www.sciencemag.org/cgi/content/full/338/6104/250/DC1
Supplementary Materials for
Sombrero Uplift Above the Altiplano-Puna Magma Body: Evidence of a
Ballooning Mid-Crustal Diapir
Yuri Fialko* and Jill Pearse
*To whom correspondence should be addressed. E-mail: [email protected]
Published 12 October 2012, Science 338, 250 (2012)
DOI: 10.1126/science.1226358
This PDF file includes:
Materials and Methods
Supplementary Text
Figs. S1 to S6
References
Materials and Methods
We analyzed all archive SAR data from ERS-1/2 tracks 3, 10, 282 and EnviSAT
tracks 10, 89 and 282 spanning the APULVZ area. Data were processed on the Lonestar
node of the TeraGrid supercomputer facility using JPL/Caltech software ROI PAC.
Unwrapped geocoded interferograms were sorted based on phase decorrelation and
atmospheric noise criteria using RMS of the detrended radar phase (29). Interferometric
pairs least affected by the atmospheric noise were stacked to obtain the mean LOS
velocity maps (Figures 1 and S2a,b). We also computed mean LOS velocities using the
SBAS method (30). Both techniques rendered essentially the same results (31). Figure S1
shows SAR acquisitions from the ascending EnviSAT track 89. Blue triangles denote
acquisitions made in the strip mode (IM6, average incidence angle of 40 degrees), and
red triangles denote acquisitions made in the wide swath (ScanSAR) mode. Acquisitions
made on 2004/11/15, 2004/12/20, 2005/6/13 and 2010/3/29 were found to be strongly
impacted by propagation effects (presumably, of ionospheric origin) and excluded from a
subsequent analysis. To improve the accuracy and temporal resolution of deformation
measurements, we combined data collected in the two imaging modes and computed
strip-mode-to-wide swath and wide-swath-to-wide swath interferograms (32,33), in
addition to conventional strip-mode-to-strip-mode interferograms. Because of the shallow
incidence angle of the IM6 mode, the inferred LOS velocity from the EnviSAT track 89
has a greater sensitivity to the horizontal component of the surface velocity field, which
provides a useful discriminant on the source geometry (10,18). Figure S2 shows the mean
LOS velocities deduced from stacking of radar interferograms from the EnviSAT tracks
282 (Figure S2a) and 89 (Figure S2b), and a single interferogram from the ERS track 282
that has a high signal-to-noise ratio (Figure S2c). A comparison of LOS velocities shown
in Figures S2a and S2c illustrates that the signal is space-time separable (i.e., the shape of
uplift is self-similar while the amplitude of uplift scales with time). A comparison of
Figures S2a,c, on one hand, and Figure S2b, on the other hand, clearly shows spatial
separation of the peak LOS velocities from different look directions. This separation
results from the contribution of horizontal displacements and provides a robust constraint
on the depth and geometry of the inflation source, as described in the main text.
Supplementary Text
Numerical modeling
We considered several classes of models to explain the deformation pattern revealed by
InSAR observations (Figure 1): (i) an inflating shallow (upper crust) source fed by melt
from APULVZ; (ii) two point sources of inflation and deflation in an elastic half-space;
(iii) a finite inflating source embedded in a distributed deflating source at the seismically
inferred depth of APULVZ; (iv) a diapir rising buoyantly through viscoelastic crust; and
(v) a diapir forming on top of APULVZ. The first model was rejected because no shallow
source was able to produce the observed separation between peak LOS velocities from
different radar look angles (see Figure S2). The second and third models are able to fit
the data reasonably well. Predictions of the best-fitting model (ii) are shown in Figure S3
for the EnviSAT track 282. The inferred source depths are 25 km for the inflating source,
2
and 80 km for the deflating source. As discussed in the main text, this model is untenable
because of the mismatch between the rates of volume change in the inflating and the
deflating sources, the decoupling effect of the APULVZ, and potentially significant
inelastic deformation below the brittle-ductile transition. In addition, an elastic half-space
model requires a steady magma supply from the source region to the inflating magma
body to explain a constant uplift rate at the surface. This in turn implies either a
permanent magma conduit connecting the two sources, or sustained frequent dike
injections from the source region. Neither scenario appears likely, as a permanent conduit
would result in unrealistically high magma overpressure in the upper magma body, and
frequent dike intrusions may reset the orientation of the least compressive stress such that
the vertical transport in magma-driven cracks is no longer possible (34). Also, it is not
clear what mechanisms may give rise to a focused production of melt in the deep source
region, and a quasi-steady evacuation of melt via small but frequent dike intrusions.
