Mathematics Colloquium: Combining Distinct Data for Improved Modeling
4:10pm SPARK 212
Measurements or observations are essential to understanding many situations in business, industry science and engineering. Typically, only a portion of the situation is observed and mathematical models are used to fill in the blanks. However, mathematical models also donâ€™t give the complete story because they often contain unknown parameters that we cannot directly observe. Inverse methods are used to find these parameters. In particular, an inverse method starts with the observations and uses the mathematical model to estimate desired parameters. Intuition suggests that inverting multiple datasets for a common set of parameters will produce better estimates than just inverting one data set. For example, simultaneous joint inversion has become quite common in geophysical applications. I will show results using Electrical Resistivity (ER) and Ground Penetrating Radar (GPR) measurements made at the surface of the Earth to image the near subsurface. The ER and GPR datasets give information across a wide range of frequencies and by combining them, we are able to produce images not possible through individual inversions. Once the inversion is complete we observe that parameter estimates using multiple data sets are better than those found with one set. This improvement can be quantified by measuring the decay rate of the singular values of the combined model operator. We have developed methodology to identify singular values of integral operators that combine multiple data types. These singular values quantify the benefit of combining distinct data types before the complicated machinery of discretizing and solving the inverse problem is implemented.