# On the estimation of statistical uncertainties on powder diffraction and small-angle scattering data from two-dimensional X-ray detectors

## Abstract

Optimal methods are explored for obtaining one-dimensional powder pattern intensities from two-dimensional planar detectors with good estimates of their standard deviations. Methods are described to estimate uncertainties when the same image is measured in multiple frames as well as from a single frame. The importance of considering the correlation of diffraction points during the integration and the resampling process of data analysis is shown. It is found that correlations between adjacent pixels in the image can lead to seriously overestimated uncertainties if such correlations are neglected in the integration process. Off-diagonal entries in the variance–covariance (VC) matrix are problematic as virtually all data processing and modeling programs cannot handle the full VC matrix. It is shown that the off-diagonal terms come mainly from the pixel-splitting algorithm used as the default integration algorithm in many popular two-dimensional integration programs, as well as from rebinning and resampling steps later in the processing. When the full VC matrix can be propagated during the data reduction, it is possible to get accurate refined parameters and their uncertainties at the cost of increasing computational complexity. However, as this is not normally possible, the best approximate methods for data processing in order to estimate uncertainties onmore »

- Authors:

- Publication Date:

- Research Org.:
- Argonne National Lab. (ANL), Argonne, IL (United States). Advanced Photon Source (APS)

- Sponsoring Org.:
- DOE - BASIC ENERGY SCIENCES

- OSTI Identifier:
- 1327049

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Applied Crystallography (Online); Journal Volume: 47; Journal Issue: 4

- Country of Publication:
- United States

- Language:
- ENGLISH

### Citation Formats

```
Yang, X., Juhás, P., and Billinge, S. J. L..
```*On the estimation of statistical uncertainties on powder diffraction and small-angle scattering data from two-dimensional X-ray detectors*. United States: N. p., 2014.
Web. doi:10.1107/S1600576714010516.

```
Yang, X., Juhás, P., & Billinge, S. J. L..
```*On the estimation of statistical uncertainties on powder diffraction and small-angle scattering data from two-dimensional X-ray detectors*. United States. doi:10.1107/S1600576714010516.

```
Yang, X., Juhás, P., and Billinge, S. J. L.. Sat .
"On the estimation of statistical uncertainties on powder diffraction and small-angle scattering data from two-dimensional X-ray detectors". United States.
doi:10.1107/S1600576714010516.
```

```
@article{osti_1327049,
```

title = {On the estimation of statistical uncertainties on powder diffraction and small-angle scattering data from two-dimensional X-ray detectors},

author = {Yang, X. and Juhás, P. and Billinge, S. J. L.},

abstractNote = {Optimal methods are explored for obtaining one-dimensional powder pattern intensities from two-dimensional planar detectors with good estimates of their standard deviations. Methods are described to estimate uncertainties when the same image is measured in multiple frames as well as from a single frame. The importance of considering the correlation of diffraction points during the integration and the resampling process of data analysis is shown. It is found that correlations between adjacent pixels in the image can lead to seriously overestimated uncertainties if such correlations are neglected in the integration process. Off-diagonal entries in the variance–covariance (VC) matrix are problematic as virtually all data processing and modeling programs cannot handle the full VC matrix. It is shown that the off-diagonal terms come mainly from the pixel-splitting algorithm used as the default integration algorithm in many popular two-dimensional integration programs, as well as from rebinning and resampling steps later in the processing. When the full VC matrix can be propagated during the data reduction, it is possible to get accurate refined parameters and their uncertainties at the cost of increasing computational complexity. However, as this is not normally possible, the best approximate methods for data processing in order to estimate uncertainties on refined parameters with the greatest accuracy from just the diagonal variance terms in the VC matrix is explored.},

doi = {10.1107/S1600576714010516},

journal = {Journal of Applied Crystallography (Online)},

number = 4,

volume = 47,

place = {United States},

year = {Sat Jul 19 00:00:00 EDT 2014},

month = {Sat Jul 19 00:00:00 EDT 2014}

}