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<title>Huddle Test of the L22 Geophones</title>
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<a accesskey="h" href="../index.html"> UP </a>
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</div><div id="content">
<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="#org446511b">1. Experimental Setup</a></li>
<li><a href="#org0689ed7">2. Signal Processing</a>
<li><a href="#org214e6d8">1. Experimental Setup</a></li>
<li><a href="#org42d6b13">2. Signal Processing</a>
<ul>
<li><a href="#org34a2d4c">2.1. Load data</a></li>
<li><a href="#org28e8648">2.2. Time Domain Data</a></li>
<li><a href="#org910f3e4">2.3. Computation of the ASD of the measured voltage</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="#org607752b">2.5. Computation of the ASD of the velocity</a></li>
<li><a href="#org0a07c74">2.6. Transfer function between the two geophones</a></li>
<li><a href="#orgdc03acb">2.7. Estimation of the sensor noise</a></li>
<li><a href="#orgf4a8298">2.1. Load data</a></li>
<li><a href="#org687ebba">2.2. Time Domain Data</a></li>
<li><a href="#org649e300">2.3. Computation of the ASD of the measured voltage</a></li>
<li><a href="#org6148805">2.4. Scaling to take into account the sensibility of the geophone and the voltage amplifier</a></li>
<li><a href="#orgfd258d8">2.5. Computation of the ASD of the velocity</a></li>
<li><a href="#org453baa3">2.6. Transfer function between the two geophones</a></li>
<li><a href="#orgd2d293a">2.7. Estimation of the sensor noise</a></li>
</ul>
</li>
<li><a href="#org58e6c46">3. Compare axis</a>
<li><a href="#orgbd963cb">3. Compare axis</a>
<ul>
<li><a href="#org8f74945">3.1. Load data</a></li>
<li><a href="#orgf13f88e">3.2. Compare PSD</a></li>
<li><a href="#orgf389e18">3.3. Compare TF</a></li>
<li><a href="#org5ae5d71">3.1. Load data</a></li>
<li><a href="#orgd12648a">3.2. Compare PSD</a></li>
<li><a href="#orgf076a23">3.3. Compare TF</a></li>
</ul>
</li>
<li><a href="#org1ca0f74">4. Appendix</a>
<li><a href="#org17f5bce">4. Appendix</a>
<ul>
<li><a href="#org9faefb2">4.1. Computation of coherence from PSD and CSD</a></li>
<li><a href="#org6babc5e">4.1. Computation of coherence from PSD and CSD</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org446511b" class="outline-2">
<h2 id="org446511b"><span class="section-number-2">1</span> Experimental Setup</h2>
<div id="outline-container-org214e6d8" class="outline-2">
<h2 id="org214e6d8"><span class="section-number-2">1</span> Experimental Setup</h2>
<div class="outline-text-2" id="text-1">
<p>
Two L22 geophones are used.
@@ -315,36 +324,46 @@ They are leveled.
<p>
The signals are amplified using voltage amplifier with a gain of 60dB.
The voltage amplifiers include a low pass filter with a cut-off frequency at 1kHz.
The voltage amplifiers includes:
</p>
<ul class="org-ul">
<li>an high pass filter with a cut-off frequency at 1.5Hz (AC option)</li>
<li>a low pass filter with a cut-off frequency at 1kHz</li>
</ul>
<div id="orgeab8098" class="figure">
<p><img src="./figs/setup.jpg" alt="setup.jpg" width="500px" />
<div id="org2edcc21" 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="orgdadbe53" class="figure">
<p><img src="./figs/geophones.jpg" alt="geophones.jpg" width="500px" />
<div id="org36e36a2" class="figure">
<p><img src="./img/geophones.jpg" alt="geophones.jpg" width="500px" />
</p>
<p><span class="figure-number">Figure 2: </span>Geophones</p>
</div>
</div>
</div>
<div id="outline-container-org0689ed7" class="outline-2">
<h2 id="org0689ed7"><span class="section-number-2">2</span> Signal Processing</h2>
<div id="outline-container-org42d6b13" class="outline-2">
<h2 id="org42d6b13"><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>.
The <code>mat</code> file containing the measurement data is accessible <a href="mat/data_001.mat">here</a>.
<a id="org7e05e6c"></a>
</p>
<div class="note">
<p>
All the files (data and Matlab scripts) are accessible <a href="data/huddle_test_signal_processing.zip">here</a>.
