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< h1 class = "title" > Huddle Test of the L22 Geophones< / h1 >
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< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#orgc7e4a9f" > 1. Experimental Setup< / a > < / li >
< li > < a href = "#org6abc0c5" > 2. Signal Processing< / a >
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< ul >
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< li > < a href = "#orgda0769c" > 2.1. Load data< / a > < / li >
< li > < a href = "#orgf362871" > 2.2. Time Domain Data< / a > < / li >
< li > < a href = "#org3a306be" > 2.3. Computation of the ASD of the measured voltage< / a > < / li >
< li > < a href = "#org0a45f4a" > 2.4. Scaling to take into account the sensibility of the geophone and the voltage amplifier< / a > < / li >
< li > < a href = "#org63db1c4" > 2.5. Computation of the ASD of the velocity< / a > < / li >
< li > < a href = "#org6df28e4" > 2.6. Transfer function between the two geophones< / a > < / li >
< li > < a href = "#org2a1c732" > 2.7. Estimation of the sensor noise< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orga1d7838" > 3. Compare axis< / a >
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< ul >
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< li > < a href = "#org1153270" > 3.1. Load data< / a > < / li >
< li > < a href = "#orgd3a0160" > 3.2. Compare PSD< / a > < / li >
< li > < a href = "#orgcd6137b" > 3.3. Compare TF< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org693291d" > 4. Appendix< / a >
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< ul >
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< li > < a href = "#org58d8ebb" > 4.1. Computation of coherence from PSD and CSD< / a > < / li >
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< / ul >
< / li >
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< / ul >
< / div >
< / div >
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< div id = "outline-container-orgc7e4a9f" class = "outline-2" >
< h2 id = "orgc7e4a9f" > < 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 >
Two L22 geophones are used.
They are placed on the ID31 granite.
They are leveled.
< / p >
< p >
The signals are amplified using voltage amplifier with a gain of 60dB.
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The voltage amplifiers includes:
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< / p >
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< 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 >
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< div id = "orga476294" class = "figure" >
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< p > < img src = "./img/setup.jpg" alt = "setup.jpg" width = "500px" / >
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< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Setup< / p >
< / div >
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< div id = "org2fb54a0" class = "figure" >
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< p > < img src = "./img/geophones.jpg" alt = "geophones.jpg" width = "500px" / >
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< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Geophones< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org6abc0c5" class = "outline-2" >
< h2 id = "org6abc0c5" > < 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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< a id = "orged15cd2" > < / a >
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< / p >
< div class = "note" >
< p >
All the files (data and Matlab scripts) are accessible < a href = "data/huddle_test_signal_processing.zip" > here< / a > .
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< / p >
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< / div >
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< / div >
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< div id = "outline-container-orgda0769c" class = "outline-3" >
< h3 id = "orgda0769c" > < 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 >
We load the data of the z axis of two geophones.
< / 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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< h3 id = "orgf362871" > < span class = "section-number-3" > 2.2< / span > Time Domain Data< / h3 >
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< div id = "orgb109da2" class = "figure" >
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< p > < img src = "figs/data_time_domain.png" alt = "data_time_domain.png" / >
< / 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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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 > ;
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< / pre >
< / div >
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< div id = "org09931c1" class = "figure" >
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< p > < img src = "figs/data_time_domain_zoom.png" alt = "data_time_domain_zoom.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > Time domain Data - Zoom< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org3a306be" class = "outline-3" >
< h3 id = "org3a306be" > < span class = "section-number-3" > 2.3< / span > Computation of the ASD of the measured voltage< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
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< p >
We first define the parameters for the frequency domain analysis.
< / p >
< div class = "org-src-container" >
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< 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 >
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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 > ;
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< / pre >
< / div >
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< p >
Then we compute the Power Spectral Density using < code > pwelch< / code > function.
< / p >
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< div class = "org-src-container" >
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< 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 > ;
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< / pre >
< / div >
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< p >
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And we plot the result on figure < a href = "#org7cf943c" > 5< / a > .
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< / 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 >
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< div id = "org7cf943c" class = "figure" >
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< 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 >
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< / div >
< / div >
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< div id = "outline-container-org0a45f4a" class = "outline-3" >
< h3 id = "org0a45f4a" > < span class = "section-number-3" > 2.4< / span > Scaling to take into account the sensibility of the geophone and the voltage amplifier< / h3 >
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< div class = "outline-text-3" id = "text-2-4" >
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< p >
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The Geophone used are L22. Their sensibility is shown on figure < a href = "#org045b636" > 6< / a > .
