Detect Sources with the Maximum Likelihood Method

 

The maximum likelihood technique works on the photon event data and   a background image. The spatial distribution of the observed photons above the background is compared with the spatial distribution of the theoretical point spread function and a likelihood that both distributions are matching is calculated. The parameters specifying the source counts, extent, x- and y-position are varied until the likelihood is maximized. The likelihood is defined as (- ln(1 - P)) where P is the probability of existence of the source. A detailed description can be found in [Cruddace et al.1988].

The maximum likelihood detection is started with the command

DETECT/MAXLIK sppar edit

  The parameter file sppar.par will be created and offered to be edited. The parameter file sppar.par can also be created by one of the commands

CREATE/PARFIL EXSP sppar ROSAT,PSPC
CREATE/PARFIL EXSP sppar ROSAT,HRI
CREATE/PARFIL EXSP sppar ROSAT,WFC
CREATE/PARFIL EXSP sppar ROSAT,SURVEY

with default values for the PSPC, HRI, and WFC detector of ROSAT in pointing mode, and for the ROSAT PSPC in Survey mode respectively. Note, that maximum likelihood source detection for WFC data does not yield reliable results yet. In Table 5.7 all parameters showing up in the parameter file sppar.par (example valid for ROSAT PSPC, pointing mode) are listed.

  
Table 5.7: Parameter file with parameters for DETECT/MAXLIK

• MISSION,DETECTOR is the experiment identifier. Allowed values are ROSAT,PSPC, ROSAT,HRI, and ROSAT,WFC.
• OBS_MODE is the observation mode. Allowed values are POINTING or SURVEY.  
• INPUT_DATASET is the name of the photon event table, file type must be PEM as the photon event table is evaluated from the small maps. Use the command MAKE/PEM to convert it from a PET to a PEM.
• AMPLITUDE_LOW is the value of the lower bound for the calibrated amplitude of the photons analysed. The range in calibrated amplitude of the background image BG_IMAGE must agree with the value of AMPLITUDE_LOW.
• AMPLITUDE_HIGH is the value of the upper bound for the calibrated amplitude of the photons analysed. The range in calibrated amplitude of the background image BG_IMAGE must agree with the value of AMPLITUDE_HIGH.
• CUT_RADIUS is the cut-out-radius (in units of the FWHM of the point spread function at the source position, evaluated at calibrated amplitude channel 30) for source photons centered at the source position. The default value for the Pointing case is 2.5 and for the Survey case is 5.0. If a smaller value is chosen, the counts determined by the maximum likelihood algorithm may be underestimated accordingly. A value of 1.0 may be chosen for pointing data in case of a source-crowded region.
• EXCLUDE_SIGMA is the threshold for exclusion of other sources within the cut radius. A ring with radius 0.4-1.0 CUT_RADIUS is divided in 8 segments. If the countrate within a segment is higher than the average countrate by more than sigma this value, it is assumed that this segment contains another source, and this segment and the adjacent two segments are excluded from the ML fit. Defaulted to 5 . Note: If the input position is offset from the position of the source to fit, this can lead to exclusion of parts of the source itself and therefore to wrong fit results. Choose a larger CUT_RADIUS or a higher EXCLUDE_SIGMA to avoid this.
• OFF_AX_MIN is the minimum off-axis-radius (in arcmin) of a source to be selected.
• OFF_AX_MAX is the maximum off-axis-radius (in arcmin) of a source to be selected (not active for Survey data).
• ML_THR is the threshold of maximum likelihood. If the existence likelihood falls above ML_THR then this source will be written to the table with the name specified under the parameter ML_LIST_ACCEP, otherwise if the source is not rejected, it will be written to the table with the name specified under the parameter ML_LIST_ALL.
• START_POSITION is the mode of input of the source positions. Two modes are possible. The first mode, merged_list means to specify source positions by the merged source table created by the command MERGE/SOURCE_TABLES. The second mode, user_list means to specify source positions by a user defined table. The name of that table has to be specified under the parameter USER_LIST.
• MERGED_LIST is the name of the merged source table, to be filled in case START_POSITION is set to merged_list. This table is created by the command MERGE/SOURCE_TABLES.
• USER_LIST is the name of the user defined source table, to be filled in case START_POSITION is set to user_list.
• FIT_POSITION determines whether the source positions specified in tables MERGED_LIST or USER_LIST should be kept fixed or free. fixed means the source positions will not be varied during the likelihood fit, free means they are allowed to be varied. In case the parameter START_POSITION is set to merged_list the parameter FIT_POSITION should always be set to free.
• BG_IMAGE_1 is the name of the background image. It is created by one of the two commands CREATE/BG_IMAGE or CREATE/SMOOTH_BG. The background image must be selected in the same calibrated amplitude range as specified in the parameters AMPLITUDE_LOW and AMPLITUDE_HIGH. The amplitude information is retrieved from the descriptor values AMP_SEL(1) and AMP_SEL(2).
• ML_LIST_ACCEP is the name of the source table with all sources accepted in the maximum likelihood detection and with an existence likelihood equal to or above the likelihood threshold ML_THR.
• ML_LIST_ALL is the name of the source table with all sources accepted in the maximum likelihood detection with any positive existence likelihood.

