Thursday, July 7, 2011

DISC course sequence

2012 SEG Distinguished Instructor Short Course

Elements of Seismic Dispersion: A somewhat practical guide to frequency-dependent phenomena

Christopher L. Liner, University of Houston


The classical meaning of the word dispersion is frequency-dependent velocity. Here we take a more general definition that includes not just wave speed, but also interference, attenuation, anisotropy, reflection characteristics and other aspects of seismic waves that show frequency-dependence. At first impression the topic seems self-evident: Of course everything is frequency dependent. But much of classical seismology and wave theory is non-dispersive: the theory of P and S waves, Rayleigh waves in a half-space, geometric spreading, reflection and transmission coefficients, head waves, etc. Yet when we look at real data, strong dispersion abounds. This course is a survey of selected frequency-dependent phenomena routinely encountered in reflection seismic data.

1. Time and frequency

The Fourier transform (FT) is a standard frequency analysis tool, but one that yields little information about combined time-frequency content. We will review the FT and its extension to short-time FT and continuous wavelet transform as representative examples of a broad class of time-frequency decomposition methods.

2. Vibroseis harmonics

The vibroseis source injects a long, slowly varying signal into the earth. We commonly find that new frequencies, or harmonics, not present in the sweep are present in the earth response. This interesting phenomena is discussed in relation to a more familiar process, that of human hearing. These harmonics are illustrative of a general property of nonlinear waves and interaction.

3. Near surface

Velocity dispersion is generally considered not to be an issue in seismic data processing. But that is a sloppy compression of reality. It is pretty nearly true for seismic body waves (P, S, and mode converted) moving around the far subsurface. In the near surface, however, velocities often show strong dispersion and the description is terribly inaccurate. This is especially the case in marine shooting over shallow water where, even in the 10-100 Hz band, velocities are observed well above and blow the speed of sound in water. This paradox arises because shallow water over an elastic earth forms a waveguide whose characteristics we will examine.

4. Anisotropy

This chapter deals with seismic velocity anisotropy and how it depends on frequency. We will restrict our comments here to velocity variation with respect to the vertical axis (VTI) in a horizontally layered earth. Of the sedimentary rock types, only shale is seen to be significantly anisotropic at the core, or intrinsic, scale. The question is how to calculate apparent anisotropic parameters of a layered medium as seen by very long waves. Backus (1962) solved this problem and his method can be applied to standard full wave sonic data. So where does dispersion come into all this? It is buried in the thorny question of the Backus averaging length.

5. Attenuation

The distinction between intrinsic and apparent frequency-dependent seismic properties is nowhere greater than in the area of attenuation. In this chapter constant Q and viscous theories of intrinsic attenuation are developed and compared to experimental intrinsic scattering attenuation data. Intrinsic attenuation is found to be compatible with the viscous theory, while constant Q yields a better explanation of scattering attenuation due to layering.

6. Interference

The preceding chapters have explored frequency-dependent phenomena related to acquisition and wave propagation, effects that would be seen and dealt with on prestack data. Data processing will remove or correct for these effects and be unseen by the interpreter. But dispersion effects (in our broad meaning) remain in the realm of final imaged data. First and foremost is the fundamental, unavoidable phenomena of interference. We will discuss selected topics in this broad field, including the thin bed, bandwidth effect on reflectivity, single frequency isolation, and reflection from a vertical transition zone.

7. Biot reflection

Many of the dispersion effects discussed previously contain information about the subsurface, but none as direct and important as the problem of reflectivity dispersion due to a poroelastic contact in the earth. We will review the nature of body waves in porous media, the characteristics of Biot reflection from an isolated interface, and wrap up with an introduction to Biot reflections in layered porous media.

Learning Goals

• Gain a broad understanding of dispersive phenomena and related investigation tools.
• Understand the fundamental difference between intrinsic and apparent dispersion phenomena.
• Improve knowledge of the reflection process beyond the classic model.
• Provide an appreciation of historical development and a deep guide to the literature for self-study.

