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huddle-test-geophones/compare_axis.m
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huddle-test-geophones/compare_axis.m
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% Matlab Init :noexport:ignore:
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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%% Initialize ans with org-babel
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ans = 0;
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% Load data
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% We first load the data for the three axis.
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z = load('mat/data_001.mat', 't', 'x1', 'x2');
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east = load('mat/data_002.mat', 't', 'x1', 'x2');
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north = load('mat/data_003.mat', 't', 'x1', 'x2');
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% Compare PSD
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% The PSD for each axis of the two geophones are computed.
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[pz1, fz] = pwelch(z.x1, hanning(ceil(length(z.x1)/100)), [], [], 1/(z.t(2)-z.t(1)));
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[pz2, ~] = pwelch(z.x2, hanning(ceil(length(z.x2)/100)), [], [], 1/(z.t(2)-z.t(1)));
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[pe1, fe] = pwelch(east.x1, hanning(ceil(length(east.x1)/100)), [], [], 1/(east.t(2)-east.t(1)));
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[pe2, ~] = pwelch(east.x2, hanning(ceil(length(east.x2)/100)), [], [], 1/(east.t(2)-east.t(1)));
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[pn1, fn] = pwelch(north.x1, hanning(ceil(length(north.x1)/100)), [], [], 1/(north.t(2)-north.t(1)));
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[pn2, ~] = pwelch(north.x2, hanning(ceil(length(north.x2)/100)), [], [], 1/(north.t(2)-north.t(1)));
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% We compare them. The result is shown on figure [[fig:compare_axis_psd]].
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figure;
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hold on;
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plot(fz, sqrt(pz1), '-', 'Color', [0 0.4470 0.7410], 'DisplayName', 'z');
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plot(fz, sqrt(pz2), '--', 'Color', [0 0.4470 0.7410], 'HandleVisibility', 'off');
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plot(fe, sqrt(pe1), '-', 'Color', [0.8500 0.3250 0.0980], 'DisplayName', 'east');
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plot(fe, sqrt(pe2), '--', 'Color', [0.8500 0.3250 0.0980], 'HandleVisibility', 'off');
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plot(fn, sqrt(pn1), '-', 'Color', [0.9290 0.6940 0.1250], 'DisplayName', 'north');
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plot(fn, sqrt(pn2), '--', 'Color', [0.9290 0.6940 0.1250], 'HandleVisibility', 'off');
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hold off;
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set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
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xlabel('Frequency [Hz]'); ylabel('PSD [m/s/sqrt(Hz)]');
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legend('Location', 'northeast');
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xlim([0, 500]);
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% Compare TF
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% 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 [[fig:compare_tf_axis]].
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[Tz, fz] = tfestimate(z.x1, z.x2, hanning(ceil(length(z.x1)/100)), [], [], 1/(z.t(2)-z.t(1)));
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[Te, fe] = tfestimate(east.x1, east.x2, hanning(ceil(length(east.x1)/100)), [], [], 1/(east.t(2)-east.t(1)));
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[Tn, fn] = tfestimate(north.x1, north.x2, hanning(ceil(length(north.x1)/100)), [], [], 1/(north.t(2)-north.t(1)));
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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plot(fz, abs(Tz), 'DisplayName', 'z');
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plot(fe, abs(Te), 'DisplayName', 'east');
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plot(fn, abs(Tn), 'DisplayName', 'north');
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hold off;
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set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
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set(gca, 'XTickLabel',[]);
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ylabel('Magnitude');
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legend('Location', 'southwest');
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ax2 = subplot(2, 1, 2);
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hold on;
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plot(fz, mod(180+180/pi*phase(Tz), 360)-180);
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plot(fe, mod(180+180/pi*phase(Te), 360)-180);
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plot(fn, mod(180+180/pi*phase(Tn), 360)-180);
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hold off;
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set(gca, 'xscale', 'log');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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xlabel('Frequency [Hz]'); ylabel('Phase');
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linkaxes([ax1,ax2],'x');
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xlim([1, 500]);
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Binary file not shown.
