Limitations on impedance inversion of band-limited reflection data

A seismic approximate impedance log is often the ultimate output in the sequence of seismic data-processing steps. In principle, the true acoustic impedance is obtainable from the inversion of full-band impulse response. Because the seismic data necessarily is band limited, its inversion obviously would produce an approximate impedance log. A question addressed is how the true and reconstructed logs are related and a mathematical relationship between the two is derived without the assumptions required in an existing derivation of the same result. The deductions also include the contribution of an individual seismic frequency in the reconstruction of the impedance and, in particular, a simple formula for the contribution of direct-current (dc) frequency, which never is recorded but is required to supplement the inversion. Analytical expressions are derived for the reconstructed impedance corresponding to any given frequency band for two cases: a simple discontinuity and a bed sandwiched between two similar half-spaces. While analyzing the results, some conclusions pertinent to seismic exploration are drawn. A criterion is formulated to decide what should be termed a thin bed within a homogeneous half-space in the context of a white band of frequencies, in contrast to a single dominant frequency which is the basis of the Widess criterion.