#!/usr/bin/env python
# -*- coding: utf-8 -*-
# -------------------------------------------------------------------
# Filename: cross_correlation.py
# Author: Moritz Beyreuther, Tobias Megies
# Email: megies@geophysik.uni-muenchen.de
#
# Copyright (C) 2008-2012 Moritz Beyreuther, Tobias Megies
# ------------------------------------------------------------------
"""
Signal processing routines based on cross correlation techniques.
:copyright:
The ObsPy Development Team (devs@obspy.org)
:license:
GNU Lesser General Public License, Version 3
(https://www.gnu.org/copyleft/lesser.html)
"""
from __future__ import (absolute_import, division, print_function,
unicode_literals)
from future.builtins import * # NOQA
from future.utils import native_str
import ctypes as C
import sys
import warnings
import numpy as np
import scipy
from obspy import Stream, Trace
from obspy.core.util.deprecation_helpers import \
DynamicAttributeImportRerouteModule
from obspy.core.util.misc import MatplotlibBackend
from obspy.signal.headers import clibsignal
from obspy.signal.invsim import cosine_taper
[docs]def xcorr(tr1, tr2, shift_len, full_xcorr=False):
"""
Cross correlation of tr1 and tr2 in the time domain using window_len.
::
Mid Sample
|
|AAAAAAAAAAAAAAA|AAAAAAAAAAAAAAA|AAAAAAAAAAAAAAA|AAAAAAAAAAAAAAA|
|BBBBBBBBBBBBBBB|BBBBBBBBBBBBBBB|BBBBBBBBBBBBBBB|BBBBBBBBBBBBBBB|
|<-shift_len/2->| <- region of support -> |<-shift_len/2->|
:type tr1: :class:`~numpy.ndarray`, :class:`~obspy.core.trace.Trace`
:param tr1: Trace 1
:type tr2: :class:`~numpy.ndarray`, :class:`~obspy.core.trace.Trace`
:param tr2: Trace 2 to correlate with trace 1
:type shift_len: int
:param shift_len: Total length of samples to shift for cross correlation.
:type full_xcorr: bool
:param full_xcorr: If ``True``, the complete xcorr function will be
returned as :class:`~numpy.ndarray`
:return: **index, value[, fct]** - Index of maximum xcorr value and the
value itself. The complete xcorr function is returned only if
``full_xcorr=True``.
.. note::
As shift_len gets higher the window supporting the cross correlation
actually gets smaller. So with shift_len=0 you get the correlation
coefficient of both traces as a whole without any shift applied. As the
xcorr function works in time domain and does not zero pad at all, with
higher shifts allowed the window of support gets smaller so that the
moving windows shifted against each other do not run out of the
timeseries bounds at high time shifts. Of course there are other
possibilities to do cross correlations e.g. in frequency domain.
.. seealso::
`ObsPy-users mailing list
<http://lists.obspy.org/pipermail/obspy-users/2011-March/000056.html>`_
and `issue #249 <https://github.com/obspy/obspy/issues/249>`_.
.. rubric:: Example
>>> tr1 = np.random.randn(10000).astype(np.float32)
>>> tr2 = tr1.copy()
>>> a, b = xcorr(tr1, tr2, 1000)
>>> a
0
>>> round(b, 7)
1.0
"""
# if we get Trace objects, use their data arrays
for tr in [tr1, tr2]:
if isinstance(tr, Trace):
tr = tr.data
# check if shift_len parameter is in an acceptable range.
# if not the underlying c code tampers with shift_len and uses shift_len/2
# instead. we want to avoid this silent automagic and raise an error in the
# python layer right here.
# see ticket #249 and src/xcorr.c lines 43-57
if min(len(tr1), len(tr2)) - 2 * shift_len <= 0:
msg = "shift_len too large. The underlying C code would silently " + \
"use shift_len/2 which we want to avoid."
raise ValueError(msg)
# be nice and adapt type if necessary
tr1 = np.ascontiguousarray(tr1, np.float32)
tr2 = np.ascontiguousarray(tr2, np.float32)
corp = np.empty(2 * shift_len + 1, dtype=np.float64, order='C')
shift = C.c_int()
coe_p = C.c_double()
res = clibsignal.X_corr(tr1, tr2, corp, shift_len, len(tr1), len(tr2),
C.byref(shift), C.byref(coe_p))
if res:
raise MemoryError
if full_xcorr:
return shift.value, coe_p.value, corp
else:
return shift.value, coe_p.value
[docs]def xcorr_3c(st1, st2, shift_len, components=["Z", "N", "E"],
full_xcorr=False, abs_max=True):
"""
Calculates the cross correlation on each of the specified components
separately, stacks them together and estimates the maximum and shift of
maximum on the stack.
