Geology 5670/6670 Inverse Theory 12 Jan 2015 © A.R. Lowry 2015 Read for Wed 14 Jan: Menke Ch 2 (15-37) Last time: Course Introduction (Cont’d) Goal is to solve for optimal model parameters from observational data … Usually involving physics of a PDE The gravity example: Linear inversion  linear algebra, and notation conventions for vectors/matrices in this class We define model parameters to be “optimal” based upon minimization of a norm of vector misfit between measurements and model predictions. The L 2 norm is commonly expressed as: or

Solve for T e … This example raises several important questions… Among them, how do we tell a “good” model from a “bad” model? Here used the L 2 norm of misfit between measured and modeled shoreline height differences: We might also ask, what is the uncertainty in the estimate? What are the “hidden” parameters? What physics did we neglect & does it matter? Gilbert Bonneville Provo

Consider another (for some) familiar example: Here, the linear operator on w is & q is the mass load… w + q T is observed Solve for T e & q ?… Part of q B observed…

Clearly the devil’s in the details. In this course we’ll examine: How to express physical relations between data and parameters to solve most easily How to deal with sparse &/or noisy data and spatially or temporally varying parameters How to evaluate the range of possible models that can fit the data within uncertainties (or “robustness” of the best-fit model), and the limitations of the measurements And we will focus on your individual research problems…

Let’s return to Example I: Gravity where G ij is a constant relating density and gravity, m j =  j is the model parameter we want to find. ( L{u} = f where L =  2 )

Note : Here we introduce some notation that will be used as a standard throughout the course. Let: d denote the observational data N be the number of observations m denote the model parameters M be the number of parameters G denote scalar ( kernel ) values relating m to d Then any linear modeling problem can be written as a set of N linear equations of the form: or in matrix notation,

So one can simply solve the set of linear equations: or in matrix notation, for i = 1, …, N equations in j = 1, …, M unknowns (observed)(known)(unknown) r2r2 r1r1 r4r4 r3r3 w

Example II is more difficult, because when parameters are inside the linear operator, the forward model (generally) will be nonlinear. In that case, (We’ll talk about how to solve this case later). For now let’s consider the linear problem: In practice, where m t are the “true” parameters and  are measurement errors. (Note this assumes physics is known and correct: No empiricisms, no simplifying approximations). Then if N = M and G -1 exists, So, if you know something about error in your measurements, you can estimate error in your model parameters…