,19 +171,18 @@ Their sensibility are shown on figure [[fig:geophone_sensibility]].
We also take into account the gain of the electronics which is here set to be $60dB$. We also take into account the gain of the electronics which is here set to be $60dB$.
The amplifiers also include a low pass filter with a cut-off frequency set at 1kHz.
#+begin_src matlab :results none #+begin_src matlab :results none
G0 = 60; % [dB] G0_db = 60; % [dB]
G = 10^(G0/20)/(1+s/2/pi/1000); G0 = 10^(60/G0_db); % [abs]
#+end_src #+end_src
We divide the ASD measured (in $\text{V}/\sqrt{\text{Hz}}$) by the transfer function of the voltage amplifier to obtain the ASD of the voltage across the geophone. We divide the ASD measured (in $\text{V}/\sqrt{\text{Hz}}$) by the gain 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 $m/s/\sqrt{Hz}$. We further divide the result by the sensibility of the Geophone to obtain the ASD of the velocity in $m/s/\sqrt{Hz}$.
#+begin_src matlab :results none #+begin_src matlab :results none
scaling = 1./squeeze(abs(freqresp(G*S, f, 'Hz'))); scaling = 1./squeeze(abs(freqresp(G0*S, f, 'Hz')));
#+end_src #+end_src
** Computation of the ASD of the velocity ** Computation of the ASD of the velocity
@ -170,7 +196,7 @@ The ASD of the measured velocity is shown on figure [[fig:psd_velocity]].
hold off; hold off;
set(gca, 'xscale', 'log'); set(gca, 'xscale', 'log');
set(gca, 'yscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('PSD [m/s/sqrt(Hz)]') xlabel('Frequency [Hz]'); ylabel('ASD of the measured Velocity $\left[\frac{m/s}{\sqrt{Hz}}\right]$')
xlim([0.1, 500]); xlim([0.1, 500]);
#+end_src #+end_src
@ -181,7 +207,7 @@ The ASD of the measured velocity is shown on figure [[fig:psd_velocity]].
#+end_src #+end_src
#+NAME: fig:psd_velocity #+NAME: fig:psd_velocity
#+CAPTION: Spectral density of the velocity #+CAPTION: Amplitude Spectral Density of the Velocity
#+RESULTS: fig:psd_velocity #+RESULTS: fig:psd_velocity
[[file:figs/psd_velocity.png]] [[file:figs/psd_velocity.png]]
@ -190,11 +216,11 @@ We also plot the ASD in displacement (figure [[fig:asd_displacement]]);
#+begin_src matlab :results none #+begin_src matlab :results none
figure; figure;
hold on; hold on;
plot(f, (pxx1.*scaling./f).^2); plot(f, (sqrt(pxx1).*scaling)./(2*pi*f));
plot(f, (pxx2.*scaling./f).^2); plot(f, (sqrt(pxx2).*scaling)./(2*pi*f));
hold off; hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('PSD [m/s/sqrt(Hz)]') xlabel('Frequency [Hz]'); ylabel('ASD of the displacement $\left[\frac{m}{\sqrt{Hz}}\right]$')
xlim([0.1, 500]); xlim([0.1, 500]);
#+end_src #+end_src
@ -205,7 +231,7 @@ We also plot the ASD in displacement (figure [[fig:asd_displacement]]);
#+end_src #+end_src
#+NAME: fig:asd_displacement #+NAME: fig:asd_displacement
#+CAPTION: Amplitude Spectral Density of the displacement as measured by the geophones #+CAPTION: Amplitude Spectral Density of the Displacement
#+RESULTS: fig:asd_displacement #+RESULTS: fig:asd_displacement
[[file:figs/asd_displacement.png]] [[file:figs/asd_displacement.png]]
@ -232,7 +258,7 @@ We also compute the coherence between the two signals (figure [[fig:coh_geophone
set(gca, 'xscale', 'log'); set(gca, 'xscale', 'log');
ylim([-180, 180]); ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]); yticks([-180, -90, 0, 90, 180]);
xlabel('Frequency [Hz]'); ylabel('Phase'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
linkaxes([ax1,ax2],'x'); linkaxes([ax1,ax2],'x');
xlim([0.1, 500]); xlim([0.1, 500]);
@ -338,7 +364,7 @@ The instrumental noise is computed below. The result in V^2/Hz is shown on figur
plot(f, pxxN, 'k--'); plot(f, pxxN, 'k--');
hold off; hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('PSD [$V^2/Hz$]'); xlabel('Frequency [Hz]'); ylabel('PSD of the measured Voltage $\left[\frac{V^2}{Hz}\right]$');
xlim([0.1, 500]); xlim([0.1, 500]);
#+end_src #+end_src
@ -362,7 +388,7 @@ This is then further converted into velocity and compared with the ground veloci
plot(f, sqrt(pxxN).*scaling, 'k--'); plot(f, sqrt(pxxN).*scaling, 'k--');
hold off; hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]'); xlabel('Frequency [Hz]'); ylabel('ASD of the Velocity $\left[\frac{m/s}{\sqrt{Hz}}\right]$');
xlim([0.1, 500]); xlim([0.1, 500]);
#+end_src #+end_src