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August 23, 2020

Scientists of all disciplines talk about models—models of how a virus works, or the universe formed, or the structure of an atom.  Models can also help volcanologists understand what is happening beneath the ground!

Yellowstone Caldera Chronicles is a weekly column written by scientists and collaborators of the Yellowstone Volcano Observatory. This week's contribution is from Dan Dzurisin, emeritus geologist with the U.S. Geological Survey.

Models have been in the news lately as a means to forecast the course of the COVID-19 pandemic. We’ve all seen the curves we’re hoping to flatten, but what exactly is a scientific model and how are they used?

Yellowstone subsurface cross-section schematic oriented SW-NE, depi...
Yellowstone subsurface cross-section schematic oriented SW-NE, depicts rise of magma beneath mantle plus heating and movement of mantle and crustal material. Credit Univ Utah. Click to enlarge.

Models take many forms in many different fields, including volcanology. For example, a model can be a representation of what a volcano looks like beneath the ground—a cartoon, essentially, that is based on data.  The depiction of Yellowstone’s magma chamber based on seismic imaging is a good example. But models can also be complex mathematical formulations. To focus our discussion, let’s consider how they’re used to analyze and gain insight into the causes of volcano deformation, which is one of the primary methods for assessing the state of a volcanic system.

Volcanologists have several powerful tools to measure subtle contortions of the ground surface at Yellowstone, including GPS, InSAR, and borehole strainmeters. But the magmatic, tectonic, and hydrothermal processes that cause deformation, although they produce measurable changes at the surface, mostly occur several kilometers beneath the landscape that’s familiar to bears and humans alike. How do volcanologists use measurements made at the surface to gain insight into what’s going on inside Earth, hidden from direct observation? Like pandemic forecasters and many other scientists, they use models.

Volcano models are based on the principle of cause and effect: From an observed effect (like surface deformation), we can infer one or more causes (for example, surface uplift might be caused by magma intrusion or the accumulation of water and gases). That’s the easy part. The next step in the modeling process is to apply what’s known about how rocks deform so that we can quantify the relationship between cause and effect. For example, the laws of physics combined with laboratory experiments on the strength of rocks allow us to write an equation that relates the amount and pattern of surface deformation to the location, amount, and pattern of pressure change within Earth. These equations are relatively straightforward when we assume that the pressure changes occur in simple shapes, such as a sphere or pipe, although the boundaries of the actual sources beneath the ground are probably irregular in the structurally complicated and heterogeneous Earth.

The best-known model for surface deformation in volcanology is the Mogi model, named after Kiyoo Mogi, the Japanese seismologist who proposed in 1958 that a mathematical formulation developed by Norio Yamakawa three years earlier could in some cases accurately reproduce surface deformation measured at volcanoes. By today’s standards, the model is overly simplified with unrealistic assumptions, such as a vanishingly small “point” source embedded in a homogeneous, flat Earth (obviously a poor assumption!). Nonetheless, the Mogi model has been a useful tool for more than 6 decades, earning it revered status among volcanologists worldwide.

Starting with a source location, depth, and pressure change, the Mogi model predicts the extent and magnitude of the resulting surface deformation pattern based on known principles and properties of rock. A shallow source produces a localized, steep-sided deformation pattern. A deeper source produces a much broader, but more gently sloped pattern. By matching model predictions to observations, scientists can gain insight into the location and depth of an unseen deformation source beneath a volcano.

Deformation that results from pressurization of a "Mogi" source model
Cross section through the Earth showing the ground surface with an embedded pressure source (red circle)—the so-called “Mogi model”—beneath the ground.  When this pressure source expands, the ground surface inflates like a balloon (the opposite occurs when the pressure in the source decreases).  Dashed black line shows, in an exaggerated way, how the shape the surface changes in response to the pressurization of the Mogi source.

There are caveats, of course. Any model is imperfect, all observations are uncertain, and Earth is complicated. Today’s models are more sophisticated than the time-honored Mogi model, allowing them to overcome some of these limitations. For example, we can dispense with the “point source” assumption and use model sources of various shapes that more nearly approximate real-Earth sources such as magma-filled dikes or sills, magma reservoirs or feeder conduits. In fact, virtually any source geometry, however complicated, can be simulated mathematically, as can multiple sources active at the same time. And we no longer assume a flat, homogenous Earth, but account for topography and Earth structure.

Modern volcano-deformation models are limited mostly by the accuracy and availability of data. Complete and accurate knowledge of both a deformation episode in four dimensions (space and time) and the subsurface properties of Earth in the deforming area would enable development of a very accurate model of the source(s) and, by inference, reliable insights into the causative processes. We can’t say with certainty that a model based on high-quality deformation data, preferably from more than one measurement technique (GPS, InSAR, borehole sensors), accurately portrays what’s happening inside Earth. But we can say that such a model accurately mimics what we can observe at the surface, and therefore it has value as an interpretive tool.

By matching possible volcanic sources to data, models help scientists turn observations into insight—and that’s what observational science is all about. Theoretical science is about turning insight into predictions, which can be tested with observations. Guided by the scientific method, those are tried-and-true paths to the same objective truth, however elusive the “truth” about a volcano might be.

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