Reference: Smith, James E., and Linda S. Heath. 2000
Considerations for interpreting probabilistic estimates of uncertainty of forest carbon. P. 102-111
In: Joyce L.A. and R. Birdsey, eds. The Impact of Climate Change on America's Forests,
USDA Forest Service, General Technical Report RMRS-GTR-59. 134 p.
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Considerations for Interpreting Probabilistic Estimates of Uncertainty of Forest Carbon
Smith and Heath
forest floor, and soil based principally on age and volume
inputs. An example of such a relationship is shown as the
solid line in figure 7.1. Here, forest floor carbon inventory is
estimated from stand age. Subsequent reference to a “FORCARB parameter” refers to this type of functional relationship. Carbon pools are then expanded to total carbon for
large areas of similar forest-type and productivity within a
region. These large areas are termed “forest management
units” (103 to 107 ha with a median of 180,000 ha for the
1990 inventory). Regional subtotals are formed and, finally,
summed to a national total. Private timberlands in the
48 contiguous states are represented by results presented
here, which include 216 forest management units. Carbon
budget projections are presented in greater detail in Heath
and Smith (2000). The basic sequence of a FORCARB simulation is illustrated in figure 7.2.
Uncertainty
Some level of uncertainty is usually a part of any
model, assessment, or decisionmaking whether or not it
is an explicitly considered part of the process. A widely
used and potentially general term such as “uncertainty”
can be confusing or misleading unless it is adequately
defined (Hattis and Burmaster 1994; Shackley and Wynne
1996). At its simplest level, uncertainty can be the state of
not knowing, or the inability to quantify something with
a single discrete value. Sources of uncertainty can vary
widely, and as a consequence, attempts to narrow the definition can require reference to variability, ignorance, systematic error, unknowns, expert opinion, semantics, or
misapplication of a model (Morgan and Henrion 1990;
Hattis and Burmaster 1994; Rowe 1994; Ferson and Ginzburg 1996; Cullen and Frey 1999). In earlier literature
(largely stemming from Knight 1921), scientists were careful to define the risk of an event by a probability based
on documented frequencies of occurrence. Risk was contrasted with uncertainty where such probabilities could
not be assigned. However, current applications employ a
range of definitions for uncertainty, including probability;
furthermore, valid definitions of probabilities can include
observed frequency or even subjective expectation (Hoffman and Hammonds 1994; Reckhow 1994; Dakins et al.
1996; Schimmelpfennig 1996; Paoli and Bass 1997; Haynes
and Cleaves 1999). We use a probabilistic definition of
uncertainty.
An unknown, but unique, inventory of carbon exists
within a given forest management unit for a particular
year. Our inability to precisely specify that value is the
general definition of uncertainty we employ here. This
concept of uncertainty implies that we can specify a range
USDA Forest Service Gen. Tech. Rep. RMRS–GTR–59. 2000.
Figure 7.1—An example of a typical functional relationship (or
FORCARB model parameter) used to project forest floor carbon
inventory based on stand age (solid line). Probability bands
illustrate our meaning and use of uncertainty in “FORCARB
parameters” for this analysis. The bands indicate the 5th, 25th,
50th (expected value), 75th, and 95th percentiles (bottom to top
respectively) of the probability distribution around the dependent
variable. (Relationship is from a Douglas fir forest management
unit.)
of possible values and an associated likelihood for values
within that range. This describes a probability distribution, or more properly, a probability density function
(PDF). Thus, we use PDFs as convenient quantitative
and graphical representations of uncertainty (Vose 1996;
Cullen and Frey 1999).
The effect of this definition of uncertainty, applied to
estimating carbon for a given subset of a forest management unit, is illustrated in figure 7.1. The broken lines
are probability bands indicating specific points on dependent variable PDFs–or uncertainties–about exact values
of carbon per unit area. These probabilities reflect uncertainty in predicting carbon from stand age. Normally
distributed PDFs were assumed to describe uncertainty
about FORCARB parameters (details in Heath and Smith,
2000). No assumption of normality was required for this
model: its use was simply a convenience for describing
assumed expected values with symmetrical distributions.
Analyses would ideally address all sources of uncertainty
relevant to policymakers’ questions about forest carbon
inventory and flux. However, as mentioned above, a pragmatic first step is to focus on uncertainty internal to FORCARB. Therefore, uncertainties presented here are limited
to this portion of the potentially much larger system of
models that describe forests.
