Short-period (SP) P waves at low signal-to-noise ratios (S/Ns) usually appear as signals of about 1 Hz or higher frequencies riding on background noise with frequencies of 0.5 Hz or less. Often bandpass filtering is used to further attenuate the noise relative to the signal to improve the S/N. However, bandpass filtering introduces significant distortion into the signal. Here the use of optimum filters is described, such filters being constructed using autocorrelation functions of the signal and noise derived from noise and signal models. The autocorrelation of the impulse response of the recording system is used to describe the signal. For the noise, a tapered cosine wave is used that has the frequency of the main noise component. To compute the filter, the relative sizes of the signal and noise correlations should be proportional to the square of the S/N. Now, the noise can be estimated by subtracting the estimated signal from the observed seismogram. This property can be used to decide by trial and error on the S/N. The optimum S/N is found by gradually reducing the trial ratio starting from too high a value, until the noise has the amplitude expected and there is no evidence of significant signal in the noise estimate. Examples are given showing how the method can lift observed signals off the background noise with less distortion to the signal than usually occurs with bandpass filtering.