## Mathematical constraints of Backus averaging

## Mathematical constraints of Backus averaging

## Mathematical constraints of Backus averaging

## Mathematical constraints of Backus averaging

## Mathematical constraints of Backus averaging

The Geomechanics Project is an interdisciplinary research group working on mathematical modelling of physical phenomena and foundation of science.

“I don’t think we’ve ever met, but I am a colleague of yours in seismology at Brown University (and previously at Caltech). I recently found your books and have enjoyed reading them, with a special appreciation for the philosophical issues you raise.”

It was created in 1997 by Marcelo Epstein and Michael Slawinski in the Department of Mechanical Engineering at the University of Calgary. Slawinski has been its director since the beginning.

Since 2001, The Project has been in the Department of Earth Sciences at Memorial University in St. John’s, where Michael became the Petro-Canada Research Chair in Applied Seismology.

The Geomechanics Project is composed of graduate students, postdoctoral fellows and research associates from physics, mathematics and computer science.

A pivotal point of the research is foundations of mathematical modelling of physical phenomena, as discussed in On foundations of seismology: Bringing idealization down to Earth, written by Slawinski with a philosopher of science from the University of Toronto. In an email to Slawinski on July 10, 2020, Professor Victor Tsai, Associate Professor of Earth, Environmental, and Planetary Sciences at Brown University writes,

### Mathematics

The philosophy of mathematics is concerned with metaphysical questions concerning the nature of mathematical objects and with epistemic questions concerning how we acquire knowledge of them. [...] The spectrum of views is very wide, from thinking mathematical objects are objectively real and independent from us (Platonism) to thinking they are somehow a human creation. It is commonly thought that the one and only source of evidence in mathematics is proof, understood to be a logical derivation from axioms or first principles. [...]

Canadian philosophers of mathematics and mathematicians who work on these philosophical issues include: [...] Michael Slawinski (nature of applied mathematics).

Seismology provides us with a possibility of formulating abstract models and examining their capacity for predictions and retrodictions of physical phenomena. Without a theory one cannot even relate measurements to their causes; this statement remains true in many disciplines.

In particular, many concepts used in seismological modelling appear—under a different guise—in theoretical formulations of physics of bicycles. Examination of either discipline allows us to study the empirical adequacy of mathematical concepts, with applicability to broader fields. Furthermore, challenging equations used therein provide an excellent material for studies in scientific computing, since an adequacy of a mathematical model commonly requires a formulation that is not soluble analytically; it requires numerical approaches.

Hence, there are two main venues to examine mathematical models: seismology and physics of bicycles; sensu lato, they share the same methodology.

Our contributions consist of providing theoretical understanding of formulations and desirable extensions of, and necessary constraints on, their applications. Apart from an intrinsic scientific value, such an approach is pertinent to applied seismology whose continuous enhancement of experimental and computational tools motivate and challenge the underlying theory.