time-distance inversions of two realistic sunspot...
TRANSCRIPT
Time-Distance Inversions of Two Realistic Sunspot Simulations
Kyle DeGrave New Mexico State University
!Jason Jackiewicz Matthias Rempel
Doug Braun
Motivation• TD used often in the past to recover flows QS and around sunspots
- (Duvall et al. 1993, Zhao et al 2001, Gizon et al. 2003, Zhao & Kosovichev 2003, Couvidat et al. 2006, Jackiewicz et al. 2008, Kosovichev & Duvall 2011, Svanda 2012 …)
• QS TD results promising in upper ~3-5 Mm - Lots of uncertainty around active regions possessing strong B-fields - Not really any general consensus about subsurface structure of sunspots (shallow, deep? flow structure?)
• Goal to shed light on the validity of current TD methods around active regions - Ultimately have confidence in real Sun - tough without validation - Comparisons between methods can be useful, but if both are wrong (active regions), it doesn’t help - Use realistic MHD simulations for which we know the real flow structure
• Disclaimer: We don’t expect to accurately recover flows near spots - show for completeness
• Filtering, tt’s, kerns, etc. (recent pub, QS analysis is same)
Sunspot Data• Use two sims (LRes, HRes)
• Provided by M. Rempel (Rempel 2009 a,b)
• Each is ~98x98x18 Mm
• HRes. of 48x48x24 km
• LRes. of 128x128x48 km
• HRes formed and maintained a penumbra, LRes had no penumbra (“pore”)
• Both contain large-scale flow structures away from spot (Fig, time avg’d vx) - vel’s high
• Surface gravity and acoustic waves
– 22 –
Fig. 1.— The two simulations used in this study. Shown are time averages (over 25 hr)
of the v
x
velocity (left column) and the B-field magnitude (|B| =pB
2
x
+B
2
y
+B
2
z
, right
column) for half of the computation domain (y � 0) for LRes (top row) and HRes (bottom
row). The horizontal slice at the top is taken at the ⌧
500
= 0.01 (z = 0) level. The contours
at the surface mark the boundaries of the spot umbra and penumbra (in the case of HRes)
as defined in the text. Figures in a given column share the same color bar.
Data Filtering• Doppler time series at tau = 0.01 layer
(45 sec) spanning 25 hr.
• Filter out waves whose travel times we would like to measure
• Tested two filter types: phase-speed (common phase-speed and ~depth) and ridge (common radial order) (Fig)
• Ridge filters f-p3 centered on solid black
• First five ps filters from Couvidat et al. (2006) (td1-td5) dashed red
• Power turns up (bound. effect reflections) - Above 12 Mm (40 km/s) dash-dot line
• Filters applied by a multiplication in the Fourier domain
Freq
uenc
y [m
Hz]
|k | [1/Mm]0 0.5 1 1.5 2 2.5
0
1
2
3
4
5
6
7
8
HRes Power
Travel Times• Two definitions: Gizon & Birch
(2002, 2004) - GB02 and GB04 - GB04 linearized version of GB02 (xcorr phase and amp) - Trivial to compute
• Travel times measured in the center-to-annulus and quadrant configurations - out-in (oi), west-east (we), north-south (ns) (Fig, ∆ inc. left to right)
• GB02: ttoi ± changes with filter type and measurement ∆ - ∆ change also seen by Braun & Birch (2008), Couvidat (2012) - GB04 almost uniformly positive for both sims (inflow)
HRes GB02 ttoi, filters
– 23 –
f
p1
p2
p3
td1
td2
td3
td4
td5
τoi
[se
c]
−40
−30
−20
−10
0
10
20
30
40
f
p1
p2
p3
td1
td2
td3
td4
td5
τoi
[se
c]
−40
−30
−20
−10
0
10
20
30
40
f
p1
p2
p3
td1
td2
td3
td4
td5
τoi
[se
c]
−40
−30
−20
−10
0
10
20
30
40
f
p1
p2
p3
td1
td2
td3
td4
td5
τoi
[se
c]
−40
−30
−20
−10
0
10
20
30
40
Fig. 2.— All LRes (top row) and HRes (bottom row) oi travel-time maps measured using
GB02 (left column) and GB04 (right column). The � for which each map is computed
increases from left to right in each plot, covering the appropriate range of values depending
on filter type as defined in the text. The f -mode travel times have been divided by a factor
of two for easier comparison. The contours mark the boundaries of the simulation umbra
and penumbra.
