Applied Radiation and Isotopes 69 (2011) 790–802

Contents lists available at ScienceDirect

Applied Radiation and Isotopes

0969-80

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/apradiso

Simulation of a method for determining one-dimensional 137Cs distributionusing multiple gamma spectroscopic measurements with an adjustablecylindrical collimator and center shield

Z.D. Whetstone, S.C. Dewey, K.J. Kearfott n

Radiological Health Engineering Laboratory, Department of Nuclear Engineering and Radiological Sciences, University of Michigan, 2355 Bonisteel Boulevard, 1906 Cooley Building,

Ann Arbor, MI 48109-2104, USA

a r t i c l e i n f o

Article history:

Received 2 April 2010

Received in revised form

8 January 2011

Accepted 14 January 2011Available online 26 January 2011

Keywords:

Germanium

Environmental measurements

Radiation detection

Positional measurements

Radionuclide distributions

In situ gamma ray spectroscopy

43/$ - see front matter & 2011 Elsevier Ltd. A

016/j.apradiso.2011.01.019

esponding author. Tel.: +1 7347639117; fax:

ail address: kearfott@umich.edu (K.J. Kearfott

a b s t r a c t

With multiple in situ gamma spectroscopic measurements obtained with an adjustable cylindrical

collimator and a circular shield, the arbitrary one-dimensional distribution of radioactive material can

be determined. The detector responses are theoretically calculated, field measurements obtained, and a

system of equations relating detector response to measurement geometry and activity distribution

solved to estimate the distribution. This paper demonstrates the method by simulating multiple

scenarios and providing analysis of the system conditioning.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The horizontal distribution of the source is often known orconstant over a large area for in situ gamma spectroscopy (Becket al., 1972; Helfer and Miller, 1988; Korun et al., 1994), but thesource distribution with depth is not. This paper introduces amethod to determine the one-dimensional (1-D) distribution of aradioactive source with no a priori activity distribution assump-tions and using a single collimator system.

The calibrated response of in situ spectroscopy systems maybe derived using a pre-determined depth distribution, such asGaussian, exponential, linear, uniform, planar, or other analyticalfunctions (Cutshall and Larsen, 1986; Helfer and Miller, 1988;Department of Energy, 1997). The parameters of each distributionare determined based upon the history of the site, other sites withsimilar properties, soil transport modeling, laboratory samplesfrom nearby sites, or by selecting typical values. However, none ofthe distributions may fit the actual distribution, leading toinaccurate total activity determinations. Previous approaches toactually determining the vertical distribution of a radioactivesource have included the multiple photopeak method, the peak to

ll rights reserved.

+1 7347634540.

).

valley ratio method, the lead plate method, and a method usingmultiple collimators.

The multiple photopeak method, limited because it appliesonly when more than one photopeak is available, was originallyproposed by Beck et al. (1972). By comparing the attenuationwith photon energy, the average photon path length can bedetermined with an estimate of the source depth (Sowa et al.,1989). The number of unknowns, or depth layers, in an analyticalrepresentation of a source distribution is restrained by thenumber of radionuclide photopeaks. Successful application ofthe technique has been made at the Chernobyl site for 140La and134Ca (Karlberg, 1990), and by comparing the primary photopeakand the X-ray lines of 137mBa and 137Cs (Rybacek et al., 1992).

Zombori et al. (1992) proposed using the ratio of the photo-peak to the adjacent scatter to determine the source depth. Themethod compares the total counts in the 662 keV 137Cs photopeakto those in the Compton edge valley from the 649 to 631 keV. Theratio of these two values depends on the average photon pathlength and therefore contains depth distribution information.Since the measurement is based on scattered photons, it requireshigher activity emitters and may be affected by interferingphotopeaks that have scattered photons into the same valleyregion. Tyler (1999) found that the method leads to littleimprovement in actual results. Chesnokov et al. (1997) added acylindrical collimator to the method, again with insignificantimprovement (Golosov et al., 2000). Korun et al. (1994)

Fig. 1. Diagram of the assumed source media distribution in the multiple layer

method. The source media is assumed to consist of multiple layers with each layer

having uniform activity within that layer.

Z.D. Whetstone et al. / Applied Radiation and Isotopes 69 (2011) 790–802 791

and Tyler (1999) used a single circular lead plate placed at varyingdistances in front of the detector to determine the coefficients of aGaussian distributed source, but did not extend this to othersource types.

By taking a series of independent measurements with varyinggeometries, which is generalizing the work done by Korun et al.(1994) and Tyler (1999), it is possible to actually calculate directlythe 1-D source distribution of an unknown functional shape bynumerically solving a set of linked equations (Benke and Kearfott,2001, 2002). Each measurement has a response that can becalculated theoretically or estimated with Monte Carlo techniques(Benke and Kearfott, 2001, 2002; Dewey et al., 2010). Thisresponse is dependent upon the source distribution and measure-ment geometry, and represents a unique, independent equationfor each distinct system geometry. The resulting set of equationscan be solved numerically. Such an approach requires no a prioriassumptions about source distribution shape, and is applicable toany radionuclide, since it does not require multiple spectralpeaks. Only the primary photopeak is used, so the poor countingstatistics of scattered photons is irrelevant.

Using such an approach, Benke and Kearfott (2001, 2002)employed a set of cylindrically symmetric collimators, each witha different field of view (FOV), to determine the depth of a multipleplanar sources as well as the 1-D distribution of a completelyunknown source. The detector response for each collimator andmeasurement geometry was calculated using commercially avail-able Monte Carlo software (In Situ Object Counting System (ISOCS)software, Canberra 800 Research Parkway, Meriden, CT 06450,USA). Unfortunately, the approach is limited because it requires aseparate prefabricated collimator for each measurement. As it isimpossible to have more independent measurements than thenumber of collimators, the measurement geometry cannot beoptimized for specific scenarios. A set of four or five collimatorswith masses �50 kg each would also result in an impractical fieldinstrument weight.

