IEEE Transactions on Nuclear Science, Vol. NS-29, No. 1, February 1982 ESTIMATION AND CORRECTION OF COUNTING DEAD-TIME BY USING INFORMATION CONTAINED IN THE TIM1E DISTRIBUTION OF THE EVENTS R.M. Holford Atomic Energy of Canada Research Company Chalk River Nuclear Laboratories Chalk River, Ontario, Canada KOJ 1JO Summary Methods for extending the high counting-rate capability of nuclear instruments which use Geiger counters, by correcting for dead-time losses, are compared. Three methods are described which allow for correction of such losses without a prior know- ledge of the dead-time. Introduction Estimation and correction of counting dead-time losses allows the high counting-rate capability of nuclear instrumentation to be extended by as much as an order of magnitude1. Methods which have been pro- posed for correction of such losses require that the value of the dead-time be known beforehand and that it be a constant, e.g., unaffected by the counting rate. In this communication, techniques are des- cribed that overcome these requirements by deriving estimates of the dead-time from the time-interval distribution of the counting events. A Geiger counter is typical of detectors exhibiting a non-extended dead-time2. Potential events (radiation interactions within the detector) occurring within a fixed interval T following a counted pulse have no observable effect and the next pulse is initiated by the first potential event occurring after this interval. The probability density function (p.d.f.) for the intervals between the pulses from such a detector is given by2 p(t) = n efn(t0T), t>T .............. (1) where n is the rate at which potential events occur. The average value of t is r + 1/n; therefore n = 1/(tav-T), the standard formiula for known T. The methods to be described for correcting for an unknown T are based on measurements of the individ- ual time intervals, ti, between successive pulses. These measurements can be made with a time-to- amplitude converter and a multi-channel analyzer2, or by direct counting of a high-frequency pulse train or 'clock', using a real-time computer to store the number of 'clock' pulses between successive detector pulses, as was done in the test examples reported herein. Method 1 The most precise estimates for the unknown parameters in a probability distribution are usually those found by the method of maximum likelihood3. This method requires that the logarithm of the joint p.d.f. of the tj be maximized with respect to the unknown parameters n and T. The joint p.d.f. for N intervals is given by p(t;. . .,tN) = nN e- nZti-T)e and the log-likelihood by log(L) = Nlog(n) - nYt. + nNT. Maximization with respect to n gives 31log(L) = O = N/n - Etj - N- O=N/nZt. N so that we again find that n = N/(T-NT) = 1/(tav - T) ..........*(2) where T = Ztj (the total counting time) and tav = T/N. The total time minus the total dead-time, T-NT, is often referred to as the 'live-time'. Since the log-likelihood increases linearly as T increases, the maximizing value of T must be the largest value compatible with the condition ti . T, and this is the minimum value of tj, tmin. When a finite number of ti are examined, the true value of T will usually be slightly smaller than tmin, however, and to calculate the bias involved in using tmin as an estimate of the dead-time some more mathematical development is needed although for large N the correction turns out to be negligible. If we sort the ti in ascending order so that T<tl<t2. . . <tN then the joint p.d.f. has the same form as in the unsorted case, but is multiplied by a factor N!4. Putting ti = tmin and substituting si for tj where sj = ti - tmin, for i > 1, gives for T<tmin<c- and O<s2 . . . <SN P(tmin,S2...SN) = N!nNe-n(tnin-T)e-nz(si+tmin-T) = nNenN(tmin-T) nN-1 (N-1)2e-n = P(tmin) - P(s2-- * sN). The p.d.f. of tmin is the same as that for a single interval (equation 1) with an effective 'rate' of nN, therefore the average value of tmin will be T + 1/nN and, from equation 2, T=(Ntmin-tav)/(N-1). The p.d.f. of the si is the same as that for the sorted tj with zero dead-time and N reduced by 1. Therefore, the p.d.f. for S = Esi = T-Ntmin (the estimated live-time) will be similar to that for T in the case where T = 0, which can be shown to be5 p(T) = n NTN-1 e-nT/(N1) Therefore p(S) = nNi1 sN-2 e--nS/(N-2) The average of S is (N-1)/n, however, the average value, r, of 1/S, which we wish to use to estimate n, is given by r = f S-lp(S)ds = n/(N-2), with a fractional standard 0 deviation of /(f S-2p(S)ds - r2)/rt = 1/(N-3). 0018-9499/82/0200-0629$00.75 / 1982 IEEE 629
Transcript
IEEE Transactions on Nuclear Science, Vol. NS-29, No. 1, February 1982
ESTIMATION AND CORRECTION OF COUNTING DEAD-TIME BYUSING INFORMATION CONTAINED IN THE TIM1E DISTRIBUTION OF THE EVENTS
R.M. HolfordAtomic Energy of Canada Research Company
Chalk River Nuclear LaboratoriesChalk River, Ontario, Canada
KOJ 1JO
SummaryMethods for extending the high counting-rate
capability of nuclear instruments which use Geigercounters, by correcting for dead-time losses, arecompared. Three methods are described which allowfor correction of such losses without a prior know-ledge of the dead-time.