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< / p >
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< div class = "org-src-container" >
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< 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 >
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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 > ;
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< / pre >
< / div >
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< div id = "org045b636" class = "figure" >
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< p > < img src = "figs/geophone_sensibility.png" alt = "geophone_sensibility.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Sensibility of the Geophone< / p >
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< / div >
< p >
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We also take into account the gain of the electronics which is here set to be \(60dB\).
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > G0_db = < span class = "org-highlight-numbers-number" > 60< / span > ; < span class = "org-comment" > % [dB]< / span >
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G0 = < span class = "org-highlight-numbers-number" > 10< / span > < span class = "org-type" > ^< / span > < span class = "org-rainbow-delimiters-depth-1" > (< / span > G0_db< span class = "org-type" > /< / span > < span class = "org-highlight-numbers-number" > 20< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ; < span class = "org-comment" > % [abs]< / span >
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< / pre >
< / div >
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< p >
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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.
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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" >
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< 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 > ;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org63db1c4" class = "outline-3" >
< h3 id = "org63db1c4" > < span class = "section-number-3" > 2.5< / span > Computation of the ASD of the velocity< / h3 >
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< div class = "outline-text-3" id = "text-2-5" >
< p >
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The ASD of the measured velocity is shown on figure < a href = "#org1aa0246" > 7< / a > .
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< / p >
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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 > 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 > ;
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hold off;
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< 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 > ;
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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 >
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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 > ;
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< / pre >
< / div >
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< div id = "org1aa0246" class = "figure" >
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< p > < img src = "figs/psd_velocity.png" alt = "psd_velocity.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Amplitude Spectral Density of the Velocity< / p >
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< / div >
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< p >
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We also plot the ASD in displacement (figure < a href = "#org2fbcc25" > 8< / a > );
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< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
hold on;
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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 > ;
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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 > ;
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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 >
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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 > ;
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< / pre >
< / div >
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< div id = "org2fbcc25" class = "figure" >
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< p > < img src = "figs/asd_displacement.png" alt = "asd_displacement.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > Amplitude Spectral Density of the Displacement< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-org6df28e4" class = "outline-3" >
< h3 id = "org6df28e4" > < span class = "section-number-3" > 2.6< / span > Transfer function between the two geophones< / h3 >
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< div class = "outline-text-3" id = "text-2-6" >
< p >
We here compute the transfer function from one geophone to the other.
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The result is shown on figure < a href = "#org008cd50" > 9< / a > .
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< / p >
< p >
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We also compute the coherence between the two signals (figure < a href = "#org47fd0f6" > 10< / a > ).
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< / p >
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< div class = "org-src-container" >
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< 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 >
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< div id = "org008cd50" class = "figure" >
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< p > < img src = "figs/tf_geophones.png" alt = "tf_geophones.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 9: < / span > Estimated transfer function between the two geophones< / p >
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< / 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 >
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< div id = "org47fd0f6" class = "figure" >
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< p > < img src = "figs/coh_geophones.png" alt = "coh_geophones.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Cohererence between the signals of the two geophones< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org2a1c732" class = "outline-3" >
< h3 id = "org2a1c732" > < span class = "section-number-3" > 2.7< / span > Estimation of the sensor noise< / h3 >
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< 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 >
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The < code > mscohere< / code > function is compared with this formula on Appendix (section < a href = "#orgd79f363" > 4.1< / a > ), it is shown that it is identical.
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< / p >
< p >
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Figure < a href = "#org7d058c1" > 11< / a > illustrate a block diagram model of the system used to determine the sensor noise of the geophone.
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< / 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 >
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< div id = "org7d058c1" class = "figure" >
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< p > < img src = "figs/huddle-test.png" alt = "huddle-test.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 11: < / span > Huddle test block diagram< / p >
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< / div >
< p >
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We here assume that each sensor has the same magnitude of instrumental noise: \(N = M\).
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We also assume that \(S_1 = S_2 = 1\).
