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Techniques and Methods 13-B1

Modeling Crustal Deformation near Active Faults and Volcanic Centers—A Catalog of Deformation Models

By Maurizio Battaglia, Peter F. Cervelli, and Jessica R. Murray


This manual provides the physical and mathematical concepts for selected models used to interpret deformation measurements near active faults and volcanic centers. The emphasis is on analytical models of deformation that can be compared with data from the Global Positioning System (GPS) receivers, Interferometric synthetic aperture radar (InSAR), leveling surveys, tiltmeters and strainmeters. Source models include pressurized spherical, ellipsoidal, and horizontal penny-shaped geometries in an elastic, homogeneous, flat half-space. Vertical dikes and faults are described following the mathematical notation for rectangular dislocations in an elastic, homogeneous, flat half-space. All the analytical expressions were verified against numerical models developed by use of COMSOL Multyphics, a Finite Element Analysis software ( In this way, typographical errors present were identified and corrected. Matlab scripts are also provided to facilitate the application of these models.

First posted July 8, 2013

dMODELS (version 1.0, June 2013)

For additional information contact:
Contact Information
Volcano Science Center - Menlo Park
U.S. Geological Survey
345 Middlefield Road, MS 910
Menlo Park, CA 94025

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Suggested citation:

Battaglia, Maurizio, Cervelli, P.F., and Murray, J.R., 2013, Modeling crustal deformation near active faults and volcanic centers—A catalog of deformation models: U.S. Geological Survey Techniques and Methods, book 13, chap. B1, 96 p.,




Geodetic Transformations Function

Spherical Source (Magma Chamber)

Prolate Spheroid Source (Magma Conduit)

Sill-Like Source

Surface Deformation and Ground Tilt for Rectangular Dikes and Faults

Internal Deformation and Strain for Rectangular Dikes and Faults

Discretized Faults

Smoothing Operator D

References Cited

Appendix 1. Mathematical Methods for Computing Displacement

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