The method presented here uses the same general approachas Benke and Kearfott (2001, 2002), but employs a singlecommercially available lead cylindrical collimator and a leadcircular shield. The arrangement is such that collimator and shieldpositions can be easily adjusted relative to the detector, resultingin a theoretically unlimited range of FOVs. An unlimited numberof mathematically independent measurements with a singlecollimator are possible, resulting in a field instrument that isboth lighter and more flexible than that of Benke and Kearfott(2001, 2002). Methods are presented here for determining thedepth and activity of a planar source, the parameters describingan exponentially distributed source, and the discretized distribu-tion of any arbitrarily shaped vertical source distribution usingsuch a collimation system.

This method differs from the ISOCS system because it uses avariable detector response, based on several measurements and achanging shielding geometry, to determine the one-dimensionaldistribution of an unknown source. ISOCS requires that the sourcedistribution and shielding geometry are known, and will providethe response function of the detector (Canberra Industries, 2009).ISOCS can be used to determine an unknown distribution throughuse of guess and check and through many possible iterations, butthe method presented here can provide an estimation of thedistribution using several measurements and equations thatdescribe the previously known detector response function in nearreal time.

Conditioning is a mathematical description of the effect of inputerrors on errors in calculated results (Trefethen and Bau, 1997;Heath, 2002). Methods for determining the source distribution relyprimarily on measuring differences in photon attenuation. Sincethis is an exponential process, small errors in measuring the

photons in the upper source depths will mask the measurementof any photons originating from deeper locations, resulting in largeerrors. This paper includes a detailed error analysis of the newmethod, examined by introducing randomized errors in measure-ments and observing the effects. Condition numbers for the systemare computed and used to optimize the measurement geometries.Conditioning improvement techniques were developed to avoidlarger errors that would limit results.

This paper is meant to serve as an introduction to this newmethod proposed here. The underlying theory and preliminarymeasurements will be discussed, as well as an attempt tocharacterize the sources of error. However, refinement of themethod will be left to future papers.

2. Methods: Theory

2.1. Detector response determination

The overall system geometry is shown in Fig. 1. The detectorresponse for a particular source distribution and collimatorposition is calculated using analytical and empirically derivedexpressions presented previously for the calibration of non-collimated in situ detectors (Dewey et al., 2010). The expressionfor the detector response to a semi-infinite volume source, Rvolume

(E, d, tmax) is

RvolumeðE,d,tmaxÞ ¼

Z tmax

0

Z p

0

SðtÞARðE,yÞRcðEÞ

2½ðdþtÞ=cosðyÞ�2

� e�mmediaðEÞt=cosðyÞe�mairðEÞ½p�t=cosðyÞ�dydt, ð1Þ

where E is the energy of the photon, d is the distance from lowerdetector face to media surface (cm), S(t) is the source activity(Bq cm�3) at depth t (cm) in the media, AR(E, y) is the angulardetector response (dimensionless) at angle y (radians), Rc(E) is thecenterline detector response (cps cm5 Bq�1), t is the depth in themedia from the surface (cm), mmedia is the media’s attenuationcoefficient (cm�1), mair is the attenuation coefficient of air (cm�1),p is the distance from the point of interest to the lowest point on

Z.D. Whetstone et al. / Applied Radiation and Isotopes 69 (2011) 790–802792

the detector centerline (cm), and tmax is the maximum depth ofinterest (cm).

To include the effect of collimation, a collimation factor CR(E, y, cd)is introduced. This is defined as the ratio of the detector responsewith no collimator to a point source at a particular angle, y, tothe detector response with the collimator to the same point source atthe same angle. The position of the collimator cd is defined as theadjustable distance from the lower face of the detector to the planeof the lower collimator face. By varying cd the FOV is changed,altering the detector response. The collimation factor for a cylindricalcollimator is not simple to calculate analytically because of thecomplex geometry, but may be measured directly using pointsources, simulated with a Monte Carlo radiation transport code, ordetermined using a combination of both approaches (Dewey et al.,2010).

The final expression of the detector photopeak response to avolume source with collimation taken into account becomes

RvolumeðE,d,tmaxÞ ¼

Z tmax

0

Z p

bmin

SpðtÞARðE,yÞCRðE,y,cdÞRcðEÞ

2½ðdþtÞ=cosðyÞ�2

� e�mmediaðEÞt=cosðyÞe�mairðEÞ½p�t=cosðyÞ�dydt, ð2Þ

where Sp is the activity of a point source and bmin is the angleoccluded by the shield and is equal to zero when a shield isnot used.

Since Eq. (2) is evaluated numerically, it is possible to solve forunknown parameters in an analytical expression of Sp(t), thesource distribution, as long as minimal assumptions are made. Insuch cases, fewer measurements would be needed to fullycharacterize the source distribution. The cases of determinationof the depth of a single planar source and an exponentialdistribution will be considered, along with the most general casefor which no prior knowledge of source distribution shape isneeded.

Fig. 2. Geometry used in performing the measurements. The vertical position of

the cylindrical collimator is varied to alter the outer field of view (FOV) of the

detector. The circular lead shield is used to alter the inner FOV.

2.2. Planar source depth

Using two measurements, the depth of an infinite planar, orthin volume, source and its activity can be estimated in anymedia. The detector response to a unit area activity planar sourcecan be calculated for each measurement for a range of depthsusing Eq. (2) above. For any two measurements:

R1ðdÞ ¼ Ar1ðdÞ,

R2ðdÞ ¼ Ar2ðdÞ, ð3Þ

where Ri represents the photopeak detector response for thecollimator position (cps), A is the activity concentration of theplanar source (Bq cm�2), and ri corresponds to the calculateddetector response (cps Bq�1 cm2) using Eq. (2) for collimatorposition i.

