Low count bias in gamma ray thickness detection and ...

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Nuclear Instruments and Methods in Physics Research A 568 (2006) 912–914 www.elsevier.com/locate/nima

Low count bias in gamma ray thickness detection and its correction Manchun Liang, Kejun Kang, Zhikang Zhang, Xinghua Lou Department of Engineering Physics, Tsinghua University, Beijing, China Received 19 June 2006; received in revised form 28 July 2006; accepted 18 August 2006 Available online 14 September 2006

Abstract In this paper we introduce the latest discovery of Low Count Bias (LCB) effect in gamma ray material thickness measurement, which adversely affects the precision of measurement. Theoretical analysis of LCB effect is presented and the correction method is proposed. In order to further prove our theory, Monte Carlo simulation is taken and the result of bias correction is given. r 2006 Elsevier B.V. All rights reserved. PACS: 29.85.+c Keywords: Low count bias; Thickness measurement; Biased estimate

1. Introduction Gamma ray thickness measurement has been broadly used in industrial application and scientiﬁc research [1]. In most cases of on-line thickness measurement, the sampling time must be kept short while maintaining good measurement accuracy [2,3]. 2. Discovery of LCB Nevertheless, short sampling time means low count, which causes a problem that always exists—some bias (offset) between the measurement and real thickness. In order to track down the measurement error, we carried out a computer based Monte Carlo Simulation. The following introduces the simulation procedure, assuming the gamma ray count I is a random variable which follows Poisson distribution. The simulation proceeds as below Step 1: Assign a speciﬁc value to the mathematical expectation of gamma ray count I (i.e. I E ) and compute its corresponding thickness d E .1 Step 2: Randomly generate the count value I following Poisson distribution. Corresponding author.

E-mail address: [email protected] (X. Lou). 1 The calculation of thickness here is performed in the conventional way which will be introduced in Eq. (3) of Section 3. 0168-9002/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2006.08.097

Step 3: Calculate the corresponding thickness value d.2 Step 4: Repeat steps 2 and 3 for N times and compute the average of thickness, as d¯ ¼ Sd=N. N should be large enough to suppress accident error. ¯ Step 5: Calculate the LCB, as Dd L ¼ d E d. Step 6: Change I E and repeat all steps above so that we can obtain the functional relationship between Dd L and I E , as shown in Fig. 1. The curve can be approximated by equation Dd L ¼

1 2mI E

(1)

where Dd L is the bias between measured thickness and the true value.

3. Correction method In this section we analyze the causality of Eq. (1), deﬁne the low count bias and give the correction method. When a mono-energy gamma ray is collimated and penetrates a thin material, it follows the exponential attenuation [4]. I E ¼ I 0 emd 2

Same as above.

(2)

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Thickness Bias −∆dE (cm−1)

M. Liang et al. / Nuclear Instruments and Methods in Physics Research A 568 (2006) 912–914

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0.7

Substitute Eq. (6) into Eq. (7),

0.6

E½ln I ln I E

0.5

Then substitute Eq. (8) into Eq. (4),

0.4

Dd L

0.3

By comparing Eq. (1) with Eq. (9), we can see that the right side of Eq. (1) is exactly the ﬁrst term at the right side of Eq. (9). This explains the result we get in the simulation before, which is apparently a coarse representation of low count bias in Eq. (9). Only when I E ! 1 could Dd L reach ZERO, namely LCB commonly exists in all radiation detection instruments but always be ignored because the count is often large enough. Now, giving the correction method for LCB (Eq. (9)), we need a better way to address it since I E is a unknown expectation. Again, we applied the Taylor expansion to expand E½1=I and E½1=I 2 , 1 1 1 2 þ þ (10) E ¼ I I E I 2E I 3E 1 1 3 (11) E 2 ¼ 2þ 3. I IE IE

0.2 0.1 0 10

20

30

40 50 60 70 80 Counting Expectation − IE

90

100 110

Fig. 1. Functional relationship between Dd L and I E .

