# Asphericity of the base of the solar convection zone

Sarbani Basu Department of Astronomy, Yale University, PO Box 208101, New
Haven, CT 06520-8101,USA  Sylvain G. Korzennik Center for Astrophysics $|$
Harvard & Smithsonian, Cambridge, MA 02138, USA

###### Abstract

We have used solar oscillation frequencies and frequency splittings obtained
over solar cycles 23 and 24 to investigate whether the base of the solar
convection zone shows any departure from spherical symmetry. We used the even-
order splitting coefficients, $a_{2}$–$a_{8}$, and estimated the contributions
from each one separately. The average asphericity over the two solar cycles
was determined using frequencies and splittings obtained with a 9216-day time-
series. We find that evidence of asphericity is, at best, marginal: the
$a_{2}$ component is consistent with no asphericity, the $a_{4}$ and $a_{6}$
components yield results at a level a little greater than $1\,\sigma$, while
the $a_{8}$ component shows a signature below $1\,\sigma$. The combined
results indicate that the time average of the departure from the spherically
symmetric position of the base of the convection zone is $\lesssim
0.0001R_{\odot}$. We have also used helioseismic data obtained from time-
series of lengths 360 days, 576 days, 1152 days, and 2304 days in order to
examine the consistency of the results and evaluate whether there is any time
variation. We find that the evidence for time variation is statistically
marginal in all cases, except for the $a_{6}$ component, for which tests
consistently yield $p$ values of less than $0.05$.

The Sun (1693) — Helioseismology (709) — Solar interior (1500) — Solar
convective zone(1998) — Solar cycle (1487)

##  1 Introduction

The base of the solar convection zone (henceforth CZ) marks the position where
the thermal gradient changes from being radiative in the interior to adiabatic
in the outer layers. The position where this change happens is known very well
(e.g., Christensen-Dalsgaard et al., 1991; Basu & Antia, 1997; Basu, 1998). It
is determined assuming that the base is spherically symmetric. In fact, the
position of the CZ base is a test of the physics used in constructing solar
models (e.g., Christensen-Dalsgaard et al., 1993).

The base of the CZ is also where the solar tachocline is located. This is a
narrow shear layer that marks the change from latitudinal differential
rotation in the CZ to a solid body rotation in the interior. The tachocline
however, is prolate in shape (Antia et al., 1998; Charbonneau et al., 1999);
at the equator this layer is centered below the CZ base, while at a latitude
of 60∘, it is above the CZ base. This raises the question, whether the base of
the CZ is indeed spherically symmetric. Spiegel & Zahn (1992) noted that the
tachocline could drive meridional flows. It has been argued (Gough &
Kosovichev, 1995) that these flows must affect the thermal balance of the
layers, and could influence where the layer becomes unstable to convection,
thereby causing the base of the CZ to deviate from spherical symmetry.

Rotation is expected to make the CZ base oblate, though the expected
oblateness is small — the surface value of the difference between equatorial
and polar radius is only a few parts in $10^{6}$ (Kuhn et al., 2012), the
rotation rate at the CZ base is not much different. Another expected cause of
any asphericity of the CZ base, is the magnetic field that might be there.
Indeed, there may be belts of magnetic fields intense enough to modify
convection. Once the magnetic fields become buoyant and rises up, convection
could be enhanced by the field, thus the CZ base would exhibit time-dependent
asphericity (see Gough & Kosovichev, 1995).

Previous investigations on the asphericity of the CZ base give results that do
not agree with one another. Using frequencies obtained by the Big Bear Solar
Observatory (Woodard & Libbrecht, 1993), Gough & Kosovichev (1995) used
combinations of frequencies and splitting coefficients to determine the
frequencies sensitive at three different latitudes, and inverted the resulting
combinations to determine the relative isothermal difference between a
spherically symmetric model and that implied by the frequency combinations.
They interpreted their results as a difference in the CZ base of no more than
$0.02R_{\odot}$ between the pole and the equator. Monteiro et al. (2001)
attempted to look for asphericity at the CZ base using data from the Michelson
Doppler Imager (MDI; Scherrer et al., 1995). They fitted the signature of the
acoustic “glitch” caused by the change in the thermal gradient at the CZ base
to determine the acoustic depth of this feature. They found a latitude-
dependent difference in the acoustic depth, of the order of 100 seconds, and a
small, time dependence at low latitudes. Like Gough & Kosovichev (1995), they
used combined frequencies and splittings to obtain frequencies that correspond
to various latitudes, but their combinations were modified by an initial
inversion to get localized latitudinal response of the combination. In
contrast, Basu & Antia (2001) found a much smaller difference, and derived an
upper limit of $0.0005R_{\odot}$ on the asphericity of the CZ base; they used
data from the Global Oscillation Network Group (GONG: Hill et al., 1996) and
the MDI and, like Gough & Kosovichev (1995), combined frequencies and
splittings to obtain frequencies that represent various latitudes. Basu &
Antia (2001) did not invert the frequencies, instead they used the technique
developed by Basu & Antia (1997) to determine the position of the CZ base.
Antia et al. (2001a, 2003) did two-dimensional inversions to determine the
latitudinal dependence of solar structure, they found distinct asphericity in
the outer layers, but nothing significant at the CZ base.

Much more helioseismic data are now available, and in this paper, we use
helioseismic data collected over solar cycles 23 and 24 to assess whether the
CZ base shows any asphericity and furthermore, whether there is any time
variation of the asphericity. We use mode frequencies and splitting obtained
by different projects, and more importantly, we use frequencies and frequency
splittings obtained using different lengths of helioseismic time-series.
Unlike all earlier attempts, we do not combine the frequencies and splittings
to determine frequencies for a given latitude, but determine the contribution
of each term separately and derive the uncertainty due to each term
separately.

The rest of the paper is organized as follows. In Section 2 we describe how
one can determine the combination of frequencies and frequency splittings in
order to be sensitive to different latitudes, and in particular we describe
our approach of determining the asphericity of the CZ base. In Section 3 we
describe the method of determining the position of the CZ base. Section 4
describes the different data sets that we use. Our results are presented in
Section 5, and our conclusions in Section 6.

##  2 Determining Asphericity

We follow the formulation presented in Antia et al. (2001b) and Basu & Antia
(2001) to determine the asphericity at the base of the CZ. We use mode
frequencies and their splittings, expressed in the usual manner as:

$\nu_{nlm}=\nu_{nl}+\sum_{j=1}^{j_{\rm max}}a_{j}(n,l)\,{\cal
P}_{j}^{(l)}(m),$ (1)

where $\nu_{nl}$, or the central frequency, is determined by the spherically
symmetric part of solar structure, $a_{j}$ are splitting coefficients and
${\cal P}_{j}$ are re-scaled Clebsch-Gordon coefficients (Schou et al., 2002).
In this decomposition, the odd-order $a_{j}$ are caused by the solar rotation,
while the even-order coefficients contain the signature of asphericity and
magnetic field.

It should be noted that with global helioseismic data, one cannot distinguish
between magnetic fields and asphericity (see e.g., Zweibel & Gough, 1995). We
chose to interpret the signature in even-order splitting coefficients as being
caused by asphericity, this is what has been done in all previous estimates of
asphericity from global helioseismic data.

We consider only axisymmetric perturbations in structure, with the symmetry
axis coinciding with the rotation axis. Since we use global modes, our
inferences, like for the rotation rate, cannot differentiate between the
northern and southern hemispheres, but return estimates for a mean hemisphere.
This allows us to use the variational principle and write the difference in
frequency between the Sun and a solar model for a mode of a given order,
spherical harmonic degree and azimuthal order ($n$, $l$, and $m$) as:

${\delta\nu_{nlm}\over\nu_{nlm}}=\int_{0}^{R}\;{\rm d}r\;\int_{0}^{2\pi}\;{\rm
d}\phi\;\int_{0}^{\pi}\sin\vartheta\;{\rm d}\vartheta\\\ \left({\cal
K}_{c^{2},\rho}^{nl}(r){\delta c^{2}\over c^{2}}(r,\vartheta)+{\cal
K}_{\rho,c^{2}}^{nl}(r){\delta\rho\over\rho}(r,\vartheta)\right)Y_{l}^{m}(Y_{l}^{m})^{*},$
(2)

where $r$ is radius, $\vartheta$ is the co-latitude,
${\delta\nu_{nlm}/\nu_{nlm}}$ is the relative frequency difference, and ${\cal
K}_{c^{2},\rho}^{nl}(r)$ and ${\cal K}_{\rho,c^{2}}^{nl}(r)$ are the kernels
for spherically symmetric perturbations (see e.g., Antia & Basu, 1994;
Kosovichev, 1999). The functions $Y_{l}^{m}$ are spherical harmonics, denoting
the angular dependence of the eigenfunctions for a spherically symmetric star,
and are normalized such that

$\int_{0}^{2\pi}\;{\rm d}\phi\;\int_{0}^{\pi}\sin\vartheta\;{\rm
d}\vartheta\;Y_{l}^{m}(Y_{l}^{m})^{*}=1.$ (3)

The angular integrals in Eq. 2 can be evaluated to give

$\int_{0}^{2\pi}\;{\rm d}\phi\;\int_{0}^{\pi}\sin\vartheta\;{\rm
d}\vartheta\;Y_{l}^{m}(Y_{l}^{m})^{*}P_{2k}(\cos\vartheta)=\\\ {1\over
l}Q_{lk}{\cal P}_{2k}^{(l)}(m)$ (4)

where $Q_{lk}$ depends only on $l,k$ and ${\cal P}_{2k}^{(l)}(m)$ are the
orthogonal polynomials defined in Eq. 1. The extra factor of $1/l$ ensures
that $Q_{lk}$ approach a constant value at large $l$.

