Prepare repository for lecture 02 #9
1002
01-Python_Introduction/Python_Crash_Course-with_solutions.ipynb
Normal file
1002
01-Python_Introduction/Python_Crash_Course-with_solutions.ipynb
Normal file
File diff suppressed because it is too large
Load Diff
544
02-FFT_and_Basic_Filtering/1-fourier_transform.ipynb
Normal file
544
02-FFT_and_Basic_Filtering/1-fourier_transform.ipynb
Normal file
@ -0,0 +1,544 @@
|
|||||||
|
{
|
||||||
|
"cells": [
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"<div style='background-image: url(\"../images/header.svg\") ; padding: 0px ; background-size: cover ; border-radius: 5px ; height: 250px'>\n",
|
||||||
|
" <div style=\"float: right ; margin: 50px ; padding: 20px ; background: rgba(255 , 255 , 255 , 0.7) ; width: 50% ; height: 150px\">\n",
|
||||||
|
" <div style=\"position: relative ; top: 50% ; transform: translatey(-50%)\">\n",
|
||||||
|
" <div style=\"font-size: xx-large ; font-weight: 900 ; color: rgba(0 , 0 , 0 , 0.8) ; line-height: 100%\">Signal Processing</div>\n",
|
||||||
|
" <div style=\"font-size: large ; padding-top: 20px ; color: rgba(0 , 0 , 0 , 0.5)\">Fourier Transformation</div>\n",
|
||||||
|
" </div>\n",
|
||||||
|
" </div>\n",
|
||||||
|
"</div>\n",
|
||||||
|
"\n",
|
||||||
|
"Seismo-Live: http://seismo-live.org\n",
|
||||||
|
"\n",
|
||||||
|
"##### Authors:\n",
|
||||||
|
"* Stefanie Donner ([@stefdonner](https://github.com/stefdonner))\n",
|
||||||
|
"* Celine Hadziioannou ([@hadzii](https://github.com/hadzii))\n",
|
||||||
|
"* Ceri Nunn ([@cerinunn](https://github.com/cerinunn))\n",
|
||||||
|
"\n",
|
||||||
|
"Some code used in this tutorial is taken from [stackoverflow.com](http://stackoverflow.com/questions/4258106/how-to-calculate-a-fourier-series-in-numpy/27720302#27720302). We thank [Giulio Ghirardo](https://www.researchgate.net/profile/Giulio_Ghirardo) for his kind permission to use his code here.\n",
|
||||||
|
"\n",
|
||||||
|
"---"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"code_folding": [
|
||||||
|
0
|
||||||
|
],
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "fragment"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 0 - Preparation: load packages, set some basic options \n",
|
||||||
|
"%matplotlib inline\n",
|
||||||
|
"from scipy import signal\n",
|
||||||
|
"from obspy.signal.invsim import cosine_taper \n",
|
||||||
|
"from matplotlib import rcParams\n",
|
||||||
|
"import numpy as np\n",
|
||||||
|
"import matplotlib.pylab as plt\n",
|
||||||
|
"plt.style.use('ggplot')\n",
|
||||||
|
"plt.rcParams['figure.figsize'] = 15, 3"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"<h1>Tutorial on Fourier transformation in 1D </h1>\n",
|
||||||
|
"<br>"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"## The Fourier transformation\n",
|
||||||
|
"\n",
|
||||||
|
"In the world of seismology, we use the *Fourier transformation* to transform a signal from the time domain into the frequency domain. That means, we split up the signal and separate the content of each frequency from each other. Doing so, we can analyse our signal according to energy content per frequency. We can extract information on how much amplitude each frequency contributes to the final signal. In other words: we get a receipt of the ingredients we need to blend our measured signal. "
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "fragment"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"The *Fourier transformation* is based on the *Fourier series*. With the *Fourier series* we can approximate an (unknown) function $f(x)$ by another function $g_n(x)$ which consists of a sum over $N$ basis functions weighted by some coefficients. The basis functions need to be orthogonal. $sin$ and $cos$ functions seem to be a pretty good choice because any signal can be filtered into several sinusoidal paths. In the period range of $[-T/2 ; T/2]$ the *Fourier series* is defined as:\n",
|
||||||
|
"\n",
|
||||||
|
"$$\n",
|
||||||
|
"f(t) \\approx g_n(t) = \\frac{1}{2} a_0 + \\sum_{k=1}^N \\left[ a_k \\cos \\left(\\frac{2\\pi k t}{T} \\right) + b_k \\sin\\left(\\frac{2\\pi k t}{T}\\right)\\right]\n",
|
||||||
|
"$$\n",
|
||||||
|
"\n",
|
||||||
|
"$$ \n",
|
||||||
|
"a_k = \\frac{2}{T} \\int_{-T/2}^{T/2} f(t) \\cos\\left(\\frac{2\\pi k t}{T}\\right)dt\n",
|
||||||
|
"$$\n",
|
||||||
|
"\n",
|
||||||
|
"$$\n",
|
||||||
|
"b_k = \\frac{2}{T} \\int_{-T/2}^{T/2} f(t) \\sin\\left(\\frac{2\\pi k t}{T}\\right)dt\n",
|
||||||
|
"$$"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"At this stage, we consider continuous, periodic and infinite functions. The more basis functions are used to approximate the unknown function, the better is the approximation, i.e. the more similar the unknown function is to its approximation. \n",
|
||||||
|
"\n",
|
||||||
|
"For a non-periodic function the interval of periodicity tends to infinity. That means, the steps between neighbouring frequencies become smaller and smaller and thus the infinite sum of the *Fourier series* turns into an integral and we end up with the integral form of the *Fourier transformation*:\n",
|
||||||
|
"\n",
|
||||||
|
"$$\n",
|
||||||
|
"F(\\omega) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} f(t) e^{-i\\omega t} dt \\leftrightarrow f(t) = \\int_{-\\infty}^{\\infty} F(\\omega)e^{i\\omega t}dt\n",
|
||||||
|
"$$\n",
|
||||||
|
"\n",
|
||||||
|
"Attention: sign and factor conventions can be different in the literature!"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "fragment"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"In seismology, we do not have continuous but discrete time signals. Therefore, we work with the discrete form of the *Fourier transformation*:\n",
|
||||||
|
"\n",
|
||||||
|
"$$\n",
|
||||||
|
"F_k = \\frac{1}{N} \\sum_{j=0}^{N-1} f_j e^{-2\\pi i k j /N} \\leftrightarrow f_k = \\sum_{j=0}^{N-1} F_j e^{2\\pi i k j /N}\n",
|
||||||
|
"$$\n",
|
||||||
|
"\n",
|
||||||
|
"Some intuitive gif animations on what the *Fourier transform* is doing, can be found [here](https://en.wikipedia.org/wiki/File:Fourier_series_and_transform.gif), [here](https://en.wikipedia.org/wiki/File:Fourier_series_square_wave_circles_animation.gif), and [here](https://en.wikipedia.org/wiki/File:Fourier_series_sawtooth_wave_circles_animation.gif).\n",
|
||||||
|
"Further and more detailed explanations on *Fourier series* and *Fourier transformations* can be found [here](https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/) and [here](www.fourier-series.com).\n",
|
||||||
|
"\n",
|
||||||
|
"---"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"### The Fourier series and its coefficients \n",
|
||||||
|
"\n",
|
||||||
|
"In the following two code cells, we first define a function which calculates the coefficients of the Fourier series for a given function. The function in the next cell does it the other way round: it is creating a function based on given coefficients and weighting factors."
