## Feature-Fraction Corrections

### Contents

(Updated 7/25/99)

### What is the feature-fraction correction?

Ideally, in measuring effective area or grating efficiency, a monochromatic source is used so that detected counts represent counts at the energy of interest (assuming background counts are removed). In practice the source of X-rays is not monochromatic and so "measured counts" consists of counts due to many energies. Likewise when using a detector with a finite aperture to measure a focussed beam, the measured counts reflect only a portion of the full spatial distribution of counts (i.e., an encircled energy correction is needed.)

By using a spectrum that has lines, detectors with some energy resolution, and analysis schemes with some finesse, it is possible to create an experimental setup with the property that the observed counts from the analysis are to first order most sensitive to the effective area or efficiency over a small range of energies. This range is given by the smaller of the width of the source line or (for a continuum source,say) the energy resolution of the detector.

If the source, optical system, detector, and analysis method can be modeled then it is possible to calculate, through simulation, a correction factor applicable to the measured data:

```                              all counts due to "the feature"
correction factor = --------------------------------------
observed counts output from analysis

```
Note that the analysis method is itself part of the system. Of course there is a chicken-and-egg problem here because the calculated correction factor depends on the very unknowns we're trying to measure (e.g., the energy dependent effective area and efficiency) but this just lets us use wonderful words like "iteration" and "self-consistent" and we keep on analyzing!

Why do I say "the feature" instead of "line"? Because some lines are broad, some lines have very close satellite lines and because the continuum under a line is at the same energy as the line, it can be tricky to specify, operationally, what "the line" is. A feature can be more clearly specified:

```     "the feature" = all photons with an energy within 1% of
the nominal line energy,
i.e., in IDL: where( ABS(energies-Eline) LT 0.01*Eline )
```
This definition has the property that it is spectrally narrow (and for many sources the feature may come from an effectively smaller range of energies) and it is unambiguous: given an incident source spectral model the "feature" photons can be counted without the complexities of continuum subtraction, satellite lines, broad lines, etc.

### MARX-Simulation for EAE Measurements

MARX simulations were carried out for most of the measurements used for zero and and plus/minus first-order eae analysis. Because of the HXDS stage tilt there may be a slight plus/minus order difference in the simultaions.

• eae_wfracs.html eae listing with links at "aperture" for those meaurements that were simulated.

The simulation includes a realistic source spectrum, finite source distance, XRCF grating geometry, and, currently, a "MARX 2.21" HRMA model. The source models are empirical and based on HSI-Grating Source Spectra. The non-flight detectors have been modelled using a flat focal-plane (the HRC-I with Ideal QE offset to the correct focal-plane location) and post-MARX processed to include the detector QE. The Y and Z locations (w.r.t. detector center) and unblurred energies of photons detected in a 12 mm by 12 mm region around the detector center are saved for analysis.

More specific details are available in the software section below under eae_sim.pro and xrcf_sim.pro.

### Description of the MARX-Simulation Plots

After a measurement is simulated, a useful summary .gif is made containing three plots. These plots are available by clicking on the aperture field in the eae_wfracs.html listing or, for some example plots, by clicking on the links in the Table below.

