Analysis of Environmental Radionuclides, Volume 11

525 Pages·2007·9.85 MB·English

by  Pavel P. Povinec

1
Foreword
TheRadioactivityintheEnvironmentseriesisanambitiousprojectcoveringrecentprogress
inthisrapidlydevelopingfield,whichhasincludedaspectssuchasthebehaviorofradionu-
clides in the environment, the use of natural and anthropogenic radionuclides as tracers of
environmentalprocesses,marineradioactivitystudies,radiationprotection,radioecology,etc.
tomentionatleastafew.
State of the art radioanalytical environmental technologies have always been a limiting
factorforenvironmentalradioactivitystudies,eitherbecausetheavailablesensitivitywasnot
high enough to get meaningful results or the required sample size was too big to carry out
suchinvestigations,veryoftenwithlimitingfinancialresources.
Therehasinrecentyearsbeengreatprogressinthedevelopmentofanalyticaltoolsrelated
tosamplingstrategies,developmentofrapidandefficientradiochemicalseparationmethods,
radiometric counting systems utilizing high sensitivity Ge detectors often working under-
ground, and mass spectrometry technologies based on ICPMS (inductively coupled plasma
mass spectrometry) and AMS (accelerator mass spectrometry) for sensitive analysis of nat-
uralandanthropogenicradionuclidesintheenvironment.
For example, in the marine environment, where research work has been heavily depen-
dentonthenewtechnologies,wehaveseenareplacementoftime-consumingandexpensive
largevolumewatersampling(500L)fromseveralkmwaterdepthsbyRosettemultisampling
systems enabling high resolution water sampling within one or two casts with 12 L bottles
only. The sampling strategies are often developed and controlled using satellite information
fortheoptimizationofthesamplingprograms.Further,thephilosophyofsamplingandlab-
oratorymeasurementshaschanged,whereappropriate,toinsituanalysisofradionuclidesin
the air, on land, in water and in the sediment, thus developing isoline maps of radionuclide
distributionsintheinvestigatedenvironment.
Inthefieldofanalyticaltechnologieswehavemovedfromsimpleradiochemicalmethods
and gas counters to robotic radiochemical technologies and sophisticated detectors working
on line with powerful computers, often situated underground or having anticosmic and/or
anti-Compton shielding to protect them against the cosmic radiation, and thus considerably
decreasing their background and increasing their sensitivity for analysis of radionuclides in
theenvironmentatverylowlevels.Thephilosophyofanalysisoflong-livedradionuclideshas
alsochangedconsiderablyfromtheoldconceptofcountingofdecays(andthuswaitingfor
them)tothedirectcountingofatoms(asiftheywerestableelements)usinghighlysensitive
mass spectrometry techniques such as AMS, ICPMS, TIMS (thermal ionization mass spec-
trometry),RIMS(resonanceionizationmassspectrometry)andSIMS(secondaryionization
massspectrometry).
Therehavealsobeenconsiderablechangesinthephilosophyandorganizationofresearch
asinstitutionalandnationalinvestigationshavebeenreplacedbyglobalinternationalprojects
2 P.P.Povinec
suchasWOCE(worldoceancirculationexperiment),CLIVAR(climatevariabilityandpre-
dictabilitystudy),PAGES(pastglobalchanges),WOMARS(worldwidemarineradioactivity
studies),GEOTRACES(globalmarinebiochemistryoftraceelementsandisotopes),SHOTS
(southernhemisphereoceantracerstudies),tomentionatleastafew.