Travel Times
– 23 –
f
p1
p2
p3
td1
td2
td3
td4
td5
τo
i [s
ec]
−40
−30
−20
−10
0
10
20
30
40
f
p1
p2
p3
td1
td2
td3
td4
td5
τo
i [s
ec]
−40
−30
−20
−10
0
10
20
30
40
f
p1
p2
p3
td1
td2
td3
td4
td5
τo
i [s
ec]
−40
−30
−20
−10
0
10
20
30
40
f
p1
p2
p3
td1
td2
td3
td4
td5
τo
i [s
ec]
−40
−30
−20
−10
0
10
20
30
40
Fig. 2.— All LRes (top row) and HRes (bottom row) oi travel-time maps measured using
GB02 (left column) and GB04 (right column). The � for which each map is computed
increases from left to right in each plot, covering the appropriate range of values depending
on filter type as defined in the text. The f -mode travel times have been divided by a factor
of two for easier comparison. The contours mark the boundaries of the simulation umbra
and penumbra.
– 25 –
5 10 15 20 25 30−1
−0.5
0
0.5
1
Corr
ela
tion
∆ [Mm]5 10 15 20 25 30
0.988
0.99
0.992
0.994
0.996
0.998
1
Co
rre
latio
n
∆ [Mm]
5 10 15 20 25 30−1
−0.5
0
0.5
1
∆ [Mm]
Corr
ela
tion
fp1p2p3td1td2td3td4td5
5 10 15 20 25 300.98
0.984
0.988
0.992
0.996
1
Co
rre
latio
n
∆ [Mm]
Fig. 4.— The 2D spatial correlation between GB02 and GB04 oi travel-time maps versus �
for simulations LRes (top tow) and HRes (bottom row) before (left column) and after (right
column) applying a circular mask of radius 25 Mm to each map to eliminate the spot and
immediate surrounding area.
• The two definitions agree pretty well away from spots (Fig 1)
• Correlate GB02 & GB04 tt’s for all filters and distances (Fig 2)
• Then apply circular mask of radius 25 Mm and correlate again (Fig) - Past 25 Mm, GB02 and GB04 show good correlation (>0.98) HRes GB02 (L), GB04 (R) ttoi
Forward-Modeled tt’s• Kernels in Born: computation of kernels depends on accurate modeling of simulation power spectra
- Find QS and spot simulation power spectra identical in the sense that they matched the model power equally well - And we don’t account for B-field in kernel computation
• Simulated data allows us to test kernel performance through forward modeling - Convolution between kernels and actual simulation flow fields
• If kerns perfect and no tt noise, FM should match measured tt’s perfectly - Always some level of noise, don’t account for B-field (might expect significant mismatch)
• FM oi tt’s computed and correlated with measured ones (Fig, masked)
• High corr., but bad corr. for p3 and td5 (same as QS)
– 29 –
5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
∆ [Mm]
Corr
ela
tion
fp1p2p3td1td2td3td4td5
5 10 15 20 25 300
0.2
0.4
0.6
0.8
1C
orr
ela
tion
∆ [Mm]
Fig. 7.— The 2D correlation between measured and forward modeled LRes (left) and HRes
(right) oi travel-time maps using the GB02 definition after applying the circular mask to
each map to eliminate the spot and immediate surrounding area.
LRes Masked HRes Masked
SOLA LRes vh Inversions• Quick - can use the already-computed
QS weights (same kernels and travel-time noise)
• (Fig) Each row shows flow maps recovered by inverting diff. set of GB02 tt’s
• Each column is a diff. depth - Bottom row are sim. flow maps
• 1 Mm - Capture much of the overall flow structure away from spot (flows only effected in umbra)
• 3 Mm - Some struct. still visible, but surrounding flows washed out by large flows in umbra
• 5 Mm - Sign change around spot (outflow now)
• None of this flow structure is visible in the actual simulation
– 30 –
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 136 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 126 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 94 m/s
y [M
m]
x [Mm]
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 148 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 197 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 107 m/s
y [M
m]
x [Mm]
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 140 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 166 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 137 m/s
y [M
m]
x [Mm]
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 190 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 145 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 103 m/s
y [M
m]
x [Mm]
Fig. 8.— LRes GB02 horizontal (vx
, v
y
) inversion flow maps for the ridge (first row), phase-
speed (second row), and ridge+phase-speed (third row) travel-time di↵erences for depths
(left to right) 1, 3 and 5 Mm. The smoothed simulation flow maps (i.e. vtgtx,y
) at these depths
are shown in the bottom row. The noise for each inversion is ⇠ 35 ms�1 and the reference
arrows represent the RMS velocity corresponding to each flow map. The 2D target function
at each depth is shown in the upper lefthand corner of the first row figures. The width
of the box corresponds to the horizontal FWHM of each target function and represents
the approximate spatial resolution of each flow map. All maps in the same column have
identical horizontal resolution. The contour marking the boundary of the spot umbra has
been overplotted.