Basically the same as :func:`~obspy.signal.cross_correlation.xcorr` but
for (normally) three components, please also take a look at the
documentation of that function. Useful e.g. for estimation of waveform
similarity on a three component seismogram.
:type st1: :class:`~obspy.core.stream.Stream`
:param st1: Stream 1, containing one trace for Z, N, E component (other
component_id codes are ignored)
:type st2: :class:`~obspy.core.stream.Stream`
:param st2: Stream 2, containing one trace for Z, N, E component (other
component_id codes are ignored)
:type shift_len: int
:param shift_len: Total length of samples to shift for cross correlation.
:type components: list of str
:param components: List of components to use in cross-correlation, defaults
to ``['Z', 'N', 'E']``.
:type full_xcorr: bool
:param full_xcorr: If ``True``, the complete xcorr function will be
returned as :class:`~numpy.ndarray`.
:return: **index, value[, fct]** - index of maximum xcorr value and the
value itself. The complete xcorr function is returned only if
``full_xcorr=True``.
"""
streams = [st1, st2]
# check if we can actually use the provided streams safely
for st in streams:
if not isinstance(st, Stream):
raise TypeError("Expected Stream object but got %s." % type(st))
for component in components:
if not len(st.select(component=component)) == 1:
msg = "Expected exactly one %s trace in stream" % component + \
" but got %s." % len(st.select(component=component))
raise ValueError(msg)
ndat = len(streams[0].select(component=components[0])[0])
if False in [len(st.select(component=component)[0]) == ndat
for st in streams for component in components]:
raise ValueError("All traces have to be the same length.")
# everything should be ok with the input data...
corp = np.zeros(2 * shift_len + 1, dtype=np.float64, order='C')
for component in components:
xx = xcorr(streams[0].select(component=component)[0],
streams[1].select(component=component)[0],
shift_len, full_xcorr=True)
corp += xx[2]
corp /= len(components)
shift, value = xcorr_max(corp, abs_max=abs_max)
if full_xcorr:
return shift, value, corp
else:
return shift, value
[docs]def xcorr_max(fct, abs_max=True):
"""
Return shift and value of maximum xcorr function
:type fct: :class:`~numpy.ndarray`
:param fct: xcorr function e.g. returned by xcorr
:type abs_max: bool
:param abs_max: determines if the absolute maximum should be used.
:return: **shift, value** - Shift and value of maximum xcorr.
.. rubric:: Example
>>> fct = np.zeros(101)
>>> fct[50] = -1.0
>>> xcorr_max(fct)
(0.0, -1.0)
>>> fct[50], fct[60] = 0.0, 1.0
>>> xcorr_max(fct)
(10.0, 1.0)
>>> fct[60], fct[40] = 0.0, -1.0
>>> xcorr_max(fct)
(-10.0, -1.0)
>>> fct[60], fct[40] = 0.5, -1.0
>>> xcorr_max(fct, abs_max=True)
(-10.0, -1.0)
>>> xcorr_max(fct, abs_max=False)
(10.0, 0.5)
"""
value = fct.max()
if abs_max:
_min = fct.min()
if abs(_min) > abs(value):
value = _min
mid = (len(fct) - 1) / 2
shift = np.where(fct == value)[0][0] - mid
return float(shift), float(value)
[docs]def xcorr_pick_correction(pick1, trace1, pick2, trace2, t_before, t_after,
cc_maxlag, filter=None, filter_options={},
plot=False, filename=None):
"""
Calculate the correction for the differential pick time determined by cross
correlation of the waveforms in narrow windows around the pick times.
For details on the fitting procedure refer to [Deichmann1992]_.
The parameters depend on the epicentral distance and magnitude range. For
small local earthquakes (Ml ~0-2, distance ~3-10 km) with consistent manual
picks the following can be tried::
t_before=0.05, t_after=0.2, cc_maxlag=0.10,
filter="bandpass", filter_options={'freqmin': 1, 'freqmax': 20}
The appropriate parameter sets can and should be determined/verified
visually using the option `plot=True` on a representative set of picks.
