# Re: Statistical errors in residual dose rates and activities and number of primaries/batches

From: Mina Nozar <nozarm_at_triumf.ca>
Date: Mon, 13 Aug 2012 13:15:08 -0700

Dear Alberto,

Hello and thank you for your response.

I am still trying to understand so please bear with me.

As I understand, the uncertainty we see in any given quantity in FLUKA is just the error in the mean of the quantity,
where the mean is the average of the quantity as calculated from the result of the batches. So, the higher the
fluctuations in a given quantity from run to run, the higher the variance and the error in the mean.

I first came across this issue when doing Stefan's two-step process. From a study I did, I found out that the error in
the dose rate, one week after EOB was on the order of 28% for 2 M primaries per run, 10 runs (20 M primaries total). I
looked at the results of the individual runs and saw large fluctuations from run to run. This is what I saw for 10 runs
(Dose rate in PSv/s):

(766.4, 986.4, 1131.9, 1785.8, 4077.8, 224.2, 1143.6, 35.4, 1511.3, 939.9)
So the result from combining all ten runs was 1260.3 +/- 28%.

When I repeated the run for 5 M primaries/run, I saw smaller fluctuations:
(907.8, 986.4, 749.2, 937.5, 587.2, 599.1, 726.8, 421.9, 985.3, 754.3)
Average: 751.5 +/- 7.9%

So obviously an improvement, which made me conclude that number of primaries per run does matter. Is this because the
total number of primaries is increased in the 2nd case (5 M prim/run *10 runs = 50 M primaries)? And that if I had run
5 M primaries/run but only 4 runs, I would have seen a similar error as in case 1?

