11institutetext: Earl P. Bellinger 22institutetext: Stellar Astrophysics
Centre, Department of Physics and Astronomy, Aarhus University, Denmark
School of Physics, UNSW Sydney, Australia
22email<EMAIL_ADDRESS>33institutetext: Sarbani Basu 44institutetext:
Department of Astronomy, Yale University, USA
44email<EMAIL_ADDRESS>55institutetext: Saskia Hekker 66institutetext:
Max Planck Institute for Solar System Research, Germany
Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus
University, Denmark
66email<EMAIL_ADDRESS>
# Inverse analysis of asteroseismic data: a review

Earl P. Bellinger  Sarbani Basu  and Saskia Hekker

###### Abstract

Asteroseismology has emerged as the best way to characterize the global and
internal properties of nearby stars. Often, this characterization is achieved
by fitting stellar evolution models to asteroseismic observations. The star
under investigation is then assumed to have the properties of the best-fitting
model, such as its age. However, the models do not fit the observations
perfectly. This is due to incorrect or missing physics in stellar evolution
calculations, resulting in predicted stellar structures that are discrepant
with reality. Through an inverse analysis of the asteroseismic data, it is
possible to go further than fitting stellar models, and instead infer details
about the actual internal structure of the star at some locations in its
interior. Comparing theoretical and observed stellar structures then enables
the determination of the locations where the stellar models have discrepant
structure, and illuminates a path for improvements to our understanding of
stellar evolution. In this invited review, we describe the methods of
asteroseismic inversions, and outline the progress that is being made towards
measuring the interiors of stars.

##  1 Introduction

In stars that are like the Sun, turbulent motions near to the stellar surface
randomly excite acoustic modes that oscillate throughout the entire star. The
oscillation frequencies of these modes depend on the star’s internal chemical
composition and structure, and particularly on the speeds at which sound
travels through the star. The sound speed at any given location in the stellar
interior is dictated by the local properties of the stellar plasma, and
specifically its density, pressure, and compressibility. Detection of a
sufficient number of these acoustic oscillation modes thus permits inferences
of the sound speed—and hence the structure—at various locations within the
star. As the structure of stars is the principal prediction made by the theory
of stellar evolution, asteroseismic measurements of stellar interiors can
therefore be used to test the theory. This analysis is the topic of the
present review.

This is the first review on asteroseismic structure inversions in the post-
_Kepler_ era (for earlier reviews, see 2003Ap&SS.284..153B, ;
2012AN….333..914C, ). Here we focus on techniques that have been developed and
applied to actual stellar data. This review is divided into several, mostly
independent sections. Section 2 gives a brief mathematical introduction to
inverse problems and defines the scope of the review. Section 3 describes how
to obtain a reference model through an evolutionary inversion, which is
generally done before performing a structure inversion. Section 4 states the
structure inversion problem and describes the difficulties and limitations
imposed by asteroseismic data. Finally, Section 5 summarizes current results
and gives an outlook on future measurements of stellar interiors.

For a general introduction to asteroseismology, we refer the reader to the two
textbooks and numerous review articles on the topic (e.g., 2010aste.book…..A,
; 2017asda.book…..B, ; 2004SoPh..220..137C, ; 2007A&ARv..14..217C, ;
2013AdSpR..52.1581H, ; 2013ARA&A..51..353C, ; 2014aste.book…60B, ;
2017A&ARv..25….1H, ; 2019arXiv190710457H, ; 2019LRSP…16….4G, ;
2019arXiv191212300A, ).

##  2 A primer on inverse problems

Finish the sequence: 1, 2, 3, 4, _.

This task cannot be done with certainty. Although the answer would naturally
seem to be 5, other solutions are also possible, such as:

  * •

6 (a sequence known as Ulam’s numbers (ulam1962some, ))

  * •

7 (numbers $n$ such that ${(68\cdot 10n+7)/3}$ is prime)

  * •

10 (counting in base 5)

  * •

17 ($n$ such that ${n\cdot\varphi(n)}$ is a palindrome, where $\varphi$ is
Euler’s totient function)

  * •

$-42\pi$ (a sequence we have just invented).

Indeed, an infinity of solutions exist, each with infinitely many possible
justifications.

This is often the situation with inverse problems. Broadly speaking, inverse
problems can be posed for every data-generation scheme, such as those listed
above. The _forward problem_ is to generate data using the scheme (for
example, listing the natural numbers). The _inverse problem_ is to determine
the generating scheme itself from the data, or to estimate the parameters of
an assumed scheme.

Inverse problems are characteristically, though not always, _ill-posed_. A
well-posed problem satisfies three requirements (hadamard, ):

  1. 1.

Existence: a solution exists,

  2. 2.

Uniqueness: there is exactly one solution, and

  3. 3.

