Elements of Seismic Dispersion: A somewhat practical guide to frequency-dependent phenomena
SEG Distinguished Instructor Short Course 2012

Chris 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. Much of classical seismology and wave theory is nondispersive: the theory of P- and S-waves, Rayleigh waves in a half-space, geometric spreading, reflection and transmission coefficients, head waves, and so forth. Yet when we look at real data, strong dispersion abounds. This course is a survey of selected frequency-dependent phenomena that routinely are encountered in reflection-seismic data.


Time and frequency

The Fourier transform (FT) is a standard frequency-analysis tool, but it 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.

Vibroseis harmonics

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

Near surface

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


In this section, we consider 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.


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


The preceding sections 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 those effects and will be unseen by the interpreter. However, dispersion effects (in our broad meaning) remain in the realm of final imaged data. First and foremost is the fundamental, unavoidable phenomenon 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.

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 resulting from a poroelastic contact in the earth. We will review the nature of body waves in porous media and the characteristics of Biot reflection from an isolated interface and will end 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 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 often are included for advanced readers. Familiarity with the Fourier transform and related topics will be beneficial. In all cases, theoretical developments are illustrated by examples or case histories.


Chris Liner

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 University 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 monitoring, 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 Seismology, 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.


Please tell us a little bit about yourself. (e.g. your education and work experience, why you became a geophysicist, etc.)

I went to college at the University of Arkansas in my recently-adopted home town of Fayetteville. Preceding me at the University of Arkansas were my older brothers Jeff (to whom my DISC book is dedicated) and Robert, both of whom majored in geology. We were really the first generation of our family to attend college, although my mother had taken some college classes. After a dead-end majoring in entomology, it seemed natural to consider geology as a major. My first meeting was with Dr. Walter Manger -- paleontologist, stratigrapher, and tireless promoter of geology -- who looked over my transcript, discussed my interests, and declared: "You are a geophysicist!" Manger became a life-long mentor, role model and friend. The University of Arkansas did not offer a geophysics degree, so I took a degree in geology with geophysics option. That meant taking advanced math and hard-core physics classes, as well as a full geology curriculum. I think this mix of geology/math/physics has shaped my interests and success ever since. In today's university, it is all too easy to specialize early and become an uber-expert in one narrow topic. I have known geophysicists with a top-school PhD who are lost in conversations about unconformities, soft-sediment deformation, or stratigraphy.

After completing my BS degree at Arkansas, I earned a Masters under Dr. Stan Laster at the University of Tulsa. Stan was a great advisor, and from him I began to absorb the lore of geophysics at the highest level: MIT and the GAG group, imaging and what was going on with Jon Claerbout at Stanford, a love of wave propagation, a deep respect for great geophysicists and great papers. Through his contact with Ken Larner (then at Western Geophysical), Stan got me an interview that turned into a research job in London. I defended my MS on Aug 15 and got on the plane the next day.

After working in London for a year, I joined Conoco in Oklahoma City as an exploration geophysicist. My geology/geophysics background was a perfect match to the job. Somewhere along the line I began asking myself this question: "Would you rather retire with 40 years experience and an MS degree, or 36 years and a PhD?" The answer lead me to the Colorado School of Mines in 1986. Whatever plans I had were instantly reset when I had lunch the first day with Norm Bleistein and Jack Cohen. They were applied mathematicians working on deep seismic problems, exactly the kind of thing Stan Laster had talked about. Bleistein and Cohen were also kind, understanding, brilliant and patient. They would need all of that to get a reformed geologist through a PhD at Colorado School of Mines. I recall Norm telling me after the PhD qualifying exam that lengthy discussion by the committee was cut short by his comment: "Come on, the guy was a geologist last week." Jack kept a slide rule duct-taped on the wall over his computer monitor with the label: Emergency Use Only. It was also Jack who stopped cold one day while deriving an expression on the hallway chalkboard, then turned to me and said "You know, mathematics for me is a social science, because it lets me spend time with people I like."

On the subject of mathematics, I should mention that in my early undergraduate days, I was weak and struggling at math. This was a hangover from high school when football and social life trumped academics. As a freshman, I failed to pass the first module of Calculus I. The way it worked back then, the professor and students who passed a module moved up together; a sort of conveyor belt of math professors. If, like me, you did not pass the module, it was retaken with the next professor in line. For me this meant I retook the module with Dr. Dennis Brewer. As if by magic, I understood what he was saying, what the math was about. I got an A in the module and followed Dr. Brewer through all of Calc I, II and III, honors differential equations, and Applied Math I. This showed me the power of one good teacher.

So, as you can see, my career was propelled by strong, benevolent, brilliant personalities. If they all think you are worth the effort, then working your butt off to accomplish something seems the least you can do.

Would you like to mention anything about your personal attributes that helped you achieve the professional status you enjoy today; was it self-belief, hard work, a mentor, or something else?

I have already covered the important topic of mentors. My family always gave me unending support and a refuge when times were hard. I don't think of myself as working harder than anyone else, but I get a lot done by organizing, prioritizing, and staying focused. I did have a range of early jobs that required very hard work. Jobs like frame carpenter and shoveler on an asphalt crew. That kind of job in the summer made college look pretty good, but it also taught me to respect and understand guys who work for a living. Perhaps because of those early experiences, one of my favorite quotes is by Voltaire, "Keeping busy is a poor substitute for accomplishment."

They may not be personal attributes, but I believe two things really helped me be successful.

