Can we perform statistical deconvolution by polynomial factorization?

We investigate the possibility of finding the source signature from multichannel seismic data by factorization of the Z-transforms of the seismic traces. In the convolutional model of the data, the source signature is the same from trace to trace within a shot gather, while the impulse response of the earth varies. In the noise‐free case, the roots of the Z-transform of the wavelet are the same from trace to trace, while the roots of the Z-transform of the impulse response of the earth must move from trace to trace. It follows that the roots of the wavelet can be found by the invariance of their positions. We demonstrate this using a simple wedge model. No assumptions about the length of the wavelet or the statistical properties of the impulse response of the earth are required. It is shown that this idea cannot work on real seismic data. There are two difficulties which we regard as insuperable. First, even without noise, a seismic trace cannot be regarded as a complete convolution, because the data are always truncated. This causes the factorization to be inexact: the wavelet roots move from trace to trace and are indistinguishable from the roots of the earth’s impulse response. Second, the addition of a small amount of noise alters the root pattern unpredictably from trace to trace and the roots of the wavelet are again indistinguishable from the roots of the earth’s impulse response. We conclude that it is impossible to identify and extract the true source signature from real seismic data using no assumptions about the statistical properties of the impulse response of the earth. We propose that the signature should be measured.