With this choice of expansion, the inversion problem can be decomposed into
independent inversions for each even splitting coefficient, $a_{2k}$, and
${\delta c^{2}/c^{2}}$ and ${\delta\rho/\rho}$ can be expressed in terms of
the even order Legendre polynomials, $P_{i}$:

${\delta c^{2}\over c^{2}}(r,\vartheta)=\sum_{k}({\delta c^{2}\over
c^{2}})_{k}(r)P_{2k}(\cos\vartheta),$ (5)

and,

${\delta\rho\over\rho}(r,\vartheta)=\sum_{k}{(\delta\rho\over\rho})_{k}(r)P_{2k}(\cos\vartheta).$
(6)

Thus, $({\delta c^{2}/c^{2}})_{k}(r)$ and $({\delta\rho/\rho})_{k}(r)$ can be
computed by inverting each of the splitting coefficient $a_{2k}$. Note that
this is similar to the “1.5D” inversions used to infer the rotation rate,
where a two-dimensional solution is obtained from a series of one dimensional
inversions:

${la_{2k}{(n,l)}\over\nu_{nl}Q_{lk}}=\\\ \int_{0}^{R}{\cal
K}_{c^{2},\rho}^{nl}c_{k}(r)\;{\rm d}r+\int_{0}^{R}{\cal
K}_{\rho,c^{2}}^{nl}\rho_{k}(r)\;{\rm d}r.$ (7)

Instead of determining the CZ base at different colatitudes from the
corresponding frequency difference, we estimate the difference in CZ over the
spherically symmetric value implied by the different orders of the even-order
coefficients separately; we do this for $a_{2}$–$a_{8}$; higher order
coefficient have large uncertainties and hence, we do not use those. In other
words, we take the combination

$\delta\nu=\delta\nu_{n\ell}+{\ell a_{2k}(n,l)\over Q_{\ell k}}$ (8)

for $k=1,$2,3, and 4 separately, determine the position of the CZ base in each
case, and then subtract the CZ base implied by $\delta\nu_{n\ell}$ to obtain
$\Delta R_{\rm CZ}(a_{2k})$ for each $a_{2k}$ under consideration. Adding
$\delta\nu_{n\ell}$ is necessary for our method (see Section 3), which relies
on determining the frequency differences between models with known CZ depth
and the Sun to work properly. Once we have estimates of $\Delta R_{\rm CZ}$
for each case, they are combined to obtain the CZ base at any given co-
latitude, i.e,

$R_{CZ}(\vartheta)=R_{\rm CZ}(\nu)+\sum_{k}\Delta R_{\rm
CZ}(a_{2k})P_{2k}(\cos\vartheta),$ (9)

where $R_{\rm CZ}(\nu)$ is the spherically symmetric part of the CZ base
obtained using the central frequencies, and $\Delta R_{\rm CZ}(a_{2}k)$ is the
contribution of a given even-order a-coefficient. We determine $\Delta R_{\rm
CZ}(a_{2}k)$ for $k=1$,2,3 and 4 separately using the data combinations in Eq.
8, and subtract the spherically symmetric part of the CZ base. We do not use
higher-order $a$ coefficients since the uncertainties in those are very large,
in fact, as we will see later in § 5, the uncertainties even in $a_{8}$ are
large enough to mask out any signs of asphericity. In the rest of the paper,
when we talk of “asphericity” we refer to, depending on the context, either
$\Delta R_{\rm CZ}(a_{2k})$ or $R_{CZ}(\vartheta)$.

Figure 1:  $P_{2k}(\cos\vartheta)$ plotted as a function of latitude $\theta$,
to show how different even-order $a$-coefficients contribute to the
asphericity of the CZ base. Thus, for example, if $\Delta R_{\rm CZ}(a_{2})$
is positive, the CZ base at the equator will be deeper than that at higher
latitudes, making it prolate.

In Fig. 1, we illustrate $P_{2k}$ as a function of latitude. The figure shows
that a positive $\Delta R_{\rm CZ}(a_{2})$ implies that the CZ base at the
equator is deeper than that at the higher latitudes, namely that the CZ base
is prolate. Of course, higher-order contributions make the shape more complex.

##  3 Estimating the position of the CZ base

Figure 2: The function $H_{1}(w)$ between a (a) solar envelope model and (b)
solar frequencies and solar envelope models with different CZ positions. The
test solar model in panel (a) has the same physics as the calibration models.
The observational data used in panel (b) is an HMI set obtained with a
2304-day time-series with a start date of 2001.08.11. Note the need for the
term $H_{s}(w)$ in Eq. 11 in order to fit the curves in panel (b) with those
in panel (a).

We use the method developed and used by Basu & Antia (1997); Basu (1998) to
determine the position of the CZ base, that can be summarized as follows. If
there are two otherwise similar solar models with different depths of the
convection zone, then just below the base of the convection zone the model
with a deeper convection zone will have a larger sound speed than the other.
This observable difference of sound speeds can be calibrated to find the
convection zone depth of a test model or the Sun, as was first done by
Christensen-Dalsgaard et al. (1991).

Asymptotically, the frequency differences between a solar model and the Sun,
or between two solar models can be written as (Christensen-Dalsgaard et al.,
1989)

$S(w){\delta\omega\over\omega}=H_{1}(w)+H_{2}(\omega),$ (10)

where $w=\omega/(l+0.5)$, $\omega=2\pi\nu$, and $S(w)$ is a known function for
a given reference model. The functions $H_{1}(w)$ and $H_{2}(\omega)$ can be
determined by a least-squares fit to the known frequency differences. The
function $H_{2}$ contains the contribution from the “surface term,” i.e., the
contribution from our inability to model the near-surface layers of the Sun
properly. $H_{1}(w)$ can be inverted to obtain the sound-speed difference,
$\delta c/c$, between the reference model and the Sun. However, Basu & Antia
(1997) showed that that is not required, since $H_{1}(w)$ itself can be used
to determine the convection-zone depth.

If $\phi(w)$ is the $H_{1}(w)$ between two solar models which differ only in
the depth of the convection zone, then $H_{1}(w)$ for any other pair of models
can be written as

$H_{1}(w)=\beta\phi(w)+H_{s}(w),$ (11)

where $H_{s}(w)$ is a smooth component of $H_{1}(w)$ which results from sound-
speed differences that arise from differences in the equation of state,
abundances, surface physics etc., and the first term is the contribution to
$H_{1}(w)$ due to the sound-speed difference caused by the difference in
$r_{b}$, the position of the base of the CZ. Thus, if $\beta$ is estimated
using a least-squares fit, the unknown $r_{b}$ of the Sun can be obtained. We
determine $\beta$ for a series of calibration models with different $r_{b}$,
and interpolate to find the CZ position for which $\beta=0$. This “null”
method allows us to determine the position of the base of the CZ to a
precision that is much better than what can be obtained with a sound-speed
inversion, and is not hampered by the limited spatial resolution of inversions
at the CZ base.

The uncertainty in our results arising from data uncertainties is evaluated by
Monte Carlo simulations. For each observed data set, we create 4000 sets of
artificial data by adding random realizations of the observed uncertainty to
the central frequency and splitting coefficients. We determine $R_{\rm eff}$
and $\Delta R_{\rm eff}$ for each of the simulated sets. The final parameters
that we adopt is the mean of the distribution of results and the standard
deviation is the uncertainty. We have found that the distribution is Gaussian
and hence using the mean and standard deviation is adequate.

As is clear from the above discussion, we need to use models with a specified
CZ depth. This can be done if we construct models of the solar envelope, for
which the position of the CZ base and the helium abundance of the solar
envelope can be specified.

Basu & Antia (1997) had shown that the largest source of systematic error in
the estimated CZ-base position is the hydrogen abundance profile. This is
because the sound speed near the base of the CZ is affected not only by the
change in the temperature gradient but also by the change in the mean
molecular weight due to gravitational settling of helium. This error does not
affect this work since we are looking at differences with respect to the
spherically symmetric component of the CZ base, and thus this ought to cancel
out. Nevertheless, to minimize the error, we use models that have $X$ and $Z$
profiles for the Sun determined by Antia & Chitre (1998). However, since the
position of the CZ base is well-known, we use calibration models with a
smaller range in CZ positions — 0.7125R⊙, 0.7130R⊙, 0.7135R⊙, 0.7140R⊙, and
0.7145R⊙. We show in Fig. 2 how $H_{1}(w)$ changes when solar models with
different CZ positions are used.

##  4 Data used

Figure 3: A visual representation of the time periods spanned by each SGK data
set. Sets have been offset vertically for clarity. The sets, from top to
bottom, are for $32\times 72$, $16\times 72$, $8\times 72$ and $5\times 72$
days respectively. Red denotes GONG, blue MDI and green HMI observations.

For this work, we use solar oscillation frequencies obtained by the ground-
based Global Oscillation Network Group (GONG; Hill et al., 1996) and the
space-based Michelson Doppler Imager (MDI) on board the Solar and Heliospheric
Observatory spacecraft (Scherrer et al., 1995) and the Helioseismic and
Magnetic Imager (HMI; Scherrer et al., 2012) on board the Solar Dynamics
Observatory. We use mode frequencies obtained from different lengths of the
time-series that cover solar cycles 23 and 24. In particular, we use data
analyzed by an alternative pipeline (Korzennik, 2004, 2008a, 2008b; Korzennik
& Eff-Darwich, 2013; Korzennik, 2017, 2018, 2023), independent of the
project’s pipeline, and henceforth referred to as the “SGK” pipeline. The mode
parameters are derived from time-series that are multiples of 72 days, we use
mode frequencies and splittings obtained for time-series lengths of
$32\times,16\times,8\times$ and $5\times 72$ days, or $\simeq 6,3,1.5$ or $1$
year, respectively. The choice of analyzing time series that are multiples of
72 days was dictated by the standard time-series length adopted by the MDI
team, and was later adopted by the HMI team. A time-series of 72 days was
considered to be long enough to produce a frequency spectrum with high signal-
to-noise ratio but short enough to maintain “good” temporal variation, since
the mode frequencies were known to vary with time on all time-scales. The MDI
and HMI teams later started analyzing 360-day (i.e., $5\times 72$-day) long
time series too.

Fig. 3 shows the temporal ranges the different data sets span. For time-series
longer than 72 days, the convention adopted by SGK is to fit time-series
separated by half the length of the time-series to increase the temporal
resolution. The exception is for the 360-day long time-series, these are
fitted every 360 day, to match what is done by the MDI project. Additionally,
to determine the average asphericity over the two solar cycles, we used a GONG
data set resulting from fitting a $128\times 72$-day time-series.

Figure 4: The figure shows the effect of the leakage matrix on $\Delta R_{CZ}$
implied by even-order splitting coefficients $a_{2}$–$a_{8}$ (Panels (a), (b),
(c) and (d) respectively) obtained for mode frequencies and splittings
obtained with $32\times 72$-day time-series data from GONG and HMI with a
start date of 2010:07:11 using different leakage matrices. The solid gray
horizontal line in each panel is the weighted mean of all the results. The
dotted gray line marks where results would lie if the CZ base were spherically
symmetric. Set 1 is the HMI pipeline set, while set 2 is the SGK pipeline
using GONG data and the JSBo=0 leakage matrix, set 3 is the SGK pipeline using
GONG and the SKBo=0 leakage matrix, set 4 is the SGK pipeline using the HMI
data and the JSBo=0 leakage matrix, set 5 is the SGK pipeline using HMI data
and the SKBo=0 leakage matrix, and set 6 is the SGK pipeline using the HMI
data and the SKBo=a2 leakage matrix.

Figure 5: Left: Deviations from sphericity of the base of the CZ implied by
different splitting coefficients $a_{2k}$ for $2k=1$–4. Right: The latitudinal
variation of the asphericity implied by the results of the individual
coefficients is shown as the red continuous line with the dotted line showing
$1\sigma$ uncertainties. The black dotted lines show the $1\sigma$ uncertainty
limit for the spherically symmetric component of the CZ base.

In some case, we also use mode parameters from the MDI and HMI projects’
pipeline. Although the primary frequency sets derived by these projects are
obtained by fitting 72-day long time-series, which is not adequate for our
work. However, there are some sets of frequencies and splittings obtained
fitting $5\times 72$-day long time-series, and also one HMI set obtained from
fitting a $32\times 72$-day long time-series.

It should be noted that the SGK pipeline predominantly obtains mode parameters
by fitting asymmetric profiles to peaks in the power spectra, though some sets
fitted using symmetric profiles are available. The MDI and HMI projects’
pipeline predominantly fits symmetric profiles, though some sets fitted using
asymmetric profiles are available.