Althoughthetopicoftheanalysisofenvironmentalradionuclideshasalreadybeencovered
in several reviews, there has not been available a book covering critical progress in recent
years. The present collection of review papers covers a wide range of topics starting with
the development of statistically based sampling strategies to study radionuclides in the en-
vironment (Chapter 1 by Scott and Dixon), followed by description of sampling techniques
andpre-concentrationof samples(Chapter 2by Macášek).Statisticalevaluationof datahas
beenacrucialpointincorrectinterpretationofmeasurements,especiallywhendealingwith
countingratesveryclosetothedetectorbackground(Chapter3byCurrie).Recentprogress
inenvironmentalstudiesisdocumentedbytheanalysisof137Cs,90SrandPuisotopesinthe
seawater column (Chapter 4 by Aoyama and Hirose). Monte Carlo simulations of detector
background characteristics have been an important pre-requisite when designing low-level
counting systems (Chapter 5 by Povinec et al.), also important when working in laborato-
ries situated hundreds of meters underground, where radioactive purity of construction ma-
terials and radon concentration in the air become dominant factors controlling the detector
background (Chapter 6 by Niese). AMS has been a revolutionary breakthrough in analyti-
cal methodologies for long-lived environmental radionuclides, as described by Jull et al. in
Chapter7forlightelements,andFifieldinChapter8forheavyelements.However,themost
widely used mass spectrometry technique for analysis of long-lived environmental radionu-
clideshasbeenICPMS,asdocumentedbyRossinChapter9.Anothernewtrendinanalytical
techniqueshasbeenanintroductionofresonanceionizationmassspectrometryforradionu-
clide analysis (Chapter 10 by Erdemann et al.), and a change from bulk sample analysis to
particle sensitive analysis, as described by Betti et al. in Chapter 11 using SIMS, scanning
electronmicroscopy(SEM),andsynchrotronbasedtechniqueslikeµ-XRFand3D-µtomog-
raphy.Neutronactivationanalysis(NAA)hasbeencontributinginspecificapplicationswith
long-lived radionuclides, and usually this is the only alternative technique for certification
of reference materials (Chapter 12 by Hou). In situ techniques represent a new approach to
analysisofenvironmentalradionuclidesandthesehavebeenrecentlywidelyappliedforsur-
face monitoring of radionuclides using either mobile gamma-ray spectrometers, helicopters
and airplanes (Chapter 13 by Tyler) or measurements carried out under the water, e.g., for
radionuclide mapping of seabed sediments and/or stationary monitoring of radionuclides in
theaquaticenvironmentasdescribedinChapter14byPovinecetal.
TheEditorwouldliketothankallauthorsfortheirfruitfulcollaborationduringpreparation
ofthiscompilationandProf.Baxter,theRadioactivityintheEnvironmentSeriesEditor,for
hispatiencewhenworkingonthisbook.Inpublishingthisbookwehopetofurtherstimulate
workintheexcitingfieldofenvironmentalradioactivityandtheuseofradionuclidesastools
forinvestigationsofenvironmentalprocesses.
PavelP.Povinec
Editor
ComeniusUniversity
Bratislava,Slovakia
3
Statistical sampling design for radionuclides
E.Marian Scotta,∗, PhilipM.Dixonb
aDepartmentofStatistics,UniversityofGlasgow,GlasgowG128QW,UK
bDepartmentofStatistics,IowaStateUniversity,Ames,IA50011-1210,USA
1. Introduction
Thischapterpresentssomeofthekeyideasfordesigningstatisticallybasedsamplingstrate-
giestostudyradionuclidesintheenvironment.Environmentalsampleswillnaturallyvaryin
theirspecificactivity,nomatterhowprecisetheradionuclidemeasurement.Thisvariabilityis
causedbynaturalvariationsintheprocessesthatcontrolradionuclidetransportanduptakein
the environment. A statistically based sampling design quantifies this variability and allows
informationfromspecificsamplestobegeneralizedtoalargerpopulation.
Thestatisticalsamplingprinciplesdiscussedherearedetailedinmanytextbooksandpapers
aboutenvironmentalsampling,suchasthegeneralsamplingtextbooksbyCochran(1977)and
Thompson (2000), the environmental statistics textbook by Gilbert (1987), and government
agency guidance documents (US EPA, 2002). A recent ICRU report (2006) presents a thor-
ough presentation of sampling issues for environmental radionuclides. This chapter draws
heavilyonthesereferenceworks.Itprovidesonlyatastertotheissues;thereaderisstrongly
encouragedtoreadmorein-depthdescriptions.
