Core Functions

msnoise.move2obspy.myCorr(data, maxlag, plot=False, nfft=None)

This function takes ndimensional data array, computes the cross-correlation in the frequency domain and returns the cross-correlation function between [-maxlag:maxlag].

Parameters:

• data (numpy.ndarray) – This array contains the fft of each timeseries to be cross-correlated.

• maxlag (int) – This number defines the number of samples (N=2*maxlag + 1) of the CCF that will be returned.

Return type:

numpy.ndarray

Returns:

The cross-correlation function between [-maxlag:maxlag]

msnoise.move2obspy.myCorr2(data, maxlag, energy, index, plot=False, nfft=None, normalized=False)

This function takes ndimensional data array, computes the cross-correlation in the frequency domain and returns the cross-correlation function between [-maxlag:maxlag].

Parameters:

• data (numpy.ndarray) – This array contains the fft of each timeseries to be cross-correlated.

• maxlag (int) – This number defines the number of samples (N=2*maxlag + 1) of the CCF that will be returned.

Return type:

numpy.ndarray

Returns:

The cross-correlation function between [-maxlag:maxlag]

msnoise.move2obspy.pcc_xcorr(data, maxlag, energy, index, plot=False, nfft=None, normalized=False)

Parameters:

• data –

• maxlag –

• energy –

• index –

• plot –

• nfft –

• normalized –

Returns:

msnoise.move2obspy.whiten(data, Nfft, delta, freqmin, freqmax, plot=False, returntime=False)

This function takes 1-dimensional data timeseries array, goes to frequency domain using fft, whitens the amplitude of the spectrum in frequency domain between freqmin and freqmax and returns the whitened fft.

Parameters:

• data (numpy.ndarray) – Contains the 1D time series to whiten

• Nfft (int) – The number of points to compute the FFT

• delta (float) – The sampling frequency of the data

• freqmin (float) – The lower frequency bound

• freqmax (float) – The upper frequency bound

• plot (bool) – Whether to show a raw plot of the action (default: False)

Return type:

numpy.ndarray

Returns:

The FFT of the input trace, whitened between the frequency bounds

msnoise.move2obspy.whiten2(fft, Nfft, low, high, porte1, porte2, psds, whiten_type)

This function takes 1-dimensional data timeseries array, goes to frequency domain using fft, whitens the amplitude of the spectrum in frequency domain between freqmin and freqmax and returns the whitened fft.

Parameters:

• data (numpy.ndarray) – Contains the 1D time series to whiten

• Nfft (int) – The number of points to compute the FFT

• delta (float) – The sampling frequency of the data

• freqmin (float) – The lower frequency bound

• freqmax (float) – The upper frequency bound

• plot (bool) – Whether to show a raw plot of the action (default: False)

Return type:

numpy.ndarray

Returns:

The FFT of the input trace, whitened between the frequency bounds

msnoise.move2obspy.smooth(x, window='boxcar', half_win=3)

some window smoothing

msnoise.move2obspy.getCoherence(dcs, ds1, ds2)

msnoise.move2obspy.mwcs(current, reference, freqmin, freqmax, df, tmin, window_length, step, smoothing_half_win=5)

The current time series is compared to the reference. Both time series are sliced in several overlapping windows. Each slice is mean-adjusted and cosine-tapered (85% taper) before being Fourier- transformed to the frequency domain. $F_{cur}(\nu )$ and $F_{ref}(\nu )$ are the first halves of the Hermitian symmetric Fourier-transformed segments. The cross-spectrum $X(\nu )$ is defined as $X(\nu )=F_{ref}(\nu )F_{cur}^{\ast }(\nu )$

in which ${}^{\ast }$ denotes the complex conjugation. $X(\nu )$ is then smoothed by convolution with a Hanning window. The similarity of the two time-series is assessed using the cross-coherency between energy densities in the frequency domain:

$C(\nu )=\frac{|\overline{X(\nu ))}|}{\sqrt{|\overline{F_{ref}(\nu ){|}^2}|\overline{F_{cur}(\nu ){|}^2}}}$

in which the over-line here represents the smoothing of the energy spectra for $F_{ref}$ and $F_{cur}$ and of the spectrum of $X$. The mean coherence for the segment is defined as the mean of $C(\nu )$ in the frequency range of interest. The time-delay between the two cross correlations is found in the unwrapped phase, $\varphi (u)$, of the cross spectrum and is linearly proportional to frequency:

${\varphi }_j=m.u_j,m=2\pi \delta t$

The time shift for each window between two signals is the slope $m$ of a weighted linear regression of the samples within the frequency band of interest. The weights are those introduced by [Clarke2011], which incorporate both the cross-spectral amplitude and cross-coherence, unlike [Poupinet1984]. The errors are estimated using the weights (thus the coherence) and the squared misfit to the modelled slope:

$e_m=\sqrt{\sum _j(\frac{w_j{\nu }_j}{\sum _i{w}_i{\nu }_i^2}{)}^2{\sigma }_{\varphi }^2}$

where $w$ are weights, $\nu$ are cross-coherences and ${\sigma }_{\varphi }^2$ is the squared misfit of the data to the modelled slope and is calculated as ${\sigma }_{\varphi }^2=\frac{\sum _j({\varphi }_j-m{\nu }_j{)}^2}{N-1}$

The output of this process is a table containing, for each moving window: the central time lag, the measured delay, its error and the mean coherence of the segment.

Warning

The time series will not be filtered before computing the cross-spectrum! They should be band-pass filtered around the freqmin-freqmax band of interest beforehand.

Parameters:

• current (numpy.ndarray) – The “Current” timeseries

• reference (numpy.ndarray) – The “Reference” timeseries

• freqmin (float) – The lower frequency bound to compute the dephasing (in Hz)

• freqmax (float) – The higher frequency bound to compute the dephasing (in Hz)

• df (float) – The sampling rate of the input timeseries (in Hz)

• tmin (float) – The leftmost time lag (used to compute the “time lags array”)

• window_length (float) – The moving window length (in seconds)

• step (float) – The step to jump for the moving window (in seconds)

• smoothing_half_win (int) – If different from 0, defines the half length of the smoothing hanning window.

Return type:

numpy.ndarray

Returns:

[time_axis,delta_t,delta_err,delta_mcoh]. time_axis contains the central times of the windows. The three other columns contain dt, error and mean coherence for each window.