Mathematical Machinery: Table of Contents. • Slepian functions and localization.
• Wavelets on the sphere. • Sparsity-promoting inversion algorithms ...
Promoting Sparsity and Localization in Geophysical Inverse Problems Frederik J Simons Princeton University
Mathematical Machinery: Table of Contents
• Slepian functions and localization
Mathematical Machinery: Table of Contents
• Slepian functions and localization • Wavelets on the sphere
Mathematical Machinery: Table of Contents
• Slepian functions and localization • Wavelets on the sphere • Sparsity-promoting inversion algorithms
Slepian functions and localization
Localization in a nutshell Functions cannot be bandlimited and spacelimited at the same time.
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Localization in a nutshell
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Functions cannot be bandlimited and spacelimited at the same time. We find a set of bandlimited functions with optimal spatial concentration, and spacelimited functions with minimal spectral leakage outside the bandwidth.
Localization in a nutshell
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Functions cannot be bandlimited and spacelimited at the same time. We find a set of bandlimited functions with optimal spatial concentration, and spacelimited functions with minimal spectral leakage outside the bandwidth. We can use these Slepian functions as windows for spectral analysis, or as a sparse basis to represent and analyze geophysical observables – on a sphere.
R1
Θ
R2
Θ
π−Θ
Not your grandpa’s boxcar — 1 α=1 λ = 1.000000
2
m=0
α=2
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α=3
λ = 0.999998
α=4
λ = 0.999320
0 −2
λ = 0.929887 λ = 1.000000
m=1
2
λ = 0.999965
λ = 0.992010
0 −2
λ = 0.685139 λ = 0.999999
m=2
2
λ = 0.999504
0 −2 λ = 0.999986
m=3
2
λ = 0.304664
λ = 0.744704
λ = 0.076183
λ = 0.397408
λ = 0.012527
λ = 0.995135
0 −2 λ = 0.999837
2
m=4
λ = 0.940854
0 −2 0°
λ = 0.966645
40°
colatitude θ
180° 0°
40°
colatitude θ
180° 0°
40°
colatitude θ
180° 0°
40°
colatitude θ
180°
Simons, Dahlen & Wieczorek, SIAM Review, 2006
Not your grandpa’s boxcar — 2 α=1 m=0
0
λ = 1.000000
m=1 m=2
λ = 0.999320
λ = 0.929887 λ = 1.000000
λ = 0.999965
λ = 0.992010
−50 −100
λ = 0.685139 λ = 0.999999
λ = 0.999504
−50 −100
0
m=3
λ = 0.999998
α=4
−100
0
λ = 0.999986
λ = 0.940854
λ = 0.304664
λ = 0.744704
λ = 0.076183
λ = 0.397408
λ = 0.012527
λ = 0.995135
−50 −100
0
m=4
α=3
−50
0
dB
α=2
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λ = 0.999837
−50 −100
λ = 0.966645
0 18
127
degree l
0 18
127
degree l
0 18
127
degree l
0 18
127
degree l Simons, Dahlen & Wieczorek, SIAM Review, 2006
A brief history of Slepian functions — 1
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In the 60s Slepian et al. solved the problem of concentrating a bandlimited signal
1 g(t) = 2π into a time interval |t|
≤ T.
Z
+W
−W
G(ω) eiωt dω
(1)
A brief history of Slepian functions — 1
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In the 60s Slepian et al. solved the problem of concentrating a bandlimited signal
1 g(t) = 2π into a time interval |t|
Z
+W
G(ω) eiωt dω
(1)
−W
≤ T . The Slepian functions optimize the concentration Z +T g 2 (t) dt λ = Z −T . (2) +∞ g 2 (t) dt −∞
A brief history of Slepian functions — 1
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In the 60s Slepian et al. solved the problem of concentrating a bandlimited signal
1 g(t) = 2π into a time interval |t|
Z
+W
G(ω) eiωt dω
(1)
−W
≤ T . The Slepian functions optimize the concentration Z +T g 2 (t) dt λ = Z −T . (2) +∞ g 2 (t) dt −∞
They are eigenfunctions of a kernel that depends on the product T W ,
Z
1
−1
sin T W (x − x ) 0 0 g(x ) dx = λg(x). 0 π(x − x ) 0
(3)
Simons, Handbook of Geomathematics, 2010
A brief history of Slepian functions — 2
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Similarly, two-dimensional Slepian functions are bandlimited,
1 g(x) = (2π)2
Z
G(k) eik·x dk,
K
and concentrate into a finite spatial region R
∈ R2 of area A.
