new assignment 10
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assignment_10/geo2.1300.HHZ
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assignment_10/multipleFilterTechnique.ipynb
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assignment_10/multipleFilterTechnique.ipynb
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"cells": [
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Multiple filter technique"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"An alternative to the moving window method is the multiple filter technique. Instead of moving a window in the time domain, a window is moved in the frequency domain. Of course, a window in the frequency domain is a filter, and hence the name of the method. Let $H_k(\\omega,\\omega_k)$ be the transfer function of the $k$-th filter centered on frequency $\\omega_k$. Applying the filter involves multiplication of the filter transfer function with the Fourier transform of the time signal and then performing an inverse Fourier transform:\n",
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"\\begin{align}\n",
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"f_k(t) = \\int_{-\\infty}^\\infty\\,H_k(\\omega,\\omega_k)F(\\omega)\\exp(i\\omega t)\\frac{d\\omega}{2\\pi} \\,.\n",
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"\\end{align}\n",
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"Basically, one could plot the resulting time series against the associated center frequency of the filter. However, $f_k(t)$ can also assume negative values. It would be better to plot the envelope of $f_k(t)$.\n",
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"\n",
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"### The analytic signal\n",
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"This can be obtained by constructing the analytic signal $z_k(t)$ of $f_k(t)$. It is a complex\n",
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"signal whose real part is the original signal and whose imaginary part is its Hilbert transform. Its absolute\n",
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"value is the envelope or instantaneous amplitude and its phase is the instantaneous phase:\n",
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"\\begin{align}\n",
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"z_k(t) = a_k(t)\\exp(i\\Phi_k(t)) = f_k(t)+iq_k(t) \\ \\mathrm{with}\\ q_k(t) = HT\\{f_k(t)\\}\\,,\n",
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"\\end{align}\n",
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"and\n",
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"\\begin{align}\n",
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"a_k(t) = \\sqrt{f_k(t)^2+q_k(t)^2},\\ \\Phi_k(t) = \\arctan\\left(\\frac{q_k(t)}{f_k(t)}\\right) \\,.\n",
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"\\end{align}\n",
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"\n",
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"The Hilbert transform can again easily be obtained because its Fourier transform is:\n",
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"\\begin{align}\n",
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"Q_k(\\omega) = i\\,\\mathrm{sign}(\\omega)F_k(\\omega)\\,.\n",
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"\\end{align}\n",
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"where $F_k(\\omega)$ is the Fourier transform of $f_k(t)$. Inverse Fourier transform of $Q_k(\\omega)$ gives the Hilbert transform of $f_k(t)$.\n",
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"\n",
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"### Work flow\n",
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"So the work flow of the multiple filter technique is as follows:\n",
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"1. Compute Fourier transform of $f(t)$: $F(\\omega)$.\n",
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"2. Multiply $F(\\omega)$ with bandpass filters $H_k(\\omega,\\omega_k)$ to obtain $F_k(\\omega)$.\n",
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"3. Compute Fourier transform of the Hilbert transform: $Q_k(\\omega)=iF_k(\\omega)$.\n",
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"4. Transform both back to the time domain.\n",
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"5. Calculate the instantaneous amplitude $a_k(t)$ and phase $\\Phi_k(t)$.\n",
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"6. Plot $a_k(t)$ in the time-frequency plane.\n",
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"\n",
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"### Gaussian filters\n",
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"One frequently made choice for the bandpass filters is a Gaussian function of the form:\n",
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"\\begin{align}\n",
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"H_k(\\omega,\\omega_k) = e^{-\\alpha\\left(\\frac{\\omega-\\omega_k}{\\omega_k}\\right)^2}\n",
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"\\end{align}\n",
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"with inverse Fourier transform (impulse response):\n",
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"\\begin{align}\n",
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"h_k(t) = \\frac{\\sqrt{\\pi}\\omega_k}{2\\alpha}e^{-\\frac{\\omega_k^2t^2}{4\\alpha}}\\cos\\omega_k t \\,.\n",
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"\\end{align}\n",
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"This Fourier transform pair demonstrates a fundamental property of spectral analysis:\n",
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"Choosing $\\alpha$ large gives a very narrow-banded filter but a very\n",
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"long impulse response. Choosing $\\alpha$ small gives a wide-banded filter but a very short \n",
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"impulse response. This is an expression of the uncertainty principle. If the frequency of a signal\n",
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"is known very well, the signal is distributed over time. On the other hand, an impulse in the time domain has\n",
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"a constant spectrum, i.e. the frequency is entirely unknown. In quantum mechanics, there is the energy-time\n",
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"uncertainty relation, where energy $E=\\hbar\\omega$.\n",
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"\n",
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"### Time and frequency resolution\n",
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"The Gaussian Fourier transform pair has a special property. If we measure the duration of the impulse response by\n",
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"\\begin{align}\n",
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"D_t^2 = \\int_{-\\infty}^\\infty t^2 |h(t)|^2 dt\n",
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"\\end{align}\n",
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"and the spread of the filter transfer function by\n",
