Hermitian hat wavelet design for singularity-detection in the PARAGUAY river level data analyses

The direct differentiation of a noisy signal ds/dt is known to be inaccurate. Differentiation can be improved by employing the Dirac delta-function introduced into a convolution product denoted by x and then integrated by parts: ds/dt = ds/dt x delta = -sx d delta/dt. The Schwartz Gaussian representation of the delta function is then explicitly used in the differentiation. It turns out that such a convolution approach to the first and the second derivatives produces a pair of mother wavelets the combination of which is the complex generalization of the Mexican hat called a Hermitian hat wavelet. It is shown that the Hermitian filter is a single oscillation wavelet having much lower frequency bandwidth than the Morlet or Garbor wavelet. As a result of Nyquist theorem, a fewer number of grid points would be needed for the discrete convolution (filter) operation. Therefore, the singularity characteristic will not be overly smeared and the noise can be smoothed away. The phase plot of the Hermitian wavelet transform in terms of the time scale and frequency domains reveal a bifurcation discontinuity of a noisy cusp singularity at the precise location of the singularity as well as the scale nature of the underlying dynamics. This phase plot is defined as theta(t/a) = tan(-1) [(ds/dt)/(-d(2)s/dt(2))] = tan(-1) [((d delta(t/a)/dt) x s)/((d(2) delta(t/a)/dt(2)) x s)] applied to a real world data of the Paraguay river levels.