@ -18,7 +18,7 @@
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#+PROPERTY: header-args:matlab+ :output-dir figs
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:END:
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* Setup
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* Experimental Setup
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Two L22 geophones are used.
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They are placed on the ID31 granite.
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They are leveled.
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@ -37,6 +37,13 @@ The voltage amplifiers include a low pass filter with a cut-off frequency at 1kH
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[[file:./figs/geophones.jpg]]
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* Signal Processing
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:PROPERTIES:
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:header-args:matlab+: :tangle signal_processing.m
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:header-args:matlab+: :comments org :mkdirp yes
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:END:
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The Matlab computing file for this part is accessible [[file:signal_processing.m][here]].
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The =mat= file containing the measurement data is accessible [[file:mat/data_001.mat][here]].
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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@ -44,6 +51,7 @@ The voltage amplifiers include a low pass filter with a cut-off frequency at 1kH
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** Load data
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We load the data of the z axis of two geophones.
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#+begin_src matlab :results none
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load('mat/data_001.mat', 't', 'x1', 'x2');
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dt = t(2) - t(1);
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@ -346,12 +354,23 @@ This is then further converted into velocity and compared with the ground veloci
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[[file:figs/intrumental_noise_velocity.png]]
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* Compare axis
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:PROPERTIES:
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:header-args:matlab+: :tangle compare_axis.m
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:header-args:matlab+: :comments org :mkdirp yes
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:END:
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The Matlab computing file for this part is accessible [[file:compare_axis.m][here]].
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The =mat= files containing the measurement data are accessible with the following links:
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- z axis: [[file:mat/data_001.mat][here]].
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- east axis: [[file:mat/data_002.mat][here]].
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- north axis: [[file:mat/data_003.mat][here]].
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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** Load data
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We first load the data for the three axis.
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#+begin_src matlab :results none
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z = load('mat/data_001.mat', 't', 'x1', 'x2');
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east = load('mat/data_002.mat', 't', 'x1', 'x2');
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@ -359,6 +378,7 @@ This is then further converted into velocity and compared with the ground veloci
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#+end_src
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** Compare PSD
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The PSD for each axis of the two geophones are computed.
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#+begin_src matlab :results none
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[pz1, fz] = pwelch(z.x1, hanning(ceil(length(z.x1)/100)), [], [], 1/(z.t(2)-z.t(1)));
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[pz2, ~] = pwelch(z.x2, hanning(ceil(length(z.x2)/100)), [], [], 1/(z.t(2)-z.t(1)));
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@ -370,7 +390,8 @@ This is then further converted into velocity and compared with the ground veloci
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[pn2, ~] = pwelch(north.x2, hanning(ceil(length(north.x2)/100)), [], [], 1/(north.t(2)-north.t(1)));
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#+end_src
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#+begin_src matlab :results none :exports none
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We compare them. The result is shown on figure [[fig:compare_axis_psd]].
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#+begin_src matlab :results none :exports none
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figure;
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hold on;
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plot(fz, sqrt(pz1), '-', 'Color', [0 0.4470 0.7410], 'DisplayName', 'z');
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@ -398,6 +419,9 @@ This is then further converted into velocity and compared with the ground veloci
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[[file:figs/compare_axis_psd.png]]
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** Compare TF
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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 [[fig:compare_tf_axis]].
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#+begin_src matlab :results none
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[Tz, fz] = tfestimate(z.x1, z.x2, hanning(ceil(length(z.x1)/100)), [], [], 1/(z.t(2)-z.t(1)));
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[Te, fe] = tfestimate(east.x1, east.x2, hanning(ceil(length(east.x1)/100)), [], [], 1/(east.t(2)-east.t(1)));
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@ -442,6 +466,7 @@ This is then further converted into velocity and compared with the ground veloci
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#+CAPTION: Compare the transfer function from one geophone to the other for the 3 axis
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#+RESULTS: fig:compare_tf_axis
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[[file:figs/compare_tf_axis.png]]
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* Appendix
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** Computation of coherence from PSD and CSD
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<<sec:coherence>>
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233
huddle-test-geophones/signal_processing.m
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233
huddle-test-geophones/signal_processing.m
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% Matlab Init :noexport:ignore:
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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%% Initialize ans with org-babel
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ans = 0;
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% Load data
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% We load the data of the z axis of two geophones.
