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<title>SpeedGoat</title>
<title>Huddle Test of the L22 Geophones</title>
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@@ -271,41 +271,41 @@ for the JavaScript code in this tag.
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<div id="content">
<h1 class="title">SpeedGoat</h1>
<h1 class="title">Huddle Test of the L22 Geophones</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org3e6baac">1. Experimental Setup</a></li>
<li><a href="#org924e227">2. Signal Processing</a>
<li><a href="#orge5c8cc5">1. Experimental Setup</a></li>
<li><a href="#org8b7c96e">2. Signal Processing</a>
<ul>
<li><a href="#org8f54945">2.1. Load data</a></li>
<li><a href="#org26953ea">2.2. Time Domain Data</a></li>
<li><a href="#orgea14a3e">2.3. Computation of the ASD of the measured voltage</a></li>
<li><a href="#org10d59fe">2.4. Scaling to take into account the sensibility of the geophone and the voltage amplifier</a></li>
<li><a href="#orgca371f4">2.5. Computation of the ASD of the velocity</a></li>
<li><a href="#orgd108d7b">2.6. Transfer function between the two geophones</a></li>
<li><a href="#orgd3b08ee">2.7. Estimation of the sensor noise</a></li>
<li><a href="#org7cb3283">2.1. Load data</a></li>
<li><a href="#org0a3a0d0">2.2. Time Domain Data</a></li>
<li><a href="#org89016b8">2.3. Computation of the ASD of the measured voltage</a></li>
<li><a href="#orga26e03c">2.4. Scaling to take into account the sensibility of the geophone and the voltage amplifier</a></li>
<li><a href="#orgfd3ab32">2.5. Computation of the ASD of the velocity</a></li>
<li><a href="#orgcd33ee8">2.6. Transfer function between the two geophones</a></li>
<li><a href="#org244b412">2.7. Estimation of the sensor noise</a></li>
</ul>
</li>
<li><a href="#orgdb15af9">3. Compare axis</a>
<li><a href="#org6beee86">3. Compare axis</a>
<ul>
<li><a href="#orgba2e891">3.1. Load data</a></li>
<li><a href="#org4fa1ea3">3.2. Compare PSD</a></li>
<li><a href="#org413e0a4">3.3. Compare TF</a></li>
<li><a href="#org53cf537">3.1. Load data</a></li>
<li><a href="#org8e12607">3.2. Compare PSD</a></li>
<li><a href="#org90e5861">3.3. Compare TF</a></li>
</ul>
</li>
<li><a href="#org47dea1f">4. Appendix</a>
<li><a href="#org7c0ad67">4. Appendix</a>
<ul>
<li><a href="#orgc970c0a">4.1. Computation of coherence from PSD and CSD</a></li>
<li><a href="#orgd75efe3">4.1. Computation of coherence from PSD and CSD</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org3e6baac" class="outline-2">
<h2 id="org3e6baac"><span class="section-number-2">1</span> Experimental Setup</h2>
<div id="outline-container-orge5c8cc5" class="outline-2">
<h2 id="orge5c8cc5"><span class="section-number-2">1</span> Experimental Setup</h2>
<div class="outline-text-2" id="text-1">
<p>
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>
<div id="org1834fe1" class="figure">
<div id="org32e95c2" class="figure">
<p><img src="./figs/setup.jpg" alt="setup.jpg" width="500px" />
</p>
<p><span class="figure-number">Figure 1: </span>Setup</p>
</div>
<div id="orgd38e8c5" class="figure">
<div id="orgdd8ca8c" class="figure">
<p><img src="./figs/geophones.jpg" alt="geophones.jpg" width="500px" />
</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 id="outline-container-org924e227" class="outline-2">
<h2 id="org924e227"><span class="section-number-2">2</span> Signal Processing</h2>
<div id="outline-container-org8b7c96e" class="outline-2">
<h2 id="org8b7c96e"><span class="section-number-2">2</span> Signal Processing</h2>
<div class="outline-text-2" id="text-2">
<p>
The Matlab computing file for this part is accessible <a href="signal_processing.m">here</a>.
