Core Functions

Whitening

move2obspy.whiten(data, Nfft, delta, freqmin, freqmax, plot=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

Correlation

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]

Moving-Window Cross-Spectral method

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 \(X(\nu) = F_{ref}(\nu) F_{cur}^*(\nu)\)

in which \({}^*\) 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, \(\phi( u)\), of the cross spectrum and is linearly proportional to frequency:

\(\phi_j = m. u_j, m = 2 \pi \delta t\)

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:

\(e_m = \sqrt{\sum_j{(\frac{w_j \nu_j}{\sum_i{w_i \nu_i^2}})^2}\sigma_{\phi}^2}\)

where \(w\) are weights, \(\nu\) are cross-coherences and \(\sigma_{\phi}^2\) is the squared misfit of the data to the modelled slope and is calculated as \(\sigma_{\phi}^2 = \frac{\sum_j(\phi_j - m \nu_j)^2}{N-1}\)

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.

WLS

move2obspy.linear_regression(xdata, ydata, weights=None, p0=None, intercept_origin=True, **kwargs)

Use linear least squares to fit a function, f, to data. This method is a generalized version of scipy.optimize.minpack.curve_fit(); allowing for Ordinary Least Square and Weighted Least Square regressions:

  • OLS through origin: linear_regression(xdata, ydata)
  • OLS with any intercept: linear_regression(xdata, ydata, intercept_origin=False)
  • WLS through origin: linear_regression(xdata, ydata, weights)
  • WLS with any intercept: linear_regression(xdata, ydata, weights, intercept_origin=False)

If the expected values of slope (and intercept) are different from 0.0, provide the p0 value(s).

Parameters:
  • xdata – The independent variable where the data is measured.
  • ydata – The dependent data - nominally f(xdata, ...)
  • weights – If not None, the uncertainties in the ydata array. These are used as weights in the least-squares problem. If None, the uncertainties are assumed to be 1. In SciPy vocabulary, our weights are 1/sigma.
  • p0 – Initial guess for the parameters. If None, then the initial values will all be 0 (Different from SciPy where all are 1)
  • intercept_origin – If True: solves y=a*x (default); if False: solves y=a*x+b.

Extra keword arguments will be passed to scipy.optimize.minpack.curve_fit().

Return type:tuple
Returns:(slope, std_slope) if intercept_origin is True; (slope, intercept, std_slope, std_intercept) if False.