We restrict ourselves to modes with frequencies between 1.6 mHz and 3.5 mHz
that have lower turning points between 0.45R⊙ and 0.95R⊙. These modes give
good coverage of the CZ base, while keeping mode uncertainties low.

The SGK pipeline produces modes and splittings using a number of different
leakage matrices. In one case, it uses the leakage matrix computed by the
MDI/HMI project (JSBo=0), in the other cases it uses a leakage matrix computed
by Korzennik (SKBo=0 & SKBo=a2). Such leakage matrix coefficients are computed
as follows: a series of images of spherical harmonics on the unit sphere are
computed, using the observational image projection, for either a heliographic
latitude at disk center, $B_{o}$ of 0, or $5.06^{\circ}$, the quadratic mean
of $B_{o}$ over a year. Each image is then spatially decomposed using the same
decomposition procedure that is used for the respective observations. However,
as can be seen in Fig. 4, the leakage matrix does not change the results to
any level of statistical significance. The results in Fig. 4 are for $32\times
72$-day data sets with a start date of 2010:07:11. Unfortunately, MDI data
does not exist for that time period. However, tests with MDI data for other
spans of time give very similar results. Consequently, for all subsequent
results reported in this paper, we only use modes obtained with the JSBo=0
leakage matrix.

Figure 6: The deviation of the CZ base from spherical, $\Delta R_{\rm CZ}$,
obtained with different lengths of time-series plotted as a function of time.
Panels (A)–(D) are results for $a_{2}$-$a_{8}$ respectively. Sub-panel (a) in
all panels shows results with GONG data, while sub-panel (b) are for MDI and
HMI data. Only results obtained with the SGK pipeline fitting asymmetric
profiles are shown.  Figure 7: A comparison of our estimates of $\Delta R_{\rm
CZ}$ obtained with long time-series data and those with averaged mode
parameters of shorter data. Only results obtained with GONG data are shown,
since the data cover both solar cycle 23 and 24. The column on the left
compares results from $16\times 72$-day time-series with those obtained with
average mode parameters of 2 $(8\times 72)$-day sets and 3 $(5\times 72)$-day
sets. The column on the right compares $32\times 72$-day results with those
obtained from the average mode parameters of 2 $(16\times 72)$-day time-
series. The rows from top to bottom show results for $a_{2}$-$a_{8}$
respectively.

##  5 Results

Our results derived from the $128\times 72$-day set are shown in Fig. 5. The
values are tabulated in Table 1 in the Appendix. We find that the departure
from sphericity, averaged over solar cycles 23 and 24 is extremely small, with
only marginal statistical significance. When the contributions of the
different components are combined to give the latitudinal difference, it is
clear that we can at most give an upper limit of $0.0001R_{\odot}$.

Earlier investigations, mentioned in Section 1 obtained much larger
deviations. This could have been because of the quality of the then available
data, or possibly because there is a time dependence to the asphericity, which
is averaged out by the long, $128\times 72$-day, time-series. To test this, we
used data sets obtained from shorter time-series. The resulting estimates of
$\Delta R_{\rm CZ}$ are shown in Fig. 6, and are tabulated in the Appendix.

A few features can be seen immediately. The scatter in the results obtained
with the shorter time-series data is larger than those with the longer time-
series data. This is, of course, expected given that the uncertainty in the
mode frequencies and splittings is lower for the longer sets. The results
obtained with GONG data agree with those of MDI and HMI data. The agreement
is, however, better between GONG and HMI than between GONG and MDI. This type
of disagreement has been seen in other helioseismic investigations, even when
using the respective projects’ pipeline data (e.g., Antia & Basu, 2022).

We can also see that the scatter in the results do not appear to be strictly
random. The right-hand panels of Fig. 7 give the impression of a possible
anti-correlation between solar activity and CZ asphericity. Tests for such
anti-correlation, where we used the 10.7cm radio flux averaged over the same
time interval as an index of solar activity, do not reveal any statistically
significant correlation. The largest anti-correlation is obtained for the
$a_{4}$ results obtained with $32\times 72$-day time series, and even that is
significant only at the 2-$\sigma$ level. A strict solar cycle dependence is
not really expected — the time variation of the tachocline is, after all, much
more complicated with a clear sign of hysteresis (Basu & Antia, 2019), hence
there may still be a time variation just like in the tachocline. This seeming
time variation is clearer for the results derived from the longer time-series,
however, the variation becomes smaller as the length of the time-series is
increased. This is to be expected if the time variation is real, since the
longer sets would average out the time variations, but this can be merely a
result of the observed uncertainties rather than a real temporal variation —
as the uncertainties get reduced, the amplitude of the inferred variations
will decrease.

We first verify whether the results obtained by the shorter time-series are
consistent with those obtained with longer time-series. For this, we compare
the results obtained with $16\times 72$-day data with those obtained with the
average mode parameters of shorter sets — we averaged mode frequencies and
splittings of 2 $8\times 72$-day sets and 3 $5\times 72$-day sets for each
$16\times 72$-day set, we also averaged mode frequencies and splittings of 2
$16\times 72$-day sets for each $32\times 72$-day set. Note that 3 $5\times
72$-day sets do not quite cover a $16\times 72$-day set, but they are close
enough. Also note that obtaining mode parameters from a long time-series is
not the same as averaging mode parameters from shorter time-series because in
the former case many more modes can rise above the noise level, while in the
latter the set of successfully fitted modes is not always the same. The
results are shown in Fig. 7. We also show results obtained by averaging
frequencies and splittings of 2 $16\times 72$-day sets and compare those with
the results of the $32\times 72$-day sets. As can be seen, averaging the mode
parameters of the shorter sets yields results that are commensurate with those
obtained with the longer sets.

Figs. 6 and 7 also shows that while $a_{2}$, $a_{4}$ and, $a_{6}$ results show
deviations from asphericity $>1\,\sigma$ at some epochs, while the
uncertainties in the $a_{8}$ results are too large to make any firm deduction
about the contribution of $a_{8}$ to the asphericity of the CZ base.

###  5.1 Tests for time variations

The different panels in Figs. 6 suggest the possibility of a time variation;
such putative variation is clearer in Fig. 7.

Given that the deviations from zero are small, we have performed additional
tests to ascertain whether there really is a non-random variation.

We performed a Wald–Wolfowitz runs test (Wald & Wolfowitz, 1940). While we
tested the results from the GONG data sets separately, we combine the MDI and
HMI sets. However, since our results have uncertainties, the results of one
runs test is not adequate. Thus, for each data set (i.e., each length of time-
series), we generated multiple sets by adding random realizations of the
uncertainties to the results at each epoch. We also create a truly random set
of results at each epoch; these are just random realizations of the
uncertainties in the results. For each sequence of the realization of results
and as well as the random set, we perform the runs test to derive a
distribution of runs.

We compare the distribution of the runs test from our results and the random
sequence. Note that since the uncertainties in the results at each epoch are
very similar, the distribution of runs in our random case can be described by
a Gaussian. We conduct the usual two-sample Kolmogorov-Smirnov test, as well
as a Student’s t test and an F test, to examine the null hypothesis that the
distribution of runs in our results is drawn from the distribution of runs for
the sample whose time variation is random. Because we are essentially dealing
with small number statistics (in terms of the number of epochs covered, in
other words the number of runs we can get), we relied on multiple measures of
differences between the distributions of runs in the results and the
distribution of runs in the random sample.

The statistical tests yield marginal evidence for a non-random time variation
in most cases. The tests generally yield $p$ values greater than 0.05 for
$a_{2}$ and $a_{4}$, however, for $a_{6}$, we consistently get $p$ values less
than $0.05$, and in some cases even more significant evidence of non-
randomness. The combination of MDI and HMI results show the same pattern,
though the significance is generally lower.

###  5.2 Examining possible sources of error

As shown above, the time-averaged signature of asphericity is very small,
evidence of time-variation is also marginal. To ensure that our results are
robust, we take a close look at possible sources of error that could cause a
spurious time dependence in our results.

####  5.2.1 The second-order effects of rotation

The even-order $a$ coefficients have a small contribution from the second-
order effects of rotation (Gough & Thompson, 1990). It is known that solar
rotation changes with time (see e.g., Antia & Basu, 2013; Howe et al., 2013;
Komm et al., 2018; Howe et al., 2018; Kosovichev & Pipin, 2019; Antia & Basu,
2022, etc.), even near the base of the solar convection zone (Basu & Antia,
2019), the region of interest for this work. We did not correct the data for
the second-order effect of rotation. In order to examine the effect of the
second-order effects, we remove those from the GONG $16\times 72$-day data-
sets; we do so using the formalism of Antia et al. (2000) and determine the
asphericity again.

Figure 8: The difference in the original estimates of asphericity and those
estimated using data sets corrected for the second order effects of rotation,
normalized by the uncertainties in the original result. These results were
obtained with GONG $16\times 72$-day data sets.

The differences between the uncorrected and corrected results, normalized by
the uncertainty in the corrected result, are shown in Fig. 8. As can be seen,
while the differences are systematic, they are a small fraction of the
uncertainty. The difference is largest for $\Delta R_{\rm CZ}(a_{2})$, but
even there, the differences are less than $0.5\,\sigma$. Furthermore, although
rotation is time-dependent, that does not affect the asphericity results.

####  5.2.2 The surface term

The surface term is the frequency-dependent frequency offset between the Sun
and a solar model. This offset increases with frequency. It is well known that
the solar surface term changes with solar cycle (Elsworth et al., 1990;
Libbrecht & Woodard, 1990; Howe et al., 2017, etc.), being largest at solar
maximum and smallest at the minimum. The surface term has a steep frequency
difference, and hence, to examine whether we remove it completely through Eq.
10, we re-determine $\Delta R_{\rm CZ}$ after reducing the upper frequency
limit from 3.5 to 3 mHz. The results are shown in Fig. 9. As can be seen,
differences between results with different upper frequency cut-off are
generally well within $1\,\sigma$. These differences do not show a solar cycle
dependence, something we would expect to see had we not removed the surface
term.

Figure 9: The change in $\Delta R_{\rm cz}$ estimates, when the splittings
were restricted to modes with frequencies $\leq 3$ mHz, normalized by their
uncertainty, obtained when using $16\times 72$-day GONG data sets. The
horizontal lines mark $0$ and $\pm 1\,\sigma$ differences. Note that most
results are well within $1\,\sigma$ and that there is no solar-cycle
dependence.

###  5.3 Asymmetry of the mode profiles

As mentioned earlier, the data sets that we use determined frequencies and
splittings by fitting an asymmetric profile to the peaks in the oscillation
power spectra. In order to discount asymmetry as the source of time variation,
we compared the $5\times 72$-day results obtained when fitting using either
symmetric or asymmetric mode profiles. The differences, normalized by the
uncertainty, are shown in Fig. 10. As can be seen, the differences are
generally within $1\sigma$ and the time variations do not look anything like
the ones seen in Figs. 6 and 7.

Figure 10: The difference in $\Delta R_{\rm cz}$ estimates, normalized by
uncertainty, obtained with mode parameters asymmetric and symmetric fits to
the data from the SGK pipeline for the $5\times 72$-day time-series. The
horizontal lines mark $0$ and $\pm 1\,\sigma$ differences.