Who Should Attend

The course is framed along the lines of acquisition, processing and interpretation to, hopefully, contain material of interest to the entire spectrum of seismic geophysicists. The mathematical level of the course is generally on the advanced undergraduate level, but deeper aspects are often included for advanced readers. Familiarity with the Fourier transform and related topics would be beneficial. In all cases, theoretical developments are illustrated by examples or case histories.


Christopher L. Liner joined the faculty of the University of Houston Department of Earth and Atmospheric Sciences in January 2008 and is now professor and associate director of the Allied Geophysical Laboratories industrial consortium. He earned a bachelor of science degree in geology from the Univer- sity of Arkansas in 1978, a master of science in geophysics from the University of Tulsa in 1980, and a Ph.D. in geophysics from the Center for Wave Phenomena at Colorado School of Mines in 1989. He began his career with Western Geophysical in London as a research geophysicist, followed by six years with Conoco. After working a year with Golden Geophysical, he served as a faculty member of the University of Tulsa Department of Geosciences from 1990 to 2004. From 2005 through 2007, Liner worked as research geophysicist with Saudi EXPEC Advanced Research Center, Dhahran, Saudi Arabia.

Liner’s research interests include petroleum reservoir characterization and monitor- ing, CO2-sequestration geophysics, advanced seismic-interpretation methods, seismic data analysis and processing, near surface, anisotropy, and seismic wave propagation. He served as editor of GEOPHYSICS in 1999–2001 and contributing editor to World Oil in 2010 and is an editorial board member for the Journal of Seismic Exploration. Liner has written many technical papers, abstracts for scientific meetings, the “Seismos” column in THE LEADING EDGE (since 1992), the “Seismos Blog,” and the textbook Elements of 3D Seismol- ogy, now in its second edition. Liner is a member of SEG, AAPG, AGU, the Seismological Society of America, and the European Academy of Sciences. In 2011, he was named an honorary member of the Geophysical Society of Houston.

Tuesday, July 5, 2011

Nonlinear wave propagation

Just a comment about harmonics and nonlinear waves. Whatever their physical origin in the machinery, vibroseis harmonics can be considered as generated by asymetric (nonlinear) up and down motion of the baseplate interacting with the earth. I have written about this elsewhere. We can say this is nonlinear interaction between the machine and the earth, although linear wave propagation occurs in the air and earth near surface. This kind of nonlinearity is interesting and common, but there is an other kind of much greater interest.

When wave equations are nonlinear, then we say the medium is nonlinear. New kinds of phenomena emerge not present in linear wave propagation. For example, when two linear waves meet the amplitude is the sum of the two waves. Not so with nonlinear waves. The primary feature of a nonlinear wave is that it's velocity depends on the peak amplitude. Linear wave velocity does not depend on peak amplitude. There are many examples, a classic being the Korteweg & de Vries equation describing solitons (non-dissapating waves) in a narrow water channel.

Why is all of this nonlinear stuff is of interest to geophysicists? Because the theory of poroelasticity (as developed originally by M. A. Biot in the 1950s) leads to wave equations that are nonlinear. when such a medium is probed by a harmonic signal (single frequency), asymmetries will develop in the propagated wave. The response will repeat the same number of times per second as the input, but because it is no longer a perfect sine or cosine a Fourier transform will reveal the original frequency and a series of harmonics. Encoded in the spacing and amplitude of the harmonics is key information about the medium, such a fluid viscosity and permeability. But it is an open question how to do in situ experiments to reveal this phenomena and, if we could, it is not known how to unravel the important information contained therein.

P.S. Steven Wolfram has written a fascinating interactive nonlinear wave equation explorer. A couple of plots are given below.

Linear waves interact by superposition just as we think they should.

Interaction of nonlinear waves is much more complicated and the details depend on the nature of the wave equation.