Stability: the solution changes continuously with changes to the
input111Consider the system of linear equations  $\displaystyle x$
$\displaystyle=a+b$ (1) $\displaystyle y$ $\displaystyle=a+b\cdot(1+\epsilon)$
(2) where $\epsilon$ is an arbitrarily small positive quantity and the rest
are real numbers. One can imagine that $x,y$ are values observed in nature
(i.e., they are the data) and $a,b$ are the hidden parameters that we wish to
infer (i.e., they form the data-generation scheme). If we observe that
$(x,y)=(2,2)$ then clearly $(a,b)=(2,0)$. Yet if $(x,y)=(2,2+\epsilon)$ then
$(a,b)=(1,1)$. An arbitrarily small change to the observed value of $y$ has
resulted in a completely different solution. The system is unstable..

Ill-posed problems—problems that violate any of these conditions—tend to be
difficult or impossible to solve. In order to obtain a solution, assumptions
must be made, or the problem must be altered. An example of a well-posed
inverse problem is to find the line corresponding to a few points belonging to
that line. An ill-posed inverse problem would be to find a line after any of
the points have been perturbed by noise. As no line will pass through all of
the points, an assumption must be made, such as assuming the noise is
Gaussian-distributed and accepting the least-squares solution.

For more information, the reader is referred to textbooks and review articles
on inverse problems (e.g., kirsch2011introduction, ; neto2012introduction, ;
tenorio2001statistical, ).

###  2.1 Inverse problems in asteroseismology

There are numerous inverse problems in asteroseismology. These include
inversions to infer a star’s:

  1. 1.

Evolutionary history: age, mass, initial composition, and evolutionary
parameters such as mixing efficiencies, assuming the theory of stellar
evolution is approximately correct (e.g., pulone1997age, ; 2016ApJ…830…31B, ;
2017EPJWC.16005003B, ; 2019A&A…622A.130B, ; 2020MNRAS.491.4752B, ;
2017ApJ…839..116A, ; AngelouAccepted, ; 2019PASP..131j8001H, ;
1994ApJ…427.1013B, ; 2004ApJ…600..419G, ; 2009ApJ…699..373M, ;
2015MNRAS.452.2127S, ; 2017ApJ…835..173S, ; 2019MNRAS.484..771R, ;
2019ApJ…887L…1B, ; 2019MNRAS.490.2890J, ; 2018ApJS..237…15A, ;
2019MNRAS.486.4612B, ; 2020MNRAS.492L..50B, ).

  2. 2.

Internal structure: specifically, the isothermal sound speed at various
locations around the stellar core (2017ApJ…851…80B, ; 2019ApJ…885..143B, ).

  3. 3.

Radial differential rotation: rates of rotation of the stellar core and
surface (e.g., 2012ApJ…756…19D, ; 2014A&A…564A..27D, ; 2015A&A…580A..96D, ;
2015MNRAS.452.2654B, ; 2019A&A…631L…6E, ; AhlbornAccepted, ).

  4. 4.

Latitudinal differential rotation: surface rotation rates at different
latitudes (2018Sci…361.1231B, ; 2018A&A…619L…9B, ; 2019A&A…623A.125B, ).

  5. 5.

Integrated quantities: any quantity that takes its value over some or all of
the stellar interior (1998MNRAS.297L..76P, ), such as the star’s

    1. (a)

Mean density: $(3/4\pi)M/R^{3}$, where $M$ is the mass of the star and $R$ is
its radius (2012A&A…539A..63R, ).

    2. (b)

Acoustic radius: $\int c^{-1}\,\text{d}r$, where $c=(\Gamma_{1}P/\rho)^{1/2}$
is the speed of sound, $\Gamma_{1}$ is the adiabatic compressibility, $P$ is
pressure, $\rho$ is density, and $r$ is the distance to the stellar center
(2015A&A…574A..42B, ).

    3. (c)

Sound speed gradient: $\int r^{-1}(\text{d}c/\text{d}r)\,\text{d}r$, which is
a proxy for stellar age on the main sequence because the sound speed gradient
depends on the mean molecular weight, which changes over time due to hydrogen
fusion (2015A&A…583A..62B, ; 2016A&A…596A..73B, ).

These problems are ill-posed and therefore difficult to solve—some due to
existence, some due to uniqueness, some due to stability, and some due to a
combination of these. This review focuses on the first two of these problems,
and especially the second. That being said, essentially all of these problems
can be solved using the techniques discussed in this review.

##  3 Evolutionary inversions

The first step for most inversions is to obtain a reference model. This is in
fact itself an inverse problem. Given observations of a star, we seek to
determine its age, internal structure, and other parameters using the theory
of stellar evolution.

Stellar evolution theory can be regarded as the function
$f_{\text{evol}}(\tau,\mathbf{x})=\mathbf{y}$, where $\tau$ is the age of the
star and $\mathbf{x}=[$mass, chemical composition, etc.$]$ are the initial
conditions. Other parameters may include anything relevant to the evolution,
such as the mixing length and overshooting parameters for convection,
diffusion coefficients, nuclear reaction rates, opacities, stellar companions,
and so on. The forward problem is to compute a stellar model222There are
numerous software implementations for computing $f_{\text{evol}}$; one open-
source and freely available one-dimensional stellar evolution code is Mesa
(2011ApJS..192….3P, ). The global oscillation modes of the resulting stellar
model is generally computed using a separate code. Two freely available codes
for computing linear adiabatic stellar oscillation frequencies are Adipls
(2008Ap&SS.316..113C, ) and Gyre (2013MNRAS.435.3406T, ).  that has
observables $\mathbf{y}=[$effective temperature, metallicity, oscillation
frequencies, etc.$]$. Other observables may include the radius, luminosity,
and so on.

