Version 1.41, 2016 July 18

Welcome! This IPython notebook (or associated python script LOSC_Event_tutorial.py ) will go through some typical signal processing tasks on strain time-series data associated with the LIGO Event data releases from the LIGO Open Science Center (LOSC):

- Find events at https://losc.ligo.org/events/.
- View the tutorial as a web page, for GW150914.
- Run this tutorial with Binder using the link on the tutorials page.
- If you are running this tutorial on your own computer, see the Download section below.
- This notebook works with nbformat version 4. If you are running version 3, pick it up from the tutorials page.
- After setting the desired "eventname" below, you can just run the full notebook.

Questions, comments, suggestions, corrections, etc: email losc@ligo.caltech.edu

This tutorial assumes that you are comfortable with python.

This tutorial also assumes that you know a bit about signal processing of digital time series data (or want to learn!). This includes power spectral densities, spectrograms, digital filtering, whitening, audio manipulation. This is a vast and complex set of topics, but we will cover many of the basics in this tutorial.

If you are a beginner, here are some resources from the web:

- http://101science.com/dsp.htm
- https://www.coursera.org/course/dsp
- https://georgemdallas.wordpress.com/2014/05/14/wavelets-4-dummies-signal-processing-fourier-transforms-and-heisenberg/
- https://en.wikipedia.org/wiki/Signal_processing
- https://en.wikipedia.org/wiki/Spectral_density
- https://en.wikipedia.org/wiki/Spectrogram
- http://greenteapress.com/thinkdsp/
- https://en.wikipedia.org/wiki/Digital_filter

And, well, lots more on the web!

If you are using a pre-configured setup (eg, in binder), great! You don't have to download or set up anything.

Otherwise, to begin, get the necessary files, by downloading the zip file and unpacking it into single directory:

This zip file contains:

- this IPython notebook LOSC_Event_tutorial.ipynb, and LOSC_Event_tutorial.py code.
- python code for reading LOSC data files: readligo.py.
- the event data files (32s sampled at 4096 Hz, in hdf5 format, for both LIGO detectors).
- Waveform templates (32s sampled at 4096 Hz, in hdf5 format, for both plus and cross polarizations).
- a parameter file in json format, O1_events.json .

You will also need a python installation with a few packages (numpy, matplotlib, scipy, h5py, json ).

- For hints on software installation, see https://losc.ligo.org/tutorial00/
- The tutorial should work on python 2.6 and above, including python 3, as well as in recent versions of Ipython.
- You
*might*see Warning or FutureWarning messages, which tend to be associated with different versions of python, Ipython, numpy, etc. Hopefully they can be ignored! - the filetype "hdf5" means the data are in hdf5 format: https://www.hdfgroup.org/HDF5/
- NOTE: GPS time is number of seconds since Jan 6, 1980 GMT. See http://www.oc.nps.edu/oc2902w/gps/timsys.html or https://losc.ligo.org/gps/

In [1]:

```
#-- SET ME Tutorial should work with most binary black hole events
#-- Default is no event selection; you MUST select one to proceed.
# eventname = ''
# eventname = 'GW150914'
eventname = 'GW151226'
# eventname = 'LVT151012'
# want plots?
make_plots = 1
plottype = "png"
#plottype = "pdf"
```

In [2]:

```
# Standard python numerical analysis imports:
import numpy as np
from scipy import signal
from scipy.interpolate import interp1d
from scipy.signal import butter, filtfilt, iirdesign, zpk2tf, freqz
import h5py
import json
# the IPython magic below must be commented out in the .py file, since it doesn't work there.
%matplotlib inline
%config InlineBackend.figure_format = 'retina'
import matplotlib.pyplot as plt
import matplotlib.mlab as mlab
# LIGO-specific readligo.py
import readligo as rl
# you might get a matplotlib warning here; you can ignore it.
```

