Supplementary Chapter A. 2-dimensional Basis
Functions for General Unstructured Grids
In constructing our 2-dimensional basis functions for VDoHS rates we have considered five types of triangular elements: (i) the Lagrange second-order quadratic element with 6 parameters per element, (ii) a cubic element, which we denote as cubic
(-), with 9 parameters per element instead of the 10 required for a general cubic function, (iii) another cubic element, which we denote as cubic(+), with 12 parameters per
element and hence 2 independent quartic factors in the shape functions, (iv) a general
quartic element with 15 parameters per element, and (v) the Argyris element which is
a general quintic element with 21 parameters per element. The common feature of
these elements is that each can be formulated with the independent parameters being
function values or first or second derivatives, specified at either corner points or midpoints of sides of the elements, as indicated in Fig. A1. Consequently, at internal
points within a grid the parameters are common to two or more triangles. This has the
obvious advantage of reducing the number of independent parameters in the grid. In a
typical triangular grid there are about half as many corner points and one and half
times as many triangle sides as there are elements. It follows from the parameter types
shown in Fig. A1 that the number of independent parameters per element for a scalar
function is about 2 for the Lagrange element, 1.5 for the cubic(-) element, 3 for the
cubic(+), and 4.5 for both the quartic and Argyris elements.
derivative
= normal
(at midpoints)
Fig. A1: The types of triangular elements considered in constructing 2-dimensional VDoHS
rate basis functions.
2
Supplementary Chapter A. 2-dimensional Basis Functions for General Unstructured Grids
The cubic(-) element, like the Lagrange second-order element, is fully accurate for
general quadratic functions, while involving fewer independent parameters within a
grid. As well, it accommodates all bar one form of cubic function within each triangle. The form that cannot be accommodated is hat-shaped, with peaks at the midpoint
and corners of the triangle, and is uniquely specified as being orthogonal within the
triangle to every quadratic function and having equal coefficient values when triangle
indices are interchanged. For the cubic(+) element, which is accurate for general cubic functions, the extra quartic terms are zero on the triangle edges and antisymmetric in two of the triangle indices—though there are three such terms only two
are independent, as the three terms sum to zero.
The big difference between the Argyris element and the other elements is that the
Argyris element is the only one of the five elements that results in functions that have
continuous derivatives everywhere along the sides where triangles meet. In this
regard, the Argyris element is the simplest extension for triangular grids of the cubic
splines used for the 1-dimensional basis functions in Figs 3 and 10. All the
2-dimensional elements result in continuous functions. With the Lagrange element,
derivatives are discontinuous everywhere around the perimeters of the triangles. With
the cubic(-) element derivatives are continuous only at corner points, while with the
cubic(+) and quartic elements derivatives are continuous at midpoints of triangle sides
as well. Given that the quartic element involves the same number of independent
parameters per element as the Argyris element, the quartic element can be ruled out as
the optimum element straight away.
The shape functions for the five elements are as follows. In these expressions,
x1, x2 , x3 are the position vectors of the corner points of the triangle and ξ1 , ξ2 , ξ3
are the spatially-linear triangle coordinates each with value 1 at the corresponding
corner point and zero at the other corner points. These coordinates are
such that ξ1 + ξ 2 + ξ3 = 1 and positions within the triangle are given
by
x = ξ1x1 + ξ 2 x 2 + ξ3x 3 .
ρ ji = ( ∇ξi ⋅∇ξ j ) / ∇ξi
2
Additional
and
quantities
κ i = (1 − 2ξi ) ξ1ξ 2ξ3
are
χ i = 1 / ∇ξ i
,
for i , j = 1, 2, 3 . The shape
functions listed below are those associated with function, first and second derivative
values
f1, ( ∇f )1 , ( ∇i∇ j f )1
rivative values f12 , ( n ⋅ ∇f
)12
at the first corner point, and function and normal deat the midpoint of the side between corner points 1
and 2. For the other corner points and midpoints the shape functions are obtained by
straightforward interchange of the subscripts in the expressions.
Supplementary Chapter A. 2-dimensional Basis Functions for General Unstructured Grids
3
Lagrange:
f1  2ξ12 − ξ1  +
(A1)
f12 [4ξ1ξ2 ] +
Cubic(-):
f1 3ξ12 − 2ξ13 + 2ξ1ξ 2ξ3  +
1


− x1 ) ⋅ ( ∇f )1 ξ12ξ 2 + ξ1ξ 2ξ 3  +
2


( x3 − x1 ) ⋅ ( ∇f )1 ξ12ξ3 + 12 ξ1ξ2ξ3  + 
(x
2
(A2)
Cubic(+):
f1 3ξ12 − 2ξ13 − 6 ( ρ12κ 2 + ρ13κ 3 )  +
(x
(x
2
3
)
− x ) ⋅ ( ∇f ) ξ
− x1 ⋅ ( ∇f )1 ξ12ξ 2 − κ 2 + (1 − ρ13 ) κ 3  +
1
1
ξ − κ 3 + (1 − ρ12 ) κ 2  +
(A3)
2
1 3
( n ⋅∇f )12 [ 4κ 3 χ3 ] + 
Quartic:
(
)
f1 ( 2ξ1 − 1) 5ξ12 − 4ξ13 − 6 ( ρ12κ 2 + ρ13κ 3 )  +
(x
(x
2
3
)
− x ) ⋅ ( ∇f ) ( 2ξ − 1) ξ
− x1 ⋅ ( ∇f )1 ( 2ξ1 − 1) ξ12ξ 2 − ρ13κ 3  +
1
1
1
f12 16ξ12ξ 2 2 + 16κ 3  +
( n ⋅∇f )12 [ 4κ 3 χ3 ] + 
ξ − ρ12κ 2  +
2
1 3
(A4)
4
Supplementary Chapter A. 2-dimensional Basis Functions for General Unstructured Grids
Argyris:
f1  6ξ15 − 15ξ14 + 10ξ13 − 30 ( ρ12ξ3 + ρ13ξ2 ) ξ12ξ2ξ3  +
(x
(x
2
3
− x1 ) ⋅ ( ∇f )1  4ξ13ξ2 − 3ξ14ξ 2 − (12 ρ13 + 5ρ 23 ) ξ12ξ2 2ξ3 − 5ξ12ξ 2ξ32  +
− x1 ) ⋅ ( ∇f )1  4ξ13ξ3 − 3ξ14ξ3 − (12 ρ12 + 5ρ 32 ) ξ12ξ2ξ32 − 5ξ12ξ2 2ξ3  +
 ( x
2
i
i
j
 ( x
2
i
i
j
 ( x
3
i
i
j
1

