def semblance(st, s, baz, winlen):
'''
Returns the semblance for a seismic array, for a beam of given slowness and backazimuth.
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)
winlen : int
Length of Hann window over which to calculate the semblance.
Returns
-------
semblance : NumPy array
The semblance at the given slowness and backazimuth, as a time series.
'''
# Check that each channel has the same number of samples, otherwise we can't construct the beam properly
assert len(set([
len(tr) for tr in st
])) == 1, "Traces in stream have different lengths, cannot stack."
nsta = len(st)
stack = linear_stack(st, s, baz)
# Taper the linear stack
stack_trace = Trace(stack)
stack_trace.taper(type='cosine', max_percentage=0.05)
stack = stack_trace.data
shifts = get_shifts(st, s, baz)
# Smooth data with sliding Hann window (i.e. convolution of signal and window function)
window = np.hanning(winlen)
shifted_st = st.copy()
for i, tr in enumerate(shifted_st):
tr.data = np.roll(tr.data,
shifts[i]) # Shift data in each trace by its offset
tr.taper(type='cosine', max_percentage=0.05) # Taper it
# Calculate the power in the beam
beampower = np.convolve(stack**2, window, mode='same')
# Calculate the summed power of each trace
tracepower = np.convolve(np.sum([tr.data**2 for tr in shifted_st], axis=0),
window,
mode='same')
# Calculate semblance
semblance = nsta * beampower / tracepower
return semblance
def semblance(st, s, baz, winlen):
"""
Returns the semblance for a seismic array, for a beam of given slowness and backazimuth.
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)
winlen : int
Length of Hann window over which to calculate the semblance.
Returns
-------
semblance : NumPy array
The semblance at the given slowness and backazimuth, as a time series.
"""
# Check that each channel has the same number of samples, otherwise we can't construct the beam properly
assert len(set([len(tr) for tr in st])) == 1, "Traces in stream have different lengths, cannot stack."
nsta = len(st)
stack = linear_stack(st, s, baz)
# Taper the linear stack
stack_trace = Trace(stack)
stack_trace.taper(type="cosine", max_percentage=0.05)
stack = stack_trace.data
shifts = get_shifts(st, s, baz)
# Smooth data with sliding Hann window (i.e. convolution of signal and window function)
window = np.hanning(winlen)
shifted_st = st.copy()
for i, tr in enumerate(shifted_st):
tr.data = np.roll(tr.data, shifts[i]) # Shift data in each trace by its offset
tr.taper(type="cosine", max_percentage=0.05) # Taper it
# Calculate the power in the beam
beampower = np.convolve(stack ** 2, window, mode="same")
# Calculate the summed power of each trace
tracepower = np.convolve(np.sum([tr.data ** 2 for tr in shifted_st], axis=0), window, mode="same")
# Calculate semblance
semblance = nsta * beampower / tracepower
return semblance
def power_vespa(st, s, baz, winlen):
'''
Returns the power vespa (i.e. the power in the linear delay-and-sum beam) for a seismic array, for a beam of given slowness and backazimuth.
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)
winlen : int
Length of Hann window over which to calculate the power.
Returns
-------
power : NumPy array
The power of the linear beam at the given slowness and backazimuth, as a time series.
'''
amplitude = linear_stack(st, s, baz)
power = np.convolve(amplitude**2, np.hanning(winlen), mode='same')
return power
def power_vespa(st, s, baz, winlen):
"""
Returns the power vespa (i.e. the power in the linear delay-and-sum beam) for a seismic array, for a beam of given slowness and backazimuth.
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)
winlen : int
Length of Hann window over which to calculate the power.
Returns
-------
power : NumPy array
The power of the linear beam at the given slowness and backazimuth, as a time series.
"""
amplitude = linear_stack(st, s, baz)
power = np.convolve(amplitude ** 2, np.hanning(winlen), mode="same")
return power
def stack_vespagram(st,
slowness_min,
slowness_max,
n_steps,
baz,
separation=0.5):
'''
Plots a stack vespagram for a seismic array over a given slowness range, for a single backazimuth.
The individual beam traces for each slowness step are plotted as a function of time (in s) and slowness (in s/km).