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"code_folding": [
|
||||||
|
1
|
||||||
|
],
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 1: code by Giulio Ghirardo \n",
|
||||||
|
"def fourier_series_coeff(f, T, N):\n",
|
||||||
|
" \"\"\"Calculates the first 2*N+1 Fourier series coeff. of a periodic function.\n",
|
||||||
|
"\n",
|
||||||
|
" Given a periodic, function f(t) with period T, this function returns the\n",
|
||||||
|
" coefficients a0, {a1,a2,...},{b1,b2,...} such that:\n",
|
||||||
|
"\n",
|
||||||
|
" f(t) ~= a0/2+ sum_{k=1}^{N} ( a_k*cos(2*pi*k*t/T) + b_k*sin(2*pi*k*t/T) )\n",
|
||||||
|
" \n",
|
||||||
|
" Parameters\n",
|
||||||
|
" ----------\n",
|
||||||
|
" f : the periodic function, a callable like f(t)\n",
|
||||||
|
" T : the period of the function f, so that f(0)==f(T)\n",
|
||||||
|
" N_max : the function will return the first N_max + 1 Fourier coeff.\n",
|
||||||
|
"\n",
|
||||||
|
" Returns\n",
|
||||||
|
" -------\n",
|
||||||
|
" a0 : float\n",
|
||||||
|
" a,b : numpy float arrays describing respectively the cosine and sine coeff.\n",
|
||||||
|
" \"\"\"\n",
|
||||||
|
" # From Nyquist theorem we must use a sampling \n",
|
||||||
|
" # freq. larger than the maximum frequency you want to catch in the signal. \n",
|
||||||
|
" f_sample = 2 * N\n",
|
||||||
|
" \n",
|
||||||
|
" # We also need to use an integer sampling frequency, or the\n",
|
||||||
|
" # points will not be equispaced between 0 and 1. We then add +2 to f_sample.\n",
|
||||||
|
" t, dt = np.linspace(0, T, f_sample + 2, endpoint=False, retstep=True)\n",
|
||||||
|
" y = np.fft.rfft(f) / t.size\n",
|
||||||
|
" y *= 2\n",
|
||||||
|
" return y[0].real, y[1:-1].real[0:N], -y[1:-1].imag[0:N]"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"code_folding": [
|
||||||
|
1
|
||||||
|
],
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 2: code by Giulio Ghirardo \n",
|
||||||
|
"def series_real_coeff(a0, a, b, t, T):\n",
|
||||||
|
" \"\"\"calculates the Fourier series with period T at times t,\n",
|
||||||
|
" from the real coeff. a0,a,b\"\"\"\n",
|
||||||
|
" tmp = np.ones_like(t) * a0 / 2.\n",
|
||||||
|
" for k, (ak, bk) in enumerate(zip(a, b)):\n",
|
||||||
|
" tmp += ak * np.cos(2 * np.pi * (k + 1) * t / T) + bk * np.sin(\n",
|
||||||
|
" 2 * np.pi * (k + 1) * t / T)\n",
|
||||||
|
" return tmp"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"Now, we can create an arbitrary function, which we use to experiment with in the following example. \n",
|
||||||
|
"1) When you re-run cell 3 several times, do you always see the same function? Why? What does it tell you about the Fourier series? "
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"scrolled": false,
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "fragment"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 3: create periodic, discrete, finite signal\n",
|
||||||
|
"\n",
|
||||||
|
"# number of samples (intial value: 3000)\n",
|
||||||
|
"samp = 3000\n",
|
||||||
|
"# sample rate (initial value: 1)\n",
|
||||||
|
"dt = 1\n",
|
||||||
|
"# period\n",
|
||||||
|
"T = 1.0 / dt\n",
|
||||||
|
"length = samp * dt\n",
|
||||||
|
"# number of coefficients (initial value: 100)\n",
|
||||||
|
"N = 100\n",
|
||||||
|
"# weighting factors for coefficients (selected randomly)\n",
|
||||||
|
"a0 = np.random.rand(1)\n",
|
||||||
|
"a = np.random.randint(1, high=11, size=N)\n",
|
||||||
|
"b = np.random.randint(1, high=11, size=N)\n",
|
||||||
|
"\n",
|
||||||
|
"t = np.linspace(0, length, samp) # time axis\n",
|
||||||
|
"sig = series_real_coeff(a0, a, b, t, T)\n",
|
||||||
|
"\n",
|
||||||
|
"# plotting\n",
|
||||||
|
"plt.plot(t, sig, 'r', label='arbitrary, periodic, discrete, finite signal')\n",
|
||||||
|
"plt.ticklabel_format(axis='y', style='sci', scilimits=(-1,1))\n",
|
||||||
|
"plt.xlabel('time [sec]')\n",
|
||||||
|
"plt.ylabel('amplitude')\n",
|
||||||
|
"plt.legend()\n",
|
||||||
|
"plt.show()"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"Now, we can play with the signal and see what happens when we try to reconstruct it with a limited number of coefficients. \n",
|
||||||
|
"2) Run the cells 4 and 5. What do you observe? \n",
|
||||||
|
"3) Increase the number of coefficients $n$ step by step and re-run cells 4 and 5. What do you observe now? Can you explain? \n",
|
||||||
|
"4) In cell 5 uncomment the lines to make a plot which is not normalized (and comment the other two) and re-run the cell. What do you see now and can you explain it?"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "fragment"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 4: determine the first 'n' coefficients of the function using the code function of cell 1\n",
|
||||||
|
"T = 1 # period\n",
|
||||||
|
"n = 100 # number of coeffs to reconstruct\n",
|
||||||
|
"a0, a, b = fourier_series_coeff(sig, T, n)\n",
|
||||||
|
"a_ = a.astype(int)\n",
|
||||||
|
"b_ = b.astype(int)\n",
|
||||||
|
"print('coefficient a0 = ', int(a0))\n",
|
||||||
|
"print('array coefficients ak =', a_)\n",
|
||||||
|
"print('array coefficients bk =', b_)"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 5: reconstruct the function using the code in cell 2\n",
|
||||||
|
"g = series_real_coeff(a0, a, b, t, dt)\n",
|
||||||
|
"\n",
|
||||||
|
"# plotting\n",
|
||||||
|
"#plt.plot(t, sig, 'r', label='original signal') # NOT normalized \n",
|
||||||
|
"#plt.plot(t, g, 'g', label='reconstructed signal')\n",
|
||||||
|
"plt.plot(t, sig/max(sig), 'r', label='original signal') # normalized \n",
|
||||||
|
"plt.plot(t, g/max(g), 'g', label='reconstructed signal')\n",
|
||||||
|
"\n",
|
||||||
|
