## 4. Multiple-data Richardson-Lucy deprojectionNow we propose the multiple-data Richardson-Lucy deprojection (MDRL) algorithm. The combination of different data sets can be achieved in three separate steps. The first step of the algorithm is to compute the three LOS integrals resulting in iterated input data , , and . In the second step we define the three integrals where , and are defined as Here , , and denote the observed input distributions. In the second step above the three integrals - for the lensing potential the integral (27a), for the X-ray surface brightness integral (27b), and for the SZ temperature decrement the integral (27c) - correspond to the integral (2b) of Richardson-Lucy's iterative inversion algorithm, while the Eqs. (26a) - (26c) correspond to the integral (2a) of the inversion algorithm. In order for the multiple-data Richardson-Lucy algorithm to work the results of the integrations have to be combined after every iteration step as , , and are weighting factors with , which can be used to determine the relative weight put on the respective input data. By means of these weighting factors, it is possible to recover the case of having just one set of measured data. As already mentioned in Sect. (3) for the X-ray case and the SZ-case the dependence on the gravitational potential is exponential. Thus it is necessary for these two cases to take the logarithm of the integrals (27b) and (27c) in order to reconstruct the gravitational potential. The factor for the X-ray integral arises due to the fact that the X-ray surface brightness depends on the electron density and not . Eq. (29) completes the multiple-data Richardson-Lucy deprojection algorithm, thus constituting the third step. In terms of the iterative inversion algorithm this step corresponds to the evaluation of the full Eq. (2b). ## 4.1. Implementation of the multiple-data Richardson-Lucy deprojectionThe implementation of the program follows closely the formulation
of the MDRL-algorithm given in the last section. For the algorithm,
two different types of integrals need to be evaluated. After reading
the observed data and assuming an initial guess for the gravitational
potential the iteration cycle is entered. The first step in the
iteration cycle is to integrate the gravitational potential
according to Eqs. (26a) - (26c),
to obtain the iterated ,
, and
. Now the results of this
Both, the input data ,
,
and the gravitational potential are
represented as discretized data on a rectangular grid. The observed
data is assumed to cover a For the first integrals of the iterative scheme (26a)-(26c) we implemented a standard LOS integral in the formulation of Eq. (9). As expected, preliminary tests with the equivalent formulation A.5 via the integral kernel as in Binney et al. (1990), did not lead to satisfactory results. We evaluate the LOS integral by first rotating the discretized
gravitational potential before computing an integral along the
The second integration (27a)-(27c) yielding , , and is performed on an ellipse as in formula A.15, that was derived for the axisymetric case in Sec 2.2. For easier reference we explicitly write down the corresponding integral for the lensing case: From Eq. 30 we see that for every given point in the plane the fraction has to be integrated over an ellipse that is shifted along the Therefore one either has to exclude all ellipses with such large -coordinates or one has to define appropriate "boundary" conditions, i.e. values for the ratio , and have to be specified for points outside of the data field . We tested in some detail the following three possible choices: -
Assume a smooth expansion of the values of the ratio *observed / iterated*data sets, e.g. for and , , and . -
Assume a perfect reconstruction, i.e. the ratio *observed / iterated*data sets equals unity: for or and or . -
Assume rotational symmetry for the values of the ratios. This can be achieved by computing the distance to the center of the data field and averaging over all values for the corresponding circle which are inside the data field. However this method is computationally relatively expensive, and the results obtained are poor.
Experimenting with these three different boundary conditions and
various sizes of the input field, we found that this boundary problem
has no significant influence on the quality of the reconstruction
achieved for the central part of the potential. A typical example of
this problem is given in Fig. 2, where a lensing based
reconstruction of the gravitational potential from a gas-dynamical
simulation (see next section) is shown for the three different
boundary conditions listed above. It is obvious from Fig. 2 that
the conditions No.2 and 3 introduce unphysical numerical artefacts for
large values of
Taking the logarithm for the X-ray and the SZ-case in Eq. (29), which is due to the exponential dependence in the two cases on the gravitational potential, is numerically a very unstable operation. Small deviations in computing the ratio and are magnified by taking the logarithm, thus preventing convergence of the algorithm. In order to avoid this problem we employed a cut-off criterion for the ratio. This is done at the expense of a slower convergence, but as the algorithm converges very quickly and is computationally inexpensive this does not pose a serious problem. © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |