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Tsunami

Tsunamis and more

mathematics applied

Tsunami group
Parameter Learning
Stochastic fluids
The convection obsession
Climate Modeling

We make use of historical anecdotal accounts from Dutch settlers and military records that documented earthquakes and tsunamis in the Indonesian archipelego to ascertain the strength, location and seismic structure of the causal earthquakes. To do this we utilize Markov Chain Monte Carlo (MCMC) methods in a Bayesian perspective to identify the likely posterior distribution for the location and magnitude of the earthquake that yielded the observed tsunami arrival times and wave heights which make up the likelihood distribution. The goal of this work is to identify the past seismic history of the region to better inform the potential future hazard risks.

  • We first introduced this approach for an earthquake and subsequent tsunami that occurred in the greater Banda Sea in Eastern Indonesia in 1852 (publication in Journal of Geophysical Research). Using random walk MCMC we found that the likely earthquake was near 8.9 Mw. The physical location of the potential earthquake was also significantly narrowed down.
  • latlonprior_vs_samples.png
    latlonprior_vs_samples.png
  • A follow up study investigates the statistical methods in the original study and identifies computational bounds on the relevant uncertainty in the posterior distribution (see preprint here).

1820 South Sulawesi earthquake and Tsunami

We investigate the potential source of an earthquake and tsunami near South Sulawesi in December 1820. This event had two potential sources: the Flores thrust or the Walanae (Selayar) Fault.

Collaborators:

  • Ron Harris (BYU, Geology): The real expert on tsunamis and Indonesia in particular.
  • Justin Krometis (Virginia Tech, Supercomputing Center)
  • Nathan Glatt-Holtz (Tulane University, Mathematics)
  • Hayden Ringer (Virginia Tech, Mathematics): a former MS student now completing his Phd.
  • Kameron Lightheart (MS Mathematics, 2021): Approximate gradients and higher order MCMC methods.
  • Claire Ashcraft (MS Geology, 2021): Seismic and geological structure of the Banda Islands
  • Taylor Paskett (MS Mathematics, 2022): Primary software developer and 1820 Sulawesi event.
  • James Griffin (MS Mathematics, 2022)
  • Jake Callahan (MS Mathematics, 2022)
  • Marja Crawford (MS Mathematics, 2022)
  • Carol Herrera (MS Mathematics, 2022)
  • Raelynn Wonnacott (MS Mathematics, 2023)
  • Chelsey Noorda (MS Mathematics, 2023)

Undergraduate students:

  • Isaac Sorensen (Mechanical Engineering)
  • Patrick Beal (Mathematics)

This project involves a lot of the following topics, but you don't have to be an expert at all of them:

  • Python programming
  • Statistics & probability (particularly Bayesian methods)
  • Geology/seismology

This is also a very large group, so you need to be comfortable working in groups and be willing to divide and conquer on different tasks. If you don't work well with others, this will not work well for you. In addition, you may not get as much 1-1 mentoring as you would on a different project so keep that in mind.

The best part of this project is that you will get to work on something that is not only very cool, but also has the potential to be extremely impactful. Understanding the seismic history of a given region is critical to being able to give assessments of potential future hazards. In addition (and this is the really important part), seismic time scales are really, really long and the instrumental record we have from the past 70 years won't be nearly enough. Understanding seismic history well enough to gauge the potential for future hazards requires careful study of these historical events.

If you are interested in learning more or would like to be included in the group, please reach out to Jared Whitehead for more details.

Group Alumni (apologies for missing anyone...we have had a lot of people that got us to where we are now):

  • Martha Morrise (BS Mathematics 2018)
  • Spencer Giddens (MS Mathematics 2020, BS Mathematics 2018)
  • Hayden Ringer (MS Mathematics 2020)
  • Garret Carver (BS Mathematics 2022)
  • Ryan Hilton (BS Mathematics 2022)
  • Ashley Avery (BS Chemical Engineering 2022)
  • McKay Harward (BS Mathematics 2020)
  • Adam Robertson (BS Mathematics 2020)
  • Cody Kessler (BS Mathematics 2020)
  • M. Hunter Klein (BS Mechanical Engineering 2020)
  • Joshua Fullwood (BS Mathematics 2019)
  • Alice Oveson Bagley (BS Mathematics 2021)
  • Regan Howell (BS Mathematics 2022)
  • Josh Lapicola (BS Mathematics 2021)
  • Camille Carter (BS Mathematics 2020)
  • Bryce Berrett (BS Civil Engineering 2020)
  • Carly Ringer (BA Geography 2019)
  • Yajing Zhao (MS Mathematics 2020)

We take a different perspective on the standard problem of assimilating data into an existing mathematical model. We extend one of the standard 'nudging' methods for dissipative systems to develop an algorithm that not only captures the unknown state of a system, but will also learn the underlying (unknown) parameters of the model itself. This is joint work with a very large group of colleagues at several different institutions.