We seek to invert this function: given observations $\mathbf{y}$, we wish to
infer $f_{\text{evol}}^{-1}(\mathbf{y})=[\tau,\mathbf{x}]$. However, this
inverse function is not guaranteed to exist because $f_{\text{evol}}$ is not
necessarily injective. Broadly speaking, there are three approaches for
solving the evolution inversion problem:

  1. 1.

Machine learning: an approximation to the inverse function
$f_{\text{evol}}^{-1}$ is learned from a large number of examples of
$\tau,\mathbf{x}$, and $\mathbf{y}$ (pulone1997age, ; 2016ApJ…830…31B, ;
2017EPJWC.16005003B, ; 2019A&A…622A.130B, ; 2020MNRAS.491.4752B, ;
2017ApJ…839..116A, ; AngelouAccepted, ; 2019PASP..131j8001H, ).

  2. 2.

Optimization: sometimes referred to as repeated forward modeling, a likelihood
function that quantifies how well a given model fits the data is maximized,
often along with priors on the parameters of the model (e.g.,
1994ApJ…427.1013B, ; 2004ApJ…600..419G, ; 2009ApJ…699..373M, ;
2015MNRAS.452.2127S, ; 2017ApJ…835..173S, ; 2019MNRAS.484..771R, ;
2019ApJ…887L…1B, ; 2019MNRAS.490.2890J, ; 2018ApJS..237…15A, ).

  3. 3.

Scaling relations: by assuming homology with the Sun (e.g.,
2019arXiv190710457H, ), the age and mass of a star can be deduced using
calibrated scaling relations (2019MNRAS.486.4612B, ; 2020MNRAS.492L..50B, ).
The chemical composition can be approximated using a galactic chemical
evolution law (e.g., 2017ApJ…835..173S, ). These parameters can then be used
with $f_{\text{evol}}$ to obtain a reference model.

In past decades, when observations consisted of only the effective
temperature, metallicity and sometimes luminosity, this problem was ill-posed
because the solution lacked uniqueness, i.e., many stellar models satisfied
the same constraints. With the advent of asteroseismology, this problem is
ill-posed because the solution lacks existence: there are no stellar models
which fit all the asteroseismic data (2017ApJ…851…80B, ; 2019ApJ…885..143B, ).
This is the case even for the Sun and models of solar evolution (e.g.,
2016LRSP…13….2B, ; 2019ApJ…881..103Z, ).

This motivates the structure inversion problem. If no stellar model has the
correct structure, then what is the correct structure?

##  4 Sound speed inversions

The problem at hand is to deduce the internal structure of a star, insofar as
it is possible, from asteroseismic measurements, without assuming the theory
of stellar evolution. This problem is generally cast as one of deducing the
differences in the internal sound speed profile between the star and a
suitably chosen reference model, which can be achieved through an analysis of
the differences in their frequencies of oscillation. This approach leads
directly to an assessment of the quality of stellar evolution models, and
particularly whether the models have the correct structure at the locations
within the star where the inverse analysis is successful. This problem is not
fundamentally different from the structure inversion problem in
helioseismology (e.g., 1991sia..book..519G, ; 2016LRSP…13….2B, ) and
geoseismology (1967GeoJ…13..247B, ; 1968geoj…16..169b, ). However, inverting
asteroseismic data comes with unique difficulties and limitations over
helioseismic inversions (2003Ap&SS.284..153B, ; 2017asda.book…..B, ).

The fact that the solar disk can be resolved permits the detection of solar
oscillation modes with spherical degrees of $\ell\lessapprox 250$
(2016LRSP…13….2B, ). In contrast, cancellation effects limit asteroseismic
detections to modes with $\ell\leq 3$ (e.g., 2010aste.book…..A, ;
1977AcA….27..203D, ; 2015MNRAS.452.2127S, ; 2017ApJ…835..172L, ). Although all
acoustic oscillation modes have upper turning points near to the stellar
surface, only the high degree modes have lower turning points in the outer
parts of the star (see Figure 4.4 of 2017ApJ…851…80B, ). Thus, unlike the Sun
(e.g., 2009ApJ…699.1403B, ), only the near-core region ($\sim 0.05-0.35R$ away
from the center) can be probed with asteroseismic inversions.

