def adsrc_tf_phase_misfit(t, data, synthetic, dt_new, width, threshold,
axis=None, colorbar_axis=None):
"""
:rtype: dictionary
:returns: Return a dictionary with three keys:
* adjoint_source: The calculated adjoint source as a numpy array
* misfit: The misfit value
* messages: A list of strings giving additional hints to what happened
in the calculation.
"""
messages = []
# Compute time-frequency representation via cross-correlation
tau_cc, nu_cc, tf_cc = time_frequency.time_frequency_cc_difference(
t, data, synthetic, dt_new, width, threshold)
# Compute the time-frequency representation of the synthetic
tau, nu, tf_synth = time_frequency.time_frequency_transform(t,
synthetic, dt_new, width, threshold)
# 2D interpolation. Use a two step interpolation for the real and the
# imaginary parts.
tf_cc_interp = RectBivariateSpline(tau_cc[0], nu_cc[:, 0], tf_cc.real,
kx=1, ky=1, s=0)(tau[0], nu[:, 0])
tf_cc_interp = np.require(tf_cc_interp, dtype="complex128")
tf_cc_interp.imag = RectBivariateSpline(tau_cc[0], nu_cc[:, 0], tf_cc.imag,
kx=1, ky=1, s=0)(tau[0], nu[:, 0])
tf_cc = tf_cc_interp
# Make window functionality
# noise taper
m = np.abs(tf_cc).max() / 10.0
weight = 1.0 - np.exp(-(np.abs(tf_cc) ** 2) / (m ** 2))
nu_t = nu.transpose()
# high-pass filter
weight *= (1.0 - np.exp((-nu_t ** 2) / (0.002 ** 2)))
nu_t_large = np.zeros(nu_t.shape)
nu_t_small = np.zeros(nu_t.shape)
thres = (nu_t <= 0.005)
nu_t_large[np.invert(thres)] = 1.0
nu_t_small[thres] = 1.0
# low-pass filter
weight *= (np.exp(-(nu_t - 0.005) ** 4 / 0.005 ** 4) *
nu_t_large + nu_t_small)
# normalisation
weight /= weight.max()
# Compute the phase difference.
DP = np.imag(np.log(eps + tf_cc / (eps + np.abs(tf_cc))))
# Attempt to detect phase jumps by taking the derivatives in time and
# frequency direction. 0.7 is an emperical value.
test_field = weight * DP / np.abs(weight * DP).max()
criterion_1 = np.abs(np.diff(test_field, axis=0)).max()
criterion_2 = np.abs(np.diff(test_field, axis=1)).max()
criterion = max(criterion_1, criterion_2)
if criterion > 0.7:
warning = "Possible phase jump detected"
warnings.warn(warning)
messages.append(warning)
# Compute the phase misfit
dnu = nu[1, 0] - nu[0, 0]
phase_misfit = np.sqrt(np.sum(weight ** 2 * DP ** 2) * dt_new * dnu)
# Sanity check. Should not occur.
if np.isnan(phase_misfit):
msg = "The phase misfit is NaN."
raise Exception(msg)
# Make kernel for the inverse tf transform
idp = weight * weight * DP * tf_synth / (eps + np.abs(tf_synth) *
np.abs(tf_synth))
# Invert tf transform and make adjoint source
ad_src, it, I = time_frequency.itfa(tau, nu, idp, width, threshold)
# Interpolate to original time axis
current_time = tau[0, :]
new_time = t[t <= current_time.max()]
ad_src = interp1d(current_time, np.imag(ad_src), kind=2)(new_time)
if len(t) > len(new_time):
ad_src = np.concatenate([ad_src, np.zeros(len(t) - len(new_time))])
# Divide by the misfit.
ad_src /= (phase_misfit + eps)
ad_src = np.diff(ad_src) / (t[1] - t[0])
# Reverse time and add a leading zero so the adjoint source has the same
# length as the input time series.
ad_src = ad_src[::-1]
ad_src = np.concatenate([[0.0], ad_src])