Environmental sampling should not be considered as a ‘recipe’ based activity. The best
(mostefficient,validandreliable)samplingschemesuseenvironmentalknowledgetoguide
the sampling. Changes in objectives (apparently small) may also lead to quite significant
changestothesamplingscheme.
1.1. Generalsamplingconceptsandprinciples
Statisticalsamplingisaprocessthatallowsinferencesaboutpropertiesofalargecollectionof
things(thepopulation)tobemadefromobservations(thesample)madeonarelativelysmall
numberofindividuals(samplingunits)belongingtothepopulation.Thepopulationistheset
ofallitemsthatcouldbesampled,suchasalldeerinaforest,allpeoplelivingintheUK,etc.
A sampling unit is a unique member of the population that can be selected as an individual
∗
Correspondingauthor.E-mailaddress:[email protected]
RADIOACTIVITYINTHEENVIRONMENT ©2008ElsevierB.V.
VOLUME11 ISSN1569-4860/DOI:10.1016/S1569-4860(07)11001-9 Allrightsreserved.
4 E.M.ScottandP.M.Dixon
sample for collection and measurement. The sample is then the set of sampling units that
aremeasured.Samplingunitsmightbeindividualdeer,individualpeople,trees,gardenplots,
or soil cores of a given dimension. An essential concept is that a statistically based sample
of a sufficient number of individual sampling units is necessary to make inferences about
the population. Statistical sampling also allows a quantification of the precision with which
inferencesorconclusionscanbedrawnaboutthepopulation.
The focus of this chapter is on statistical sampling design, namely how to select specific
samplingunitsfromapopulationorsamplinglocationswithinalargerarea,andhowtode-
termine the number of individual units to collect. Sampling has many purposes, including
estimationofthedistribution(andmean)concentrationofaradionuclide(Bql−1)inariver,
or in fruit in a region (Bqkg−1), or a map of radionuclide deposition (Bqm−2). Different
purposesrequiredifferentsamplingstrategiesanddifferentsamplingeffortsinordertobeef-
fectiveandefficient,soitisimportantthatthepurpose(s)ofthesamplingprogrambeclearly
specified.Theenvironmentalcontextalsoplaysanimportantpartindeterminingthechoiceof
samplingmethod.Statisticalsamplingrequiresinformationaboutthenatureofthepopulation
andcharacteristicstobedescribed.
1.2. Methodsofsampling
Astatisticalsamplingdesignisbasedonprobabilitysampling,inwhicheverysamplingunit
hasaknownandnon-zeroprobabilityofbeingselected.Theactualsample(setofsampling
unitstobemeasured)ischosenbyrandomization,usingpublishedtablesofrandomnumbers
or computer algorithms. Selecting a probability sample is easy when the population can be
enumerated. As a simple example, imagine sampling 10 adults from a specified geographic
area for whole body monitoring. We could use an electoral register or census information
to enumerate all individuals. Suppose that the population comprised 972 such individuals,
then we could generate 10 random numbers lying between 1 and 972, such as 253, 871,
15, 911,520, 555,106, 83, 614,932to identifythe 10individuals.If thesame numberwas
generatedmorethanonce,thenwewouldsimplycontinuetheprocesstillwehad10unique
random numbers and these would then identify the individuals to be called for monitoring.
Theactualnumbers(253,871,etc.)arereadfromrandomnumbertablesormaybegenerated
bystatisticalsoftware.
Therearemanysamplingdesigns.Wedescribesimplerandomsampling,stratifiedrandom
samplingandsystematicsamplingbecausethesearethethreemostcommoninenvironmental
studies.Foreachdesign,wediscusshowtoselectasampleandhowtoestimatethepopula-
tionmeananditssamplingerror.Abriefreviewoftheadvantagesanddisadvantagesofthe
differentmethodsisalsoincluded.MoredetailcanbefoundinICRU(2006).