The MDI and HMI project pipelines also produce data sets from $5\times 72$-day
time-series mainly by fitting symmetric profiles, but some sets have also been
fit using asymmetric profile. This allows us to compare results obtained with
the project pipeline data with the SGK pipeline data, as shown in Fig. 11.

Figure 11: Asphericity parameter $\Delta R_{\rm CZ}$ obtained with $5\times
72$-day MDI and HMI project pipeline data as well as with SGK pipeline data
fitted using both symmetric and asymmetric peak profiles. The rows from top to
bottom are for $a_{2}$–$a_{8}$ respectively. While Fig. 10 shows the
normalized difference between two sets, this figure shows the actual values
for four sets of data.

##  6 Discussion and Conclusions

We have used global helioseismic data, frequencies and $a_{2}$–$a_{8}$ even-
order splitting coefficients, available from various sources and derived using
different methodologies as well as time-series of various lengths, all
collected over solar cycles 23 and 24, to determine whether the base of the
solar convection zone is aspherical.

Using a data set obtained from a $128\times 72$-day time-series, we find that
the time-averaged asphericity of the CZ base is small, with an upper limit of
$0.0001R_{\odot}$ on the deviation from sphericity.

Our new estimate is smaller than those derived by earlier work. There are a
number of reasons this could be the case. For one, all previous estimates used
data sets from much shorter time-series, which means larger uncertainties that
results in a larger uncertainty on the amplitude. Alternatively, if there is
indeed a small variation, using a shorter time-series could make it more
visible. And yet another possible reason for the discrepancy is the method by
which asphericity is determined.

Our results are much smaller than the estimates of Gough & Kosovichev (1995).
It is possible that the Gough & Kosovichev (1995) inversion results (for the
isothermal sound speed $u$ with the helium abundance $Y$ as the second
parameter) have errors because calculating ($u,Y$) kernels requires one to
assume that the equation of state of solar material is known and is the same
as that of the reference model; Basu & Christensen-Dalsgaard (1997) have shown
that this leads to systematic errors in the inversion results both above and
below the CZ base and also results in unrealistically small uncertainties.

It is more difficult to compare our results with those of Monteiro et al.
(2001) since they do not translate their acoustic depth differences into
differences in radius. An acoustic depth difference of 100 seconds at the CZ
base translates to a radius difference of approximately 0.025 R⊙, again much
larger than what we find. One of the possible reasons for this is that
Monteiro et al. (2001) only fit the acoustic glitch at the CZ base and neglect
the contribution from the helium ionization zone, and additionally do not take
the surface-term into account. The glitch signature from the helium ionization
zone is known to vary with solar cycle (Basu & Mandel, 2004; Watson & Basu,
2020), and the surface term is not only time dependent, the time-dependence
depends on latitude (Antia et al., 2001b; Basu et al., 2007). Not taking the
latitudinal dependence of the surface term into account could lead to a large
asphericity, while the time-varying signature from the helium ionization zone
could add to the time-dependence of their results. Our results are smaller
than, but in line with, those of Basu & Antia (2001). Basu & Antia (2001) used
shorter time-series data and averaged the results over slightly more than four
years, but that is still a much shorter time interval than 25-years, which is
what we use for our main results.

We have tested our results to examine whether there is a non-random time
dependence in the results. Only the $a_{6}$ component gives a $p$-value of
less than $0.05$. Our efforts to determine whether there is a real time-
dependence are hampered by the small number of data sets we have. Results
obtained with data from short time-series (like the usual 72-day sets from the
project pipelines) have unfortunately very large uncertainties, but increasing
the length of the time-series decreases the number of sets covering the two
solar cycles, and additionally average out any underlying time dependence.

Determining the cause of the asphericity, and any possible changes thereof, is
beyond the scope of the paper. However, assuming that the asphericity is an
indirect result of magnetic fields and the resulting modification of the
thermal structure, a shift in the CZ base of $0.0001R_{\odot}$ implies a
magnetic field of $\sim 300$ kG. This is consistent with the limits on the
magnetic fields at the base of the CZ of a few hundred kilo-Gauss (e.g.,
D’Silva & Choudhuri, 1993; Basu, 1997; Antia et al., 2000, 2003).

## Acknowledgements

We thank H. M. Antia for his comments on an earlier draft of the paper that
led us to conduct statistical tests to ascertain the significance of the
temporal variations and for his help in calculating the second-order effects
of rotation. This work is supported by NASA grant 80NSSC23K0563 to SB and NASA
grants 80NSSC22K0516 and NNH18ZDA001N-DRIVE to SGK. This work utilizes data
from the National Solar Observatory Integrated Synoptic Program, which is
operated by the Association of Universities for Research in Astronomy, under a
cooperative agreement with the National Science Foundation and with additional
financial support from the National Oceanic and Atmospheric Administration,
the National Aeronautics and Space Administration, and the United States Air
Force. The GONG network of instruments is hosted by the Big Bear Solar
Observatory, High Altitude Observatory, Learmonth Solar Observatory, Udaipur
Solar Observatory, Instituto de Astrofísica de Canarias, and Cerro Tololo
Interamerican Observatory. This work also uses provided by the SOHO/MDI
consortium. SOHO is a project of international cooperation between ESA and
NASA. We also use data from the Helioseismic and Magnetic Imager on board the
Solar Dynamics Observatory. HMI data are courtesy of NASA/SDO and the AIA,
EVE, and HMI science teams.

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We tabulate our results for the SGK asymmetric sets in Tables 1, 2, 3, 4 and
5. Note that the start date of each set is in YYYYMMDD format.