The structure inversion problem is complicated by the fact that frequencies of
stellar oscillations do not depend exclusively on the sound speed; the effects
of a second variable, such as the density or adiabatic compressibility, must
be considered as well (e.g., 2010aste.book…..A, , Section 3.3.3.2).
Unfortunately, asteroseismic mode sets are too limited to resolve (or
suppress) the effects of a second variable (2017asda.book…..B, , Chapter 10).
A path forward was realized very early (1993ASPC…40..541G, ). By assuming an
equation of state, it is possible to explain oscillation frequencies in terms
of the abundance of helium and any function of pressure and density, such as
the squared isothermal sound speed $u=P/\rho$. The reason this is helpful is
because the helium abundance only substantially affects the oscillation
frequencies in ionization zones, which are located very near to the stellar
surface where asteroseismic inversions are not sensitive anyway. Hence,
asteroseismic data probes the $u$ profile of the stellar core.

A further difficulty concerns the fundamental parameters of the star. The
solar fundamental parameters, and particularly the solar mass and radius, are
precise, accurate, and known independently of helioseismology. For stars,
these parameters must be estimated, generally also from the same asteroseismic
data with which one wishes to perform the inverse analysis. The relative
uncertainties of asteroseismic mass and radius estimates are about 4% and 2%,
respectively (e.g., 2019A&A…622A.130B, ). Thus, the model we choose as
reference may not have the same mass and radius as the star. This has two
consequences. The first is in the mode frequencies themselves: the frequencies
scale with the stellar root mean density, and so a difference there results in
a constant relative difference between the frequencies of the star and of the
stellar model. Secondly, a difference in mass or radius affects the $u$
profile itself. From a dimensional analysis of the equations of stellar
structure, one finds ${P\propto M^{2}/R^{4}}$ and ${\rho\propto M/R^{3}}$, and
hence ${u\propto M/R}$. Thus, asteroseismic structure inversions are actually
sensitive to the dimensionless structure $u^{\prime}=uR/M$
(2003Ap&SS.284..153B, , Figures 9 and 10). Hereinafter, $u^{\prime}$ will be
referred to simply as the sound speed.

A final complication is that theoretical and observed mode frequencies are
well-known to systematically differ due to deficiencies in modeling the near-
surface layers of stars (e.g., 2017EPJWC.16002001B, ). Generally, these
differences are merely fit and subtracted. Though clearly not the best
solution, it does not otherwise present a major hurdle here.

Once a suitable reference model is selected, one can perturb the equations of
adiabatic stellar oscillations and linearize them around the reference model
(e.g., 2003Ap&SS.284..153B, ; 2010aste.book…..A, ; 1991sia..book..519G, ;
2011LNP…832….3K, ; 2016LRSP…13….2B, ; 2017asda.book…..B, ;
buldgen2017development, ; 2018PhDT…….125B, ; 2018ASSP…49…75R, ). From this,
one obtains the central equations of the asteroseismic structure inversion—a
set of equations, one for each observed oscillation mode, that express how the
mode frequencies depend on the stellar structure and composition:

$\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color@gray@fill{.25}\underbrace{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{\delta\nu_{i}}{\nu_{i}}}_{\mathrlap{\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color<EMAIL_ADDRESS>in oscillation
frequency}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=\int\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color@gray@fill{.25}\overbrace{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}K_{i}^{(u^{\prime},Y)}\,\frac{\delta
u^{\prime}}{u^{\prime}}+K_{i}^{(Y,u^{\prime})}\,\delta
Y}^{\mathclap{\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color<EMAIL_ADDRESS>in structure \&
composition}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\,\text{d}r+\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color@gray@fill{.25}\underbrace{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}F_{\text{surf}}(\nu_{i})}_{\mathclap{\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color<EMAIL_ADDRESS>term}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color@gray@fill{.25}\overbrace{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\epsilon_{i},}^{\mathclap{\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color<EMAIL_ADDRESS>errors}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\qquad
i=1\ldots N.$ (3)

Here the index $i$ refers to a mode whose frequency $\nu_{i}$ has been
measured. The difference in any quantity between the star and the reference
model is denoted by $\delta$. The kernels $K_{i}$ for a particular mode are
functions derived from the perturbation analysis that relate $u^{\prime}$ and
the fractional helium abundance $Y$ to that mode’s frequency (for examples,
see 2018PhDT…….125B, , Figures 1.21 and 1.22). The surface term
$F_{\text{surf}}$ is some function of frequency (e.g., 2014A&A…568A.123B, ).
The unknown measurement error, given by $\epsilon_{i}$, is assumed to be drawn
from normal distribution with zero mean and standard deviation $\sigma_{i}$.

The principal assumption of a structure inversion is that the mode kernels of
the reference model serve as sufficient approximations to the actual mode
kernels of the star. As mode kernels do not differ much between even rather
different models (2000ApJ…529.1084B, ), the inversion results can be
considered independent of evolutionary considerations, despite usually using a
model calculated using stellar evolution theory as reference.

The problem now at hand is to determine the $\delta u^{\prime}/u^{\prime}$
(or, equivalently, $u^{\prime}$) profile from the data using these $N$
equations. This problem is ill-posed for two reasons. The first is uniqueness:
infinitely many sound-speed profiles fit the data equally well (e.g.,
1991sia..book..519G, ). These include nonphysical solutions, such as those
with negative sound speeds. The second is stability: as the modes trace
similar paths through the star, the kernels are nearly linearly dependent, and
so a small perturbation to the mode frequencies can lead to a disproportionate
change in the result (recall Equations 1 and 2). Thus the problem must be
changed. A way forward was discovered by geologists Backus and Gilbert
(1967GeoJ…13..247B, ; 1968geoj…16..169b, ): the method of _optimally localized
averages_ (OLA).