1.2.1. Simplerandomsampling
In a simple random sample, every sampling unit in the population has an equal probability
ofbeingincludedinthesampleandallpairsofsamplingunitshavethesameprobabilityof
beingincludedinthesample.Onewaytoselectasimplerandomsampleistoenumerateall
sampling units in the population, then use random numbers to select the desired number of
sampling units. Simple random sampling is easy to describe but may be difficult to achieve
Statisticalsamplingdesignforradionuclides 5
in practice. Some common problems include lack of response from some individuals, inac-
cessibilityof some plots of ground,and long traveltimes betweensamplinglocationswhen
samplinglargeareas.
Example:Estimationoftheaveragebaseline14Clevelinthefood-chain
An estimate of the dose to the general public due to 14C in the food-chain is an important
radiological quantity for regulatory impact assessment since many nuclear power stations
discharge 14CO which is rapidly taken up. At a given station, the task would be to select
2
representative environmental samples that enter the food-chain, e.g., root or cereal crops or
fruits.Forthisparticularproblem,definitionofthepopulationshouldincludeidentificationof
thespeciesandinformationonwhereandwhenitgrewanditsspatialcontext.Forthechoice
ofspecies,itshouldbewidelyavailableandasuitablematerialfor14Cassay;amaterialsuch
assoftfruit,mushroomsorgrainwouldbeideal.Theanalysisrequirementswouldthendefine
how much material needed to be collected for each sampling unit. In terms of the temporal
extent, it would be logical for the samples to be selected from a single growing season and
in a specific year such as 2004. This results in a clear definition of the population, namely
allselectedcropinthevicinityofthesitegrowinginaspecificyearandofasamplingunit,
namelyabulkedsampleofberries,vegetablesorwheatharvestedfromaspecificlocation.
Thenextsteprequiresidentificationofthelocationsatwhichthesampleswillbecollected
and determination of how many sampling units will be required to satisfy the objectives of
thestudy.Amapofthevicinityintermsofalllocationswherethecropsgrow,wouldallow
thenumberingofallthelocationsfrom1toN,andrandomnumberswouldthenbeusedto
identifywhichactuallocationswouldbesampled.Thereadermightcaretoconsiderwhether
therearemoreefficientbutequallyvalidsamplingapproachestothisproblem.
Analysisoftheresultsfromasimplerandomsample
Supposethatthe14Cactivitydensityismeasuredineachofthensamplingunitsinthesample.
Thevaluefromsamplingunitiisdenotedasy .Thesampleaverage,y¯,givenbyEquation(1)
i
is a good (unbiased) estimate of the population mean 14C activity density and the sample
variance,s2,givenbyEquation(2)isagoodestimateofthepopulationvariance:
(cid:2)
y
y¯ = i, (1)
n
and
(cid:2)
(y −y¯)2
s2 = i . (2)
n−1
Thesampleaverageisarandomquantity;itwillbeadifferentnumberifdifferentsampling
units were chosen for the sample, because of the natural variation in 14C activity densities
among sampling units. The uncertainty in the sample average is quantified by the sampling
variance, given by Equation (3) or its square root, the estimated standard error, e.s.e. The
samplingfraction,f,isthefractionofthepopulationincludedinthesample,n/N,whichis
usuallyverysmall.
(cid:3) (cid:4)
1−f
Var(y¯)=s2 . (3)
n
6 E.M.ScottandP.M.Dixon
Inthe14Cexample,thereisnoenvironmentalevidencetobelievethatthepopulationofcrop
isnon-homogeneous.Thismeansthatwehavenoreasontoexpectanysub-groupswithdis-
tinctlydifferent14Clevelsandsosimplerandomsamplingisareasonablesamplingstrategy.
However, there may be spatial information which could prove important, such as distance
anddirectionfromthestackandwhichmightleadthescientisttobelievethatthepopulation
isheterogeneousandsoamoredirectedsamplingschememightusecontextualinformation
such as wind rose data to determine a different sampling scheme. Next, we consider an ex-
ample with a similar objective, but where the environmental context would suggest that the
population could be non-homogeneous, hence a different sampling scheme could be better
(Dixonetal.,2005).