Mina

On 12-08-11 09:33 AM, Alberto Fasso' wrote:
> Dear Mina,
>
> you results are exactly what should be expected.
> The error of the average is inversely proportional to the square root of
> the TOTAL number of primaries, independent of how you divide this total
> number over the batches. The only limitations about the number of of
> batches and the number of primaries per batch are two:
> 1) The batches need to be at least 4 or 5 (better 5)
> 2) Detector scoring should be >> 0 in each batch
> With these two limitations, the batches don't even need to be all of the same
> size.
> For a same total number of primaries, increasing the number of primaries per
> batch is not particularly useful, unless it is necessary to satisfy the
> condition 2).
>
> Kind regards,
>
> Alberto
>
>
> On Fri, 10 Aug 2012, Mina Nozar wrote:
>
>> Hello everyone,
>>
>> I was recently wondering about how to reduce the statistical fluctuations in residual dose rate calculations and thought
>> that if I increase number or primaries per run for a total number of primaries, I would see a reduction in statistical
>> errors but this is not what I see. As I understand it, the values from different batches give the variance of the mean
>> for a given quantity.
>>
>> In the study I had two sets of runs, each totalling 100 M primaries.
>> First set: 1 M primaries * 100 batches
>> Second set: 10 M primaries * 10 batches
>>
>> Residual dose rate results:
>>
>> Time Tot. response (pSv/s)
>> First set Second set
>> --------------------------------------------------------------------------
>> 1h_SOB 7.3209280E+07 +/- 0.2788494 % 7.3209280E+07 +/- 0.1696288 %
>> 1d_SOB 3.9039216E+08 +/- 0.1965424 % 3.9039216E+08 +/- 0.1532526 %
>> 3d_SOB 4.5325405E+08 +/- 0.1940250 % 4.5325405E+08 +/- 0.1533875 %
>> 10d_SOB 4.6606554E+08 +/- 0.1911391 % 4.6606554E+08 +/- 0.1504155 %
>> 0s_EOB 4.7146954E+08 +/- 0.1893668 % 4.7146954E+08 +/- 0.1492323 %
>> 1h_EOB 4.0237386E+08 +/- 0.2029038 % 4.0237386E+08 +/- 0.1707704 %
>> 1d_EOB 7.9427664E+07 +/- 0.3225139 % 7.9427664E+07 +/- 0.3200963 %
>> 10d_EOB 6036197. +/- 0.3228723 % 6036197. +/- 0.3079922 %
>> 40d_EOB 190254. +/- 0.2063927 % 2190254. +/- 0.1794188 %
>> 1y_EOB 358294.0 +/- 0.3270685 % 358294.0 +/- 0.3348824 %
>> 2y_EOB 214956.4 +/- 0.3732046 % 214956.4 +/- 0.3926361 %
>> 3y_EOB 147807.1 +/- 0.3735141 % 147807.1 +/- 0.3969630 %
>> 5y_EOB 71939.66 +/- 0.3682632 % 71939.66 +/- 0.4093317 %
>>
>> Why do I not see a reduction in the variance of the mean in the second set? When I look at the dominant activities in
>> different regions (1y, 3y, and 5y after EOB), I don't see much difference in the errors between each set. So doesn't
>> seem to be advantageous to run a larger set of primaries per batch.
>>
>>
>> How are the errors in activities determined? Looking at production rates and time evolution of activities, seems as if
>> the errors are determined from production rates and remain the same for most but not all isotopes (see Lu,172 below).
>> This might have to do with the grow-in contribution from decay of other isotopes.
>>
>> Also, sometimes I see larger errors for smaller activities (on the order of 10^2 Bq) that for larger activities (on the
>> order of 10^4 Bq). So I don't think the fluctuations in activities depend on the production rates alone.
>>
>> Thanks and best wishes,
>> Mina
>>
>> Here, I am listing production rates and activities in a Tantalum target at different times after End of Beam a subset of
>> isotopes:
>>
>> Production rates (nuclei/prim):
>> Mn,56,25, 9.7495E-22 +/- 66.68 %
>> Co,60,27, 4.4082E+04 +/- 99.00 %
>> Se,75,34, 8.1350E+04 +/- 99.00 %
>> Y, 88,39, 2.6066E+04 +/- 55.28 %
>> Lu,173,71, 3.6126E+08 +/- 0.24 %
>> Lu,172,71, 1.8536E+09 +/- 2.110 %
>> Hf,175,72 1.6301E+10 +/- 0.09%
>> Hf,172,72, 4.8303E+08 +/- 0.3%
>>
>>
>> Activities (Bq):
>>
>> 0 sec after EOB:
>> Mn,56,25, 9.99e+06 +/- 66.67 %
>> Co,60,27, 4.47e+04 +/- 99.00 %
>> Se,75,34, 6.73e+05 +/- 99.00 %
>> Y, 88,39, 3.00e+06 +/- 55.28 %
>> Lu,173,71, 3.90e+10 +/- 0.24 %
>> Lu,172,71, 2.53e+10 +/- 1.46 %
>> Hf,175,72, 7.21e+11 +/- 0.09 %
>> Hf,172,72, 1.23e+10 +/- 0.30 %
>>
>> 10 days after EOB:
>> Mn,56,25, 9.75e-22 +/- 66.68 %
>> Co,60,27, 4.46e+04 +/- 99.00 %
>> Se,75,34, 6.35e+05 +/- 99.00 %
>> Y,88,39, 2.81e+06 +/- 55.28 %
>> Lu,173,71, 4.11e+10 +/- 0.24 %
>> Lu,172,71, 1.22e+10 +/- 1.07 %
>> Hf,175,72, 6.73e+11 +/- 0.09 %
>> Hf,172,72, 1.22e+10 +/- 0.30 %
>>
>> 40 days after EOB:
>> Co,60,27, 4.41e+04 +/- 99.00 %
>> Se,75,34, 5.34e+05 +/- 99.00 %
>> Y,88,39, 2.31e+06 +/- 55.28 %
>> Lu,173,71, 3.94e+10 +/- 0.24 %
>> Lu,172,71, 5.15e+09 +/- 0.29 %
>> Hf,175,72, 5.00e+11 +/- 0.09 %
>> Hf,172,72, 1.19e+10 +/- 0.30 %
>>
>> 1 year after EOB:
>> Co,60,27, 3.92e+04 +/- 99.00 %
>> Se,75,34, 8.14e+04 +/- 99.00 %
>> Y,88,39, 2.79e+05 +/- 55.28 %
>> Lu,173,71, 2.51e+10 +/- 0.24 %
>> Lu,172,71, 3.47e+09 +/- 0.30 %
>> Hf,175,72, 2.00e+10 +/- 0.09 %
>> Hf,172,72, 8.52e+09 +/- 0.30 %
>>
>> 3 years after EOB:
>> Co,60,27, 3.01e+04 +/- 99.00 %
>> Se,75,34, 1.19e+03 +/- 99.00 %
>> Y,88,39, 2.43e+03 +/- 55.28 %
>> Lu,173,71, 9.14e+09 +/- 0.24 %
>> Lu,172,71, 1.65e+09 +/- 0.30 %
>> Hf,175,72, 1.45e+07 +/- 0.09 %
>> Hf,172,72, 4.06e+09 +/- 0.30 %
>>
>> 5 years after EOB:
>> Co,60,27, 2.32e+04 +/- 99.00 %
>> Se,75,34, 1.74e+01 +/- 99.00 %
>> Y,88,39, 2.11e+01 +/- 55.28 %
>> Lu,173,71, 3.32e+09 +/- 0.24 %
>> Lu,172,71, 7.87e+08 +/- 0.30 %
>> Hf,175,72, 1.05e+04 +/- 0.09 %
>> Hf,172,72, 1.93e+09 +/- 0.30 %
>>
>>
>>
>
Received on Mon Aug 13 2012 - 23:54:49 CEST

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