IntroductionEstimation and correction of counting dead-time
losses allows the high counting-rate capability ofnuclear instrumentation to be extended by as much asan order of magnitude1. Methods which have been pro-posed for correction of such losses require that thevalue of the dead-time be known beforehand and thatit be a constant, e.g., unaffected by the countingrate. In this communication, techniques are des-cribed that overcome these requirements by derivingestimates of the dead-time from the time-intervaldistribution of the counting events.
A Geiger counter is typical of detectorsexhibiting a non-extended dead-time2. Potentialevents (radiation interactions within the detector)occurring within a fixed interval T following acounted pulse have no observable effect and the nextpulse is initiated by the first potential eventoccurring after this interval. The probabilitydensity function (p.d.f.) for the intervals betweenthe pulses from such a detector is given by2
p(t) = n efn(t0T), t>T ..............(1)
where n is the rate at which potential events occur.The average value of t is r + 1/n; thereforen = 1/(tav-T), the standard formiula for known T.
The methods to be described for correcting foran unknown T are based on measurements of the individ-ual time intervals, ti, between successive pulses.These measurements can be made with a time-to-amplitude converter and a multi-channel analyzer2,or by direct counting of a high-frequency pulse trainor 'clock', using a real-time computer to store thenumber of 'clock' pulses between successive detectorpulses, as was done in the test examples reportedherein.
Method 1The most precise estimates for the unknown
parameters in a probability distribution are usuallythose found by the method of maximum likelihood3.This method requires that the logarithm of the jointp.d.f. of the tj be maximized with respect to theunknown parameters n and T. The joint p.d.f. for Nintervals is given by
p(t;. . .,tN) = nN e-nZti-T)eand the log-likelihood by
log(L) = Nlog(n) - nYt. + nNT.
Maximization with respect to n gives
31log(L) = O = N/n - Etj - N-O=N/nZt. N
so that we again find that
n = N/(T-NT) = 1/(tav - T) ..........*(2)where T = Ztj (the total counting time) andtav = T/N. The total time minus the total dead-time,T-NT, is often referred to as the 'live-time'.
Since the log-likelihood increases linearly asT increases, the maximizing value of T must be thelargest value compatible with the condition ti . T,and this is the minimum value of tj, tmin. When afinite number of ti are examined, the true value ofT will usually be slightly smaller than tmin, however,and to calculate the bias involved in using tmin asan estimate of the dead-time some more mathematicaldevelopment is needed although for large N thecorrection turns out to be negligible.
If we sort the ti in ascending order so thatT<tl<t2. . . <tN then the joint p.d.f. has the sameform as in the unsorted case, but is multiplied by afactor N!4. Putting ti = tmin and substituting sifor tj where sj = ti - tmin, for i > 1, gives forT<tmin<c- and O<s2 . . . <SN
P(tmin,S2...SN) = N!nNe-n(tnin-T)e-nz(si+tmin-T)
= nNenN(tmin-T) nN-1 (N-1)2e-n
= P(tmin) - P(s2--* sN).The p.d.f. of tmin is the same as that for a
single interval (equation 1) with an effective 'rate'of nN, therefore the average value of tmin will beT + 1/nN and, from equation 2, T=(Ntmin-tav)/(N-1).
The p.d.f. of the si is the same as that for thesorted tj with zero dead-time and N reduced by 1.Therefore, the p.d.f. for S = Esi = T-Ntmin (theestimated live-time) will be similar to that for Tin the case where T = 0, which can be shown to be5
The average of S is (N-1)/n, however, the averagevalue, r, of 1/S, which we wish to use to estimaten, is given by
r = f S-lp(S)ds = n/(N-2), with a fractional standard0
deviation of /(f S-2p(S)ds - r2)/rt = 1/(N-3).
0018-9499/82/0200-0629$00.75 / 1982 IEEE 629
An unbiased estimate of the 'true' counting rate istherefore n = (N-2)/(T-Ntmin), and for large N we canuse the approximation n Z 1/(tav - tmin) with stand-ard error n/Aff.