3

− xi1 )( x 2j − x1j ) ( ∇i ∇ j f )  ξ13ξ2 2 −  ρ13 + ρ 23  ξ12ξ2 2ξ3  +
1 2
2



(A5)
− xi1 )( x 3j − x1j ) ( ∇i ∇ j f ) ξ13ξ2ξ3 − ξ12ξ 2 2ξ3 − ξ12ξ 2ξ32  +
1
1

3

− xi1 )( x 3j − x1j ) ( ∇i∇ j f )  ξ13ξ32 −  ρ12 + ρ 32  ξ12ξ2ξ32  +
1 2
2



( n ⋅ ∇f )12 16ξ12ξ2 2ξ3 χ3  + 
Schematically, for each type of element the function representation in a triangular
grid is
f ( x ) =  pkψ k ( x ).
(A6)
k
Here the parameters
pk are the independent function, first derivative and second
derivative values at the corner points and midpoints of sides for all the triangles in the
grid, and for each
pk the corresponding function ψ k ( x ) is comprised of the asso-
pk as one of the parameters. We now
want to separate the parameters into distinct sets: S0 is the set of function values at
our basis function sites (see Fig. 5), S f consists of function values at other corner
ciated shape functions in all the triangles with
points of triangles and all other parameters that are not second derivative values, and
S s (used only for the Argyris element) contains the second derivative values. In constructing our VDoHS rate basis functions, our aim is to reduce the set of free parameters to just the function values at our basis function sites in S0 . For the set S f we do
this by finding the parameter values in S f that minimise
the parameter values in
S0 .
 ∇f ( x )
2
dS in terms of
Supplementary Chapter A. 2-dimensional Basis Functions for General Unstructured Grids
5
The presence of second derivative values among the parameters for the Argyris
element means that we have to undertake a preliminary step, as minimising the spatial
integral of the square of the magnitudes of the first derivatives of f ( x ) is not a
valid way to estimate second derivative values. On the other hand, minimising
  ∇ ∇ f ( x )
i
i
2
j
dS is appropriate. Writing
j
f (x) =

k∈S0 ∪ S f
pkψ k ( x ) +  pksψ ks ( x ),
(A7)
k∈Ss
s
and minimising this integral with respect to the parameter values pk in
tain the following solution. With G
ss
S s we ob-
denoting the matrix with elements
(
)(
)
Gkqss =   ∇i ∇ jψ ks ( x ) ∇i ∇ jψ qs ( x ) dS
i
s
s
for pk and pq in
j
(A8)
S s , G s denoting the matrix with elements
Gkqs =   ∇i ∇ jψ ks ( x ) ( ∇i ∇ jψ q ( x ) )dS
i
s
for pk in
j
(
)
(A9)
S s and pq in S0 or S f , and introducing P s = − ( G ss ) G s , the
−1
parameter values pk are given by pk =
s
s

Pkqs pq .
q∈S0 ∪ S f
s
By substituting in these values for the parameters pk , we get an interim expression for f ( x ) involving only parameter values in
f (x) =

k∈S0 ∪ S f
S0 and S f :


pk ψ k ( x ) +  Pqks ψ qs ( x ) .
q∈S s


(A10)
This expression for the Argyris element is equivalent in form to the expressions for
the other four types of element, which don’t involve second derivatives as parameters.
The only difference here is that for the Argyris element
ψ k (x)
is replaced by
ψ k ( x ) +  P ψ ( x ) . To keep expressions below as simple as possible, we will
s
qk
s
q
q∈Ss
use
ψ k (x)
to mean
ψ k ( x ) +  Pqks ψ qs ( x )
q∈Ss
in the Argyris case.
6
Supplementary Chapter A. 2-dimensional Basis Functions for General Unstructured Grids
 ∇f ( x )
The minimisation of
2
dS proceeds in the same way to obtain expresf
sions for the corresponding parameter values pk in S f . Writing
 p ψ ( x ) +  p ψ ( x ),
f (x) =
0
k
0
k
f
k
k∈S0
f
k
(A11)
k∈S f
f
and minimising the integral with respect to the parameter values p k in S f , the
solution is as follows. With G
ff
(
denoting the matrix with elements
)(
)
Gkqff =  ∇ψ kf ( x ) ⋅ ∇ψ qf ( x ) dS
f
f
for p k and pq in S f , G
f
(A12)
denoting the matrix with elements
(
)
Gkqf =  ∇ψ kf ( x ) ⋅ ( ∇ψ q ( x ) ) dS
f
for pk in S f and pq in
f
values p k are given by
(A13)
( )
S0 , and introducing P f = − G ff
pkf =
P
f
kq
−1
G f , the parameter
pq .
q∈S0
Furthermore, we have reduced the expression for f ( x ) to the form
f ( x) =