The ratio of R1 and R2 is dependent upon the source depth inthe media but not upon the source activity, as shown by

ratioðdÞ ¼R1 dð Þ

R2 dð Þ¼

Ar1 dð Þ

Ar2 dð Þ¼

r1 dð Þ

r2 dð Þ, ð4Þ

which has the single unknown, d, the planar source depth. Thisdepth can be solved applying iterative numerical methods toEqs. (2) and (4). Alternatively, the planar source depth may bedetermined by graphing the ratio for a range of unit activitysource depths and determining the depth from the plot. Once thedepth is known then the source activity per unit area can becalculated by solving Eq. (3) for the activity concentration.

2.3. Exponentially distributed source

A common model for a radioactive material source distributionin soil is an exponential, i.e.

SpðtÞ ¼ S0 e�at , ð5Þ

where a is the inverse of the relaxation length of the one-dimensional activity distribution and S0 is the surface activityconcentration. Since the response will vary linearly with S0, theresults of the two measurements are

R1ðaÞ ¼ S0r1ðaÞ,R2ðaÞ ¼ S0r2ðaÞ, ð6Þ

where Ri is the detector response for the collimator position (cps),S0 is the activity concentration of the source at the media surface(Bq cm�3), and ri is the calculated detector response using Eq. (2)for the collimator in position i (cps cm3 Bq�1).

The ratio of the measurement from two collimator positionscan be taken for each a, namely:

ratioðaÞ ¼ R1ðaÞR2ðaÞ

¼S0r1ðaÞS0r2ðaÞ

¼r1ðaÞr2ðaÞ

: ð7Þ

By comparing the ratio of two measurements to the calculatedresults, a can be determined. A non-linear numerical method canalso be used to solve Eqs. (2) and (7) for a.

This overall approach could be applied to any 1-D distributionthat is described by an analytical equation employing two ormore variables.

2.4. Arbitrary source

Often, a source distribution shape may not be known or mayvary arbitrarily with depth and cannot be modeled with a simpleanalytical expression. Under those circumstances this method hasgreat advantages over others. The distribution of such sources canbe estimated by discretizing the source into smaller separatelayers with assumed uniform activity, say from ti to ti +1. The totalvolume source is then assumed to consist of several such layers,as shown in Fig. 2. If the source is divided into m multiple layersand there are n measurements, each with the collimator andshield uniquely positioned, then if nZm the system is consideredto be well-defined. In such cases, the detector response to auniform volume source of unit volume activity at a depth from tj

to tj + 1 can be calculated using Eq. (2). The detector response toeach discrete layer for each measurement geometry can beexpressed as

Ri ¼ Ajri,j, ð8Þ

where Ri is the detector response in a single position. Aj is theuniform activity distribution, in Bq cm�3, of the jth layer, which isthe unknown to be determined. The term ri,j is the calculatedresponse contribution from the jth uniform volume layer to the

Z.D. Whetstone et al. / Applied Radiation and Isotopes 69 (2011) 790–802 793

ith measurement. For a single measurement, the total detectorresponse to the total source consisting of m layers is

Ri ¼ A1ri,1þA2r2,iþ � � � þAmRi,m: ð9Þ

The results of n multiple measurements of a source consistingof m uniform layers results in the following system of equations:

R1

^

Rn

0B@

1CA¼

r1,1 � � � r1,m

^ � � � ^

rn,1 � � � rn,m

0B@

1CA

A1

^

Am

0B@

1CA

R¼ rA ð10Þ

where R is a vector of the detector response at each collimatorposition used, r is an n by m matrix of the calculated unit detectorresponses with a collimator in position j to a volume source ofunit volume activity i, and A is a vector of the activity concentra-tions present in each layer.

Since all of the factors of the r matrix can be calculated using Eq.(2), and the R vector entries are the measured detector responses,the only unknowns are the values of the A vector. This linearsystem of equations can be solved using a variety of numericalmethods (Trefethen and Bau, 1997; Heath, 2002). The only a prioriassumption is that the source has horizontally uniform activity,thus this method can be applied for determining the 1-D distribu-tion of any arbitrary source. Several scenarios were designed todemonstrate the efficacy of this method, all of which had nZm.These included a three-layer system and four-layer system thatwere exactly (n¼m) and over- (n4m) determined.

2.5. Design optimization

The selection of measurement geometries plays a significant rolein the system error properties. The measurement system, in apractical sense, determines the source distribution by comparingthe mean photon path length through the source media that reachthe detector. For example, for a planar source near the surface,photons will all travel an equal distance of zero through the sourcemedia before reaching the detector. For a deeper source, photonsfarther from the detector centerline will travel a larger mean distancethrough the media than photons produced closer to the centerline.The system with ideal error properties would have measurementsthat have the greatest dependence on source depth. This is accom-plished by having the first measurement with the primary source ofphotons generated near the detector centerline and the secondmeasurement with the primary source of photons generated farfrom the centerline. The collimator controls which peripheralphotons reach the detector. The shield controls whether the center-line photons reach the detector, thus allowing for measurementswhere the primary source of photons are from sources far from thedetector centerline. The more the ratio of the two measurementsvaries with source depth the better the system error properties are.

It can be hypothesized that the ideal measurement for a three-measurement system consists of three-measurement configura-tions. In the first, photons from the periphery are the only photonsthat reach the detector. These photons have the longest paththrough the attenuating media to reach the detector. The secondmeasurement configuration will only allow photons from thedetector centerline to reach the detector, and have the shortestpath through the attenuating media. The third measurementwould consist of some measurement configuration between thesetwo extremes that minimizes the system condition number. Thisassumes, however, that the measurement errors for each mea-surement are equal. This is not true as the detector’s efficiency foreach measurement will be dependent upon the collimator andshield positions.

