Genetic Forward modeling applied to helioseismic inversion
of the Sun's internal differential rotation functions as
follows.
The data consists of a discrete set of frequency splittings
a*(n,l,s) with associated errors epsilon_nls. The data is
related to the internal rotation profile through the
following integral equation:
Which expresses the fact that the measured frequency splitting
for a pair of prograde/retrograde modes of identical radial degree
*n* and angular degree *l*
is a global measure of the rotation rate encountered by the
mode as it travels through the solar interior.
In solving for the solar internal differential rotation,
an underlying solar structural model is assumed given so
that the integration Kernels *K(a)_nls* are known
quantities.
The solution procedure is a direct transcription of the general
genetic forward modeling algorithm outlined in section 1 above:
It is assumed here that the
rotation rate (Omega) is defined by a discrete set of
parameters, to be solved for. This could be done through
a direct 2-D spatial discretization on a pre-defined mesh, or through some
specific functional relationship.
Note that the structure of the algorithm is not affected by the form of
the relationship between the rotation rate and the adopted goodness-of-fit
measure. The use of a chi-squared above is merely illustrative; any other
statistical estimator can be substituted without altering the
overall algorithm. The goodness-of-fit measure need not even be
differentiable with respect to the defining parameters,
as no gradient information is required by the algorithm.
Note finally that in contrast to formal inversion methods
(such as SOLA or regularized least-squares), the procedure simply
involves the computation
of the integral appearing on the RHS of eq.~(1), as opposed to discretization
and inversion of the RHS.