k∈S0


pk ψ k ( x ) +  Pqkf ψ qf ( x )  =  pkϕk ( x ),
q∈S f

 k∈S0
in which the functions
ϕk ( x ) = ψ k ( x ) +  Pqkf ψ qf ( x )
(A14)
are recognisable as being
q∈S1
our basis functions. A property that may not be obvious is that each
ϕk (x )
has value
1 at the corresponding basis function site and is zero at the other basis function sites.
This follows from the original
ψ k (x)
for the function value at the basis function site
having this property, plus all the shape functions that contribute to the forms
ψ kf ( x )
and
ψ ks ( x )
being zero at all the basis function sites.
Another property of the basis functions
integral
 ∇ϕ ( x )
k
2
ϕk (x )
is that for each element type the
dS has a nearly uniform value that is effectively the same
for all grids, independent of the spacing and arrangement of the basis function sites.
Supplementary Chapter A. 2-dimensional Basis Functions for General Unstructured Grids
7
Basis Function Integrals
a)
Int(gradient2)
4
3
2
20
40
60
80
100
40
60
80
100
basis function site
log(Int(function2))
b)
2.5
2
1.5
20
 ∇ϕ ( x )
Fig. A2: The integrals
k
basis function site
2
dS
(top panel a) and
 ϕ (x )
k
2
dS
(bottom panel
b) for basis functions constructed using Lagrange (green), Cubic(-) (blue), Cubic(+) (red),
Quartic (magenta) and Argyris (black) elements for the grid in Fig. 5. Basis function numbers
on the horizontal axes are ordered according to the x values of the positions in Fig. 5 of the
basis function sites. The integral values in the top panel are unit-less, whereas the units in the
bottom panel are km2 and the scale is logarithmic, with the values of the integrals spanning
over one and a half orders of magnitude.
The integral
 ϕ (x )
k
2
dS , on the other hand, is highly variable and is propor-
tional to the area spanned by each basis function. The values of these two integrals for
the grid in Fig. 5 are shown in Fig. A2, with the second integral
 ϕ (x )
k
2
dS being
shown on a logarithmic scale because of its wide range of values. The basis function sites
are ordered from left to right according to their x positions in Fig. 5. To understand the
8
Supplementary Chapter A. 2-dimensional Basis Functions for General Unstructured Grids
integral values, note that the four basis functions illustrated in Fig. 6 have their largest
0
values in the regions surrounding the corresponding basis function sites x . Note also
that three in particular resemble Gaussians in those regions of the form
 1
0 2
x
−
x
 with different values of σ . Such Gaussian func2
 2σ