"plt.ticklabel_format(axis='y', style='sci', scilimits=(-1,1))\n",
|
||||||
|
"plt.xlabel('time [sec]')\n",
|
||||||
|
"plt.ylabel('amplitude')\n",
|
||||||
|
"plt.legend()\n",
|
||||||
|
"plt.show()"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"### Fourier series, convergence and Gibb's phenomenon\n",
|
||||||
|
"\n",
|
||||||
|
"As seen above the convergence of the *Fourier series* can be tricky due to the fact that we work with signals of finite length. To analyse this effect in a bit more detail, we define a square wave in cell 6 and try to reconstruct it in cell 7. \n",
|
||||||
|
"5) First, we use only 5 coefficients to reconstruct the wave. Describe what you see. \n",
|
||||||
|
"6) Increase the number of coefficients $n$ in cell 7 step by step and re-run the cell. What do you see now? Can you explain it?"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 6: define a square wave of 5 Hz\n",
|
||||||
|
"freq = 5.\n",
|
||||||
|
"npts = 3000\n",
|
||||||
|
"dt_ = 0.002\n",
|
||||||
|
"length = npts * dt_\n",
|
||||||
|
"t_ = np.linspace(0, length, npts, endpoint=False)\n",
|
||||||
|
"square = signal.square(2 * np.pi * freq * t_)\n",
|
||||||
|
"\n",
|
||||||
|
"plt.plot(t_, square)\n",
|
||||||
|
"plt.xlabel('time [sec]')\n",
|
||||||
|
"plt.ylabel('amplitude')\n",
|
||||||
|
"plt.xlim(0, 1.05)\n",
|
||||||
|
"plt.ylim(-1.2, 1.2)\n",
|
||||||
|
"plt.show()"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "fragment"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"You may replace the square function by something else. What about a sawtooth function?"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "fragment"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"signal.sawtooth?"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 7: reconstruct signal using convergence criterion\n",
|
||||||
|
"n = 150 # number of coefficients (initial: 5)\n",
|
||||||
|
"T_ = 1/freq # period of signal\n",
|
||||||
|
"\n",
|
||||||
|
"# determine coefficients\n",
|
||||||
|
"a0 = 0\n",
|
||||||
|
"a = []\n",
|
||||||
|
"b = []\n",
|
||||||
|
"for i in range(1,n):\n",
|
||||||
|
" if (i%2 != 0):\n",
|
||||||
|
" a_ = 4/(np.pi*i)\n",
|
||||||
|
" else:\n",
|
||||||
|
" a_ = 0\n",
|
||||||
|
" a.append(a_)\n",
|
||||||
|
" b_ = (2*np.pi*i)/T_\n",
|
||||||
|
" b.append(b_)\n",
|
||||||
|
"\n",
|
||||||
|
"# reconstruct signal\n",
|
||||||
|
"g = np.ones_like(t_) * a0\n",
|
||||||
|
"for k, (ak, bk) in enumerate(zip(a, b)):\n",
|
||||||
|
" g += ak * np.sin(bk*t_)\n",
|
||||||
|
"\n",
|
||||||
|
"# plotting\n",
|
||||||
|
"plt.plot(t_, square, 'r', label='original signal') \n",
|
||||||
|
"plt.plot(t_, g, 'g', label='Reihenentwicklung')\n",
|
||||||
|
"plt.ticklabel_format(axis='y', style='sci', scilimits=(-1,1))\n",
|
||||||
|
"plt.xlabel('time [sec]')\n",
|
||||||
|
"plt.ylabel('amplitude')\n",
|
||||||
|
"#plt.ylim(-1.1,1.1)\n",
|
||||||
|
"plt.legend()\n",
|
||||||
|
"plt.show()"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"### Fourier transformation\n",
|
||||||
|
"\n",
|
||||||
|
"Let us now do the Fourier transformation of the signal created in cell 3 and have a look on the amplitude spectra. In computer science the transformation is performed as fast Fourier transformation (FFT). \n",
|
||||||
|
"\n",
|
||||||
|
"7) Why do we need to taper the signal before we perform the FFT? \n",
|
||||||
|
"8) How do you interpret the plot of the amplitude spectra? \n",
|
||||||
|
"9) Which frequency contributes most to the final signal? "
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 8: FFT of signal\n",
|
||||||
|
"# number of sample points need to be the same as in cell 3\n",
|
||||||
|
"print('samp =',samp,' Need to be the same as in cell 3.')\n",
|
||||||
|
"# number of sample points need to be the same as in cell 3\n",
|
||||||
|
"print('T =',T,' Need to be the same as in cell 3.')\n",
|
||||||
|
"# percentage of taper applied to signal (initial: 0.1)\n",
|
||||||
|
"taper_percentage = 0.1\n",
|
||||||
|
"taper = cosine_taper(samp,taper_percentage)\n",
|
||||||
|
"\n",
|
||||||
|
"sig_ = square * taper\n",
|
||||||
|
"Fsig = np.fft.rfft(sig_, n=samp)\n",
|
||||||
|
"\n",
|
||||||
|
"# prepare plotting\n",
|
||||||
|
"xf = np.linspace(0.0, 1.0/(2.0*T), (samp//2)+1)\n",
|
||||||
|
"rcParams[\"figure.subplot.hspace\"] = (0.8)\n",
|
||||||
|
"rcParams[\"figure.figsize\"] = (15, 9)\n",
|
||||||
|
"rcParams[\"axes.labelsize\"] = (15)\n",
|
||||||
|
"rcParams[\"axes.titlesize\"] = (20)\n",
|
||||||
|
"rcParams[\"font.size\"] = (12)"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"scrolled": true,
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"#plotting\n",
|
||||||
|
"plt.subplot(311)\n",
|
||||||
|
"plt.title('Time Domain')\n",
|
||||||
|
"plt.plot(t, square, linewidth=1)\n",
|
||||||
|
"plt.xlabel('Time [s]')\n",
|
||||||
|
"plt.ylabel('Amplitude')\n",
|
||||||
|
"\n",
|
||||||
|
"plt.subplot(312)\n",
|
||||||
|
"plt.title('Frequency Domain')\n",
|
||||||
|
"plt.plot(xf, 2.0/npts * np.abs(Fsig))\n",
|
||||||
|
"#plt.xlim(0, 0.04) \n",
|
||||||
|
"plt.xlabel('Frequency [Hz]')\n",
|
||||||
|
"plt.ylabel('Amplitude')\n",
|
||||||
|
"plt.show()"
|
||||||
|
]
|
||||||
|
}
|
||||||
|
],
|
||||||
|
"metadata": {
|
||||||
|
"anaconda-cloud": {},
|
||||||
|
"celltoolbar": "Slideshow",
|
||||||
|
"kernelspec": {
|
||||||
|
"display_name": "Python 3",
|
||||||
|
"language": "python",
|
||||||
|
"name": "python3"
|
||||||
|
},
|
||||||
|
"language_info": {
|
||||||
|
"codemirror_mode": {
|
||||||
|
"name": "ipython",
|
||||||
|
"version": 3
|
||||||
|
},
|
||||||
|
"file_extension": ".py",
|
||||||
|
"mimetype": "text/x-python",
|
