In recent years methods have been developed in the field of high-order statistics that can reliably estimate a mixed-phase wavelet from the noisy output of a convolutional process. These methods use high-order covariance functions of the data called cumulants, which retain phase information and allow recovery of the wavelet. The assumption is that the reflection coefficient series is a non-Gaussian, stationary, and statistically independent random process.The method described in this paper uses the fourth-order cumulant of the data, and a moving-average, noncausal parametric model for the wavelet. The fourth-order moment function of this wavelet matches the fourth-order cumulant of the data in a minimum mean-squared error sense. Numerical simulations demonstrate that the method can accurately estimate mixed-phase wavelets, even when the reflectivity has a distribution close to Gaussian.Three seismic data examples demonstrate possible uses of the method. In the first two, an average source wavelet is estimated from marine shot records acquired with air gun and water gun sources, respectively. These wavelets compare favorably with recorded far-field signatures modified for receiver ghosting. In the third example, a wavelet estimate from marine stacked data is used to correct the phase of the stacked section, resulting in zero-phase water bottom and salt top reflections.