ϕ ( x ) = exp  −
 ∇ϕ ( x )
tions have integral values
2
dS = π and
 ϕ (x )
2
dS = πσ 2 . The
exception in Fig. 6a is the basis function associated with the function site close to the
left-hand boundary. Zero-normal-derivative boundary conditions are applied at the
left-hand and right-hand boundaries. This results in smaller values of
 ∇ϕ ( x )
2
k
dS for basis functions near those boundaries, which have their integral
values plotted at the left-hand and right-hand ends in Fig. A2.
The basis functions using the Argyris element are especially Gaussian-like, with
values of
 ∇ϕ ( x )
k
2
dS close to 3. Those using the quartic element are similar
with slightly smaller values of both
 ∇ϕ ( x )
k
2
dS and
 ϕ (x)
k
2
dS . The basis
functions using the Lagrange element have comparatively spikey, less-rounded peaks,
 ∇ϕ ( x )
which result in values of
 ϕ (x )
k
2
k
2
dS close to 2 and also smaller values of
dS than for the other elements. On the other hand, the basis functions
using the cubic(-) and cubic(+) elements have comparatively flat peaks and steeper
flanks. Their
 ∇ϕk ( x ) dS values are less consistent and the range of values
2
varies for different grids, with
 ∇ϕ ( x )
k
2
dS and
 ϕ (x)
k
2
dS being always
larger than for the Argyris element. For the grid in Fig. 5 the average value of
 ∇ϕ ( x )
k
2
dS is about 4 for the cubic(+) element and slightly less than 4 for the
cubic(-) element.
The Argyris element is the best choice of the five types of element we have tested,
based primarily on continuity of first derivatives. The quartic element follows closely
behind, except regarding continuity of first derivatives. Given that the Argyris and quartic elements involve virtually the same number of independent parameters within a grid,
the lack of complete continuity of first derivatives for the quartic element means it is
inferior to the Argyris element. In the remaining figures in this Chapter we demonstrate
that whichever element is chosen makes very little difference when applying the
basis functions in our inversion methodology. Even so, there is a clear progression.
Supplementary Chapter A. 2-dimensional Basis Functions for General Unstructured Grids
9
Rift Example: Different Elements for 120 GPS
log VDoHS rates (1e-6 /km/yr)
Lagrange
a)
Cubic(-)
b)
−1.6
−1.6
80
80
−1.8
−1.8
60
60
−2
−2
40
40
−2.2
distance (km)
−2.2
20
20
−2.4
−2.4
0
0
−2.6
−2.6
−20
−20
−2.8
−2.8
−40
−40
−3
−3
−60
−60
−3.2
−3.2
−80
−80
−3.4
−3.4
−100
−50
c)
0
−100
50
Cubic(+)
−50
d)
0
50
Quartic
−1.6
80
−1.8
60
−2
distance (km)
40
−2.2
20
−2.4
0
−2.6
−20
−1.6
80
−1.8
60
−2
40
−2.2
20
−2.4
0
−2.6
−20
−2.8
−40
−2.8
−40
−3
−60
−3
−60
−3.2
−80
−3.2
−80
−3.4
−100
−50
0
distance (km)
50
−3.4
−100
−50
0
distance (km)
50
Fig. A3: VDoHS rates scaled by shear modulus from inversions for our rift example using
different element types in constructing the VDoHS rate basis functions for the dataset with 120
GPS sites for the case of velocity standard error 0.2 mm/yr. The corresponding results using the
Argyris element are in Fig. 27a.
10
Supplementary Chapter A. 2-dimensional Basis Functions for General Unstructured Grids
Rift Example: Difference from Argyris for 120 GPS
log VDoHS rate difference (1e-6 /km/yr)
Lagrange
a)
b)
Cubic(-)
−2.6
−2.4
80
−2.6
distance (km)
60
−2.8
80
−2.8
60
−3
−3.2
40
−3
40
20
−3.2
20
0
−3.4
0
−20
−3.6 −20
−40
−3.8
−4
−60
−3.4
−3.6
−3.8
−40
−4
−60
−4.2
−4.2
−80
−80
−4.4
−4.4
−100
−50
0
−100
50
Cubic(+)
c)
−50
0
50
Quartic
d)
−2.6
80
−2.8
60
−3
distance (km)
40
−3.2
20
−3.4
0
−3.6
−20
−2.5
80
60
−3
40
20
0
−3.5
−20
−3.8
−40
−40
−4
−60
−4
−60
−4.2
−80
−80
−4.4
−100
−50
0
distance (km)
50
−100
−50
0
distance (km)
50
−4.5
Fig. A4: Magnitudes of VDoHS rate differences scaled by shear modulus from inversions for
our rift example using different element types in constructing the VDoHS rate basis functions
for the dataset with 120 GPS sites for the case of velocity standard error 0.2 mm/yr. The differences plotted are relative to the VDoHS rates using the Argyris element in Fig. 27a, and the
scales are logarithmic. Note that the differences are typically an order of magnitude smaller
than the VDoHS rates.
Supplementary Chapter A. 2-dimensional Basis Functions for General Unstructured Grids
11
Rift Example: Different Elements for 400 GPS
log VDoHS rates (1e-6 /km/yr)
Lagrange
a)
Cubic(-)
b)
80
−1.4
80
−1.5
60
60
40
40
−1.6
distance (km)
−1.8
−2
−2
20
20
0
0
−2.4
−20
−2.6
−40
−2.8
−60
−3
−2.5
−20
−40
−3
−60
−80
−80
−100
−3.5 −100
−2.2
−3.2
−3.4
−50
0
50
Cubic(+)
c)
−50
d)
−1.4
80
−1.6
60
0
50
Quartic
80
−1.5
60
−1.8
distance (km)
40
40
−2
20
−2.2
0
−2.4
−2
20
0
−20
−2.6 −20
−40
−2.8 −40
−60
−3
−3.2
−80
−2.5
−3
−60
−80
−3.4
−100
−50
0
distance (km)
50
−100
−50
0
50
−3.5
distance (km)
Fig. A5: VDoHS rates scaled by shear modulus from inversions for our rift example using
different element types in constructing the VDoHS rate basis functions for the dataset with 400
GPS sites for the case of velocity standard error 0.2 mm/yr. The corresponding results using the
Argyris element are in Fig. 27c.