ridge
LRes ans
ps
comb
1 Mm 3 Mm 5 Mm
HRes vh Inversion Results– 32 –
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 106 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 158 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 133 m/s
y [M
m]
x [Mm]
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 182 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 206 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 181 m/s
y [M
m]
x [Mm]
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 155 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 230 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 179 m/s
y [M
m]
x [Mm]
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 232 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 140 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 103 m/s
y [M
m]
x [Mm]
Fig. 10.— HRes GB02 horizontal (vx
, v
y
) inversion flow maps for the ridge (first row), phase-
speed (second row), and ridge+phase-speed (third row) travel-time di↵erences for depths (left
to right) 1, 3 and 5 Mm. The smoothed simulation flow maps (i.e. vtgtx,y
) at these depths are
shown in the bottom row. The noise for each inversion is ⇠ 35 ms�1 and the reference arrows
represent the RMS velocity corresponding to each flow map. The 2D target function at each
depth is shown in the upper lefthand corner of the first row figures. The width of the box
corresponds to the horizontal FWHM of each target function and represents the approximate
spatial resolution of each flow map. All maps in the same column have identical horizontal
resolution. The contours marking the boundaries of the spot umbra and penumbra have
been overplotted.
– 33 –
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 115 m/s
y [
Mm
]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 155 m/s
y [
Mm
]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 166 m/s
y [
Mm
]
x [Mm]
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 119 m/s
y [
Mm
]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 169 m/s
y [
Mm
]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 114 m/s
y [
Mm
]
x [Mm]
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 104 m/s
y [
Mm
]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 230 m/s
y [
Mm
]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 223 m/s
y [
Mm
]
x [Mm]
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 232 m/s
y [
Mm
]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 140 m/s
y [
Mm
]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 103 m/s
y [
Mm
]
x [Mm]
Fig. 11.— HRes GB04 horizontal (vx
, v
y
) inversion flow maps for the ridge (first row), phase-
speed (second row), and ridge+phase-speed (third row) travel-time di↵erences for depths (left
to right) 1, 3 and 5 Mm. The smoothed simulation flow maps (i.e. vtgtx,y
) at these depths are
shown in the bottom row. The noise for each inversion is ⇠ 35 ms�1 and the reference arrows
represent the RMS velocity corresponding to each flow map. The 2D target function at each
depth is shown in the upper lefthand corner of the first row figures. The width of the box
corresponds to the horizontal FWHM of each target function and represents the approximate
spatial resolution of each flow map. All maps in the same column have identical horizontal
resolution. The contours marking the boundaries of the spot umbra and penumbra have
been overplotted.
Both GB02 and GB04 shown here
HRes see moat flow - Hard to tell both do okay away from spot GB02 gets moat flow (not GB04), inflow seen in umbra
ridge
HRes ans
ps
comb
1 Mm 3 Mm 5 Mm
HRes Correlations• I said they agree well outside of the spots - mask and correlate inv’s
and answer (Fig) - Filter type is given by line color, but don’t worry about that now - Solid lines are GB02 and dashed are GB04 (agree well in upper 3 Mm) - Inverted forward-modeled correlations are very high (kind of weird?)