Table 1: Results for the $128\times 72$-day set Start Date |  $\Delta R_{\rm CZ}$/R⊙
---|---
of set | $a_{2}$ | $a_{4}$ | $a_{6}$ | $a_{8}$
19960501 |  $-$9.4590E$-$07 $\pm$ 3.032E$-$05 |  $-$6.913E$-$05 $\pm$ 5.007E$-$05 |  $-$1.159E$-$04 $\pm$ 6.921E$-$05 |  $-$6.173E$-$05 $\pm$ 8.067E$-$05
Table 2: Results for the $32\times 72$-day sets Start Date |  $\Delta R_{\rm CZ}$/R⊙
---|---
of set | $a_{2}$ | $a_{4}$ | $a_{6}$ | $a_{8}$
| GONG
19960501 |  $-$4.738E$-$05 $\pm$ 5.718E$-$05 |  $-$1.002E$-$04 $\pm$ 9.185E$-$05 |  $-$2.444E$-$04 $\pm$ 1.333E$-$04 |  $-$1.163E$-$04 $\pm$ 3.116E$-$04
19990627 |  $-$3.263E$-$05 $\pm$ 5.430E$-$05 |  $-$2.082E$-$04 $\pm$ 8.657E$-$05 |  $-$4.283E$-$04 $\pm$ 1.279E$-$04 |  $-$1.265E$-$04 $\pm$ 2.975E$-$04
20020822 |  $-$8.575E$-$06 $\pm$ 5.061E$-$05 |  $-$1.016E$-$04 $\pm$ 8.703E$-$05 |  $-$3.006E$-$04 $\pm$ 1.172E$-$04 |  $-$8.281E$-$06 $\pm$ 2.824E$-$04
20051017 |  $-$5.070E$-$05 $\pm$ 5.268E$-$05 |  $-$3.764E$-$05 $\pm$ 9.057E$-$05 |  $-$9.180E$-$05 $\pm$ 1.219E$-$04 |  $-$1.601E$-$04 $\pm$ 2.911E$-$04
20081212 |  $-$5.876E$-$05 $\pm$ 5.171E$-$05 |  $-$1.637E$-$04 $\pm$ 8.471E$-$05 |  $-$2.272E$-$04 $\pm$ 1.165E$-$04 |  $-$1.718E$-$04 $\pm$ 2.881E$-$04
20100711 |  $-$4.541E$-$05 $\pm$ 5.081E$-$05 |  $-$7.202E$-$05 $\pm$ 8.235E$-$05 |  $-$1.693E$-$04 $\pm$ 1.152E$-$04 |  $-$9.342E$-$05 $\pm$ 2.830E$-$04
20120207 |  $-$9.347E$-$05 $\pm$ 5.085E$-$05 |  $-$1.363E$-$04 $\pm$ 8.488E$-$05 |  $-$2.298E$-$04 $\pm$ 1.161E$-$04 |  $-$1.971E$-$04 $\pm$ 2.869E$-$04
20150404 |  $-$7.865E$-$05 $\pm$ 5.255E$-$05 |  $-$3.237E$-$06 $\pm$ 8.627E$-$05 |  $-$4.353E$-$05 $\pm$ 1.204E$-$04 |  $-$6.526E$-$05 $\pm$ 2.908E$-$04
| MDI
19960501 |  $-$7.867E$-$05 $\pm$ 6.447E$-$05 |  $-$1.385E$-$04 $\pm$ 1.051E$-$04 |  $-$1.907E$-$04 $\pm$ 1.518E$-$04 |  $-$2.438E$-$04 $\pm$ 4.312E$-$04
19990627 |  $-$2.981E$-$05 $\pm$ 6.308E$-$05 |  $-$1.783E$-$04 $\pm$ 1.025E$-$04 |  $-$2.384E$-$04 $\pm$ 1.443E$-$04 |  $-$7.186E$-$05 $\pm$ 4.077E$-$04
20020822 |  $-$6.684E$-$05 $\pm$ 6.030E$-$05 |  $-$9.508E$-$05 $\pm$ 9.902E$-$05 |  $-$2.753E$-$04 $\pm$ 1.419E$-$04 |  $-$1.137E$-$04 $\pm$ 3.954E$-$04
| HMI
20100711 |  $-$4.329E$-$05 $\pm$ 5.089E$-$05 |  $-$7.144E$-$05 $\pm$ 8.622E$-$05 |  $-$1.255E$-$04 $\pm$ 1.221E$-$04 |  $-$4.889E$-$05 $\pm$ 3.299E$-$04
20120207 |  $-$5.962E$-$05 $\pm$ 5.164E$-$05 |  $-$9.135E$-$05 $\pm$ 8.593E$-$05 |  $-$2.892E$-$04 $\pm$ 1.213E$-$04 |  $-$6.755E$-$05 $\pm$ 3.368E$-$04
20150404 |  $-$4.604E$-$05 $\pm$ 5.262E$-$05 |  $-$3.790E$-$05 $\pm$ 8.924E$-$05 |  $-$5.797E$-$05 $\pm$ 1.240E$-$04 |  $-$1.780E$-$04 $\pm$ 3.445E$-$04
Table 3: Results for the $16\times 72$-day sets Start Date |  $\Delta R_{\rm CZ}$/R⊙
---|---
of set | $a_{2}$ | $a_{4}$ | $a_{6}$ | $a_{8}$
| GONG
19960501 |  $-$9.578E$-$06 $\pm$ 7.559E$-$05 |  $-$2.472E$-$04 $\pm$ 1.247E$-$04 |  $-$4.292E$-$04 $\pm$ 1.695E$-$04 |  $-$5.980E$-$04 $\pm$ 3.937E$-$04
19971128 |  $-$3.080E$-$05 $\pm$ 7.665E$-$05 |  $-$4.067E$-$05 $\pm$ 1.254E$-$04 |  $-$2.863E$-$04 $\pm$ 1.721E$-$04 |  $-$2.182E$-$04 $\pm$ 4.119E$-$04
19990627 |  $-$3.039E$-$05 $\pm$ 7.607E$-$05 |  $-$1.888E$-$05 $\pm$ 1.256E$-$04 |  $-$1.395E$-$04 $\pm$ 1.717E$-$04 |  $-$4.238E$-$04 $\pm$ 4.263E$-$04
20010123 |  $-$8.732E$-$06 $\pm$ 7.203E$-$05 |  $-$1.294E$-$04 $\pm$ 1.196E$-$04 |  $-$1.026E$-$04 $\pm$ 1.630E$-$04 |  $-$1.114E$-$04 $\pm$ 4.160E$-$04
20020822 |  $-$4.753E$-$06 $\pm$ 6.905E$-$05 |  $-$4.011E$-$04 $\pm$ 1.150E$-$04 |  $-$5.198E$-$04 $\pm$ 1.572E$-$04 |  $-$3.296E$-$04 $\pm$ 3.920E$-$04
20040320 |  $-$1.501E$-$04 $\pm$ 7.010E$-$05 |  $-$2.298E$-$04 $\pm$ 1.168E$-$04 |  $-$6.399E$-$04 $\pm$ 1.588E$-$04 |  $-$7.852E$-$05 $\pm$ 3.892E$-$04
20051017 |  $-$1.015E$-$04 $\pm$ 7.376E$-$05 |  $-$1.206E$-$04 $\pm$ 1.230E$-$04 |  $-$1.313E$-$04 $\pm$ 1.660E$-$04 |  $-$1.512E$-$04 $\pm$ 4.082E$-$04
20070516 |  $-$1.252E$-$04 $\pm$ 7.478E$-$05 |  $-$3.250E$-$04 $\pm$ 1.248E$-$04 |  $-$2.841E$-$04 $\pm$ 1.684E$-$04 |  $-$2.288E$-$04 $\pm$ 4.118E$-$04
20081212 |  $-$2.202E$-$04 $\pm$ 7.036E$-$05 |  $-$6.091E$-$05 $\pm$ 1.172E$-$04 |  $-$1.707E$-$04 $\pm$ 1.603E$-$04 |  $-$5.158E$-$04 $\pm$ 3.947E$-$04
20100711 |  $-$2.534E$-$04 $\pm$ 6.866E$-$05 |  $-$1.315E$-$04 $\pm$ 1.133E$-$04 |  $-$3.695E$-$04 $\pm$ 1.567E$-$04 |  $-$7.099E$-$04 $\pm$ 3.859E$-$04
20120207 |  $-$1.009E$-$04 $\pm$ 6.826E$-$05 |  $-$1.183E$-$04 $\pm$ 1.133E$-$04 |  $-$1.037E$-$04 $\pm$ 1.557E$-$04 |  $-$2.133E$-$04 $\pm$ 3.893E$-$04
20130905 |  $-$2.129E$-$04 $\pm$ 6.937E$-$05 |  $-$2.424E$-$04 $\pm$ 1.162E$-$04 |  $-$1.825E$-$04 $\pm$ 1.587E$-$04 |  $-$4.772E$-$04 $\pm$ 3.979E$-$04
20150404 |  $-$4.178E$-$05 $\pm$ 6.884E$-$05 |  $-$1.570E$-$04 $\pm$ 1.153E$-$04 |  $-$1.379E$-$04 $\pm$ 1.571E$-$04 |  $-$2.948E$-$04 $\pm$ 3.890E$-$04
20161031 |  $-$5.547E$-$05 $\pm$ 7.416E$-$05 |  $-$2.548E$-$04 $\pm$ 1.240E$-$04 |  $-$6.474E$-$05 $\pm$ 1.689E$-$04 |  $-$3.919E$-$04 $\pm$ 4.142E$-$04
20180530 |  $-$1.997E$-$04 $\pm$ 7.548E$-$05 |  $-$1.588E$-$04 $\pm$ 1.278E$-$04 |  $-$5.430E$-$04 $\pm$ 1.733E$-$04 |  $-$2.581E$-$04 $\pm$ 4.238E$-$04
| MDI
19960501 |  $-$1.333E$-$04 $\pm$ 8.700E$-$05 |  $-$3.274E$-$04 $\pm$ 1.439E$-$04 |  $-$3.134E$-$04 $\pm$ 2.064E$-$04 |  $-$1.311E$-$05 $\pm$ 5.777E$-$04
19971128 |  $-$8.289E$-$05 $\pm$ 8.848E$-$05 |  $-$5.004E$-$05 $\pm$ 1.460E$-$04 |  $-$5.322E$-$04 $\pm$ 2.104E$-$04 |  $-$1.935E$-$04 $\pm$ 5.817E$-$04
19990627 |  $-$2.209E$-$05 $\pm$ 8.436E$-$05 |  $-$5.543E$-$05 $\pm$ 1.396E$-$04 |  $-$9.628E$-$05 $\pm$ 1.983E$-$04 |  $-$4.837E$-$04 $\pm$ 5.509E$-$04
20010123 |  $-$3.061E$-$05 $\pm$ 8.254E$-$05 |  $-$2.247E$-$04 $\pm$ 1.365E$-$04 |  $-$1.705E$-$04 $\pm$ 1.933E$-$04 |  $-$1.451E$-$04 $\pm$ 5.396E$-$04
20020822 |  $-$9.667E$-$06 $\pm$ 8.087E$-$05 |  $-$4.345E$-$04 $\pm$ 1.344E$-$04 |  $-$4.311E$-$04 $\pm$ 1.901E$-$04 |  $-$5.783E$-$04 $\pm$ 5.295E$-$04
20040320 |  $-$1.466E$-$04 $\pm$ 8.305E$-$05 |  $-$2.539E$-$04 $\pm$ 1.373E$-$04 |  $-$5.356E$-$04 $\pm$ 1.957E$-$04 |  $-$1.133E$-$04 $\pm$ 5.460E$-$04
20051017 |  $-$2.067E$-$04 $\pm$ 9.270E$-$05 |  $-$2.023E$-$04 $\pm$ 1.536E$-$04 |  $-$8.969E$-$05 $\pm$ 2.179E$-$04 |  $-$3.453E$-$04 $\pm$ 6.127E$-$04
20070516 |  $-$1.721E$-$04 $\pm$ 9.550E$-$05 |  $-$2.101E$-$04 $\pm$ 1.586E$-$04 |  $-$1.006E$-$04 $\pm$ 2.252E$-$04 |  $-$4.621E$-$04 $\pm$ 6.324E$-$04
| HMI
