# Jonathan Ajo-Franklin's Research Techniques

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Computational Geophysics
Tomographic Reconstruction Methods I have a long-standing interest in the inversion of seismic data including tomographic reconstruction algorithms using traveltimes and recently the entire seismic waveform. As part of my doctoral work, I developed an adaptive traveltime tomography package (w. Jaime Urban) based on greedy mesh refinement constrained by the properties of the model resolution matrix. The forward and inverse formulations we used exploited unstructured trigonal meshes to allowrefinement in any region of the model. I'm currently developing a regularized least-squares waveform inversion algorithm in the frequency-domain efficient enough to solve small 3D problems. The figure to the right shows a synthetic 2D mono-frequency inversion result with 2nd order Tikhonov regularization. I'm also working on techniques for incorporating constraints on tomography which mimic the physical process responsible for property variations. My most recent experiments have used compactness or minimal support constraints in an IRLS framework.

Dynamic (Time-Lapse) Inverse Theory Time-lapse geophysical measurements, and seismic imaging methods in particular, are powerful techniques for monitoring changes in subsurface properties. Traditional time-lapse processing methods treat each dataset as an independent unit and estimate changes in reservoir state through either differencing separate inversion results or differencing the acquired data. I'm investigating a general least-squares approach to jointly inverting time-varying property models through use of spatio-temporal coupling operators. Originally developed within the medical imaging community (Brooks et.al. 1999), this extension of traditional Tikhonov regularization allows us to constrain the way in which models vary in time, thereby reducing artifacts observed in traditional time-lapse imaging formulations. The same methodology can also accommodate changes in experiment geometry as a function of time thus allowing inversion of surveys different spatial extents. In this case, temporal resolution is traded for improved spatial coverage at individual timesteps. I use seismic traveltime tomography as a model problem although almost any geophysical inversion task can be posed within this formalism, including full waveform techniques. For some related ideas, see the work of Fred Day-Lewis et.al. (2002, Geophysics) on bayesian approaches for the same problem.

Advanced Forward Modeling The core of any effective imaging algorithm is a suitable description and solution to the forward problem. My current focus is on the development of finite-difference solvers capable of generating frequency domain Green's functions, including traditional FDFD methods using sparse LU decomposition (ala Pratt) and more advanced techniques based on analysis of time-domain solutions. Hybrid time/frequency solvers offer hope for extending waveform inversion to true 3D geometries. These solvers form the basis for my regularized waveform inversion codes. The figure to the left depicts a subvolume of SCEC's Community Velocity model (top) and the 3D frequency domain wavepath for a source/receiver pair at the model's surface (bottom). The subvolume corresponds to the zone imaged by the LARSE II active source experiment.

I first became involved in full wavefield modeling when I was trying to complete a series of sensitivity studies for geophysical DNAPL detection (1999). Since then, I have written 2D and 3D codes for solving the acoustic wave equation on various types of stretched cartesian meshes and a second set of codes for modeling GPR using a 2nd order TE-mode wave equation with a conductivity term. I have also spent some time parallelizing (shot distribution) this library of codes and optimizing the inner loops for better cache performance. I have also worked on the efficient calculation of sesmic (and radar) traveltimes for solving tomographic imaging problems since 1997. My earliest investigations explored graph theoretic techniques (Shortest Path Raytracing) for isotropic and anisotropic medium, finite-difference methods for solving the eikonal equation (Fast Marching), and a variety of more traditional initial value and boundary value approaches to the ray equations.

Large Scale Parallel Computation Waveform inversion algorithms, particularly when implemented in three dimensions, require tremendous computational resources from both a time and memory perspective. Solution of large 2D and small 3D problems is virtually impossible on typical workstations and requires use of parallel machines. Since many geophysical problems are easy to parallelize and have limited communication requirements, inexpensive distributed memory clusters have been the architecture of choice. During my graduate work I built a 16 node Athlon cluster (picture to the right) which our group used for wavefield modeling and traveltime tomography. I'm currently developing codes for use on EAP's ACES cluster. I'm also interested in developing flexible programming tools, scripting languages, and scheduling systems to enable easy parallel extension of serial codes.

Applications & Field Methods
## Dynamic Subsurface Processes

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## Near-Surface/Environmental Imaging

My graduate work focused on integrating crosswell seismic and crosswell radar tomography for use in detecting andcharacterizing zones of Dense Non-Aqueous Phase Liquid (DNAPL) contamination in the subsurface, more specifically at the former Pinellas DOE site. Work on integration was motivated by the petrophysical non-uniqueness problems inherent in using only acoustic or dielectric properties to diagnose the presence of DNAPLs. As part of this project, I was involved in data acquisition, tomographic processing, and now, development of dual acoustic/dielectric rock-physics models to allow fusion of the two datasets.

## Borehole Geophysical Measurements

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Experimental & Theoretical Rock Physics

## Acoustic Properties Of Unconsolidated Materials

The properties of granular materials at low pressure are a critically important yet sadly neglected area of rock-physics. Understanding the acoustic properties of clean sand is still in many ways an open problem even after 60 years of investigation. Most of my work with unconsolidated sediments has focussed on the effects of dense contaminants on the acoustic properties of samples from my field site. I have also investigated poroelastic and contact theories to understand the effects of DNAPLs on elastic properties. I worked with Jil Geller (LBNL) to understand ultrasonic measurements which we acquired during the Pinellas experiment.

## Soil Dielectric Properties

On the experimental side, I have built a series of DNAPL compatible coaxial cells for TDR measurements on aquifer sediments partially saturated with TCE. To the right you can see the coax cell, constructed from modified high vacuum components, I used for most of the dielectric measurements shown in my thesis.

## Rock Properties Integration

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