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.