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< body >
< 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" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#org6392291" > 1. Experimental Setup< / a > < / li >
< li > < a href = "#orgb55c46d" > 2. Signal Processing< / a >
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< ul >
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< li > < a href = "#org016f692" > 2.1. Load data< / a > < / li >
< li > < a href = "#orgc616389" > 2.2. Time Domain Data< / a > < / li >
< li > < a href = "#org8794282" > 2.3. Computation of the ASD of the measured voltage< / a > < / li >
< li > < a href = "#org09670e4" > 2.4. Scaling to take into account the sensibility of the geophone and the voltage amplifier< / a > < / li >
< li > < a href = "#org4e5224a" > 2.5. Computation of the ASD of the velocity< / a > < / li >
< li > < a href = "#orge366f60" > 2.6. Transfer function between the two geophones< / a > < / li >
< li > < a href = "#org4cacf62" > 2.7. Estimation of the sensor noise< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org559b5b9" > 3. Compare axis< / a >
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< ul >
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< li > < a href = "#org24b196e" > 3.1. Load data< / a > < / li >
< li > < a href = "#org00c635c" > 3.2. Compare PSD< / a > < / li >
< li > < a href = "#org16927e1" > 3.3. Compare TF< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgf4800f0" > 4. Appendix< / a >
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< ul >
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< li > < a href = "#org556257c" > 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-org6392291" class = "outline-2" >
< h2 id = "org6392291" > < 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.
The voltage amplifiers include a low pass filter with a cut-off frequency at 1kHz.
< / p >
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< div id = "orgddb9341" class = "figure" >
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< p > < img src = "./figs/setup.jpg" alt = "setup.jpg" width = "500px" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Setup< / p >
< / div >
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< div id = "org8077b6d" class = "figure" >
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< p > < img src = "./figs/geophones.jpg" alt = "geophones.jpg" width = "500px" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Geophones< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb55c46d" class = "outline-2" >
< h2 id = "orgb55c46d" > < 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 >
The Matlab computing file for this part is accessible < a href = "signal_processing.m" > here< / a > .
The < code > mat< / code > file containing the measurement data is accessible < a href = "mat/data_001.mat" > here< / a > .
< / p >
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< / div >
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< h3 id = "org016f692" > < 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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< h3 id = "orgc616389" > < span class = "section-number-3" > 2.2< / span > Time Domain Data< / h3 >
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< p > < span class = "figure-number" > Figure 3: < / span > Time domain Data< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > Time domain Data - Zoom< / p >
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< h3 id = "org8794282" > < 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 >
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< pre class = "src src-matlab" > 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 > < 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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< 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 >
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< div id = "outline-container-org09670e4" class = "outline-3" >
< h3 id = "org09670e4" > < span class = "section-number-3" > 2.4< / span > Scaling to take into account the sensibility of the geophone and the voltage amplifier< / h3 >
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< div class = "outline-text-3" id = "text-2-4" >
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< p >
The Geophone used are L22.
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Their sensibility are shown on figure < a href = "#orgdbecac7" > 5< / 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 >
S = < span class = "org-rainbow-delimiters-depth-1" > (< / span > s< span class = "org-type" > /< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > f0< span class = "org-rainbow-delimiters-depth-1" > )< / span > < span class = "org-type" > /< / span > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > f0< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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< / pre >
< / div >
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< div id = "orgdbecac7" 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 5: < / 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 \(60dB\).
The amplifiers also include a low pass filter with a cut-off frequency set at 1kHz.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > G0 = < span class = "org-highlight-numbers-number" > 60< / span > ; < span class = "org-comment" > % [dB]< / span >
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G = < span class = "org-highlight-numbers-number" > 10< / span > < span class = "org-type" > ^< / span > < span class = "org-rainbow-delimiters-depth-1" > (< / span > G0< span class = "org-type" > /< / span > < span class = "org-highlight-numbers-number" > 20< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > < span class = "org-type" > /< / span > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-highlight-numbers-number" > 1< / span > < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > < span class = "org-highlight-numbers-number" > 1000< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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< / pre >
< / div >
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< p >
We divide the ASD measured (in \(\text{V}/\sqrt{\text{Hz}}\)) by the transfer function of the voltage amplifier to obtain the ASD of the voltage across the geophone.
We 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 > G< span class = "org-type" > *< / span > S, f, < span class = "org-string" > 'Hz'< / span > < span class = "org-rainbow-delimiters-depth-3" > )< / span > < span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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< / pre >
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< div id = "outline-container-org4e5224a" class = "outline-3" >
< h3 id = "org4e5224a" > < 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 = "#orgf6e310e" > 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" > < 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 > ;
xlabel< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'Frequency < / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-2" > [< / span > < / span > < span class = "org-string" > Hz< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-2" > ]< / span > < / span > < span class = "org-string" > '< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-1" > )< / span > < / span > < span class = "org-string" > ; ylabel< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < / span > < span class = "org-string" > 'PSD < / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-2" > [< / span > < / span > < span class = "org-string" > m/s/sqrt< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-3" > (< / span > < / span > < span class = "org-string" > Hz< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-3" > )< / span > < / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-2" > ]< / span > < / span > < span class = "org-string" > '< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span >
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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 = "orgf6e310e" 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 6: < / span > Spectral density of the velocity< / p >
< / div >
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< p >
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We also plot the ASD in displacement (figure < a href = "#org0ea2712" > 7< / 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, < span class = "org-rainbow-delimiters-depth-2" > (< / span > pxx1< span class = "org-type" > .*< / span > scaling< span class = "org-type" > ./< / span > f< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > .^< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
plot< span class = "org-rainbow-delimiters-depth-1" > (< / span > f, < span class = "org-rainbow-delimiters-depth-2" > (< / span > pxx2< span class = "org-type" > .*< / span > scaling< span class = "org-type" > ./< / span > f< span class = "org-rainbow-delimiters-depth-2" > )< / span > < span class = "org-type" > .^< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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" > 'PSD < / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-2" > [< / span > < / span > < span class = "org-string" > m/s/sqrt< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-3" > (< / span > < / span > < span class = "org-string" > Hz< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-3" > )< / span > < / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-2" > ]< / span > < / span > < span class = "org-string" > '< / span > < span class = "org-rainbow-delimiters-depth-1" > )< / span >
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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 >
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< div id = "org0ea2712" class = "figure" >
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< p > < img src = "figs/asd_displacement.png" alt = "asd_displacement.png" / >
< / p >
< p > < span class = "figure-number" > Figure 7: < / span > Amplitude Spectral Density of the displacement as measured by the geophones< / p >
< / div >
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< / div >
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< div id = "outline-container-orge366f60" class = "outline-3" >
< h3 id = "orge366f60" > < 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 = "#orgba6f718" > 8< / a > .