20100711 |  $-$1.855E$-$04 $\pm$ 6.960E$-$05 |  $-$1.310E$-$05 $\pm$ 1.160E$-$04 |  $-$3.097E$-$04 $\pm$ 1.631E$-$04 |  $-$8.094E$-$04 $\pm$ 4.522E$-$04
20120207 |  $-$8.665E$-$07 $\pm$ 7.001E$-$05 |  $-$1.380E$-$04 $\pm$ 1.169E$-$04 |  $-$1.291E$-$04 $\pm$ 1.629E$-$04 |  $-$1.249E$-$04 $\pm$ 4.519E$-$04
20130905 |  $-$1.444E$-$04 $\pm$ 6.895E$-$05 |  $-$3.080E$-$04 $\pm$ 1.159E$-$04 |  $-$1.568E$-$04 $\pm$ 1.612E$-$04 |  $-$2.675E$-$04 $\pm$ 4.466E$-$04
20150404 |  $-$3.823E$-$05 $\pm$ 6.993E$-$05 |  $-$8.838E$-$05 $\pm$ 1.177E$-$04 |  $-$2.527E$-$04 $\pm$ 1.646E$-$04 |  $-$1.293E$-$05 $\pm$ 4.568E$-$04
20161031 |  $-$3.855E$-$05 $\pm$ 7.470E$-$05 |  $-$2.291E$-$04 $\pm$ 1.261E$-$04 |  $-$4.941E$-$06 $\pm$ 1.759E$-$04 |  $-$8.129E$-$04 $\pm$ 4.902E$-$04
20180530 |  $-$9.789E$-$05 $\pm$ 7.637E$-$05 |  $-$1.044E$-$04 $\pm$ 1.278E$-$04 |  $-$4.850E$-$04 $\pm$ 1.784E$-$04 |  $-$1.443E$-$04 $\pm$ 4.982E$-$04
Table 4: Results for the $8\times 72$-day sets Start Date |  $\Delta R_{\rm CZ}$/R⊙
---|---
of set | $a_{2}$ | $a_{4}$ | $a_{6}$ | $a_{8}$
| GONG
19950718 |  $-$2.768E$-$05 $\pm$ 1.121E$-$04 |  $-$5.896E$-$07 $\pm$ 1.862E$-$04 |  $-$4.227E$-$04 $\pm$ 2.477E$-$04 |  $-$2.671E$-$04 $\pm$ 5.791E$-$04
19960501 |  $-$1.740E$-$04 $\pm$ 1.078E$-$04 |  $-$8.954E$-$04 $\pm$ 1.813E$-$04 |  $-$9.408E$-$04 $\pm$ 2.446E$-$04 |  $-$6.814E$-$04 $\pm$ 5.705E$-$04
19970213 |  $-$1.695E$-$06 $\pm$ 1.036E$-$04 |  $-$2.947E$-$04 $\pm$ 1.682E$-$04 |  $-$6.202E$-$05 $\pm$ 2.302E$-$04 |  $-$1.127E$-$04 $\pm$ 5.302E$-$04
19971128 |  $-$2.398E$-$06 $\pm$ 1.025E$-$04 |  $-$4.251E$-$04 $\pm$ 1.717E$-$04 |  $-$1.401E$-$04 $\pm$ 2.354E$-$04 |  $-$5.582E$-$04 $\pm$ 5.335E$-$04
19980912 |  $-$1.036E$-$04 $\pm$ 1.042E$-$04 |  $-$5.010E$-$04 $\pm$ 1.676E$-$04 |  $-$1.571E$-$04 $\pm$ 2.347E$-$04 |  $-$2.426E$-$04 $\pm$ 5.557E$-$04
19990627 |  $-$9.036E$-$05 $\pm$ 1.069E$-$04 |  $-$7.658E$-$04 $\pm$ 1.747E$-$04 |  $-$3.574E$-$04 $\pm$ 2.406E$-$04 |  $-$8.071E$-$04 $\pm$ 5.870E$-$04
20000410 |  $-$7.886E$-$05 $\pm$ 1.030E$-$04 |  $-$3.777E$-$04 $\pm$ 1.675E$-$04 |  $-$2.984E$-$06 $\pm$ 2.337E$-$04 |  $-$2.935E$-$04 $\pm$ 5.651E$-$04
20010123 |  $-$2.371E$-$04 $\pm$ 1.019E$-$04 |  $-$2.336E$-$04 $\pm$ 1.668E$-$04 |  $-$2.481E$-$04 $\pm$ 2.289E$-$04 |  $-$6.042E$-$04 $\pm$ 5.790E$-$04
20011107 |  $-$7.796E$-$05 $\pm$ 9.848E$-$05 |  $-$3.725E$-$04 $\pm$ 1.656E$-$04 |  $-$6.139E$-$04 $\pm$ 2.302E$-$04 |  $-$1.621E$-$05 $\pm$ 5.961E$-$04
20020822 |  $-$1.382E$-$04 $\pm$ 9.195E$-$05 |  $-$1.558E$-$04 $\pm$ 1.596E$-$04 |  $-$1.240E$-$04 $\pm$ 2.159E$-$04 |  $-$9.486E$-$04 $\pm$ 5.505E$-$04
20030606 |  $-$4.546E$-$04 $\pm$ 9.708E$-$05 |  $-$2.729E$-$04 $\pm$ 1.606E$-$04 |  $-$4.267E$-$05 $\pm$ 2.180E$-$04 |  $-$2.217E$-$05 $\pm$ 5.547E$-$04
20040320 |  $-$5.259E$-$05 $\pm$ 9.476E$-$05 |  $-$4.382E$-$04 $\pm$ 1.549E$-$04 |  $-$1.462E$-$03 $\pm$ 2.164E$-$04 |  $-$2.752E$-$04 $\pm$ 5.265E$-$04
20050102 |  $-$3.309E$-$04 $\pm$ 9.830E$-$05 |  $-$1.548E$-$04 $\pm$ 1.611E$-$04 |  $-$5.587E$-$04 $\pm$ 2.261E$-$04 |  $-$3.355E$-$04 $\pm$ 5.552E$-$04
20051017 |  $-$2.970E$-$05 $\pm$ 1.003E$-$04 |  $-$4.462E$-$04 $\pm$ 1.667E$-$04 |  $-$7.517E$-$04 $\pm$ 2.292E$-$04 |  $-$6.905E$-$04 $\pm$ 5.653E$-$04
20060801 |  $-$5.539E$-$05 $\pm$ 1.009E$-$04 |  $-$4.082E$-$05 $\pm$ 1.709E$-$04 |  $-$2.453E$-$04 $\pm$ 2.309E$-$04 |  $-$2.289E$-$04 $\pm$ 5.714E$-$04
20070516 |  $-$4.547E$-$04 $\pm$ 9.964E$-$05 |  $-$2.595E$-$04 $\pm$ 1.663E$-$04 |  $-$7.486E$-$05 $\pm$ 2.259E$-$04 |  $-$1.291E$-$04 $\pm$ 5.659E$-$04
20080228 |  $-$4.131E$-$04 $\pm$ 1.005E$-$04 |  $-$8.335E$-$04 $\pm$ 1.720E$-$04 |  $-$1.025E$-$03 $\pm$ 2.331E$-$04 |  $-$1.958E$-$03 $\pm$ 5.736E$-$04
20081212 |  $-$3.793E$-$04 $\pm$ 1.002E$-$04 |  $-$6.164E$-$04 $\pm$ 1.647E$-$04 |  $-$6.337E$-$04 $\pm$ 2.261E$-$04 |  $-$9.614E$-$04 $\pm$ 5.635E$-$04
20090926 |  $-$1.014E$-$04 $\pm$ 9.360E$-$05 |  $-$4.171E$-$04 $\pm$ 1.573E$-$04 |  $-$3.940E$-$04 $\pm$ 2.151E$-$04 |  $-$1.164E$-$04 $\pm$ 5.285E$-$04
20100711 |  $-$3.666E$-$04 $\pm$ 9.888E$-$05 |  $-$1.948E$-$04 $\pm$ 1.608E$-$04 |  $-$1.967E$-$05 $\pm$ 2.228E$-$04 |  $-$1.424E$-$03 $\pm$ 5.518E$-$04
20110425 |  $-$3.511E$-$04 $\pm$ 9.564E$-$05 |  $-$3.026E$-$04 $\pm$ 1.563E$-$04 |  $-$4.268E$-$04 $\pm$ 2.187E$-$04 |  $-$4.378E$-$04 $\pm$ 5.429E$-$04
20120207 |  $-$2.882E$-$04 $\pm$ 9.331E$-$05 |  $-$2.004E$-$04 $\pm$ 1.564E$-$04 |  $-$9.971E$-$04 $\pm$ 2.156E$-$04 |  $-$8.493E$-$06 $\pm$ 5.303E$-$04
20121121 |  $-$1.074E$-$04 $\pm$ 9.441E$-$05 |  $-$1.453E$-$04 $\pm$ 1.572E$-$04 |  $-$2.014E$-$04 $\pm$ 2.152E$-$04 |  $-$1.176E$-$04 $\pm$ 5.446E$-$04
20130905 |  $-$2.840E$-$04 $\pm$ 9.456E$-$05 |  $-$7.684E$-$04 $\pm$ 1.584E$-$04 |  $-$3.281E$-$04 $\pm$ 2.160E$-$04 |  $-$4.680E$-$04 $\pm$ 5.513E$-$04
20140620 |  $-$1.505E$-$04 $\pm$ 9.491E$-$05 |  $-$5.927E$-$04 $\pm$ 1.566E$-$04 |  $-$3.403E$-$05 $\pm$ 2.193E$-$04 |  $-$2.929E$-$04 $\pm$ 5.468E$-$04
20150404 |  $-$1.182E$-$04 $\pm$ 9.221E$-$05 |  $-$3.787E$-$04 $\pm$ 1.523E$-$04 |  $-$8.066E$-$05 $\pm$ 2.127E$-$04 |  $-$1.006E$-$03 $\pm$ 5.242E$-$04
20160117 |  $-$1.371E$-$04 $\pm$ 9.953E$-$05 |  $-$2.947E$-$04 $\pm$ 1.642E$-$04 |  $-$1.361E$-$04 $\pm$ 8.816E$-$05 |  $-$7.683E$-$04 $\pm$ 5.628E$-$04
20161031 |  $-$7.077E$-$06 $\pm$ 9.976E$-$05 |  $-$2.267E$-$04 $\pm$ 1.658E$-$04 |  $-$3.942E$-$04 $\pm$ 2.248E$-$04 |  $-$1.029E$-$03 $\pm$ 5.638E$-$04
20170815 |  $-$7.155E$-$05 $\pm$ 1.019E$-$04 |  $-$6.435E$-$04 $\pm$ 1.672E$-$04 |  $-$5.155E$-$04 $\pm$ 2.345E$-$04 |  $-$1.014E$-$03 $\pm$ 5.626E$-$04
20180530 |  $-$1.711E$-$04 $\pm$ 1.031E$-$04 |  $-$6.856E$-$04 $\pm$ 1.710E$-$04 |  $-$2.455E$-$04 $\pm$ 2.409E$-$04 |  $-$3.270E$-$06 $\pm$ 5.821E$-$04
20190314 |  $-$1.832E$-$04 $\pm$ 1.026E$-$04 |  $-$5.598E$-$04 $\pm$ 1.703E$-$04 |  $-$1.281E$-$04 $\pm$ 2.407E$-$04 |  $-$2.135E$-$04 $\pm$ 5.726E$-$04
20191227 |  $-$1.297E$-$04 $\pm$ 1.017E$-$04 |  $-$2.802E$-$04 $\pm$ 1.699E$-$04 |  $-$2.404E$-$04 $\pm$ 2.363E$-$04 |  $-$8.210E$-$04 $\pm$ 5.696E$-$04
20201010 |  $-$9.323E$-$05 $\pm$ 1.017E$-$04 |  $-$5.041E$-$04 $\pm$ 1.701E$-$04 |  $-$1.027E$-$04 $\pm$ 2.334E$-$04 |  $-$7.132E$-$04 $\pm$ 5.813E$-$04
| MDI
19960501 |  $-$1.679E$-$04 $\pm$ 3.243E$-$05 |  $-$6.696E$-$04 $\pm$ 1.995E$-$04 |  $-$1.122E$-$04 $\pm$ 2.893E$-$04 |  $-$4.351E$-$04 $\pm$ 7.790E$-$04
19970213 |  $-$8.866E$-$05 $\pm$ 1.185E$-$04 |  $-$1.727E$-$04 $\pm$ 1.990E$-$04 |  $-$6.224E$-$05 $\pm$ 2.829E$-$04 |  $-$4.827E$-$04 $\pm$ 7.841E$-$04
