radiochemistry webinars statistics in nuclear forensics... · 2016-12-02 · statistics in nuclear...

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Statistics in Nuclear Forensics

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Meet the Presenters… Dr. Luther McDonald

Contact Information Phone: 801-581-7768 Email: [email protected]

and the Nuclear Engineering Program. His research lab focuses on three primary topic areas: environmental radiochemistry, spent nuclear fuel reprocessing, and nuclear forensics. While these topics may seem quite diverse, they are all interrelated and devoted to our core research philosophy of improving our fundamental knowledge of basic radionuclide material properties to aid in the rapid detection of hostile nuclear materials, remediate

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and dispose of radioactive materials in an environmentally responsible manner, and develop advanced processes for the recycling of spent nuclear fuel. Since joining the University, Dr. McDonald has been awarded over $5M in funding to improve nuclear forensics capabilities in the U.S. This funding is supporting 10 new Ph.D. students in areas relevant to nuclear forensics including: nuclear engineering, environmental engineering, and computer science. Prior to joining the University of Utah, McDonald performed a post-doctoral fellowship at Pacific Northwest National Laboratory in National Technical Nuclear Forensics, worked as a visiting scientist at the Commissariat a lenergie atomique, and completed his Ph.D. at Washington State University in Radiochemistry. He currently serves at the Secretary of the American Chemical Society’s Division of Nuclear Chemistry and Technology.

Professor McDonald joined the University of Utah in January 2014 as an Assistant Professor in the Department of Civil and Environmental Engineering

Statistics in Nuclear Forensics Professor Luther McDonald IV University of Utah

Department of Civil and Environmental Engineering and Nuclear Engineering Program

National Analytical Management Program (NAMP)

U.S. Department of Energy Carlsbad Field Office

TRAINING AND EDUCATION SUBCOMMITTEE

Goal

Investigate and determine processing history,

intended use, and geographical origin of

interdicted nuclear materials.

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Objectives

1. Identify unique signatures

2. Validate analytical methods and measurements

3. Apply statistical data interpretation

The Dilemma

How do we reduce the technical nuclear forensics timeline while maintaining good accuracy and precision?

How do we combine multiple signatures (physical, chemical, and isotopic measurements) to yield a definitive solution?

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The Solution

It’s all in the statistics!

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Introduction to Statistics1 –

Types of Error

Sources of Error

–Systematic Error

• Error that has the same magnitude and sign

• Typically arises from design flaw in the experiment

or equipment

–Random Error

• Error of varying magnitude and sign

• Arises from uncontrollable variables in the

experiment

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Systematic Vs. Random Error

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Systematic Error Random Error

• Systematic error can be detected and corrected using

standard reference materials!

• Random error cannot be eliminated, only reduced in

better experiment planning.

Introduction to Statistics1–

Precision and Accuracy

Precision = Reproducibility

Accuracy = Nearness to the truth

• A measurement can be very precise, but far from accurate.

• Standard reference materials are used to verify the accuracy of a measurement.

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Error, Precision, and Accuracy

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Low Accuracy Low Precision

High Accuracy Low Precision

Low Accuracy High Precision

High Accuracy High Precision


Probability Distributions If error from repeating an experiment multiple times is purely random, then the results will cluster around the average value

– This is known as a Gaussian distribution

– The more replicates, the closer to achieving a perfectly smooth curve.

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Standard Deviation

Mean


Sample Mean • The most common result from a

measurement performed repeatedly.

• It is an estimate of the population mean from the average value of a finite set of measurements –It is calculated using:

𝑥 =1

𝑛 𝑥𝑖𝑖

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Standard Deviation

• Distribution about the mean from the repeated measurements

𝑠 =1

𝑛 − 1 (𝑥𝑖−𝑥 )

2

𝑖

• It has the same units as the mean

• The larger the standard deviation, the greater the dispersion in the measured values

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Sample Size and Population1

For an infinite data set, the mean (𝑥 ) and standard deviation (s) are designated as µ and σ, respectively.

