*Astron. Astrophys. 364, 377-390 (2000)*
## 6. -based determination of inclination angle *i*
Clearly in addition to a quantitative assessment of the quality of
the reconstruction a systematic procedure for determining the input
parameters for the MDRL algorithm is highly desirable.
As explained in Sect. 2.2 the MDRL-algorithm in its current
formulation needs to be provided with the inclination angle *i*
as input parameter and a choice for the weight factors
,
and for the three different
contributions has to be made. Once all these parameters have been
specified, the MDRL algorithm yields a reconstructed estimate
for the potential, which in turn
determines the best estimates ,
, and
for the observed input distributions
(lensing potential), (X-ray surface
brightness), and
(Sunyaev-Zel'dovich temperature decrement).
The idea is now to minimize an appropriate
function, e.g.
to obtain the best parameters within the framework of the model
assumptions underlying the presented MDRL algorithm. Here
is the number of grid points, with
the *i*-th grid point given by a vector
in the *xy*-plane onto which
the observables are projected. The desired quantitative assessment of
the qualitity of the reconstruction could then be given in terms of a
subsequent goodness-of-fit (GoF) evaluation, which would tell us how
likely the reconstruction within this model is for the best set of
parameters found before.
This idea is illustrated in the following using the cluster example
from Sect. 5. We recall that the true 3-dimensional gravitational
cluster potential is observed under
an angle of , and assume that we do
not know the proper angle *i* in advance. For setting up the
statistics we compute a series of
single data and multiple data reconstructions with various values for
the angles *i* and the weight factors. We then use Eq. 32 to
evaluate the corresponding function,
and obtain a value for every set of
input parameters.
The minimum over all the values
then indicates the best choice for the inclination angle *i* and
the weight factors. According to our qualitative analysis from
Sect. 5 we would expect that the inclination angle *i*
indeed should be very close to the true value.
Work on the quantitative estimation of the inclination angle and
the weight factors ,
and
using the approach sketched above
is currently in progress. Preliminary results indicate that with the
simulation data available in this study the main contribution to the
for a fixed inclination angle
*i* does not result from the mismatch between reconstructed
potential and true potential, but from discretization effects due to
the finite grid. The question, if the above sketched determination of
the inclination angle *i* and the weight factors
,
and is feasible for realistic
observational data sets of finite sizes thus requires further
investigation.
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001
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