1.2.2. Stratifiedsampling
Stratifiedsamplingdesignsprovidetwoimportantadvantagesoversimplerandomsampling
designs,namely,efficiencyandimprovedestimatesformeaningfulsubdivisionsofthepopu-
lation.Wemustassumethatthepopulationcanbedividedintostrata,eachofwhichismore
homogeneous than the entire population. In other words, the individual strata have charac-
teristics that allow them to be distinguished from the other strata, and such characteristics
are known to affect the measured attribute of interest, namely the radioactivity. Usually, the
proportionofsampleobservationstakenineachstratumissimilartothestratumproportion
ofthepopulation,butthisisnotarequirement.Stratifiedsamplingismorecomplexandre-
quiresmorepriorknowledgethansimplerandomsampling,andestimatesofthepopulation
quantitiescanbebiasedifthestratumproportionsareincorrectlyspecified.
Example:60Coactivityinanestuary
Mappingradioactivecontaminationinspecificlocationsorareasisacommonobjectiveinra-
dioecologicalinvestigations(ICRU,2001).Supposeonewishedtomap60Cointhesediments
of an estuary. The population could be conceived to be all possible sediment cores (depth
30 cm, diameter 10 cm) (N in total) within the estuary; a sampling unit would be a single
core.Asimplerandomsample,i.e.arandomselectionofnoftheN possiblecorelocations,
is a valid sampling design, but it may not be the best. Stratified sampling, using additional
information about the estuary and the environmental behavior of 60Co, can provide a more
preciseestimateofthemeanactivitydensityintheestuary.
The distribution of 60Co in the estuary may not be homogeneous because sediment type
andparticlesizedistributionareassociatedwith60Coactivity.Ifamapofthesedimenttype
withintheestuaryisavailable,itwillindicateareasofmud,sand,etc.,eachofwhichwould
beexpectedtohaveadifferent60Coactivitydensity.Astratifiedrandomsampleestimatesthe
mean60Coactivityineachsedimenttype.Estimatesfromeachstratumandtheareaofeach
sedimenttypeintheestuaryarecombinedtoestimatetheoverallmeanactivityfortheestuary
andthesamplingerror.
Instratifiedsampling,thepopulationisdividedintotwoormorestratathatindividuallyare
morehomogeneousthantheentirepopulation,andasamplingmethodisusedtoestimatethe
propertiesofeachstratum.Usually,theproportionofsampleobservationsineachstratumis
similartothestratumproportioninthepopulation.Intheexampleabove,wemightconsider
theestuaryasbeingcomposedofareasofmud,sandandrock.Thesewouldthendefinethe
Statisticalsamplingdesignforradionuclides 7
strata.Thisisanexamplewhereenvironmentalknowledgeandtheproblemcontextleadtoa
bettersamplingscheme.
In stratified sampling, the population of N units is first divided into sub-populations of
N ,N ,...,N units representing the sampling units in each of the different strata. These
1 2 L
sub-populationsarenon-overlappingandtogethercomprisethewholepopulation.Theyneed
not have the same number of units, but, to obtain the full benefit of stratification, the sub-
population sizes or areas must be known. In stratified sampling, a sample of units is drawn
fromeachofthestrata.Often,simplerandomsamplingisusedineachstratum.Thenumber
ofsamplingunitsallocatedtoastratumisoftenproportionaltothepopulationsizeorareaof
thatstratum.Wheneachstratumhasthesamewithin-stratumvariance,proportionalallocation
leadstothemostpreciseestimateofthepopulationmean(Cochran,1977).
For the sediment example, the strata might be defined as distinct sediment types. Knowl-
edge of the fractional areas of each sediment type within the estuary would be needed to
ensure appropriate sampling fractions within each stratum. Simple random samples of size
n ,n ,...,n would be taken from each strata. Thus if the estuary was 60% sand, 30% silt
1 2 l
and10%mud,then60%ofthesamplingunitswouldbeselectedinthesandyareas,30%in
thesiltyareasand10%inthemuddyareas.
Toestimatetheaverageandvarianceofeachstratum,onewoulduseEquations(1)and(2).