# Plot if required.
if axis:
import matplotlib.cm as cm
import matplotlib.pyplot as plt
weighted_phase_difference = (DP * weight).transpose()
abs_diff = np.abs(weighted_phase_difference)
max_val = abs_diff.max()
mappable = axis.pcolormesh(tau, nu,
weighted_phase_difference, vmin=-max_val, vmax=max_val,
cmap=cm.RdBu_r)
axis.set_xlabel("Seconds since event")
axis.set_ylabel("TF Phase Misfit: Frequency [Hz]")
# Smart scaling for the frequency axis.
temp = abs_diff.max(axis=1) * (nu[1] - nu[0])
ymax = len(temp[temp > temp.max() / 1000.0])
ymax *= nu[1, 0] - nu[0, 0]
ymax *= 2
axis.set_ylim(0, ymax)
if colorbar_axis:
cm = plt.gcf().colorbar(mappable, cax=colorbar_axis)
else:
cm = plt.gcf().colorbar(mappable, ax=axis)
cm.set_label("Phase difference in radian")
axis.set_title("Weighted phase difference")
ax2 = axis.twinx()
ax2.plot(t, data, color="black", alpha=1.0)
ax2.plot(t, synthetic, color="red", alpha=1.0)
min_value = min(data.min(), synthetic.min())
max_value = max(data.max(), synthetic.max())
value_range = max_value - min_value
axis.twin_axis = ax2
ax2.set_ylim(min_value - value_range, max_value + 0.05 * value_range)
ax2.set_ylabel("Waveforms: Amplitude [m/s]")
axis.set_xlim(0, tau[:, -1][-1])
ax2.set_xlim(0, tau[:, -1][-1])
text = "Misfit: %.4f" % phase_misfit
axis.text(x=0.99, y=0.02, s=text, transform=axis.transAxes,
bbox=dict(facecolor='orange', alpha=0.8),
verticalalignment="bottom",
horizontalalignment="right")
if messages:
message = "\n".join(messages)
axis.text(x=0.99, y=0.98, s=message, transform=axis.transAxes,
bbox=dict(facecolor='red', alpha=0.8),
verticalalignment="top",
horizontalalignment="right")
ret_dict = {
"adjoint_source": ad_src,
"misfit": phase_misfit,
"messages": messages}
return ret_dict
def adsrc_tf_phase_misfit(t, data, synthetic, min_period, max_period,
axis=None, colorbar_axis=None):
"""
:rtype: dictionary
:returns: Return a dictionary with three keys:
* adjoint_source: The calculated adjoint source as a numpy array
* misfit: The misfit value
* messages: A list of strings giving additional hints to what happened
in the calculation.
"""
messages = []
# Compute time-frequency representations ----------------------------------
# compute new time increments and Gaussian window width for the
# time-frequency transforms
dt_new = float(int(min_period / 3.0))
width = 2.0 * min_period
# Compute time-frequency representation of the cross-correlation
tau_cc, nu_cc, tf_cc = time_frequency.time_frequency_cc_difference(
t, data, synthetic, dt_new, width)
# Compute the time-frequency representation of the synthetic
tau, nu, tf_synth = time_frequency.time_frequency_transform(t, synthetic,
dt_new, width)
# 2D interpolation to bring the tf representation of the correlation on the
# same grid as the tf representation of the synthetics. Uses a two-step
# procedure for real and imaginary parts.
tf_cc_interp = RectBivariateSpline(tau_cc[0], nu_cc[:, 0], tf_cc.real,
kx=1, ky=1, s=0)(tau[0], nu[:, 0])
tf_cc_interp = np.require(tf_cc_interp, dtype="complex128")
tf_cc_interp.imag = RectBivariateSpline(tau_cc[0], nu_cc[:, 0], tf_cc.imag,
kx=1, ky=1, s=0)(tau[0], nu[:, 0])
tf_cc = tf_cc_interp
# compute tf window and weighting function --------------------------------
# noise taper: downweigh tf amplitudes that are very low
m = np.abs(tf_cc).max() / 10.0
weight = 1.0 - np.exp(-(np.abs(tf_cc) ** 2) / (m ** 2))
nu_t = nu.transpose()
# highpass filter (periods longer than max_period are suppressed
# exponentially)
weight *= (1.0 - np.exp(-(nu_t * max_period) ** 2))
# lowpass filter (periods shorter than min_period are suppressed
# exponentially)
nu_t_large = np.zeros(nu_t.shape)
nu_t_small = np.zeros(nu_t.shape)
thres = (nu_t <= 1.0 / min_period)
nu_t_large[np.invert(thres)] = 1.0
nu_t_small[thres] = 1.0
weight *= (np.exp(-10.0 * np.abs(nu_t * min_period - 1.0)) * nu_t_large +
nu_t_small)
# normalisation
weight /= weight.max()