3. Methods: Error analysis

3.1. Error analysis by random perturbation method

Gamma spectroscopy measurements have inherent errorsfrom counting statistics in background and total photopeakcounts, peak area calculations, variations in background levels,calibration sources, and calculated detector response. The effectsof these uncertainties on the results of a measurement methodcan be examined by perturbing the mathematical system inputswith Gaussian random variables that represent errors in mea-sured values, repeating this multiple times, and determining thestandard deviations of the results (Kearfott et al., 1995).

This paper uses this approach to analyze the error associatedwith the proposed method. The detector responses are calculatedusing Eq. (2), which represents the ‘‘true’’ or exact measurement.For actual measurements, there is some error present in eachmeasured response R. For counting errors, this error is Gaussian innature and is represented by the variable e. Each measurement hasa distinct e that is determined by generating a random numberthat has a Gaussian distribution. This perturbation is then added toeach exact measurement. The source activity, depth, parameters ofthe analytical function describing the distribution, or the 1-Ddistribution is then calculated using the perturbed measurementsto obtain a perturbed result. This perturbed result is then com-pared to the actual results to determine the effect of the addedinput errors in the measured values on the calculated solution. Theprocess is repeated numerous times using different randomvariables, and a standard deviation for the activity distributionor parameters which describe it is then computed. The number ofrepetitions applied, 10,000–20,000 for this work, is such thatsmooth results histograms are obtained. Standard deviations inthe measurements of 5% and 10% were assumed for most cases,but varied between 0.1% and 20% for others.

3.2. Condition numbers

The condition number of a matrix is a measure of the errorproperties of a matrix when it is used in the solution of a linearsystem of equations of the form Ax¼b (Heath, 2002). If errors areintroduced into the vector b then the equation can be expressedas A(x+Dx)¼b+Db, where errors in the vector b represent errorsin the measured detector response. The following equation holds(Heath, 2002):

:Dx::x:

rcondðAÞ:Db::b:

, ð11Þ

where :x:, :b:, :Dx:, and :Db: are the 2-norms of the vectors,measures of vector length. The condition number of the matrix A

is a measure of the system susceptibility to errors. A highcondition number implies that small errors in inputs, in this casemeasured detector responses, will lead to large errors in calcu-lated results, namely the calculated activity distribution. Such asystem is referred to as a poorly conditioned system.

3.3. Three-measurement system condition number

Assuming for a three-measurement system the measurementtimes are selected such that the counting errors from each measure-ment are equal, then from Eq. (11) it can be shown that:

ZmaxrcondðrÞY, ð12Þ

where Zmax is the maximum possible percent error in the calculatedactivity, and Y is the percent error in each of the measurements.The addition of a shield will decrease the measurement efficiency.

Z.D. Whetstone et al. / Applied Radiation and Isotopes 69 (2011) 790–802794

The measurement standard deviation due to counting statistics willvary in proportion to the inverse of the square root of the totalphotopeak counts. As the size of the shield increases, the systemcondition number decreases. Also, the measurement errors increasesince the detection system efficiency decreases. If no shield is usedwhile making the first measurement, but then it is used while takinga second measurement, the ratio of the maximum percent error inthe calculated activity for each measurement scenario will be:

Zmax,2

Zmax,1r

condðrÞ2condðrÞ1

ffiffiffiffiffie1

e2

r, ð13Þ

where ei is the efficiency of each measurement. Thus the pointwhere the above ratio is minimized is the point where the lowestoutput errors would be obtained.

For systems using more than two measurements, the selectionof the third measurement geometry has a significant effect on thesystem error properties. For linear systems, the measurement thatproduces an r matrix that is closest to orthogonal produces thelowest condition number. For each measurement system, thecondition numbers of a variety of measurement geometries werecalculated to select the optimal measurement geometries.

3.4. Condition numbers for exponential attenuation

Since exponential attenuation is a primary contributor to poorconditioning, the maximum depth was varied from 10 to 30 cmand condition numbers determined. Although activity at greaterdepths may not contribute a significant amount to the totalresponse of a detector, it may contribute a significant fraction tothe total site activity. However, some locations may only haveconcentration in the top layers of soil. Previous studies demon-strated that even after 10 years cesium deposited on the soilsurface was primarily concentrated in the top 5 cm of untilled soil(Ritchie and McHenry, 1973). Additionally, since in most casesactivity below 30 cm has no significant impact on detectorresponse, it can be neglected in determining detector response.

Fig. 3. Ratio of the calculated detector photopeak count rates from two measure-

ments for a series of semi-infinite 662 keV planar sources from 0 to 30 cm depth in

soil. The ratio is the detector response with the collimator in the 11 cm position to

the detector response with no collimator.

4. Simulation methods

4.1. System and sources

For the adjustable collimation approach, a spectroscopicdetector with sufficient energy resolution is used either bare orwith a collimator at a variable position relative to the detectorand an optional lead shield, as shown in Fig. 1. For this work, anHPGe detector, which is 8.2 cm in diameter and 9.0 cm in height,is simulated with a commercially available 20 cm long, 5.5 cminner radius, 1.6 cm wall thickness lead collimator (Ortec SHD-1Shield 801 South Illinois Avenue Oak Ridge, TN 37830, USA). The20 cm in radius shield is 2.5 cm thick. The collimator position cd,is adjustable from 0 to 11 cm. The position of the shield for anygiven measurement was specified by its angle, y. The detector wasa distance, d, of 1 m above the source media surface.

Standard 1.6 g cm�2 soil (Eckerman and Ryman, 1993) wasselected for simulation, with 137Cs, having a 662 keV gamma,chosen as the test radionuclide. Normalized values of activityconcentration were used.

