874 lines
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874 lines
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<title>Huddle Test of the L22 Geophones</title>
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</head>
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<body>
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<div id="content">
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<h1 class="title">Huddle Test of the L22 Geophones</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#org446511b">1. Experimental Setup</a></li>
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<li><a href="#org0689ed7">2. Signal Processing</a>
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<ul>
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<li><a href="#org34a2d4c">2.1. Load data</a></li>
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<li><a href="#org28e8648">2.2. Time Domain Data</a></li>
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<li><a href="#org910f3e4">2.3. Computation of the ASD of the measured voltage</a></li>
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<li><a href="#org4e984e1">2.4. Scaling to take into account the sensibility of the geophone and the voltage amplifier</a></li>
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<li><a href="#org607752b">2.5. Computation of the ASD of the velocity</a></li>
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<li><a href="#org0a07c74">2.6. Transfer function between the two geophones</a></li>
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<li><a href="#orgdc03acb">2.7. Estimation of the sensor noise</a></li>
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</ul>
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</li>
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<li><a href="#org58e6c46">3. Compare axis</a>
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<ul>
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<li><a href="#org8f74945">3.1. Load data</a></li>
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<li><a href="#orgf13f88e">3.2. Compare PSD</a></li>
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<li><a href="#orgf389e18">3.3. Compare TF</a></li>
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</ul>
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</li>
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<li><a href="#org1ca0f74">4. Appendix</a>
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<ul>
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<li><a href="#org9faefb2">4.1. Computation of coherence from PSD and CSD</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org446511b" class="outline-2">
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<h2 id="org446511b"><span class="section-number-2">1</span> Experimental Setup</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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Two L22 geophones are used.
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They are placed on the ID31 granite.
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They are leveled.
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</p>
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<p>
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The signals are amplified using voltage amplifier with a gain of 60dB.
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The voltage amplifiers include a low pass filter with a cut-off frequency at 1kHz.
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</p>
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<div id="orgeab8098" class="figure">
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<p><img src="./figs/setup.jpg" alt="setup.jpg" width="500px" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Setup</p>
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</div>
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<div id="orgdadbe53" class="figure">
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<p><img src="./figs/geophones.jpg" alt="geophones.jpg" width="500px" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Geophones</p>
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</div>
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</div>
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</div>
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<div id="outline-container-org0689ed7" class="outline-2">
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<h2 id="org0689ed7"><span class="section-number-2">2</span> Signal Processing</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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The Matlab computing file for this part is accessible <a href="signal_processing.m">here</a>.
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The <code>mat</code> file containing the measurement data is accessible <a href="mat/data_001.mat">here</a>.
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</p>
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</div>
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<div id="outline-container-org34a2d4c" class="outline-3">
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<h3 id="org34a2d4c"><span class="section-number-3">2.1</span> Load data</h3>
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<div class="outline-text-3" id="text-2-1">
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<p>
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We load the data of the z axis of two geophones.
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</p>
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<div class="org-src-container">
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<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>;
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dt = t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-type">-</span> t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org28e8648" class="outline-3">
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<h3 id="org28e8648"><span class="section-number-3">2.2</span> Time Domain Data</h3>
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<div class="outline-text-3" id="text-2-2">
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-type">figure</span>;
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hold on;
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plot<span class="org-rainbow-delimiters-depth-1">(</span>t, x1<span class="org-rainbow-delimiters-depth-1">)</span>;
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plot<span class="org-rainbow-delimiters-depth-1">(</span>t, x2<span class="org-rainbow-delimiters-depth-1">)</span>;
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hold off;
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xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Time </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">s</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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ylabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Voltage </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">V</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span>, t<span class="org-rainbow-delimiters-depth-3">(</span>end<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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</pre>
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</div>
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<div id="orgbc90092" class="figure">
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<p><img src="figs/data_time_domain.png" alt="data_time_domain.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Time domain Data</p>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
hold on;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>t, x1<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>t, x2<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
hold off;
|
|
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Time </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">s</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
ylabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Voltage </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">V</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span> <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgbb22470" 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>
|
|
</div>
|
|
</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 class="outline-text-3" id="text-2-3">
|
|
<p>
|
|
We first define the parameters for the frequency domain analysis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">Fs = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>dt; <span class="org-comment">% [Hz]</span>
|
|
|
|
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>
|
|
</div>
|
|
|
|
<p>
|
|
Then we compute the Power Spectral Density using <code>pwelch</code> function.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>pxx1, f<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>x1, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-rainbow-delimiters-depth-1">[</span>pxx2, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>x2, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</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 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 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>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">S0 = <span class="org-highlight-numbers-number">88</span>; <span class="org-comment">% Sensitivity [V/(m/s)]</span>
|
|
f0 = <span class="org-highlight-numbers-number">2</span>; <span class="org-comment">% Cut-off frequnecy [Hz]</span>
|
|
|
|
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>
|
|
</div>
|
|
|
|
|
|
<div id="org0a867d9" 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>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
We also take into account the gain of the electronics which is here set to be \(60dB\).