These efforts were originally motivated by two key investigations that extended the use of the standard nudging data assimilation algorithm to systems where key parameters were unknown. We numerically and rigorously justified the nudging algorithm for Rayleigh-Bénard convection when only the temperature variable is observed (see https://doi.org/10.1137/19M1248327). We found that observations of the temperature field alone were sufficient so long as the non-dimensional Prandtl number was infinite, or at least 'large'. As part of this investigation, we also considered the impact of observations taken from a system where the Prandtl number was distinct from the modeled system itself, and analyzed the error in this setting. Surprisingly the rigorous error bounds on the parameter differences numerically appear to be sharp (in contrast to the other error bounds produced from the analysis).

Simultaneous to this study of Rayleigh-Bénard convection, a different subset of our group considered a similar question where they evaluated the effectiveness of the nudging algorithm on the 2D Navier-Stokes equations when the viscosity is an unknown parameter (see https://doi.org/10.1137/19M1248583). Finding almost the same type of results (the rigorous error bounds on the parameter error in the synchronization process appear to be sharp), they took this a step further, and created an algorithm that will 'learn' the unknown parameters of the system using these parameter error estimates.

Most recently we have extended the parameter learning to multiple unknown parameters in the context of the Kuramoto-Sivashinsky equation (see https://arxiv.org/pdf/2106.06069.pdf). This generalized algorithm also computationally recovered which coefficients of generalized linear terms were nonzero, and showed remarkable robustness to different aspects of the underlying algorithm. We also found clear evidence that this algorithm will work for all dissipative systems for which the observations are sufficiently broad.

In a concurrent effort, we also studied parameter learning and data assimilation on the finite dimensional 3D Lorenz equations (see https://arxiv.org/pdf/2108.08354.pdf). In this finite dimensional setting we established rigorous convergence of the parameter learning algorithm to any proper subset of the three parameters of the Lorenz system. In addition to these rigorous results, we computationally demonstrated that this approach is robust to a host of potential complications including: discrete measurements in time, noise in the observation, noise in the dynamics of the actual system, and several other potential hazards of the method. Not only were these results promising, but the approach taken to establish the rigorous results seems tenable to extend to dissipative infinite dimensional systems (such as the Navier-Stokes or Kuramoto-Sivashinsky systems).

Collaborators:

  • Adam Larios (Nebraska, Math)
  • Vincent Martinez (CUNY- Hunter College, Math)
  • Joshua Hudson (Sandia National Laboratory)
  • Elizabeth Carlson (University of Victoria, Math)
  • Shane McQuarrie (University of Texas, Oden Institute)
  • Benjamin Pachev (University of Texas, Oden Institute)
  • Eunice Ng (Stonybrook, Math)
  • Jacob Murri (BYU MS Math)
  • Tuan Pham (BYU, Math)

Motivated in part by the groundbreaking work of Hairer and Mattingly, we have investigated several questions related to the ergodicity and invariance of statistical measures for stochastically forced hydrodynamic systems. A large portion of this work is still in progress, and there is quite a bit left to be done.

Continuing some previous work that investigated the ergodicity of the 2D Boussinesq system, we investigated the ergodicity of a stochastically forced Rayleigh-Bénard convective system (see https://iopscience.iop.org/article/10.1088/0951-7715/29/11/3309/meta). In this same article we also showed that the heat transport could be bounded rigorously even in the presence of an additional stochastic heat source.

Most recently we have considered the problem of hydrodynamic stability when a stochastic source is present. This has led to a novel concept of nonlinear stability for stochastically driven hydrodynamic systems. For Rayleigh-Bénard convection driven by stochastic heating either through an internal source, or via the boundary conditions, we find mild stochastic heating has a stabilizing effect. At a certain point, the stochastic heating will destabilize the system entirely however, introducing instabilities slightly different from those seen in the standard deterministic setting.

Rigorous upper bounds on the transport of heat in the Rayleigh-Bénard convective setting have been an area