def get_shifts(st, s, baz):
'''
Calculates the shifts (as an integer number of samples in the time series) for every station in a stream of time series seismograms for a slowness vector of given magnitude and backazimuth.
The shift is that which needs to be applied in order to align an arrival (arriving with slowness s and backazimuth baz) with the same arrival at the array reference point (the location of the station that makes up the first trace in the stream).
Parameters
----------
st : ObsPy Stream object
Stream of SAC format seismograms for the seismic array, length K = no. of stations in array
s : float
Magnitude of slowness vector, in s / km
baz : float
Backazimuth of slowness vector, (i.e. angle from North back to epicentre of event)
Returns
-------
shifts : list
List of integer delays at each station in the array, also length K
'''
theta = [
] # Angular position of each station, measured clockwise from North
r = [] # Distance of each station
# First station is reference point, so has zero position vector
theta.append(0.0)
r.append(0.0)
geometry = get_station_coordinates(st) / 1000. # in km
# For each station, get distance from array reference point (first station), and the angular displacement clockwise from north
for station in geometry[1:]:
r_x = station[0] # x-component of position vector
r_y = station[1] # y-component of position vector
# theta is angle c/w from North to position vector of station; need to compute diffently for each quadrant
if r_x >= 0 and r_y >= 0:
theta.append(np.degrees(np.arctan(r_x / r_y)))
elif r_x > 0 and r_y < 0:
theta.append(180 + np.degrees(np.arctan(r_x / r_y)))
elif r_x <= 0 and r_y <= 0:
theta.append(180 + np.degrees(np.arctan(r_x / r_y)))
else:
theta.append(360 + np.degrees(np.arctan(r_x / r_y)))
r.append(np.sqrt(r_x**2 + r_y**2))
# Find angle between station position vector and slowness vector in order to compute dot product
# Angle between slowness and position vectors, measured clockwise
phi = [180 - baz + th for th in theta]
sampling_rate = st[0].stats.sampling_rate
shifts = []
# Shift is dot product. The minus sign is because a positive time delay needs to be corrected by a negative shift in order to stack
for i in range(0, len(st)):
shifts.append(
-1 *
int(round(r[i] * s * np.cos(np.radians(phi[i])) * sampling_rate)))
return shifts
def get_shifts(st, s, baz):
"""
Calculates the shifts (as an integer number of samples in the time series) for every station in a stream of time series seismograms for a slowness vector of given magnitude and backazimuth.
The shift is that which needs to be applied in order to align an arrival (arriving with slowness s and backazimuth baz) with the same arrival at the array reference point (the location of the station that makes up the first trace in the stream).
Parameters
----------
st : ObsPy Stream object
Stream of SAC format seismograms for the seismic array, length K = no. of stations in array
s : float
Magnitude of slowness vector, in s / km
baz : float
Backazimuth of slowness vector, (i.e. angle from North back to epicentre of event)
Returns
-------
shifts : list
List of integer delays at each station in the array, also length K
"""
theta = [] # Angular position of each station, measured clockwise from North
r = [] # Distance of each station
# First station is reference point, so has zero position vector
theta.append(0.0)
r.append(0.0)
geometry = get_station_coordinates(st) / 1000.0 # in km
# For each station, get distance from array reference point (first station), and the angular displacement clockwise from north
for station in geometry[1:]:
r_x = station[0] # x-component of position vector
r_y = station[1] # y-component of position vector
# theta is angle c/w from North to position vector of station; need to compute diffently for each quadrant
if r_x >= 0 and r_y >= 0:
theta.append(np.degrees(np.arctan(r_x / r_y)))
elif r_x > 0 and r_y < 0:
theta.append(180 + np.degrees(np.arctan(r_x / r_y)))
elif r_x <= 0 and r_y <= 0:
theta.append(180 + np.degrees(np.arctan(r_x / r_y)))
else:
theta.append(360 + np.degrees(np.arctan(r_x / r_y)))
r.append(np.sqrt(r_x ** 2 + r_y ** 2))
# Find angle between station position vector and slowness vector in order to compute dot product
# Angle between slowness and position vectors, measured clockwise
phi = [180 - baz + th for th in theta]
sampling_rate = st[0].stats.sampling_rate
shifts = []
# Shift is dot product. The minus sign is because a positive time delay needs to be corrected by a negative shift in order to stack
for i in range(0, len(st)):
shifts.append(-1 * int(round(r[i] * s * np.cos(np.radians(phi[i])) * sampling_rate)))
return shifts
def fk_analysis(st, smax, fmin, fmax, tmin, tmax, stat='power'):
'''
Performs frequency-wavenumber space (FK) analysis on a stream of time series data for a given slowness range, frequency band and time window.