Method 2
If N = 2, T = tl + t2 and tmin = t or t2. Ineither case s = T-2tmin = It2 - t1l, and p(s) = nens.This has the form given by equation 1 with T = 0.Therefore, we can take a long event train containingN successive intervals and form the sum
N
i-2 I - tj 1l
which has an average value of (N-1)/n and provides analternative estimate of the live-time and thereforeof n. Because of correlation between successiveterms in the summation, the standard deviation of thisestimate (as found by computer simulation usingexponentially distributed random numbers) is about30% greater than for method 1. The loss of precisionmay not be significant in high counting-rateapplications, however, and this method has theadvantage of being less sensitive than method 1 tointerference from any extraneous pulses which mightoccur during the dead-time of the detector,
This method is also simpler to implement usingavailable digital integrated circuits since therequirement for absolute differences can be replacedby a requirement to measure only positive differences(dim), defined by
dim(ti,tiij)=ti-tiij if ti>ti-1, = 0 if tj<t; 1.
The relationship between the differences is
|ti-ti-11+ti-ti- l= 2 dim (ti,ti-1), therefore
NE ti-tijI =2ZdiM(ti ,ti 1)-Z(ti-ti_1)
i=2= 2Mdim(ti,ti-l) - (tN-tl).
The remainder term, tN-tl, has an average value ofzero and makes only a small contribution to thevariance. Hence, for long pulse trains, twice thesum of the positive differences can replace the sumof the absolute differences with negligible loss ofprecision in estimating n. This permits the designof a relatively straight-forward correction circuitdescribed in the Appendix.
Method 3
This method requires some knowledge of the actualvalue of the dead-time, so that a cut-off time, T',can be selected which is known to be greater than T.The true counting rate, n, is estimated from theaverage of those intervals which exceed T'.
t'av = f nten(tTl)dt/f ne-n(t-T)dtT
- TI + 1/n
n= 1/(t'av-T').This method is equivalent to adding a second,
externally defined, dead-time t', of the extendedtype, to the counting circuit - a technique whichhas previously been proposed for precise correctionof dead-time losses2.
Experimental Tests
Time interval distributions were obtained atfour different counting rates using a small computerand a radiation survey meter based on a Philips type18545 Geiger counter6, which is specified by themanufacturer as having a dead-time of 200 ,us. Whileaccumulating the time-interval distribution thecomputer was also used to obtain a spectrum of thepositive differences between successive intervals, toallow for an estimate of counting rate using method 2.
The low-level time-interval distribution, Figure1, has the form predicted by equation 1 with a dead-time of about 160 us. At higher counting rates thespectrum is more complex (Figures 2,3,4) and the dead-time appears to decrease to about a third of the low-rate value. The spectrum of the differences shouldbe a straight line on a log plot with a slope equalto the 'true' counting rate. This form is indeedseen in Figures 2 and 3, but there is a markedcurvature to the line at low time-intervals in Figure4.
Table 1 shows the results obtained by calculatingthe 'true' counting rate (n) using the three methodsdescribed above and also the result of using a fixeddead-time of 160 ps, the experimentally determinedlow-rate value. Method 1 was modified to simulatethe effect of taking the intervals in groups of 100,rather than as a single group of a million or more,by taking tmin as the 1% point of the observedcumulative time-interval distribution.
The 'best estimate' was calculated from the slopeof the time-interval spectrum, and of the differencespectrum, for large intervals; except at the lowestrate where equation 2 was used with the observedvalue of T (157 ps from Figure 1). At the highestrate a second estimate was available, based on know-ing the approximate ratio of the radiation fieldintensities at the two highest rates. The value inthe table is the average of these two estimates, whichagreed to within 20%.
Discussion
The 18545 counter provides a severe test of anycorrection method because of the extensive changesin its dead-time behaviour as the counting rateincreases. At the higher rates the average timeinterval between pulses is less than the low-ratedead-time, therefore the correction method based ona fixed dead-time (equation 2) fails completely.Method 3 probably gives the most accurate resultsat all but the highest rate, but it too fails whenthe rate is so high that hardly any of the pulseintervals exceed the externally defined threshold, T'.Methods 1 and 2 can be used at any counting rate,however, Method 1 is much more vulnerable than Method2 to departures of the time-interval spectrum fromthe ideal form described by Equation 1. Method 2gives good results except at the highest counting-ratewhere its prediction is about 50% low. Thiscorresponds to the departure from the ideal seen inthe difference spectrum shown in Figure 4. A moreaccurate estimate of the true rate in Figure 4 can beobtained from the slope of the difference curve above20 us, that is by a combination of Methods 2 and 3.