Parameters
----------
st : ObsPy Stream object
Stream of SAC format seismograms for the seismic array, length K = no. of stations in array
slowness_min : float
Minimum magnitude of the slowness vector, in s/km
slowness_max : float
Maximum magnitude of the slowness vector, in s/km
n_steps : int
Maximum backazimuth, in degrees
baz : float
Backazimuth of slowness vector, (i.e. angle from North back to epicentre of event)
winlen : int
Length of Hann window over which to calculate the power.
separation : float
Fraction of a single beam that overlaps with its neighbours. Smaller separations make a denser vespagram.
'''
slowness_range = np.linspace(slowness_min, slowness_max, n_steps)
vespas = [linear_stack(st, slowness, baz) for slowness in slowness_range]
times = st[0].times()
plt.figure(figsize=(14, 10))
# Find maximum and minimum amplitudes of the beam in order to set y-axes for beam traces.
max_y = np.max(np.array(vespas))
min_y = np.min(np.array(vespas))
trace_height = 1. / (1 + (n_steps - 1) * separation)
# Plot beam trace for each slowness value
for i, trace in enumerate(vespas):
# rect = [left, bottom, width, height]
trace_bottom = (
1 - trace_height) + (i - n_steps + 1) * separation * trace_height
rect = [0, trace_bottom, 1, trace_height]
ax = plt.axes(rect, axisbg=None, frameon=False)
ax.get_xaxis().set_visible(False)
ax.get_yaxis().set_visible(False)
plt.plot(times, trace, alpha=0.6)
plt.ylim(min_y, max_y)
plt.xlim(0, 350)
# Create external axes to contain beam traces, and set its slowness range
plt.axes([0, 0, 1, 1], axisbg='none')
main_ax_slowness_max = slowness_max / (1 - 0.5 * trace_height)
main_ax_slowness_min = slowness_min - 0.5 * trace_height * main_ax_slowness_max
plt.ylim(main_ax_slowness_min, main_ax_slowness_max)
plt.xlim(0, 350)
plt.xlabel('Time (s)')
plt.yticks(fontsize=14)
plt.xticks(fontsize=14)
plt.ylabel('Slowness (s / km)')
plt.show()
def stack_vespagram(st, slowness_min, slowness_max, n_steps, baz, separation=0.5):
'''
Plots a stack vespagram for a seismic array over a given slowness range, for a single backazimuth.
The individual beam traces for each slowness step are plotted as a function of time (in s) and slowness (in s/km).
Parameters
----------
st : ObsPy Stream object
Stream of SAC format seismograms for the seismic array, length K = no. of stations in array
slowness_min : float
Minimum magnitude of the slowness vector, in s/km
slowness_max : float
Maximum magnitude of the slowness vector, in s/km
n_steps : int
Maximum backazimuth, in degrees
baz : float
Backazimuth of slowness vector, (i.e. angle from North back to epicentre of event)
winlen : int
Length of Hann window over which to calculate the power.
separation : float
Fraction of a single beam that overlaps with its neighbours. Smaller separations make a denser vespagram.
'''
slowness_range = np.linspace(slowness_min, slowness_max, n_steps)
vespas = [linear_stack(st, slowness, baz) for slowness in slowness_range]
times = st[0].times()
plt.figure(figsize=(14, 10))
# Find maximum and minimum amplitudes of the beam in order to set y-axes for beam traces.
max_y = np.max(np.array(vespas))
min_y = np.min(np.array(vespas))
trace_height = 1. / (1 + (n_steps - 1) * separation)
# Plot beam trace for each slowness value
for i, trace in enumerate(vespas):
# rect = [left, bottom, width, height]
trace_bottom = (1 - trace_height) + (i - n_steps + 1) * separation * trace_height
rect = [0, trace_bottom, 1, trace_height]
ax = plt.axes(rect, axisbg=None, frameon=False)
ax.get_xaxis().set_visible(False)
ax.get_yaxis().set_visible(False)
plt.plot(times, trace, alpha=0.6)
plt.ylim(min_y, max_y);
plt.xlim(0, 350);
# Create external axes to contain beam traces, and set its slowness range
plt.axes([0, 0, 1, 1], axisbg='none')
main_ax_slowness_max = slowness_max / (1 - 0.5 * trace_height)
main_ax_slowness_min = slowness_min - 0.5 * trace_height * main_ax_slowness_max
plt.ylim(main_ax_slowness_min, main_ax_slowness_max)
plt.xlim(0, 350)
plt.xlabel('Time (s)')
plt.yticks(fontsize=14);
plt.xticks(fontsize=14);
plt.ylabel('Slowness (s / km)')
plt.show()