where d is the real thickness of material; I E is the count expectation corresponding to d; I 0 is the count without materials; m is the linear attenuation coefﬁcient. The material thickness can be calculated as dm ¼

ln I 0 ln I m

(3)

where d m is the measured thickness of material and I is the count for a single sampling [1]. In real situation, the thickness will be calculated as a multiple sampling expectation and low count bias is deﬁned as the difference between the expectation and real thickness, Dd L ¼ E½d m d

(4)

where, E½ is the expectation operator. By substituting Eq. (4) with Eqs. (2) and (3) we get ln I E E½ln I . Dd L ¼ m

Eq. (5) clearly explains the cause of LCB: ln I E aE½ln I. In other words, it is biased estimate of ln I E using expectation of ln I. Before moving further in evaluating Dd L , we need draw some conclusion ﬁrst. Since count I follows Poisson distribution, the following equations can be derived [5]: 8 E½I I E ¼ 0 > > > > < E½ðI I E Þ2 ¼ I E (6) E½ðI I E Þ3 ¼ I E > > > > : E½ðI I E Þ4 ¼ 3I 3 þ I E : E Expand E½ln I using Taylor expansion and keep the ﬁrst four terms, we have E½ln I ln I E þ þ

E½I I E E½ðI I E Þ2 IE 2I 2E

E½ðI I E Þ3 E½ðI I E Þ4 . 3I 3E 4I 4E

ð7Þ

1 1 1 þ þ . 2mI E 12mI 2E 4mI 3E

From Eqs. (10), (11) and (8) we can get 1 1 E ln I E E½ln I. 2I 12I 2

(8)

(9)

(12)

Substitute Eqs. (13) and (4) into Eq. (5), we will ﬁnally get the correction formula: dC ¼

(5)

1 1 1 . 2I E 12I 2E 4I 3E

ln I 0 ln I ð1=2IÞ þ ð1=12I 2 Þ m

(13)

where d C approximates the unbiased estimate of real thickness. Sometimes Eq. (2) is replaced with I E ¼ I 0 BðdÞemd for better precision, where BðdÞ is the build-up factor. Under this circumstance, d C could be retrived by solving equation Eq. (14) which is a revised version of Eq. (13). Considering that the build-up factor BðdÞ is known value, Eq. (14) could be easily solved using Newton–Raphson method. dC ¼

lnðI 0 Bðd C ÞÞ ln I ð1=2IÞ þ ð1=12I 2 Þ . m

(14)

4. Correction effect To further prove our theory, we carried out another Monte Carlo simulation which is very similar to the one introduced in Section 1. The only difference is that Eq. (13) is applied to compute the thickness instead of Eq. (3). The result is shown in Fig. 2. It is quite obvious that when

ARTICLE IN PRESS M. Liang et al. / Nuclear Instruments and Methods in Physics Research A 568 (2006) 912–914

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5. Conclusions

x 10−5

Thickness Bias − ∆dE (cm−1)

2

LCB is a general phenomenon in radiation detection system and it becomes obtrusive where count is low. We discussed the fundamental cause of LCB and presented a correction method which has been successfully applied to our thickness measurement system and brought dramatic improvement to its precision as well as performance. For example for 1 ms sampling duration, the relative error is 0.1% vs. 2% when the correction method is disabled.

0 −2 −4 −6 −8

References −10 10

20

30

40 50 60 70 80 Counting Expectation − IE

90

100 110

Fig. 2. Functional relationship between Dd L and I E after correction.

count exceeds 10 the bias is close to ZERO, which means the correction method is effective.

[1] [2] [3] [4]

Y. Shirakawa, J. Appl. Radiat. Isot. 53 (2000) 581. J.A. Oyedele, J. Appl. Radiat. Isot. 33 (1982) 1465. J.A. Oyedele, J. Appl. Radiat. Isot. 38 (1987) 527. G.F. Knoll, Radiation Detection and Measurement, third ed., Wiley, New York, 2000. [5] A. Papoulis, S.U. Pillai, Probability, Random Variables and Stochastic Processes, third ed., McGraw-Hill, New York, 1991.