||||||
|
"name": "python",
|
||||||
|
"nbconvert_exporter": "python",
|
||||||
|
"pygments_lexer": "ipython3",
|
||||||
|
"version": "3.7.6"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"nbformat": 4,
|
||||||
|
"nbformat_minor": 4
|
||||||
|
}
|
||||||
601
02-FFT_and_Basic_Filtering/2-filter_basics.ipynb
Normal file
601
02-FFT_and_Basic_Filtering/2-filter_basics.ipynb
Normal file
@ -0,0 +1,601 @@
|
|||||||
|
{
|
||||||
|
"cells": [
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"<div style='background-image: url(\"../images/header.svg\") ; padding: 0px ; background-size: cover ; border-radius: 5px ; height: 250px'>\n",
|
||||||
|
" <div style=\"float: right ; margin: 50px ; padding: 20px ; background: rgba(255 , 255 , 255 , 0.7) ; width: 50% ; height: 150px\">\n",
|
||||||
|
" <div style=\"position: relative ; top: 50% ; transform: translatey(-50%)\">\n",
|
||||||
|
" <div style=\"font-size: xx-large ; font-weight: 900 ; color: rgba(0 , 0 , 0 , 0.8) ; line-height: 100%\">Signal Processing</div>\n",
|
||||||
|
" <div style=\"font-size: large ; padding-top: 20px ; color: rgba(0 , 0 , 0 , 0.5)\">Filtering Basics</div>\n",
|
||||||
|
" </div>\n",
|
||||||
|
" </div>\n",
|
||||||
|
"</div>\n",
|
||||||
|
"\n",
|
||||||
|
"Seismo-Live: http://seismo-live.org\n",
|
||||||
|
"\n",
|
||||||
|
"##### Authors:\n",
|
||||||
|
"* Stefanie Donner ([@stefdonner](https://github.com/stefdonner))\n",
|
||||||
|
"* Celine Hadziioannou ([@hadzii](https://github.com/hadzii))\n",
|
||||||
|
"* Ceri Nunn ([@cerinunn](https://github.com/cerinunn))\n",
|
||||||
|
"\n",
|
||||||
|
"\n",
|
||||||
|
"---"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "fragment"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 0 - Preparation: load packages, set some basic options \n",
|
||||||
|
"#%matplotlib inline\n",
|
||||||
|
"from obspy import *\n",
|
||||||
|
"from obspy.clients.fdsn import Client\n",
|
||||||
|
"import numpy as np\n",
|
||||||
|
"import matplotlib.pylab as plt\n",
|
||||||
|
"plt.style.use('ggplot')\n",
|
||||||
|
"plt.rcParams['figure.figsize'] = 15, 4\n",
|
||||||
|
"plt.rcParams['lines.linewidth'] = 0.5"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"# Basics in filtering"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"## The Filter\n",
|
||||||
|
"\n",
|
||||||
|
"In seismology, filters are used to correct for the instrument response, avoid aliasing effects, separate 'wanted' from 'unwanted' frequencies, identify harmonic signals, model a specific recording instrument, and much more ...\n",
|
||||||
|
"\n",
|
||||||
|
"There is no clear classification of filters. Roughly speaking, we can distinguish linear vs. non-linear, analog (circuits, resistors, conductors) vs. digital (logical components), and continuous vs. discrete filters. In seismology, we generally avoid non-linear filters, because their output contains frequencies which are not in the input signal. Analog filters can be continuous or discrete, while digital filters are always discrete. Discrete filters can be subdivided into infinite impulse response (IIR) filters, which are recursive and causal, and finite impulse response (FIR) filters, which are non-recursive and causal or acausal. We will explain more about these types of filters below.\n",
|
||||||
|
"Some filters have a special name, such as Butterworth, Chebyshev or Bessel filters, but they can also be integrated in the classification described above."
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"### Response Funtion\n",
|
||||||
|
"\n",
|
||||||
|
"A filter is characterised by its frequency response function $T(j\\omega)$ which relate the Fourier transformation of the output signal $Y(j\\omega)$ and the Fourier transformation of the input signal $X(j\\omega)$:\n",
|
||||||
|
"$$ Y(j\\omega) = T(j\\omega)X(j\\omega)$$\n",
|
||||||
|
"\n",
|
||||||
|
"For a simple lowpass filter it is given as:\n",
|
||||||
|
"$$ |T(j\\omega)| = \\sqrt{ \\frac{1}{1+(\\frac{\\omega}{\\omega_c})^{2n}} } $$\n",
|
||||||
|
"\n",
|
||||||
|
"with $\\omega$ indicating the frequency samples, $\\omega_c$ the corner frequency of the filter, and $n$ the order of the filter (also called the number of corners of the filter). For a lowpass filter, all frequencies lower than the corner frequency are allowed to pass the filter. This is the pass band of the filter. On the other hand, the range of frequencies above the corner frequency is called stop band. In between lies the transition band, a small band of frequencies in which the passed amplitudes are gradually decreased to zero. The steepness of the slope of this transition band is defined by the order of the filter: the higher the order, the steeper the slope, the more effectively 'unwanted' frequencies get removed."
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"### Time Domain vs. Frequency Domain\n",
|
||||||
|
"\n",
|
||||||
|
"In the time domain, filtering means to convolve the data with the impulse response function of the filter. Doing this operation in the time domain is mathematically complex, computationally expensive and slow. Therefore, the digital application of filters is almost always done in the frequency domain, where it simplifies to a much faster multiplication between data and filter response. The procedure is as follows: transfer the signal into the frequency domain via FFT, multiply it with the filter's frequency response function (i.e. the FFT of the impulse response function), and transfer the result back to the time-domain. As a consequence, when filtering, we have to be aware of the characteristics and pit-falls of the Fourier transformation."