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"\\begin{align}\n",
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"D_\\omega^2 = \\int_{-\\infty}^\\infty \\omega^2 |H(\\omega)|^2 \\frac{d\\omega}{2\\pi}\\,,\n",
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"\\end{align}\n",
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"then the product\n",
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"\\begin{align}\n",
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"D_t D_\\omega \\ge \\frac{1}{2} \\,.\n",
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"\\end{align}\n",
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"For the Gaussian filter, equality holds. Hence, the Gaussian filter is the one with the\n",
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"best compromise between bandwidth and duration."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"scrolled": true
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},
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"outputs": [],
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"source": [
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"# Preparation: load packages\n",
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"%matplotlib inline\n",
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"from obspy.core import read\n",
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"from obspy.core import UTCDateTime\n",
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"import numpy as np\n",
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"import matplotlib.pylab as plt\n",
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"from setupFigure import SetupFigure"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"scrolled": true
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},
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"outputs": [],
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"source": [
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"def multipleFilterAnalysis(data, alfa, cfreq, dt, ndec):\n",
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" \"\"\"\n",
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" Perform a multiple filter analysis of data.\n",
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" :param data: Array of detrended and demeaned data \n",
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" :param alfa: Width parameter of Gaussian bandpass filter\n",
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" :param cfreq: Array of center frequencies of Gaussian filter\n",
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" :param dt: sampling interval of data\n",
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" :param ndec: decimation factor for instantaneous amplitude output\n",
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" \"\"\"\n",
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" nd = len(data)\n",
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" ftd = np.fft.rfft(data,nd) # Fourier transform of entire data set (pos. freq.)\n",
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" freq = np.fft.rfftfreq(nd,dt) # Fourier frequencies (positive frequencies)\n",
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" jf = 0 # center frequency counter\n",
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" mfa = np.zeros((len(cfreq),nd//ndec)) # numpy array for MFA result\n",
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" for cf in cfreq:\n",
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" hg = np.exp(-alfa*((freq-cf)/cf)**2) # Gaussian filter (use f instead of omega here)\n",
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" fk = hg*ftd # multiply FT of data with filter\n",
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" qk = 1j*fk # FT of Hilbert transform of filtered data\n",
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" ftk = np.fft.irfft(fk) # filtered data\n",
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" qtk = np.fft.irfft(qk) # Hilbert transform of filtered data\n",
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" at = np.sqrt(ftk**2+qtk**2) # instantaneous amplitude\n",
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" mfa[jf,:] = at[0:nd:ndec] # store decimated original result\n",
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" jf = jf+1 # increase center frequency count\n",
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" return mfa"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"def normalizeMFT(mfa, mode = 'full', exp = 0.5):\n",
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" \"\"\"\n",
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" Normalize the result of the mutiple filtering operation.\n",
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" :param mfa: array with envelopes versus frequency (mfa(f,t))\n",
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" :param mode: normalization mode: \n",
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" if 'time': normalize timewise\n",
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" if 'freq': normalize frequencywise\n",
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" else: normalize to absolute maximum of entire array\n",
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" :param exp: exponent for modifying envelope using a power\n",
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" \"\"\"\n",
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" nf, nt = mfa.shape\n",
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" if mode == 'freq':\n",
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" for jf in range(0,nf):\n",
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" mfamax = np.amax(mfa[jf,:])\n",
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" mfa[jf,:] = np.power(mfa[jf,:]/mfamax+1.e-10, exp) \n",
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" elif mode == 'time':\n",
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" for jt in range(0,nt):\n",
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" mfamax = np.amax(mfa[:,jt])\n",
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" mfa[:,jt] = np.power(mfa[:,jt]/mfamax+1.e-10, exp)\n",
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" else: \n",
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" mfamax = np.amax(mfa)\n",
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" mfa = np.power(mfa/mfamax+1.e-10, exp)\n",
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" # mfa = np.log10(mfa/mfamax+1.e-10)\n",
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" return mfa"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"record = '1400' # either use record starting at 1300 or 1400\n",
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"datapath = 'geo2.' + record+ '.HHZ'\n",
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"\n",
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"if record =='1300':\n",
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" stime = UTCDateTime('2013-01-23 13:16:00Z') # use record starting at 1300\n",