In [3]:

```
# Read the event properties from a local json file
fnjson = "O1_events.json"
try:
events = json.load(open(fnjson,"r"))
except IOError:
print("Cannot find resource file "+fnjson)
print("You can download it from https://losc.ligo.org/s/events/"+fnjson)
print("Quitting.")
quit()
# did the user select the eventname ?
try:
events[eventname]
except:
print('You must select an eventname that is in '+fnjson+'! Quitting.')
quit()
```

In [4]:

```
# Extract the parameters for the desired event:
event = events[eventname]
fn_H1 = event['fn_H1'] # File name for H1 data
fn_L1 = event['fn_L1'] # File name for L1 data
fn_template = event['fn_template'] # File name for template waveform
fs = event['fs'] # Set sampling rate
tevent = event['tevent'] # Set approximate event GPS time
fband = event['fband'] # frequency band for bandpassing signal
print("Reading in parameters for event " + event["name"])
print(event)
```

We will make use of the data, and waveform template, defined above.

In [5]:

```
#----------------------------------------------------------------
# Load LIGO data from a single file.
# FIRST, define the filenames fn_H1 and fn_L1, above.
#----------------------------------------------------------------
try:
# read in data from H1 and L1, if available:
strain_H1, time_H1, chan_dict_H1 = rl.loaddata(fn_H1, 'H1')
strain_L1, time_L1, chan_dict_L1 = rl.loaddata(fn_L1, 'L1')
except:
print("Cannot find data files!")
print("You can download them from https://losc.ligo.org/s/events/"+eventname)
print("Quitting.")
quit()
```

**NOTE** that in general, LIGO strain time series data has gaps (filled with NaNs) when the detectors are not taking valid ("science quality") data. Analyzing these data requires the user to
loop over "segments" of valid data stretches.

**In this tutorial, for simplicity, we assume there are no data gaps - this will not work for all times!** See the
notes on segments for details.

In [6]:

```
# both H1 and L1 will have the same time vector, so:
time = time_H1
# the time sample interval (uniformly sampled!)
dt = time[1] - time[0]
# Let's look at the data and print out some stuff:
print('time_H1: len, min, mean, max = ', \
len(time_H1), time_H1.min(), time_H1.mean(), time_H1.max() )
print('strain_H1: len, min, mean, max = ', \
len(strain_H1), strain_H1.min(),strain_H1.mean(),strain_H1.max())
print( 'strain_L1: len, min, mean, max = ', \
len(strain_L1), strain_L1.min(),strain_L1.mean(),strain_L1.max())
#What's in chan_dict? (See also https://losc.ligo.org/tutorials/)
bits = chan_dict_H1['DATA']
print("For H1, {0} out of {1} seconds contain usable DATA".format(bits.sum(), len(bits)))
bits = chan_dict_L1['DATA']
print("For L1, {0} out of {1} seconds contain usable DATA".format(bits.sum(), len(bits)))
```

In [7]:

```
# plot +- deltat seconds around the event:
# index into the strain time series for this time interval:
deltat = 5
indxt = np.where((time >= tevent-deltat) & (time < tevent+deltat))
print(tevent)
if make_plots:
plt.figure()
plt.plot(time[indxt]-tevent,strain_H1[indxt],'r',label='H1 strain')
plt.plot(time[indxt]-tevent,strain_L1[indxt],'g',label='L1 strain')
plt.xlabel('time (s) since '+str(tevent))
plt.ylabel('strain')
plt.legend(loc='lower right')
plt.title('Advanced LIGO strain data near '+eventname)
plt.savefig(eventname+'_strain.'+plottype)
```

**low frequency noise**; there is no way to see a signal here, without some signal processing.

Plotting these data in the Fourier domain gives us an idea of the frequency content of the data. A way to visualize the frequency content of the data is to plot the amplitude spectral density, ASD.

The ASDs are the square root of the power spectral densities (PSDs), which are averages of the square of the fast fourier transforms (FFTs) of the data.

They are an estimate of the "strain-equivalent noise" of the detectors versus frequency, which limit the ability of the detectors to identify GW signals.

