of 8
submitted to
Geophys. J. Int.
SUPPLEMENTARY MATERIAL for:
1
A Bayesian 3D Linear Gravity Inversion for Complex
2
Density Distributions: Application to the Puysegur
3
Subduction System
4
Erin Hightower
1
, Michael Gurnis
1
⇤⇤
, and Harm van Avendonk
2
⇤⇤⇤
1
Seismological Laboratory, California Institute of Technology, Pasadena, CA
2
Institute for Geophysics, University of Texas, Austin, TX
Corresponding Author: email: ehightow@caltech.edu, phone: (713) 419-2198
⇤⇤
email: gurnis@caltech.edu
⇤⇤⇤
email: harm@ig.utexas.edu
5
6
2
E. Hightower et al.
SUPPLEMENTARY FIGURES
7
Here we present the results from each of the synthetic tests. For each metric shown in Figures
8
S1-S4, the combinations of Tikhonov order and priors tested are organized into panels. Rows corre-
9
spond to the order(s) of Tikhonov tested and columns to the combination of priors used. Each of the
10
figures also shows the
,
combination that gives the lowest mean absolute error on the gravity and
11
the lowest mean absolute error on the model parameters as compared to the true model for compar-
12
ison. Discussion of these results is provided in the main text of the paper. We also show additional
13
cross-sections of the model results for different combinations of
and
and different orientations to
14
illustrate how the quality of the final model changes with different regularization strengths across the
15
different tests and how the model changes lateral in both the x and y directions.
16
Supplementary Material
3
Figure S1.
Mean absolute error between the gravity from the true density model and that predicted by the
inversion for each combination of
and
, which are labeled for every other value. Panel rows represent either
first or second order Tikhonov regularization or a combination of the two. Panel columns represent, from left
to right, inversion with no priors, inversion with priors only on prisms that fall within the ocean, inversion
with priors on prisms in the ocean and crustal rocks, and inversion with priors on all prisms, including the
mantle. Red circles mark the
,
combination corresponding to the minimum MAE on the gravity; red squares
mark the
,
combination corresponding to the minimum MAE on the model parameters relative to the true
model. Colorbar is saturated at 25 mGal. Gray regions correspond to
,
combinations that yield unstable or
unreasonable results.
4
E. Hightower et al.
Figure S2.
Mean absolute error between the predicted model parameter values and the known model parameter
values from the synthetic model for each combination of
and
, which are labeled for every other value. Panel
rows represent either first or second order Tikhonov regularization or a combination of the two. Panel columns
represent, from left to right, inversion with no priors, inversion with priors only on prisms that fall within the
ocean, inversion with priors on prisms in the ocean and crustal rocks, and inversion with priors on all prisms,
including the mantle. Red circles mark the
,
combination corresponding to the minimum MAE on the gravity;
red squares mark the
,
combination corresponding to the minimum MAE on the model parameters relative
to the true model. Colorbar is saturated at
800
kg/m
3
. Gray regions correspond to
,
combinations that yield
unstable or unreasonable results.
Supplementary Material
5
Figure S3.
Mean standard deviation on the model parameters as determined from the diagonal of the covariance
matrix
C
for each combination of
and
, which are labeled for every other value. Panel rows represent either
first or second order Tikhonov regularization or a combination of the two. Panel columns represent, from left
to right, inversion with no priors, inversion with priors only on prisms that fall within the ocean, inversion with
priors on prisms in the ocean and crustal rocks, and inversion with priors on all prisms, including the mantle. Red
circles mark the
,
combination corresponding to the minimum MAE on the gravity; red squares mark the
,
combination corresponding to the minimum MAE on the model parameters relative to the true model. Colorbar
is saturated at
800
kg/m
3
. Gray regions correspond to
,
combinations that yield unstable or unreasonable
results.
6
E. Hightower et al.
Figure S4.
Mean resolution of the model parameters as determined from the diagonal of the resolution matrix
R
for each combination of
and
, which are labeled for every other value. Panel rows represent either first or
second order Tikhonov regularization or a combination of the two. Panel columns represent, from left to right,
inversion with no priors, inversion with priors only on prisms that fall within the ocean, inversion with priors
on prisms in the ocean and crustal rocks, and inversion with priors on all prisms, including the mantle. Red
circles mark the
,
combination corresponding to the minimum MAE on the gravity; red squares mark the
,
combination corresponding to the minimum MAE on the model parameters relative to the true model. Gray
regions correspond to
,
combinations that yield unstable or unreasonable results. Lower resolution means that
model parameters are determined more by the prior than they are the gravity data itself. Resolution values of 1
or near 1 mean model parameter values are resolved more by the gravity data than the prior.