To get the corrected differential pick time calculate: ``((pick2 +
pick2_corr) - pick1)``. To get a corrected differential travel time using
origin times for both events calculate: ``((pick2 + pick2_corr - ot2) -
(pick1 - ot1))``
:type pick1: :class:`~obspy.core.utcdatetime.UTCDateTime`
:param pick1: Time of pick for `trace1`.
:type trace1: :class:`~obspy.core.trace.Trace`
:param trace1: Waveform data for `pick1`. Add some time at front/back.
The appropriate part of the trace is used automatically.
:type pick2: :class:`~obspy.core.utcdatetime.UTCDateTime`
:param pick2: Time of pick for `trace2`.
:type trace2: :class:`~obspy.core.trace.Trace`
:param trace2: Waveform data for `pick2`. Add some time at front/back.
The appropriate part of the trace is used automatically.
:type t_before: float
:param t_before: Time to start cross correlation window before pick times
in seconds.
:type t_after: float
:param t_after: Time to end cross correlation window after pick times in
seconds.
:type cc_maxlag: float
:param cc_maxlag: Maximum lag/shift time tested during cross correlation in
seconds.
:type filter: str
:param filter: `None` for no filtering or name of filter type
as passed on to :meth:`~obspy.core.Trace.trace.filter` if filter
should be used. To avoid artifacts in filtering provide
sufficiently long time series for `trace1` and `trace2`.
:type filter_options: dict
:param filter_options: Filter options that get passed on to
:meth:`~obspy.core.Trace.trace.filter` if filtering is used.
:type plot: bool
:param plot: If `True`, a plot window illustrating the alignment of the two
traces at best cross correlation will be shown. This can and should be
used to verify the used parameters before running automatedly on large
data sets.
:type filename: str
:param filename: If plot option is selected, specifying a filename here
(e.g. 'myplot.pdf' or 'myplot.png') will output the plot to a file
instead of opening a plot window.
:rtype: (float, float)
:returns: Correction time `pick2_corr` for `pick2` pick time as a float and
corresponding correlation coefficient.
"""
# perform some checks on the traces
if trace1.stats.sampling_rate != trace2.stats.sampling_rate:
msg = "Sampling rates do not match: %s != %s" % \
(trace1.stats.sampling_rate, trace2.stats.sampling_rate)
raise Exception(msg)
if trace1.id != trace2.id:
msg = "Trace ids do not match: %s != %s" % (trace1.id, trace2.id)
warnings.warn(msg)
samp_rate = trace1.stats.sampling_rate
# don't modify existing traces with filters
if filter:
trace1 = trace1.copy()
trace2 = trace2.copy()
# check data, apply filter and take correct slice of traces
slices = []
for _i, (t, tr) in enumerate(((pick1, trace1), (pick2, trace2))):
start = t - t_before - (cc_maxlag / 2.0)
end = t + t_after + (cc_maxlag / 2.0)
duration = end - start
# check if necessary time spans are present in data
if tr.stats.starttime > start:
msg = "Trace %s starts too late." % _i
raise Exception(msg)
if tr.stats.endtime < end:
msg = "Trace %s ends too early." % _i
raise Exception(msg)
if filter and start - tr.stats.starttime < duration:
msg = "Artifacts from signal processing possible. Trace " + \
"%s should have more additional data at the start." % _i
warnings.warn(msg)
if filter and tr.stats.endtime - end < duration:
msg = "Artifacts from signal processing possible. Trace " + \
"%s should have more additional data at the end." % _i
warnings.warn(msg)
# apply signal processing and take correct slice of data
if filter:
tr.data = tr.data.astype(np.float64)
tr.detrend(type='demean')
tr.data *= cosine_taper(len(tr), 0.1)
tr.filter(type=filter, **filter_options)
slices.append(tr.slice(start, end))
# cross correlate
shift_len = int(cc_maxlag * samp_rate)
_cc_shift, cc_max, cc = xcorr(slices[0].data, slices[1].data,
shift_len, full_xcorr=True)
cc_curvature = np.concatenate((np.zeros(1), np.diff(cc, 2), np.zeros(1)))
cc_convex = np.ma.masked_where(np.sign(cc_curvature) >= 0, cc)
cc_concave = np.ma.masked_where(np.sign(cc_curvature) < 0, cc)
# check results of cross correlation
if cc_max < 0:
msg = "Absolute maximum is negative: %.3f. " % cc_max + \
"Using positive maximum: %.3f" % max(cc)
warnings.warn(msg)
cc_max = max(cc)
if cc_max < 0.8:
msg = "Maximum of cross correlation lower than 0.8: %s" % cc_max
warnings.warn(msg)
# make array with time shifts in seconds corresponding to cc function
cc_t = np.linspace(-cc_maxlag, cc_maxlag, shift_len * 2 + 1)