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">G0_db = <span class="org-highlight-numbers-number">60</span>; <span class="org-comment">% [dB]</span>
|
|
|
|
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>
|
|
</div>
|
|
|
|
<p>
|
|
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 \(m/s/\sqrt{Hz}\).
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">scaling = <span class="org-highlight-numbers-number">1</span><span class="org-type">./</span>squeeze<span class="org-rainbow-delimiters-depth-1">(</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>freqresp<span class="org-rainbow-delimiters-depth-3">(</span>G0<span class="org-type">*</span>S, f, <span class="org-string">'Hz'</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</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 class="outline-text-3" id="text-2-5">
|
|
<p>
|
|
The ASD of the measured velocity is shown on figure <a href="#orgd9a4009">7</a>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
hold on;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>pxx1<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.*</span>scaling<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>pxx2<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.*</span>scaling<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
hold off;
|
|
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'yscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'</span>ASD of the measured Velocity $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span>m<span class="org-type">/</span>s<span class="org-rainbow-delimiters-depth-3">}{</span><span class="org-type">\</span>sqrt<span class="org-rainbow-delimiters-depth-4">{</span>Hz<span class="org-rainbow-delimiters-depth-4">}</span><span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>
|
|
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgd9a4009" 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>);
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
hold on;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, <span class="org-rainbow-delimiters-depth-2">(</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span>pxx1<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">.*</span>scaling<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">./</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>f<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, <span class="org-rainbow-delimiters-depth-2">(</span>sqrt<span class="org-rainbow-delimiters-depth-3">(</span>pxx2<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">.*</span>scaling<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">./</span><span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">2</span><span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>f<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
hold off;
|
|
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; set</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">gca, 'yscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'</span>ASD of the displacement $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span>m<span class="org-rainbow-delimiters-depth-3">}{</span><span class="org-type">\</span>sqrt<span class="org-rainbow-delimiters-depth-4">{</span>Hz<span class="org-rainbow-delimiters-depth-4">}</span><span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>
|
|
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgc0b2ca5" 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>
|
|
</div>
|
|
</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 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>.
|
|
</p>
|
|
|
|
<p>
|
|
We also compute the coherence between the two signals (figure <a href="#org628544f">10</a>).
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>T12, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = tfestimate<span class="org-rainbow-delimiters-depth-1">(</span>x1, x2, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org7d8ea2b" class="figure">
|
|
<p><img src="figs/tf_geophones.png" alt="tf_geophones.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 9: </span>Estimated transfer function between the two geophones</p>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>coh12, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = mscohere<span class="org-rainbow-delimiters-depth-1">(</span>x1, x2, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org628544f" 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>
|
|
</div>
|
|
</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 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>.
|
|
</p>
|
|
|
|
<p>
|
|
The coherence between signals \(X\) and \(Y\) is defined as follow
|
|
\[ \gamma^2_{XY}(\omega) = \frac{|G_{XY}(\omega)|^2}{|G_{X}(\omega)| |G_{Y}(\omega)|} \]
|
|
where \(|G_X(\omega)|\) is the output Power Spectral Density (PSD) of signal \(X\) and \(|G_{XY}(\omega)|\) is the Cross Spectral Density (CSD) of signal \(X\) and \(Y\).
|
|
</p>
|
|
|
|
<p>
|
|
The PSD and CSD are defined as follow:
|
|
</p>
|
|
\begin{align}
|
|
|G_X(\omega)| &= \frac{2}{n_d T} \sum^{n_d}_{n=1} \left| X_k(\omega, T) \right|^2 \\
|
|
|G_{XY}(\omega)| &= \frac{2}{n_d T} \sum^{n_d}_{n=1} [ X_k^*(\omega, T) ] [ Y_k(\omega, T) ]
|
|
\end{align}
|
|
<p>
|
|
where:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>\(n_d\) is the number for records averaged</li>
|
|
<li>\(T\) is the length of each record</li>
|
|
<li>\(X_k(\omega, T)\) is the finite Fourier transform of the kth record</li>
|
|
<li>\(X_k^*(\omega, T)\) is its complex conjugate</li>
|
|
</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.