# computation of phase difference, make quality checks and misfit ---------
# Compute the phase difference.
# DP = np.imag(np.log(m + tf_cc / (2 * m + np.abs(tf_cc))))
DP = np.angle(tf_cc)
# Attempt to detect phase jumps by taking the derivatives in time and
# frequency direction. 0.7 is an emperical value.
test_field = weight * DP / np.abs(weight * DP).max()
criterion_1 = np.sum([np.abs(np.diff(test_field, axis=0)) > 0.7])
criterion_2 = np.sum([np.abs(np.diff(test_field, axis=1)) > 0.7])
criterion = np.sum([criterion_1, criterion_2])
# criterion_1 = np.abs(np.diff(test_field, axis=0)).max()
# criterion_2 = np.abs(np.diff(test_field, axis=1)).max()
# criterion = max(criterion_1, criterion_2)
if criterion > 7.0:
warning = ("Possible phase jump detected. Misfit included. No "
"adjoint source computed.")
warnings.warn(warning)
messages.append(warning)
# Compute the phase misfit
dnu = nu[1, 0] - nu[0, 0]
phase_misfit = np.sqrt(np.sum(weight ** 2 * DP ** 2) * dt_new * dnu)
# Sanity check. Should not occur.
if np.isnan(phase_misfit):
msg = "The phase misfit is NaN."
raise LASIFAdjointSourceCalculationError(msg)
# compute the adjoint source when no phase jump detected ------------------
if criterion <= 7.0:
# Make kernel for the inverse tf transform
idp = weight * weight * DP * tf_synth / (m + np.abs(tf_synth) *
np.abs(tf_synth))
# Invert tf transform and make adjoint source
ad_src, it, I = time_frequency.itfa(tau, nu, idp, width)
# Interpolate to original time axis
current_time = tau[0, :]
new_time = t[t <= current_time.max()]
ad_src = interp1d(current_time, np.imag(ad_src), kind=2)(new_time)
if len(t) > len(new_time):
ad_src = np.concatenate([ad_src, np.zeros(len(t) - len(new_time))])
# Divide by the misfit and change sign.
ad_src /= (phase_misfit + eps)
ad_src = -1.0 * np.diff(ad_src) / (t[1] - t[0])
# Reverse time and add a leading zero so the adjoint source has the
# same length as the input time series.
ad_src = ad_src[::-1]
ad_src = np.concatenate([[0.0], ad_src])
else:
# Criterion failed, no misfit and adjoint source calculated.
raise LASIFAdjointSourceCalculationError(
"Criterion failed, no misfit has been calculated.")
# Plot if required. -------------------------------------------------------
if axis:
import matplotlib.cm as cm
import matplotlib.pyplot as plt
# Primary axis: plot weighted phase difference. -----------------------
weighted_phase_difference = (DP * weight).transpose()
abs_diff = np.abs(weighted_phase_difference)
mappable = axis.pcolormesh(tau, nu, weighted_phase_difference,
vmin=-1.0, vmax=1.0, cmap=cm.RdBu_r)
axis.set_xlabel("Seconds since event")
axis.set_ylabel("Frequency [Hz]")