19971128 |  $-$2.465E$-$04 $\pm$ 1.566E$-$04 |  $-$3.129E$-$04 $\pm$ 2.105E$-$04 |  $-$1.275E$-$05 $\pm$ 2.979E$-$04 |  $-$1.032E$-$03 $\pm$ 8.467E$-$04
19980912 |  $-$2.922E$-$04 $\pm$ 1.198E$-$04 |  $-$2.801E$-$04 $\pm$ 1.999E$-$04 |  $-$8.307E$-$04 $\pm$ 2.939E$-$04 |  $-$5.026E$-$04 $\pm$ 8.066E$-$04
19990627 |  $-$1.646E$-$04 $\pm$ 1.166E$-$04 |  $-$8.052E$-$04 $\pm$ 1.975E$-$04 |  $-$8.777E$-$04 $\pm$ 2.783E$-$04 |  $-$3.687E$-$05 $\pm$ 7.673E$-$04
20000410 |  $-$1.817E$-$04 $\pm$ 1.176E$-$04 |  $-$4.883E$-$04 $\pm$ 1.907E$-$04 |  $-$1.037E$-$04 $\pm$ 2.718E$-$04 |  $-$8.150E$-$04 $\pm$ 7.486E$-$04
20010123 |  $-$1.237E$-$04 $\pm$ 1.132E$-$04 |  $-$5.303E$-$04 $\pm$ 1.885E$-$04 |  $-$1.721E$-$04 $\pm$ 2.701E$-$04 |  $-$9.066E$-$04 $\pm$ 7.568E$-$04
20011107 |  $-$1.415E$-$04 $\pm$ 1.136E$-$04 |  $-$1.612E$-$04 $\pm$ 1.935E$-$04 |  $-$4.518E$-$04 $\pm$ 2.736E$-$04 |  $-$6.942E$-$05 $\pm$ 7.623E$-$04
20020822 |  $-$2.985E$-$04 $\pm$ 1.139E$-$04 |  $-$1.852E$-$04 $\pm$ 1.876E$-$04 |  $-$6.602E$-$04 $\pm$ 2.712E$-$04 |  $-$1.729E$-$03 $\pm$ 7.419E$-$04
20030606 |  $-$3.188E$-$04 $\pm$ 1.124E$-$04 |  $-$3.285E$-$04 $\pm$ 1.881E$-$04 |  $-$3.241E$-$04 $\pm$ 2.704E$-$04 |  $-$1.118E$-$03 $\pm$ 7.515E$-$04
20040320 |  $-$6.623E$-$05 $\pm$ 1.138E$-$04 |  $-$3.604E$-$04 $\pm$ 1.832E$-$04 |  $-$8.670E$-$04 $\pm$ 2.714E$-$04 |  $-$1.889E$-$04 $\pm$ 7.413E$-$04
20050102 |  $-$2.914E$-$04 $\pm$ 1.182E$-$04 |  $-$1.887E$-$04 $\pm$ 1.905E$-$04 |  $-$3.699E$-$04 $\pm$ 2.837E$-$04 |  $-$5.695E$-$04 $\pm$ 7.536E$-$04
20051017 |  $-$8.274E$-$05 $\pm$ 1.172E$-$04 |  $-$4.241E$-$04 $\pm$ 1.966E$-$04 |  $-$7.149E$-$04 $\pm$ 2.828E$-$04 |  $-$7.244E$-$04 $\pm$ 8.036E$-$04
20060801 |  $-$1.746E$-$04 $\pm$ 1.259E$-$04 |  $-$1.810E$-$04 $\pm$ 1.966E$-$04 |  $-$5.586E$-$04 $\pm$ 2.922E$-$04 |  $-$3.703E$-$05 $\pm$ 7.920E$-$04
20070516 |  $-$3.081E$-$04 $\pm$ 1.235E$-$04 |  $-$4.014E$-$04 $\pm$ 2.011E$-$04 |  $-$4.883E$-$04 $\pm$ 3.002E$-$04 |  $-$5.064E$-$04 $\pm$ 8.360E$-$04
20080228 |  $-$3.176E$-$04 $\pm$ 1.261E$-$04 |  $-$3.337E$-$04 $\pm$ 2.074E$-$04 |  $-$8.664E$-$04 $\pm$ 3.022E$-$04 |  $-$6.559E$-$04 $\pm$ 8.489E$-$04
20081212 |  $-$3.450E$-$04 $\pm$ 1.265E$-$04 |  $-$8.905E$-$04 $\pm$ 2.091E$-$04 |  $-$3.180E$-$04 $\pm$ 3.004E$-$04 |  $-$1.078E$-$03 $\pm$ 8.400E$-$04
| HMI
20100711 |  $-$2.960E$-$04 $\pm$ 1.005E$-$04 |  $-$9.257E$-$05 $\pm$ 1.687E$-$04 |  $-$7.059E$-$05 $\pm$ 2.342E$-$04 |  $-$6.507E$-$04 $\pm$ 6.662E$-$04
20110425 |  $-$3.035E$-$04 $\pm$ 9.867E$-$05 |  $-$3.076E$-$04 $\pm$ 1.622E$-$04 |  $-$6.568E$-$04 $\pm$ 2.388E$-$04 |  $-$1.052E$-$03 $\pm$ 6.354E$-$04
20120207 |  $-$2.004E$-$04 $\pm$ 9.656E$-$05 |  $-$2.827E$-$04 $\pm$ 1.613E$-$04 |  $-$1.092E$-$03 $\pm$ 2.259E$-$04 |  $-$1.454E$-$04 $\pm$ 5.285E$-$05
20121121 |  $-$2.966E$-$05 $\pm$ 9.677E$-$05 |  $-$2.004E$-$04 $\pm$ 1.602E$-$04 |  $-$4.489E$-$04 $\pm$ 2.236E$-$04 |  $-$1.435E$-$04 $\pm$ 6.175E$-$04
20130905 |  $-$1.784E$-$04 $\pm$ 9.671E$-$05 |  $-$8.528E$-$04 $\pm$ 1.614E$-$04 |  $-$5.878E$-$04 $\pm$ 2.201E$-$04 |  $-$1.844E$-$04 $\pm$ 6.199E$-$04
20140620 |  $-$2.003E$-$05 $\pm$ 9.364E$-$05 |  $-$5.574E$-$04 $\pm$ 1.646E$-$04 |  $-$8.529E$-$05 $\pm$ 2.323E$-$04 |  $-$5.331E$-$04 $\pm$ 6.363E$-$04
20150404 |  $-$8.478E$-$05 $\pm$ 9.577E$-$05 |  $-$3.624E$-$04 $\pm$ 1.616E$-$04 |  $-$1.534E$-$05 $\pm$ 2.305E$-$04 |  $-$6.690E$-$04 $\pm$ 6.253E$-$04
20160117 |  $-$2.328E$-$04 $\pm$ 1.001E$-$04 |  $-$2.673E$-$04 $\pm$ 1.685E$-$04 |  $-$4.211E$-$04 $\pm$ 2.341E$-$04 |  $-$2.158E$-$04 $\pm$ 6.560E$-$04
20161031 |  $-$1.006E$-$04 $\pm$ 1.021E$-$04 |  $-$1.180E$-$04 $\pm$ 1.744E$-$04 |  $-$2.347E$-$04 $\pm$ 2.401E$-$04 |  $-$1.175E$-$03 $\pm$ 6.647E$-$04
20170815 |  $-$9.367E$-$05 $\pm$ 1.057E$-$04 |  $-$1.250E$-$04 $\pm$ 1.735E$-$04 |  $-$3.774E$-$04 $\pm$ 2.456E$-$04 |  $-$1.029E$-$03 $\pm$ 6.871E$-$04
20180530 |  $-$1.398E$-$04 $\pm$ 1.079E$-$04 |  $-$5.611E$-$04 $\pm$ 1.824E$-$04 |  $-$4.173E$-$04 $\pm$ 2.494E$-$04 |  $-$2.271E$-$04 $\pm$ 7.161E$-$04
20190314 |  $-$9.100E$-$05 $\pm$ 1.120E$-$04 |  $-$4.868E$-$04 $\pm$ 1.874E$-$04 |  $-$8.267E$-$06 $\pm$ 2.726E$-$04 |  $-$8.871E$-$05 $\pm$ 7.282E$-$04
20191227 |  $-$1.880E$-$04 $\pm$ 1.035E$-$04 |  $-$5.511E$-$05 $\pm$ 1.711E$-$04 |  $-$3.143E$-$04 $\pm$ 2.511E$-$04 |  $-$3.282E$-$04 $\pm$ 6.994E$-$04
20201010 |  $-$1.053E$-$04 $\pm$ 1.014E$-$04 |  $-$3.848E$-$04 $\pm$ 1.690E$-$04 |  $-$2.648E$-$05 $\pm$ 2.373E$-$04 |  $-$1.121E$-$03 $\pm$ 6.684E$-$04
Table 5: Results for the $5\times 72$-day sets Start Date |  $\Delta R_{\rm CZ}$/R⊙
---|---
of set | $a_{2}$ | $a_{4}$ | $a_{6}$ | $a_{8}$
| GONG
19960501 |  $-$3.588E$-$05 $\pm$ 1.319E$-$04 |  $-$9.555E$-$04 $\pm$ 2.174E$-$04 |  $-$1.704E$-$03 $\pm$ 3.081E$-$04 |  $-$2.839E$-$04 $\pm$ 7.079E$-$04
19970426 |  $-$5.235E$-$05 $\pm$ 1.254E$-$04 |  $-$2.939E$-$04 $\pm$ 2.077E$-$04 |  $-$3.811E$-$04 $\pm$ 2.894E$-$04 |  $-$3.486E$-$04 $\pm$ 6.605E$-$04
19980421 |  $-$2.454E$-$04 $\pm$ 1.224E$-$04 |  $-$4.944E$-$04 $\pm$ 1.958E$-$04 |  $-$2.373E$-$04 $\pm$ 2.715E$-$04 |  $-$3.828E$-$04 $\pm$ 6.463E$-$04
19990416 |  $-$3.982E$-$04 $\pm$ 1.248E$-$04 |  $-$7.308E$-$05 $\pm$ 2.007E$-$04 |  $-$4.238E$-$04 $\pm$ 2.924E$-$04 |  $-$1.002E$-$03 $\pm$ 6.957E$-$04
20000410 |  $-$3.672E$-$04 $\pm$ 1.254E$-$04 |  $-$3.735E$-$04 $\pm$ 2.032E$-$04 |  $-$9.070E$-$04 $\pm$ 2.842E$-$04 |  $-$3.152E$-$04 $\pm$ 6.923E$-$04
20010405 |  $-$2.928E$-$04 $\pm$ 1.199E$-$04 |  $-$4.916E$-$05 $\pm$ 1.961E$-$04 |  $-$2.728E$-$04 $\pm$ 2.752E$-$04 |  $-$9.851E$-$04 $\pm$ 6.963E$-$04
20020331 |  $-$1.104E$-$04 $\pm$ 1.232E$-$04 |  $-$4.174E$-$04 $\pm$ 1.926E$-$04 |  $-$1.592E$-$03 $\pm$ 2.699E$-$04 |  $-$8.070E$-$04 $\pm$ 6.928E$-$04
20030326 |  $-$2.632E$-$04 $\pm$ 1.083E$-$04 |  $-$4.787E$-$04 $\pm$ 1.849E$-$04 |  $-$8.017E$-$04 $\pm$ 2.524E$-$04 |  $-$1.207E$-$03 $\pm$ 6.362E$-$04
20040320 |  $-$7.737E$-$05 $\pm$ 1.132E$-$04 |  $-$5.069E$-$04 $\pm$ 1.866E$-$04 |  $-$1.369E$-$03 $\pm$ 2.646E$-$04 |  $-$9.853E$-$04 $\pm$ 6.463E$-$04
20050315 |  $-$5.549E$-$04 $\pm$ 1.141E$-$04 |  $-$8.013E$-$04 $\pm$ 1.935E$-$04 |  $-$1.379E$-$03 $\pm$ 2.636E$-$04 |  $-$5.958E$-$04 $\pm$ 6.638E$-$04
20060310 |  $-$2.420E$-$04 $\pm$ 1.196E$-$04 |  $-$1.808E$-$04 $\pm$ 2.008E$-$04 |  $-$3.567E$-$04 $\pm$ 2.713E$-$04 |  $-$9.843E$-$04 $\pm$ 6.710E$-$04
