Analytical volcano deformation source models

Abstract

8.1 Introduction 279 8.2 The elastic half-space: a first approximation of the Earth 280 8.2.1 Properties of an isotropic linearly elastic solid 280 8.2.2 Elastic constraints 280 8.3 Notation 281 8.3.1 Coordinate system and displacements 281 8.3.2 Stress and strain 281 8.3.3 Tilt 8.4 Surface loads 282 8.4.1 Deformation from point, uniform disk, and uniform rectangular surface loads 282 8.5 Point force, pipes, and spheroidal pressure sources 285 8.5.1 Spheroidal cavities and pipes: model elements for inflating and deflating magma chambers and vertical conduits 286 8.5.2 Point pressure source 288 8.5.3 Finite spherical pressure source 290 8.5.4 Closed pipe: a model for a plugged conduit or a cigar-shaped magma chamber 292 8.5.5 Closed pip tilt and strain components 293 8.5.6 Open pipe: a composite model for the filling of an open conduit 294 8.5.7 Sill-like magma chambers 8.6 Dipping point and finite rectangular tension cracks 297 8.7 Gravity change 300 8.8 Relationship between subsurface and surface volume changes 300 8.9 Topographic corrections to modeled deformation 301 8.9.1 Reference elevation model 302 8.9.2 Varying depth model 302 8.9.3 Topographically corrected model 303 8.10 Inversion of source parameters from deformation data 303 8.10.1 Non-linear inversion and model parameter error estimates 303 8.10.2 Choosing the best source model 304 8.1 INTRODUCTION "Forces applied to solids cause deformation, and forces applied to liquids cause flow." (Fung, 1977) Primary volcanic landforms are created by the ascent and eruption of magma. The ascending magma displaces and interacts with surrounding rock and fluids as it creates new pathways, flows through cracks or conduits, vesiculates, and accumulates in underground reservoirs. The formation of new pathways and pressure changes within existing conduits and reservoirs stress and deform the surrounding rock. Eruption products load the crust. The pattern and rate of surface deformation around volcanoes reflect the tectonic and volcanic processes transmitted to the surface through the mechanical properties of the crust. Mathematical models, based on solid and fluid mechanics, have been developed to approximate deformation from tectonic and volcanic activity. Knowledge of the concepts and limitations of continuum mechanics is helpful to understanding this chapter. The models predict surface deformation from forces acting, or displacements occurring, within the Earth. These subterranean forces or displacements are referred to as sources of deformation. Quantitative estimates of their location, geometry, and dynamics are inferred by comparing or fitting surface observations to the predictions from these idealized mathematical models. We do not derive the equations that relate forces or displacements at the source to deformation at the surface, but we do provide an overview of the methods and references for such derivations. These source models are mathematical abstractions and, as a result, this chapter is filled with equations that may be daunting, but we use plots and tables to describe important characteristics of the predicted deformation. Volcanic deformation sources include inflating, deflating, and growing bodies of various shapes and sizes, which are collectively known as volumetric sources. The opening or closing of a cavity or crack is distinct from the typical tectonic source, such as a strike-slip or dip-slip fault, where the two sides of a fault slide by one another. Of course, there are composite sources that include both tensile and shear movements, such as a leaky transform fault. Volumetric sources grow and shrink through the movement of fluids and, in some cases, include both sources and sinks. For example, magma filling a growing dike is drawn from (deflates) an adjacent magma chamber. Observed surface deformation can be fit to the predictions of the source models. The modeling of surface deformation, however, does not provide a unique description of the source causing the deformation. Even with a perfect description of the surface deformation, we could find many different ways to account for it. Model assumptions, simplifications, and data uncertainty further complicate interpretation. Nevertheless, much can be learned from non-unique modeling of sparse and imprecise data. We are interested in predicting or fitting geodetic data: station displacements, line length changes, tilt, and strain. We limit our discussion to the slow static changes that occur over long periods of time and permanent offsets associated with volcanic or tectonic events. We do not discuss oscillatory, high-frequency ground motions, such as the dynamic strains that accompany earthquakes, even though they sometimes excite volcanic systems. Earth scientists do not like to label any change as static, so they often refer to these very low frequency ground movements as quasi-static ground deformation. This chapter is more elementary than a previous discussion on modeling ground deformation in volcanic areas by De Natale and Pingue (1996), although there is some overlap. Both chapters include a short introduction to the theory of elementary strain sources in an elastic medium, a summary of spheroidal pressure sources, and a discussion of the ambiguities inherent in modeling surface deformation. DeNatale and Pingue extend their discussion to modifications needed to make the elastic half-space models more realistic. These include inversion techniques for non-uniform pressure distributions and crack openings, and the effects of inhomogeneity, plastic rheologies, and structural discontinuities. The simplifications that make analytical models tractable can, particularly in the case of structural discontinuities, result in misleading volcanological interpretations. All equations, calculations, and most figures in this chapter are included in a Mathematica notebook, although some of the chapter figures are modified versions of those created in the notebook. Mathematica is one of several mathematics software packages capable of symbolic mathematics. Mathematica allows entry of equations in familiar typeset forms, rather than the more cryptic inline expressions typical of programming languages such as FORTRAN or C. In general, equations for surface displacements for each model are entered directly into Mathematica and derivatives, such as tilt and strain, are calculated directly. The Mathematica notebook that forms the basis of this chapter and a free reader to access the notebook are included in the DVD that accompanies this book."