HRes Masked
– 35 –
1 2 3 4 50
0.2
0.4
0.6
0.8
1
Corr
ela
tion
Depth [Mm]1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
Corr
ela
tion
Depth [Mm]
1 2 3 4 5−1
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Depth [Mm]
Co
rre
latio
n
GB02 ridgeGB02 phase−speedGB02 combinedGB04 ridgeGB04 phase−speedGB04 combined
1 2 3 4 50
0.2
0.4
0.6
0.8
1
Corr
ela
tion
Depth [Mm]
Fig. 13.— The 2D correlation between horizontal inversion flow maps, vinvx,y
, and simulation
target flow maps, vtgtx,y
for LRes (top row) and HRes (bottom row) before (left column) and
after (right column) applying the circular mask to each map to eliminate the spot. All
values are computed from the mean of the individual vinvx
and v
inv
y
correlations, as these
flow components di↵er substantially in several cases. The solid lines represent the GB02
inversions, while the dashed lines represent the GB04 inversions. Also plotted using the
dash-dot (- .) lines are the correlations found using the inverted forward-modeled travel
times. Line color represents the particular filtering scheme used in the inversions as defined
in the legend.
Spot Conclusions• Able to recover flow structure in upper 3 Mm, but only at
sufficiently large distances from these magnetic features (~30 Mm)
• Consistently underestimate flow amplitudes (like quiet Sun sims) - Could be non-linearities due to strong flows? (many >500 m/s)
• No single “best” filtering scheme - Ridge, ps, comb filtering - ps and comb give comparable results, ridge worst
• Inversions of GB02 tt’s are more robust (suggested before)
• Inversions of forward-modeled tt’s outperform those of measured tt’s in terms of horizontal flow correlations - Probably highlights our inability to accurately measure travel times around strong magnetic features
Sensitivity Kernels• Need to relate surface tt measurements to
subsurface conditions present in simulations
!
• Represent the sensitivity of wave travel times to flows
• This is a linear relationship between tt's and flows - Assumes tt’s vary linearly with the subsurface perturbation
• Set of kernels in Born approximation (Birch 2007) - One set per filter, per ∆ (Fig, hor. avg.)
• Note: B-field perturbations not accounted for in kernel computation - Significant consequences
– 32 –
−10 −8 −6 −4 −2 00
0.2
0.4
0.6
0.8
1
No
rma
lize
d S
en
sitiv
ity
Depth [Mm]
fp1p2p3
−10 −8 −6 −4 −2 00
0.2
0.4
0.6
0.8
1
No
rma
lize
d S
en
sitiv
ity
Depth [Mm]
td1td2td3td4td5
Fig. 5.— Horizontally integrated ‘we’ sensitivity kernels as a function of depth. The top
panel shows examples of kernels for ridge filters and the bottom panel shows those for phase-
speed filters. In each case, the absolute value is shown, and each profile has been normalized
by the largest sensitivity value in that particular plot. The kernel distances for which these
are computed correspond to the mid-range � value for each filter.
– 11 –
5. Sensitivity Kernels
To infer flows, we relate our travel-time measurements to subsurface conditions present
in the simulations. The relationship between travel-time perturbations and flow velocity,
v, developed in the GB02 and GB04 formulation, is given in the form of a linear integral
equation
�⌧
a(r) = h
r
h
z
X
ij
K
a(ri
� r, z
j
) · v(ri
, z
j
) +N
a(r) (4)
where h
r
= h
x
h
y
and h
z
is the vertical grid spacing, Ka are three-dimensional vector-
valued kernels describing the sensitivity of wave travel times to flows for each particular
measurement geometry, filter, and �, captured in the a index. N
a represents the noise
for travel-time measurement a. The summation over i represents a horizontal spatial
convolution.
A set of kernels for flows {Kv
x
, K
v
y
, K
v
z
} in the +x̂ direction was computed in the
single scattering Born approximation (Birch & Gizon 2007). The model power spectra
that are needed for the kernel computation (as prescribed in Gizon & Birch (2002)) are
filtered exactly as the initial data power spectra used to derive the travel times for which
we are modeling. These “point-to-point” kernels are then combined in an appropriate
way to be consistent with the ‘oi’, ‘we’, and ‘ns’ travel-time geometries. Figure 5 shows
the approximate depth sensitivity of a selection of kernels computed using ridge and
phase-speed filtering. All the kernels are strongly peaked at the top of the box with
lower-amplitude peaks slightly below this region. Only one set of kernels and covariance
matrices is subsequently used in the inversions, as the power spectra of the simulations are
indistinguishable for the modes of interest.
