Measurement of uncertainty in orientation distribution function

calculations

A. Creuziger,a,⇑ K. Syedb and T. Gna ̈upel-Herolda

a

National Institute of Standards and Technology, 100 Bureau Dr., Gaithersburg, MD 20899, USA b

University of Maryland, College Park, MD 20742, USA

Received 17 September 2013; accepted 18 October 2013

Available online 26 October 2013

A technique to quantify the uncertainty in an estimated orientation distribution function (ODF) by propagating the uncertainty

from measured pole figures is demonstrated in this paper using Monte Carlo techniques and the Matlab toolbox for quantitative

texture analysis. Unlike the uncertainty in the measured pole figures, where small values of intensity correlate to high uncertainty,

uncertainty as a function of orientation in an ODF is not directly related to intensity.

Published by Elsevier Ltd. on behalf of Acta Materialia Inc.

Keywords: Neutron diffraction; Texture; Monte Carlo simulation; Ferritic steels

The field of texture analysis has been described as

“quantitative” for over 30 years [1–4]. Techniques to

measure crystallographic texture have been widely used

to measure preferred crystallographic orientations. The

crystallographic information is often described in pole

figures or orientation distribution functions (ODFs).

Pole figures display the concentration of orientations of

a fixed (hkl) crystal axis as a function of spatial orientation. Pole figure descriptions are incomplete in that the

rotation about the fixed axis is not shown. ODFs present

a way to show the complete orientation data.

Two major classifications of techniques to estimate

ODFs exist: direct and indirect. Direct methods for estimating an ODF use a series of measurements of the

complete orientation information spatially (such as electron backscatter diffraction). Indirect methods typically

rely on comparisons between pole figures provided as an

input to a solver and pole figures generated from an estimated ODF. Some examples of indirect methods are

harmonic1 [5], WIMV [6], ADC [7] and the Matlab toolbox for quantitative texture analysis (MTEX) [8].

Estimates of the precision and accuracy in crystallographic texture measurements such as pole figures and

ODFs have not been widely used. Uncertainty in pole

figure measurements have been previously investigated

by Nikolayev et al. [9] and Lychagina et al. [10]. These

are based on time-of-flight (TOF) neutron-diffraction

measurements to estimate the effect of counting time

on the pole-figure quality. The article by Mu ̈cklich and

Klimanek [11] summarizes some of the sources of uncertainty in quantitative texture analysis, particularly for

X-ray-based diffraction measurements.

Prior attempts to assess uncertainty in ODF measurements have focused on errors in the solver techniques.

For indirect ODF estimation methods, the effect of a

finite number of pole figures [4], truncation [4,8] and

“ghost correction” [12] are widely known sources of

uncertainty. Similarly, bias errors and disagreement between solvers are widely known issues [1]. One approach

to assessing the difference between models is to check the

solver using standard orientation distribution functions

[13,14] or known texture functions [15]. For direct solvers, other problems, such as the effect of grain statistics,

the non-equiaxed nature of grain shape and spatial correlations between grains, have been considered theoretically [16], but more experimental work is needed.

As described, a variety of sources of uncertainty exist

in estimated ODFs: random errors from the pole figure

measurements, truncation errors from the finite number

of pole figures or the finite range of accessible reflections

(X-ray), and differences between ODF solver techniques.

1359-6462/$ - see front matter Published by Elsevier Ltd. on behalf of Acta Materialia Inc.

http://dx.doi.org/10.1016/j.scriptamat.2013.10.017

⇑ Corresponding author. Tel.: +1 3019756015; e-mail: adam.

creuziger@nist.gov

1Certain commercial equipment, instruments or materials are identified in this paper in order to specify the experimental procedure

adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and

Technology, nor is it intended to imply that the materials or

equipment identified are necessarily the best available for the

purpose.

Available online at www.sciencedirect.com

ScienceDirect

Scripta Materialia 72–73 (2014) 55–58

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The focus of this work is assessing the first source; specifically, how random errors in the intensity data of pole

figures from counting statistics propagate into the orientation distribution function.

Estimates of the uncertainty of an ODF are crucial for

many reasons. ODFs are commonly used as a qualitative

comparison metric. An assessment of uncertainty is a

necessary component to determine if the difference between two (or more) estimated ODFs is significant.

Uncertainty for extracted values such as volume fractions are also important as the volume fraction of a given

orientation is often used when comparing two ODFs.

Monte Carlo techniques are used to propagate the

uncertainty in computational models, and rely on taking

pseudo-random draws from a probability distribution

[17,18]. When employed in error analysis or uncertainty

analysis, the input variables are perturbed in a manner

consistent with their uncertainty and the perturbed input variables are used to create a new output value.

The input variables are repeatedly perturbed in this

manner to create an ensemble of output values. The

ensemble of output values can then be used to summarize the variation in the output. Morawiec has explored

this approach for ODFs where model pole figures are

perturbed in a similar manner [15]. However, in Ref.