(4)
A brief history of Slepian functions — 2
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Similarly, two-dimensional Slepian functions are bandlimited,
1 g(x) = (2π)2
Z
G(k) eik·x dk,
K
and concentrate into a finite spatial region R
Z λ= Z
∈ R2 of area A. They maximize
g 2 (x) dx
R +∞
−∞
(4)
g 2 (x) dx
.
(5)
A brief history of Slepian functions — 2
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Similarly, two-dimensional Slepian functions are bandlimited,
1 g(x) = (2π)2
Z
G(k) eik·x dk,
K
and concentrate into a finite spatial region R
Z λ= Z
(4)
∈ R2 of area A. They maximize
g 2 (x) dx
R +∞
g 2 (x) dx
.
(5)
−∞
These are then also eigenfunctions of the integral equation
Z R
1 (2π)2
Z
ik·(x−x0 )
e
dk g(x0 ) dx0 = λg(x).
(6)
K
Simons, Handbook of Geomathematics, 2010
A brief history of Slepian functions — 3
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On the sphere, Slepian functions are the bandlimited
g(ˆ r) =
L X l X
glm Ylm (ˆ r)
l=0 m=−l
that are concentrated within a region R
∈ Ω.
(7)
A brief history of Slepian functions — 3
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On the sphere, Slepian functions are the bandlimited
g(ˆ r) =
L X l X
glm Ylm (ˆ r)
(7)
l=0 m=−l
that are concentrated within a region R
Z λ = ZR Ω
∈ Ω. They optimize the energy ratio g 2 (ˆ r) dΩ g 2 (ˆ r) dΩ
.
(8)
A brief history of Slepian functions — 3
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On the sphere, Slepian functions are the bandlimited
g(ˆ r) =
L X l X
glm Ylm (ˆ r)
(7)
l=0 m=−l
that are concentrated within a region R
Z λ = ZR Ω
∈ Ω. They optimize the energy ratio g 2 (ˆ r) dΩ g 2 (ˆ r) dΩ
.
(8)
They are eigenfunctions of
Z "X L 2l + 1 R
l=0
4π
# Pl (ˆ r ·ˆ r0 ) g(ˆ r0 ) dΩ0 = λg(ˆ r).
(9)
Simons, Handbook of Geomathematics, 2010
A bandlimited basis for: Africa
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Simons, Dahlen & Wieczorek, SIAM Review, 2006
Summary of the theory, on the sphere
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Spherical harmonics Ylm form an orthonormal basis on Ω:
Z Ω
Ylm Yl0 m0 dΩ = δll0 δmm0 .
(10)
Summary of the theory, on the sphere
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Spherical harmonics Ylm form an orthonormal basis on Ω:
Z Ω
Ylm Yl0 m0 dΩ = δll0 δmm0 .
(10)
The spherical harmonics Ylm are not orthogonal on R:
Z R
Ylm Yl0 m0 dΩ = Dlm,l0 m0 .
(11)
Summary of the theory, on the sphere
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Spherical harmonics Ylm form an orthonormal basis on Ω:
Z Ω
Ylm Yl0 m0 dΩ = δll0 δmm0 .
(10)
The spherical harmonics Ylm are not orthogonal on R:
Z R
Ylm Yl0 m0 dΩ = Dlm,l0 m0 .
(11)
The Slepian functions are the eigenfunctions of the matrix D. They form a bandlimited localized basis orthogonal on R and also on Ω.
Summary of the theory, on the sphere
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Spherical harmonics Ylm form an orthonormal basis on Ω:
Z Ω
Ylm Yl0 m0 dΩ = δll0 δmm0 .
(10)
The spherical harmonics Ylm are not orthogonal on R:
Z R
Ylm Yl0 m0 dΩ = Dlm,l0 m0 .
(11)
The Slepian functions are the eigenfunctions of the matrix D. They form a bandlimited localized basis orthogonal on R and also on Ω. The Shannon number
A K = (L + 1) , 4π 2
is the effective dimension of the space for which the bandlimited g(ˆ r) are a basis.