# Smart scaling for the frequency axis.
temp = abs_diff.max(axis=1) * (nu[1] - nu[0])
ymax = len(temp[temp > temp.max() / 1000.0])
ymax *= nu[1, 0] - nu[0, 0]
ymax *= 2
axis.set_ylim(0, 2.0 / min_period)
if colorbar_axis:
cm = plt.gcf().colorbar(mappable, cax=colorbar_axis)
else:
cm = plt.gcf().colorbar(mappable, ax=axis)
cm.set_label("Phase difference in radian")
# Secondary axis: plot waveforms and adjoint source. ------------------
ax2 = axis.twinx()
ax2.plot(t, ad_src, color="black", alpha=1.0)
min_value = min(ad_src.min(), -1.0)
max_value = max(ad_src.max(), 1.0)
value_range = max_value - min_value
axis.twin_axis = ax2
ax2.set_ylim(min_value - 2.5 * value_range,
max_value + 0.5 * value_range)
axis.set_xlim(0, tau[:, -1][-1])
ax2.set_xlim(0, tau[:, -1][-1])
ax2.set_yticks([])
text = "Misfit: %.4f" % phase_misfit
axis.text(x=0.99, y=0.02, s=text, transform=axis.transAxes,
bbox=dict(facecolor='orange', alpha=0.8),
verticalalignment="bottom",
horizontalalignment="right")
if messages:
message = "\n".join(messages)
axis.text(x=0.99, y=0.98, s=message, transform=axis.transAxes,
bbox=dict(facecolor='red', alpha=0.8),
verticalalignment="top",
horizontalalignment="right")
ret_dict = {
"adjoint_source": ad_src,
"misfit_value": phase_misfit,
"details": {"messages": messages}
}
return ret_dict
def adsrc_tf_phase_misfit(t,
data,
synthetic,
min_period,
max_period,
axis=None,
colorbar_axis=None):
"""
:rtype: dictionary
:returns: Return a dictionary with three keys:
* adjoint_source: The calculated adjoint source as a numpy array
* misfit: The misfit value
* messages: A list of strings giving additional hints to what happened
in the calculation.
"""
messages = []
# Compute time-frequency representations ----------------------------------
# compute new time increments and Gaussian window width for the
# time-frequency transforms
dt_new = float(int(min_period / 3.0))
width = 2.0 * min_period
# Compute time-frequency representation of the cross-correlation
tau_cc, nu_cc, tf_cc = time_frequency.time_frequency_cc_difference(
t, data, synthetic, dt_new, width)
# Compute the time-frequency representation of the synthetic
tau, nu, tf_synth = time_frequency.time_frequency_transform(
t, synthetic, dt_new, width)
# 2D interpolation to bring the tf representation of the correlation on the
# same grid as the tf representation of the synthetics. Uses a two-step
# procedure for real and imaginary parts.
tf_cc_interp = RectBivariateSpline(tau_cc[0],
nu_cc[:, 0],
tf_cc.real,
kx=1,
ky=1,
s=0)(tau[0], nu[:, 0])
tf_cc_interp = np.require(tf_cc_interp, dtype="complex128")
tf_cc_interp.imag = RectBivariateSpline(tau_cc[0],
nu_cc[:, 0],
tf_cc.imag,
kx=1,
ky=1,
s=0)(tau[0], nu[:, 0])
tf_cc = tf_cc_interp
# compute tf window and weighting function --------------------------------
# noise taper: downweigh tf amplitudes that are very low
m = np.abs(tf_cc).max() / 10.0
weight = 1.0 - np.exp(-(np.abs(tf_cc)**2) / (m**2))
nu_t = nu.transpose()
# highpass filter (periods longer than max_period are suppressed
# exponentially)
weight *= (1.0 - np.exp(-(nu_t * max_period)**2))
# lowpass filter (periods shorter than min_period are suppressed
# exponentially)
nu_t_large = np.zeros(nu_t.shape)
nu_t_small = np.zeros(nu_t.shape)
thres = (nu_t <= 1.0 / min_period)
nu_t_large[np.invert(thres)] = 1.0
nu_t_small[thres] = 1.0
weight *= (np.exp(-10.0 * np.abs(nu_t * min_period - 1.0)) * nu_t_large +
nu_t_small)
# normalisation
weight /= weight.max()
# computation of phase difference, make quality checks and misfit ---------
# Compute the phase difference.
# DP = np.imag(np.log(m + tf_cc / (2 * m + np.abs(tf_cc))))
DP = np.angle(tf_cc)
# Attempt to detect phase jumps by taking the derivatives in time and
# frequency direction. 0.7 is an emperical value.
test_field = weight * DP / np.abs(weight * DP).max()
criterion_1 = np.sum([np.abs(np.diff(test_field, axis=0)) > 0.7])
criterion_2 = np.sum([np.abs(np.diff(test_field, axis=1)) > 0.7])
criterion = np.sum([criterion_1, criterion_2])
# criterion_1 = np.abs(np.diff(test_field, axis=0)).max()
# criterion_2 = np.abs(np.diff(test_field, axis=1)).max()
# criterion = max(criterion_1, criterion_2)
if criterion > 7.0:
warning = ("Possible phase jump detected. Misfit included. No "
"adjoint source computed.")