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"## Filter types\n",
|
||||||
|
"\n",
|
||||||
|
"There are 4 main types of filters: a lowpass, a highpass, a bandpass, and a bandstop filter. Low- and highpass filters only have one corner frequency, allowing frequencies below and above this corner frequency to pass the filter, respectively. In contrast, bandpass and bandstop filters have two corner frequencies, defining a frequency band to pass and to stop, respectively.\n",
|
||||||
|
"Here, we want to see how exactly these filters act on the input signal. In Cell 1, the vertical component of the M$_w\\,$9.1 Tohoku earthquake, recorded at Bochum - Germany, is downloaded and pre-processed. In Cell 2, the four basic filters are applied to these data and plotted together with the filter functions and the resulting amplitude spectrum.\n",
|
||||||
|
"\n",
|
||||||
|
"1) Look at the figure and explain what the different filters do.\n",
|
||||||
|
"\n",
|
||||||
|
"2) Change the order of the filter (i.e the number of corners). What happens and why?"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 1: prepare data from Tohoku-Oki earthquake. \n",
|
||||||
|
"client = Client(\"BGR\")\n",
|
||||||
|
"t1 = UTCDateTime(\"2011-03-11T05:00:00.000\")\n",
|
||||||
|
"st = client.get_waveforms(\"GR\", \"BUG\", \"\", \"LHZ\", t1, t1 + 6 * 60 * 60, \n",
|
||||||
|
" attach_response = True)\n",
|
||||||
|
"st.remove_response(output=\"VEL\")\n",
|
||||||
|
"st.detrend('linear')\n",
|
||||||
|
"st.detrend('demean')\n",
|
||||||
|
"st[0].plot();"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "skip"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"print(st[0].stats)"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 2 - filter types\n",
|
||||||
|
"npts = st[0].stats.npts # number of samples in the trace\n",
|
||||||
|
"dt = st[0].stats.delta # sample interval\n",
|
||||||
|
"fNy = 1. / (2. * dt) # Nyquist frequency \n",
|
||||||
|
"time = np.arange(0, npts) * dt # time axis for plotting\n",
|
||||||
|
"freq = np.linspace(0, fNy, npts // 2 + 1) # frequency axis for plotting\n",
|
||||||
|
"corners = 4 # order of filter\n",
|
||||||
|
"\n",
|
||||||
|
"# several filter frequencies for the different filter types\n",
|
||||||
|
"f0 = 0.04\n",
|
||||||
|
"fmin1 = 0.04\n",
|
||||||
|
"fmax1 = 0.07\n",
|
||||||
|
"fmin2 = 0.03\n",
|
||||||
|
"fmax2 = 0.07\n",
|
||||||
|
"\n",
|
||||||
|
"# filter functions\n",
|
||||||
|
"LP = 1 / ( 1 + (freq / f0) ** (2 * corners))\n",
|
||||||
|
"HP = 1 - 1 / (1 + (freq / f0) ** (2 * corners))\n",
|
||||||
|
"wc = fmax1 - fmin1\n",
|
||||||
|
"wb = 0.5 * wc + fmin1\n",
|
||||||
|
"BP = 1/(1 + ((freq - wb) / wc) ** (2 * corners))\n",
|
||||||
|
"wc = fmax2 - fmin2\n",
|
||||||
|
"wb = 0.5 * wc + fmin2\n",
|
||||||
|
"BS = 1 - ( 1 / (1 + ((freq - wb) / wc) ** (2 * corners)))"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "fragment"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# filtered traces\n",
|
||||||
|
"stHP = st.copy()\n",
|
||||||
|
"stHP.filter('highpass', freq=f0, corners=corners, zerophase=True)\n",
|
||||||
|
"stLP = st.copy()\n",
|
||||||
|
"stLP.filter('lowpass', freq=f0, corners=corners, zerophase=True)\n",
|
||||||
|
"stBP = st.copy()\n",
|
||||||
|
"stBP.filter('bandpass', freqmin=fmin1, freqmax=fmax1, corners=corners, zerophase=True)\n",
|
||||||
|
"stBS = st.copy()\n",
|
||||||
|
"stBS.filter('bandstop', freqmin=fmin2, freqmax=fmax2, corners=corners, zerophase=True)\n",
|
||||||
|
"\n",
|
||||||
|
"# amplitude spectras\n",
|
||||||
|
"Ospec = np.fft.rfft(st[0].data)\n",
|
||||||
|
"LPspec = np.fft.rfft(stLP[0].data)\n",
|
||||||
|
"HPspec = np.fft.rfft(stHP[0].data)\n",
|
||||||
|
"BPspec = np.fft.rfft(stBP[0].data)\n",
|
||||||
|
"BSspec = np.fft.rfft(stBS[0].data)"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# ---------------------------------------------------------------\n",
|
||||||
|
"# plot\n",
|
||||||
|
"plt.rcParams['figure.figsize'] = 17, 17\n",
|
||||||
|
"tx1 = 3000\n",
|
||||||
|
"tx2 = 8000\n",
|
||||||
|
"fx2 = 0.12\n",
|
||||||
|
"\n",
|
||||||
|
"fig = plt.figure()\n",
|
||||||
|
"\n",
|
||||||
|
"ax1 = fig.add_subplot(5,3,1)\n",
|
||||||
|
"ax1.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n",
|
||||||
|
"plt.plot(time, st[0].data, 'k')\n",
|
||||||
|
"plt.xlim(tx1, tx2)\n",
|
||||||
|
"plt.title('time-domain data')\n",
|
||||||
|
"plt.ylabel('original data \\n amplitude [ms$^-1$]')\n",
|
||||||
|
"\n",
|
||||||
|
"ax3 = fig.add_subplot(5,3,3)\n",
|
||||||
|
"plt.plot(freq, abs(Ospec), 'k')\n",
|
||||||
|
"plt.title('frequency-domain data \\n amplitude spectrum')\n",
|
||||||
|
"plt.ylabel('amplitude')\n",
|
||||||
|
"plt.xlim(0,fx2)\n",
|
||||||
|
"\n",
|
||||||
|
"ax4 = fig.add_subplot(5,3,4)\n",
|
||||||
|
"ax4.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n",
|
||||||
|
"plt.plot(time, stLP[0].data, 'k')\n",
|
||||||
|
"plt.xlim(tx1, tx2)\n",
|
||||||
|
"plt.ylabel('LOWPASS \\n amplitude [ms$^-1$]')\n",
|
||||||
|
"\n",
|
||||||
|
"ax5 = fig.add_subplot(5,3,5)\n",
|
||||||
|
"plt.plot(freq, LP, 'k', linewidth=1.5)\n",
|
||||||
|
"plt.xlim(0,fx2)\n",
|
||||||
|
"plt.ylim(-0.1,1.1)\n",
|
||||||
|
"plt.title('filter function')\n",