Because we are working with synthetic rather than real solar data, we have an advantage
Spot Intro• A major goal in solar physics is to understand the subsurface
structure of sunspots, as they are a dominant feature at the solar surface
• Despite abundant surface observations, there is no general consensus about their subsurface properties
• Helioseismic results regarding spot structure are often mixed regarding depth (shallow, deep?) and flow structure
• Goal: use two of the most realistic spot simulations we have available to us to test performance of TD helioseismic methods (flows)
• Disclaimer: We don’t expect to accurately recover flows near spot (see what happens)
• Identical to QS in terms of filtering, kernels, tt’s, inversions (weights)
Travel Times
HRes GB02 HRes GB04
– 26 –
0 10 20 30 40 50−50
−25
0
25
50
75
100
Azi
mu
tha
lly−
Ave
rag
ed
tt o
i [s
ec]
Distance from Spot Center [Mm]
30 40 50
−10
0
10
0 10 20 30 40 50−50
−25
0
25
50
75
100
Azi
mu
tha
lly−
Ave
rag
ed
tt o
i [s
ec]
Distance from Spot Center [Mm]
30 40 50
−10
0
10
0 10 20 30 40 50−100
−75
−50
−25
0
25
50
75
100
125
Azi
mu
tha
lly−
Ave
rag
ed
tt o
i [s
ec]
Distance from Spot Center [Mm]
fp1p2p3td1td2td3td4td5
30 40 50
−10
0
10
0 10 20 30 40 50−100
−75
−50
−25
0
25
50
75
100
125
Azi
mu
tha
lly−
Ave
rag
ed
tt o
i [s
ec]
Distance from Spot Center [Mm]
30 40 50
−10
0
10
Fig. 5.— Azimuthally-averaged GB02 (left column) and GB04 (right column) oi travel times
for LRes (top row) and HRes (bottom row). The f -mode travel times have been divided
by a factor of two before averaging for easier comparison. The vertical lines represent the
boundaries of the simulation umbra, penumbra, and the circular mask. The travel-time
distances for which these profiles are computed correspond to the mid-range � value for
each filter. The figure inset is a zoom-in of the profiles at a radius � 25 Mm.
• Illustrate comparison another way: Azim-avg oi tt’s around spot center rad profile (Fig)
• Dashed lines are boundaries of umb., pen., 25 Mm mask
• Inset plots >25 Mm
• Effect of B-field evident near spot (sign diff. between GB02 and GB04)
• Agreement at 25-30Mm, then small values
HRes Correlations• I said they agree well outside of the spots - mask and correlate inv’s and answer (Fig 1)
- Filter type is given by line color, but don’t worry about that now - Solid lines are GB02 and Dashed are GB04 (agree well in upper 3 Mm) - Inverted forward-modeled correlations are very high (kind of weird?)
• Another test: correlate annuli of increasing radii (but same area) (Fig 2) - See how corr. changes with distance from spot center - Solid lines are GB02 and dashed are GB04 - Find agreement after ~30 Mm
– 36 –
0 10 20 30 40 50−1
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Distance from Spot Center [Mm]
Corr
ela
tion
GB02 1MmGB02 3MmGB02 5MmGB04 1MmGB04 3MmGB04 5Mm
0 10 20 30 40 50−1
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Distance from Spot Center [Mm]
Corr
ela
tion
Fig. 14.— The 2D correlation between LRes (left) and HRes (right) v
tgt
x,y
and phase-speed
v
inv
x,y
flow maps at each depth as a function of distance from spot center. These correlations
were computed over annuli of increasing inner and outer radii of roughly equal area centered
about each spot. Values shown here are the mean of the individual vx
and v
y
flow component
correlations. The distance from spot center is taken to be the average of the inner and outer
radii for a particular annulus. The vertical dashed lines represent the boundaries of the
simulation umbra, penumbra, and the circular mask.
– 35 –
1 2 3 4 50
0.2
0.4
0.6
0.8
1
Co
rre
latio
n
Depth [Mm]1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
Co
rre
latio
n
Depth [Mm]
1 2 3 4 5−1
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Depth [Mm]
Co
rre
latio
n
GB02 ridgeGB02 phase−speedGB02 combinedGB04 ridgeGB04 phase−speedGB04 combined
1 2 3 4 50
0.2
0.4
0.6
0.8
1
Co
rre
latio
n
Depth [Mm]
Fig. 13.— The 2D correlation between horizontal inversion flow maps, vinvx,y
, and simulation
target flow maps, vtgtx,y
for LRes (top row) and HRes (bottom row) before (left column) and
after (right column) applying the circular mask to each map to eliminate the spot. All
values are computed from the mean of the individual vinvx
and v
inv
y
correlations, as these
flow components di↵er substantially in several cases. The solid lines represent the GB02
inversions, while the dashed lines represent the GB04 inversions. Also plotted using the
dash-dot (- .) lines are the correlations found using the inverted forward-modeled travel
times. Line color represents the particular filtering scheme used in the inversions as defined
in the legend.