103
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THROUGHAPPLICATIONOFTHECENTRALLIMITTHEOREM-ORGAN
AND(ENRION#ULLENAND&REY4HISASSUMES
RELATIVELYBALANCEDCONTRIBUTIONSAMONGEACHOFTHE0$&S
SUMMED AND INDEPENDENCE AMONG 0$&S 5NDER THESE
CONDITIONS THE SUM IS EXPECTED TO BE NORMALLY DISTRIB
UTEDANDTHEVARIANCEOFTHESUMISEQUALTOTHESUMOF
THEVARIANCES
$ISPARITY AMONG SIZE OF THE FOREST MANAGEMENT
UNIT CARBON INVENTORY POOLS CAN INmUENCE CONTROL OVER
TOTAL CARBON AND TOTAL UNCERTAINTY )F MOST OF THE TOTAL
CARBONINVENTORYISATTRIBUTABLETOAFEWLARGEFORESTUNITS
THENRESEARCHTOIMPROVETHEPARAMETERESTIMATESOFTHESE
UNITSWILLUSUALLYCONTRIBUTEMORETOIMPROVEESTIMATES
OFTOTALCARBONINVENTORYTHANIMPROVINGTHEPARAMETERS
OFSMALLERFORESTMANAGEMENTUNITS4HELARGERPERCENT
OF THE PRIVATE TIMBERLAND UNITS SIMULATED FOR THIS STUDY
ACCOUNTFORMORETHANTWOTHIRDSOFTHETOTALCARBONlG
4HAT IS ONLY PERCENT OF THE MANAGEMENT UNITS
EXCEED 0G # THE SECOND SIZE CLASS IN lG YET
THEYACCOUNTFOROVERTWOTHIRDSOFTHETOTAL#INVENTORY
4HEUNCERTAINTYOFPARAMETERSOFTHESMALLERUNITSWOULD
HAVE TO BE EXTREMELY LARGE TO PRODUCE GREATER ABSOLUTE
UNCERTAINTY THAN THE LARGE UNITS 4HE DISPARITY IN SIZE
AMONG THE FOREST MANAGEMENT UNITS SUGGESTS THAT
THE 0$& OF AN AGGREGATE TOTAL COULD NOT BE DETERMINED
THROUGHSIMPLEAPPLICATIONOFTHECENTRALLIMITTHEOREM
$ETERMINATIONOFINDEPENDENCEORCONVERSELYDEPEN
DENCE AMONG 0$&S DEPENDS ON BOTH PRIOR KNOWLEDGE
OF THE VALUES AND THE MODELING PROCESS 4HE MEANING
ASSIGNED TO UNCERTAINTY OF INPUT 0$&S OR &/2#!2"
PARAMETERS BECOMES CRITICAL AS THE SEPARATE POOLS ARE
SUMMED 7E USE UNCERTAINTY AS AN EXPRESSION OF OUR
EXPECTED LEVEL OF IGNORANCE &OR EXAMPLE UNCERTAINTY
INCLUDES OUR INABILITY TO TRANSLATE AN INDEPENDENT VARI
ABLE SUCH AS AN EXACT VOLUME OF TIMBER ON AN EXACTLY
SPECIlED AREA OF LAND TO A PRECISE QUANTITY OF CARBON
INTHESYSTEM)FOURABILITYTOMAKETHATESTIMATEISSIM
53$!&OREST3ERVICE'EN4ECH2EP2-23n'42n
&IGUREˆ(ISTOGRAMSILLUSTRATINGTHEDISPARITYINSIZEOFCARBON
INVENTORIESAMONGTHEFORESTMANAGEMENTUNITSCONTRIBUTING
TOTHENATIONALESTIMATEFORTHEYEARINTERMSOFANUMBER
OFFORESTMANAGEMENTUNITSPERSIZECATEGORYANDBTOTALCARBON
SUMOFUNITSPERSIZECATEGORYBILLIONMETRICTONS
ILAR ACROSS FOREST TYPES AND REGIONS THEN THE ESTIMATES
OF UNCERTAINTY WOULD BE JOINTLY RELATED OR HIGHLY CORRE
LATED(OWEVERASTHEESTIMATESBECOMEMOREDEPENDENT
ONELEMENTSOFBIOLOGYMANAGEMENTECOLOGYORBIOGEO
CHEMISTRYOFTHERESPECTIVEFORESTSTHEDEGREEOFINDEPEN
DENCEAMONGTHESEPARATEESTIMATESWILLTENDTOINCREASE