|
||||||
|
"plt.ylabel('amplitude [%]')\n",
|
||||||
|
"\n",
|
||||||
|
"ax6 = fig.add_subplot(5,3,6)\n",
|
||||||
|
"plt.plot(freq, abs(LPspec), 'k')\n",
|
||||||
|
"plt.ylabel('amplitude ')\n",
|
||||||
|
"plt.xlim(0,fx2)\n",
|
||||||
|
"\n",
|
||||||
|
"ax7 = fig.add_subplot(5,3,7)\n",
|
||||||
|
"ax7.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n",
|
||||||
|
"plt.plot(time, stHP[0].data, 'k')\n",
|
||||||
|
"plt.xlim(tx1, tx2)\n",
|
||||||
|
"plt.ylabel('HIGHPASS \\n amplitude [ms$^-1$]')\n",
|
||||||
|
"\n",
|
||||||
|
"ax8 = fig.add_subplot(5,3,8)\n",
|
||||||
|
"plt.plot(freq, HP, 'k', linewidth=1.5)\n",
|
||||||
|
"plt.xlim(0,fx2)\n",
|
||||||
|
"plt.ylim(-0.1,1.1)\n",
|
||||||
|
"plt.ylabel('amplitude [%]')\n",
|
||||||
|
"\n",
|
||||||
|
"ax9 = fig.add_subplot(5,3,9)\n",
|
||||||
|
"plt.plot(freq, abs(HPspec), 'k')\n",
|
||||||
|
"plt.ylabel('amplitude ')\n",
|
||||||
|
"plt.xlim(0,fx2)\n",
|
||||||
|
"\n",
|
||||||
|
"ax10 = fig.add_subplot(5,3,10)\n",
|
||||||
|
"ax10.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n",
|
||||||
|
"plt.plot(time, stBP[0].data, 'k')\n",
|
||||||
|
"plt.xlim(tx1, tx2)\n",
|
||||||
|
"plt.ylabel('BANDPASS \\n amplitude [ms$^-1$]')\n",
|
||||||
|
"\n",
|
||||||
|
"ax11 = fig.add_subplot(5,3,11)\n",
|
||||||
|
"plt.plot(freq, BP, 'k', linewidth=1.5)\n",
|
||||||
|
"plt.xlim(0,fx2)\n",
|
||||||
|
"plt.ylim(-0.1,1.1)\n",
|
||||||
|
"plt.ylabel('amplitude [%]')\n",
|
||||||
|
"\n",
|
||||||
|
"ax12 = fig.add_subplot(5,3,12)\n",
|
||||||
|
"plt.plot(freq, abs(BPspec), 'k')\n",
|
||||||
|
"plt.ylabel('amplitude ')\n",
|
||||||
|
"plt.xlim(0,fx2)\n",
|
||||||
|
"\n",
|
||||||
|
"ax13 = fig.add_subplot(5,3,13)\n",
|
||||||
|
"ax13.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n",
|
||||||
|
"plt.plot(time, stBS[0].data, 'k')\n",
|
||||||
|
"plt.xlim(tx1, tx2)\n",
|
||||||
|
"plt.xlabel('time [sec]')\n",
|
||||||
|
"plt.ylabel('BANDSTOPP \\n amplitude [ms$^-1$]');\n",
|
||||||
|
"\n",
|
||||||
|
"ax14 = fig.add_subplot(5,3,14)\n",
|
||||||
|
"plt.plot(freq, BS, 'k', linewidth=1.5)\n",
|
||||||
|
"plt.xlim(0,fx2)\n",
|
||||||
|
"plt.ylim(-0.1,1.1)\n",
|
||||||
|
"plt.ylabel('amplitude [%]')\n",
|
||||||
|
"plt.xlabel('frequency [Hz]')\n",
|
||||||
|
"\n",
|
||||||
|
"ax15 = fig.add_subplot(5,3,15)\n",
|
||||||
|
"plt.plot(freq, abs(BSspec), 'k')\n",
|
||||||
|
"plt.xlabel('frequency [Hz]')\n",
|
||||||
|
"plt.ylabel('amplitude ')\n",
|
||||||
|
"plt.xlim(0,fx2)\n",
|
||||||
|
"\n",
|
||||||
|
"plt.subplots_adjust(wspace=0.3, hspace=0.4)\n",
|
||||||
|
"plt.show()"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"## Causal vs. acausal\n",
|
||||||
|
"\n",
|
||||||
|
"Filters can be causal or acausal. The output of a causal filter depends only on past and present input, while the output also depends on future input. Thus, an acausal filter is always symmetric and a causal one not. In this exercise, we want to see the effects of such filters on the signal. In Cell 3, the example seismogram of ObsPy is loaded and lowpass filtered several times with different filter order $n$ and causality.\n",
|
||||||
|
"\n",
|
||||||
|
"3) Explain the effects of the different filters. You can also play with the order of the filter (variable $ncorners$).\n",
|
||||||
|
"\n",
|
||||||
|
"4) Zoom into a small window around the first onset (change the variables $start$, $end$ and $amp$). Which filter would you use for which purpose?"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 3 - filter effects\n",
|
||||||
|
"stf = read() # load example seismogram\n",
|
||||||
|
"tr = stf[0] # select the first trace in the Stream object\n",
|
||||||
|
"tr.detrend('demean') # preprocess data\n",
|
||||||
|
"tr.detrend('linear')\n",
|
||||||
|
"tr.filter(\"highpass\", freq=2) # removing long-period noise\n",
|
||||||
|
"print(tr)\n",
|
||||||
|
"t = tr.times() # time vector for x axis\n",
|
||||||
|
"\n",
|
||||||
|
"f = 15.0 # frequency for filters (intial: 10 Hz)\n",
|
||||||
|
"start = 4 # start time to plot in sec (initial: 4)\n",
|
||||||
|
"end = 8 # end time to plot in sec (initial: 8)\n",
|
||||||
|
"amp = 1500 # amplitude range for plotting (initial: 1500) \n",
|
||||||
|
"ncorners = 4 # number of corners/order of the filter (initial: 4)\n",
|
||||||
|
"\n",
|
||||||
|
"tr_filt = tr.copy() # causal filter / not zero phase. Order = 2\n",
|
||||||
|
"tr_filt.filter('lowpass', freq=f, zerophase=False, corners=2)\n",
|
||||||
|
"tr_filt2 = tr.copy() # causal filter / not zero phase. Order = set by ncorners\n",
|
||||||
|
"tr_filt2.filter('lowpass', freq=f, zerophase=False, corners=ncorners)\n",
|
||||||
|
"tr_filt3 = tr.copy() # acausal filter / zero phase. Order = set by ncorners\n",
|
||||||
|
"tr_filt3.filter('lowpass', freq=f, zerophase=True, corners=ncorners)"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "skip"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# plot - comment single lines to better see the remaining ones\n",
|
||||||
|
"plt.rcParams['figure.figsize'] = 15, 4\n",
|
||||||
|
"plt.plot(t, tr.data, 'k', label='original', linewidth=1.)\n",
|
||||||
|
"plt.plot(t, tr_filt.data, 'b', label='causal, n=2', linewidth=1.2)\n",
|
||||||
|