Model (iii) shows that the observed sombrero pattern of central uplift and flanking
subsidence can be explained by volume changes within the seismically imaged APULVZ.
However, due to its kinematic nature, model (iii) provides little insight into physical
processes responsible for the observed deformation. In particular, a magma body with
interconnected melt fraction cannot sustain localized overpressure next to regions of
magma underpressure, as the resulting pressure gradients will cause fluid flow and
pressure equili- bration. The model (iii) also ignores the effects of inelastic deformation.
Given that such deformation is likely to occur below the brittle-ductile transition on time
scale of years and decades, we investigated to which extent it might bias results of
inversions based on purely elastic half-space models. In particular, we performed
numerical experiments for the three candidate intrusion shapes - spherical, vertical
prolate and horizontal oblate pressurized sources in a layered viscoelastic medium.
Calculations were conducted using the finite element code Abaqus/Simulia (26). In these
experiments we assumed the source depth of 16 km, immediately above the seismically
imaged boundary of the APULVZ. The sides and the bottom of the computational
domain were prescribed zero normal displacement and zero shear stress boundary
conditions, and the top was stress-free. The top 10-km thick layer was assumed to be
elastic, and the substrate was assumed to be viscoelastic with linear Maxwell rheology. A
constant excess pressure boundary condition was prescribed at the surface of the
intrusion. Figure S4 shows the ratio of horizontal to vertical displacements at the Earth’s
surface for the three source types as a function of time (normalized by the Maxwell
relaxation time of the viscoelastic substrate). The first data point of each curve (at zero
time) represents an elastic response. Results shown in Figure S4 show that the ratio of
horizontal to vertical surface displacements decreases with time for all considered source
types. For each source type, a decrease in horizontal to vertical displacement ratio with
time will bias inversions that are based on elastic half-space models, in that they will
make the source geometry appear more oblate than it actually is. Given that elastic
models assuming oblate source geometry already under-predict the observed separation
between peaks in the LOS velocities from different orbits (Figure S2), a possible
contribution of ductile deformation strengthens our inference of the prolate source
geometry and the source depth corresponding to the APULVZ depth. For the assumed
linear rheology, calculations shown in Figure S4 indicate that if viscoelastic deformation
is taking place on the timescale of observations (18 years), the inferred prolate source in
3
the middle crust may have an aspect ratio greater than 2:1.
Models (iv) and (v) were subsequently designed to investigate surface deformation
associated with buoyant spheroidal magma bodies (i.e., magmatic diapirs) in the Earth’s
crust. A range of material properties and spheroid geometries and depths was explored.
We found that models of type (iv) are able to produce the amplitude and wavelength of
surface uplift that are in reasonable agreement with observations (Figure 1), but fail to
produce a noticeable peripheral subsidence. Based on results obtained in models (iii) and
(iv), we considered a model of a buoyant diapir originating from the center of the
APULVZ (Figure 1). In this model the only driving force is the density contrast between
the diapir and the ambient crust. Our model domain consisted of a 12-km thick elastic
crust underlain by 188 km thick viscoelastic substrate. The latter was assumed to obey a
temperature-dependent power-law rheology, !! = Aσn exp(−Q/RT ), where !! and σ are
the uniaxial equivalents of strain rate and deviatoric stress, respectively, Q is the
activation energy, R is the universal gas constant, T is the absolute temperature, and A is a
pre-multiplying constant. In out simulations we assumed rheological properties
intermediate between those of dry and wet granite (35): A = 9.5 × 10−6 MPa−ns−1, Q =
1.6 × 105 J mole−1, and n = 2.6. The upper crust has the Young modulus of 45 GPa (36),