–µ and σ cannot be experimentally measured

–As the number of replicates increases, (𝑥 ) and s approach µ and σ.

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Standard Deviation and Probability1

A Gaussian curve is defined by:

𝑦 =1

𝜎 2𝜋𝑒−𝑥−𝜇 2

2𝜎2

• The standard deviation defines the width of the curve.

– Smaller standard deviation = more narrow curve

– In a Gaussian curve, 68.3% of the measurements are expected to lie within one standard deviation of the mean (µ ± 1σ)

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Range % of Measurements

µ ± 1σ 68.3

µ ± 2σ 95.5

µ ± 3σ 99.7


Rejecting Data (Q-Test)

• What if I measure many replicates, but one datum is

far from the rest of the series?

• A way of statistically rejecting outliers far from the

mean

𝑄𝑐𝑎𝑙𝑐 =𝑔𝑎𝑝

𝑟𝑎𝑛𝑔𝑒

• When 𝑄𝑐𝑎𝑙𝑐 > 𝑄𝑡𝑎𝑏, then the outlier can be rejected

• 𝑄𝑡𝑎𝑏 is a tabulated value at many different

confidence levels.

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Comparing Multiple Data Sets

• F-test – Determines if the

variance from two data sets are the same.

𝐹𝑐𝑎𝑙𝑐 =𝑠𝑎2

𝑠𝑏2

– Where a and b are chosen so that 𝐹𝑐𝑎𝑙𝑐 > 1

– If 𝐹𝑐𝑎𝑙𝑐 < 𝐹𝑡𝑎𝑏 then “there is no statistical difference between the distributions at the specific confidence interval.”

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Comparing Multiple Data Sets1

• t-test – Using results from the F-test, the t-test can be used to

determines if two means from different data sets are the same.

– If the F-test passed, then:

𝑡𝑐𝑎𝑙𝑐 =𝑥 𝑎 − 𝑥 𝑏

𝑠𝑝𝑜𝑜𝑙𝑒𝑑1𝑛𝑎+1𝑛𝑏

𝑤ℎ𝑒𝑟𝑒 𝑠𝑝𝑜𝑜𝑙𝑒𝑑 =𝑛𝑎 − 1 𝑠𝑎

2 + 𝑛𝑏 − 1 𝑠𝑏2

𝑛𝑎 + 𝑛𝑏 − 2

Where 𝑛𝑎 and 𝑛𝑏 are the number of replicates from each experiment.

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Comparing Multiple Data Sets1

• t-test – The t-test can also

be used to compare the mean of an experimental value to the known value (μ)

𝑡𝑐𝑎𝑙𝑐 =𝑛

𝑠𝜇 − 𝑥

– If 𝑡𝑐𝑎𝑙𝑐 < 𝑡𝑡𝑎𝑏 then there is no statistical difference between 𝑥 and 𝜇 within the specified confidence interval.

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Examples of Statistics in Nuclear

Forensics

Statistics in Nuclear Forensics2,3,4 –

Chronometry

• Chronometry (or age dating) is a long established process in archaeological sciences but also has many applications in nuclear forensics

–Age of enrichment or purification

• U/Th, Pu/U, and Am/Pu isotope ratios

–Dating nuclear explosions for verification of the Comprehensive Nuclear Test-Ban Treaty (CTBT)

• 140Ba/140La isotope ratio

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Chronometry

• There are many sources of error in chronometry

–Chemical separations

–Analytical measurements

• Radiometric counting

• Mass spectrometry

–Half-life uncertainty

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Chronometry

Very robust, precise, and accurate methods for chemical separations and analytical measurements have been developed.

