The stationary wavelet transform (SWT) is a variation of the traditional discrete wavelet transform (DWT) that preserves the original length of the data trace at each wavelet scale. With an appropriate choice of wavelet basis, a seismic signal can have a sparse impulsive representation in the wavelet domain. Signals spiking above some picked noise floor threshold can then be boosted, or noise below a threshold attenuated. The compact time-frequency localization of the wavelet domain provides more opportunity for signal and noise to cleanly separate, while the oversampled transformed domain of the SWT makes this application practical. The process of thresholding to remove noise can zero out noisy wavelet scales, where the signal never rises above the noise floor. Fortunately, events on seismic traces typically encompass mutiple wavelet scales. In the SWT, the original time sampling is preserved at every wavelet scale, making them directly comparable. We can thus estimate the missing signal at a strongly noise-contaminated wavelet scale from adjacent scales that are less contaminated by noise, resulting in an increase in the effective signal bandwidth. We demonstrate signal enhancement in the SWT domain using synthetic data. We expect that the method should be more robust against noise than traditional Fourier-domain techniques, which boost signal and noise at the same frequency by the same amount.