4.2. Detector responses and system measurements

The detector responses were calculated using Monte CarloN-Particle Transport Code (MCNP) (MCNP Version 5, Los AlamosNational Laboratory, P.O. Box 1663, Los Alamos, NM 87545, USA).A sufficient number of particle histories were used so that MCNPreported uncertainties less than 5%. Point sources were simulated

at different angles with smaller intervals at smaller angles. A fullrange of collimator positions was modeled and source energiessimulated from 50 to 1000 keV.

An adaptive Simpson’s quadrature method (McKeeman, 1962;Gander and Gautschi, 2000) was implemented using commer-cially available mathematics software (MATLAB version 2006, TheMathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, USA)to numerically integrate Eq. (2) to obtain the system response, ormeasurement, due to any source distribution and detector-colli-mator combination. The effect of including the optional centerlineshield was included by limiting the range of numerical integrationof y in the integration. This limiting value will be used to describethe shield position.

It was further assumed that all photons which originated froma specific point source and reached the detector passed throughthe same distance of air and attenuating source media and thatphotons interacting in the detector travel the exact same distance.Therefore, the minor path length differences for individualphotons were neglected in order to simplify the calculations.

When appropriate, data were fit to two-dimensional cubicsplines, because these provide accurate representations of shapesfor large data sets using only low degree polynomials (Trefethenand Bau, 1997; Heath, 2002).

4.3. Planar source depth

System response simulations included a series of planar sourcesbetween 0 and 30 cm. Approaches studied included the pairs ofmeasurements chosen from combinations of geometries whichincluded: (a) no collimator or shield, (b) the collimator at 11 cm,(c) the shield without a collimator, and (d) the shield with thecollimator at 11 cm. The shield positions were varied from 0 to p/4.

4.4. Exponentially distributed source

Values of the exponential parameter a were varied from 0 to 100,corresponding to a uniformly distributed source to a very thinsource at the surface, and simulated. One pair of measurementsstudied was the bare detector and with the collimator at 11 cm. The

Z.D. Whetstone et al. / Applied Radiation and Isotopes 69 (2011) 790–802 795

other primary pair of measurements studied were with the p/4shield and no collimator, and the collimator at 11 cm with no shield.

4.5. Arbitrary source

For these simulations, the source was first considered toconsist of three equal thickness layers, namely from 0 to 10 cm,10 to 20 cm and 20 to 30 cm. Disequal layers of 0–5, 5–15, and15–30 cm were then studied. Maximum depths of 10, 20, and30 cm were also considered. Similarly four equal 7.5 cm thicklayers were studied along with four disequal thickness layers.

Measurement geometries were chosen from combinations of:(a) the bare detector; (b) the collimator in positions ranging from1 to 11 cm, and (c) with and without a shield with view ranging

Fig. 4. Ratio of measured results from two measurements for various size circular

lead center shields for semi-infinite planar sources 0–30 cm depth in soil. The ratio

of measurements is detector response with 11 cm collimator position to detector

response with the circular shield of various sizes.

Fig. 5. Example of a histogram of calculated depth of source in soil of a 25 cm depth

uniform layers with standard Gaussian random number deviation of 5%.

between 0 and p/3 radians. The three-layer system was studiedwith three measurements, but also with six measurements tostudy the effects of over-determination. The four-layer systemwas studied with four measurements.

5. Results

5.1. Depth of planar source

The ratio of the detector responses for two-measurementgeometries, which may be used for determining the planar sourcedepth when measurements are taken with the bare detector andthe detector with the collimator in the 11 cm position, are shownin Fig. 3. This same relationship is portrayed in Fig. 4 for variousshield sizes, showing increasing measurement ratio with shieldsize. For a planar source at 25 cm depth in soil and a random

semi-infinite planar source. The configuration consisted of no center shield and

Fig. 6. Ratio of the detector photopeak count rate from two-measurement

configurations to exponential sources with a range of values of a.

Z.D. Whetstone et al. / Applied Radiation and Isotopes 69 (2011) 790–802796

input perturbation of 5%, the standard deviation of the calculatedsource depth was 10.2% with a mean depth of 25.2 cm. Using ap/4 shield, the calculated depth was 25.0 cm with a standarddeviation of 5.99%, an improvement of a factor of 1.7. When thesystem input perturbations were increased to 10% with the shieldstill in place, the coefficient of variation of the calculated depthshowed only a minimal increase to 6.02%. An example of thecalculated depth distribution from the random perturbationmethod is shown in Fig. 5 for an arrangement with no centershield and uniform layers with a Gaussian random numberstandard deviation of 5%. This shows the expected normaldistribution of results because of the linear nature of this system.

When the Monte Carlo method was used to calculate the depthof sources simulated at 5, 10, and 20 cm in soil, using the baredetector and the collimator in the 11 cm position, the source depthswere determined to within �13%, 3%, and 11% of their actual depth,respectively. Furthermore, each planar source had a uniform activityof 1 Bq/cm2 and the calculated activities were 1.21, 0.929, and1.49 Bq/cm2 at depths of 5, 10, and 20 cm, respectively.

Fig. 7. Error properties of multiple measurements system used to determine the a in a

perturbation of 5% and consisted of a measurement with no collimation and a measurem

for a geometry with (a) no circular shield and (b) using a p/4 radians circular shield.

5.2. Exponentially distributed source

For a system designed to measure a source known to be of thistype, one measurement was made with the bare detector and the11 cm collimator, and the second with a p/4 shield and nocollimator. The ratio of the two measurements is shown inFig. 6 for exponential source parameters. This graph could beused to directly determine the exponential distribution from fieldmeasurements. The results of representative random perturbationstudies with a 5% standard deviation appear in Fig. 7, for a¼0.2.These demonstrate the expected lognormal distribution.