12
Supplementary Chapter A. 2-dimensional Basis Functions for General Unstructured Grids
Rift Example: Difference from Argyris for 400 GPS
log VDoHS rate difference (1e-6 /km/yr)
a)
Cubic(-)
b)
Lagrange
−2
−2.4
80
−2.6
60
−2.8
40
−3
20
−3.2
0
0
−3.4
−20
−3.5 −20
−3.6
−40
−40
80
60
−2.5
distance (km)
40
20
−3
−60
−4
−3.8
−4
−60
−4.2
−80
−80
−100
−4.5 −100
−4.4
−50
c)
0
Cubic(+)
50
−50
0
50
Quartic
d)
−2.4
80
−2.6
distance (km)
60
−2.4
80
−2.6
−2.8
60
40
−3
40
20
−3.2
20
−3.2
0
−3.4
0
−3.4
−3.6 −20
−3.6
−3.8
−3.8
−20
−40
−4
−60
−2.8
−3
−40
−4
−60
−4.2
−80
−4.2
−80
−4.4
−100
−50
0
distance (km)
50
−4.4
−100
−50
0
distance (km)
50
Fig. A6: Magnitudes of VDoHS rate differences scaled by shear modulus from inversions for
our rift example using different element types in constructing the VDoHS rate basis functions
for the dataset with 400 GPS sites for the case of velocity standard error 0.2 mm/yr. The differences plotted are relative to the VDoHS rates using the Argyris element in Fig. 27c, and the
scales are logarithmic. Note that the differences are typically an order of magnitude smaller
than the VDoHS rates.
Supplementary Chapter A. 2-dimensional Basis Functions for General Unstructured Grids
13
The quartic element giving results that are closest to those with the Argyris element.
Next come the cubic(+) and cubic(-) elements. The Lagrange element gives results that
are least satisfactory and most obviously mesh dependent. Of the other elements, the
Argyris is clearly the best at giving results that show little or no influence due to the
location of triangle boundaries.
In Fig. A3 we show VDoHS rate results from inversions for our rift example in
Fig. 25 for the GPS dataset with 120 sites and data standard error 0.2 mm/yr in
Fig. 19. Here the Lagrange, cubic(-), cubic(+) and quartic elements are used, and the
results are to be compared with the corresponding results using the Argyris element in
the top left panel of Fig. 27. All five VDoHS rate solutions are very similar. This is
emphasised in Fig. A4. There the magnitudes of the differences from the Argyris
results are plotted. The corresponding results for the GPS dataset with 400 sites are
shown in Figs A5 and A6. In this case, the reference results using the Argyris element
are those in the bottom left panel of Fig. 27. Overall, the differences from the Argyris
results are about an order of magnitude smaller than the magnitudes of the VDoHS
rates. The largest differences are for the Lagrange element, with the cubic(-), cubic(+)
and quartic elements having approximately the same biggest differences. The feature
in Figs A4 and A6 that shows the quartic element gives the best agreement with Argyris results is that big differences for the cubic(-) and cubic(+) elements are spread over
a much larger area.
Supplementary Chapter B. Supplementary Figures
There are 14 supplementary figures in addition to those in Supplementary Chapter A.
They are referenced in the following sections of the book.
Fig. B1 is referenced in Section 4.4.
Figs B2-B8 are referenced in Section 5.4.
Figs B9-B12 are referenced in Section 6.3.
Figs B13-B14 are referenced in Section 3.2, and relate to the rift example in
Chapter 6.
Supplementary Chapter B. Supplementary Figures
15
u (mm/yr)
Random Additions to GPS Velocities
1
1
0
0
−1
−1
u (mm/yr)
−100
u (mm/yr)
50
100
−100
1
0
0
−1
−1
−50
0
50
100
−100
1
1
0
0
−1
−1
−100
u (mm/yr)
0
1
−100
−50
0
50
100
−100
1
1
0
0
−1
−1
−100
u (mm/yr)
−50
−50
0
50
100
−100
1
1
0
0
−1
−1
−100
−50
0
distance (km)
50
100
−100
−50
0
50
100
−50
0
50
100
−50
0
50
100
−50
0
50
100
0
50
100
−50
distance (km)
Fig. B1: The random additions for the 50 datasets of GPS velocities used in the inversions in
Figs 10-12, shown as 10 groups of 5 datasets. Locations of the 20 GPS sites in each dataset are
shown, and the error bars show the randomly chosen data standard errors, taken from a triangular probability distribution between 0.2 mm/yr and 0.6 mm/yr with average value 1/3 mm/yr.
The random additions themselves are taken from Gaussian probability distributions with zero
means and standard deviations equal to the standard error values.
16
Supplementary Chapter B. Supplementary Figures
Dip-Slip
Strike-Slip
Data Resolution
Profile 1
Data Resolution
20
20
0.8
0.5
0.7
0.4
0.3
10
0.2
15
prediction
prediction
15
0.5
0.4
10
0.3
0.2
0.1
5
0.6
5
0.1
0
5
10
15
0
20
5
10
2
0
−2
−100
−50
0
20
observation
velocity (mm/yr)
velocity (mm/yr)
observation
15
50
100
150
200
2
0
−2
−100
−50
0
50
100
150
Model Resolution
Model Resolution
0.8
12
12
0.8
10
0.6
8
0.5
0.4
6
0.3
4
10
prediction
prediction
0.7
0.1
2
0.6
8
0.4
6
4
0.2
0.2
2
0
0
2
4
6
8
10
12
2
4
8
10
12
10
VDoHS rate
(10-9 /yr/km)
10
VDoHS rate
(10-9 /yr/km)
6
input
input
5
0
−5
−100
200
distance (km)
distance (km)
−50
0
50
100
150
200
0
−10
−100
−50
0
50
100
150
200
distance (km)
distance (km)
Fig. B2: Data and model resolution for profile 1, for dip-slip (left) and strike-slip (right). Top
panels show the data resolution matrix
Rd , second panels d − Rd d
