Time-delay estimation using the Wigner-Ville distribution

Abstract

Accurately calculating time delays between signals is pivotal in many modern physics applications. One approach to estimating these delays is computing the cross-spectrum in the time-frequency domain. Linear time-frequency representations, such as the continuous wavelet transform (CWT), are widely used to construct these cross-spectra. However, it is well known that the frequency resolution is inherently limited by the localized nature of the convolving wavelet. Moreover, the functional form of the CWT cross-spectrum is not a proper correlation measure and typically requires post-processing smoothing. Conversely, quadratic representations achieve joint time-frequency resolution approaching the Gabor-Heisenberg limit while also providing an adequate measure of similarity between the signals. Motivated by these advantages, we propose a time-delay estimation method based on the Wigner-Ville Distribution (WVD). Considering nonstationary signals arising from two typical wave-physics scenarios, we show that the WVD yields more accurate time-delay estimates with lower uncertainty, particularly in the most energetic frequency bands.

Figures (6)

• Figure 1: Stochastic medium with spatial correlations considered in the first example. The nonstationary signals (panel b) were recorded by the receiver (green triangle in panel a), which was excited by a source (purple star in panel a). There is no amplitude prior to 8 s. The inset in panel (b) emphasizes that the velocity perturbation in the current medium induces very small time shifts between the signals, as it is also reflected in almost identical normalized frequency spectra (panel c).
• Figure 2: Comparison between the cross-spectrum (shown in log scale) computed using the CWT [Eq. \ref{['eq:cross_cwt']}] and WVD [Eq. \ref{['eq:wvd']}]. Panels in the bottom row show the corresponding time-frequency time-delay maps obtained with these methods. The central frequency of the Morlet wavelet in the CWT case was $\omega_{0} = 6$ Hz.
• Figure 3: Marginalization of the time-delay maps (see Fig. \ref{['fig:2']}) for three frequency bands (2.7 - 3.7 Hz, 6.0 - 8.0 Hz, 2.0 - 8.0 Hz). The shaded areas correspond to the standard deviation within each frequency interval. The WVD (orange dashed lines) yields more accurate time delays with lower uncertainty than the CWT (blue solid lines) across these frequency bands.
• Figure 4: A cropped version of the Marmousi velocity model (panel a) used in the second example. The surface acquisition geometry is represented by a source (red star) and a receiver (green triangle). The current signal is constructed from the reference, including a nonlinear time delay (panel b). This nonlinear time delay induces a pronounced difference in the corresponding frequency spectra (panel c).
• Figure 5: Time-delay maps estimated with the CWT (panel a) and WVD (panel b). The CWT was computed with a wavelet of central frequency $\omega_{0} = 2$ Hz. The WVD more accurately reproduces the trend of the true delay in the most energetic interval and consistently exhibits reduced artifacts.