Instructor: Charles J. Ammon 
This is primarily a reading and discussion course and most of your time will be spent reading the literature. The number of papers on tomography is substantial and we cannot discuss them all, but we'll try to sample a broad cross section of published material. To help fill in gaps that we have not had the opportunity to cover in other courses or to set the context for the readings and explain important details, I will present a few of lectures. Planned lectures include an introduction to :
 Seismic Tomography
 Seismic Ray Theory
 Geophysical Inverse Theory
 Seismic Velocities and Rocks
The value of this course will be proportional to your efforts in it.
Exercise One: OneDimensional Tomography and LeastSquares InversionExercise Two: TwoDimensional Tomography and LeastSquares Inversion
Iyer, H. M., and K. Hirahara, Seismic Tomography  Theory and Practice, Chapman and Hall, London, 842 pages, 1993. Nolet, G., Seismic Tomography, D. Reidel Publishing Co., Dordrecht, Holland, 386 pages, 1987.
Lay, T., and T. C. Wallace, Modern Global Seismology, Academic Press, New York, 517 pages, 1995 (Section 7.1.3  pages 243  249). Lee, W. H. K., and V. Pereyra, Mathematical introduction to seismic tomography, in Seismic Tomography  Theory and Practice, H. M. Iyer and K. Hirahara Ed., Chapman & Hall, London, 922, 1993.
Lay, T., and T. C. Wallace, Modern Global Seismology, Academic Press, New York, 517 pages, 1995 (Chapter 3, sections 3.13.5, pages 7396). Virieux, J., Seismic Ray Tracing, in Seismic Modeling of Earth Structure, E. Boschi, G. Ekstrom and A. Morelli Ed., Instituto Nazionale di Geofisica, Rome, 223304, 1996.
Objectives: To provide an overview of a tomographic imaging problem using one of the simplest propagation "systems" (an incident plane wave). Pay attention to issues such as resolution (vertical and horizontal), uncertainty, data quality issues, relative versus absolute variations, inversion stability including issues such as damping and smoothing, the size of perturbations, etc.
Reading Set Two (in suggested reading order):
Aki, K., A. Christofferson and E. S. Husbeye, Determination of threedimensional seismic structure of the lithosphere, J. Geophys. Res., 82, 277296, 1977. Zandt, G., Seismic Images of the deep structure of the San Andreas fault system, central Coast Ranges, California, J. Geophys. Res., 86, 50395052, 1982.
Benz, H. M., and G. Zandt, Teleseismic tomography: Lithospheric structure of the San Andreas fault system in northern and central California, in Seismic Tomography  Theory and Practice, H. M. Iyer and K. Hirahara Ed., Chapman & Hall, London, 441465, 1993.
Evans, J., and U. Achauer, Teleseismic velocity tomography using the ACH method: theory and application to continentalscale studies, in Seismic Tomography  Theory and Practice, H. M. Iyer and K. Hirahara Ed., Chapman & Hall, London, 319360, 1993.
Dueker, K., E. Humphreys and G. Biasi, Teleseismic imaging of the western United States upper mantle structure using the simultaneous iterative reconstruction technique, in Seismic Tomography  Theory and Practice, H. M. Iyer and K. Hirahara Ed., Chapman & Hall, London, 265298, 1993.
Here's a list of questions and activities to help you focus on our objectives while reading the articles.
Summarize the observations and the methods used to invert the traveltime / slowness equations and the method(s) selected by the different authors to investigate resolution of their solutions. Make a table of the year of publication, the number of observations, and the method used to invert the equations. What are the range of velocity perturbations that these authors have estimated? How does that agree with the assumptions they made regarding ray paths?
Why do the researchers use relative arrival times as opposed to absolute times?
Why didn't the researchers include teleseismic Swaves in their analyses?
In a regional teleseismic bodywave tomography study, what's better, horizontal or vertical resolution? Why? Discuss how the lateral resolution varies with depth in teleseismic Pvelocity tomography.
Compare and contrast the study by Zandt (1981) and Benz and Zandt (1993). Try to quantify the advances in computing power that occurred during the 1980's.
Objectives: To wade deeper into the mathematics of tomography which is common to all imaging problems and many other inverse problems in the geosciences. Also to explore the differences between linear and nonlinear tomography and review some of the methods used by researchers to stabilize poorly conditioned numerical problems.
Reading Set Three (in suggested reading order):
Jackson, D. D., Interpretation of inaccurate, insufficient, and inconsistent data, Geophys. J. R. Astron. Soc., 28, 97109, 1972. Aki, K. and P. G. Richards, Quantitative Seismology  Theory and Methods, W. H. Freeman and Company, San Francisco, 1980 (Volume II, Section 12.3, 675717).
Berryman, J. G., Weighted leastsquares criteria for seismic traveltime tomography, IEEE Trans. Geosci. and Remote Sensing, 27, 302309, 1989.
Scales, J. A., Tomographic inversion via the the conjugate gradient method, Geophysics, 52, 179185, 1987.
Phillips, W. S., and M. C. Fehler, Traveltime tomography: A comparison of popular methods, Geophysics, 56, 16391649, 1990.
Aki, K., A. Christofferson and E. S. Husbeye, Determination of threedimensional seismic structure of the lithosphere, J. Geophys. Res., 82, 277296, 1977 (again).