|
|
</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.
|
|
</p>
|
|
|
|
<p>
|
|
Two geophones are mounted side by side to ensure that they are exposed by the same motion input \(U\).
|
|
</p>
|
|
|
|
<p>
|
|
Each sensor has noise \(N\) and \(M\).
|
|
</p>
|
|
|
|
|
|
<div id="org9b31b02" 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>
|
|
</div>
|
|
|
|
<p>
|
|
We here assume that each sensor has the same magnitude of instrumental noise (\(N = M\)).
|
|
We also assume that \(S_1 = S_2 = 1\).
|
|
</p>
|
|
|
|
<p>
|
|
We then obtain:
|
|
</p>
|
|
\begin{equation}
|
|
\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}
|
|
\end{equation}
|
|
|
|
<p>
|
|
Since the input signal \(U\) and the instrumental noise \(N\) are incoherent:
|
|
</p>
|
|
\begin{equation}
|
|
\label{org845ba9b}
|
|
|G_X(\omega)| = |G_N(\omega)| + |G_U(\omega)|
|
|
\end{equation}
|
|
|
|
<p>
|
|
From equations \eqref{orgf197c52} and \eqref{org845ba9b}, we finally obtain
|
|
</p>
|
|
<div class="important">
|
|
\begin{equation}
|
|
\label{org0941f4e}
|
|
|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>.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">pxxN = pxx1<span class="org-type">.*</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span> <span class="org-type">-</span> coh12<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
hold on;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, pxx1, <span class="org-string">'-'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, pxx2, <span class="org-string">'-'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, pxxN, <span class="org-string">'k--'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
hold off;
|
|
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; set</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">gca, 'yscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'</span>PSD of the measured Voltage $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span>V<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">}{</span>Hz<span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orged7b0f2" 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>)
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
hold on;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>pxx1<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.*</span>scaling, <span class="org-string">'-'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>pxx2<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.*</span>scaling, <span class="org-string">'-'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, sqrt<span class="org-rainbow-delimiters-depth-2">(</span>pxxN<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.*</span>scaling, <span class="org-string">'k--'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
hold off;
|
|
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; set</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">gca, 'yscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'</span>ASD of the Velocity $<span class="org-type">\</span>left<span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">\</span>frac<span class="org-rainbow-delimiters-depth-3">{</span>m<span class="org-type">/</span>s<span class="org-rainbow-delimiters-depth-3">}{</span><span class="org-type">\</span>sqrt<span class="org-rainbow-delimiters-depth-4">{</span>Hz<span class="org-rainbow-delimiters-depth-4">}</span><span class="org-rainbow-delimiters-depth-3">}</span><span class="org-type">\</span>right<span class="org-rainbow-delimiters-depth-2">]</span>$'<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">0</span>.<span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="org3b9b556" 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>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org58e6c46" class="outline-2">
|
|
<h2 id="org58e6c46"><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:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>z axis: <a href="mat/data_001.mat">here</a>.</li>
|
|
<li>east axis: <a href="mat/data_002.mat">here</a>.</li>
|
|
<li>north axis: <a href="mat/data_003.mat">here</a>.</li>
|
|
</ul>
|
|
</div>
|
|
|
|
<div id="outline-container-org8f74945" class="outline-3">
|
|
<h3 id="org8f74945"><span class="section-number-3">3.1</span> Load data</h3>
|
|
<div class="outline-text-3" id="text-3-1">
|
|
<p>
|
|
We first load the data for the three axis.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">z = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_001.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
east = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_002.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
north = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_003.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
</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 class="outline-text-3" id="text-3-2">
|
|
<p>
|
|
The PSD for each axis of the two geophones are computed.