Energy vs. Y Plot
This plot summarizes the spatial-spectral nature of the data and the feature-fraction analysis. The plot's X-axis is the photon's location along XRCF Facility Y, i.e., the nominal dispersion direction for the LETG. The "Photon Energy" is the MARX photon energy (not blurred by the detector.) Solid and dashed lines indicate several regions:
• solid lines - the Feature - The energy region of the spectral feature is marked by the vertical separation of the two (closely spaced) horizontal solid lines.
• solid lines - the Order region - The horizontal extent of these lines is the smaller of the simulation width (+/- 6 mm) or +/- 1/3 of the "order spacing" for the grating in use. The LEG, 1st order example below shows why this is necessary: the feature photons at 3.444+/-0.034 keV extend over the full range from zero to first order. Where do we define "first order" photons? The +/- 1/3 order region seems to reduce the "spillover" from zero to first.
• Number of all feature counts in the order is shown on the right just above the solid feature lines.
• vert dashed lines - the Aperture - The Y extent of the circular aperture is indicated by the vertical dashed lines.
• horz dashed lines - the ROI - The Region of Interest for the PHA analysis is indicated in keV (approximately) by the horizontal dashed lines. For the SSD, the ROI is as shown (based on 5.6 eV/channel). For the FPCs, where the gain was varied, the ROI values in keV are approximations based on how the ROIs were set for analysis. (See the guts of feat_fraction.pro for specifics.)
• Number of ROI counts is shown at the bottom of the dashed box. These are all counts in the ROI energy range AND falling in the circular aperture (radius < diameter/2).
• Feature fraction - this is the ratio of:
```                        number of all feature counts in the order
feature-fraction = -------------------------------------------
number of ROI counts
```
and it explicitly includes any "encircled energy" correction. (Though the correction is referenced to the Order region diameter.)
Feature in Spectrum Plot
This plot is produced to give an idea of just what the energy "feature" is (and for checking that the +/- 1% is sensible - so far looks good!) A histogram of all counts in the Order region is made over a small range centered on the spectral feature. The feature is highlighted with a histogram-type plot, non-feature bins are plotted with a dot only. Note that this plot does not depend on the aperture or the energy ROI, but does include the mirror, grating, and detector QE effects.
Y-Z Plot of Events
This is perhaps the most straight-forward plot: the events are plotted in Y, Z at the detecor aperture plane. The circular aperture is drawn as well. The EE correction is the ratio of all feature counts in the Order region to those within the aperture.
 Table of Sn-La,Snx2 ROI Feature-fraction Examples
Grating
Order
TRW_ID/date/run_id/iteration Feature-fraction =
`fracNom`
Observed
fraction
`ObsFeatFrac`
* EE correction
`fracNom /ObsFeatFrac`
df/f error
NONE D-IXF-3D-22.017/970124/110081/i0 0.4238 <-- click! 0.4179 1.014 13.9 %
MEG, 0 D-HXF-3D-22.019/970124/110084/i1 0.4130 <-- click! 0.4065 1.016 14.6 %
LEG, 1 D-LXF-3D-22.018a/970124/110083/i0 0.4424 <-- click! 0.4250 1.041 13.6 %
MEG, 1 D-HXF-3D-22.019/970124/110084/i0 0.6137 <-- click! 0.6017 1.020 28.1 %
HEG, 1 D-HXF-3D-22.020/970124/110085/i0 0.9109 <-- click! 0.9037 1.008 1.6 %

### Feature-fractions and Error Estimates

The feature-fraction analysis produces the file feature_fractions.rdb, with the following columns (capitalization added for readability):
```eaeIndex       - the zero-based index (line number) of the eae file
fracNom        - the feature fraction at nominal aperture size/location
fracDplus      - the feature fraction for diameter = diameter + 0.025 mm
fracDminus     - the feature fraction for diameter = diameter - 0.025 mm
fracYplus      - the feature fraction for Y_center = Y_center + 0.050 mm
fracYminus     - the feature fraction for Y_center = Y_center - 0.050 mm
fracZplus      - the feature fraction for Y_center = Y_center + 0.050 mm
fracZminus     - the feature fraction for Y_center = Y_center - 0.050 mm
Nfeature       - the number of events in the full feature
ObsFeatFrac    - the  observed feature fraction is the ratio:
number of feature events in the aperture
----------------------------------------
number of ROI events
(EE correction)- The EE correction is not explicitly given in the file but
can be calculated as:
EE correction  =  fracNom / ObsFeatFrac
```
The additional columns beyond fracNom are present to allow some estimates of an error for the value of fracNom.

The value of the feature-fraction depends on the full simulation used to generate it and hence on all aspects of modeling the measurement. The procedure feat_frac_err.pro calculates a "df/f" fractional error from the feature_fractions.rdb file and other internal parameters - see the code for specifics. The error terms currently included are described briefly here:

• Simulation statistics - a finite number of events is used to "calculate" the feature fraction and this gives rise to an error in the fraction value. Nfeature is used in these calculations and MARX simulations can be extended inorder to reduce this error source.
• Correction Systematics - the fraction values largely come from the detailed source spectral modeling based on contemporary HSI images. The ratios of bright lines in these complexes may only be good to, say, 10%. Thus a feature-fraction value of 0.4 may have an error of 10% of 0.6 = 0.06. The resulting fractional error on the 0.4 is then 0.06/0.4 = 15%, hence the numbers in the table above.
• Aperture errors - the additional columns (fracDplus, fracDminus, fracYplus, fracYminus, fracZplus, fracZminus) explore the variation of the feature fraction with aperture diameter and Y,Z position.
These errors are then combined in quatrature to arrive at a final "df/f" error. This df/f error is used to assign systematic errors to the grating efficiencies (the spikey error lines on the plots.)

### Results File from "Feature-fraction" Calculations

feature_fractions.rdb

### Software for "Feature-fraction" Calculation

feat_fraction.pro
feat_frac_err.pro
eae_sim.pro
xrcf_sim.pro
xss_sim.pro