Physical‐statistical modeling in geophysics

Two powerful formulas have been available to scientists for more than two centuries: Newton's second law, providing a foundation for classical physics, and Bayes's theorem, prescribing probabilistic learning about unknown quantities based on observations. For the most part the use of these formulas has been separated, with Newton being the more dominant in geophysics. This separation is arguably surprising since numerous sources of uncertainty arise in the application of classical physics in complex situations. One explanation for the separation is the difficulty in implementing Bayesian analysis in complex settings. However, recent advances in both modeling strategies and computational tools have contributed to a significant change in the scope and feasibility of Bayesian analysis. This paradigm provides opportunities for the combination of physical reasoning and observational data in a coherent analysis framework but in a fashion which manages the uncertainties in both information sources. A key to the modeling is the hierarchical viewpoint, in which separate statistical models are developed for the process variables studied and for the observations conditional on those variables. Modeling process variables in this way enables the incorporation of physics across a spectrum of levels of intensity, ranging from a qualitative use of physical reasoning to a strong reliance on numerical models. Selected examples from this spectrum are reviewed. So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality.Albert Einstein (1921)