Basis I: spherical harmonics
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40° 20° 0° −20°
l = 2, m = 1
l = 28, m = 23
l = 14, m = 4
l = 14, m = 9
l = 35, m = 9
l = 11, m = 6
l = 1, m = 1
l = 8, m = 6
l = 18, m = 16
l = 11, m = 7
l = 14, m = 10
l = 4, m = 1
l = 30, m = 12
l = 14, m = 12
l = 25, m = 22
40° 20° 0° −20°
40° 20° 0° −20°
0°
20° 40°
0°
20° 40°
0°
20° 40°
0°
20° 40°
0°
20° 40°
Simons, Hawthorne & Beggan, SPIE, 2009
Basis I: spherical harmonics
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5040 (5027) spherical harmonic coefficients 90°
degree l
72
0°
0 −72
−90° −120°
−60°
0°
60°
−1/2 max(abs(value))
120°
36
180°
0
−36
0 order m
36
72
1/2 max(abs(value))
A global basis, good for global problems.
Simons, Hawthorne & Beggan, SPIE, 2009
Basis I: spherical harmonics
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5329 (4181) spherical harmonic coefficients 72
degree l
30°
0°
−30° −50°
−25°
0°
25°
50°
−1/2 max(abs(value))
36
0 −72
75°
0
−36
0 order m
36
72
1/2 max(abs(value))
A global basis, bad for local problems.
Simons, Hawthorne & Beggan, SPIE, 2009
Spherical harmonics → Slepian functions
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An orthogonal transform by the eigenmatrix of D introduces welcome sparsity.
spherical harmonic degree and order
1
229
457
685
913
1141
1369 1
229 457 685 913 1141 rank of the Slepian functions
1369
1
229 457 685 913 1141 rank of the Slepian functions
1369
Simons, Dahlen & Wiecorek, SIAM Review, 2006
Basis II: Slepian functions
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40° 20° 0° −20°
λ = 1.000
λ = 1.000
λ = 1.000
λ = 1.000
λ = 1.000
λ = 1.000
λ = 1.000
λ = 1.000
λ = 1.000
λ = 1.000
λ = 1.000
λ = 1.000
λ = 1.000
λ = 1.000
λ = 1.000
40° 20° 0° −20°
40° 20° 0° −20°
0°
20° 40°
0°
20° 40°
0°
20° 40°
0°
20° 40°
0°
20° 40°
Simons, Hawthorne & Beggan, SPIE, 2009
Basis II: Slepian functions
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130 (41) Slepian coefficients
R−average mse 1.18e−09% 36 rank + order α + |m|
30°
0°
−30° −50°
−25°
0°
25°
50°
−1/2 max(abs(value))
18
0 −36
75°
0
−18
0 order m
18
36
1/2 max(abs(value))
A local basis, good for local problems. Sparsity!
Simons, Hawthorne & Beggan, SPIE, 2009
That was us — how about nature?
We know how to design spatially localized Slepian basis functions, from plain old spherical harmonics, using optimization.
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That was us — how about nature?
We know how to design spatially localized Slepian basis functions, from plain old spherical harmonics, using optimization.
This is being put to good use on the Earth and planets, e.g. for the analysis of localized geomagnetic or gravitational anomalies.
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That was us — how about nature?
We know how to design spatially localized Slepian basis functions, from plain old spherical harmonics, using optimization.
This is being put to good use on the Earth and planets, e.g. for the analysis of localized geomagnetic or gravitational anomalies.
Are there processes in geophysics that are particularly suited for such a representation?
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Is geophysics sparse? √ M = [1 0 -1 0 0 0]/ 2
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√ M = [0 0 0 1 0 0]/ 2
√ M = [0 1 -1 0 0 0]/ 2
15° 10° 5° 0° −5°
a
c
b λ1 = 0.999944
λ2 = 0.998609
λ3 = 0.998609
15° 10° 5° 0° −5°
d 85°
e 90°
95°
100° 105°
85°
f 90°
95°
100° 105°
85°
90°
95°
100° 105°
Simons, Hawthorne & Beggan, SPIE, 2009
latitude from 5N
Gravity signature of the Sumatra earthquake −10° 0°
10°
−10° 0°
10°
−10° 0°
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10°
10° 0° −10° −4
x 10
a
b
c
d
e
f
0
−2.5 −4
2.5
x 10
0
−2.5 2003 latitude from 5N
geoid anomaly (m)
2.5
2005
2007
2009
2003
2005
2007
2009
2003
2005
2007
2009
10° 0° −10° −10° 0° 10° longitude from 95E
−10° 0° 10° longitude from 95E
−10° 0° 10° longitude from 95E Simons, Hawthorne & Beggan, SPIE, 2009
Slepian functions: Conclusions
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Spherical harmonics are a poor choice to invert for, represent or analyze geophysical processes without perfect global coverage. Slepian functions afford spatial and spectral selectivity.