They are in units of strain/rt(Hz). So, if you want to know the root-mean-square (rms) strain noise in a frequency band, integrate (sum) the squares of the ASD over that band, then take the square-root.

There's a signal in these data! For the moment, let's ignore that, and assume it's all noise.

In [8]:

```
make_psds = 1
if make_psds:
# number of sample for the fast fourier transform:
NFFT = 4*fs
Pxx_H1, freqs = mlab.psd(strain_H1, Fs = fs, NFFT = NFFT)
Pxx_L1, freqs = mlab.psd(strain_L1, Fs = fs, NFFT = NFFT)
# We will use interpolations of the ASDs computed above for whitening:
psd_H1 = interp1d(freqs, Pxx_H1)
psd_L1 = interp1d(freqs, Pxx_L1)
# Here is an approximate, smoothed PSD for H1 during O1, with no lines. We'll use it later.
Pxx = (1.e-22*(18./(0.1+freqs))**2)**2+0.7e-23**2+((freqs/2000.)*4.e-23)**2
psd_smooth = interp1d(freqs, Pxx)
if make_plots:
# plot the ASDs, with the template overlaid:
f_min = 20.
f_max = 2000.
plt.figure(figsize=(10,8))
plt.loglog(freqs, np.sqrt(Pxx_L1),'g',label='L1 strain')
plt.loglog(freqs, np.sqrt(Pxx_H1),'r',label='H1 strain')
plt.loglog(freqs, np.sqrt(Pxx),'k',label='H1 strain, O1 smooth model')
plt.axis([f_min, f_max, 1e-24, 1e-19])
plt.grid('on')
plt.ylabel('ASD (strain/rtHz)')
plt.xlabel('Freq (Hz)')
plt.legend(loc='upper center')
plt.title('Advanced LIGO strain data near '+eventname)
plt.savefig(eventname+'_ASDs.'+plottype)
```

NOTE that we only plot the data between f_min = 20 Hz and f_max = 2000 Hz.

Below f_min, the data **are not properly calibrated**. That's OK, because the noise is so high below f_min that LIGO cannot sense gravitational wave strain from astrophysical sources in that band.

The sample rate is fs = 4096 Hz (2^12 Hz), so the data cannot capture frequency content above the Nyquist frequency = fs/2 = 2048 Hz. That's OK, because our events only have detectable frequency content in the range given by fband, defined above; the upper end will (almost) always be below the Nyquist frequency. We set f_max = 2000, a bit below Nyquist.

You can see strong spectral lines in the data; they are all of instrumental origin. Some are engineered into the detectors (mirror suspension resonances at ~500 Hz and harmonics, calibration lines, control dither lines, etc) and some (60 Hz and harmonics) are unwanted. We'll return to these, later.

You can't see the signal in this plot, since it is relatively weak and less than a second long, while this plot averages over 32 seconds of data. So this plot is entirely dominated by instrumental noise.

The smooth model is hard-coded and tuned by eye; it won't be right for arbitrary times. We will only use it below for things that don't require much accuracy.

A standard metric that LIGO uses to evaluate the sensitivity of our detectors, based on the detector noise ASD, is the BNS range.

This is defined as the distance to which a LIGO detector can register a BNS signal with a signal-to-noise ratio (SNR) of 8, averaged over source direction and orientation.

We take each neutron star in the BNS system to have a mass of 1.4 times the mass of the sun, and negligible spin.

We have found from experience (supported by careful calculations) that detector noise produces false signals with SNRs that fall off exponentially; only above an SNR of around 8 is it possible to identify a signal above the detector noise.

GWs from BNS mergers are like "standard sirens"; we know their amplitude at the source from theoretical calculations. The amplitude falls off like 1/r, so their amplitude at the detectors on Earth tells us how far away they are. This is great, because it is hard, in general, to know the distance to astronomical sources.

