Estimation of the Three-component Impulse Response of the Earth

All direct imaging methods assume that the input data have been converted to a band-limited estimate of the impulse response of the medium. In modern passive array imaging the primary data are receiver functions computed by deconvolving teleseismic P wave data recorded on the radial component with the data recorded on the vertical (longitudinal) component. Standard receiver function estimates are founded on an assumption that the earth is transparent to P waves as shown by the fact the actual output of the deconvolution operator applied to the vertical data is a band-limited delta function at zero lag. I show that the problem should be recast with the incident wavelet at the base of the imaging volume as an unknown in the problem. This changes the problem from the simple, linear-inverse problem of deconvolution to a nonlinear inverse problem with an Earth model and an unknown wavelet as fundamental components of the solution. I show how synthetics derived from an Earth model can be used to provide an improved estimate of the three-component impulse response of the medium. I demonstrate the validity of this theoretical framework using synthetic seismograms computed from radially symmetric Earth models with various input wavelets. I examined the sensitivity of this new algorithm to errors in the assumed (radially symmetric) Earth model. I find the wavelet estimate has a weak dependence on the Earth model, but the impulse response estimates have a strong dependence on the assumed model. I find when a wavelet is estimated from one data component alone, the estimated impulse response of necessity always comes close to matching the synthetics computed for the trial model.. This suggest any scheme for finding an optimal Earth model must use statistical averages of data from multiple components and stations to resolve this ambiguity.