@@ -343,8 +343,8 @@ The <code>mat</code> file containing the measurement data is accessible <a href=
</p>
</div>
<div id="outline-container-org8f54945" class="outline-3">
<h3 id="org8f54945"><span class="section-number-3">2.1</span> Load data</h3>
<div id="outline-container-org7cb3283" class="outline-3">
<h3 id="org7cb3283"><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.
@@ -358,8 +358,8 @@ dt = t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-high
</div>
</div>
<div id="outline-container-org26953ea" class="outline-3">
<h3 id="org26953ea"><span class="section-number-3">2.2</span> Time Domain Data</h3>
<div id="outline-container-org0a3a0d0" class="outline-3">
<h3 id="org0a3a0d0"><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>;
@@ -374,7 +374,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="orge43c98a" class="figure">
<div id="org09ad879" 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>
@@ -394,7 +394,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="orgb1d4c24" class="figure">
<div id="org04c8d74" 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>
@@ -402,8 +402,8 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
</div>
<div id="outline-container-orgea14a3e" class="outline-3">
<h3 id="orgea14a3e"><span class="section-number-3">2.3</span> Computation of the ASD of the measured voltage</h3>
<div id="outline-container-org89016b8" class="outline-3">
<h3 id="org89016b8"><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.
@@ -422,12 +422,12 @@ Fs = <span class="org-highlight-numbers-number">1</span><span class="org-type">/
</div>
</div>
<div id="outline-container-org10d59fe" class="outline-3">
<h3 id="org10d59fe"><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-orga26e03c" class="outline-3">
<h3 id="orga26e03c"><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 are shown on figure <a href="#org542fee1">5</a>.
Their sensibility are shown on figure <a href="#orgb86dfe6">5</a>.
</p>
<div class="org-src-container">
@@ -438,7 +438,7 @@ S = <span class="org-rainbow-delimiters-depth-1">(</span>s<span class="org-type"
</div>
<div id="org542fee1" class="figure">
<div id="orgb86dfe6" class="figure">
<p><img src="figs/geophone_sensibility.png" alt="geophone_sensibility.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Sensibility of the Geophone</p>
@@ -469,11 +469,11 @@ We further divide the result by the sensibility of the Geophone to obtain the AS
</div>
</div>
<div id="outline-container-orgca371f4" class="outline-3">
<h3 id="orgca371f4"><span class="section-number-3">2.5</span> Computation of the ASD of the velocity</h3>
<div id="outline-container-orgfd3ab32" class="outline-3">
<h3 id="orgfd3ab32"><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="#orga342d1d">6</a>.
The ASD of the measured velocity is shown on figure <a href="#org204dcef">6</a>.
</p>
<div class="org-src-container">
@@ -490,14 +490,14 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="orga342d1d" class="figure">
<div id="org204dcef" class="figure">
<p><img src="figs/psd_velocity.png" alt="psd_velocity.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Spectral density of the velocity</p>
</div>
<p>
We also plot the ASD in displacement (figure <a href="#orgb224e73">7</a>);
We also plot the ASD in displacement (figure <a href="#org9967b3b">7</a>);
</p>
<div class="org-src-container">
@@ -513,7 +513,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="orgb224e73" class="figure">
<div id="org9967b3b" class="figure">
<p><img src="figs/asd_displacement.png" alt="asd_displacement.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Amplitude Spectral Density of the displacement as measured by the geophones</p>
@@ -521,16 +521,16 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
</div>
<div id="outline-container-orgd108d7b" class="outline-3">
<h3 id="orgd108d7b"><span class="section-number-3">2.6</span> Transfer function between the two geophones</h3>
<div id="outline-container-orgcd33ee8" class="outline-3">
<h3 id="orgcd33ee8"><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="#org9006692">8</a>.