with observational stan-
dard errors (error bars) and plus and minus one standard error for the fitted model
Rd d
 −m
 (thick
Rm m
 (grey curves).
black curve) with plus and minus one standard error for the fitted model m
(curves), third panels the model resolution matrix
Rm ,
bottom panels
Locations of GPS sites (second panels) and basis function sites for VDoHS rate (bottom panels)
are shown as circles.
Supplementary Chapter B. Supplementary Figures
Dip-Slip
Strike-Slip
Data Resolution
Data Resolution
17
Profile 2
0.8
0.5
20
20
0.7
15
0.3
10
0.2
0.6
prediction
prediction
0.4
15
0.4
10
0.3
0.2
0.1
5
0.5
5
0.1
0
5
10
15
0
20
5
10
2
0
−2
−100
−50
0
50
2
100
150
0
−2
−100
200
−50
0
distance (km)
50
100
150
Model Resolution
16
16
0.8
14
0.8
14
0.6
10
0.4
8
12
prediction
12
0.6
10
0.4
8
6
6
0.2
0.2
4
4
2
2
0
5
10
0
15
5
10
input
20
VDoHS rate
(10-9 /yr/km)
VDoHS rate
(10-9 /yr/km)
15
input
10
0
−10
−100
200
distance (km)
Model Resolution
prediction
20
observation
velocity (mm/yr)
velocity (mm/yr)
observation
15
−50
0
50
100
150
200
0
−20
−100
−50
distance (km)
0
50
100
150
200
distance (km)
Fig. B3: Data and model resolution for profile 2, for dip-slip (left) and strike-slip (right). Top
panels show the data resolution matrix
Rd , second panels d − Rd d
with observational stan-
dard errors (error bars) and plus and minus one standard error for the fitted model
Rd d
 −m
 (thick
Rm m
 (grey curves).
black curve) with plus and minus one standard error for the fitted model m
(curves), third panels the model resolution matrix
Rm ,
bottom panels
Locations of GPS sites (second panels) and basis function sites for VDoHS rate (bottom panels)
are shown as circles.
18
Supplementary Chapter B. Supplementary Figures
Dip-Slip
Strike-Slip
Data Resolution
Data Resolution
18
18
0.8
16
16
0.7
14
12
0.5
10
0.4
8
0.3
6
0.6
10
0.4
8
0.2
4
0.1
2
12
6
0.2
4
0.8
14
0.6
prediction
prediction
Profile 3
2
0
0
5
10
15
5
10
observation
2
velocity (mm/yr)
velocity (mm/yr)
observation
0
−2
−4
−100
−50
0
50
100
150
200
2
0
−2
−4
−100
−50
0
distance (km)
50
100
150
Model Resolution
8
6
0.8
10
0.6
8
0.4
prediction
10
4
0.8
0.6
6
0.4
4
0.2
0.2
2
2
0
0
2
4
6
8
10
2
4
input
8
10
10
VDoHS rate
(10-9 /yr/km)
VDoHS rate
(10-9 /yr/km)
6
input
20
10
0
−10
−100
200
distance (km)
Model Resolution
prediction
15
−50
0
50
100
150
200
0
−10
−20
−100
−50
distance (km)
0
50
100
150
200
distance (km)
Fig. B4: Data and model resolution for profile 3, for dip-slip (left) and strike-slip (right). Top
panels show the data resolution matrix
Rd , second panels d − Rd d
with observational stan-
dard errors (error bars) and plus and minus one standard error for the fitted model
Rd d
 −m
 (thick
Rm m
 (grey curves).
black curve) with plus and minus one standard error for the fitted model m
(curves), third panels the model resolution matrix
Rm ,
bottom panels
Locations of GPS sites (second panels) and basis function sites for VDoHS rate (bottom panels)
are shown as circles.
Supplementary Chapter B. Supplementary Figures
Dip-Slip
Strike-Slip
Data Resolution
19
Profile 5
Data Resolution
0.8
20
20
0.7
0.8
15
0.5
0.4
10
0.3
prediction
prediction
0.6
15
0.6
10
0.4
5
0.2
0.2
5
0.1
0
5
10
15
0
20
5
10
2
0
−2
−100
−50
0
50
20
observation
2
velocity (mm/yr)
velocity (mm/yr)
observation
15
100
150
0
−2
−100
200
−50
0
distance (km)
50
100
150
Model Resolution
Model Resolution
16
16
0.8
14
14
12
0.8
12
8
0.4
6
prediction
prediction
0.6
10
0.6
10
8
0.4
6
0.2
4
0.2
4
2
2
0
0
5
10
15
5
input
15
20
VDoHS rate
(10-9 /yr/km)
VDoHS rate
(10-9 /yr/km)
10
input
20
10
0
−10
−100
200
distance (km)
−50
0
50
100
150
200
10
0
−10
−100
distance (km)
−50
0
50
100
150
200
distance (km)
Fig. B5: Data and model resolution for profile 5, for dip-slip (left) and strike-slip (right). Top
panels show the data resolution matrix
Rd , second panels d − Rd d
with observational stan-
dard errors (error bars) and plus and minus one standard error for the fitted model
Rd d
 −m
 (thick
Rm m
 (grey curves).
black curve) with plus and minus one standard error for the fitted model m
(curves), third panels the model resolution matrix
Rm ,
bottom panels
Locations of GPS sites (second panels) and basis function sites for VDoHS rate (bottom panels)
are shown as circles.
20
Supplementary Chapter B. Supplementary Figures
Dip-Slip
Strike-Slip
Data Resolution
18
0.8
18
16
0.7
16
14
0.6
14
12
0.5
10
0.4
8
0.3
6
prediction
prediction
Data Resolution
0.4
8
6
0.2
4
2
0
10
0.6
10
0.1
2
0.8
12
0.2
4
5
0
15
5
2
0
−2
−50
0
10
15
observation
velocity (mm/yr)
velocity (mm/yr)
observation
−4
−100
Profile 6
50
100
150
200
2
0
−2
−100
−50
0
distance (km)
50
100
150
Model Resolution
Model Resolution
10
10
0.8
0.8
0.7
8
8
0.5
0.4
0.3
4
0.6
prediction
prediction
0.6
6
6
0.4
4
0.2
0.2
0.1
2
2
0
0
2
4
6
8
10
2
4
input
input
6
8
10
5
VDoHS rate
(10-9 /yr/km)
VDoHS rate
(10-9 /yr/km)
10
5
0
−5
−100
200
distance (km)
−50
0
50
100
150
0
−5
−100
200
−50
distance (km)
0
50
100
150
200
distance (km)
Fig. B6: Data and model resolution for profile 6, for dip-slip (left) and strike-slip (right). Top