For two measurements without a shield, a had a standarddeviation of 8.6% for measurement perturbations of 5%. With theaddition of a p/4 shield, the system had a standard deviation of6.8%, an improvement for these idealized simulations. Table 1is a summary of the results. For the different values of a, theaddition of a shield reduced the standard deviation by a factor of1.27–1.31. With the use of a shield there was an approximateimprovement in error of 30%.

n exponentially distributed source. For this source a was set to 0.2 with a random

ent with the cylindrical collimator in the 11 cm position. The calculated results are

Table 1Errors in determining a for an exponentially distributed source. The error analysis

method was used to examine the effect of the addition of a lead circular shield

upon the determination of a in an exponentially distributed system. The error

analysis method was applied to two-measurement systems. One of which

included the use of the lead circular shield and one of which did not.

a Standard deviation

with no circular shield

Standard deviation

with circular shield

Relative improvement

in standard deviation

0 0.0264 0.0201 1.31

0.2 0.0861 0.0678 1.27

0.5 0.2643 0.1910 1.38

Fig. 8. Detector photopeak count rate for various collimator positions to a 30 cm

thick uniformly distributed 662 keV 137Cs source in soil. Collimator position is

defined as the distance from the detector face to the lowest plane of the

collimator.

Table 2Detector responses and calculated condition numbers for a variety of measure-

ment geometries for a three-layer, three-measurement system. The source is a

uniform semi-infinite 30 cm thick volume source of 662 keV 137Cs in soil. The

three measurements consisted of the detector with no collimator, the detector

with the collimator in the 11 cm position, and the detector in the position

indicated in the table. No circular center shield was used.

Collimator

position (cm)

Detector response contribution (cps cm5 Bq�1)Condition

number

of ‘ri,j’ matrix

Depth

0–10 (cm)

Depth

10–20 (cm)

Depth

20–30 (cm)

1 22.184 4.3209 0.97118 1539.0

2 17.916 3.8015 0.88541 1748.2

3 14.706 3.3325 0.80435 2110.6

4 12.058 2.8728 0.71570 2643.2

5 10.083 2.4915 0.63693 3370.8

6 8.4564 2.1544 0.56448 4619.7

7 7.2321 1.8827 0.50250 6418.3

8 6.1964 1.6416 0.44518 9572.0

9 5.3663 1.4395 0.39543 15,820

10 4.6985 1.2715 0.35273 30,419

11 4.1011 1.1190 0.31394 NA

No coll. 64.128 6.9221 1.3218 NA

Z.D. Whetstone et al. / Applied Radiation and Isotopes 69 (2011) 790–802 797

5.3. Optimization of measurement configuration for a

three-measurement, three-layer system

The calculated detector response for a variety of collimatorconfigurations to a 30 cm thick semi-infinite uniform source inthe three-layer system is shown in Fig. 8. The calculated con-tribution to the detector response of each layer for each collima-tor position is shown in Table 2. The condition number of themeasurement system ranged from 1540–30,400. The systemusing the 1 cm collimator position had the lowest conditionnumber.

The contribution to the detector from the source at variousdepths in soil was also investigated. A 10 cm thick semi-infinitevolume source was placed 0–10, 10–20, and 20–30 cm deep in thesoil. The response of the detector with varying collimator posi-tions was determined. As can be seen in Fig. 9, the detectorresponse dropped by about a factor of five each time the sourcemoved down a layer. Due to the shielding effect of the soil, the toplayer of a distributed source contributes a disproportionally highcount rate to the detector and must be accounted for wheninterpreting measurement results. Furthermore, the shape of thecurve when the collimator is varied changed depending on thesource depth. This could be used to help determine the sourcedepth that is being detected.

5.4. Effects of employing variable thickness layers for three-layer

system

For computed layer boundaries set to 0–5, 5–15, and 15–30 cm,designed to minimize the effects of decreasing sensitivity to thedeeper source layers, the results are displayed in Table 3. Thismeasurement configuration also included the use of a p/4 shieldwhen no collimator was used. The condition numbers of themeasurement system ranged from 451 to 10,850, with the 1 cmcollimator position having the lowest condition number. Thiscondition number was decreased by a factor of 3.4 when variablelayer thicknesses were implemented.

5.5. Effects of shield

With equal layer thicknesses, and measurements consisting ofthe collimator in the 1 cm position and the 11 cm positioncoupled with a measurement with the shield varied from 0 top/3 radians, the system condition number ranged from 820 to1530, as shown in Fig. 10a. The condition number decreased withincreasing shield size. For the same measurement setup, but withvarying layer thicknesses, the condition number varied from 255to 450, as shown in Fig. 10b. The condition number continued todecrease with increasing shield size. The corresponding relativechange in maximum potential error in the calculated activity isshown in Fig. 11. The maximum relative error decreased withincreasing shadow shield size to a minimum of 0.72 with a 7p/18shield, the maximum shield size studied.

5.6. Effects of employing maximum depth of interest on three-layer

system

In many cases it may be known that the source is onlydeposited to a certain depth. By limiting the maximum depththat is examined, errors in the calculated distribution may bereduced.

Table 4 demonstrates the effect of the maximum depth oncondition number. The maximum depth examined varied from 10to 30 cm. With no shield, the condition number varied from 861to 1540, with the lowest condition number for a 15 cm maximumdepth.

Fig. 9. Detector response to 662 keV 137Cs volume sources at various depths in soil using various collimator positions for sources located at soil depths of (a) 0–10 cm,

(b) 10–20 cm, and (c) 20–30 cm.