Ammon, C. J., and J. E. Vidale, Tomography without Rays, Bull. Seism. Soc. Am., 83, 509528, 1993.
Nolet, G., Solving large linearized tomographic problems, in Seismic Tomography  Theory and Practice, H. M. Iyer and K. Hirahara Ed., Chapman & Hall, London, 227247, 1993.
Spakman, W., Iterative strategies for nonlinear travel time tomography using global earthquake data, in Seismic Tomography  Theory and Practice, H. M. Iyer and K. Hirahara Ed., Chapman & Hall, London, 190226, 1993.
Here are some questions and activities to help you focus on our objectives while reading the articles.
What is the difference between linear and nonlinear tomography? What is regularization and why is it necessary?
What kind of constraints can you place on tomographic problems?
What is "data weighting" and how improtant is it to the results of tomographic imaging? How would you calculate the weights for different observations?
What are coverage and hit count? How do they affect resolution?
What mathematical tools are available for solving matrix algebra associated with tomographic problems?
Construct a onedimensional tomography problem with five cells and four observations (and ray paths). Construct the equations necessary to invert for the structure. Add noise to the observations and invert the equations.
Objectives: To examine a more complex tomographic problem where the precise location and origin time of the source is unknown. In comparison with the teleseismic studies from section one, the ray tracing and inversion in this problem is more challenging. Our goals are to study both basic earthquake location and seismic imaging problems.
Reading Set Four (in suggested reading order):
First Week: Lay, T., and T. C. Wallace, Modern Global Seismology, Academic Press, New York, 517 pages, 1995 (Sections 6.3 and 6.4  pages 217223).
Aki, K. and W. H. K. Lee, Determination of threedimensional velocity anamalies under a seismic array using first P arrival times from local earthquakes, J. Geophys. Res., 81, 43814399, 1976.
Thurber, C. H. and K. Aki, Threedimensional seismic imaging, Ann. Rev. Earth Planet. Sci., 15, 11539, 1987.
Kissling, E. Geotomography with local earthquake data, Rev. Geophys., 26, 659698, 1988.
EberhartHillips, D., Local earthquake tomography: earthquake source regions, in Seismic Tomography  Theory and Practice, H. M. Iyer and K. Hirahara Ed., Chapman & Hall, London, 613643, 1993.
Second Week:
Chiarabba, C. A. Amato, and M Meghraoui, Tomographic images of the El Asnam fault zone and the evolution of a seismogenic thrustrelated fold, J. Geophys. Res., 102, 24,48524,498, 1997.
Ghose, S. , M. W. Hamburger, and J. Virieux, Threedimensional veloicty structure and earthquake locations beneath the northern Tien Shan of Kyrgystan, central Asia, J. Geophys. Res., 27252748, 1998.
Thurber, C., Local earthquake tomography: velocities and Vp/Vs  theory, in Seismic Tomography  Theory and Practice, H. M. Iyer and K. Hirahara Ed., Chapman & Hall, London, 563583, 1993.
Sambridge, M. S., Nonlinear arrival time inversion: Constraining velocity anomalies by seeking smooth models in 3D, Geophys. J. Int., 94, 653677, 1990.
Pavlis, G. L., and J. R. Booker, The mixed discrete continuous inverse problem: Application to the simultaneous determination of hypocenters and velocity structure, J. Geophys. Res., 85, 480110.
Here are some questions and activities to help you focus on our objectives while reading the articles.
Construct a table of the number of observations (earthquakes and stations), the cell size or nodespacing, the inversion method, the regularization method, etc. in each study. What is the maximum resolution in any of the studies? What controls the spatial resolution?
What are typical values of misfit reduction obtained in local earthquake tomography?
Sometimes some earthquakes can be located better than others, how is this fact incorporated into the inversion for veloicty structure?
How do the researchers investigate resolution in the inverse problem?
What is the difference between linear and nonlinear tomography when applied to local earthquake tomographi imaging?
Construct a onedimensional tomography problem with five cells and four observations (and ray paths). Construct the equations necessary to invert for the structure. Add noise to the observations and invert the equations.
Objectives: These two papers are designed as an introduction to global tomography. We will expand our reading next week. The review by Romanowicz is almost ten years old, but describes much of the previous work without much mathematics. The Morelli and Dziewonski introduces an alternative way to parameterize earth models  which is commonly used in more recent inversions.
Reading Set Five (in suggested reading order):
Romanowicz, B., Seismic Tomography of the Earth's mantle, Annu. Rev. Earth Planet. Sci., 19, 7799, 1991. Morelli, A., and A. M. Dziewonski, The harmonic expansion approach to the retrieval of deep Earth Structure, in Seismic Tomography, edited by G. Nolet, D. Reidel Publishing Company, 251274, 1987.
Here are some questions and activities to help you focus on our objectives while reading the articles.
Construct a table of the number of observations (earthquakes and stations), the cell size, nodespacing, or number of expansion coefficients, the inversion method, the regularization method, the variance reduction, etc. in each study. What kind of obervations are used in global/mantle tomography?
List the different parameterizations used to study deep earth structure?
Where do you expect to find larger heterogeneities, in the upper of lower mantle?