###  4.1 The method of optimally localized averages

OLA is a method that can be used to measure the sound speed at various
locations within a star. The method works by combining the mode kernels
belonging to the observed modes into a new kernel, called the _averaging
kernel_ , which by design is only sensitive to one region of the star, called
the _target radius_. If the creation of an averaging kernel is successful, it
can be used as an instrument to measure the sound speed at the target radius.
However, the creation of an averaging kernel must be balanced against the
amplification of the data errors and the contamination of the effects of a
differing helium abundance. It is important to note that the OLA method does
not involve fitting the data; in fact, the construction of the averaging
kernel does not make use of the oscillation frequencies at all; it uses only
their uncertainties.

The averaging kernel $\mathscr{K}$ is formed by taking a linear combination
$\mathbf{c}$ of the mode kernels:
$\mathscr{K}=\sum_{i}^{N}c_{i}K_{i}^{(u^{\prime},Y)}$. If this linear
combination is chosen such that $\mathscr{K}$ integrates to unity and only has
amplitude around the target radius, then $\int\mathscr{K}\delta
u^{\prime}/u^{\prime}\,\text{d}r$ provides a localized average, denoted
$\langle\delta u^{\prime}/u^{\prime}\rangle$, of the sound speed at the target
radius. Now consider a sum over all the equations in Equation 3:

$\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color@gray@fill{.25}\underbrace{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left\langle\frac{\delta
u^{\prime}}{u^{\prime}}\right\rangle}_{\mathrlap{\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color<EMAIL_ADDRESS>average}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\simeq\sum_{i}^{N}\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color@gray@fill{.25}\overbrace{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c_{i}\frac{\delta\nu_{i}}{\nu_{i}}}^{\mathclap{\text{observed
data}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=\int\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color@gray@fill{.25}\underbrace{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathscr{K}\,\frac{\delta
u^{\prime}}{u^{\prime}}}_{\mathclap{\text{averaging
term}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\,\text{d}r+\int\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color@gray@fill{.25}\overbrace{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathscr{C}\,\delta
Y}^{\mathclap{\text{cross
term}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\,\text{d}r+\sum_{i}^{N}\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color@gray@fill{.25}\underbrace{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c_{i}\,F_{\text{surf}}(\nu_{i})}_{\mathclap{\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color<EMAIL_ADDRESS>effects}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\sum_{i}^{N}\color[rgb]{.25,.25,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.25,.25,.25}\pgfsys@color@gray@stroke{.25}\pgfsys@color@gray@fill{.25}\overbrace{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c_{i}\,\epsilon_{i}.}^{\mathclap{\text{errors}}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
(4)

The same linear combination that forms the averaging kernel is applied to all
terms, such as the cross-term kernel
$\mathscr{C}=\sum_{i}c_{i}K_{i}^{(Y,u^{\prime})}$. Thus, $\mathbf{c}$ must be
chosen in such a way that suppresses these other effects.

A common way of solving the OLA problem is known as Subtractive OLA (SOLA,
1994a&a…281..231p, ). This method defines a _target kernel_ , which is
typically a Gaussian function peaked at the target radius. Free parameters
control the width of the target kernel, the suppression of the cross-term
kernel, and the amplification of uncertainty (e.g., 1999MNRAS.309…35R, ). SOLA
then finds the $\mathbf{c}$ that makes the averaging kernel resemble the
target kernel, insofar as it is possible (e.g., 2018PhDT…….125B, ). Examples
of averaging kernels constructed using SOLA with _Kepler_ data
(2017ApJ…835..172L, ) are given in Figure 1. The final averaging kernel is
reasonably well-localized and suffices to estimate $\delta
u^{\prime}/u^{\prime}$ at the target radius.

Figure 1: Averaging kernels at a target radius of $0.1R$ constructed using
four differently sized mode sets from the solar-like oscillator 16 Cyg A
(2017ApJ…851…80B, ). The target kernel is indicated by the gray line.

##  5 Summary of current results and future outlook

Prior to the widespread detection of oscillations in solar-like stars by space
missions such as _Kepler_ , a number of early works investigated the prospect
of asteroseismic inversions using faux data generated from stellar models
(1993ASPC…40..541G, ; 1998mons.proc…33G, ; 2001ESASP.464..407B, ;
2001ESASP.464..411B, ; 2002ESASP.485..337R, ; 2002ESASP.485..249B, ;
2002ESASP.485…95T, ; 2002ESASP.485…75R, ). These works are summarized in the
first review on asteroseismic inversions (2003Ap&SS.284..153B, ). Generally,
these early attempts were optimistic about the number of mode frequencies that
would be observed and the uncertainties on those observations. Most
critically, the assumed uncertainties on the mass and radius of the target
star were generally too small.