Z.D. Whetstone et al. / Applied Radiation and Isotopes 69 (2011) 790–802798

5.7. Gaussian random variable perturbation of a three-layer,

three-measurement system

The error properties of a three-layer, three-measurementsystem using equally spaced layers from 0 to 10, 10 to 20, 20to 30 cm deep were determined. The standard deviationof the calculated activity present was 23.0% for the top layer,407% for the middle layer, and 1210% for the bottom layer.An example of the error properties for the top layer can beseen in Fig. 12. If non-uniform layers from 0 to 3, 3 to 10, and10 to 20 cm and a lead shield are used, the standard deviation ofthe top layer is 53%, the middle layer 145%, and the bottomlayer 222%.

The standard deviation of the calculated activity in the firstlayer as a function of the standard deviation in the measurementerror is shown in Fig. 13. The standard deviation of the calculatedactivity varied linearly with the standard deviation in the inputerror, in accordance with y¼10.26x+0.00005.

5.8. Over determined system

For a system with three layers 10 cm thick from 0 to 30 cm andsix measurements, the standard deviation of the calculatedactivity was 62% for the top layer, 175% for the middle layer,and 277% for the bottom layer.

Table 3Detector responses and calculated condition numbers for a variety of measure-

ment geometries for a three-layer, three-measurement system using a circular

center shield. The source is a 30 cm thick uniform semi-infinite volume source of

662 keV 137Cs in soil. The three measurements consisted of the detector with no

collimator and a p/4 radians circular center shield, the detector with the

collimator in the 11 cm position, and the detector in the position indicated in

the table.

Collimator

position (cm)

Detector response contribution

(cps cm5 Bq�1)

Condition number

of ‘ri,j’ matrix

Depth

0–5 cm

Depth

5–15 cm

Depth

15–30 cm

1 15.607 9.5259 2.3427 450.69

2 12.375 8.1165 2.1114 528.85

3 10.025 6.9212 1.8962 657.92

4 8.1395 5.8385 1.6680 832.52

5 6.7622 4.9789 1.4708 1077.4

6 5.6411 4.2423 1.2920 1489.4

7 4.8070 3.6680 1.1424 2079.3

8 4.1070 3.1694 1.0062 3135.4

9 3.5517 2.7595 0.88973 5412.0

10 3.1068 2.4253 0.79081 10,845

11 2.7097 2.1237 0.70080 NA

No coll. 50.968 18.044 3.3609 NA

Fig. 10. Condition number of a linear system using three layers and three

Z.D. Whetstone et al. / Applied Radiation and Isotopes 69 (2011) 790–802 799

5.9. Error properties of the four-layer, four-measurement system

Fig. 14 shows the conditions numbers for measurementgeometries where the collimator varied from 1 to 11 cm. Thecondition numbers of the system ranged from 11,000 to 110,000.The measurement system with the collimator in the 3 cm positionhad the lowest condition number.

The distributions of the calculated activity in each uniformlayer were determined. The four measurements taken include thedetector with no collimator and a p/4 shield and the detector withthe collimator in the 1, 3, and 11 cm positions and no centershield. The standard deviation for the 0–7.5 cm deep layer was72.0%, for the 7.5–15 cm layer it was 1990%, for the 15–22.5 cmlayer it was 10,400%, and for the 22.5–30 cm layer it was 14,800%.The calculated activity distributions for non-uniform thicknesslayers were determined. Measurements were simulated with nocylindrical collimator using a p/4 radians center shield, thecollimator in the 1 cm positions with a 2p/45 radians centershield, and with the collimator in the 11 and 3 cm positions withno center shield. The standard deviation of the calculated activityfor the 0–3 cm deep layer was 904%, for the 3–7 cm layer it was5950%, for the 7–12 cm layer it was 12,700% and for the 12–20 cmlayer it was 8040%.

measurements. Three measurements were calculated using the 11 cm collimator

position, the 1 cm collimator position, and a detector with no collimator. The

condition numbers are shown for a system using (a) no center shield and uniform

thickness layers, and (b) a circular shield and non-uniform layers.

6. Discussion

The method can be used to accurately determine, in compar-ison to other errors present, planar source depth. In the typicalscenario studied, the depth of a thin or planar source in soil couldbe determined to less than 710% error. The use of a lead shieldreduced the error from 10.2% to 6.0% allowing for an even moreaccurate depth determination. The estimate derived from MCNP,without the use of a shield, closely matched the theoreticallycalculated results with a maximum error of 13% in calculating thesource depth.

For uniformly distributed exponential distributions wherea¼0, with this method the a term can be estimated with astandard deviation of 2%. For a44 the results of the methodbecome extremely inaccurate with errors greater than 1000% indetermining a. However, for such cases the distribution ispractically approximated as a thin layer on the media surface

and the source distribution is actually determined relativelyaccurately. For a¼4, greater than 99% of the radioactive materialis deposited in the first 1 cm. The detector response varies littlesince the distribution is all concentrated very near the surface.Very large values of a thus result in small changes in the detectorresponse, which would be obscured by background and limited bynoise, making it difficult to determine values of a greater than 4.

The addition of a shield was demonstrated to significantlyimprove the system error properties for determining source depthby reducing the standard deviation of the results by a factor ofapproximately 1.3 for each of the exponential measurementscenarios examined.

This method could easily be applied to any source distributionthat was accurately described by a known analytical expression,

Fig. 11. Relative change in the maximum possible error in the calculated activity

for each layer for a range of circular shield sizes. This error is calculated using

Eq. (16) and accounts for the change in detector efficiency and the change in

condition number of the system for different sized shields.

Table 4The condition number of a three-layer, three-measurement system using various

maximum depths. Uniform layers were used and no center shield was employed.

Maximum depth (cm) Condition number

10 972.53

15 860.97

20 953.28

25 1176.5

30 1539

Z.D. Whetstone et al. / Applied Radiation and Isotopes 69 (2011) 790–802800

such as: Gaussian distributions, linear, multiple Gaussian distri-butions, or any other practical analytical expression. For systemswith more than two unknowns, more measurements could beadded to fully determine the system. Further work with otherexpressions is needed.