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load('mat/data_001.mat', 't', 'x1', 'x2');
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dt = t(2) - t(1);
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% Time Domain Data
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figure;
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hold on;
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plot(t, x1);
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plot(t, x2);
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hold off;
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xlabel('Time [s]');
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ylabel('Voltage [V]');
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xlim([t(1), t(end)]);
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% #+NAME: fig:data_time_domain
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% #+CAPTION: Time domain Data
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% #+RESULTS: fig:data_time_domain
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% [[file:figs/data_time_domain.png]]
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figure;
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hold on;
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plot(t, x1);
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plot(t, x2);
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hold off;
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xlabel('Time [s]');
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ylabel('Voltage [V]');
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xlim([0 1]);
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% Computation of the ASD of the measured voltage
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% We first define the parameters for the frequency domain analysis.
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win = hanning(ceil(length(x1)/100));
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Fs = 1/dt;
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[pxx1, f] = pwelch(x1, win, [], [], Fs);
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[pxx2, ~] = pwelch(x2, win, [], [], Fs);
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% Scaling to take into account the sensibility of the geophone and the voltage amplifier
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% The Geophone used are L22.
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% Their sensibility are shown on figure [[fig:geophone_sensibility]].
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S0 = 88; % Sensitivity [V/(m/s)]
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f0 = 2; % Cut-off frequnecy [Hz]
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S = (s/2/pi/f0)/(1+s/2/pi/f0);
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figure;
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bodeFig({S});
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ylabel('Amplitude [V/(m/s)]')
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% #+NAME: fig:geophone_sensibility
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% #+CAPTION: Sensibility of the Geophone
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% #+RESULTS: fig:geophone_sensibility
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% [[file:figs/geophone_sensibility.png]]
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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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% The amplifiers also include a low pass filter with a cut-off frequency set at 1kHz.
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G0 = 60; % [dB]
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G = G0/(1+s/2/pi/1000);
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% 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.
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% We further divide the result by the sensibility of the Geophone to obtain the ASD of the velocity in $m/s/\sqrt{Hz}$.
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scaling = 1./squeeze(abs(freqresp(G, f, 'Hz')))./squeeze(abs(freqresp(S, f, 'Hz')));
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% Computation of the ASD of the velocity
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% The ASD of the measured velocity is shown on figure [[fig:psd_velocity]].
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figure;
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hold on;
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plot(f, sqrt(pxx1)./scaling);
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plot(f, sqrt(pxx2)./scaling);
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hold off;
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set(gca, 'xscale', 'log');
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set(gca, 'yscale', 'log');
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xlabel('Frequency [Hz]'); ylabel('PSD [m/s/sqrt(Hz)]')
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xlim([2, 500]);
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% Transfer function between the two geophones
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% We here compute the transfer function from one geophone to the other.
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% The result is shown on figure [[fig:tf_geophones]].
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% We also compute the coherence between the two signals (figure [[fig:coh_geophones]]).
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[T12, ~] = tfestimate(x1, x2, win, [], [], Fs);
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figure;
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ax1 = subplot(2, 1, 1);
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plot(f, abs(T12));
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set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
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set(gca, 'XTickLabel',[]);
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ylabel('Magnitude');
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ax2 = subplot(2, 1, 2);
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plot(f, mod(180+180/pi*phase(T12), 360)-180);
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set(gca, 'xscale', 'log');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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xlabel('Frequency [Hz]'); ylabel('Phase');
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linkaxes([ax1,ax2],'x');
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xlim([1, 500]);
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% #+NAME: fig:tf_geophones
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% #+CAPTION: Estimated transfer function between the two geophones
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% #+RESULTS: fig:tf_geophones
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% [[file:figs/tf_geophones.png]]
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[coh12, ~] = mscohere(x1, x2, win, [], [], Fs);
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figure;
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plot(f, coh12);
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set(gca, 'xscale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Coherence');
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ylim([0,1]); xlim([1, 500]);
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% Estimation of the sensor noise
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% The technique to estimate the sensor noise is taken from cite:barzilai98_techn_measur_noise_sensor_presen.