• Current research in this area focuses on:

– Shortening the time required for chemical separations

– Reducing systematic error in the analytical measurements

• Development of standard reference materials

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Chronometry Example Number of parent atoms over time:

𝑃 𝑡 = 𝑃 0 𝑒−λ𝑝𝑡

Number of daughter atoms over time:

𝐷 𝑡 = 𝐷 0 𝑒−λ𝐷𝑡 + 𝑃 0λ𝑃λ𝐷 − λ𝑃

(𝑒−λ𝑃𝑡−𝑒−λ𝐷𝑡)

The ratio daughter to parent atoms at any time greater than 0 is:

𝑅 𝑡 =𝐷 𝑡

𝑃(𝑡)= 𝑅 0 𝑒− λ𝐷−λ𝑃 𝑡 +

λ𝑃λ𝐷 − λ𝑃

1 − 𝑒− λ𝐷−λ𝑃 𝑡

All of these equations assume that no daughter was present at t = 0.

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Chronometry Example

The age of time that has lapsed is calculated from measured values of the ratio of daughter to parent and from literature values of the decay constant:

𝐴𝑔𝑒 =1

λ𝑃 − λ𝐷ln 1 − 𝑅 𝑡

λ𝐷 − λ𝑃λ𝑃

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Chronometry Example

Propagating the uncertainty from the decay constants and atom ratio yields: 𝜎 𝑡

𝑡

2

=λ𝐷 − λ𝑃λ𝑃

𝑇

𝑡−λ𝑃λ𝐷

2𝜎 λ𝑃λ𝑃

2

+−λ𝐷λ𝑃 −λ𝐷

𝑇

𝑡− 1

2𝜎 λ𝐷λ𝐷

2

+𝑇

𝑡

2 𝜎 𝑅

𝑅

2

Where T is:

𝑇

𝑡=𝑒− λ𝐷−λ𝑃 𝑡 − 1

λ𝐷 − λ𝑃 𝑡

And the relative uncertainties of the decay constants is: 𝜎 λ

λ= −𝜎(𝑇1 2 )

𝑇1 2

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Chronometry Example

Let’s break this down into different components. 𝜎 𝑡

𝑡

2


𝑇

𝑡−λ𝑃λ𝐷

2𝜎 λ𝑃λ𝑃

2


𝑇

𝑡− 1

2𝜎 λ𝐷λ𝐷

2

+𝑇

𝑡

2 𝜎 𝑅

𝑅

2

This first term is specifically addressing the uncertainty in…

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Where T is:

𝑇




λ= −𝜎(𝑇1 2 )

𝑇1 2


Chronometry Example

Let’s break this down into different components. 𝜎 𝑡

𝑡

2


𝑇

𝑡−λ𝑃λ𝐷

2𝜎 λ𝑃λ𝑃

2


𝑇

𝑡− 1

2𝜎 λ𝐷λ𝐷

2

+𝑇

𝑡

2 𝜎 𝑅

𝑅

2

This second term is specifically addressing the uncertainty in…

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Where T is:

𝑇




λ= −𝜎(𝑇1 2 )

𝑇1 2


Chronometry Example

Let’s break this down into different components.

𝜎 𝑡

𝑡

2


𝑇

𝑡−λ𝑃λ𝐷

2𝜎 λ𝑃λ𝑃

2


𝑇

𝑡− 1

2𝜎 λ𝐷λ𝐷

2

+𝑇

𝑡

2 𝜎 𝑅

𝑅

2

This third term is specifically addressing the uncertainty in…

• Uncertainty in R comes primary from uncertainty in the analytical measurement.

• When R = 1, the uncertainty is expected to be smallest, but it will increase with log (𝑅) when the parent and daughter concentrations differ by orders of magnitude.

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𝑅 𝑡 =𝐷 𝑡

𝑃(𝑡)= 𝑅 0 𝑒− λ𝐷−λ𝑃 𝑡 +

λ𝑃λ𝐷 − λ𝑃

1 − 𝑒− λ𝐷−λ𝑃 𝑡


Chronometry Example

Propagating the uncertainty from the decay constants and atom ratio yields: 𝜎 𝑡

𝑡

2


𝑇

𝑡−λ𝑃λ𝐷

2𝜎 λ𝑃λ𝑃

2


𝑇

𝑡− 1

2𝜎 λ𝐷λ𝐷

2

+𝑇

𝑡

2 𝜎 𝑅

𝑅

2

If there is an incomplete chemical separation at time t = 0, then R(t) will increase relative to:

𝑅 0

𝑅 𝑡 𝑒−(𝜆𝐷−λ𝑃)𝑡≠ 0

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Why is all this important?