Thepopulationmeanactivity,A ,anditssamplingerror,Var(A ),areweightedaveragesof
c c
the average, y¯ , and variance, s2, for each stratum, l. The weights, W, are the fractions of
1 l l
eachstratuminthepopulation,i.e.W =N/N.
l l
(cid:2)
(Ny¯ )
A = l l l , (4)
c
N
and
(cid:6) (cid:7)
(cid:5) s2
Var(A )= W2 l (1−f) . (5)
c l n l
l
l
Theequationforthesamplingerror(5)assumesthatW,thestratumweight,isknown.This
l
would be the case when the strata are areas on a map. When proportional allocation for the
samplingfractionisused(i.e.,n/n=N/N),theninEquation(5),N isreplacedbyn andN
l l l l
isreplacedbyn.Itisnotnecessaryforthesedimentmaptobeaccurate,butinaccuracyinthe
definitionofthestrataincreasesthesamplingerroranddecreasesthebenefitofstratification.
Stratified random sampling will, with appropriate use, provide more precise (i.e. less un-
certain)estimatesthansimplerandomsampling,butmoreinformationisrequiredbeforethe
specific strategy can be carried out. Simple random and stratified random sampling may be
impractical, say in the sediment example, if finding precise sampling locations is difficult.
Amorepracticalmethodofsamplingmightinvolvecoveringtheareainasystematicmanner,
say in parallel-line transects and this final sampling method, systematic sampling, which is
ofteneasiertoexecutethansimpleorstratifiedrandomsampling,andwhichinsomecasesis
morerepresentativethana randomsampleis describedbelow.Onedisadvantageofsystem-
aticsamplingisthattheanalysisoftheresultsisoftenmorecomplex.Againthisexamplealso
illustrateshowthetheoreticaldescriptionofthesamplingschemeneedstobemodifiedbythe
practicalrealityofsamplingintheenvironment.
8 E.M.ScottandP.M.Dixon
1.2.3. Systematicsampling
Systematic sampling is probably the most commonly used method for field sampling. It is
generallyunbiasedaslongasthestartingpointisrandomlyselectedandthesystematicrules
arefollowedwithcare.Transectsandtwodimensionalgridsarespecifictypesofsystematic
samples. Consider sampling sediment in an estuary. One possible systematic sample of 15
locationsmightbeobtainedbyrandomlychoosingatransectacrosstheestuaryandtaking5
coresamplesalongthistransect,andthenplacingafurthertwotransectsequallyspacedalong
the estuary. Systematic sampling is often more practical than random sampling because the
proceduresarerelativelyeasytoimplementinpractice,butthisapproachmaymissimportant
featuresifthequantitybeingsampledvarieswithregularperiodicity.
Systematic sampling differs from the methods of random sampling in terms of practical
implementationandintermsofcoverage.Again,assumethereareN (=nk)unitsinthepop-
ulation.Thentosamplenunits,aunitisselectedforsamplingatrandom.Then,subsequent
samplesaretakenateverykunits.Systematicsamplinghasanumberofadvantagesoversim-
ple random sampling, not least of which is convenience of collection. A systematic sample
isthusspreadevenlyoverthepopulation.Inaspatialcontextsuchasthesedimentsampling
problem,thiswouldinvolvelayingoutaregulargridofpoints,whicharefixeddistancesapart
inbothdirectionswithinaplanesurface.
Data from systematic designs are more difficult to analyze, especially in the most com-
moncaseofasinglesystematicsample(Gilbert,1987;Thompson,2000).Considerfirstthe
simplercaseofmultiplesystematicsamples.Forexample,60Coactivityinestuarysediment
couldbesampledusingtransectsacrosstheestuaryfromoneshorelinetotheother.Samples
arecollectedevery5malongthetransect.Thelocationsofthetransectsarerandomlychosen.