[15] only the uncertainty in the overall texture function

was considered; the effect on individual orientations in

the ODF was not.

Complete pole figures for an undeformed SAE 1008

steel sample were collected using the BT-8 residual stress

diffractometer at NIST [19]. Neutron diffraction was

chosen so that complete pole figures could be measured,

nullifying the effect of incomplete pole figures as a source

of uncertainty. SAE 1008 steel is a ferritic steel with a

body-centered cubic crystal structure. The (2 0 0), (1 1 0)

and (2 1 1) reflection planes (h) were measured using a

hexagonal mesh approximately equivalent to a 5angular spacing [20]. Each point in the pole figure (a,b) is

determined by stereographic projection of the reflection

vector (r) onto a two-dimensional surface. The number

of diffracted neutrons (counts) at each point (a; b) on

the pole figure is referred to as the peak counts Np, determined by subtracting the integrated intensity of the background Nb from that of the total number of counts Nt:

Np 1⁄4 Nt Nb ð1Þ

The pole figure data, P(h,r), is therefore a plot of the

number of peak counts as a function of the reflection

plane and the orientation of the sample, calculated from

the total and background counts:

Pðh;rÞ 1⁄4 Ptðh;rÞ Pbðh;rÞ ð2Þ

The numbers of total and background counts are

both expected to follow a Poisson distribution. The relative standard deviation of the number of peak counts is

then:

rp 1⁄4

ffiffiffiffi

N p t þ ffiffiffiffiN p b

Nt Nb

ð3Þ

The program PeakFit (PF) [21] was used to analyze

the experimental pole figure data. At each point on the

pole figure, the diffraction peak was fitted with a

Gaussian peak and a constant background. The total

number of counts and the background number of counts

at each point in the pole figure measurement were calculated and exported to the crystallographic texture analysis package MTEX.

MTEX was used to estimate the ODFs. Using the

nomenclature of MTEX, a set of Euler angles was defined as an “orientation” (g). The orientation was then

used to determine the “center” of a kernel function.

For the analysis presented here, the de la Vallee–Poussin

kernel and 5half width were used. The choice of half

width resulted in 1232 radially symmetric kernel functions distributed to fill Euler space.

MTEX uses a finite mixture of a uniform texture distribution (gu) and radially symmetric kernel functions

(gi) to approximate the ODF f(g). The radially symmetric kernel functions are centered on a grid in the ODF

space (Bunge Euler angles: u1, U, u2) in defined locations determined by the half-width of the kernel function. In order to indicate the relative proportion of

each of the radially symmetric kernel functions, each

radially symmetric kernel function has a corresponding

“weight” value (ai). The ODF is then represented as a

series of the uniform and radially symmetric kernel

functions:

f ðgÞ 1⁄4 augu þXi

1

a1g1 þ a2g2 ... þ aigi ð4Þ

The term “intensity” for an ODF is the ratio of

weight at a given orientation relative to the weight of

the same orientation in a uniform (or random) texture

distribution, with units of multiples of uniform distribution (MUD), like the pole figures. The details of the calculation from pole figure to ODF are described by

Hielscher and Schaeben [8].

A Monte Carlo technique was used to estimate the

uncertainty in the estimated ODF due to the uncertainty

in the pole figure measurements. This technique was applied by perturbing each point in the pole figure P(h,r).

A Poisson pseudo-random number generator (poissrnd)

was applied to the total and background counts at each

point and then subtracted to generate the perturbed pole

figure data P^N ðh;rÞ:

P^N ðh;rÞ 1⁄4 poissrndðPtðh;rÞÞ poissrndðPbðh;rÞÞ ð5Þ

The observed counts were used as the Poisson mean

in the poissrnd because they are the maximum likelihood estimators [22]. This method of perturbing observed data for the purpose of quantifying uncertainty

is known generally as the parametric bootstrap [23].

These perturbed pole figures were then used to calculate

a perturbed ODF ^f N ðgÞ.

An ensemble of perturbed pole figures and perturbed

ODFs of size N were calculated. The intensity values of

each perturbed ODF were evaluated on a 2.5regular

grid in Euler space. From the ensemble, the mean, absolute standard deviation and relative standard deviation

of the intensities were calculated at each grid point

and plotted.

The experimental pole figures are shown in Figure 1a.

The pole figures are typical of rolled ferritic steel,

showing orthotropic sample symmetry from the rolling

56 A. Creuziger et al. / Scripta Materialia 72–73 (2014) 55–58
process. The relative uncertainty in the pole figure measurements are shown in Figure 1b, calculated using Eq.

(3). The relative uncertainty ranges from 1.6% to 13.5%.

As expected, regions of higher intensities in Figure 1a

correspond to lower relative uncertainty values in Figure 1b, and vice versa. The mean values of the relative

uncertainty in the pole figure measurements were

3.76%, 6.64% and 3.26% for the (1 1 0), (2 0 0) and

(2 1 1) reflections, respectively.

