# Thread: nuclear waste analysis problem

1. ## nuclear waste analysis problem

I am working in the nuclear waste analysis. I would really appreciate your input. We measure a content of Pu-239 in grams (radioactive element) in a long crate. There were 2 or 3 measurements on each side. Each measurement has the uncertainty.
The total mass (in grams) is measured as the sum of the masses divided by 2 since we assume that the crate is transparent that means that we measure the same mass from both sides, so we just average it.

I have a problem with the way how a total error is reported
The total error is measured as the error propagation in the quadrature, that means that the second powers of errors are added and then the square is calculated. I do not like this approach because it looks that the total error depends on the number of measurements rather than on the geometry, statistics etc. I think this approach would make sense if the measurements were completely independent, if the errors would come for instance from statistics, geometry etc. I think this is wrong since the measurements are not completely independent since we measure the same mass from both sides. The more measurements we do the higher the error.

We assume that the detector sees only a half (pr 1/3) of the container length in each measurement because of the field of view.

2. ## Re: nuclear waste analysis problem

To combine independent measures of the same quantity, the value is obtained by weighting each measurment by its inverse error, and the total error is obtained by combining the individual errors in inverse quadrature:

This means that more measurements produces a smaller error, decreasing by ~1/sqrt(n), as you would expect.

But it sounds like maybe each of your measurements is not actually of the same quantity, but rather contributes an nth of the total. In that case you do add each part to get the whole, and add the errors in quadrature:

It also sounds like you may not believe that your n measurements are actually independent. Why is that? In any case, you can test whether the measurements are independent by computing the corrleation coefficient between them and seeing whether it is non-zero at a significant level.

3. ## Re: nuclear waste analysis problem

Thank you so much for your fast and very good answer. So that means that the errors for independent measurements are actually going down, while for dependent are summarized.

I am attaching the table with data. You can see the scheme of the crate. Pu gram content is measured in 2 different parts for each side. Theoretically the detector sees only the half of the crate, the field of view is attached but as you can see the field of view is getting broaden on the other side.

We assume that the detector sees the same gram value from both sides (A1 and B1) and (A2 and B2), therefore the gram table has the total sum divided by two.

As you can see the error is calculated as a total error in quadrature and it is very high. Below is the picture
http://culture.polishsite.us/images-10/table1.pdf

What approach would be the best for this project?

4. ## Re: nuclear waste analysis problem

If I understand your diagam and explanation correctly, A1 and B1 are supposed to be measuring the same quantity: the Pu-239 in the left half of the crate. Let's call that quantity M1. Similiary, A2 and B2 are supposed to both be measurements of another quantity: the Pu-239 in the right half of the crate. Let's call that quantity M2. You are interested in obtaining a value, with uncertainty, in the sum M = M1 + M2. Is that right?

So there are really two questions here: (1) Given this model, how do we calculate a value with uncertainty for M? (2) How good a model is this of reality? Since they are taken from different sides, do we really expect A1 and B1 to be measuring the same quantity? Since there is some overlap in the solid angles viewed by the detector looking at each half of the crate, is the sum of those two quantities really a good approximation to the Pu-239 in the whole crate?

(1) is straightforward, and involves using both the forumulas I gave above. If A1 = 51.9 +/- 19.2 and B1 = 343 +/- 129 are really two measurments of the same quantity M1, then M1 = 89.6 +/- 19.0 by the formula I gave for combining measurments of the same quantity. (Note that the result is dominated by A1 because it has a much lower uncertainty that B1.) Similiarly, if A2 = 187 +/- 72.7 and B2 = 65.9 +/- 24.4 are really two measurements of the same quantity M2, then M2 = 96.3 +/- 23.1. The sum M = M1 + M2 = 185.9 +/- 29.9 can be computed by the formula I gave for sums of quantities. So that's the answer, assuming the model is correct.

(2) is less straightforward. Looking at the divergence A1 and B1 and between A2 and B2 certainly makes me suspect that they might not be really measuring the same quantity. B1, for example, is nearly 2-sigma from our best value for M1. But there aren't enough measurements here to be very confident that they are not, either. But given that you also have physical grounds for suspecting that the model might be flawed, you might want to spend some time with an engineer who understands the geometry and physics of your detector to see if you can't come up with a better model of your system, and test it.

5. ## Re: nuclear waste analysis problem

Hi Ichbin,

I was writing my reply so long... that I got disconnected.

Thank you so much for your help!!!
I went through all the calculations and it works.

I agree that we have problems with FOI (field of view of detector) and relatively big difference in masses for the independent measurements of the same quantities but we expect it. We are happy if the values are of the same order of magnitude.

I have a question:

1. Could you give me a good website reference so that I can convince my guys to use your approach?

2. I think what they are doing is too simplistic - just adding the 4 masses and dividing by 2 for a total mass and calculating the error as sqrt(error1^2+... error4^4).
how can I convince them that this is not a right approach?

6. ## Re: nuclear waste analysis problem

Since I was disconnected during my last post, my reply was short, I had to re-create it. Let me add a couple of words.
We expect errors to be relatively high since:
the radioactive material is in a huge crate.... so there is lots of absorption going on inside it and the material may not be homogeneous, therefore we measure it on both sides.

The error is also big since Pu-239 is relatively long living isotope and has a low activity.

By the way, are you German? I used to speak German pretty well, not anymore since I moved to the US. I still need to work on my English and a sentences clarity

thanks again!

7. ## Re: nuclear waste analysis problem

Taylor's classic "An Introduction to Error Analysis" covers this in Chapter 7. Here is a Google Books link to the relevant section: http://books.google.com/books?id=giF...page&q&f=false.

Knoll's "Radiation Detection and Measurement" is another classic for problems in this area. It has a good chapter on statistics, as well as introductions to the analysis of many standard detector setups.

I started out in nuclear physics before switching fields. I'm not German, but I lived and worked there for many years, and I do speak German (thus the alias).

8. ## Re: nuclear waste analysis problem

Originally Posted by ichbin
Taylor's classic "An Introduction to Error Analysis" covers this in Chapter 7.
Thanks for the link. Unfortunately this is not available through interlibrary loan for me, but I can use the title to find other books that are.

[BTW, I use my own version of German to put off panhandlers - it almost always works.]

/s/ IchBinNicht

9. ## Re: nuclear waste analysis problem

Ichbin,

thank you so much. I have a Knoll book. I forgot that it has such a good chapter on error analysis. I actually participated once with Knoll's lectures organized by IAEA in Greece.

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