To date, asteroseismic structure inversions have been applied to three actual
stars: the solar analogs 16 Cyg A and B (2017ApJ…851…80B, ) and a star with a
small convective core, KIC 6225718 (2019ApJ…885..143B, ). In the former case,
the dimensional $u$ profile was sought by inverting many reference models
spanning the uncertainties in mass and radius. A procedure called inversions-
for-agreement was devised, which had the benefit of using the data to select
the parameters of the inversion.

In each of the three cases, there was tension between the predicted and
observed internal stellar structures. The model of the close solar analog 16
Cyg B performed the best, while the star with a convective core—the least
solar-like star—showed the greatest differences. For this star, alternate
models employing differing physics in the evolution of the star were
calculated. The input physics of convective core overshoot and atomic
diffusion were tested against the asteroseismic observations. However, no set
of input physics was found that could explain the observed structure.

The field of asteroseismic inversions is still in its infancy with many
interesting results yet to come. Several more stars from legacy _Kepler_ data
are still amenable to inverse analysis; this is work in preparation from the
present authors. By analyzing a large number of stars, it will be possible to
rigorously test theories of stellar evolution across a variety of non-solar
regimes and pinpoint the areas of stellar modeling that are currently
deficient. Future space missions will furthermore yield orders-of-magnitude
more detections of solar-like oscillations, at which point inverse analyses
will likely need to be applied at a large scale rather than star-by-star.

As a final note, the stars discussed here are all firmly on the main sequence.
Structure inversions of evolved solar-like stars, whose non-radial acoustic
modes couple with gravity modes in the core and give insights into the very
deep interiors of stars, are sure to be a promising source of discovery.

###### Acknowledgements.

These proceedings are dedicated to the memory of Michael J. Thompson. Funding
for the Stellar Astrophysics Centre is provided by The Danish National
Research Foundation (Grant agreement no.: DNRF106). S.H. acknowledges funding
from the European Research Council under the European Community’s Seventh
Framework Programme (FP7/2007-2013) / ERC grant agreement no 338251
(StellarAges). S.B. acknowledges partial support from NSF grant AST-1514676
and NASA grant NNX13AE70G.