"plt.plot(t, tr_filt2.data, 'r', label='causal, n=%s' % ncorners, linewidth=1.2)\n",
|
||||||
|
"plt.plot(t, tr_filt3.data, 'g', label='acausal, n=%s' % ncorners, linewidth=1.2)\n",
|
||||||
|
"\n",
|
||||||
|
"plt.xlabel('time [s]')\n",
|
||||||
|
"plt.xlim(start, end) \n",
|
||||||
|
"plt.ylim(-amp, amp)\n",
|
||||||
|
"plt.ylabel('amplitude [arbitrary]')\n",
|
||||||
|
"plt.legend(loc='lower right')\n",
|
||||||
|
"\n",
|
||||||
|
"plt.show();"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "slide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"### Frequency ranges - bandpass filter\n",
|
||||||
|
"\n",
|
||||||
|
"We will now look at an event which took place in Kazakhstan on July 8, 1989. We want to see how filtering helps to derive information from a signal. The signal has been recorded by a Chinese station and is bandpassed in several different frequency bands.\n",
|
||||||
|
"\n",
|
||||||
|
"5) What do you see in the different frequency bands?\n",
|
||||||
|
"\n",
|
||||||
|
"6) Play with the channel used for filtering in Cell 5. What do you not see? Can you guess what kind of event it is?"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 4 - get + preprocess data\n",
|
||||||
|
"c = Client(\"IRIS\")\n",
|
||||||
|
"tmp1 = UTCDateTime(\"1989-10-19T09:50:00.0\")\n",
|
||||||
|
"tmp2 = UTCDateTime(\"1989-10-19T10:20:00.0\")\n",
|
||||||
|
"dat = c.get_waveforms(\"CD\", \"WMQ\", \"\", \"BH*\", tmp1, tmp2, attach_response = True)\n",
|
||||||
|
"dat.detrend('linear')\n",
|
||||||
|
"dat.detrend('demean')\n",
|
||||||
|
"dat.remove_response(output=\"VEL\")\n",
|
||||||
|
"dat.detrend('linear')\n",
|
||||||
|
"dat.detrend('demean')\n",
|
||||||
|
"print(dat)\n",
|
||||||
|
"dat.plot();"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"# Cell 5 - filter data in different frequency ranges\n",
|
||||||
|
"\n",
|
||||||
|
"chanel = 2\n",
|
||||||
|
"tm = dat[chanel].times()\n",
|
||||||
|
"xmin = 0\n",
|
||||||
|
"xmax = 700\n",
|
||||||
|
"\n",
|
||||||
|
"dat1 = dat[chanel].copy()\n",
|
||||||
|
"dat2 = dat[chanel].copy()\n",
|
||||||
|
"dat2.filter(type=\"bandpass\", freqmin=0.01, freqmax=0.05)\n",
|
||||||
|
"dat3 = dat[chanel].copy()\n",
|
||||||
|
"dat3.filter(type=\"bandpass\", freqmin=0.05, freqmax=0.1)\n",
|
||||||
|
"dat4 = dat[chanel].copy()\n",
|
||||||
|
"dat4.filter(type=\"bandpass\", freqmin=0.1, freqmax=0.5)\n",
|
||||||
|
"dat5 = dat[chanel].copy()\n",
|
||||||
|
"dat5.filter(type=\"bandpass\", freqmin=0.5, freqmax=1)\n",
|
||||||
|
"dat6 = dat[chanel].copy()\n",
|
||||||
|
"dat6.filter(type=\"bandpass\", freqmin=1., freqmax=5.)\n",
|
||||||
|
"dat7 = dat[chanel].copy()\n",
|
||||||
|
"dat7.filter(type=\"bandpass\", freqmin=5., freqmax=9.99)"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "code",
|
||||||
|
"execution_count": null,
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "subslide"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"outputs": [],
|
||||||
|
"source": [
|
||||||
|
"plt.rcParams['figure.figsize'] = 17, 21\n",
|
||||||
|
"fig = plt.figure()\n",
|
||||||
|
"ax1 = fig.add_subplot(7,1,1)\n",
|
||||||
|
"ax1.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n",
|
||||||
|
"plt.plot(tm, dat1.data, 'k')\n",
|
||||||
|
"plt.xlim(xmin, xmax)\n",
|
||||||
|
"plt.title('unfiltered')\n",
|
||||||
|
"plt.ylabel('amplitude \\n [m/s]')\n",
|
||||||
|
"ax2 = fig.add_subplot(7,1,2)\n",
|
||||||
|
"ax2.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n",
|
||||||
|
"plt.plot(tm, dat2.data, 'k')\n",
|
||||||
|
"plt.xlim(xmin, xmax)\n",
|
||||||
|
"plt.title('0.01 - 0.05 Hz')\n",
|
||||||
|
"plt.ylabel('amplitude \\n [m/s]')\n",
|
||||||
|
"ax3 = fig.add_subplot(7,1,3)\n",
|
||||||
|
"ax3.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n",
|
||||||
|
"plt.plot(tm, dat3.data, 'k')\n",
|
||||||
|
"plt.xlim(xmin, xmax)\n",
|
||||||
|
"plt.title('0.05 - 0.1 Hz')\n",
|
||||||
|
"plt.ylabel('amplitude \\n [m/s]')\n",
|
||||||
|
"ax4 = fig.add_subplot(7,1,4)\n",
|
||||||
|
"ax4.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n",
|
||||||
|
"plt.plot(tm, dat4.data, 'k')\n",
|
||||||
|
"plt.xlim(xmin, xmax)\n",
|
||||||
|
"plt.title('0.1 - 0.5 Hz')\n",
|
||||||
|
"plt.ylabel('amplitude \\n [m/s]')\n",
|
||||||
|
"ax5 = fig.add_subplot(7,1,5)\n",
|
||||||
|
"ax5.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n",
|
||||||
|
"plt.plot(tm, dat5.data, 'k')\n",
|
||||||
|
"plt.xlim(xmin, xmax)\n",
|
||||||
|
"plt.title('0.5 - 1.0 Hz')\n",
|
||||||
|
"plt.ylabel('amplitude \\n [m/s]')\n",
|
||||||
|
"ax6 = fig.add_subplot(7,1,6)\n",
|
||||||
|
"ax6.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n",
|
||||||
|
"plt.plot(tm, dat6.data, 'k')\n",
|
||||||
|
"plt.xlim(xmin, xmax)\n",
|
||||||
|
"plt.title('1.0 - 5.0 Hz')\n",
|
||||||
|
"plt.ylabel('amplitude \\n [m/s]')\n",
|
||||||
|
"ax7 = fig.add_subplot(7,1,7)\n",
|
||||||
|
"ax7.ticklabel_format(style='sci', axis='y', scilimits=(-1,1))\n",
|
||||||
|
"plt.plot(tm, dat7.data, 'k')\n",
|
||||||
|
"plt.xlim(xmin, xmax)\n",
|
||||||
|
"plt.title('5.0 - 10.0 Hz')\n",
|
||||||
|
"plt.xlabel('time [sec]')\n",
|
||||||
|
"plt.ylabel('amplitude \\n [m/s]')\n",
|
||||||
|
"\n",
|
||||||
|
"plt.subplots_adjust(hspace=0.3)\n",
|
||||||
|