# take the subportion of the cross correlation around the maximum that is
# convex and fit a parabola.
# use vertex as subsample resolution best cc fit.
peak_index = cc.argmax()
first_sample = peak_index
# XXX this could be improved..
while first_sample > 0 and cc_curvature[first_sample - 1] <= 0:
first_sample -= 1
last_sample = peak_index
while last_sample < len(cc) - 1 and cc_curvature[last_sample + 1] <= 0:
last_sample += 1
if first_sample == 0 or last_sample == len(cc) - 1:
msg = "Fitting at maximum lag. Maximum lag time should be increased."
warnings.warn(msg)
# work on subarrays
num_samples = last_sample - first_sample + 1
if num_samples < 3:
msg = "Less than 3 samples selected for fit to cross " + \
"correlation: %s" % num_samples
raise Exception(msg)
if num_samples < 5:
msg = "Less than 5 samples selected for fit to cross " + \
"correlation: %s" % num_samples
warnings.warn(msg)
# quadratic fit for small subwindow
coeffs, residual = scipy.polyfit(
cc_t[first_sample:last_sample + 1],
cc[first_sample:last_sample + 1], deg=2, full=True)[:2]
# check results of fit
if coeffs[0] >= 0:
msg = "Fitted parabola opens upwards!"
warnings.warn(msg)
if residual > 0.1:
msg = "Residual in quadratic fit to cross correlation maximum " + \
"larger than 0.1: %s" % residual
warnings.warn(msg)
# X coordinate of vertex of parabola gives time shift to correct
# differential pick time. Y coordinate gives maximum correlation
# coefficient.
dt = -coeffs[1] / 2.0 / coeffs[0]
coeff = (4 * coeffs[0] * coeffs[2] - coeffs[1] ** 2) / (4 * coeffs[0])
# this is the shift to apply on the time axis of `trace2` to align the
# traces. Actually we do not want to shift the trace to align it but we
# want to correct the time of `pick2` so that the traces align without
# shifting. This is the negative of the cross correlation shift.
dt = -dt
pick2_corr = dt
# plot the results if selected
if plot is True:
with MatplotlibBackend(filename and "AGG" or None, sloppy=True):
import matplotlib.pyplot as plt
fig = plt.figure()
ax1 = fig.add_subplot(211)
tmp_t = np.linspace(0, len(slices[0]) / samp_rate, len(slices[0]))
ax1.plot(tmp_t, slices[0].data / float(slices[0].data.max()), "k",
label="Trace 1")
ax1.plot(tmp_t, slices[1].data / float(slices[1].data.max()), "r",
label="Trace 2")
ax1.plot(tmp_t - dt, slices[1].data / float(slices[1].data.max()),
"g", label="Trace 2 (shifted)")
ax1.legend(loc="lower right", prop={'size': "small"})
ax1.set_title("%s" % slices[0].id)
ax1.set_xlabel("time [s]")
ax1.set_ylabel("norm. amplitude")
ax2 = fig.add_subplot(212)
ax2.plot(cc_t, cc_convex, ls="", marker=".", color="k",
label="xcorr (convex)")
ax2.plot(cc_t, cc_concave, ls="", marker=".", color="0.7",
label="xcorr (concave)")
ax2.plot(cc_t[first_sample:last_sample + 1],
cc[first_sample:last_sample + 1], "b.",
label="used for fitting")
tmp_t = np.linspace(cc_t[first_sample], cc_t[last_sample],
num_samples * 10)
ax2.plot(tmp_t, scipy.polyval(coeffs, tmp_t), "b", label="fit")
ax2.axvline(-dt, color="g", label="vertex")
ax2.axhline(coeff, color="g")
ax2.set_xlabel("%.2f at %.3f seconds correction" % (coeff, -dt))
ax2.set_ylabel("correlation coefficient")
ax2.set_ylim(-1, 1)
ax2.legend(loc="lower right", prop={'size': "x-small"})
# plt.legend(loc="lower left")
if filename:
fig.savefig(filename)
else:
plt.show()
return (pick2_corr, coeff)
[docs]def templates_max_similarity(st, time, streams_templates):
"""
Compares all event templates in the streams_templates list of streams
against the given stream around the time of the suspected event. The stream
that is being checked has to include all trace ids that are included in
template events. One component streams can be checked as well as multiple
components simultaneously. In case of multiple components it is made sure,
that all three components are shifted together. The traces in any stream
need to have a reasonable common starting time. The stream to check should
have some additional data to left/right of suspected event, the event
template streams should be cut to the portion of the event that should be
compared. Also see :func:`obspy.signal.trigger.coincidence_trigger` and the
corresponding example in the
`Trigger/Picker Tutorial
<https://tutorial.obspy.org/code_snippets/trigger_tutorial.html>`_.