HRes Masked Annuli
Quiet Sun Helioseismology
• Before jumping in to the active regions, first validate using quiet-Sun simulations - if promising, then go on to more complicated
• Couple of previous QS validation studies: Zhao et al. (2007) inversions in ray approximation of convective sims by Benson et al. - Good horizontal flow correlation, but bad vertical
• Svanda et al. (2011) invert FM travel times - Good horizontal and improved vertical flow correlations - Implemented regularization parameters to minimize x-talk
• How do our inversions compare?
• Some detail about data filtering, tt’s, kerns, etc. (sunspot analysis is same)
QS-Simulations• Use two QS-sims (QS1, QS2) - 20 hrs
• Provided by M. Rempel (Rempel 2009 a,b)
• Each is ~98x98x18 Mm - QS1 64x64x32 km - QS2 128x128x48 km (Fig, vx)
• Periodic boundary conditions, and both are magnetic
• Both contain large-scale supergranule-sized flows (fast) which extend deeply through domains
• Surface gravity and acoustic waves
– 28 –
Fig. 1.— The two simulations used in this study. Shown are time averages (over ⇡ 20 hr) of
the vx
velocity for half of the computation domain (y � 0) for QS1 (top) and QS2 (bottom).
The horizontal slice at the top is taken at the ⌧500 = 0.01 (z = 0) level. The color scale is
the same in each image.
Travel Times• Wave travel times meas. by finding
difference between measured cross-covariances between pairs of points and a reference cross-covariance function
• Two definitions: Gizon & Birch (2002, 2004) - GB02 and GB04
• Travel times measured in the center-to-annulus and quadrant configurations
• Measured over 11-29 Mm inc. of 4 Mm
• One tt-map per distance, per filter (117 measurements) (Fig, large flows)
• Corr > 0.99, GB04 10% larger
y [M
m]
−40
−20
0
20
40
y [M
m]
x [Mm]−40−20 0 20 40
−40
−20
0
20
40
x [Mm]
−40−20 0 20 40
toi [sec]−20 −10 0 10 20
GB04 ttoi QS1, QS2, p2 (L), td3 (R)
Sensitivity Kernels• Need to relate surface tt measurements to
subsurface conditions present in simulations
• Represent the sensitivity of wave travel times to flows
!
• This is a linear relationship between tt perturbation and flows
• Set of kernels in Born approximation (Fig, hor. avg.)
• Can compute FM tt’s by convolving kernels and sim flows - compare to measured ones
• High corr. between meas. and modeled tt’s, except p3 and td5 (modeling issue?) (Fig)
−10 −8 −6 −4 −2 00
0.2
0.4
0.6
0.8
1
Nor
mal
ized
Sen
sitiv
ity
Depth [Mm]
td1td2td3td4td5
– 11 –
5. Sensitivity Kernels
To infer flows, we relate our travel-time measurements to subsurface conditions present
in the simulations. The relationship between travel-time perturbations and flow velocity,
v, developed in the GB02 and GB04 formulation, is given in the form of a linear integral
equation
�⌧
a(r) = h
r
h
z
X
ij
K
a(ri
� r, z
j
) · v(ri
, z
j
) +N
a(r) (4)
where h
r
= h
x
h
y
and h
z
is the vertical grid spacing, Ka are three-dimensional vector-
valued kernels describing the sensitivity of wave travel times to flows for each particular
measurement geometry, filter, and �, captured in the a index. N
a represents the noise
for travel-time measurement a. The summation over i represents a horizontal spatial
convolution.