Figure 2 shows the estimated ODF constructed from

the set of experimental pole figures in Figure 1a. The difference between the observed pole figures and those derived from the estimated ODF, reported by the MTEX

pole figure to ODF calculation, is 0.83%. The ODF

shown in Figure 2 is typical of rolled mild steel. Most

of the grain orientations are aligned with the c fiber

(normal direction parallel to the h111i crystal axis;

/1 = 0to 90, U = 55.4, /2 = 45), indicated by high

MUD values along those orientations.

An ensemble size of N = 5000 was used in the Monte

Carlo analysis. The mean ODF calculated from the

ensemble (not shown) was similar to that estimated from

experimental pole figures, as shown in Figure 2. Figure 3

shows a difference plot between the mean of the ODF

ensemble and the ODF from experimental pole figures.

Figure 4 shows the relative uncertainty in the estimated ODF. The mean relative uncertainty is 1.61%

(range 4.01% to 0.52%). The R-Goss orientation

{0 1 1}h11 0i (90, 45, 0) shows the highest relative

uncertainty, while orientations along the gamma fiber

have the lowest values.

Three orientations were investigated in more detail:

the R-Goss orientation {0 1 1}h11 0i (90, 45, 0), the

c1 orientation {1 1 1}h11 0i (0, 55, 45) and the brass

orientation {0 1 1}h2 1 0i (35, 45, 0). The mean and

relative uncertainty values are shown in Table 1. The

N = 5000 ensemble was broken into two subsets of

N = 2500 to evaluate the effect of the number of

elements in the ensemble.

The mean relative uncertainty in the estimated ODF,

shown in Figure 4, is less than that in any of the pole

figures shown in Figure 1b. The uncertainty in the pole

figure measurements calculated from this technique is

Figure 2. ODF constructed from the set of the three input pole figures

shown in Figure 1a. Intensity is displayed in units of MUD with Bunge

Euler angles. The ODF plot is represented as 18 sections in increments

of u2 = 5from 0to 85, while u1 (along the x-axis) and U (along the

y-axis) range from 0to 90in each section.

Figure 3. Difference between the ODF estimated from experimental

pole figures and the mean ODF from ensemble N = 5000. The scale is

in units of MUD.

Figure 4. Representation of relative uncertainty (%) in the mean ODF

Figure 1. (a) (top) Pole figures from (1 1 0), (2 0 0) and (2 1 1) reflections using the Euler angle conventions of Figure 2.

(left to right) generated from neutron diffraction for undeformed SAE

1008 steel. The rolling direction (RD) is the vertical axis and the

transverse direction (TD) is horizontal. Pole figures are displayed as

number of counts in the peak. (b) (bottom) Relative uncertainty pole

figures (%) generated from neutron diffraction data for undeformed

SAE 1008 steel.

A. Creuziger et al. / Scripta Materialia 72–73 (2014) 55–58 57
similar to that observed by Lychagina et al. [10] using

the TOF technique. To our knowledge, this is the first

estimate of how the uncertainty varies from orientation

to orientation in the estimated ODF.

While, in general, there is a trend that larger intensity

in the mean ODF results in lower relative uncertainty,

this is not always the case. Comparing the R-Goss and

brass orientations shows that the brass orientation, despite having a mean intensity of nearly one-third of

the R-Goss orientation, has a smaller relative

uncertainty.

In summary, a technique to quantify the uncertainty

in an estimated ODF by propagating the uncertainty in

measured pole figures through the calculation of the

estimated ODF was demonstrated. By quantifying

uncertainty in an estimated ODF, results can be compared to determine if the difference between two estimated ODFs is statistically significant and

comparisons between ODFs can be more rigorous.

The authors acknowledge following people and

institutions for their assistance: Dr. Ralf Hielscher at the

Chemnitz University of Technology for developing,

maintaining and assistance using MTEX; Dr. Antonio

Possolo of the Statistical Engineering Division at NIST

for introducing the authors to the Monte Carlo technique; the NIST Center for Neutron Research (NCNR)

for providing the neutron research facilities used in this

work; Dr. William Luecke and Dr. Adam Pintar for

their helpful comments; and personnel in the NIST Center for Automotive Lightweighting for helpful discussions. K.S. was supported in part by the Summer

Undergraduate Research Fellowship (SURF) program

at NIST, together with the National Science Foundation

and the NIST-ARRA Undergraduate Fellowship

Program.

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Table 1. Mean and relative uncertainty values at selected orientations.

N = 5000 N = 2500 N = 2500

Mean (MUD) Relative uncertainty (%) Mean (MUD) Relative uncertainty (%) Mean (MUD) Relative uncertainty (%)

R-Goss 0.702 4.01 0.701 4.02 0.702 4.00

c1 6.379 0.74 6.380 0.72 6.378 0.75

Brass 0.266 2.22 0.266 2.22 0.266 2.22

58 A. Creuziger et al. / Scripta Materialia 72–73 (2014) 55–58