## References

  * (1) S. Basu, 284(1), 153 (2003)
  * (2) J. Christensen-Dalsgaard, Astronomische Nachrichten 333(10), 914 (2012).  DOI 10.1002/asna.201211836
  * (3) C. Aerts, J. Christensen-Dalsgaard, and D. W. Kurtz, _Asteroseismology_ (Springer, 2010)
  * (4) S. Basu and W. J. Chaplin, _Asteroseismic Data Analysis: Foundations and Techniques_ (Princeton University Press, 2017)
  * (5) J. Christensen-Dalsgaard, 220(2), 137 (2004).  DOI 10.1023/B:SOLA.0000031392.43227.7d
  * (6) M. S. Cunha, C. Aerts, J. Christensen-Dalsgaard, et al., 14(3-4), 217 (2007).  DOI 10.1007/s00159-007-0007-0
  * (7) S. Hekker, Advances in Space Research 52(9), 1581 (2013).  DOI 10.1016/j.asr.2013.08.005
  * (8) W. J. Chaplin and A. Miglio, 51(1), 353 (2013).  DOI 10.1146/annurev-astro-082812-140938
  * (9) T. R. Bedding, _Solar-like oscillations: An observational perspective_ (Cambridge University Press, 2014), p. 60
  * (10) S. Hekker and J. Christensen-Dalsgaard, 25(1), 1 (2017).  DOI 10.1007/s00159-017-0101-x
  * (11) S. Hekker, arXiv e-prints arXiv:1907.10457 (2019)
  * (12) R. A. García and J. Ballot, Living Reviews in Solar Physics 16(1), 4 (2019).  DOI 10.1007/s41116-019-0020-1
  * (13) C. Aerts, arXiv e-prints arXiv:1912.12300 (2019)
  * (14) S. Ulam, in _Proceedings of Symposia in Applied Mathematics_ , vol. 14 (Am. Math. Soc. Vol. 14, Providence, 1962), pp. 215–224
  * (15) J. Hadamard, Princeton University Bulletin 13, 49 (1902)
  * (16) A. Kirsch, _An introduction to the mathematical theory of inverse problems_ (Springer, 2011)
  * (17) F. D. M. Neto and A. J. d. S. Neto, _An introduction to inverse problems with applications_ (Springer, 2012)
  * (18) L. Tenorio, SIAM Review 43(2), 347 (2001)
  * (19) L. Pulone and R. Scaramella, in _Neural Nets WIRN VIETRI-96_ (Springer, 1997), pp. 231–236
  * (20) E. P. Bellinger, G. C. Angelou, S. Hekker, et al., 830, 31 (2016).  DOI 10.3847/0004-637X/830/1/31
  * (21) E. P. Bellinger, G. C. Angelou, S. Hekker, et al., in _European Physical Journal Web of Conferences_ , vol. 160 (2017), p. 05003.  DOI 10.1051/epjconf/201716005003
  * (22) E. P. Bellinger, S. Hekker, G. C. Angelou, et al., 622, A130 (2019).  DOI 10.1051/0004-6361/201834461
  * (23) E. P. Bellinger, S. M. Kanbur, A. Bhardwaj, et al., 491(4), 4752 (2020).  DOI 10.1093/mnras/stz3292
  * (24) G. C. Angelou, E. P. Bellinger, S. Hekker, et al., 839(2), 116 (2017).  DOI 10.3847/1538-4357/aa6a54
  * (25) G. C. Angelou, E. P. Bellinger, S. Hekker, et al., accepted (2020)
  * (26) L. Hendriks and C. Aerts, 131(1004), 108001 (2019).  DOI 10.1088/1538-3873/aaeeec
  * (27) T. M. Brown, J. Christensen-Dalsgaard, B. Weibel-Mihalas, et al., 427, 1013 (1994).  DOI 10.1086/174208
  * (28) D. B. Guenther and K. I. T. Brown, 600(1), 419 (2004).  DOI 10.1086/379644
  * (29) T. S. Metcalfe, O. L. Creevey, and J. Christensen-Dalsgaard, 699(1), 373 (2009).  DOI 10.1088/0004-637X/699/1/373
  * (30) V. Silva Aguirre, G. R. Davies, S. Basu, et al., 452(2), 2127 (2015).  DOI 10.1093/mnras/stv1388
  * (31) V. Silva Aguirre, M. N. Lund, H. M. Antia, et al., 835(2), 173 (2017).  DOI 10.3847/1538-4357/835/2/173
  * (32) B. M. Rendle, G. Buldgen, A. Miglio, et al., 484(1), 771 (2019).  DOI 10.1093/mnras/stz031
  * (33) E. P. Bellinger and J. Christensen-Dalsgaard, 887(1), L1 (2019).  DOI 10.3847/2041-8213/ab43e7
  * (34) A. C. S. Jørgensen and G. C. Angelou, 490(2), 2890 (2019).  DOI 10.1093/mnras/stz2825
  * (35) C. Aerts, G. Molenberghs, M. Michielsen, et al., 237(1), 15 (2018).  DOI 10.3847/1538-4365/aaccfb
  * (36) E. P. Bellinger, 486(4), 4612 (2019).  DOI 10.1093/mnras/stz714
  * (37) E. P. Bellinger, 492(1), L50 (2020).  DOI 10.1093/mnrasl/slz178
  * (38) E. P. Bellinger, S. Basu, S. Hekker, et al., 851(2), 80 (2017).  DOI 10.3847/1538-4357/aa9848
  * (39) E. P. Bellinger, S. Basu, S. Hekker, et al., 885(2), 143 (2019).  DOI 10.3847/1538-4357/ab4a0d
  * (40) S. Deheuvels, R. A. García, W. J. Chaplin, et al., 756(1), 19 (2012).  DOI 10.1088/0004-637X/756/1/19
  * (41) S. Deheuvels, G. Doğan, M. J. Goupil, et al., 564, A27 (2014).  DOI 10.1051/0004-6361/201322779
  * (42) S. Deheuvels, J. Ballot, P. G. Beck, et al., 580, A96 (2015).  DOI 10.1051/0004-6361/201526449
  * (43) O. Benomar, M. Takata, H. Shibahashi, et al., 452(3), 2654 (2015).  DOI 10.1093/mnras/stv1493
  * (44) P. Eggenberger, J. W. den Hartogh, G. Buldgen, et al., 631, L6 (2019).  DOI 10.1051/0004-6361/201936348
  * (45) F. Ahlborn, E. P. Bellinger, S. Hekker, et al., accepted (2020)