−→ Simons, Dahlen & Wieczorek, SIAM Review 2006.
Slepian functions: Conclusions
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Spherical harmonics are a poor choice to invert for, represent or analyze geophysical processes without perfect global coverage. Slepian functions afford spatial and spectral selectivity.
−→ Simons, Dahlen & Wieczorek, SIAM Review 2006.
For inverse problems linear in the data, express the unknown model in terms of a truncated Slepian basis. This improves the conditioning of the inverse problem and allows for the recovery of local or regional structure in an orthogonal basis that suffers minimally from spectral leakage.
−→ Simons & Dahlen, GJI 2006.
For inverse problems quadratic in the data, e.g. power-spectral estimation, window the data by a set of Slepian functions and form multitaper estimates. Spectral leakage is kept under control, the estimation variance is lowered dramatically, and the spatial bias is small. This is completely equivalent with modern methods of one-dimensional power-spectral analysis.
−→ Dahlen & Simons, GJI 2008.
Wavelets on the sphere
The wavelet basis — 1 A signal can be decomposed into a set of wavelet and scaling coefficients:
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The wavelet basis — 1
23/53
A signal can be decomposed into a set of wavelet and scaling coefficients: Scaling functions
Φj,k (t) = 2j/2 Φ(2j t − k)
(12)
The wavelet basis — 1
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A signal can be decomposed into a set of wavelet and scaling coefficients: Scaling functions
Φj,k (t) = 2j/2 Φ(2j t − k)
(12)
Ψj,k (t) = 2j/2 Ψ(2j t − k)
(13)
Wavelets
The wavelet basis — 1
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A signal can be decomposed into a set of wavelet and scaling coefficients: Scaling functions
Φj,k (t) = 2j/2 Φ(2j t − k)
(12)
Ψj,k (t) = 2j/2 Ψ(2j t − k)
(13)
Wavelets
Multiresolution approximation J X X X x= hx, Ψj,k iΨj,k + hx, ΦJ,k iΦJ,k j=1
k
k
(14)
The wavelet basis — 2
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Wavelets on the sphere ?
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Cartesian wavelets are “easy” — well understood, fast algorithms, and proper accounting of boundaries (wavelets on the interval).
x
H0 H1
↓2
a1
↓2
d1
H0 H1
H0
H1
↓2 ↓2
a2 d2
↓2
aJ [n]
hx, ΦJ,n i
↓2
dJ [n]
hx, ΨJ,n i hx, Ψ2,n i hx, Ψ1,n i
Wavelets on the sphere ?
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Cartesian wavelets are “easy” — well understood, fast algorithms, and proper accounting of boundaries (wavelets on the interval).
x
H0 H1
↓2
a1
↓2
d1
H0 H1
H0
H1
↓2 ↓2
a2 d2
↓2
aJ [n]
hx, ΦJ,n i
↓2
dJ [n]
hx, ΨJ,n i hx, Ψ2,n i hx, Ψ1,n i
If we could treat the Earth like a three-dimensional Cartesian box, we would.
Wavelets on the sphere ?
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Cartesian wavelets are “easy” — well understood, fast algorithms, and proper accounting of boundaries (wavelets on the interval).
x
H0 H1
↓2
a1
↓2
d1
H0 H1
H0
H1
↓2 ↓2
a2 d2
↓2
aJ [n]
hx, ΦJ,n i
↓2
dJ [n]
hx, ΨJ,n i hx, Ψ2,n i hx, Ψ1,n i
If we could treat the Earth like a three-dimensional Cartesian box, we would. We’d take the Haar basis with depth and the Daubechies basis on the surface.
Wavelets on the sphere ?
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Cartesian wavelets are “easy” — well understood, fast algorithms, and proper accounting of boundaries (wavelets on the interval).
x
H0 H1
↓2
a1
↓2
d1
H0 H1
H0
H1
↓2 ↓2
a2 d2
↓2
aJ [n]
hx, ΦJ,n i
↓2
dJ [n]
hx, ΨJ,n i hx, Ψ2,n i hx, Ψ1,n i
If we could treat the Earth like a three-dimensional Cartesian box, we would. We’d take the Haar basis with depth and the Daubechies basis on the surface. We’d account for the edges with preconditioners and special boundary filters.
Wavelets on the sphere ?
