The S‐transform with windows of arbitrary and varying shape

The S‐transform is an invertible time‐frequency spectral localization technique which combines elements of wavelet transforms and short‐time Fourier transforms. In previous usage, the frequency dependence of the analyzing window of the S‐transform has been through horizontal and vertical dilations of a basic functional form, usually a Gaussian. In this paper, we present a generalized S‐transform in which two prescribed functions of frequency control the scale and the shape of the analyzing window, and apply it to determining P‐wave arrival time in a noisy seismogram. The S‐transform is also used as a time‐frequency filter; this helps in determining the sign of the P arrival.