1974 – 1978

In 1984 I moved to Los Angeles for a brief stint at the University of Southern California, thanks to Professor Blum and a recomendation of Gene Golub. From there I moved to Caltech to start one of the most flourishing times of my career. The atmosphere at the Applied Mathematics Department, where I worked closely with Herb Keller, was very exciting and stimulating. Lots of visitors, good courses and seminars, and excellent contact with applied disciplines, were all conducive to innovation.

An attempt to write a book only got to the report stage:

A couple of papers of deferred corrections applied to ordinary boundary value problems.

I started collaborating with Herb in several projects of mutual interest that resulted in some publications:


This work combined the use of MACSYMA (accessed remotely at MIT over the internet! Remember acoustic phone coupling and 300 baud modems?), to generate symbolically high order finite difference formulas, with deferred corrections. Very recently I discovered that current high order methods for solving the wave equations use some of our formulas.

Also at this time I joined forces with Olof Widlund and W. Proskurowski to apply deferred corrections to the solution of Laplace’s equation in general 2D regions. The main issue here was the complexity of the asymptotic expansions for the truncation error near the boundary. An unpublished theorem of Kreiss pointed the way and we worked for quite a while to reconstruct a proof and implement a successful algorithm. Oddly enough, Godela Scherer, my undergraduate student and co-author, worked on her Ph D thesis with H. O. Kreiss on a related problem, although I did not realized it until many years after, when my colleague Jose Castillo discover it as relevant to his research.

O. Widlund

W. H. K. Lee

M. Lentini

In the summer of 1976 I accepted an invitation from Leslie Fox and spent a wonderful time at Oxford and traveling in Europe. Then later in 1976, while I was in one of my many visits to Stanford, I met a seismologist, W.H.K. Lee, of the US Geological Survey in Menlo Park, who was trying to use our boundary value code to solve some problems in seismology. I helped him out to set the problem up and forgot about it. To my surprise, the next year he had a nice report with some encouraging results and he was eager to collaborate and do a more thorough job. This was my first exposure to seismic ray tracing.

Upon my return at Caltech I decided to educate myself on the subject and found to my delight that Herb Keller had been, together with his brother Joe, one of the US early workers in this area, having written very clear, concise and precise expositions on the role of rays in wave propagation. That initiated a new career that has lasted 25 years and has resulted in numerous papers, large software developments and several fundamental changes in my life.

Since Willie Lee had been a student of K. Aki and had collaborated with him in some of the original work on Seismic tomography, that led me naturally to the subject of travel time inversion. I found there a subject that linked naturally most of my life work and experience and therefore decided to take a stab at it. The early work in two-point ray tracing and inversion was done while at Caltech and published in:

I had also the opportunity of helping some of Herb’s PhD students, that by this time were working on the same subject of two point ray tracing, noticeably, D. Perozzi and J. Fawcett.