- computes cross correlation on each component (one stream serves as
template, one as a longer search stream)
- stack all three and determine best shift in stack
- normalization is a bit problematic so compute the correlation coefficient
afterwards for the best shift to make sure the result is between 0 and 1.
>>> from obspy import read, UTCDateTime
>>> import numpy as np
>>> np.random.seed(123) # make test reproducible
>>> st = read()
>>> t = UTCDateTime(2009, 8, 24, 0, 20, 7, 700000)
>>> templ = st.copy().slice(t, t+5)
>>> for tr in templ:
... tr.data += np.random.random(len(tr)) * tr.data.max() * 0.5
>>> print(templates_max_similarity(st, t, [templ]))
0.922536411468
:param time: Time around which is checked for a similarity. Cross
correlation shifts of around template event length are checked.
:type time: :class:`~obspy.core.utcdatetime.UTCDateTime`
:param st: One or multi-component Stream to check against event templates.
Should have additional data left/right of suspected event (around half
the length of template events).
:type st: :class:`~obspy.core.stream.Stream`
:param streams_templates: List of streams with template events to check for
waveform similarity. Each template has to include data for all
channels present in stream to check.
:type streams_templates: list of :class:`~obspy.core.stream.Stream`
:returns: Best correlation coefficient obtained by the comparison against
all template events (0 to 1).
"""
values = []
for st_tmpl in streams_templates:
ids = [tr.id for tr in st_tmpl]
duration = st_tmpl[0].stats.endtime - st_tmpl[0].stats.starttime
st_ = st.slice(time - (duration * 0.5),
time + (duration * 1.5))
cc = None
for id_ in reversed(ids):
if not st_.select(id=id_):
msg = "Skipping trace %s in template correlation " + \
"(not present in stream to check)."
warnings.warn(msg % id_)
ids.remove(id_)
# determine best (combined) shift of multi-component data
for id_ in ids:
tr1 = st_.select(id=id_)[0]
tr2 = st_tmpl.select(id=id_)[0]
if len(tr1) > len(tr2):
data_short = tr2.data
data_long = tr1.data
else:
data_short = tr1.data
data_long = tr2.data
data_short = (data_short - data_short.mean()) / data_short.std()
data_long = (data_long - data_long.mean()) / data_long.std()
tmp = np.correlate(data_long, data_short, native_str("valid"))
try:
cc += tmp
except TypeError:
cc = tmp
except ValueError:
cc = None
break
if cc is None:
msg = "Skipping template(s) for station %s due to problems in " + \
"three component correlation (gappy traces?)"
warnings.warn(msg % st_tmpl[0].stats.station)
break
ind = cc.argmax()
ind2 = ind + len(data_short)
coef = 0.0
# determine correlation coefficient of best shift as the mean of all
# components
for id_ in ids:
tr1 = st_.select(id=id_)[0]
tr2 = st_tmpl.select(id=id_)[0]
if len(tr1) > len(tr2):
data_short = tr2.data
data_long = tr1.data
else:
data_short = tr1.data
data_long = tr2.data
coef += np.corrcoef(data_short, data_long[ind:ind2])[0, 1]
coef /= len(ids)
values.append(coef)
if values:
return max(values)
else:
return 0
# Remove once 0.11 has been released.
sys.modules[__name__] = DynamicAttributeImportRerouteModule(
name=__name__, doc=__doc__, locs=locals(),
original_module=sys.modules[__name__],
import_map={},
function_map={
"xcorrPickCorrection":
"obspy.signal.cross_correlation.xcorr_pick_correction",
"xcorr_3C": "obspy.signal.cross_correlation.xcorr_3c",
"templatesMaxSimilarity":
"obspy.signal.cross_correlation.templates_max_similarity"
})
if __name__ == '__main__':
import doctest
doctest.testmod(exclude_empty=True)