The conjugate gradient method

The conjugate gradient method can be used to solve many large linear geophysical problems — for example, least-squares parabolic and hyperbolic Radon transform, traveltime tomography, least-squares migration, and full-waveform inversion (FWI) (e.g., Witte et al., 2018). This tutorial revisits the “Linear inversion tutorial” (Hall, 2016) that estimated reflectivity by deconvolving a known wavelet from a seismic trace using least squares. This tutorial solves the same problem using the conjugate gradient method. This problem is easy to understand, and the concepts apply to other applications. The conjugate gradient method is often used to solve large problems because the least-squares algorithm is much more expensive — that is, even a large computer may not be able to find a useful solution in a reasonable amount of time.