A set of kernels for flows {Kv
x
, K
v
y
, K
v
z
} in the +x̂ direction was computed in the
single scattering Born approximation (Birch & Gizon 2007). The model power spectra
that are needed for the kernel computation (as prescribed in Gizon & Birch (2002)) are
filtered exactly as the initial data power spectra used to derive the travel times for which
we are modeling. These “point-to-point” kernels are then combined in an appropriate
way to be consistent with the ‘oi’, ‘we’, and ‘ns’ travel-time geometries. Figure 5 shows
the approximate depth sensitivity of a selection of kernels computed using ridge and
phase-speed filtering. All the kernels are strongly peaked at the top of the box with
lower-amplitude peaks slightly below this region. Only one set of kernels and covariance
matrices is subsequently used in the inversions, as the power spectra of the simulations are
indistinguishable for the modes of interest.
Because we are working with synthetic rather than real solar data, we have an advantage
SOLA Inversions (vh) QS2– 35 –
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 209 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 116 m/sy
[M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 60 m/s
y [M
m]
x [Mm]
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 155 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 171 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 90 m/s
y [M
m]
x [Mm]
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 197 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 166 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 101 m/s
y [M
m]
x [Mm]
−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 278 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 196 m/s
y [M
m]
x [Mm]−40 −20 0 20 40
−40
−30
−20
−10
0
10
20
30
40 127 m/s
y [M
m]
x [Mm]
Fig. 8.— QS2 horizontal (vx
, v
y
) inversion flow maps for the ridge (first row), phase-speed
(second row), and ridge+phase-speed (third row) travel-time di↵erences for depths (left to
right) 1, 3 and 5 Mm. The smoothed simulation flow maps (i.e. v
tgtx,y
) at these depths are
shown in the bottom row. The noise for each inversion is ⇠35 ms�1 and the reference arrows
represent the RMS velocity corresponding to each flow map. The 2D target function at each
depth is shown in the upper lefthand corner of the first row figures. The width of the box
corresponds to the horizontal FWHM of each target function and represents the approximate
spatial resolution of each flow map. All maps in the same column have identical horizontal
resolution. Other parameters of each inversion are presented in Table 1, inversion set 1.
Correlation coe�cients found between each inversion and the simulation are presented in
Table 2.
1 Mm 3 Mm 5 Mm
ridge
QS2 ans
(targ conv)
ps
comb
noise = 35 m/s
• Each row inv’s by inverting diff. set of tt’s
• Each column is a diff. depth
• Results reliable in upper 3 Mm - Corrs. 0.9, 0.8, 0.6
• Underestimate flow vel’s by 50%
• ps slightly better, ridge is the worst
QS2 vz Inversions• Implemented weight and x-talk constraints Svanda et al. (2011)
(Fig)
• Phase-speed: QS2 okay at 1 Mm, all others very bad
• QS1 bad at all depths
– 43 –
y [
Mm
]
−40 −20 0 20 40
−40
−20
0
20
40
v z [m
/s]
−40
−20
0
20
40
60
y [M
m]
−40 −20 0 20 40
−40
−20
0
20
40
v z [m
/s]
−600
−400
−200
0
200
400
600
y [M
m]
−40 −20 0 20 40
−40
−20
0
20
40
v z [m
/s]
−1500
−1000
−500
0
500
1000
y [M
m]
x [Mm]
−40 −20 0 20 40
−40
−20
0
20
40
v ztgt −
<v ztg
t > [m
/s]
−40
−30
−20
−10
0
10
20
30
40
50
y [M
m]
x [Mm]
−40 −20 0 20 40
−40
−20
0
20
40
v ztgt −
<v ztg
t > [m
/s]
−40
−20
0
20
40
60
y [M
m]
x [Mm]
−40 −20 0 20 40
−40
−20
0
20
40
v ztgt −
<v ztg
t > [m
/s]
−60
−40
−20
0
20
40
Fig. 16.— QS2 vertical velocity inversion maps for the phase-speed filter case (top row) for
depths (left to right) of 1, 3 and 5 Mm. The smoothed simulation flow maps (i.e. vtgtz
) at these
depths are shown in the bottom row. The parameters of each inversion (i.e. target FWHM,
inversion error, etc.) are presented in Table 1, inversion set 3. Correlation coe�cients found
between each inversion and the simulation are presented in Table 2.
ps inversions best cols: 1, 3, 5 Mm
– 41 –
1 2 3 4 5 6 7 8 9−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1C
orr
ela
tion
Depth [Mm]
vmn
inv
div invv
mn sim
div sim
Fig. 14.— Correlations between the QS2 flows at 1 Mm depth and those at deeper layers.