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< / p >
< p >
We then obtain:
< / p >
\begin{equation}
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\label{org68c59ea}
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\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}
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\label{org6525ac9}
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|G_X(\omega)| = |G_N(\omega)| + |G_U(\omega)|
\end{equation}
< p >
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From equations \eqref{org68c59ea} and \eqref{org6525ac9}, we finally obtain
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< / p >
< div class = "important" >
\begin{equation}
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\label{orgcac9782}
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|G_N(\omega)| = |G_X(\omega)| \left( 1 - \sqrt{\gamma_{XY}^2(\omega)} \right)
\end{equation}
< / div >
< p >
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The instrumental noise is computed below. The result in V< sup > 2< / sup > /Hz is shown on figure < a href = "#org0f85822" > 12< / a > .
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< / 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 > ;
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< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
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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;
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< 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 > ;
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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 > ;
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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 > ;
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< / pre >
< / div >
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< div id = "org0f85822" class = "figure" >
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< p > < img src = "figs/intrumental_noise_V.png" alt = "intrumental_noise_V.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 12: < / span > Instrumental Noise and Measurement in \(V^2/Hz\)< / p >
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< / div >
< p >
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This is then further converted into velocity and compared with the ground velocity measurement. (figure < a href = "#org81c8985" > 13< / a > )
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< / 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 > ;
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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 > ;
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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 > ;
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< / pre >
< / div >
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< div id = "org81c8985" class = "figure" >
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< p > < img src = "figs/intrumental_noise_velocity.png" alt = "intrumental_noise_velocity.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 13: < / span > Instrumental Noise and Measurement in \(m/s/\sqrt{Hz}\)< / p >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orga1d7838" class = "outline-2" >
< h2 id = "orga1d7838" > < span class = "section-number-2" > 3< / span > Compare axis< / h2 >
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< div class = "outline-text-2" id = "text-3" >
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< p >
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< a id = "org73446c8" > < / a >
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< / p >
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< 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 >
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< / div >
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< div id = "outline-container-org1153270" class = "outline-3" >
< h3 id = "org1153270" > < span class = "section-number-3" > 3.1< / span > Load data< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
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< p >
We first load the data for the three axis.
< / p >
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< 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 >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgd3a0160" class = "outline-3" >
< h3 id = "orgd3a0160" > < span class = "section-number-3" > 3.2< / span > Compare PSD< / h3 >
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< div class = "outline-text-3" id = "text-3-2" >
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< p >
The PSD for each axis of the two geophones are computed.
< / p >
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< 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 > ;
< / pre >
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< / div >
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< p >
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We compare them. The result is shown on figure < a href = "#org16d65ef" > 14< / a > .
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< / p >
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< div id = "org16d65ef" class = "figure" >
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< p > < img src = "figs/compare_axis_psd.png" alt = "compare_axis_psd.png" / >
< / p >
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< 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 >
< / div >
< / div >
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< div id = "outline-container-orgcd6137b" class = "outline-3" >
< h3 id = "orgcd6137b" > < span class = "section-number-3" > 3.3< / span > Compare TF< / h3 >
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< div class = "outline-text-3" id = "text-3-3" >
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< p >
The transfer functions from one geophone to the other are also computed for each axis.
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The result is shown on figure < a href = "#orgdcda32f" > 15< / a > .
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< / p >
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< 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 >
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< div id = "orgdcda32f" class = "figure" >
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< p > < img src = "figs/compare_tf_axis.png" alt = "compare_tf_axis.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 15: < / span > Compare the transfer function from one geophone to the other for the 3 axis< / p >
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< / div >
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< div id = "outline-container-org693291d" class = "outline-2" >
< h2 id = "org693291d" > < span class = "section-number-2" > 4< / span > Appendix< / h2 >
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< div class = "outline-text-2" id = "text-4" >
< / div >
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< div id = "outline-container-org58d8ebb" class = "outline-3" >
< h3 id = "org58d8ebb" > < span class = "section-number-3" > 4.1< / span > Computation of coherence from PSD and CSD< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
< p >
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< a id = "orgd79f363" > < / a >
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< / 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 > ;
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< div id = "orgc65c7db" class = "figure" >
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< p > < img src = "figs/comp_coherence_formula.png" alt = "comp_coherence_formula.png" / >
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< p > < span class = "figure-number" > Figure 16: < / span > Comparison of < code > mscohere< / code > and manual computation< / p >
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< 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 >
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< div id = "postamble" class = "status" >
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< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2019-05-15 mer. 10:08< / p >
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< p class = "validation" > < a href = "http://validator.w3.org/check?uri=referer" > Validate< / a > < / p >
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