"plt.show();"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "skip"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"# Answers"
|
||||||
|
]
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"cell_type": "markdown",
|
||||||
|
"metadata": {
|
||||||
|
"slideshow": {
|
||||||
|
"slide_type": "skip"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"source": [
|
||||||
|
"1) The lowpass removes frequencies below the corner frequency of the filter, while a highpass removes all frequencies above the corner frequency. The bandpass allows all frequencies within the two corner frequencies, while a bandstop allows all frequencies outside this range.\n",
|
||||||
|
"In the filtered signals not only the frequency content is changed (see plot of spectrum) but also the maximum amplitudes (see scaling of y-axis in time-domain plot.) In the frequency-domain plot, we see that each frequency contributes with a distinct amount of amplitude to the final signal. Depending on which frequencies remain after filtering, we have altered the amplitudes as well as the frequencies of the filtered signal.\n",
|
||||||
|
"Be aware: \"high-\" and \"low-\"pass refers to the unit 'frequency'! When you think in terms of 'period', you have to switch your mind.\n",
|
||||||
|
"\n",
|
||||||
|
"2) When setting the order of the filter very high, the slopes of the filter functions turn into steps and the amplitude spectra stop abruptly. Due to the harsh steps in the filter functions, we risk Gibbs effects during filtering. However, when setting the order of the filter lower, the slopes become too gentle and the unwanted frequencies are not removed effectively. Thus, we always have to balance between minimizing the Gibbs effects (low order) and maximizing the effective removal of frequencies (high order).\n",
|
||||||
|
"\n",
|
||||||
|
"3) The difference between the two causal filters (red and blue) are minor, even when the order of the filter is increased. But the acausal filter (green) shows a phase shift relative to the original data. This phase shift increases drastically when increasing the order of the filter.\n",
|
||||||
|
"\n",
|
||||||
|
"4) Zooming into a time window from 4.5 s to 5 s and setting the maximum amplitude to 400, we see that the phase shift is also shifting the first onset. We would pick it incorrectly. Therefore, whenever the scientific task involves the correct time picking of phases, we have to work with causal filters. When only the frequency content is important, we can also work with acausal filters.\n",
|
||||||
|
"\n",
|
||||||
|
"5) In the first two bands (0.01-0.05 Hz) we see Love waves with different amplitudes between 200 and 400 seconds. At 0.1-0.5 Hz we see Rayleigh waves after around 350 seconds and some scatter in the earlier times. At 0.5-1 Hz it is nearly all scattered energy. At 1-5 Hz we see a clear P-wave at around 25 seconds and some later scatter. The last band between 5 and 10 Hz removes the scatter completely with a very nice P-wave coda remaining. We can even distinguish the P-wave of a much smaller event at around 390 seconds.\n",
|
||||||
|
"\n",
|
||||||
|
"6) There is no clear S-wave visible. Also in the horizontal components (channel 0 and 1) almost no S-wave energy is visible. This is a clear hint for an explosive type of event. Indeed, this is the recording of a nuclear test explosion. The recording with exactly these bandpass filters was used as cover figure for the book 'Quantitative Seismology' by Keiiti Aki and Paul G. Richards, 2nd edition."
|
||||||
|
]
|
||||||
|
}
|
||||||
|
],
|
||||||
|
"metadata": {
|
||||||
|
"celltoolbar": "Slideshow",
|
||||||
|
"kernelspec": {
|
||||||
|
"display_name": "Python 3",
|
||||||
|
"language": "python",
|
||||||
|
"name": "python3"
|
||||||
|
},
|
||||||
|
"language_info": {
|
||||||
|
"codemirror_mode": {
|
||||||
|
"name": "ipython",
|
||||||
|
"version": 3
|
||||||
|
},
|
||||||
|
"file_extension": ".py",
|
||||||
|
"mimetype": "text/x-python",
|
||||||
|
"name": "python",
|
||||||
|
"nbconvert_exporter": "python",
|
||||||
|
"pygments_lexer": "ipython3",
|
||||||
|
"version": "3.7.6"
|
||||||
|
}
|
||||||
|
},
|
||||||
|
"nbformat": 4,
|
||||||
|
"nbformat_minor": 4
|
||||||
|
}
|
||||||
@ -8,4 +8,8 @@ The content of this repository is licensed under Creative Commons Attribution 4.
|
|||||||
|
|
||||||
### 01 - Python Introduction
|
### 01 - Python Introduction
|
||||||
|
|
||||||
This has been forked from [Seismo Live](http://seismo-live.org). The source code is available at https://github.com/krischer/seismo_live (licensed under a ***CC BY-NC-SA 4.0 License***. © 2015-2019 Lion Krischer).
|
This has been forked from [Seismo Live](http://seismo-live.org). The source code is available at https://github.com/krischer/seismo_live (licensed under a ***CC BY-NC-SA 4.0 License***. © 2015-2019 Lion Krischer).
|
||||||
|
|
||||||
|
### 02 - Fourier-Transform and Basic Filtering
|
||||||
|
|
||||||
|
This has been forked from [Seismo Live](http://seismo-live.org). The source code is available at https://github.com/krischer/seismo_live (licensed under a ***CC BY-NC-SA 4.0 License***. © 2015-2019 Lion Krischer).
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user