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>pz1, fz<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>z.x1, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>z.x1<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-rainbow-delimiters-depth-1">[</span>pz2, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>z.x2, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>z.x2<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
|
|
<span class="org-rainbow-delimiters-depth-1">[</span>pe1, fe<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>east.x1, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>east.x1<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>east.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>east.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-rainbow-delimiters-depth-1">[</span>pe2, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>east.x2, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>east.x2<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>east.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>east.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
|
|
<span class="org-rainbow-delimiters-depth-1">[</span>pn1, fn<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>north.x1, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>north.x1<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>north.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>north.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-rainbow-delimiters-depth-1">[</span>pn2, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>north.x2, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>north.x2<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>north.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>north.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
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|
</pre>
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</div>
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|
|
|
<p>
|
|
We compare them. The result is shown on figure <a href="#orgdfcdc16">14</a>.
|
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</p>
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|
|
|
<div id="orgdfcdc16" 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>
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|
</div>
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</div>
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|
</div>
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|
|
|
<div id="outline-container-orgf389e18" class="outline-3">
|
|
<h3 id="orgf389e18"><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>.
|
|
</p>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>Tz, fz<span class="org-rainbow-delimiters-depth-1">]</span> = tfestimate<span class="org-rainbow-delimiters-depth-1">(</span>z.x1, z.x2, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>z.x1<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>z.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-rainbow-delimiters-depth-1">[</span>Te, fe<span class="org-rainbow-delimiters-depth-1">]</span> = tfestimate<span class="org-rainbow-delimiters-depth-1">(</span>east.x1, east.x2, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>east.x1<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>east.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>east.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
<span class="org-rainbow-delimiters-depth-1">[</span>Tn, fn<span class="org-rainbow-delimiters-depth-1">]</span> = tfestimate<span class="org-rainbow-delimiters-depth-1">(</span>north.x1, north.x2, hanning<span class="org-rainbow-delimiters-depth-2">(</span>ceil<span class="org-rainbow-delimiters-depth-3">(</span>length<span class="org-rainbow-delimiters-depth-4">(</span>north.x1<span class="org-rainbow-delimiters-depth-4">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-2">(</span>north.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">-</span>north.t<span class="org-rainbow-delimiters-depth-3">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgdd8cabb" 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>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org1ca0f74" class="outline-2">
|
|
<h2 id="org1ca0f74"><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 class="outline-text-3" id="text-4-1">
|
|
<p>
|
|
<a id="orgc9ed210"></a>
|
|
</p>
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_001.mat', 't', 'x1', 'x2'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
dt = t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span> <span class="org-type">-</span> t<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
Fs = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>dt;
|
|
win = hanning<span class="org-rainbow-delimiters-depth-1">(</span>ceil<span class="org-rainbow-delimiters-depth-2">(</span>length<span class="org-rainbow-delimiters-depth-3">(</span>x1<span class="org-rainbow-delimiters-depth-3">)</span><span class="org-type">/</span><span class="org-highlight-numbers-number">100</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab">pxy = cpsd<span class="org-rainbow-delimiters-depth-1">(</span>x1, x2, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
pxx = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>x1, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
pyy = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>x2, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
coh = mscohere<span class="org-rainbow-delimiters-depth-1">(</span>x1, x2, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
<div class="org-src-container">
|
|
<pre class="src src-matlab"><span class="org-type">figure</span>;
|
|
hold on;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, abs<span class="org-rainbow-delimiters-depth-2">(</span>pxy<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">.^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">./</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>pxx<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">./</span>abs<span class="org-rainbow-delimiters-depth-2">(</span>pyy<span class="org-rainbow-delimiters-depth-2">)</span>, <span class="org-string">'-'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, coh, <span class="org-string">'--'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
hold off;
|
|
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'Coherence'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
|
|
</pre>
|
|
</div>
|
|
|
|
|
|
<div id="orgea88bec" 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>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<p>
|
|
|
|
<h1 class='org-ref-bib-h1'>Bibliography</h1>
|
|
<ul class='org-ref-bib'><li><a id="barzilai98_techn_measur_noise_sensor_presen">[barzilai98_techn_measur_noise_sensor_presen]</a> <a name="barzilai98_techn_measur_noise_sensor_presen"></a>Aaron Barzilai, Tom VanZandt & Tom Kenny, Technique for Measurement of the Noise of a Sensor in the Presence of Large Background Signals, <i>Review of Scientific Instruments</i>, <b>69(7)</b>, 2767-2772 (1998). <a href="https://doi.org/10.1063/1.1149013">link</a>. <a href="http://dx.doi.org/10.1063/1.1149013">doi</a>.</li>
|
|
</ul>
|
|
</p>
|
|
</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="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
|
</div>
|
|
</body>
|
|
</html>
|