</p>
</div>
</div>
<div id="outline-container-org34a2d4c" class="outline-3">
<h3 id="org34a2d4c"><span class="section-number-3">2.1</span> Load data</h3>
<div id="outline-container-orgf4a8298" class="outline-3">
<h3 id="orgf4a8298"><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 +377,8 @@ dt = t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-high
</div>
</div>
<div id="outline-container-org28e8648" class="outline-3">
<h3 id="org28e8648"><span class="section-number-3">2.2</span> Time Domain Data</h3>
<div id="outline-container-org687ebba" class="outline-3">
<h3 id="org687ebba"><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 +393,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="orgbc90092" class="figure">
<div id="org20233d2" 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 +413,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="orgbb22470" class="figure">
<div id="org1b7bc10" 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 +421,8 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
</div>
<div id="outline-container-org910f3e4" class="outline-3">
<h3 id="org910f3e4"><span class="section-number-3">2.3</span> Computation of the ASD of the measured voltage</h3>
<div id="outline-container-org649e300" class="outline-3">
<h3 id="org649e300"><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.
@@ -425,7 +444,7 @@ Then we compute the Power Spectral Density using <code>pwelch</code> function.
</div>
<p>
And we plot the result on figure <a href="#orgbf77081">5</a>.
And we plot the result on figure <a href="#org3ce04bf">5</a>.
</p>
<div class="org-src-container">
@@ -442,7 +461,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="orgbf77081" class="figure">
<div id="org3ce04bf" 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>
@@ -450,11 +469,11 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
</div>
<div id="outline-container-org4e984e1" class="outline-3">
<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 id="outline-container-org6148805" class="outline-3">
<h3 id="org6148805"><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="#org0a867d9">6</a>.
The Geophone used are L22. Their sensibility is shown on figure <a href="#orgd7b0965">6</a>.
</p>
<div class="org-src-container">
@@ -466,7 +485,7 @@ S = S0<span class="org-type">*</span><span class="org-rainbow-delimiters-depth-1
</div>
<div id="org0a867d9" class="figure">
<div id="orgd7b0965" 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>
@@ -496,11 +515,11 @@ We further divide the result by the sensibility of the Geophone to obtain the AS
</div>
</div>
<div id="outline-container-org607752b" class="outline-3">
<h3 id="org607752b"><span class="section-number-3">2.5</span> Computation of the ASD of the velocity</h3>
<div id="outline-container-orgfd258d8" class="outline-3">
<h3 id="orgfd258d8"><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="#orgd9a4009">7</a>.
The ASD of the measured velocity is shown on figure <a href="#org3cb06de">7</a>.
</p>
<div class="org-src-container">
@@ -517,14 +536,14 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="orgd9a4009" class="figure">
<div id="org3cb06de" 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="#orgc0b2ca5">8</a>);
We also plot the ASD in displacement (figure <a href="#org2012a56">8</a>);
</p>
<div class="org-src-container">
@@ -540,7 +559,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="orgc0b2ca5" class="figure">
<div id="org2012a56" 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>
@@ -548,16 +567,16 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
</div>
<div id="outline-container-org0a07c74" class="outline-3">
<h3 id="org0a07c74"><span class="section-number-3">2.6</span> Transfer function between the two geophones</h3>
<div id="outline-container-org453baa3" class="outline-3">
<h3 id="org453baa3"><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="#org7d8ea2b">9</a>.
The result is shown on figure <a href="#org8ca997a">9</a>.
</p>
<p>
We also compute the coherence between the two signals (figure <a href="#org628544f">10</a>).
We also compute the coherence between the two signals (figure <a href="#org4366ab4">10</a>).
</p>
<div class="org-src-container">
@@ -566,7 +585,7 @@ We also compute the coherence between the two signals (figure <a href="#org62854
</div>
<div id="org7d8ea2b" class="figure">
<div id="org8ca997a" 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>
@@ -578,7 +597,7 @@ We also compute the coherence between the two signals (figure <a href="#org62854
</div>
<div id="org628544f" class="figure">
<div id="org4366ab4" 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>
@@ -586,8 +605,8 @@ We also compute the coherence between the two signals (figure <a href="#org62854
</div>
</div>
<div id="outline-container-orgdc03acb" class="outline-3">
<h3 id="orgdc03acb"><span class="section-number-3">2.7</span> Estimation of the sensor noise</h3>
<div id="outline-container-orgd2d293a" class="outline-3">
<h3 id="orgd2d293a"><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>.
@@ -617,11 +636,11 @@ where:
</ul>
<p>
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.
The <code>mscohere</code> function is compared with this formula on Appendix (section <a href="#org956da99">4.1</a>), it is shown that it is identical.
</p>
<p>
Figure <a href="#org9b31b02">11</a> illustrate a block diagram model of the system used to determine the sensor noise of the geophone.
Figure <a href="#orgc9be925">11</a> illustrate a block diagram model of the system used to determine the sensor noise of the geophone.
</p>
<p>
@@ -633,7 +652,7 @@ Each sensor has noise \(N\) and \(M\).
</p>
<div id="org9b31b02" class="figure">
<div id="orgc9be925" 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>
@@ -648,7 +667,7 @@ We also assume that \(S_1 = S_2 = 1\).
