In a shallow water waveguide, the low-frequency acoustic field can be viewed as a sum of normal modes. Warping transform provides an effective tool to filter the normal modes from the received signal of a single hydrophone, which can be used for source ranging and geoacoustic inversion. However, it should be noted that the conventional warping operator h(t)=√t2+t2r is only valid for a signal consisting of reflection dominated modes, where r represents the source range. In a waveguide with a strong thermocline or a surface channel where refracted modes dominate the received sound field, the dispersive characteristics of the waveguide become different and the performance of the warping operator h(t) = √t2+t2r will be significantly degraded. In this paper, the dispersive characteristics and warping transform of the refractive normal modes in a waveguide with a linearly decreased sound speed profile are discussed. The formulae for the horizontal wavenumber, the phase in frequency domain and the instantaneous phase in time domain of the refractive mode are deduced. Based on these formulae, the time warping and frequency warping operators verified by the simulated data are presented. Through time-axis stretching or compression, the time warping operator h(t)=tr−t−2, where tr = r/c(h) and c(h) represents the bottom sound speed, can transform the refracted modes into single-tone components of frequencies determined by source range, sound speed gradient of water, bottom sound speed and mode number. The frequency warping operator h(f)=Df3, where D is a constant, can transform the refracted modes into separable impulsive sequences through frequency-axis stretching or compression and the time delay of the impulsive sequences changes linearly with the source range. As the warped modes are separated in time domain or frequency domain, these two operators can be used for filtering the refracted normal modes from the received signal. The theories in this paper are also applicable for refractive modes in the waveguide with a linearly increased sound speed profile or a linear variation of the square of the index of refraction (n2-linear sound speed profile).
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