First, as a teen I was a voracious reader of science fiction, following the example of my older brothers. So many of those great books had a professor-type who got called off somewhere in the world (or beyond) to give advice or to solve a problem. I wanted to be that guy, and now 40 years later, I suppose I am. The DISC, for example, has me going around the world to share my thoughts with thousands of SEG members, many of whom are great friends.

The second thing that has helped me is related to the first. Maybe it came from all that early reading, but I like to write. My son Dave (a geologist) once told me: "I think you are just like a lot of other geophysicists, but you are a good writer." I will pass on further comment and let the reader decide.

Why did you choose this lecture topic? Why is it important?

This is a tough question. This all started when Tad Smith, then chairman of the DISC committee, took one of my classes at U of Houston. Little did I know he was also there to get a good look at my style as a possible DISC presenter. I was offered the DISC and then asked what topic I wanted to present. Now, most DISC presenters are chosen to build the course around a lifetime of focused work on a tight and well-known subject (anisotropy, AVO, etc.). My work has been all over the map -- wave propagation, attributes, near surface, reservoir characterization, deconvolution -- not something that fits into a neat category. But even before the DISC offer, it was dawning on me that there was one over-arching theme in my interests and published work. That is frequency-dependence, the broad and varied collection of seismic phenomena that vary with frequency. This has been central to my work since the mid-1990s thanks to former University of Tulsa PhD student Chun-Feng Li (now professor at Tongji University). He introduced me to wavelet transforms and time-frequency methods in general and his doctoral work lead to the Spice attribute. The DISC, then, is my chance to bring many far-flung topics together under a single heading of Seismic Dispersion. To my knowledge, this is the first attempt to build a coherent narrative of the various aspects of dispersion in a general sense. The Greeks used to say that all things are imperfect at the first try, and so I suspect it will be with my DISC.

The topic is important because these phenomena exist in our data. I think of it like anisotropy before Leon Thomsen's landmark paper in 1986. From a hydrocarbon exploration point of view, if we ignore dispersive effects present in the data and someone else figures out how to use them to create better images or more detailed interpretations, then they get a competitive advantage.

Could you tell us in a few sentences what your course objectives are?

The idea is to build an appreciation and understanding of the many seismic characteristics that depend strongly on frequency. For example, sound in water is generally understood to be about 1500 m/s for any frequency you care to test. True, so long as we are not doing the test in shallow water. In that case, the wave speed becomes a very complicated topic. The idea of phase and group velocity pops up, as well as the vexing question of how 'shallow' is shallow?

Another example is reflection itself. We are all familiar with the normal incidence reflection coefficient and Zoeppritz's equations for angular incidence. But these assume reflection from an elastic interface, an idealization that ignores porosity, pore fluid motion, and permeability. Reflection from a poor-elastic contact leads to a frequency-dependent reflection coefficient. How does it behave? Is the variation important? Can it be observed in real data? These are the really interesting questions.

So, briefly, the course objectives are to explore a broad group of frequency-dependent phenomena related to acquisition, processing and interpretation. As we come to understand them, it also becomes clear that many dispersive phenomena contain information of interest to us.

Are there any more specific areas that you want to emphasize?

It is my hope that working geophysicists and students will not be daunted by the apparently theoretical nature of this course. Just the terminology of dispersive phenomena is a barrier to understanding. But that is why we have a DISC program. The DISC participants and I will go methodically through selected topics, bringing in a bit of math where it illuminates the subject and always working to draw out useful conclusions and applications. As a professor of 20 years, I have a pretty good eye on the audience and will work at the pace that makes sense for the group at hand.

What do you hope people will have learned after they attend your lecture?

DISC attendees should go away with a general knowledge of time-frequency methods and an appreciation of frequency-dependent phenomena. I hope attendees will be more comfortable in a meeting when someone says "the harmonics are killing us" or "wish we knew the seafloor shear wave speed" or "should be run a full-wave sonic?"

You have quite a busy year ahead. Do you enjoy traveling? Will it be difficult to balance the tour with your work?

I do enjoy traveling, but you may want to ask me this question again in a year. As for balance with work, I taught an extra course in Fall 2011 to clear out this semester, and as an advisor, I have not accepted new graduate students since mid-2011 (but I still have 12). My DISC travel is scheduled around the Fall 2012 semester, so it is back to work as usual. With planning, almost anything can be accommodated.

Would you share with us one or two of your most exciting successes?

I'm not sure 'exciting' is the right term, but I would say being Editor of Geophysics (1999-2001) was a remarkable honor and recognition. Being named 2012 DISC instructor is in the same category. Not sure, but I might be the only former-editor also to be DISC. Interesting.

How about a couple of disappointments?

Like everyone in the seismic business I have applied for a few schools and jobs that did not work out. But in reality, I think we land where we really ought to be. Looking back at my student days, I would rather be at a grad school that was a good fit than somehow squeeze into one a bit beyond my abilities. My attitude about grad school is 'follow the money' and go to the school that wants you enough to offer an assistantship, that is where you belong. In the work world, every company has a different culture. The ones I worked with -- Western Geophysical, Conoco, Amoco, Saudi Aramco -- were all great companies and a pleasure to be with.

What advice would you give to geophysics students and professionals just starting out in the industry?

While an undergrad, do a double major or a minor.

Learn geology; the kind of geophysics we do is aimed at imaging geology and pore fluids.

If you get the chance, work for a major oil company for 5 years. You get great training and you will know where your career is heading with that company. If you like it, stay. If not, jump to another (likely smaller) company and head for the career you want. Join several societies (SEG, AAPG, SPE, etc.) and volunteer. The people you get to know this way cut across company and discipline lines. Most likely, you will meet lifelong friends.

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