Locating the source of an acoustic wave equation using likelihood estimates from the kalman filter applied to surface readings

Abstract

Cardiovascular disease (CVD) was the second-largest cause of death in the United Kingdom in 2014 [1], accounting for 32% of all deaths in 2009 [2]. CVD encompasses many diseases, one of which is coronary artery disease (CAD), otherwise known as atherosclerosis. Atherosclerosis is the build-up of fatty material, called plaque, inside the wall of the artery. Over time, this plaque will grow too large or break o , causing a blockage resulting in a heart attack. Currently, mortality from CAD has decreased by 72% between 1979 and 2013 [3]. However, predictions show that if the increasing trend of Body Mass Index (BMI) continues, then mortality from CAD could start increasing again [4]. There are several di erent methods currently available to the National Health Service (NHS) to diagnose CAD. However, there are long waiting lists and expensive costs associated with current diagnosis methods. Our aim is to look at a non-invasive approach of diagnosing CAD. We have limited our investigation to simple model problems. Therefore, further work would be required to consider more complex cases which align with the real-world application. In this thesis, we consider both 1-dimensional (1D) and 2-dimensional (2D) problems modelled by an acoustic wave equation with a forcing function which attempts to emulate a localised disturbance caused by CAD.We use an explicit nite di erence method (FDM) to approximate the solution in our partial di erential equation (PDE) and discard the disturbance location used. Having added noise to these approximations in an attempt to mimic noise from real readings, we record these approximations at speci c locations on the surface of our domains to imitate data collected from actual sensors. Using this data in the Kalman lter (KF), where guesses for the disturbance location are made, we can estimate the approximation of u throughout our domain. Using data generated by the KF, we compute likelihood estimates for each guess made and obtain the most probable disturbance location used to generate our sensor readings.