The dashed lines are for the inferred flows from the inversions (vinvx,y
), and the solid lines are
directly from the simulation (vtgtx,y
). Each case shows the mean correlation found using the
individual flow components vx
and v
y
, as well as the correlation found using the horizontal
divergence, rh · vh.
Spot Intro• A major goal in solar physics is to understand the subsurface structure of sunspots, as
they are a dominant feature at the solar surface
• Despite abundant surface observations, there is no general consensus about their subsurface properties - Mechanism of formation - How they are assembled and transported through the convections zone - What their 3D subsurface structure is
• They are of particular importance as they are a product of the solar dynamo, and are associated with energetic solar events like flares and coronal mass ejections
• Helioseismic results regarding spot structure are often mixed regarding depth (shallow, deep?) and flow structure
• Goal: use two of the most realistic spot simulations we have available to us to test performance of TD helioseismic methods (flows)
• Disclaimer: We don’t expect to accurately recover flows near spot (see what happens)
• Identical to QS in terms of filtering, kernels, tt’s, inversions (weights)
10) Quick QS Conclusions• Horizontal inversion results correlate well with
simulation flow maps in upper 3 Mm of domains (0.8-0.9), but worse deeper
• We consistently underestimate flow amplitudes - Possibly suggests some non-linearity of the forward problem due to strong flows in the sims (>500 m/s)
• Phase-speed filtering scheme slightly outperformed the others in terms of flow correlations in the upper 5 Mm
• Vertical inversions show very poor correlation with simulation flow maps at nearly every depth
– 34 –
0 10 20 30 40 50−2
−1.5
−1
−0.5
0
0.5
1
Azi
mu
tha
lly−
Ave
rag
ed
Ra
dia
l Ve
loci
ty
[m/s
]
Distance from Spot Center [Mm]
0
500
1000
1500
2000
2500
0 10 20 30 40 50−7
−6
−5
−4
−3
−2
−1
0
1
2
Azi
muth
ally
−A
vera
ged R
adia
l Velo
city
[m
/s]
Distance from Spot Center [Mm]
0
500
1000
1500
2000
2500
3000
3500
4000
0 10 20 30 40 50−8
−7
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
Distance from Spot Center [Mm]
Azi
muth
ally
−A
vera
ged R
adia
l Velo
city
[m
/s]
0
1000
2000
3000
4000
5000
SimGB02 phase−speedGB02 ridgeGB02 combGB04 phase−speedGB04 ridgeGB04 comb
0 10 20 30 40 50−1
−0.5
0
0.5
1
Azi
mu
tha
lly−
Ave
rag
ed
Ra
dia
l Ve
loci
ty
[m/s
]
Distance from Spot Center [Mm]
0
500
1000
1500
2000
2500
3000
3500
4000
0 10 20 30 40 50−3
−2
−1
0
1
2
Azi
muth
ally
−A
vera
ged R
adia
l Velo
city
[m
/s]
Distance from Spot Center [Mm]
0
1000
2000
3000
4000
5000
6000
0 10 20 30 40 50−4
−3
−2
−1
0
1
2
3
Distance from Spot Center [Mm]
Azi
muth
ally
−A
vera
ged R
adia
l Velo
city
[m
/s]
0
1000
2000
3000
4000
5000
6000
7000
8000
SimGB02 phase−speedGB02 ridgeGB02 combGB04 phase−speedGB04 ridgeGB04 comb
Fig. 12.— Azimuthally-averaged radial velocity profiles computed from the v
inv
x,y
flow maps
for LRes (top row) and HRes (bottom row) at depths (left to right) 1, 3 and 5 Mm. Re-
sults obtained using both GB02 and GB04 travel-time definitions are shown together in the
same figures. The solid black lines correspond to the azimuthally-averaged radial velocity
computed from the smoothed simulation flow maps (i.e. v
tgt
x,y
) at each depth. All profiles
have been normalized by the largest absolute velocity value of the simulation profile in each
figure for easier comparison. The gray dashed lines represent the magnetic field profile (i.e.
|B| =pB
2
x
+B
2
y
+B
2
z
) at each depth as a function of distance from spot center corre-
sponding to the right-most y-axis in units of Gauss. The vertical dashed lines represent the
boundaries of the simulation umbra, penumbra, and the circular mask.