panels show the data resolution matrix
Rd ,
second panels
d − Rd d
with observational
standard errors (error bars) and plus and minus one standard error for the fitted model
Rd d
Rm m − m (thick
 (grey curves).
black curve) with plus and minus one standard error for the fitted model m
(curves), third panels the model resolution matrix
Rm ,
bottom panels
Locations of GPS sites (second panels) and basis function sites for VDoHS rate (bottom panels)
are shown as circles.
Supplementary Chapter B. Supplementary Figures
Dip-Slip
Strike-Slip
Data Resolution
16
0.8
14
14
12
12
0.6
10
0.4
8
prediction
prediction
Profile 7
Data Resolution
16
6
0.8
0.6
10
8
0.4
6
0.2
4
0.2
4
2
2
0
0
5
10
15
5
10
observation
velocity (mm/yr)
velocity (mm/yr)
0
−2
−100
−50
0
50
100
150
200
1
0
−1
−100
−50
0
distance (km)
50
100
150
Model Resolution
0.8
10
8
0.6
8
0.4
prediction
10
6
4
0.8
0.6
6
0.4
4
0.2
0.2
2
2
0
0
2
4
6
8
10
2
4
input
6
8
10
input
10
VDoHS rate
(10-9 /yr/km)
10
VDoHS rate
(10-9 /yr/km)
200
distance (km)
Model Resolution
prediction
15
observation
2
0
−10
−100
21
−50
0
50
100
150
200
0
−10
−100
−50
distance (km)
0
50
100
150
200
distance (km)
Fig. B7: Data and model resolution for profile 7, for dip-slip (left) and strike-slip (right). Top
panels show the data resolution matrix
Rd , second panels d − Rd d
with observational stan-
dard errors (error bars) and plus and minus one standard error for the fitted model
Rd d
 −m
 (thick
Rm m
 (grey curves).
black curve) with plus and minus one standard error for the fitted model m
(curves), third panels the model resolution matrix
Rm ,
bottom panels
Locations of GPS sites (second panels) and basis function sites for VDoHS rate (bottom panels)
are shown as circles.
22
Supplementary Chapter B. Supplementary Figures
Dip-Slip
Strike-Slip
Data Resolution
Data Resolution
0.7
30
Profile 8
30
0.8
0.6
25
0.5
20
0.4
0.6
prediction
prediction
25
20
15
0.3
10
0.2
10
5
0.1
5
0.4
15
0.2
0
0
5
10
15
20
25
30
5
10
2
0
−2
−100
−50
0
50
25
30
2
100
150
0
−2
−100
200
−50
0
distance (km)
50
100
150
Model Resolution
Model Resolution
16
0.8
14
0.7
14
12
0.6
12
10
0.5
0.3
6
0.8
0.6
prediction
0.4
8
10
8
0.4
6
0.2
4
2
2
0
10
0.2
4
0.1
5
0
15
5
input
15
10
VDoHS rate
(10-9 /yr/km)
VDoHS rate
(10-9 /yr/km)
10
input
10
0
−10
−100
200
distance (km)
16
prediction
20
observation
velocity (mm/yr)
velocity (mm/yr)
observation
15
−50
0
50
100
150
200
0
−10
−100
distance (km)
−50
0
50
100
150
200
distance (km)
Fig. B8: Data and model resolution for profile 8, for dip-slip (left) and strike-slip (right). Top
panels show the data resolution matrix
Rd , second panels d − Rd d
with observational stan-
dard errors (error bars) and plus and minus one standard error for the fitted model
Rd d
 −m
 (thick
Rm m
 (grey curves).
black curve) with plus and minus one standard error for the fitted model m
(curves), third panels the model resolution matrix
Rm ,
bottom panels
Locations of GPS sites (second panels) and basis function sites for VDoHS rate (bottom panels)
are shown as circles.
Supplementary Chapter B. Supplementary Figures
23
Rift Example: Effect of Sampling
log VDoHS rates (1e-6 /km/yr)
model exact sampling
a)
model 5 km sampling
b)
−1
80
80
−1
60
60
distance (km)
40
−1.5
40
−1.5
20
20
−2
0
−20
−2
0
−20
−2.5
−2.5
−40
−40
−60
−3
−80
−100
−60
−3
−80
−50
c)
0
50
model 10 km sampling
−3.5 −100
−50
d)
0
50
model 20 km sampling
−3.5
−1.5
80
80
60
−2.2
60
−2.4
distance (km)
−2
40
40
20
20
0
−2.5
−2.6
0
−20
−20
−40
−40
−2.8
−3
−3
−60
−60
−80
−80
−100
−3.5 −100
−50
0
distance (km)
50
−3.2
−3.4
−50
0
distance (km)
50
Fig. B9: VDoHS rates scaled by shear modulus for the rift example from regular sampling of
the exact velocities on rectangular grids with sampling distances 5, 10 and 20 km (panels b-d),
compared with the exact values (panel a). Plotting of the VDoHS rates is the same as in Fig.
25d, and results for intermediate sampling distances 7 and 14 km are shown in Figs 27d and b.
24
Supplementary Chapter B. Supplementary Figures
Rift Example: Strain Rates with 0.2mm/yr Standard Error
a) area strain rates (1e-6 /yr), 120 GPS
b) log shear strain rates (1e-6 /yr), 120 GPS
−1
0.08
80
80
0.07
60
60
0.06
distance (km)
40
40
−1.5
20
0.05
20
0
0.04
0
−20
0.03
−20
−40
0.02
−2
−40
−60
−60
0.01
−80
−80
−100
0
−50
0
−100
50
−50
0
50
−2.5
d) log shear strain rates (1e-6 /yr), 400 GPS
c) area strain rates (1e-6 /yr), 400 GPS
0.12
0.1
60
distance (km)
40
0.08
60
40
−1.5
20
20
0.06
0
−20
0.04
0
−20
−2
−40
−40
0.02
−60
−80
−100
−1
80
80
0
−50
0
distance (km)
50
−60
−80
−100
−50
0
distance (km)
50
−2.5
Fig. B10: Strain rates for the rift example from inversions for the cases of 0.2 mm/yr standard
errors. Plotting of the area strain rates and shear strain rates is the same as in Figs 25b and c
respectively. Note that the only source clearly imaged is the magma chamber, in the plot of area
strain rate for the GPS dataset with 400 sites.
Supplementary Chapter B. Supplementary Figures
25
Rift Example: Strain Rates with 0.4mm/yr Standard Error
a) area strain rates (1e-6 /yr), 120 GPS