There is a set of extended parameters which normally does not appear in the parameter file. They are not needed in standard processing. If, however, one of these parameters needs to be set, they can be accessed via WRITE/PARFIL or CREATE/PARFIL with the extended option. Some of them are listed below.

• MIN_ML_EXIST is the lower bound in likelihood of existence (retrieved from the column :SIGNA of the merged source table) of a source to be selected.
• MAX_ML_EXIST is the upper bound in likelihood of existence (retrieved from the column :SIGNA of the merged source table) of a source to be selected.
• NUM_IMAGES is the number of energy bands analysed by the maximum likelihood method. Presently only one energy band is allowed.
• BG_SCALE is the scaling factor applied to the data of the background image. This factor should always be 1.0 in spatial analysis with EXSAS version 96JAN or later. For newer background images, this factor is contained in descriptor BG_SCALE. If this descriptor is present, the value specified in the parameter file will be overwritten.
• MAX_ITER is the maximum number of iterations in the maximum likelihood fit algorithm. The default value is 100 and should not be modified.
• ACCURACY_ITER is the iteration accuracy. This parameter determines the convergence of the fit. The default value of 0.00001 should not be modified.

In the standard maximum likelihood detection the source positions passed to the maximum likelihood task are the positions from the merged source table; there is a second way to define pre-selected positions. This is done by a so-called user defined source table (cf. Table 5.8).

  
Table 5.8: Extract of sppar.par for user specified input source table

Note:The coordinates in the merged source table are given in image pixels, the coordinates in the user defined source table in equatorial coordinates right ascension (degrees) and declination (degrees) for equinox 2000.

In the parameter file for the maximum likelihood source detection a minimum and a maximum off-axis-angle OFF_AX_MIN and OFF_AX_MAX (in arc minutes) is specified. Only sources lying in this range will be analysed. A cut-radius in units of the FWHM of the experiment CUT_RADIUS is given. Photons lying within that cut-radius centered at the source position will be checked by the maximum likelihood technique. The parameters of the selected sources will be written to the table file ML_LIST_ACCEP in case the source is accepted and the likelihood of existence is equal to or falls above the value ML_THR and they will be written to the table file ML_LIST_ALL in case the source is accepted for any positive value of the likelihood of existence. The background counts per image pixel at the source position are determined from the background image BG_IMAGE_1.

The information written to the source tables ML_LIST_ACCEP and ML_LIST_ALL is described in Table 5.29. The best fit maximum likelihood positions are given in the columns :X_IMA and :Y_IMA in units of image pixels and can be converted to sky pixel and astronomical coordinates by the command TRANSFORM/COORDINATES applied to the source table. Attention, the position specified in the columns :X_POS and :Y_POS are the input positions and are in units of image pixels or sky pixels depending whether the parameter START_POSITION was set to merged_list or user_list, respectively.