## Atmospheric dispersion

I work on multiple industrial projects relating to atmospheric dispersion of pollutants. I am mainly interested in the inverse problem of estimating the emissions of particulates from indirect measurements of concentration in the farfield. My interest in this topic stems from a collaboration with Teck Operations in Trail , British Columbia, Canada. I have worked on different aspects of atmospheric dispersion ranging from 3D finite volume solvers to Bayesian inversion frameworks.

### Dispersion of particulates in the atmosphere

My MSc thesis project focused on the development of a 3D finite volume solver for modelling the dispersion of pollutants in the atmosphere. Here we model the atmospheric dispersion process as an advection diffusion PDE with coefficients that depend on time and space. The time dependence of the coefficients is due to variations in the wind field and the spacial dependence is due to the Monin-Obukhov similarity theory that models the variations of eddy diffusivity in the atmosphere. We constructed a Fortran code that is based on the CLAWPACK package and solves the 3D PDE using Godunov splitting.

### Estimation of emissions

Estimation of emissions of particulates into the atmosphere is a difficult inverse problem. The problem is linear in nature however one has to deal with positivity constraints on the emission rates. More importantly the available data is highly noisy and there are large discrepancies between the output of the models and the actual measured data. I studied this problem as a Bayesian inverse problem using a Gaussian plume model as well as the PDE model that was mentioned above. In the case of the Gaussian plume model it is possible for us to solve the inverse problem with direct sampling and impose the positivity constraints by altering the forward map. A particularly challenging case of this problem is the construction of sources that are varying in time. This problem is infinite dimensional in nature and results in a high dimensional inverse problem after discretization. For more detail see BHJS’16.

Solving the source inversion problem using the direct PDE model is more challenging since the finite volume solver is expensive to evaluate and so direct sampling is no longer feasible. In this case we restrict ourselves to the case of sources with constant emissions in time and construct the forward map by running the finite volume code for each source separately. This way we can reduce the size of the parameter space significantly by exploiting the linear dependence of the PDE on the source term. For more detail see our articles BHJS’16 and JGBHJS’19.

## High intensity focused ultrasound

This project is part of an ongoing collaboration with researchers from Philips Healthcare Canada, Thunder Bay Regional Research Institute and The Hospital for Sick Children. We studied the problem of refocusing an ultrasound beam for treatment of tissue in the brain. Maintaining a focused beam is a key challenge during treatment that impacts the duration, cost and effectiveness of the procedure. Current refocusing techniques in the literature require the patient to remain in an MRI machine for up to two hours. We cast the refocusing problem as that of estimating the acoustic aberrations due to the skull bone and solved this problem using a Bayesian algorithm. With an accurate estimate of the aberrations at hand we compensate for the phase shift at the transducer and refocus the beam within the tissue. Further detail can be found in the manuscript BHetal’19.