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< / p >
< p >
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We also compute the coherence between the two signals (figure < a href = "#org251a7af" > 9< / 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 = "orgba6f718" 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 8: < / 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 = "org251a7af" 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 9: < / span > Cohererence between the signals of the two geophones< / p >
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< / div >
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< / div >
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< div id = "outline-container-org4cacf62" class = "outline-3" >
< h3 id = "org4cacf62" > < 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 = "#org1e77eec" > 4.1< / a > ), it is shown that it is identical.
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< / p >
< p >
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Figure < a href = "#org8c23b33" > 10< / 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 = "org8c23b33" 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 10: < / span > Huddle test block diagram< / p >
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< / div >
< p >
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{org3a8a59f}
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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{org221f31e}
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|G_X(\omega)| = |G_N(\omega)| + |G_U(\omega)|
\end{equation}
< p >
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From equations \eqref{org3a8a59f} and \eqref{org221f31e}, we finally obtain
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< / p >
< div class = "important" >
\begin{equation}
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\label{org3574759}
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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 = "#orge1676ed" > 11< / 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 > 'Frequency < span class = "org-rainbow-delimiters-depth-2" > [< / span > Hz< span class = "org-rainbow-delimiters-depth-2" > ]< / span > '< span class = "org-rainbow-delimiters-depth-1" > )< / span > ; ylabel< span class = "org-rainbow-delimiters-depth-1" > (< / span > 'PSD < span class = "org-rainbow-delimiters-depth-2" > [< / span > $V< span class = "org-type" > ^< / span > < span class = "org-highlight-numbers-number" > 2< / span > < span class = "org-type" > /< / span > Hz$< span class = "org-rainbow-delimiters-depth-2" > ]< / span > '< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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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 = "orge1676ed" 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 11: < / 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 = "#org30e84b0" > 12< / 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 > ;
xlabel< span class = "org-rainbow-delimiters-depth-1" > (< / span > < span class = "org-string" > 'Frequency < / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-2" > [< / span > < / span > < span class = "org-string" > Hz< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-2" > ]< / span > < / span > < span class = "org-string" > '< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-1" > )< / span > < / span > < span class = "org-string" > ; ylabel< / span > < span class = "org-string" > < span class = "org-rainbow-delimiters-depth-1" > (< / span > < / span > < span class = "org-string" > '< / span > PSD < span class = "org-rainbow-delimiters-depth-2" > [< / span > $m< span class = "org-type" > /< / span > s< span class = "org-type" > /\< / span > sqrt< span class = "org-rainbow-delimiters-depth-3" > {< / span > Hz< span class = "org-rainbow-delimiters-depth-3" > }< / span > $< span class = "org-rainbow-delimiters-depth-2" > ]< / span > '< span class = "org-rainbow-delimiters-depth-1" > )< / span > ;
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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 = "org30e84b0" 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 12: < / 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-org559b5b9" class = "outline-2" >
< h2 id = "org559b5b9" > < 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 >
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 >
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< / div >
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< div id = "outline-container-org24b196e" class = "outline-3" >
< h3 id = "org24b196e" > < 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-org00c635c" class = "outline-3" >
< h3 id = "org00c635c" > < 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 = "#org6734ed2" > 13< / a > .
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< / p >
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< div id = "org6734ed2" 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 13: < / span > Compare the measure PSD of the two geophones for the three axis< / p >
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< div id = "outline-container-org16927e1" class = "outline-3" >
< h3 id = "org16927e1" > < 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 = "#org8df4d36" > 14< / 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 > ;
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< div id = "org8df4d36" 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 14: < / span > Compare the transfer function from one geophone to the other for the 3 axis< / p >
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< div id = "outline-container-orgf4800f0" class = "outline-2" >
< h2 id = "orgf4800f0" > < span class = "section-number-2" > 4< / span > Appendix< / h2 >
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< div class = "outline-text-2" id = "text-4" >
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< div id = "outline-container-org556257c" class = "outline-3" >
< h3 id = "org556257c" > < span class = "section-number-3" > 4.1< / span > Computation of coherence from PSD and CSD< / h3 >
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< p >
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< a id = "org1e77eec" > < / 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 > ;
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< 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 >
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< 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 >
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< div id = "org190093c" class = "figure" >
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< p > < img src = "figs/comp_coherence_formula.png" alt = "comp_coherence_formula.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 15: < / 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" >
< p class = "author" > Author: Thomas Dehaeze< / p >
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< p class = "date" > Created: 2019-05-03 ven. 08:40< / p >
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< p class = "validation" > < a href = "http://validator.w3.org/check?uri=referer" > Validate< / a > < / p >
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