We then obtain:
</p>
\begin{equation}
\label{orgf197c52}
\label{orgc4ca458}
\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}
@@ -656,23 +675,23 @@ We then obtain:
Since the input signal \(U\) and the instrumental noise \(N\) are incoherent:
</p>
\begin{equation}
\label{org845ba9b}
\label{orgb9a5b79}
|G_X(\omega)| = |G_N(\omega)| + |G_U(\omega)|
\end{equation}
<p>
From equations \eqref{orgf197c52} and \eqref{org845ba9b}, we finally obtain
From equations \eqref{orgc4ca458} and \eqref{orgb9a5b79}, we finally obtain
</p>
<div class="important">
\begin{equation}
\label{org0941f4e}
\label{org618c850}
|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="#orged7b0f2">12</a>.
The instrumental noise is computed below. The result in V<sup>2</sup>/Hz is shown on figure <a href="#org8fc8f62">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>;
@@ -693,14 +712,14 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="orged7b0f2" class="figure">
<div id="org8fc8f62" 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="#org3b9b556">13</a>)
This is then further converted into velocity and compared with the ground velocity measurement. (figure <a href="#orgaf005ac">13</a>)
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
@@ -716,7 +735,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="org3b9b556" class="figure">
<div id="orgaf005ac" 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>
@@ -725,22 +744,23 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
</div>
<div id="outline-container-org58e6c46" class="outline-2">
<h2 id="org58e6c46"><span class="section-number-2">3</span> Compare axis</h2>
<div id="outline-container-orgbd963cb" class="outline-2">
<h2 id="orgbd963cb"><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>.
The <code>mat</code> files containing the measurement data are accessible with the following links:
<a id="org04574c0"></a>
</p>
<ul class="org-ul">
<li>z axis: <a href="mat/data_001.mat">here</a>.</li>
<li>east axis: <a href="mat/data_002.mat">here</a>.</li>
<li>north axis: <a href="mat/data_003.mat">here</a>.</li>
</ul>
<div class="note">
<p>
All the files (data and Matlab scripts) are accessible <a href="data/huddle_test_compare_axis.zip">here</a>.
</p>
</div>
</div>
<div id="outline-container-org8f74945" class="outline-3">
<h3 id="org8f74945"><span class="section-number-3">3.1</span> Load data</h3>
<div id="outline-container-org5ae5d71" class="outline-3">
<h3 id="org5ae5d71"><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.
@@ -754,8 +774,8 @@ north = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="or
</div>
</div>
<div id="outline-container-orgf13f88e" class="outline-3">
<h3 id="orgf13f88e"><span class="section-number-3">3.2</span> Compare PSD</h3>
<div id="outline-container-orgd12648a" class="outline-3">
<h3 id="orgd12648a"><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.
@@ -773,10 +793,10 @@ The PSD for each axis of the two geophones are computed.
</div>
<p>
We compare them. The result is shown on figure <a href="#orgdfcdc16">14</a>.
We compare them. The result is shown on figure <a href="#orgbd316c4">14</a>.
</p>
<div id="orgdfcdc16" class="figure">
<div id="orgbd316c4" 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>
@@ -784,12 +804,12 @@ We compare them. The result is shown on figure <a href="#orgdfcdc16">14</a>.
</div>
</div>
<div id="outline-container-orgf389e18" class="outline-3">
<h3 id="orgf389e18"><span class="section-number-3">3.3</span> Compare TF</h3>
<div id="outline-container-orgf076a23" class="outline-3">
<h3 id="orgf076a23"><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="#orgdd8cabb">15</a>.
The result is shown on figure <a href="#org1278c1f">15</a>.
</p>
<div class="org-src-container">
@@ -800,7 +820,7 @@ The result is shown on figure <a href="#orgdd8cabb">15</a>.
</div>
<div id="orgdd8cabb" class="figure">
<div id="org1278c1f" 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>
@@ -809,15 +829,15 @@ The result is shown on figure <a href="#orgdd8cabb">15</a>.
</div>
</div>
<div id="outline-container-org1ca0f74" class="outline-2">
<h2 id="org1ca0f74"><span class="section-number-2">4</span> Appendix</h2>
<div id="outline-container-org17f5bce" class="outline-2">
<h2 id="org17f5bce"><span class="section-number-2">4</span> Appendix</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-org9faefb2" class="outline-3">
<h3 id="org9faefb2"><span class="section-number-3">4.1</span> Computation of coherence from PSD and CSD</h3>
<div id="outline-container-org6babc5e" class="outline-3">
<h3 id="org6babc5e"><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="orgc9ed210"></a>
<a id="org956da99"></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>;
@@ -848,7 +868,7 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</div>
<div id="orgea88bec" class="figure">
<div id="orgdd25190" 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>
@@ -865,8 +885,8 @@ xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbo
</p>
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-05-03 ven. 11:32</p>
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2019-05-10 ven. 14:24</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>