Journal of Geophysical Research
Supporting Information for
Shallow crustal mechanics from volumetric strain data: Insights from Soufrière
Hills Volcano, Montserrat
N. K. Young and J. Gottsmann, School of Earth Sciences
University of Bristol, Bristol, BS8 1RJ, United Kingdom
Contents of this file
Text S1 including Figures S1 to s3 and Tables S1 to S2
Additional Supporting Information (Files uploaded separately)
ds01.pdf: Basic Model Data set. Modeled in COMSOL Multiphysics 5.0 as per set-up explanation in
Figure 3a and Table 1 and 2 (main text). A key is provided with geometric and material properties. Data
for SBM, TBM, HBM and CBM are shown for varying properties of conduit length and pressure drops
where predictions for AIRS fit the χ2 criteria.
ds02.pdf: Crustal Advanced Models Data. Modeled in COMSOL Multiphysics 5.0 as per explanation in
Figure 3a and Table 1 and 2 (main text). Model inputs and results are shown here. A key is given
referencing domain 1 and 2 and χ2 quality of fit. Details are provided for the geometry of the domains
and their material properties. Within the results tables, the Young’s modulus E of domain 2, conduit
length l and conduit radius rc are changed in turn (1-10 GPa, 1000-2000 m, and 15-50 m respectively)
until all combinations have been explored for pressure drops within 1-10 MPa. The resultant strain for
AIRS is shown. Resultant strain for TRNT and GERD can also be shown. Finally the χ2 values are
shown for AIRS and TRNT, and those combinations, which attain the AIRS and, TRNT χ2 quality of fit
are highlighted.
ds03.pdf: Halo Advanced Models Data. Modeled in COMSOL Multiphysics 5.0 as per explanation in
Figure 3a and Table 1 and 2 (main text). Model inputs and results are shown here for SHAM and
DHAM with halo radii 2 x, 5 x and 10 x the conduit radius. A key is given referencing domain 1, 2 and
3, and χ2 quality of fit. Details are provided for the geometry of the domains and their material
properties Within the results tables, the Young’s modulus E of domain 2, conduit length l and conduit
radius rc are changed in turn (1-10 GPa, 1000-2000 m, and 15-50 m respectively) until all combinations
have been explored for pressure drops within 1-10 MPa This is done for a domain 3 with Young’s
Modulus E = 0.5 GPa and = 1 GPa. The resultant strain for AIRS is shown. Resultant strain for TRNT
and GERD can also be shown. Finally the χ2 values are shown for AIRS and TRNT, and those
combinations, which attain the AIRS and TRNT χ2 quality of fit are highlighted.
1
ds04.pdf: IDHAM Inversion Data. Modeled in COMSOL Multiphysics 5.0 as per set-up for DHAM.
Starting inputs for each inversion are given for conduit radius rc, conduit length l, and halo radius rh
(geometrical properties), Domain 2 E and Domain 3 E (material properties) and pressure drop.
Inversions are run iteratively for geometry then material properties. The best geometry (in red, bottom
of IDHAM Geom 1,2 and 3, values within the standard deviation) is then used to invert for material
properties. The best-fit material properties (in red, bottom of IDHAM Material 1 and 2) are used to run
a new inversion for geometry. The starting conditions for geometry in this case do not matter as long as
they are in the range the algorithm explores.
ds05.pdf: ISHAM Inversion Data. Modeled in COMSOL Multiphysics 5.0 as per set-up for SHAM.
Starting inputs for each inversion are given for conduit radius rc, conduit length l, and halo radius rh
(geometrical properties), Domain 2 E and Domain 3 E (material properties) and pressure drop. The
best geometry (in red, bottom of ISHAM Geom 1, 2 and 3, values within the standard deviation) is
then used to invert for material properties. The best-fit material properties (in red, bottom of
ISHAM Material 1 and 2) are used to run a new inversion for geometry. The starting conditions for
geometry in this case do not matter as long as they are in the range the algorithm explores.
Introduction
These auxiliary files contain a full description of the benchmarking of analytical and numerical models
prior to the main modelling sequence and a description of the inverse modeling set-up. Additionally full
tables of results from modeling are provided in ds01-05.
This data was created from 2013-2015 using the commercially available software COMSOL
Multiphysics (5.0).
Text S1
a) Benchmarking
ai) Analytical Conduit Model
The conduit model was set up according to the model by Yang et al. [1988]. The Yang model is for
a shallowly dipping finite prolate ellipsoidal body in a HOHS. It produces vertical and horizontal
displacements. The model was adapted from the original script to produce 2D output graphs for
displacement and strain. Strain was calculated using equations S1-3 presented by Lisowski [2008],
derived from Fung
[1977] where εθθ is tangential strain, εrr is radial strain and εv is volumetric strain.
