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Astron. Astrophys. 364, 377-390 (2000)

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4. Multiple-data Richardson-Lucy deprojection

Now 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

[EQUATION]

[EQUATION]

[EQUATION]

resulting in iterated input data [FORMULA], [FORMULA], and [FORMULA]. In the second step we define the three integrals

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA], [FORMULA] and [FORMULA] are defined as

[EQUATION]

Here [FORMULA], [FORMULA], and [FORMULA] denote the observed input distributions.

In the second step above the three integrals - for the lensing potential [FORMULA] the integral (27a), for the X-ray surface brightness [FORMULA] integral (27b), and for the SZ temperature decrement [FORMULA] 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

[EQUATION]

[FORMULA], [FORMULA], and [FORMULA] are weighting factors with [FORMULA], 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 [FORMULA] 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 [FORMULA] for the X-ray integral [FORMULA] arises due to the fact that the X-ray surface brightness depends on the electron density [FORMULA] and not [FORMULA].

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 deprojection

The 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 [FORMULA] according to Eqs. (26a) - (26c), to obtain the iterated [FORMULA], [FORMULA], and [FORMULA]. Now the results of this first integration can be used to compute the integrals (27a) - (27c) that constitute the second integration. The last step of the iteration cycle is to compute the new estimate for the gravitational potential [FORMULA] from Eq. (29). This new, improved estimate of the gravitational potential is used to reenter the iteration cycle. The whole algorithm is stopped after a few iterations; usually 7-8 iterations lead to satisfactory results for the potential when compared to the original potential as obtained from cluster simulations.

Both, the input data [FORMULA], [FORMULA], [FORMULA] and the gravitational potential [FORMULA] are represented as discretized data on a rectangular grid. The observed data is assumed to cover a finite data field [FORMULA], and the reconstructed potential thus covers a corresponding field [FORMULA], in cluster coordinates [FORMULA].

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 [FORMULA] 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 z-axis.

The second integration (27a)-(27c) yielding [FORMULA], [FORMULA], and [FORMULA] 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:

[EQUATION]

From Eq. 30 we see that for every given point in the [FORMULA] plane the fraction [FORMULA] has to be integrated over an ellipse

[EQUATION]

that is shifted along the y axis by [FORMULA] and contracted in the y-direction by a factor of [FORMULA]. Due to the finite range of values for x and y we are faced with the problem that some parts of the ellipse, and thus of our integration path, may not be covered by the input data. This is most likely the case if both R and [FORMULA] take medium to large values. Even worse, if either R or Z becomes very large, then the full ellipse will be outside the input range for x and y.

Therefore one either has to exclude all ellipses with such large [FORMULA]-coordinates or one has to define appropriate "boundary" conditions, i.e. values for the ratio [FORMULA], [FORMULA] and [FORMULA] have to be specified for points [FORMULA] outside of the data field [FORMULA]. We tested in some detail the following three possible choices:

  1. Assume a smooth expansion of the values of the ratio observed / iterated data sets, e.g. for [FORMULA] and [FORMULA] [FORMULA], [FORMULA], and [FORMULA].

  2. Assume a perfect reconstruction, i.e. the ratio observed / iterated data sets equals unity: [FORMULA] for [FORMULA] or [FORMULA] and [FORMULA] or [FORMULA].

  3. 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 R that have nothing to do with the true potential. Furthermore, condition No.3 is computationally relatively expensive. Thus we have decided to exclusively use boundary condition number No.1 in the remainder of this chapter.

[FIGURE] Fig. 2. Gravitational potential of a cluster from a gas-dynamical simulation reconstructed from lensing data using different boundary conditions in the second integration of the axissymmetric RL algorithm. The enumeration of the boundary conditions is as in Sect. 4.1.

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 [FORMULA] and [FORMULA] 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.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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