Geophysical inverse theory is used to obtain information about the structure of the solid Earth based on a handful of measurements at the surface. In 3D, this is a multidimensional, computationally expensive and ill-posed inverse problem.
I will focus on a numerical approach to electromagnetic geophysical inverse problems, which employs Maxwell's equations to estimate electrical conductivity of the solid Earth. Electrical conductivity is a physical parameter that allows us to constrain water / melt content and distribution in the Earth's crust and mantle, and is therefore widely useful in geophysics.
The mapping from model parameters to observations is non-linear, and the problem is best addressed using gradient-based inversion methods such as the non-linear conjugate gradients. I will discuss a framework for Jacobian computations that allows us to address this problem numerically, and give some use examples.
We shall also see that Earth models obtained through geophysical inversion are non-unique. Quantifying uncertainty in these inverse models is an important open problem.