𝑢
𝑟
𝜕𝑢
𝜀𝑟𝑟 =
𝜕𝑟
1 − 2𝜈
𝜀𝑣 =
(𝜀 + 𝜀𝜃𝜃 )
1 − 𝜈 𝑟𝑟
𝜀𝜃𝜃 =
(S1)
(S2)
(S3)
aii) Numerical Conduit Model
2
An axisymmetric half-space was built on a scale large enough to prevent the boundary conditions
from interfering with results, but small enough to be computationally quick (Hickey and
Gottsmann, 2014). This half-space obeys Hooke’s Law and has a planar surface which extends
infinitely in all directions. An infinite element domain was created on the lower and right bounds of
the half-space as shown in Figure 3a (main text), with the following boundary conditions (Hickey
and Gottsmann, 2014); (1) a fixed constraint on the bottom boundary to prevent deformation, (2) a
roller on the right boundary to prevent deformation perpendicular to the boundary and (3) a free top
boundary (Hickey and Gottsmann, 2014). An oval cavity was embedded in the half-space and
numerical rotation during computation causes the oval to become an ellipsoid. The domain was
built as an HOHS. A positive pressure change was imposed on the boundary of this cavity. The
horizontal and vertical displacement and the volumetric strain were computed.
aiii) Conduit Benchmark
The Yang model for a slightly dipping prolate spheroid was used to benchmark the conduit. The
Yang model source code cannot accept a 0 value for the angle of dip. As the SHV conduit is
thought to be vertical this problem was overcome by using a value extremely close to 90° so as to
render negligible the error produced by the code. These model dimensions were then set at the
dimensions of the SHV conduit for realism. The desired depth to model the top of the conduit at
was -1m (as the SHV conduit begins at the surface) but unfortunately, the Yang script did not
permit using any depths above 10 m. Model parameters are as in Tables S1 and S2. Residuals from
benchmarking of horizontal and vertical displacement and volumetric strain are shown in Figures
S1-3.
b) Inverse Modeling
We used the optimisation module in COMSOL 5.0 to invert for geometrical properties of conduit
length l, conduit radius rc and halo radius rh, and pressure drops, and material properties of domain
2 Young’s Modulus E and domain 3 Young’s Modulus E.
The optimisation was set to minimise the objective function, which was set up as follows;
AIRS = AIRS predicted
TRNT = TRNT predicted
AIRS_strain = AIRS observed
TRNT_strain = TRNT observed
AIRS_error = 1σ error on AIRS
TRNT_error= 1σ error on TRNT
a1=AIRS/AIRS_error
a2 = TRNT/TRNT_error
asum=a1+a2
weight1 = a1/asum
weight2=a2/asum
functionweight1= (AIRS_strain*10e8-AIRS*10e8*weight1)2
functionweight2= (TRNT_strain*10e8-TRNT*10e8*weight2)2
functionweightsum = functionweight1+functionweight2 (objective function)
3
We first inverted for geometrical properties as described above and the pressure drop. The
algorithm was permitted to search for these properties within set ranges as given in Table 2 (main
text) while material properties remained constant.
Results within our σ error for AIRS and TRNT for this inversion were selected and averaged. The
average geometric properties (or within their standard deviation) were made constant in the next
iteration, which then inverted for material properties within the ranges in Table 2 (main text). This
iteration process continued until the results converged or remained within their standard deviations.
Table S1: Analytical Conduit Model Benchmarking Parameters
Using the dipping prolate spheroid model according to Yang et al. (1988)
Depth
(km)
Semi-Major
Axis (km)
Semi-Minor
Axis (km)
Poisson’s
Ratio
Young’s
Modulus
(Pa)
Density
(kg/m-3)
Pressure
Change
(MPa)
0.01
0.5
0.015
0.25
8x109
2800
10
Table S2: Numerical Conduit Model Benchmarking Parameters
Domain
Area (km)
Depth
(km)
Semi-Major
Axis (km)
SemiMinor Axis
(km)
Poisson’s
Ratio
Young’s
Modulus (Pa)
Pressure Change
(MPa)
50x50
0.01
0.5
0.015
0.25
8x109
10
4
Figure S1: Horizontal displacement residual
Figure S2: Vertical displacement residual
5
Figure S3: Volumetric strain residual
6
REFERENCES
Yang, X., P. M. Davis, J., H., Dieterich (1988), Deformation from inflation of a dipping finite
prolate spheroid in an elastic half-space as a model for volcanic stressing, Journal of
Geophysical Research: Solid Earth, (93), 4249-4257, doi:10.1029/JB093iB05p04249.
Lisowski, M. (2006), Analytical volcano deformation source models, in Volcano
deformation, 279-304, Springer Berlin Heidelberg, doi: 10.1007/978-3-540-49302-0_8
Fung, Y. C., (1977), A first course in continuum mechanics 2nd edition, Englewood Cliffs,
N.J., Prentice-Hall, Inc., 1977.~351 p.
7