Well, the error from the half-life alone, can contribute much error to the chronometric calculation!

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Let’s look specifically at the 230Th/234U

230Th/234U is a very common chronometer in nuclear forensics5.

𝑈234𝑡1/2=2.455𝑥10

5𝑦𝑇ℎ230𝑡1/2=7.54𝑥10

4𝑦𝑅𝑎226

• During processing of uranium for nuclear fuel or weapons, the U is purified from everything else.

– This is time zero

• Since 𝑈234 has a relatively long half-life, not much 𝑇ℎ230 will be present for age-dating.

– Typically only 10-10 – 10-7 g per gram of U depending on the enrichment and age.

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State of the art techniques are needed to measure this small quantity, precisely and accurately5.

• A chemical/isotopic standard did not exist for the measurement of this material

– Traditionally, other reference materials were used to verify the accuracy of the measurement

– To provide a more solid scientific and legal defense, a 230Th/234U standard was needed

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• The challenge is that the 230Th/234U chronometer is very sensitive to the initial purity. A separation factor of more than 107 is needed to eliminate any positive bias5.

• Recall that if a chemical separation is incomplete at time t = 0, then R(t) will increase relative to:

𝑅 0

𝑅 𝑡 𝑒−(𝜆𝐷−λ𝑃)𝑡≠ 0

• Starting with zero 230Th, enables the amount of 230Th detected after some fixed time, to be solely dependent on the amount produced from the radioactive decay of 234U.

𝑁 𝑇ℎ230 = 𝑁 𝑈234λ 𝑈234

λ 𝑇ℎ230 − λ 𝑈234𝑒−λ

𝑈234𝑡− 𝑒−λ

𝑇ℎ230 𝑡


In 2015, a round robin exercise was completed to

compare the preparation and validation of a 230Th/234U standard based on radiochemical

separations and subsequent mass spectrometry

analysis of 230Th/234U in Highly Enriched

Uranium (HEU), Low Enriched Uranium (LEU),

and Natural Uranium (NU) 5.

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Laboratory Production Date – HEU

Production Date – LEU

Production Date - NU

Lab A 1,877 ± 85 days

Lab B 1,916 ± 26 days 1,919 ± 22 days 1,917 ± 23 days

Lab C 1,941 ± 22 days 1,854 ± 34 days 2,109 ± 80 days

Lab D 1,910 ± 57 days 1,955 ± 71 days 1,836 ± 46 days

Lab E 1,914 ± 15 days 1,918 ± 42 days 1,911 ± 35 days

Average 1,912 ± 23 days 1,912 ± 42 days 1,944 ± 117 days

The Result4:


• How do we know if these results are the same or different?

–F-test and t-test!

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Laboratory Production Date – HEU

Production Date – LEU

Production Date - NU

Lab A 1,877 days ± 85 days

Lab B 1,916 days ± 26 days 1,919 days ± 22 days 1,917 days ± 23 days

Lab C 1,941 days ± 22 days 1,854 days ± 34 days 2,109 days ± 80 days

Lab D 1,910 days ± 57 days 1,955 days ± 71 days 1,836 days ± 46 days

Lab E 1,914 days ± 15 days 1,918 days ± 42 days 1,911 days ± 35 days

Average 1,912 days ± 23 days 1,912 days ± 42 days 1,944 days ± 117 days

How can we reduce the error5?

Improved measurement of the 234U half-life to reduce the error in chronometric calculations!