The amplitude at the source is computed in the post-Newtonian "quadrupole approximation". This is valid for the inspiral phase only, and is approximate at best; there is no simple expression for the post-inspiral (merger and ringdown) phase. So this won't work for high-mass binary black holes like GW150914, which have a lot of signal strength in the post-inspiral phase.

But, in order to use them as standard sirens, we need to know the source direction and orientation relative to the detector and its "quadrupole antenna pattern" response to such signals. It is a standard (if non-trivial) computation to average over all source directions and orientations; the average amplitude is 1./2.2648 times the maximum value.

This calculation is described in Appendix D of: FINDCHIRP: An algorithm for detection of gravitational waves from inspiraling compact binaries B. Allen et al., PHYSICAL REVIEW D 85, 122006 (2012) ; http://arxiv.org/abs/gr-qc/0509116

In [9]:

```
BNS_range = 1
if BNS_range:
#-- compute the binary neutron star (BNS) detectability range
#-- choose a detector noise power spectrum:
f = freqs.copy()
# get frequency step size
df = f[2]-f[1]
#-- constants
# speed of light:
clight = 2.99792458e8 # m/s
# Newton's gravitational constant
G = 6.67259e-11 # m^3/kg/s^2
# one parsec, popular unit of astronomical distance (around 3.26 light years)
parsec = 3.08568025e16 # m
# solar mass
MSol = 1.989e30 # kg
# solar mass in seconds (isn't relativity fun?):
tSol = MSol*G/np.power(clight,3) # s
# Single-detector SNR for detection above noise background:
SNRdet = 8.
# conversion from maximum range (horizon) to average range:
Favg = 2.2648
# mass of a typical neutron star, in solar masses:
mNS = 1.4
# Masses in solar masses
m1 = m2 = mNS
mtot = m1+m2 # the total mass
eta = (m1*m2)/mtot**2 # the symmetric mass ratio
mchirp = mtot*eta**(3./5.) # the chirp mass (FINDCHIRP, following Eqn 3.1b)
# distance to a fiducial BNS source:
dist = 1.0 # in Mpc
Dist = dist * 1.0e6 * parsec /clight # from Mpc to seconds
# We integrate the signal up to the frequency of the "Innermost stable circular orbit (ISCO)"
R_isco = 6. # Orbital separation at ISCO, in geometric units. 6M for PN ISCO; 2.8M for EOB
# frequency at ISCO (end the chirp here; the merger and ringdown follow)
f_isco = 1./(np.power(R_isco,1.5)*np.pi*tSol*mtot)
# minimum frequency (below which, detector noise is too high to register any signal):
f_min = 20. # Hz
# select the range of frequencies between f_min and fisco
fr = np.nonzero(np.logical_and(f > f_min , f < f_isco))
# get the frequency and spectrum in that range:
ffr = f[fr]
# In stationary phase approx, this is htilde(f):
# See FINDCHIRP Eqns 3.4, or 8.4-8.5
htilde = (2.*tSol/Dist)*np.power(mchirp,5./6.)*np.sqrt(5./96./np.pi)*(np.pi*tSol)
htilde *= np.power(np.pi*tSol*ffr,-7./6.)
htilda2 = htilde**2
# loop over the detectors
dets = ['H1', 'L1']
for det in dets:
if det is 'L1': sspec = Pxx_L1.copy()
else: sspec = Pxx_H1.copy()
sspecfr = sspec[fr]
# compute "inspiral horizon distance" for optimally oriented binary; FINDCHIRP Eqn D2:
D_BNS = np.sqrt(4.*np.sum(htilda2/sspecfr)*df)/SNRdet
# and the "inspiral range", averaged over source direction and orientation:
R_BNS = D_BNS/Favg
print(det+' BNS inspiral horizon = {0:.1f} Mpc, BNS inspiral range = {1:.1f} Mpc'.format(D_BNS,R_BNS))
```

NOTE that, since mass is the source of gravity and thus also of gravitational waves, systems with higher masses (such as the binary black hole merger GW150914) are much "louder" and can be detected to much higher distances than the BNS range. We'll compute the BBH range, using a template with specific masses, below.

