Module eacf

source code

Computes the envelope auto-correlation function. This function is estimated by computing the power spectrum of a smoothed power spectrum of a time series. It is ideally suited to discover periodicities in the power spectrum, as is often the case for solar-like oscillators (cf the asymptotic relation of Tassoul), and can thus be used to estimate the mean large separation (mean frequency spacing between two consecutive radial modes).

As an example we take the Kepler red giant KIC3744043, which shows a beautiful spectrum of solar-like oscillations:

]include figure]]powerspec_KIC3744043.png]

To compute the EACF, we first import:

>>> import numpy as np
>>> from ivs.timeseries.eacf import eacf

Red giants have spacings in the power spectrum between, say, 1.0 and 15.0, so this is the interval in which we will compute the EACF:

>>> spacings = np.arange(1.0, 15.0, 0.005)

The figure above shows that the oscillations are only present between (roughly) 80 microHz and 135 microHz, so we will limit ourselves to this frequency interval:

>>> minFreq, maxFreq = 80.0, 135.0

The EACF is now computed as:

>>> autoCorrelation, croppedFreqs, smoothedSpectrum = eacf(freq, spec, spacings, 2.0, minFreq, maxFreq)

The value '2.0' is related to the FWHM of the (Hann) kernel that is used to smooth the power spectrum of the signal, on which we will come back later.

The EACF looks as follows:

]include figure]]eacf_smooth15.png]

The highest peak is caused by the main periodicity in the power spectrum. This does not, however, correspond directly to the mean large separation, but to half the large separation. The reason is that almost in the middle between each two radial modes there is a dipole mode. To compute the large separation we therefore have to enter:

>>> meanLargeSeparation = 2.0 * spacings[np.argmax(autoCorrelation)]
>>> meanLargeSeparation
9.8599999999998325

How did we choose the value 2.0 (microHz) for the FWHM of the smoothing kernel? The smoothing is done by convolving the power spectrum with a Hann kernel with a certain width. If you take the width too large, the spectrum will be too heavily smoothed so that no periodicities can be detected. If you take the width too small, the EACF will contain too many features, which makes it difficult to recognize the right peak. As a rule-of-thumb you can take the kernel width to be 1/8 of your initial estimate of the mean large separation. If you don't have such an initial estimate, you can estimate nu_max, which is the frequency of maximum power (= the location of the gaussian envelope of the power excess), and derive from that value a first estimate for the large separation:

>>> nuMax = 120.0                                      # for KIC3744043
>>> initialEstimateLargeSep = 0.24 * nuMax**0.78       # general approximation for red giants
>>> kernelWidth = initialEstimateLargeSep / 8.0        # rule of thumb
>>> autoCorrelation, croppedFreqs, smoothedSpectrum = eacf(freq, spec, spacings, kernelWidth, minFreq, maxFreq)