warnings.warn(warning)
messages.append(warning)
# Compute the phase misfit
dnu = nu[1, 0] - nu[0, 0]
phase_misfit = np.sqrt(np.sum(weight**2 * DP**2) * dt_new * dnu)
# Sanity check. Should not occur.
if np.isnan(phase_misfit):
msg = "The phase misfit is NaN."
raise LASIFAdjointSourceCalculationError(msg)
# compute the adjoint source when no phase jump detected ------------------
if criterion <= 7.0:
# Make kernel for the inverse tf transform
idp = weight * weight * DP * tf_synth / (
m + np.abs(tf_synth) * np.abs(tf_synth))
# Invert tf transform and make adjoint source
ad_src, it, I = time_frequency.itfa(tau, nu, idp, width)
# Interpolate to original time axis
current_time = tau[0, :]
new_time = t[t <= current_time.max()]
ad_src = interp1d(current_time, np.imag(ad_src), kind=2)(new_time)
if len(t) > len(new_time):
ad_src = np.concatenate([ad_src, np.zeros(len(t) - len(new_time))])
# Divide by the misfit and change sign.
ad_src /= (phase_misfit + eps)
ad_src = -1.0 * np.diff(ad_src) / (t[1] - t[0])
# Reverse time and add a leading zero so the adjoint source has the
# same length as the input time series.
ad_src = ad_src[::-1]
ad_src = np.concatenate([[0.0], ad_src])
else:
# Criterion failed, no misfit and adjoint source calculated.
raise LASIFAdjointSourceCalculationError(
"Criterion failed, no misfit has been calculated.")
# Plot if required. -------------------------------------------------------
if axis:
import matplotlib.cm as cm
import matplotlib.pyplot as plt
# Primary axis: plot weighted phase difference. -----------------------
weighted_phase_difference = (DP * weight).transpose()
abs_diff = np.abs(weighted_phase_difference)
mappable = axis.pcolormesh(tau,
nu,
weighted_phase_difference,
vmin=-1.0,
vmax=1.0,
cmap=cm.RdBu_r)
axis.set_xlabel("Seconds since event")
axis.set_ylabel("Frequency [Hz]")
# Smart scaling for the frequency axis.
temp = abs_diff.max(axis=1) * (nu[1] - nu[0])
ymax = len(temp[temp > temp.max() / 1000.0])
ymax *= nu[1, 0] - nu[0, 0]
ymax *= 2
axis.set_ylim(0, 2.0 / min_period)
if colorbar_axis:
cm = plt.gcf().colorbar(mappable, cax=colorbar_axis)
else:
cm = plt.gcf().colorbar(mappable, ax=axis)
cm.set_label("Phase difference in radian")
# Secondary axis: plot waveforms and adjoint source. ------------------
ax2 = axis.twinx()
ax2.plot(t, ad_src, color="black", alpha=1.0)
min_value = min(ad_src.min(), -1.0)
max_value = max(ad_src.max(), 1.0)
value_range = max_value - min_value
axis.twin_axis = ax2
ax2.set_ylim(min_value - 2.5 * value_range,
max_value + 0.5 * value_range)
axis.set_xlim(0, tau[:, -1][-1])
ax2.set_xlim(0, tau[:, -1][-1])
ax2.set_yticks([])
text = "Misfit: %.4f" % phase_misfit
axis.text(x=0.99,
y=0.02,
s=text,
transform=axis.transAxes,
bbox=dict(facecolor='orange', alpha=0.8),
verticalalignment="bottom",
horizontalalignment="right")
if messages:
message = "\n".join(messages)
axis.text(x=0.99,
y=0.98,
s=message,
transform=axis.transAxes,
bbox=dict(facecolor='red', alpha=0.8),
verticalalignment="top",
horizontalalignment="right")
ret_dict = {
"adjoint_source": ad_src,
"misfit_value": phase_misfit,
"details": {
"messages": messages
}
}
return ret_dict
