Felix comments

Author: mloubout

FWI_forward_adjoint/FWIpart1.md

@@ -15,7 +15,7 @@ Corresponding author: mloubout@eoas.ubc.ca
 
 ## Introduction
 
-Since its re-introduction by Pratt (1999), Full-waveform inversion (FWI) has gained a lot of attention in geophysical exploration because of its ability to build high resolution velocity models more or less automatically in areas of complex geology. While there is an extensive and growing literature on this topic, publications focus mostly on technical aspects, making this topic inaccessible for a broader audience due to the lack of simple introductory resources for newcomers to geophysics. This is part one of three tutorials that together attempt to provide an introduction to the technique and software to help people get started. We hope to accomplish this by providing a hands-on walkthrough of FWI using Devito [2], a system based on domain-specific languages that automatically generates code for time-domain finite-differences. In this capacity, Devito provides a concise and straightforward computational framework for discretizing wave equations, which underlie all FWI frameworks. We will show that it generates verifiable executable code at run time for wave propagators associated with forward and adjoint (in part two) wave equations. Devito frees the user from the recurrent and time-consuming development of performant time-stepping codes and allows the user to concentrate on the geophysics of the problem rather than on low-level implementation details of wave-equation simulators. This tutorial covers the conventional adjoint-state formulation of full-waveform tomography [5] that underlies most of the current methods referred to as full-waveform inversion [Matthias add ref]. While other formulations have been developed to improve the convergence of FWI for poor starting models, in these tutorials we will concentrate on the standard formulation that relies on the combination of a forward/adjoint pair of propagators and a correlation-based gradient. In part one of this tutorial, we discuss how to set up wave simulations for inversion, including how to express the wave equation in Devito symbolically and how to deal with the acquisition geometry.
+Since its re-introduction by Pratt (1999), Full-waveform inversion (FWI) has gained a lot of attention in geophysical exploration because of its ability to build high resolution velocity models more or less automatically in areas of complex geology. While there is an extensive and growing literature on this topic, publications focus mostly on technical aspects, making this topic inaccessible for a broader audience due to the lack of simple introductory resources for newcomers to geophysics. This is part one of three tutorials that together attempt to provide an introduction to the technique and software to help people get started. We hope to accomplish this by providing a hands-on walkthrough of FWI using Devito [2], a system based on domain-specific languages that automatically generates code for time-domain finite-differences. In this capacity, Devito provides a concise and straightforward computational framework for discretizing wave equations, which underlie all FWI frameworks. We will show that it generates verifiable executable code at run time for wave propagators associated with forward and adjoint (in part two) wave equations. Devito frees the user from the recurrent and time-consuming development of performant time-stepping codes and allows the user to concentrate on the geophysics of the problem rather than on low-level implementation details of wave-equation simulators. This tutorial covers the conventional adjoint-state formulation of full-waveform tomography [5] that underlies most of the current methods referred to as full-waveform inversion [6]. While other formulations have been developed to improve the convergence of FWI for poor starting models, in these tutorials we will concentrate on the standard formulation that relies on the combination of a forward/adjoint pair of propagators and a correlation-based gradient. In part one of this tutorial, we discuss how to set up wave simulations for inversion, including how to express the wave equation in Devito symbolically and how to deal with the acquisition geometry.
 
 
 Full-waveform inversion tries to iteratively minimize the difference between data that was acquired in a seismic survey and synthetic data that is generated from a wave simulator with an estimated (velocity) model of the subsurface. As such, each FWI framework essentially consists of a wave simulator for forward modeling the predicted data and an adjoint simulator for calculating a model update from the data misfit. The first part of this tutorial is dedicated to the forward modeling part and demonstrates how to discretize and implement the acoustic wave equation using Devito. This tutorial is accompanied by a Jupyter notebook - **`forward_modeling.ipynb`** -, in which we describe how to simulate synthetic data for a specified source and receiver setup and how to save the corresponding wavefields and shot records. In part two of this series, we will address how to calculate model updates, i.e. gradients of the FWI objective function, via adjoint modeling. Finally, in part three we will demonstrate how to use this gradient as part of an optimization framework for inverting an unknown velocity model. 
@@ -79,9 +79,8 @@ We can now use this symbolic representation of our wavefield to
 generate simple discretized stencil expressions for finite-difference
 derivative approximations using shorthand expressions, such as `u.dx` and `u.dx2` to
 denote $\frac{\partial u}{\partial x}$ and $\frac{\partial^2
-u}{\partial x^2}$ respectively.
+u}{\partial x^2}$ respectively. We show the symbolic finite-difference for these two derivatives in the following code snippet for the previously defined Devito object `u`.
 
-[This piece of Python code is not connected to anything. Like an equation it needs to be connected to the text.)
 
 ```python
   In []: u
@@ -135,7 +134,7 @@ Since the source appears on the right-hand side in the original equation (Equati
 
 Since sources and receivers are interspersed sparsely and often located off the computational grid, Devito provides a separate symbolic type specifically designed for sparse objects called `SparseFunction` and that handles interpolations onto the model grid. Just like we defined wavefields as `TimeFunction` objects and model and damping terms as `Function` objects, we can construct `SparseFunction` objects for sources and receiver and add them to our `stencil` expression.
 
-Devito also provides a special function for setting up a Ricker wavelet called `RickerSource`, which acts as a wrapper around `SparseFunction` objects and automatically creates an instance of a `SparseFunction` object for a Ricker wavelet with a specified peak frequency `f0` and origin time `t_0` [Is this correct????] and source coordinates `src_coords`:
+Devito also provides a special function for setting up a Ricker wavelet called `RickerSource`, which acts as a wrapper around `SparseFunction` objects and automatically creates an instance of a `SparseFunction` object for a Ricker wavelet with a specified peak frequency `f0` and source coordinates `src_coords`:
 
 ```	
 	# define source object with Ricker wavelet and inject
@@ -249,3 +248,5 @@ Python. PeerJ Computer Science 3:e103 https://doi.org/10.
 [4] Pratt, R. G., 1999, Seismic waveform inversion in the frequency domain, part 1: Theory and verification in a physical scale model: GEOPHYSICS, 64, 888–901. doi:10.1190/1.1444597
 
 [5] Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation: GEOPHYSICS, 49, 1259–1266. doi:10.1190/1.1441754
+
+[6] Virieux, J., and Operto, S., 2009, An overview of full-waveform inversion in exploration geophysics: GEOPHYSICS, 74, WCC1–WCC26. doi:10.1190/1.3238367