Eachtransectisasinglesystematicsample.Eachsampleisidentifiedbythetransectnumber
and the location along the transect. Suppose there are i = 1,...,t systematic samples (i.e.
transectsintheestuaryexample)andtheyij isthejthobservationontheithsys(cid:2)tematicsam-
pleforj =1,...,n .Theaverageofthesamplesfromtheithtransectisy¯ = ni y /n .
i i j=1 ij i
Thepopulationmeanisestimatedby
(cid:2) (cid:2) (cid:2)
y¯ = (cid:2)ti=1niy¯i = ti=(cid:2)1 nj=i 1yij. (6)
sy t n t n
i=1 i i=1 i
The estimator of the population mean from a systematic sample is exactly the same as the
estimatorforasimplerandomsamplebutitismoredifficulttoestimatethevariance.
Whentherearemultiplesystematicsamples,eachwithnobservations,thevarianceofthe
meancanbeestimatedby
1−t/T (cid:5)t
Var(y¯ )= (y¯ −y¯ )2, (7)
sy t(t −1) i sy
i=1
whereT isthenumberoftransectsinthepopulation(Gilbert,1987;Thompson,2000).The
term in the numerator, 1−t/T, is a finite population correction factor that can be ignored
if the number of transects in the systematic sample, t, is small relative to the number in the
population.
The variance estimator given by Equation (7) cannot be used in the common case of a
single systematic sample, i.e. when t = 1. Many different estimators have been proposed
Statisticalsamplingdesignforradionuclides 9
(summarized in Cochran, 1977; Gilbert, 1987). If the population can be assumed to be in
random order, then the variance can be estimated as if the systematic sample were a simple
randomsample,i.e.usingEquation(3).Thatequationisnotappropriatewhenthepopulation
hasanyformofnon-randomstructure.Moredetailsoftheseandotherproblemsaregivenin
Cochran(1977),Gilbert(1987)andThompson(2000),andinICRU(2006).
Othersamplingschemesexist,buttheyareoftenintendedforratherspecializedsituations.
These include cluster sampling, double sampling and adaptive sampling. These are beyond
thescopeofthischapterbutdetailscanbefoundinICRU(2006).
Althoughestimationoftheaverageisprobablythemostcommonobjectiveforasampling
campaign,thereareotherquantitiesthatareofinterestinthepopulation,andthebasicsam-
plingdesignsareequallyapplicable.Perhapsoneofthemostcommonsamplingpurposesis
to map the spatial extent of a pollutant, or to estimate the spatial pattern. This is described
in more detail in the next section since most radionuclide problems have a spatial context
and there is growing use of geographic information systems (GIS) within the radioecology
community.
2. Samplingtoestimatespatialpattern
2.1. Introduction
In many sampling problems, especially in the environmental context, we must consider the
spatial nature of the samples collected. It is only common sense that samples, e.g., plants,
animals, or soil that are close together are more similar to each other than other samples
thatarefartherapart.Euclideandistancebetweenthemcanbemeasuredinonedimensionif
samplesaretakenalongasingletransect,orintwodimensionsifsamplesaretakenoveran
area.Spatialsamplingmethodsusethepossiblerelationshipbetweennearbysamplingunits
as additional information to better estimate the quantities of interest. One other important
considerationinspatialsamplingconcernsthe‘spatialsize’ofthesamplingunit,e.g.,asoil
sampleisofsmall‘spatialsize’inthecontextofmappingavalley.Inremotesensingapplica-
tions,thespatialsize’ofthesamplingunitmaybeseveralhundredsquaremetersandsmall
scalefeaturesandvariationwithinasamplingunitwouldnotbeobservable.Inenvironmental
radioactivity,asingeneralspatialproblems,therearetwoquitedifferentgeneralcases:
CASE1. Weassumethatinprincipleitispossibletomeasuretheradionuclideatanylocation
definedbycoordinates(x,y)overthedomainorareaofinterest.Thiscasewouldgenerally
beappropriateforradionuclidesinsoil,waterandair,suchasmappingtheChernobylfallout,
wherex andy wouldtypicallybelatitudeandlongitudeorsomeotherpositioningmetric.
CASE 2. We assume that in principle it is not possible to measure the radionuclide at all
locationsdefinedbycoordinates(x,y)overthedomainorareaofinterest,butthatitcanbe
measured only at specific locations. For example, consider 137Cs concentrations in trees. It
canonlybemeasuredatlocationsoftrees.