the Poisson ratio of 0.25, and the density of 2.8×103 kg m−3. The viscoelastic substrate
has the Young modulus of 60 GPa and the same Poisson ratio and density as the upper
crust. The density contrast between the diapir and the ambient crust is 0.4×103 kg m−3
(5,37). We prescribed a distribution of temperature with depth as follows: T(z) = 900
arctan(z/20)+273, where z is depth in kilometers. The assumed geotherm reaches a
temperature of 950 K (close to solidus of granite) at depth of 19 km (top of the
APULVZ). The finite element mesh consisted of linear tetrahedral elements, increasing
in size from less than 1 km in the vicinity of the diapir to 10-40 km on the far-field sides
of the domain (Figure S5). At the beginning of the simulation we applied the lithostatic
stress distribution and allowed it to equilibrate with body forces due to gravity to avoid
an initial mesh distortion. A time-dependent solution was obtained for the surface
deformation resulting from buoyant ascent of the diapir and the entrainment of lowviscosity material from the APULVZ. Figure S6 shows the irreversible strain resulting
from the ballooning diapir after 40 years of deformation. Note that we did not prescribe
any pressure boundary conditions inside the hypothesized magma bodies. Because we are
interested in a quasi-steady response of the crust to a finite perturbation in density and do
not consider the initiation of a diapir, we gradually reduced the viscosity of the diapir and
the ULVZ from high values at the beginning of simulation (thereby preventing rapid flow
in response to a sudden change in density within the diapir) using the following effective
rheology: !! = Bσtm, where t is time in years, m = 3, and B = 10−8MPa−1 yr−(m+1). This
results in the effective viscosity of the diapir and the ULVZ of 1018 and 1.5×1016 Pa s 10
and 40 years after the beginning of the simulation, respectively. Figure 3 in the main text
shows the predicted surface velocity at time t = 18 years (solid red line), corresponding to
the time span of InSAR observations (see Figure 2 in the main text). The modeled uplift
pattern remains fairly constant over tens of years, slowly decelerating and broadening
with time. The accuracy of the solution was verified using mesh refinement. The highest
resolution model used in our simulations had the element size of 0.2 km at the boundary
4
between the diapir and the host rocks, and a total of 2.5 million elements. There are tradeoffs between the assumed material properties and the volume of the diapir. In particular,
lower elastic moduli, lower effective viscosities of the lower crust and the diapir/ULVZ,
and higher density contrasts would require a smaller diapir to produce the same uplift rate
at the surface.
5
0.7
2003
2004
2005
2007
0.5
2008
2009
2010
EnviSAT IM6
EnviSAT WideSwath
05
0.6
25
0
0540
149 0 06
72 108
10
3 82 1
7 1
01
12
05
0
1
1
02 0
14
18
04
06
28
02 0 07
81 07
09
110
15 1
20
12
4
04
05
13
06
18
22
08
10
1311
05
09
01
02
0138
22
052
093
12
0.2
04
0.3
19 05
24
0.4
0.1
−0.2
−0.3
10
16
−0.1
12
20 0 01
22 24
06 8
13
0
09
22
02
09
−0.7
11
20
−0.6
04
04
−0.5
01
29
−0.4
11
15
Perpendicular baseline, km
2006
009
71
8
0.8
−0.8
J M S J M S J M S J M S J M S J M S J M S J M S
Time
Fig. S1: SAR acquisitions and interferograms used in the analysis of data from the
EnviSAT track 89. Blue symbols denote acquisitions in strip mode (IM6) and red symbols
denote acquisitions in Wide Swath mode. Numerical labels indicate month and day of
Fig.
S1
the respective
acquisition. Green lines denote interferometric pairs used in the timeseries
analysis (Figure 2) and black lines denote interferometric pairs used to calculate the
SAR
acquisitions
and interferograms
used
in the
analysis
from theaxis
EnviSAT
average
LOS velocity
shown in Figure
S2b.
Letters
on of
thedata
horizontal
denotetrack
months
89.
Blue
symbols
denote
acquisitions
in
strip
mode
(IM6)
and
red
symbols
denote
of the year (January, May, September).
acquisitions in Wide Swath mode. Numerical labels indicate month and day of the
respective acquisition. Green lines denote interferometric pairs used in the timeseries
analysis (Figure 2) and black lines denote interferometric pairs used to calculate the
average LOS velocity shown in Figure S2b. Letters on the horizontal axis denote months
of the year (January, May, September).
