Fully simulating our seismic experiment with all its acquisition variables, medium properties and the physical behavior of waves to reproduce the observed seismic data is an ultimate objective we all seek to achieve. Not with standing the computational limitations and the physical approximations, inverting for the medium properties (i.e. seismic velocities) is at the heart of waveform inversion. Thus, full seismic waveform inversion (FWI) is fast becoming the premiere research focus in Geophysics, with many academic and industry labs devoting a lot of resources to the problem. Many of the current tools used to perform FWI were known since the 1980’s, but only recently did our computational capability allow us to implement FWI with higher resolution and speed, albeit we are still focused on 2D acoustic media models. Despite the recent advances, the remaining challenges are three fold: 1) The highly nonlinear nature of the objective (data misfit) function due to the sinusoidal nature of the wavefield and the complex Earth reflectivity, which renders gradient methods useless in some cases as we get trapped in local minima, or requiring a very accurate initial velocity model, 2) The acoustic, isotropic and 2D simplification of the medium, which unlike imaging, can cause tremendous problems to FWI, taking into account that the alternative can induce tremendous Null space into the problem, 3) Finally, the high computation cost of FWI as each iteration is equivalent to 3-5 imaging steps, and the iterations to go up to the hundreds.

In this Graduate level course, we will focus on Full waveform inversion (FWI) starting by covering it’s fundamentals, including an introduction to inverse theory, and getting an understanding on the philosophy behind FWI, trying to pick the mind of it’s introducer to our field Albert Tarantola. We next look at the components of FWI, including the model and data space, their connection, and definition, including the physics of modeling and medium considerations. The objective of FWI is simply given by reducing the misfit between the observed data from an experiment (in our case a seismic one), and what we can generate using our computing devices. If we got the physics and the experiment parameters right, then most probably the model used to obtain the synthetic data that fits the observed (field) data is accurate. We will look at how do we measure such misfit, and what we do when the synthetic data do not resemble the observed data: the model update process. This will include covering the calculation of the gradient (the Frechet derivative), and for more advanced convergence components, the Hessian. The course will discuss the challenges we face in FWI and potential solutions to such challenges, and the uncertainties involved. In summary, the course starts by introducing the fundamentals of full waveform inversion (FWI) starting from it's basic definition. It soon focuses on the model update issues and provides analysis of it's probable success in converging to a plausible model. In the course, we will discuss the many challenges we face in applying FWI on seismic data, and introduce modern day proposed solutions to these challenges.

In FWI a very simple, but important, fact resonates: We can only invert what the simulation (many approximate) assumptions and the acquired sinusoidal nature of our data with limited regional coverage allow us to estimate, so we have to make sure that we take these limitations into account. This challenge, despite the many advances, will remain a topic of research for the industry and academia alike, including a topic attractive to graduate student’s research and dissertation projects.