For an input stream of length K, the output is a K x K array with values of the chosen statistic calculated on a slowness grid in the x and y spatial dimensions. This statistic can be one of:-
* 'power' - the power in frequency-domain stack
* 'semblance' - the semblance calculated in the frequency domain
* 'F' - the F-statistic calculated in the frequency domain
Before the FK analysis is performed, the seismograms are cut to a time window between tmin and tmax, and the data is bandpass-filtered between frequencies fmin and fmax.
Parameters
----------
st : ObsPy Stream object
Stream of SAC format seismograms for the seismic array, length K = no. of stations in array
smax : float
Maximum magnitude of slowness, used for constructing the slowness grid in both x and y directions
fmin : float
Lower end of frequency band to perform FK analysis over
fmax : float
Upper end of frequency band to perform FK analysis over
tmin : float
Start of time window, seismograms are cut between tmin and tmax before FK starts
tmax : int
End of time window, seismograms are cut between tmin and tmax before FK starts
stat : string
Statistic that is to be calculated over the slowness grid, either 'power', 'semblance', or 'F'
Returns
-------
fk : NumPy array
The chosen statistic calculated at each point in a K x K grid of slowness values in the x and y directions
'''
assert (stat == 'power' or stat == 'semblance' or stat == 'F'), "Argument 'stat' must be one of 'power', 'semblance' or 'F'"
st = st.copy().trim(starttime=tmin, endtime=tmax)
# Retrieve metadata: time step and number of stations to be stacked
delta = st[0].stats.delta
nbeam = len(st)
# Pre-process, and filter to frequency window
st.detrend()
st.taper(type='cosine', max_percentage=0.05)
st = st.copy().filter("bandpass", freqmin=fmin, freqmax=fmax)
npts = st[0].stats.npts
# Computer Fourier transforms for each trace
fft_st = np.zeros((nbeam, (npts / 2) + 1), dtype=complex) # Length of real FFT is only half that of time series data
for i, tr in enumerate(st):
fft_st[i, :] = np.fft.rfft(tr.data) # Only need positive frequencies, so use rfft
freqs = np.fft.fftfreq(npts, delta)[0:(npts / 2) + 1]
# Slowness grid
slow_x = np.linspace(-smax, smax, nbeam)
slow_y = np.linspace(-smax, smax, nbeam)
# Array geometry
x, y = np.split(get_station_coordinates(st)[:, :2], 2, axis=1)
# convert to km
x /= 1000.
y /= 1000.
# Calculate the F-K spectrum
fk = np.zeros((nbeam, nbeam))
for ii in range(nbeam):
for jj in range(nbeam):
dt = slow_x[jj] * x + slow_y[ii] * y
beam = np.sum(np.exp(-1j * 2 * np.pi * np.outer(dt, freqs)) * fft_st / nbeam, axis=0)
fk[ii, jj] = np.vdot(beam, beam).real
if stat == 'semblance' or stat == 'F':
tracepower = np.vdot(fft_st, fft_st).real
if stat == 'semblance':
fk_semb = nbeam * fk / tracepower
return fk_semb
elif stat == 'F':
fk_F = (nbeam - 1)* nbeam * fk / (tracepower - nbeam * fk)
return fk_F
else:
return fk