The use of the differences between successivetime-intervals, Method 2, therefore shows promise as
a way of extending the counting-rate range of Geigercounters upward by at least an order of magnitude over
that obtainable with correction methods which rely on
a known, constant, dead-time, but a fuller understand-
Observed Rate Fixed D.T. Method 1 Method 2 Method 3 Best Estimate Av. D.T.Counts/sec (T=160 pIs) (M=100) (T'=160 Ps) (see text)
16.1 x 10 00 4.27 x 103! 93.8 x 103 I* 1X180 x 103 56 us
7.65 x 103 0 12.5 x 1031 23.3 x 10i 23.7 x 10i 23.5 x 101 88 Ps
830 957 929 926 925 925 124 Ps
17.7 17.75 -+ -+ 17.75 157 us
* Too few intervals >160 ps to give an average 1/observed rate - 1/true rate+
Average time interval exceeded the scaling caoacity of the test circuit
ing of the factors determining the observed pulse-interval distributions at high rates is requiredbefore this method can be used with confidence inpractical instruments. For instance, the amplitudeof the pulses from a Geiger counter decreases at highcounting rates, due to the recovery effect, and as aresult some 'events' within the counter may fail toregister as 'counts' in the external circuit. If anappreciable proportion of 'events' were missed inthis way, the effective dead-time would exhibitpartially extended behaviour and the observed ratewould tend to zero rather than to a constant value asthe intensity of the radiation field was increasedwithout limit.
Acknowledgement
This idea grew out of the work described in Ref.1 and I wish to thank my colleague and co-author onthat paper, A.R. Jones, for helpful discussions andencouragement in preparing this communication. Iwould also like to thank L. Shankland and P. Bungefor their technical assistance.
References
1. Jones, A.R. and Holford, R.M., Nuclear Instrumentsand Methods (in press).
2. MUller, J.W., Nuclear Instruments and Methods,112 (1973) 47.
3. Lindgren, B.W., Statistical. Theory (Macmillan,1968, p. 280 f.f.
4. op. cit. p.402.5. op. cit. p.223, example 4-38.6. Jones, A.R., Two Survey Meters for Measuring Low
y-ray Dose Rates (1979), Atomic Energy of CanadaLimited Report No. AECL-6407.
Appendi xAn Electronic Circuit for Dead-Time Correction Using
Method 2
This circuit is shown as a block diagram inFigure A.1. Pulses from a high frequency 'clock' arecontinually counted by the 'up' counter until an'event', a pulse from the Geiger counter, occurs. Atthis point in time the 'down' counter is preset tothe current state of the 'up' counter and the 'up'counter is reset. The 'down' counter also receivesthe clock pulses and, if it reaches zero before thenext event occurs, a gate is opened to admit the clockpulses to the live-time prescaler until the next eventoccurs. The number of clock pulses transmitted to thelive-time prescaler is thus a measure of the positivedifference between successive time intervals; if the
difference between two intervals is negative the 'dowrycounter does not reach zero during the second intervaland no pulses are admitted to the live-time circuit.As shown in the main text, the sum of the positivedifferences between time intervals should be equal tohalf the live-time of the detector; the factor of ahalf is allowed for in the live-time pre-scaler whichis set to divide by 5000 rather than 10,000 whichwould be needed to convert a 10 MHz pulse train into1 ms intervals.
At low counting-rates dead-time correction isunnecessary and, to avoid the loss of precision itintroduces, the circuit can be switched to normaloperation by replacing the gated live-time pulseswith a steady pulse train into the live-time pre-scaler at half the clock frequency. In either casecounting proceeds for a fixed time or for a fixednumber of events, and the counting rate is estimatedin the usual way as the number of events divided bythe accumulated time. In Figure A.1 this function isperformed by a commercial counter-timer (Fluke Model1953A), which also provides a convenient source forthe 10 MHz 'clock' signal. The instrument is usedin 'FREQ A/B' mode with the live-time signal (atapproximately 1 kHz) applied to its 'B' input and thepulses from the Geiger counter applied to its 'A'input. In this mode the instrument accumulates 'A'and 'B' input pulses until a preset number of 'B'pulses (a power of ten selected by a front panelcontrol switch) is reached. It then displays thenumber of 'A' pulses accumulated with the decimalpoint adjusted to give the average number of 'A'pulses per 'B' pulse, which in this case representscounts per millisecond or thousands of counts persecond. The instrument can be set to perform thisoperation repetitively and therefore provides acontinuous display of the counting rate with aresponse time determined by the number of 'B' pulsesrequired in each accumulation interval.
ea d4Btim c Orueti oiruiuin CorritionLve TM
eCounder|Clecke m i 50002
\|/ ~Preset| mlieod'
*- T 1~~~~~0MH Counter-Timer\ Reset 24 Bit CokOt set to FREa A/B'
-oray 'Up Counter / Fluke model 1953A)
Figure Al
Dead-time correction circuit using positivedifference between time intervals (method 2)