Supplementary Material
7
True Gravity
1
st
Order
2
nd
Order
1
st
and 2
nd
Order
-100
0
100
Gravity (mGal)
0
100
200
300
Distance (km)
No Priors
5
10
15
20
25
True Model
Depth (km)
0
100
200
300
Distance (km)
Prior on Ocean
0
100
200
300
Distance (km)
Prior on
Ocean + Crust
0
100
200
300
Distance (km)
Prior On
All Parameters
1500
2000
2500
3000
Density (kg/m
3
)
5
10
15
20
25
1
st
Order
Depth (km)
= 1e+05
= 1e+08
= 1e+04
= 1e+04
= 1e+01
= 1e+00
= 1e+01
= 1e-03
5
10
15
20
25
2
nd
Order
Depth (km)
= 1e+06
= 1e+04
= 1e+08
= 1e+08
= 1e+05
= 1e+03
= 1e+04
= 1e+01
5
10
15
20
25
2
nd
Order in x,y
1
st
Order in z
Depth (km)
= 1e+08
= 1e+02
= 1e+08
= 1e+04
= 1e+05
= 1e+00
= 1e+04
= 1e-01
Figure S5.
Representative cross section in the x-direction of the 3D inversion results for the
and
combina-
tions that produced the minimum MAE on the model parameters for each of the regularization order and prior
combinations, as determined from the test results depicted in Figs. S1-S4.
Row 1
: gravity profiles for each of
the three cases depicted in the panels below. Dark blue line: true gravity produced by the synthetic model, with
noise; gray line: gravity from inversion using only first order Tikhonov; light blue line: gravity from inversion
using only second order Tikhonov; orange line: gravity from inversion using second order Tikhonov in the hor-
izontal and first order in the vertical.
Row 2
: cross-sections of the density model recovered from using only first
order Tikhonov for the cases of no priors, priors only on the ocean water parameters, priors on the ocean and
crustal parameters, and priors on all parameters, each with their respective minimum model parameter MAE
,
combinations.
Row 3
: cross-sections of the density model recovered from using only second order Tikhonov
for each of the different prior cases.
Row 4
: cross-sections of the density model recovered from using a com-
bination of first and second order Tikhonov for each of the different prior cases.
Row 5
: cross-section of true
synthetic density model for comparison.
8
E. Hightower et al.
True Gravity
1
st
Order
2
nd
Order
1
st
and 2
nd
Order
-60
-40
-20
0
Gravity (mGal)
0
100
200
300
400
Distance (km)
No Priors
5
10
15
20
25
True Model
Depth (km)
0
100
200
300
400
Distance (km)
Prior on Ocean
0
100
200
300
400
Distance (km)
Prior on
Ocean + Crust
0
100
200
300
400
Distance (km)
Prior On
All Parameters
1500
2000
2500
3000
Density (kg/m
3
)
5
10
15
20
25
1
st
Order
Depth (km)
= 1e+05
= 1e+08
= 1e+04
= 1e+04
= 1e+01
= 1e+00
= 1e+01
= 1e-03
5
10
15
20
25
2
nd
Order
Depth (km)
= 1e+06
= 1e+04
= 1e+08
= 1e+08
= 1e+05
= 1e+03
= 1e+04
= 1e+01
5
10
15
20
25
2
nd
Order in x,y
1
st
Order in z
Depth (km)
= 1e+08
= 1e+02
= 1e+08
= 1e+04
= 1e+05
= 1e+00
= 1e+04
= 1e-01
Figure S6.
Representative cross-section in the y-direction of the inversion results for
and
combinations that
produced the minimum MAE on the model parameters for each of the regularization order and prior combina-
tions, as determined by comparing the test results depicted in Figures S1-S4.
Row 1
: gravity profiles for each of
the three cases depicted in the panels below. Dark blue line: true gravity produced by the synthetic model, with
noise; gray line: gravity from inversion using only first order Tikhonov; light blue line: gravity from inversion
using only second order Tikhonov; orange line: gravity from inversion using second order Tikhonov in the hor-
izontal and first order in the vertical.
Row 2
: cross-sections of the density model recovered from using only first
order Tikhonov for the cases of no priors, priors only on the ocean water parameters, priors on the ocean and
crustal parameters, and priors on all parameters, each with their respective minimum model parameter MAE
,
combinations.
Row 3
: cross-sections of the density model recovered from using only second order Tikhonov
for each of the different prior cases.
Row 4
: cross-sections of the density model recovered from using a com-
bination of first and second order Tikhonov for each of the different prior cases.
Row 5
: cross-section of true
synthetic density model for comparison.