Inverse Problems at CAAM

Seismic imaging consists in processing this data to generate images of the earth. This process requires sophisticated mathematical techniques, as can be seen in this overview of seismic imaging. The CAAM professors most involved with seismic imaging are William Symes, Liliana Borcea. Students currently working with these faculty on seismic imaging topics are Eric Dussaud, Jintan Li, and Sichao Chen. We also have among us Alexandre Khoury, a visiting researcher from France sponsored by Total Exploration and Production USA.

How to image when there is clutter in the medium?
Read an overview of the research done at CAAM.

In many important applications, waves propagate through a cluttered medium. Traditional methods fail because in this case we do not know the medium in a pointwise manner.

We invite you to read further information about the exciting applications of Array Imaging in Cluttered Media .

Take for example electrical impedance tomography (EIT) where we wish to determine the electrical properties inside a body from measurements of electric currents and voltages at the surface. See Fig 1

(*** must fabricate Figure 1. Something like a compact body with electrodes in its surface.).

This problem has applications in medical imaging (for example for detection of pulmonary emboli, monitoring of heart function and blood flow, etc.), in environmental sciences (e.g. detecting leaks in underground storage tanks), in oil extraction and environmental cleaning (e.g. monitoring flow of injected fluids in the Earth), etc.

Mathematically, EIT is an example of a severely ill-posed inverse problem because very small noise in our measurements is typically translated into unpredictable and huge variations in the image, unless proper care is taken during the reconstruction process.

Our group, formed by Dr. Borcea, graduate student Fernando Guevara and external collaborator V. Druskin (Schlumberger-Doll Research Center) seeks to stabilize the inversion process by choosing a proper parametrization of the unknown.

Optimal grid with 171 parameters.

### we can explain this picture better

We could get away from this instability by estimating one single parameter -- an average conductivity for instance. For obvious reasons, this is not helpful in medical diagnostic. Fundamentally we want to estimate the largest number of parameters used to represent the medical image that can be obtained in a stable way from the measurements. The spatial placement of these parameters is also crucial: one can gain resolution by using more parameters close to the skin of the patient than deep inside. PhD student Fernando Guevara Vasquez under the supervision of his advisor Dr. Borcea, is working on this problem.

*** must include the following two pictures:

1. Image with a fine grid and noisy data (put noise that is small enough to give some optimal grid with 2 layers)
2. Image an optimal grid properly chosen. Show grid too.