# Inversion methods for data modeling

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

Historically, the term *seismic inversion* often has been used within the context of acoustic impedance estimation from a broad-band time-migrated CMP-stacked data. This narrow meaning commonly is referred to as trace inversion. In practice, however, seismic inversion has a broader scope of applications which can be grouped in two categories — *data modeling* and *earth modeling.*

What we do in seismic data processing described in fundamentals of signal processing through 3-D seismic exploration is based largely on data modeling. An observed seismic wavefield can be described in two parts — traveltimes and amplitudes. Seismic amplitudes are more prone to the detrimental effects of noise as compared to traveltimes. Hence, in seismic inversion, we almost always treat traveltimes and amplitudes separately. When modeling the observed data, we either model the traveltimes or amplitudes. When modeling the earth, again, we use the traveltimes, as in structural inversion (structural inversion), or amplitudes, as in stratigraphic inversion (reservoir geophysics).

It is most appropriate to provide a summary list of applications of seismic inversion for data modeling that are spread throughout fundamentals of signal processing to 3-D seismic exploration.

- Deconvolution is based on modeling a one-dimensional (1-D) seismogram by optimum Wiener filtering for a minimum-phase estimate of the source wavelet, to predict multiples, and obtain an estimate of white reflectivity series (the convolutional model).
- We model traveltime deviations on moveout-corrected CMP gathers to estimate surface-consistent shot and receiver residual statics (residual statics corrections).
- We model refracted arrival times to estimate, again, surface-consistent shot and receiver intercept time anomalies, and thus obtain shot and receiver refraction statics (refraction statics corrections).
- One type of formulation of the discrete Radon transform is by generalized linear inversion. The discrete Radon transform is used to model a CMP gather so as to attenuate multiples and random noise, while compensating for missing data and finite cable length in recording (the radon transform).
- We model the seismic signal represented by reflection events assumed to be linear from trace to trace and attenuate random noise uncorrelated from trace to trace by using spatial prediction filters (linear uncorrelated noise attenuation).
- Based on the same data modeling concept, we design spatial prediction filters to perform trace interpolation (processing of 3-D seismic data).
- Data modeling also can be used in the design of a three-dimensional (3-D) dip-moveout correction operator which accounts for irregular spatial sampling and undersampling of recorded data (processing of 3-D seismic data).

Most data modeling applications are based on the theory of *generalized linear inversion.* Although each of the applications listed above is treated in the appendixes of the respective chapters, for completeness, a mathematical summary based on the generalized linear inversion theory is provided in Sections J.1, J.2 and J.3.

## See also

- Introduction to earth modeling in depth
- Inversion procedures for earth modeling
- Velocity-depth ambiguity
- Model representation and visualization