Before revision:  The motivation to introduce these new GPWs, amplitude-based, lies in the poor behavior of the phase-based GPWs in the pre-asymptotic regime, which will be addressed by avoiding high degree polynomials within an exponential.
After revision:  The motivation to introduce these new GPWs, amplitude-based, lies in the poor behavior of the phase-based GPW approximations in the pre-asymptotic regime, which will be addressed by avoiding high degree polynomials within an exponential.