b) log shear strain rates (1e-6 /yr), 120 GPS
0.06
80
80
60
0.05
−1.2
60
−1.4
40
40
distance (km)
0.04
−1.6
20
20
0
0.03
0
−1.8
−20
−20
0.02
−2
−40
−40
−60
0.01
−2.2
−60
−80
−80
−2.4
0
−100
−50
0
−100
50
−50
0
50
d) log shear strain rates (1e-6 /yr), 400 GPS
c) area strain rates (1e-6 /yr), 400 GPS
−1
80
0.07
60
0.06
80
60
40
40
distance (km)
0.05
−1.5
20
20
0.04
0
0
−20
0.03
−20
−40
0.02
−40
−60
0.01
−2
−60
−80
−80
0
−100
−50
0
distance (km)
50
−100
−50
0
distance (km)
50
−2.5
Fig. B11: Strain rates for the rift example from inversions for the cases of 0.4 mm/yr standard
errors. Plotting of the area strain rates and shear strain rates is the same as in Figs 25b and c
respectively.
26
Supplementary Chapter B. Supplementary Figures
Rift Example: Velocities with 0.1mm/yr Standard Error
a)
b)
data & misfit: 120 GPS
model & misfit: 120 GPS
distance (km)
10 mm/yr
80 1 mm/yr
60
60
40
40
20
20
0
0
−20
−20
−40
−40
−60
−60
−80
−80
−100
−50
c)
distance (km)
10 mm/yr
80 1 mm/yr
0
50
data & misfit: 400 GPS
−100
−50
d)
0
10 mm/yr
80 1 mm/yr
10 mm/yr
80 1 mm/yr
60
60
40
40
20
20
0
0
−20
−20
−40
−40
−60
−60
−80
−80
−100
−50
0
distance (km)
50
50
model & misfit: 400 GPS
−100
−50
0
distance (km)
50
Fig. B12: Velocities (blue) and misfits (red) at GPS sites for the two random rift-example datasets with 120 and 400 sites for the case of data standard error 0.1 mm/yr, plotted in the same
way as the 0.2 mm/yr and 0.4 mm/yr results in Figs 26 and 28.
Supplementary Chapter B. Supplementary Figures
ux for x VDoHS rate (km2)
a)
14
uy for x VDoHS rate (km2)
b)
80
80
1
12
60
60
distance (km)
40
10
40
8
0
0
6
−20
0
−20
−40
−40
−0.5
4
−60
−60
2
−80
−80
−100
−50
0
0
50
ux for y VDoHS rate (km2)
c)
0.5
20
20
−100
27
−1
−50
0
50
uy for y VDoHS rate (km2)
d)
15
80
1
60
60
distance (km)
40
0.5
20
40
10
20
0
0
−20
0
−20
−40
−0.5
−60
5
−40
−60
−80
−100
80
−1
−50
0
distance (km)
50
−80
−100
−50
0
distance (km)
50
0
28
Supplementary Chapter B. Supplementary Figures
Fig. B13: (previous page): Example of
uix ( x )
(top panels) and
uiy ( x )
functions determined from a 2-dimensional VDoHS rate basis function
dataset
for
the
rift
example
in
Chapter
5.
The
basis
(top panels) velocity
ϕi ( x )
for the 400 GPS
function
site
is
at
( x, y ) = ( −31.3, −23.2) km. For the 400 GPS dataset, all the basis function sites shown as
black dots coincide with GPS sites, as all the GPS sites are separated by more than the minimum
spacing of 4 km. That is, in this case the GPS observations are also at the black dots. These velocity functions are solved for using finite elements with a much finer mesh than the mesh used in
constructing the VDoHS rate basis functions. Zero-velocity boundary conditions are applied on
the top and bottom boundaries and the left and right boundaries are reflecting boundaries. The
uix ( x )
values at the GPS sites form one column of the A-matrix relating GPS velocities to
VDoHS rates for this 2-dimensional example, corresponding to the A-matrix for the 1dimensional case in the top panel of Fig. 4,. The
uiy ( x )
values at the GPS sites form another
column of the A-matrix.
Fig. B14: (next page): Entries in the inverse of the A-matrix corresponding to the entries in the
A-matrix shown in Fig. B13. The inverse of the A-matrix relates VDoHS rates at the basis
function sites to velocities at the GPS sites for the hypothetical case of error-free data. The
points shown as black dots are again the coinciding basis function and GPS sites for the 400
GPS dataset for the rift example in Chapter 5. The functions plotted here are constructed as
follows. The VDoHS rate values at the basis function sites are those in the two columns of the
inverse of the A-matrix for the GPS site at
( x, y ) = ( −31.3, −23.2) km. One column is
for x component of velocity (top panels) and the other column is for the y component of
velocity (bottom panels). These correspond to the columns of the inverse of the A-matrix for
the 1-dimensional case in the bottom panel of Fig. 4. The VDoHS rate functions are completed
by multiplying the value at each basis function site by the associated basis function
ϕi ( x )
and summing the contributions at each point. Note how the VDoHS rate functions are much
more localized around the GPS site than the velocity functions in Fig. B13 are about the coinciding basis function site. This corresponds in 2 dimensions to the 1-dimensional localization
close to the diagonal of the inverse of the A-matrix in the bottom panel of Fig. 4.
Supplementary Chapter B. Supplementary Figures
x VDoHS rate for ux (km-2)
a)
29
y VDoHS rate for ux (km-2)
b)
0.03
0.3
80
0.02
0.25
60
60
0.01
0.2
40
distance (km)
80
40
0.15
20
0.1
0
0
20
0
−0.01
−0.02
−20
0.05
−20
−40
0
−40
−60
−0.05
−60
−80
−0.1
−80
−0.03
−0.04
−0.05
−100
−50
0
−100
50
x VDoHS rate for uy (km-2)
c)
−50
0
50
y VDoHS rate for uy (km-2)
d)
0.3
0.03
80
80
0.02
60
0.25
60
distance (km)
0.2
40
0.01
40
20
0
20
0
−0.01
−20
0.15
0.1
0
−20
0.05
−0.02
−40
−40
0
−0.03
−60
−60
−0.04
−80
−100
−50
0
distance (km)
50
−0.05
−80
−0.05 −100
−0.1
−50
0
distance (km)
50