Because seismic reflection data are band limited, acoustic impedance profiles derived from them are nonunique. The conventional inversion methods counter the nonuniqueness either by stabilizing the answer with respect to an initial model or by imposing mathematical constraints such as sparsity of the reflection coefficients. By making a nominal assumption of an earth model locally consisting of a stack of homogeneous and horizontal layers, we have formulated a set of linear equations in which the reflection coefficients are the unknowns and the recursively integrated seismic trace constitute the data. Drawing only on first principles, the Zoeppritz equation in this case, the approach makes a frontal assault on the problem of reconstructing reflection coefficients from band-limited data. The local layer-cake assumption and the strategy of seeking a singular value decomposition solution of the linear equations counter the nonuniqueness, provided that the objective is to reconstruct a smooth version of the impedance profile that includes only its crude structures. Tests on synthetic data generated from elementary models and from measured logs of acoustic impedance demonstrated the efficacy of the method, even when a significant amount of noise was added to the data. The emergence of consistent estimates of impedance, approximating the original impedance, from synthetic data generated for several frequency bands has inspired our confidence in the method. The other attractive outputs of the method are as follows: (1) an accurate estimate of the impedance mean, (2) an accurate reconstruction of the direct-current (DC) frequency of the reflectivity, and (3) an acceptable reconstruction of the broad outline of the original impedance profile. These outputs can serve as constraints for either more refined inversions or geologic interpretations. Beginning from the restriction of band-limited data, we have devised a method that neither requires a starting input model nor imposes mathematical constraints on the earth reflectivity and still yielded significant and relevant geologic information.
Design Exploration of Hybrid CMOS and Memristor Circuit by New Modified Nodal Analysis
