Time delay estimation as a function of frequency for dispersive elastic waves

Ultrasonic materials characterization requires measurements of the elastic wave's velocity to within 0.1%. When the specimen is dispersive, conventional techniques used to gain such accuracy fail due to the changing signal shape as the ultrasonic wave propagates. Given two measurements of the ultrasonic signal, we explore the problem of estimating the time delay between these measured signals. We treat this problem as the estimation of the phase spectrum of an unknown filter, which represents the transfer function caused by the specimen. We address several different methods for estimating the time delay as a function of frequency. These methods include the classic method of using the phase difference between the measurement spectrums; several least squares approaches; and a maximum likelihood estimate of the time delay as a function of frequency. We show that the uncertainty of our time delay estimation is directly related to the uncertainty of our measurements, reducing our problem to the classical problem of power spectrum estimation. Next, we use the knowledge that our ultrasonic measurements are time and bandwidth limited. We prove that we can approximate our elastic wave with a finite dimensional basis. Then we provide a method for constructing an “optimal” basis, where the optimality of the basis is shown by the centering of the basis within the space of effectively time and band limited sequences. We apply the approximation of the signal with the finite dimensional basis to the time delay estimation. Next we illustrate the time delay estimators with three examples: (1) A computer simulation of dispersive propagation, (2) an ultrasonic measurement in a nondispersive steel specimen and (3) an ultrasonic measurement of a dispersive surface acoustic wave. In all three examples, we see that these estimates are biased.