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Example Study of Morphology6

Summary of Key Features U3O8 600°C to 800°C

1 μm 1 μm 2 μm 3 μm 4 μm

U3O8 600°C U3O8 800°C U3O8 650°C U3O8 700°C U3O8 750°C

Increasing grain uniformity

Increasing surface uniformity

Increasing grain suturing and annealing

Increasing grain rounding

Reduction in intergrain porosity

Euhedral, angular grains dominant Sub-rounded grains dominant

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• Morphological Analysis for Materials

–MAMA Software

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Statistical Analysis of U3O8

Segmentation with MAMA

Use the segmentation tool and select “smoothest.”

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Is the image in focus? (Edge Detection)

Use ImageJ Process (Find Edges)

Yes No

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Is the Particle Clearly a Single

Particle

YES: Label Particle

NO: Next Step

Adapted from a poster by Adam Olsen showing flowsheet of particle analysis

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Fully Segmented Image

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σ

Mean Circularity of Grains

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σ

Mean Area of Grains

The Challenge

If basing the forensic analysis off only a single particle, then statistically none of the particles could be distinguished based on their calcination temperature.

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The Challenge

That doesn’t mean they are not statistically different!

We have to determine a minimum population size to compare the overall trend!

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Comparing Distributions

• The total population of particles show very different trends

– But how do we compare particle distributions and not just individual particles?

• To quantify, we use the two sample Kolmogorov-Smirnov Test (K-S Test) – This test compares populations of data as opposed to single data

points.

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K-S test

• The K-S statistic (Dmax) is calculated from:

Dmax = maxxFn1 x − Fn2 x

– Where Fn1 x and Fn2 x are the cumulative distribution

functions with n1 and n2 being the sample sizes of the respective distributions.

• One way of evaluating the K-S test statistic, Dmax, is to compare it to a Dcritical value.

Dcritical = c(α)n1 + n2n1n2

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K-S test

• If Dmax is greater than

Dcritical then the two

distributions are

statistically different.

• The K-S test was used to

show that a minimum of

750 particles in a

population are needed to

prove statistical

differences at the 99%

confidence interval

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How can we improve statistical

analysis in the future?

1. Develop chemical and isotopic standards

2. Develop databases of data for the greater community to compare results

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What’s Next for Nuclear Forensics

• Nuclear forensics is starting to encounter the “big data” problem.

• We have many advanced techniques producing more data than we can process using our traditional ways

–Machine learning will start to take a bigger role in helping us process nuclear forensics data

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References

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1. Harris, Daniel C. Quantitative chemical analysis. Macmillan, 2010.

2. Pommé, S., S. M. Jerome, and C. Venchiarutti. "Uncertainty propagation in nuclear forensics." Applied Radiation and Isotopes 89 (2014): 58-64.

3. Pommé, S. "The uncertainty of the half-life." Metrologia 52.3 (2015): S51.

4. Varga, Z., et al. "Validation of reference materials for uranium radiochronometry in the frame of nuclear forensic investigations." Applied Radiation and Isotopes 102 (2015): 81-86.

5. Varga, Zsolt, et al. "Remeasurement of 234U Half-Life." Analytical chemistry (2016): 2763-2769.

6. Adam Olsen, Bryony Richards, Ian Schwerdt, Sean Heffernan, Robert Lusk, Braxton Smith, Elizabeth Jurrus, Christy Ruggiero, Luther W. McDonald IV, “Quantifying Morphological Features with Images Analysis for Nuclear Forensics” Submitted to Analytical Chemistry

Contact Information

Luther McDonald

• [email protected]

• 801-581-7768

• http://mcdonald-radiochemistry-research.com

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mailto:[email protected]

http://mcdonald-radiochemistry-research.com/





MAMA Software Contacts

• Jeff Bloch, [email protected] or Christy Ruggiero ([email protected]) at Los Alamos National Laboratory

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Upcoming Webinars

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radiochemistry webinars statistics in nuclear forensics... · 2016-12-02 · statistics in nuclear...

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