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Cartesian wavelets are “easy” — well understood, fast algorithms, and proper accounting of boundaries (wavelets on the interval).
x
H0 H1
↓2
a1
↓2
d1
H0 H1
H0
H1
↓2 ↓2
a2 d2
↓2
aJ [n]
hx, ΦJ,n i
↓2
dJ [n]
hx, ΨJ,n i hx, Ψ2,n i hx, Ψ1,n i
If we could treat the Earth like a three-dimensional Cartesian box, we would. We’d take the Haar basis with depth and the Daubechies basis on the surface. We’d account for the edges with preconditioners and special boundary filters. And then we’d map the box onto a sphere — the cubed sphere.
Cohen, Daubechies & Vial (1993)
Cartesian → sphere: The cubed sphere
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Simons et al., GJI, 2011
Basis III: cubed-sphere wavelets
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a
b
c
d
scale 4
scale 4
scale 4
scale 4
e
f
g
h
i
scale 3
scale 3
scale 3
scale 2
scale 2
i f
g
c d a b
e
h
Simons et al., GJI, 2011
Basis IV: cubed-sphere wavelets
28/53
a
b
c
d
scale 4
scale 4
scale 4
scale 4
e
f
g
h
i
scale 3
scale 3
scale 3
scale 2
scale 2
i f
g
c d a b
e
h
Simons et al., GJI, 2011
Earth’s topography in cubed-sphere wavelets
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rmse 0.0% | 0%ile | 0% zero 5
0
−5 −1 −707
−6 D4 coefficients (m)
708
−5104
−1327 D4 reconstruction (m)
0.8 2 3.2 log10(D4 coefficients)
4.8
1257
Simons et al., GJI, 2011
Sparsity
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The sparsity appears after thresholding:
s |{z}
the signal
→
W[s] | {z }
wavelet transform
→
Sλ {W[s]} | {z }
thresholded transform
Simons et al., GJI, 2011
Thresholded cubed-spherical wavelets
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rmse 0.4% | 25%ile | 25% zero 5
0
−5 −1 −934
−42 D4 coefficients (m)
920
−5105
−1328 D4 reconstruction (m)
0.8 2 3.2 log10(D4 coefficients)
4.8
1255
Simons et al., GJI, 2011
Thresholded cubed-spherical wavelets
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rmse 1.2% | 50%ile | 50% zero 4 2 0 −2 −4 −1 −1469
−102 D4 coefficients (m)
1382
−5106
−1325 D4 reconstruction (m)
0.8 2 3.2 log10(D4 coefficients)
4.8
1254
Simons et al., GJI, 2011
Thresholded cubed-spherical wavelets
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rmse 4.9% | 85%ile | 85% zero 4 2 0 −2 −4 −1 −11949
477 D4 coefficients (m)
3748
−5098
−1319 D4 reconstruction (m)
0.8 2 3.2 log10(D4 coefficients)
4.8
1243
Simons et al., GJI, 2011
All Earth models are sparse
34/53
Simons et al., GJI, 2011
The scale length of heterogeneity — 1 3
z+
η
y+
1
y−
η ξ
6
35/53
ξ
2
x−
max = 6089 m N=7
med = −3392 m η
L = 256 5
ξ
x+
η
η
4
z−
η ξ
min = −8012 m
ξ
−7500 ξ
−3750
0 topography (m)
3750
Simons, Loris et al., GJI, 2011
scale 3 (wavs) scale 4 (wavs) scale 4 (scals)
5 N America
4 S America
6 Africa
R 0.07
R 0.02
R 0.13
R 0.17
R 0.07
R 0.28
2 Antarctica
36/53
3 Asia
1 Pacific
1 0 −1
R 0.09
1 Ritsema (2010) S wave model at 0 km
scale 2 (wavs)
scale 1 (wavs)
The scale length of heterogeneity — 2
0 −1
R 0.09
1
R 0.27
S/P 8.07
S/P 7.33
0 −1
R 0.27
R 0.22
1
R 0.46
R 0.24
S/P 4.64
R 0.40
R 0.10
S/P 4.59
S/P 4.16
0 −1
R 0.29
R 0.64
1
R 0.27
R 0.65
S/P 3.99
S/P 7.99
R 0.17
R 0.58
S/P 4.64
S/P 5.30
S/P 6.03
0 R 0.66
−1 −1
R 0.70
0
1 −1
R 0.76
0
1 −1
R 0.81
0
1 −1
R 0.55
0
1 −1
R 0.26
0
1 −1
0
1
Montelli (2006) P wave model at 0 km
Simons, Loris et al., GJI, 2011
Sparsity-promoting algorithms
Geophysical Inverse Theory
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What can we learn from noisy observations of a geophysical signal?