20070305 |  $-$1.442E$-$05 $\pm$ 1.206E$-$04 |  $-$6.549E$-$04 $\pm$ 1.963E$-$04 |  $-$6.800E$-$04 $\pm$ 2.784E$-$04 |  $-$5.827E$-$04 $\pm$ 6.745E$-$04
20080228 |  $-$9.281E$-$04 $\pm$ 1.246E$-$04 |  $-$4.970E$-$04 $\pm$ 2.050E$-$04 |  $-$2.762E$-$04 $\pm$ 2.874E$-$04 |  $-$1.660E$-$03 $\pm$ 6.899E$-$04
20090222 |  $-$1.614E$-$04 $\pm$ 1.193E$-$04 |  $-$1.082E$-$03 $\pm$ 2.054E$-$04 |  $-$1.009E$-$03 $\pm$ 2.777E$-$04 |  $-$4.417E$-$04 $\pm$ 6.982E$-$04
20100217 |  $-$4.107E$-$04 $\pm$ 1.185E$-$04 |  $-$3.724E$-$04 $\pm$ 1.983E$-$04 |  $-$2.199E$-$04 $\pm$ 2.767E$-$04 |  $-$2.806E$-$04 $\pm$ 6.732E$-$04
20110212 |  $-$2.066E$-$04 $\pm$ 1.129E$-$04 |  $-$1.228E$-$04 $\pm$ 1.814E$-$04 |  $-$3.439E$-$04 $\pm$ 2.537E$-$04 |  $-$1.859E$-$03 $\pm$ 6.089E$-$04
20120207 |  $-$4.621E$-$04 $\pm$ 1.126E$-$04 |  $-$2.408E$-$05 $\pm$ 1.842E$-$04 |  $-$1.372E$-$03 $\pm$ 2.652E$-$04 |  $-$1.059E$-$04 $\pm$ 6.187E$-$05
20130201 |  $-$3.435E$-$04 $\pm$ 1.105E$-$04 |  $-$1.916E$-$04 $\pm$ 1.863E$-$04 |  $-$3.637E$-$04 $\pm$ 2.588E$-$04 |  $-$2.188E$-$04 $\pm$ 6.373E$-$04
20140127 |  $-$6.314E$-$04 $\pm$ 1.101E$-$04 |  $-$5.387E$-$04 $\pm$ 1.837E$-$04 |  $-$1.264E$-$03 $\pm$ 2.585E$-$04 |  $-$1.069E$-$03 $\pm$ 6.439E$-$04
20150122 |  $-$2.463E$-$04 $\pm$ 1.116E$-$04 |  $-$6.898E$-$04 $\pm$ 1.846E$-$04 |  $-$8.648E$-$04 $\pm$ 2.553E$-$04 |  $-$3.754E$-$04 $\pm$ 6.317E$-$04
20160117 |  $-$4.519E$-$05 $\pm$ 1.129E$-$04 |  $-$5.044E$-$05 $\pm$ 1.867E$-$04 |  $-$3.817E$-$04 $\pm$ 2.592E$-$04 |  $-$1.344E$-$03 $\pm$ 6.463E$-$04
20170111 |  $-$2.313E$-$04 $\pm$ 1.156E$-$04 |  $-$3.874E$-$04 $\pm$ 1.969E$-$04 |  $-$9.711E$-$04 $\pm$ 2.686E$-$04 |  $-$1.930E$-$03 $\pm$ 6.802E$-$04
20180106 |  $-$2.746E$-$06 $\pm$ 1.232E$-$04 |  $-$7.786E$-$04 $\pm$ 2.075E$-$04 |  $-$4.994E$-$05 $\pm$ 2.776E$-$04 |  $-$3.548E$-$04 $\pm$ 6.902E$-$04
20190101 |  $-$1.264E$-$04 $\pm$ 1.227E$-$04 |  $-$8.592E$-$04 $\pm$ 2.064E$-$04 |  $-$1.348E$-$05 $\pm$ 2.897E$-$04 |  $-$2.098E$-$03 $\pm$ 7.146E$-$04
20191227 |  $-$9.851E$-$05 $\pm$ 1.184E$-$04 |  $-$2.520E$-$04 $\pm$ 8.238E$-$05 |  $-$5.347E$-$04 $\pm$ 2.786E$-$04 |  $-$1.272E$-$03 $\pm$ 6.917E$-$04
20201221 |  $-$1.857E$-$04 $\pm$ 1.194E$-$04 |  $-$7.613E$-$04 $\pm$ 1.960E$-$04 |  $-$4.214E$-$04 $\pm$ 2.781E$-$04 |  $-$1.551E$-$04 $\pm$ 6.788E$-$04
| MDI
19960501 |  $-$3.249E$-$05 $\pm$ 1.496E$-$04 |  $-$6.739E$-$04 $\pm$ 2.465E$-$04 |  $-$5.510E$-$04 $\pm$ 3.652E$-$04 |  $-$6.758E$-$04 $\pm$ 1.002E$-$03
19970426 |  $-$1.408E$-$04 $\pm$ 1.413E$-$04 |  $-$8.259E$-$05 $\pm$ 2.230E$-$04 |  $-$8.475E$-$04 $\pm$ 3.428E$-$04 |  $-$2.433E$-$04 $\pm$ 9.325E$-$04
19980421 |  $-$4.957E$-$04 $\pm$ 1.669E$-$04 |  $-$1.159E$-$03 $\pm$ 2.623E$-$04 |  $-$1.588E$-$03 $\pm$ 3.901E$-$04 |  $-$1.131E$-$03 $\pm$ 1.081E$-$03
19990416 |  $-$6.330E$-$04 $\pm$ 1.397E$-$04 |  $-$2.152E$-$04 $\pm$ 2.252E$-$04 |  $-$5.035E$-$04 $\pm$ 3.297E$-$04 |  $-$1.589E$-$03 $\pm$ 8.885E$-$04
20000410 |  $-$4.040E$-$04 $\pm$ 1.355E$-$04 |  $-$5.631E$-$05 $\pm$ 2.295E$-$04 |  $-$1.165E$-$03 $\pm$ 3.199E$-$04 |  $-$2.085E$-$04 $\pm$ 8.908E$-$04
20010405 |  $-$3.047E$-$04 $\pm$ 1.399E$-$04 |  $-$5.622E$-$04 $\pm$ 2.298E$-$04 |  $-$3.100E$-$04 $\pm$ 3.347E$-$04 |  $-$6.868E$-$04 $\pm$ 9.124E$-$04
20020331 |  $-$1.789E$-$04 $\pm$ 1.372E$-$04 |  $-$5.572E$-$04 $\pm$ 2.242E$-$04 |  $-$1.441E$-$03 $\pm$ 3.222E$-$04 |  $-$5.479E$-$04 $\pm$ 8.969E$-$04
20030326 |  $-$3.194E$-$04 $\pm$ 2.223E$-$04 |  $-$3.193E$-$04 $\pm$ 2.273E$-$04 |  $-$2.665E$-$04 $\pm$ 3.130E$-$04 |  $-$8.599E$-$04 $\pm$ 8.816E$-$04
20040320 |  $-$2.402E$-$04 $\pm$ 1.355E$-$04 |  $-$6.730E$-$04 $\pm$ 2.291E$-$04 |  $-$7.335E$-$04 $\pm$ 3.338E$-$04 |  $-$3.220E$-$04 $\pm$ 9.001E$-$04
20050315 |  $-$3.596E$-$04 $\pm$ 1.345E$-$04 |  $-$1.755E$-$04 $\pm$ 2.169E$-$04 |  $-$1.810E$-$03 $\pm$ 3.311E$-$04 |  $-$1.319E$-$04 $\pm$ 9.100E$-$04
20060310 |  $-$8.495E$-$05 $\pm$ 1.465E$-$04 |  $-$4.389E$-$05 $\pm$ 2.351E$-$04 |  $-$2.662E$-$04 $\pm$ 3.444E$-$04 |  $-$8.014E$-$04 $\pm$ 9.261E$-$04
20070305 |  $-$1.004E$-$05 $\pm$ 1.484E$-$04 |  $-$7.712E$-$04 $\pm$ 2.398E$-$04 |  $-$6.664E$-$04 $\pm$ 3.555E$-$04 |  $-$3.810E$-$04 $\pm$ 9.607E$-$04
20080228 |  $-$1.021E$-$04 $\pm$ 1.549E$-$04 |  $-$1.166E$-$04 $\pm$ 2.602E$-$04 |  $-$1.027E$-$03 $\pm$ 3.755E$-$04 |  $-$8.020E$-$04 $\pm$ 1.028E$-$03
20090222 |  $-$2.454E$-$04 $\pm$ 1.513E$-$04 |  $-$1.410E$-$04 $\pm$ 2.515E$-$04 |  $-$3.212E$-$04 $\pm$ 3.520E$-$04 |  $-$1.850E$-$04 $\pm$ 9.895E$-$04
20100217 |  $-$4.135E$-$04 $\pm$ 1.466E$-$04 |  $-$6.335E$-$04 $\pm$ 2.361E$-$04 |  $-$1.217E$-$05 $\pm$ 3.362E$-$04 |  $-$1.275E$-$04 $\pm$ 9.530E$-$04
| HMI
20110212 |  $-$2.221E$-$04 $\pm$ 1.158E$-$04 |  $-$3.019E$-$04 $\pm$ 1.966E$-$04 |  $-$3.283E$-$05 $\pm$ 2.749E$-$04 |  $-$5.365E$-$04 $\pm$ 7.576E$-$04
20120207 |  $-$8.597E$-$05 $\pm$ 1.199E$-$04 |  $-$2.730E$-$04 $\pm$ 2.011E$-$04 |  $-$1.516E$-$03 $\pm$ 2.815E$-$04 |  $-$1.766E$-$04 $\pm$ 7.678E$-$04
20130201 |  $-$3.320E$-$04 $\pm$ 1.133E$-$04 |  $-$3.490E$-$04 $\pm$ 1.913E$-$04 |  $-$3.782E$-$04 $\pm$ 2.728E$-$04 |  $-$5.435E$-$04 $\pm$ 7.725E$-$04
20140127 |  $-$4.768E$-$04 $\pm$ 1.144E$-$04 |  $-$5.917E$-$04 $\pm$ 1.920E$-$04 |  $-$1.209E$-$03 $\pm$ 2.717E$-$04 |  $-$4.920E$-$04 $\pm$ 7.298E$-$04
20150122 |  $-$2.282E$-$04 $\pm$ 1.143E$-$04 |  $-$8.276E$-$04 $\pm$ 1.904E$-$04 |  $-$4.310E$-$04 $\pm$ 2.738E$-$04 |  $-$3.211E$-$04 $\pm$ 7.434E$-$04
20160117 |  $-$1.630E$-$05 $\pm$ 1.192E$-$04 |  $-$2.743E$-$05 $\pm$ 1.970E$-$04 |  $-$2.686E$-$04 $\pm$ 2.685E$-$04 |  $-$8.003E$-$04 $\pm$ 7.849E$-$04
20170111 |  $-$5.495E$-$05 $\pm$ 1.165E$-$04 |  $-$3.503E$-$04 $\pm$ 2.067E$-$04 |  $-$8.565E$-$04 $\pm$ 2.853E$-$04 |  $-$1.195E$-$03 $\pm$ 7.689E$-$04
20180106 |  $-$1.701E$-$04 $\pm$ 1.305E$-$04 |  $-$1.054E$-$04 $\pm$ 2.117E$-$04 |  $-$2.592E$-$04 $\pm$ 2.929E$-$04 |  $-$4.161E$-$04 $\pm$ 8.183E$-$04
20190101 |  $-$1.138E$-$04 $\pm$ 1.274E$-$04 |  $-$6.896E$-$04 $\pm$ 2.201E$-$04 |  $-$3.924E$-$04 $\pm$ 3.210E$-$04 |  $-$1.611E$-$03 $\pm$ 8.639E$-$04
20191227 |  $-$2.661E$-$04 $\pm$ 1.332E$-$04 |  $-$2.377E$-$04 $\pm$ 2.241E$-$04 |  $-$1.229E$-$04 $\pm$ 3.164E$-$04 |  $-$8.825E$-$04 $\pm$ 8.766E$-$04
20201221 |  $-$2.669E$-$04 $\pm$ 1.245E$-$04 |  $-$3.865E$-$04 $\pm$ 2.142E$-$04 |  $-$2.735E$-$04 $\pm$ 3.101E$-$04 |  $-$5.221E$-$04 $\pm$ 8.384E$-$04