The Continuous wavelet transform is given by

(1)

where the * represents the complex conjugate of the function. The CWT is simply a correlation of the signal, f(t), with a scaled basis function y*(t). In general, the basis function is complex and is termed the “mother wavelet”. The wavelet coefficient, W(a,t) is also complex and can be represented in the Fourier domain as

. (2)

The CWT is a linear operation and has an exact inverse transform given by the double integral

(3)

where C is found from a Parseval-like integral

(4)

that requires the basis function to have zero mean for this integral to be bounded at . We use Morlet’s wavelet (Goupillaud et al. 1985) in which

. (5)

As can be seen, Morlet’s wavelet is a heavily damped sinusoid around t=0 allowing the scale parameter to be approximately interpreted as Fourier period (inverse frequency) in the CWT domain since we use .

Hard Thresholding

An estimate of the noise power is made in a time window before the first arrival of the event. This is then used with the following criterion to remove wavelet coefficients from the map. Coefficients are kept if they are greater than the mean plus 3 standard deviations (99.8% confidence level; Starck et al. 2010), otherwise they are set to zero. Mathematically, this is represented by

(6)

where

(7)

(8)

and

. (9)

The limits t2 and t1 represent the limits of the noise time lag window.

Soft Thresholding

Alternatively, thresholding can be done in a somewhat less severe manner by taking into account the amplitude of the noise in a “keep or shrink” process where the coefficients that survive are modified by the inferred noise level:

(10)

where

. (11)

Donoho (1995) suggested a thresholding factor for the standard deviation that is equal to , where n is the number of CWT coefficients. This number is often close to the value of 3 as shown in equation (9) for realistic time series.

References

Goupillaud, P., A. Grossmann, and J. Morlet, 1985, Cycle-octave and related transforms in seismic signal analysis, Geoexploration, 23, pp. 85-102.

Starck, J.-L., F. Murtagh, and J. M. Fadili, 2010, Sparse Image and Signal Processing, Cambridge University Press, New York.

Donoho, D. L., 1995, De-Noising by soft-thresholding, IEEE Trans. Inf. Theory, 41, 613-627.