The result is shown on figure <a href="#org5faacba">8</a>.
</p>
<p>
We also compute the coherence between the two signals (figure <a href="#org9200232">9</a>).
We also compute the coherence between the two signals (figure <a href="#orgf374e25">9</a>).
</p>
<div class="org-src-container">
@@ -539,7 +539,7 @@ We also compute the coherence between the two signals (figure <a href="#org92002
</div>
<div id="org9006692" class="figure">
<div id="org5faacba" class="figure">
<p><img src="figs/tf_geophones.png" alt="tf_geophones.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Estimated transfer function between the two geophones</p>
@@ -551,7 +551,7 @@ We also compute the coherence between the two signals (figure <a href="#org92002
</div>
<div id="org9200232" class="figure">
<div id="orgf374e25" class="figure">
<p><img src="figs/coh_geophones.png" alt="coh_geophones.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Cohererence between the signals of the two geophones</p>
@@ -559,8 +559,8 @@ We also compute the coherence between the two signals (figure <a href="#org92002
</div>
</div>
<div id="outline-container-orgd3b08ee" class="outline-3">
<h3 id="orgd3b08ee"><span class="section-number-3">2.7</span> Estimation of the sensor noise</h3>
<div id="outline-container-org244b412" class="outline-3">
<h3 id="org244b412"><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>.
@@ -590,11 +590,11 @@ where:
</ul>
<p>
The <code>mscohere</code> function is compared with this formula on Appendix (section <a href="#org550fd32">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="#orge316ad4">4.1</a>), it is shown that it is identical.
</p>
<p>
Figure <a href="#org72885b4">10</a> illustrate a block diagram model of the system used to determine the sensor noise of the geophone.
Figure <a href="#org57a5459">10</a> illustrate a block diagram model of the system used to determine the sensor noise of the geophone.
</p>
<p>
@@ -606,7 +606,7 @@ Each sensor has noise \(N\) and \(M\).
</p>
<div id="org72885b4" class="figure">
<div id="org57a5459" class="figure">
<p><img src="figs/huddle-test.png" alt="huddle-test.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Huddle test block diagram</p>
@@ -614,14 +614,14 @@ Each sensor has noise \(N\) and \(M\).
<p>
We here assume that each sensor has the same magnitude of instrumental noise (\(N = M\)).
We also assume that \(H_1 = H_2 = 1\).
We also assume that \(S_1 = S_2 = 1\).
</p>
<p>
We then obtain:
</p>
\begin{equation}
\label{orgb78feac}
\label{org5b2976f}
\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}
@@ -629,23 +629,23 @@ We then obtain:
Since the input signal \(U\) and the instrumental noise \(N\) are incoherent:
</p>
\begin{equation}
\label{org8419580}
\label{orgb493a2c}
|G_X(\omega)| = |G_N(\omega)| + |G_U(\omega)|
\end{equation}
<p>
From equations \eqref{orgb78feac} and \eqref{org8419580}, we finally obtain
From equations \eqref{org5b2976f} and \eqref{orgb493a2c}, we finally obtain
</p>
<div class="important">
\begin{equation}
\label{orgbe05867}
\label{org4f59d30}
|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="#org40db312">11</a>.
The instrumental noise is computed below. The result in V<sup>2</sup>/Hz is shown on figure <a href="#org9584668">11</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>;
@@ -666,14 +666,14 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="org40db312" class="figure">
<div id="org9584668" class="figure">
<p><img src="figs/intrumental_noise_V.png" alt="intrumental_noise_V.png" />
</p>
<p><span class="figure-number">Figure 11: </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="#org2a811b0">12</a>)
This is then further converted into velocity and compared with the ground velocity measurement. (figure <a href="#orgdd66b8f">12</a>)
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
@@ -689,7 +689,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="org2a811b0" class="figure">
<div id="orgdd66b8f" class="figure">
<p><img src="figs/intrumental_noise_velocity.png" alt="intrumental_noise_velocity.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Instrumental Noise and Measurement in \(m/s/\sqrt{Hz}\)</p>
@@ -698,8 +698,8 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
</div>
<div id="outline-container-orgdb15af9" class="outline-2">
<h2 id="orgdb15af9"><span class="section-number-2">3</span> Compare axis</h2>
<div id="outline-container-org6beee86" class="outline-2">
<h2 id="org6beee86"><span class="section-number-2">3</span> Compare axis</h2>
<div class="outline-text-2" id="text-3">
<p>
The Matlab computing file for this part is accessible <a href="compare_axis.m">here</a>.