s(x, t) | {z }
the truth
←
F[s(x, t)] + n | {z } the data
Ockham — Tikhonov
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Minimize the data fit subject to quadratic constraints:
ˆ s = arg min kd − F sk2 s | {z }
+ λksk2 | {z } quadratic misfit `2 penalty
` — Tao — Romberg Candes
Minimize the data fit subject to sparsity constraints:
ˆ s = arg min kd − F sk2 s | {z }
+ λksk0 | {z } quadratic misfit `0 penalty
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` — Tao — Romberg Candes
Minimize the data fit subject to sparsity constraints:
ˆ s = arg min kd − F sk2 s | {z }
+ λksk0 | {z } quadratic misfit `0 penalty
Sparse is what has few nonzero components in a certain basis:
P ksk0 or, by extension ksk1 = i |si | is small |{z} | {z } `0 norm `1 norm
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Faceoff – 1
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Let this be s, the (certainly sparse) truth:
0.8
0.6
0.4
0.2
20
40
60
80
100
Loris (2008)
Faceoff – 2
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Let this be F, the forward model:
The noisy data are d
= Fs + n
Loris (2008)
Faceoff – 3
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The Ockham-Tikhonov solution
ˆ s = arg min kd − F sk2 + λksk2 0.8
0.6
0.4
0.2
20
40
60
80
100
Loris (2008)
Faceoff – 4
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` The Candes-Tao solution
ˆ s = arg min kd − F sk2 + 2λksk1 0.8
0.6
0.4
0.2
20
40
60
80
100
Loris (2008)
Conclusion:
We used to solve noisy inverse problems with insufficient or inconsistent data with quadratic regularization.
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Conclusion:
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We used to solve noisy inverse problems with insufficient or inconsistent data with quadratic regularization.
But if the signal is sparse in some basis, we should regularize towards sparsity.
Conclusion:
45/53
We used to solve noisy inverse problems with insufficient or inconsistent data with quadratic regularization.
But if the signal is sparse in some basis, we should regularize towards sparsity. Sparsity is `0 . Its combination with `2 data fitting is intractable. Equivalent solutions are obtained iteratively when the `1 norm is used.
The algorithm
The iterative algorithm is fast and avoids matrix inversion per se:
W[sn+1 ] = | {z } iterate
Sλ |{z}
threshold
n
h
W sn + FT (d − F sn ) | {z }
io
|
}
→
ˆ s
gradient descent
{z
wavelet transform
Simons et al., GJI, 2011
Faceoff – 1
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Let this be s, the truth (sparse or not):
Simons et al., GJI, 2011
Faceoff – 2
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Let this be F, the forward model:
The noisy data are d
= Fs + n Simons et al., GJI, 2011
Faceoff – 3
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The cubed-sphere-thresholded-wavelet-transform solution
ˆ s = arg min kd − F sk2 + 2λkW[s]k1
Simons et al., GJI, 2011
Faceoff – 4
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The known input s (again, for comparison)
ˆ s = arg min kd − F sk2 + 2λkWsk1
Simons et al., GJI, 2011
Faceoff – 5
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Behavior of the algorithm
χ2/N misfit norm
ˆ s = arg min kd − F sk2 + 2λkWsk1
1
10
0
10
1
10
2
10
3
10
ℓ 1 nor m of t he mo del co effi cie Simons et al., GJI, 2011
Conclusions
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Recent results in applied mathematics can be brought to bear fruitfully on problems in geophysical inverse theory and spectral estimation, both in geophysics and cosmology.
Conclusions
52/53
Recent results in applied mathematics can be brought to bear fruitfully on problems in geophysical inverse theory and spectral estimation, both in geophysics and cosmology. Using spatiospectral localization techniques and basis projection we are recovering subtle changes in Earth’s gravitational and magnetic fields from noisy and incomplete satellite data. This provides complementary information to study large earthquakes and melting icecaps from time-variable gravity, among other things.
Conclusions
52/53
Recent results in applied mathematics can be brought to bear fruitfully on problems in geophysical inverse theory and spectral estimation, both in geophysics and cosmology. Using spatiospectral localization techniques and basis projection we are recovering subtle changes in Earth’s gravitational and magnetic fields from noisy and incomplete satellite data. This provides complementary information to study large earthquakes and melting icecaps from time-variable gravity, among other things. Using sparsity-promoting iterative inversion algorithms and novel wavelet constructions on the sphere we can tackle the largest-scale multi-resolution tomographic data sets of today. Chief workhorse in this context is a set of algorithms to handle mixed-norm penalty functions.