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" etime = UTCDateTime('2013-01-23 13:25:00Z')\n",
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" ttitle = ', depth: 36.5 m'\n",
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"else:\n",
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" stime = UTCDateTime('2013-01-23 14:14:00Z') # use record starting at 1400 (14:14-14:23)\n",
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" etime = UTCDateTime('2013-01-23 14:23:00Z')\n",
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" ttitle = ', depth: 56.5 m'\n",
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" \n",
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"st = read(datapath) # read file using obspy read\n",
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"\n",
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"st.trim(stime,etime) # trim data stream to desired start and end time\n",
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"st.detrend('linear') # do a linear detrending\n",
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"st.detrend('demean') # subtract mean value"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"scrolled": true
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},
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"outputs": [],
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"source": [
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"tr = st[0] # extract first trace from stream (there is only one)\n",
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"tr.plot() # plot trace"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# prepare multiple filtering\n",
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"alfa = 10. # filter width parameter\n",
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"nd = tr.stats.npts # number of samples in time series\n",
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"dt = tr.stats.delta # sampling interval\n",
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"fmax = 1./(2.*dt) # Nyquist frequency\n",
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"cfreq = np.linspace(1., fmax, 1000) # array of filter center frequencies\n",
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"freq = np.fft.rfftfreq(nd, dt) # Fourier frequencies"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"scrolled": true
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},
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"outputs": [],
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"source": [
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"ndec = 100 # decimation factor for envelope\n",
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"mfa = multipleFilterAnalysis(tr.data, alfa, cfreq, dt, ndec)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"mfa = normalizeMFT(mfa, mode='full', exp = 0.5)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# plot the result\n",
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"extent = (0., nd*dt, fmax, 1.) # extent of matrix in true time and frequency\n",
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"fig1, ax1 = SetupFigure(15, 6, \"Time [s]\", \"Center frequency [Hz]\", \"Multiple filter analysis\", 14)\n",
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"ax1.imshow(mfa, extent = extent, aspect=1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# plot Gaussian filter\n",
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"cf = 30\n",
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"fig2, ax2 = SetupFigure(15, 6, \"Center frequency [Hz]\", \"Amplitude\", \"Gaussian filter\", 14)\n",
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"arg = (freq-cf)/cf\n",
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"hg = np.exp(-alfa*((freq-cf)/cf)**2) # Gaussian filter (use f instead of omega here)\n",
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"ax2.plot(freq, hg)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# plot filter response\n",
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"fig3, ax3 = SetupFigure(15, 6, \"Time [s]\", \"Amplitude\", \"Filter response\", 14)\n",
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"wk = cf*2.*np.pi\n",
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"ts = np.linspace(-2.5,2.5,5000)\n",
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"hres = wk/np.sqrt(np.pi*alfa)*np.exp(-(0.25*(wk*ts)**2/alfa))*np.cos(wk*ts) \n",
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"ax3.plot(ts, hres)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Tasks"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"1. Study the effect of $\\alpha$ on the result of the multiple filtering and on the form of the filter impulse response.\n",
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"2. Up to which value of $\\alpha$ do you need to go to achieve a frequency resolution comparable to the moving window method\n",
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"3. Think about the numerical effort of MFT in comparison with the moving window technique.\n",
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"4. Try different scalings of the MFT result using different powers and also the logarithm.\n",
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"5. Try timewise and frequenc-wise normalization.\n",
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"6. Do you have some ideas why the results look different? What role play scaling and normalization? \n",
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"7. Plot the Gaussian filter transfer function for different values of $\\alpha$.\n",
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"8. Plot the filter responses for different values of $\\alpha$."
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.12.4"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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15
assignment_10/setupFigure.py
Normal file
15
assignment_10/setupFigure.py
Normal file
@ -0,0 +1,15 @@
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|
import matplotlib.pyplot as plt
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|
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|
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|
def SetupFigure(wx, wy, xlabel, ylabel, title, tls):
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|
"""
|
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|
Create a new figure frame
|
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|
"""
|
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|
fig = plt.figure(figsize=(wx,wy))
|
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|
ax = fig.add_subplot(1,1,1)
|
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|
ax.set_title(title, fontsize=tls+2)
|
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|
ax.grid(which='major', axis='both', ls=':')
|
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|
ax.set_ylabel(ylabel, fontsize=tls)
|
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|
ax.set_xlabel(xlabel, fontsize=tls)
|
||||||
|
ax.tick_params(labelsize=tls)
|
||||||
|
return fig, ax
|
||||||
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