8
6
−21.8
(a)
(b)
(c)
−22
−22.2
−22.4
−22.6
−22.8
−68
10
6
4
2
0
−2
−67.5
5
0
−5
−67 −67.6 −67.4 −67.2
−67
−66.8 −68
−67.5
−67
Fig. S2: LOS velocities from different satellite tracks: (a) descending EnviSAT track
282, image mode 2 (average incidence angle 23 deg.), epoch 2003-2010; (b) ascending
EnviSAT track 89, image mode 6 (average incidence angle 40 deg.), epoch 2003-2010;
(c) descending ERS track 282 (average incidence angle 23 deg.), interferogram Aug. 12,
1995-Jul. 31, 2005 (not used in calculation of the average velocity field shown in Figure
1). S2
A pink square denotes a peak in the LOS velocity for the descending track 282, a
Fig.
pink triangle denotes a peak in LOS velocity for the ascending track 89, and a pink star
denotes
a reference
point used
to calculate
thedescending
LOS displacement
LOS
velocities
from different
satellite
tracks: (a)
EnviSATtimeseries
track 282, (Figure
image 2).
mode 2 (average incidence angle 23 deg.), epoch 2003-2010; (b) ascending EnviSAT
track 89, image mode 6 (average incidence angle 40 deg.), epoch 2003-2010; (c)
descending ERS track 282 (average incidence angle 23 deg.), interferogram Aug. 12,
1995-Jul. 31, 2005 (not used in calculation of the average velocity field shown in Figure
1). Arrows denote the satellite heading. A pink square denotes a peak in the LOS velocity
for the descending track 282, a pink triangle denotes a peak in LOS velocity for the
ascending track 89, and a pink star denotes a reference point used to calculate the LOS
displacement timeseries (Figure 2).
9
7
−21˚00'
−21˚30'
APULVZ
−22˚00'
−22˚30'
mm/yr
−23˚00'
6
4
2
0
−2
−23˚30'
−68˚30'
−68˚00'
−67˚30'
−67˚00'
Fig.
Fig.S3S3: Predicted surface velocity from a model involving two Mogi sources: an inflating
source at depth of 25 km with the rate of volume change of 3.8 × 10−2 km3 /yr, and a
−1
Predicted
involving
sources:
source
deflating surface
source velocity
at depthfrom
of 80a model
km with
the ratetwo
of Mogi
volume
changeanofinflating
1.6 × 10
km3 /yr.
at depth of 25 km with the rate of volume change of 3.8 × 10−2 km3/yr, and a deflating
source at depth of 80 km with the rate of volume change of 1.6 × 10−1 km3/yr.
10
8
0.6
Mogi source, D=22 km
Sill, D=18 km, R=23 km
Yang source, D=16 km
Umax
/Umax
r
z
0.5
0.4
0.3
0.2
0.1
0
10
20
30
40
50
60
Non−dimensional time, t/tm
70
80
Fig. S4: Predicted ratio of maximum horizontal to maximum vertical displacements as
a function of time (normalized by the Maxwell relaxation time), for three generic source
types in viscoelastic middle crust. The upper crust in these simulations is assumed to be
elastic
Fig.
S4 and has thickness of 10 km. Solid line corresponds to an isotropic volume change
(Mogi source) at depth of 22 km, dashed line corresponds to a horizontal penny-shaped
crack having
radius
of 23 km
and depth
of 18 km,vertical
and dotted
line corresponds
to a vertical
Predicted
ratio of
maximum
horizontal
to maximum
displacements
as a function
spheroid (Yang
aspecttime),
ratio for
of 2:1
and
centroid
depth
ofin16 km.
ofprolate
time (normalized
by the source)
Maxwellwith
relaxation
three
generic
source
types
viscoelastic middle crust. The upper crust in these simulations is assumed to be elastic
and has thickness of 10 km. Solid line corresponds to an isotropic volume change (Mogi
source) at depth of 22 km, dashed line corresponds to a horizontal penny-shaped crack
having radius of 23 km and depth of 18 km, and dotted line corresponds to a vertical
prolate spheroid (Yang source) with aspect ratio of 2:1 and centroid depth of 16 km.
11
9
Fig. S5
A cross-section through the finite element mesh used in numerical simulations. Colors
denote the prescribed temperature distribution, in degrees Celsius.
10
Fig. S6
Maximum principal component of ductile strain due to a buoyant diapir rising from the
top of the ULVZ (simulation time t = 40 years).
11
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