  * (46) O. Benomar, M. Bazot, M. B. Nielsen, et al., Science 361(6408), 1231 (2018).  DOI 10.1126/science.aao6571
  * (47) M. Bazot, M. B. Nielsen, D. Mary, et al., 619, L9 (2018).  DOI 10.1051/0004-6361/201834251
  * (48) M. Bazot, O. Benomar, J. Christensen-Dalsgaard, et al., 623, A125 (2019).  DOI 10.1051/0004-6361/201834594
  * (49) F. P. Pijpers, 297(3), L76 (1998).  DOI 10.1046/j.1365-8711.1998.01801.x
  * (50) D. R. Reese, J. P. Marques, M. J. Goupil, et al., 539, A63 (2012).  DOI 10.1051/0004-6361/201118156
  * (51) G. Buldgen, D. R. Reese, M. A. Dupret, et al., 574, A42 (2015).  DOI 10.1051/0004-6361/201424613
  * (52) G. Buldgen, D. R. Reese, and M. A. Dupret, 583, A62 (2015).  DOI 10.1051/0004-6361/201526390
  * (53) G. Buldgen, S. J. A. J. Salmon, D. R. Reese, et al., 596, A73 (2016).  DOI 10.1051/0004-6361/201628773
  * (54) B. Paxton, L. Bildsten, A. Dotter, et al., 192(1), 3 (2011).  DOI 10.1088/0067-0049/192/1/3
  * (55) J. Christensen-Dalsgaard, 316(1-4), 113 (2008).  DOI 10.1007/s10509-007-9689-z
  * (56) R. H. D. Townsend and S. A. Teitler, 435(4), 3406 (2013).  DOI 10.1093/mnras/stt1533
  * (57) S. Basu, Living Reviews in Solar Physics 13(1), 2 (2016).  DOI 10.1007/s41116-016-0003-4
  * (58) Q.-S. Zhang, Y. Li, and J. Christensen-Dalsgaard, 881(2), 103 (2019).  DOI 10.3847/1538-4357/ab2f77
  * (59) D. O. Gough and M. J. Thompson, _The inversion problem._ (University of Arizona Press, 1991), pp. 519–561
  * (60) G. E. Backus and J. F. Gilbert, Geophysical Journal 13(1-3), 247 (1967).  DOI 10.1111/j.1365-246X.1967.tb02159.x
  * (61) G. Backus and F. Gilbert, Geophysical Journal 16, 169 (1968).  DOI 10.1111/j.1365-246X.1968.tb00216.x
  * (62) W. Dziembowski, 27, 203 (1977)
  * (63) M. N. Lund, V. Silva Aguirre, G. R. Davies, et al., 835(2), 172 (2017).  DOI 10.3847/1538-4357/835/2/172
  * (64) S. Basu, W. J. Chaplin, Y. Elsworth, et al., 699(2), 1403 (2009).  DOI 10.1088/0004-637X/699/2/1403
  * (65) D. O. Gough and A. G. Kosovichev, _Initial asteroseismic inversions_ (ASP Conference Series, 1993), _ASPCS_ , vol. 40, p. 541
  * (66) W. H. Ball, in _European Physical Journal Web of Conferences_ , vol. 160 (2017), p. 02001.  DOI 10.1051/epjconf/201716002001
  * (67) A. G. Kosovichev, _Advances in Global and Local Helioseismology: An Introductory Review_ (Springer-Verlag, 2011), vol. 832, p. 3.  DOI 10.1007/978-3-642-19928-8˙1
  * (68) G. Buldgen, Development of inversion techniques in asteroseismology.  Ph.D. thesis, Université de Liège (2017).  URL http://hdl.handle.net/2268/213523
  * (69) E. P. Bellinger, Inverse Problems in Asteroseismology.  Ph.D. thesis, Max Planck Institute for Solar System Research (2018).  URL http://hdl.handle.net/11858/00-1735-0000-002E-E5BB-A
  * (70) D. R. Reese, in _Asteroseismology and Exoplanets_ , vol. 49, ed. by T. L. Campante, N. C. Santos, and M. J. P. F. G. Monteiro (2018), p. 75.  DOI 10.1007/978-3-319-59315-9˙4
  * (71) W. H. Ball and L. Gizon, 568, A123 (2014).  DOI 10.1051/0004-6361/201424325
  * (72) S. Basu, M. H. Pinsonneault, and J. N. Bahcall, 529(2), 1084 (2000).  DOI 10.1086/308302
  * (73) F. P. Pijpers and M. J. Thompson, 281, 231 (1994)
  * (74) M. C. Rabello-Soares, S. Basu, and J. Christensen-Dalsgaard, 309(1), 35 (1999).  DOI 10.1046/j.1365-8711.1999.02785.x
  * (75) D. O. Gough, in _The First MONS Workshop: Science with a Small Space Telescope_ , ed. by H. Kjeldsen and T. R. Bedding (1998), p. 33
  * (76) S. Basu, J. Christensen-Dalsgaard, M. J. P. F. G. Monteiro, et al., in _SOHO 10/GONG 2000 Workshop_ , vol. 464, ed. by A. Wilson and P. L. Pallé (2001), pp. 407–410
  * (77) G. Berthomieu, T. Toutain, G. Gonczi, et al., in _SOHO 10/GONG 2000 Workshop_ , vol. 464, ed. by A. Wilson and P. L. Pallé (2001), pp. 411–414
  * (78) I. W. Roxburgh and S. V. Vorontsov, in _Stellar Structure and Habitable Planet Finding_ , vol. 485, ed. by B. Battrick, F. Favata, I. W. Roxburgh, et al. (2002), pp. 337–339
  * (79) S. Basu, J. Christensen-Dalsgaard, and M. J. Thompson, in _Stellar Structure and Habitable Planet Finding_ , vol. 485, ed. by B. Battrick, F. Favata, I. W. Roxburgh, et al. (2002), pp. 249–252
  * (80) M. J. Thompson and J. Christensen-Dalsgaard, in _Stellar Structure and Habitable Planet Finding_ , vol. 485, ed. by B. Battrick, F. Favata, I. W. Roxburgh, et al. (2002), pp. 95–101
  * (81) I. W. Roxburgh, in _Stellar Structure and Habitable Planet Finding_ , vol. 485, ed. by B. Battrick, F. Favata, I. W. Roxburgh, et al. (2002), pp. 75–85