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% The coherence between signals $X$ and $Y$ is defined as follow
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% \[ \gamma^2_{XY}(\omega) = \frac{|G_{XY}(\omega)|^2}{|G_{X}(\omega)| |G_{Y}(\omega)|} \]
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% 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 $Y$.
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% The PSD and CSD are defined as follow:
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% \begin{align}
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% |G_X(\omega)| &= \frac{2}{n_d T} \sum^{n_d}_{n=1} \left| X_k(\omega, T) \right|^2 \\
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% |G_{XY}(\omega)| &= \frac{2}{n_d T} \sum^{n_d}_{n=1} [ X_k^*(\omega, T) ] [ Y_k(\omega, T) ]
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% \end{align}
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% where:
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% - $n_d$ is the number for records averaged
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% - $T$ is the length of each record
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% - $X_k(\omega, T)$ is the finite Fourier transform of the kth record
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% - $X_k^*(\omega, T)$ is its complex conjugate
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% The =mscohere= function is compared with this formula on Appendix (section [[sec:coherence]]), it is shown that it is identical.
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% Figure [[fig:huddle_test]] illustrate a block diagram model of the system used to determine the sensor noise of the geophone.
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% Two geophones are mounted side by side to ensure that they are exposed by the same motion input $U$.
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% Each sensor has noise $N$ and $M$.
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% #+NAME: fig:huddle_test
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% #+CAPTION: Huddle test block diagram
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% [[file:figs/huddle-test.png]]
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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 $H_1 = H_2 = 1$.
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% We then obtain:
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% #+NAME: eq:coh_bis
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% \begin{equation}
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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}
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% \end{equation}
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% Since the input signal $U$ and the instrumental noise $N$ are incoherent:
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% #+NAME: eq:incoherent_noise
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% \begin{equation}
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% |G_X(\omega)| = |G_N(\omega)| + |G_U(\omega)|
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% \end{equation}
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% From equations [[eq:coh_bis]] and [[eq:incoherent_noise]], we finally obtain
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% #+begin_important
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% #+NAME: eq:noise_psd
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% \begin{equation}
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% |G_N(\omega)| = |G_X(\omega)| \left( 1 - \sqrt{\gamma_{XY}^2(\omega)} \right)
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% \end{equation}
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% #+end_important
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% The instrumental noise is computed below. The result in V^2/Hz is shown on figure [[fig:intrumental_noise_V]].
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pxxN = pxx1.*(1 - coh12);
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figure;
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hold on;
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plot(f, pxx1, '-');
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plot(f, pxx2, '-');
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plot(f, pxxN, 'k--');
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hold off;
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set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
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xlabel('Frequency [Hz]'); ylabel('PSD [$V^2/Hz$]');
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xlim([1, 500]);
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% #+NAME: fig:intrumental_noise_V
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% #+CAPTION: Instrumental Noise and Measurement in $V^2/Hz$
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% #+RESULTS: fig:intrumental_noise_V
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% [[file:figs/intrumental_noise_V.png]]
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% This is then further converted into velocity and compared with the ground velocity measurement. (figure [[fig:intrumental_noise_velocity]])
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figure;
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hold on;
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plot(f, sqrt(pxx1).*scaling, '-');
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plot(f, sqrt(pxx2).*scaling, '-');
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plot(f, sqrt(pxxN).*scaling, 'k--');
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hold off;
|
||||
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('PSD [$m/s/\sqrt{Hz}$]');
|
||||
xlim([1, 500]);
|
||||
Loading…
Reference in New Issue
Block a user