From the ASD above, we can see that the data are very strongly "colored" - noise fluctuations are much larger at low and high frequencies and near spectral lines, reaching a roughly flat ("white") minimum in the band around 80 to 300 Hz.

We can "whiten" the data (dividing it by the noise amplitude spectrum, in the fourier domain), suppressing the extra noise at low frequencies and at the spectral lines, to better see the weak signals in the most sensitive band.

Whitening is always one of the first steps in astrophysical data analysis (searches, parameter estimation). Whitening requires no prior knowledge of spectral lines, etc; only the data are needed.

To get rid of remaining high frequency noise, we will also bandpass the data.

The resulting time series is no longer in units of strain; now in units of "sigmas" away from the mean.

We will plot the whitened strain data, along with the signal template, after the matched filtering section, below.

In [10]:

```
# function to whiten data
def whiten(strain, interp_psd, dt):
Nt = len(strain)
freqs = np.fft.rfftfreq(Nt, dt)
# whitening: transform to freq domain, divide by asd, then transform back,
# taking care to get normalization right.
hf = np.fft.rfft(strain)
white_hf = hf / (np.sqrt(interp_psd(freqs) /dt/2.))
white_ht = np.fft.irfft(white_hf, n=Nt)
return white_ht
whiten_data = 1
if whiten_data:
# now whiten the data from H1 and L1, and the template (use H1 PSD):
strain_H1_whiten = whiten(strain_H1,psd_H1,dt)
strain_L1_whiten = whiten(strain_L1,psd_L1,dt)
# We need to suppress the high frequency noise (no signal!) with some bandpassing:
bb, ab = butter(4, [fband[0]*2./fs, fband[1]*2./fs], btype='band')
strain_H1_whitenbp = filtfilt(bb, ab, strain_H1_whiten)
strain_L1_whitenbp = filtfilt(bb, ab, strain_L1_whiten)
```

Now let's plot a short time-frequency spectrogram around our event:

In [11]:

```
if make_plots:
# index into the strain time series for this time interval:
indxt = np.where((time >= tevent-deltat) & (time < tevent+deltat))
# pick a shorter FTT time interval, like 1/8 of a second:
NFFT = int(fs/8)
# and with a lot of overlap, to resolve short-time features:
NOVL = int(NFFT*15./16)
# and choose a window that minimizes "spectral leakage"
# (https://en.wikipedia.org/wiki/Spectral_leakage)
window = np.blackman(NFFT)
# the right colormap is all-important! See:
# http://matplotlib.org/examples/color/colormaps_reference.html
# viridis seems to be the best for our purposes, but it's new; if you don't have it, you can settle for ocean.
#spec_cmap='viridis'
spec_cmap='ocean'
# Plot the H1 spectrogram:
plt.figure(figsize=(10,6))
spec_H1, freqs, bins, im = plt.specgram(strain_H1[indxt], NFFT=NFFT, Fs=fs, window=window,
noverlap=NOVL, cmap=spec_cmap, xextent=[-deltat,deltat])
plt.xlabel('time (s) since '+str(tevent))
plt.ylabel('Frequency (Hz)')
plt.colorbar()
plt.axis([-deltat, deltat, 0, 2000])
plt.title('aLIGO H1 strain data near '+eventname)
plt.savefig(eventname+'_H1_spectrogram.'+plottype)
# Plot the L1 spectrogram:
plt.figure(figsize=(10,6))
spec_H1, freqs, bins, im = plt.specgram(strain_L1[indxt], NFFT=NFFT, Fs=fs, window=window,
noverlap=NOVL, cmap=spec_cmap, xextent=[-deltat,deltat])
plt.xlabel('time (s) since '+str(tevent))
plt.ylabel('Frequency (Hz)')
plt.colorbar()
plt.axis([-deltat, deltat, 0, 2000])
plt.title('aLIGO L1 strain data near '+eventname)
plt.savefig(eventname+'_L1_spectrogram.'+plottype)
```