# Acoustic impedance estimation

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

The goal in amplitude inversion is to obtain a broadband impedance model of a reservoir zone from a band-limited time-migrated CMP-stacked data. Alternatively, amplitude inversion can be applied to the CRP stack derived from prestack time migration or the intercept AVO attribute section. The trace amplitudes on a migrated CMP-stack, CRP stack, or the intercept section represent the one-dimensional (1-D), primary, P-wave reflectivity series associated with vertical incidence. Care must be given in processing to preserve relative amplitudes and increase vertical and lateral resolution of the seismic data. Reflection amplitude variations on any of these sections may indicate changes in some rock parameters, such as porosity or fluid saturation, within the reservoir unit.

By considering a sparse-spike reflectivity model of the earth, we first obtain a broad-band reflectivity section. Mathematically, the process involves minimizing the quantity that is equal to the sum of the absolute values of the trace amplitudes on the migrated CMP stack. Computation of the impedance series at a CMP location then involves simple integration of the broadband reflectivity series.

Recall from analysis of amplitude variation with offset that we assume a seismic source that generates a compressional plane wave propagating down into the earth at an angle from the vertical. When this incident plane wave encounters a layer boundary, which is assumed to be flat with no curvature, it is partitioned into four components — reflected and refracted P- and S-waves. The Zoeppritz equations (12a,12b,12c,12d) describe the amplitudes of the partitioned wave components. Aside from the layer parameters — density, P- and S-wave velocities, the wave amplitudes depend upon the angle of incidence (analysis of amplitude variation with offset). Implicit to CMP stacking, we must assume that the reflected P-wave amplitude variation with angle of incidence measured along the normal-moveout trajectory associated with the reflection event itself on the CMP gather is negligible. This assumption becomes critical for shallow targets at large source-receiver offsets. A test of the offset range used in stacking for the time window of interest is crucial for the meaningful interpretation of a poststack amplitude inversion result. By making the vertical-incidence assumption, we also assume that there is no P-to-S conversion within the offset range used in CMP stacking.

 ${\displaystyle \cos \varphi _{1}A_{1}+{\frac {\alpha _{1}}{\beta _{1}}}\sin \psi _{1}B_{1}+{\frac {\alpha _{1}}{\alpha _{2}}}\cos \varphi _{2}A_{2}-{\frac {\alpha _{1}}{\beta _{2}}}\sin \psi _{2}B_{2}=\cos \varphi _{1},}$ (12a)

 ${\displaystyle -\sin \varphi _{1}A_{1}+{\frac {\alpha _{1}}{\beta _{1}}}\cos \psi _{1}B_{1}+{\frac {\alpha _{1}}{\alpha _{2}}}\sin \varphi _{2}A_{2}+{\frac {\alpha _{1}}{\beta _{2}}}\cos \psi _{2}B_{2}=\sin \varphi _{1},}$ (12b)

 ${\displaystyle -\cos 2\psi _{1}A_{1}-\sin 2\psi _{1}B_{1}+{\frac {\rho _{2}}{\rho _{1}}}\cos 2\psi _{2}A_{2}-{\frac {\rho _{2}}{\rho _{1}}}\sin 2\psi _{2}B_{2}=\cos 2\psi _{1},}$ (12c)

 ${\displaystyle \sin 2\varphi _{1}A_{1}-{\frac {\alpha _{1}^{2}}{\beta _{1}^{2}}}\cos 2\psi _{1}B_{1}+{\frac {\rho _{2}\beta _{2}^{2}\alpha _{1}^{2}}{\rho _{1}\beta _{1}^{2}\alpha _{2}^{2}}}\sin 2\varphi _{2}A_{2}+{\frac {\rho _{2}\alpha _{1}^{2}}{\rho _{1}\beta _{1}^{2}}}\cos 2\psi _{2}B_{2}=\sin 2\varphi _{1}.}$ (12d)

 ${\displaystyle c={\frac {A_{1}}{A_{0}}}={\frac {\rho _{2}\alpha _{2}-\rho _{1}\alpha _{1}}{\rho _{2}\alpha _{2}+\rho _{1}\alpha _{1}}}.}$ (9)

The special case of the Zoeppritz equation for vertically incident P-to-P reflection amplitude takes a simple form in which the amplitude depends only on the impedance values of the layer above and the layer below the reflecting boundary (equation 9). Based on this special case of the Zoeppritz equation, computing the impedance series from a reflectivity series involves a simple integration of the latter. The assumption of vertical incidence also requires that the stacked section input to inversion must be migrated.

Results of poststack amplitude inversion must be viewed within the realm of the underlying critical assumptions we make about seismic amplitudes. Amplitude inversion of poststack time-migrated data should be limited to earth models with low-relief structures and stratigraphic targets. Many of the assumptions indicated above are violated in the presence of reflectors with dip and curvature. To account for the dip and curvature effects, at least kinematically, amplitude inversion must be applied to prestack time-migrated data. Moreover, limitations in the removal of multiple reflections, linear and random noise must be taken into consideration when interpreting anomalies observed on an acoustic impedance section.