Introduction

This tutorial provides a step-by-step guide to performing Markov Chain Monte Carlo (MCMC) inversion for geophysical applications, with a focus on helping students understand both the conceptual framework and the practical implementation of the method.

The original code used in this tutorial was developed by Elizabeth M. Berg (e.g., Berg et al., 2018, 2020), with later improvements and extensions contributed by Liu et al. (2021). This work is associated with the research efforts of Fan-Chi Lin’s group at the University of Utah. The earlier implementations mainly focused on using B-spline parameterization to smooth the model. In contrast, this tutorial adopts a gradient-based approach, which is more suitable for the current application and provides a clearer interpretation of model variations with depth.

MCMC inversion is a powerful technique that allows us to estimate model parameters, such as subsurface velocity structures, by sampling a range of possible solutions rather than finding a single best-fit model. This probabilistic approach helps quantify uncertainties and provides deeper insight into the reliability of inversion results.

However, for many beginners, the main difficulty is not the theory itself, but understanding:

  • how to prepare and organize input data files

  • how to prepare control files

  • and how to execute the program.

This tutorial is still evolving, and your feedback plays an important role in improving it for future users. If you come across anything that is confusing, incomplete, or could be explained more clearly, we would really appreciate hearing from you. Your suggestions and comments help us refine both the content and the structure of the tutorial. Please feel free to reach out through the “Give us feedback” section in the Appendix.

The main code directory and examples can be found here:

https://github.com/IES-ESLab/MCMC_1D_gradient

References

Berg, E. M., Lin, F.-C., Allam, A., Qiu, H., Shen, W., & Ben-Zion, Y. (2018). Tomography of Southern California via Bayesian joint inversion of Rayleigh wave ellipticity and phase velocity from ambient noise cross-correlations. Journal of Geophysical Research: Solid Earth, 123, 9933–9949. https://doi.org/10.1029/2018JB016269

Berg, E. M., Lin, F.-C., Allam, A., Schulte-Pelkum, V., Ward, K. M., & Shen, W. (2020). Shear velocity model of Alaska via joint inversion of Rayleigh wave ellipticity, phase velocities, and receiver functions across the Alaska transportable Array. Journal of Geophysical Research: Solid Earth, 125, e2019JB018582. https://0120100299/10.1029/2019JB018582

Liu, C.-N., Lin, F.-C., Huang, H.-H., Wang, Y., Berg, E. M., & Lin, C.-H. (2021). High-resolution 3-D shear wave velocity model of Northern Taiwan via Bayesian joint inversion of Rayleigh wave ellipticity and phase velocity with Formosa array. Journal of Geophysical Research: Solid Earth, 126, e2020JB021610. https://doi.org/10.1029/2020JB021610