However, the strength of the approach is that an arbitrary 1-Dsource distribution consisting of multiple layers can be estimated.For an optimized approach divided into three layers using theerror reduction techniques discussed in this paper, it was foundthat the standard deviation in the calculated activity ranged from53% to 222%, with the lowest standard deviation in the superficiallayers. It was found that the error reduction techniques greatlyimprove the system error properties reducing the standarddeviation of the results from 1210% to 222% for the deepest layer,an almost six fold decrease in the standard deviation.

By careful application of the conditioning reduction techni-ques, the system condition number was reduced by a factor of119. This reduction included the use of a shield, the use of non-uniform thickness in the layer boundaries, and limiting themaximum depth of interest. The matrices are not well-condi-tioned, but if the measurement techniques are chosen carefully,the method provides enough information to help determine, oreliminate, particular vertical distributions of the radionuclide.This is done without the use of core sampling or other slow andexpensive techniques.

Over-determination of the system did not improve its errorproperties. For the shield, with increasing shield size up to 7p/18,it was found that the reduction in condition number for larger

shield sizes was greater than the increase in error from thereduced detector’s efficiency; thus it is desirable to use a shieldof at least 7p/18 for the three-measurement, three-layer system.

A system consisting of three layers and six measurements haderror properties that were inferior to the three-measurementsystem. This may not be the case for all measurements geome-tries, however, as only a limited number of additional geometrieswere examined. This is an area requiring further research. Forexample, having additional measurements with no collimatorusing the shield in different positions or of different sizes mayimprove the system error properties.

A four-layer system was tested and it was found that thesystem error properties are such that it could only be used fororder of magnitude estimates of source distribution. Errors, withthe use of the error reduction techniques described, are especiallyhigh for the deeper layers when compared to others errorspresent for in situ measurements. Calculated results may be offby a factor of 50 times the actual activity present. This may beuseful for making rough distribution estimates of a completelyunknown source or a source primarily located on the surface, butthe system is poorly conditioned. Unless further error reductiontechniques can be developed and applied, the use of four or morelayers may have limited practical application.

For all of these methods, except for the use of additionalmeasurements beyond the number of unknown variables, errorreduction techniques, such as the selection of the optimalsampling geometry, the use of a shield, using non-uniform layers,and limiting the maximum depth of interest, significantlyimprove the system error properties. The addition of a shield formeasurements where no collimator was used reduced the stan-dard deviation in the calculated results by 30–500% dependingupon the system. In many cases, error reduction techniques arerequired for a practical system.

When selecting measurement geometries to reduce the sys-tem’s condition number, other factors must be considered. Thelarger the center shield and the higher the collimator position, thelower the detector’s efficiency. A detector with lower efficiencymeans a longer measurement must be made to achieve the samecounting errors, which results in larger input errors. Thus, a tradeoff exists between detector efficiency and system error properties.

Testing of the method in the environment using actual equip-ment should be performed to further verify the method and developthe system into a practical field instrument. Further investigation isalso possible into additional methods for error reduction, especiallyfor the discrete four-layer system. There are other possibilities thatcould also be examined, such as allowing each layer to have somesimple distribution such as linear, instead of being uniform. Anothermethod would be to impose a boundary condition on the layers andforce the activity concentration at the boundaries to be continuous.Alternatively, multiple measurements could be performed and theresults fit to a series of analytical equations to determine whichequation best fits the data. This could then be assumed the bestrepresentation of the source distribution. Forcing any negativeactivity concentrations to zero may further improve the systemerror properties. The exponential nature of attenuation and theinherent errors in counting statistics make it difficult to derive anyadditional information about the vertical distribution from thesystem. However, even a small reduction in error properties mayallow for a four-layer system to be more practical in a greatervariety of circumstances.

The method assumes a homogenous media for the source.Whether the media is soil, concrete, or other media this assump-tion is rarely 100% correct. Soil may vary in density or composi-tion. Concrete may contain steel rods for reinforcement. Smallvariations or large variations over very small areas will not have asignificant impact on the results since the detector response is the

Fig. 12. An example of error properties of a three-layer, three-measurement system of 662 keV 137Cs in a soil matrix. The calculated activity layer was located from 0 to

10 cm depth with no center shield.

Fig. 13. Standard deviation in the first layer of a three-layer system versus the

standard deviation of the input error used.

Fig. 14. The condition numbers of a four-layer, four-measurement system. Mea-

surements were made using the collimator in varying positions from 2 to 10 cm.

Z.D. Whetstone et al. / Applied Radiation and Isotopes 69 (2011) 790–802 801

measurement of a source over a very large area; however, thequestion needs to be answered as to at what point do suchvariations have an impact on results. Furthermore, this methodcould be applied to other gamma rays sources. The 662 keVgamma rays were useful for investigating the method, but thesystem will have to be studied for photons of varying energies upto several MeV.

7. Conclusions

The presented approach can be applied to in situ gammaspectroscopy measurement scenarios to better characterize a 1-Dsource distribution. An enhanced distribution characterizationallows more accurate in situ activity measurements and thus moreaccurate site characterization. This method requires fewer and lessrestrictive a priori assumptions and is more flexible in its application

than previous methods. It allows for characterization of sites witheither partially known or no source distribution information.

Through the careful selection of measurement geometries andthe application of error reduction techniques, the conditioning ofmany practical measurement scenarios can be improved so thaterrors in calculated activity or source depth of less than 10% arepossible. By using a circular lead shield, layers of varying thick-ness, and limiting depths of interest, the error properties ofmeasurement system were improved by a factor of 120. Althoughthe method requires field testing, initial simulations have shownit to be effective in determining a 1-D source distribution.

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