@@ -712,8 +712,8 @@ The <code>mat</code> files containing the measurement data are accessible with t
</ul>
</div>
<div id="outline-container-orgba2e891" class="outline-3">
<h3 id="orgba2e891"><span class="section-number-3">3.1</span> Load data</h3>
<div id="outline-container-org53cf537" class="outline-3">
<h3 id="org53cf537"><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.
@@ -727,8 +727,8 @@ north = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="or
</div>
</div>
<div id="outline-container-org4fa1ea3" class="outline-3">
<h3 id="org4fa1ea3"><span class="section-number-3">3.2</span> Compare PSD</h3>
<div id="outline-container-org8e12607" class="outline-3">
<h3 id="org8e12607"><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.
@@ -746,10 +746,10 @@ The PSD for each axis of the two geophones are computed.
</div>
<p>
We compare them. The result is shown on figure <a href="#org8d0a71b">13</a>.
We compare them. The result is shown on figure <a href="#org4b31119">13</a>.
</p>
<div id="org8d0a71b" class="figure">
<div id="org4b31119" class="figure">
<p><img src="figs/compare_axis_psd.png" alt="compare_axis_psd.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Compare the measure PSD of the two geophones for the three axis</p>
@@ -757,12 +757,12 @@ We compare them. The result is shown on figure <a href="#org8d0a71b">13</a>.
</div>
</div>
<div id="outline-container-org413e0a4" class="outline-3">
<h3 id="org413e0a4"><span class="section-number-3">3.3</span> Compare TF</h3>
<div id="outline-container-org90e5861" class="outline-3">
<h3 id="org90e5861"><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="#org167ce04">14</a>.
The result is shown on figure <a href="#org76b2565">14</a>.
</p>
<div class="org-src-container">
@@ -773,7 +773,7 @@ The result is shown on figure <a href="#org167ce04">14</a>.
</div>
<div id="org167ce04" class="figure">
<div id="org76b2565" class="figure">
<p><img src="figs/compare_tf_axis.png" alt="compare_tf_axis.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Compare the transfer function from one geophone to the other for the 3 axis</p>
@@ -782,15 +782,15 @@ The result is shown on figure <a href="#org167ce04">14</a>.
</div>
</div>
<div id="outline-container-org47dea1f" class="outline-2">
<h2 id="org47dea1f"><span class="section-number-2">4</span> Appendix</h2>
<div id="outline-container-org7c0ad67" class="outline-2">
<h2 id="org7c0ad67"><span class="section-number-2">4</span> Appendix</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-orgc970c0a" class="outline-3">
<h3 id="orgc970c0a"><span class="section-number-3">4.1</span> Computation of coherence from PSD and CSD</h3>
<div id="outline-container-orgd75efe3" class="outline-3">
<h3 id="orgd75efe3"><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="org550fd32"></a>
<a id="orge316ad4"></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>;
@@ -821,7 +821,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="org229a280" class="figure">
<div id="org5c602bf" class="figure">
<p><img src="figs/comp_coherence_formula.png" alt="comp_coherence_formula.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Comparison of <code>mscohere</code> and manual computation</p>
@@ -839,7 +839,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-04-18 jeu. 17:24</p>
<p class="date">Created: 2019-05-02 jeu. 14:11</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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