3IMILARLYIFWEVIEWUNCERTAINTYASSIMPLYRANDOMVARI
ABILITY THEN THE SEPARATE ESTIMATES MADE FOR DIFFERENT
FORESTTYPESWOULDALSOBECONSIDEREDINDEPENDENT
!SSUMPTIONSABOUTCOVARIABILITYAMONGSEPARATELY
DETERMINED FOREST CARBON POOLS CAN HAVE A TREMENDOUS
EFFECTONTHEAPPARENTUNCERTAINTYOFTHETOTAL&/2#!2"
SIMULATIONS IN (EATH AND 3MITH REmECTED A REL
ATIVELY HIGH DEGREE OF JOINT CORRELATIONnGENERALLY WITH
COEFlCIENTS OF CORRELATION BETWEEN AND &IGURE
SHOWS THE POSSIBLE EFFECTS OF COVARIABILITY AMONG
THEFORESTMANAGEMENTUNITS4HEDISTRIBUTIONSWERE
SPECIlEDASHAVINGJOINTCORRELATIONSWITHCOEFlCIENTSOF
CORRELATIONOFAPPROXIMATELYANDLOWTO
HIGHCOVARIABILITYBASEDONMODIFYINGTHEIRRANKORDERS
FROM THE -ONTE #ARLO SIMULATION )MAN AND #ONOVER
4HISWASSIMPLYANUMERICALMANIPULATIONTODEM
3MITHAND(EATH
#ONSIDERATIONSFOR)NTERPRETING0ROBABILISTIC%STIMATESOF5NCERTAINTYOF&OREST#ARBON
&IGURE ˆ(YPOTHETICAL ESTIMATE OF CARBON INVENTORY BILLION
METRICTONSOFPRIVATETIMBERLANDSFORTHEYEARASAFFECTED
BYCOVARIABILITYAMONG0$&SFOREACHOFTHEFORESTMANAGE
MENTUNITS"EFORESUMMINGTHESEPARATE0$&SCORRELATIONCOEF
lCIENTSWERESETATAPPROXIMATELYANDTOPRODUCE
NARROWESTTOWIDESTDISTRIBUTIONSFORTHETOTALRESPECTIVELY
ONSTRATE THE EFFECT OF COVARIABILITY ON APPARENT UNCER
TAINTY 4HE PROBABILITIES OF HIGHVALUED SAMPLES ARE
LARGELY CANCELED OUT BY LOWVALUED SAMPLES WHEN THE
SUMMED DISTRIBUTIONS ARE CONSIDERED LARGELY INDEPEN
DENTWITHCOEFlCIENTOFCORRELATIONR4HISCENTRAL
TENDENCY PRODUCES A RELATIVELY NARROW DISTRIBUTION IN
CONTRAST TO HIGH CORRELATION WHERE FACTORS LEADING TO
HIGHERVALUED SAMPLES OF CARBON IN ONE SYSTEM WOULD
ALSOLEADTOHIGHERVALUEDSAMPLESINANOTHER4HEINTER
VAL BETWEEN THE TH AND TH PERCENTILES WAS TIMES
GREATERWITHRTHANWITHR7EEMPHASIZETHAT
MANIPULATIONS DONE HERE WERE SIMPLY A MEANS OF DEM
ONSTRATINGTHECONSEQUENCESOFCOVARIANCETERMSANDTHE
IMPORTANCEOFANYASSUMPTIONABOUTINDEPENDENCE
!VERAGEANNUALCARBONmUXISBASEDONTHEDIFFERENCE
BETWEEN 0$&S REPRESENTING CARBON INVENTORY ESTIMATES
lG (ERE TOO THE VALUE OF THE COVARIANCE TERM IS
IMPORTANT7ITHINDEPENDENCEBETWEENTHETWOINVENTO
RIES UNCERTAINTY OF THE mUX ESTIMATE IS DIRECTLY PROPOR
TIONAL TO UNCERTAINTY IN THE TWO DISTRIBUTIONS (OWEVER
NONZERO COVARIANCE AFFECTS THE SIZE OF THE mUX 0$& AS
ILLUSTRATEDINlGUREBYMANIPULATIONSOFTHECOEFlCIENT
OF CORRELATION BETWEEN THE TWO INVENTORY 0$&S )N GEN
ERALRANGEOFUNCERTAINTYINESTIMATEDAVERAGEANNUALmUX
&IGUREˆ%XAMPLESOFTHEEFFECTSOFCOVARIABILITYBETWEENESTIMATESOFCARBONINVENTORYMILLIONMETRICTONSONAVERAGEANNUAL
NETmUXMILLIONMETRICTONSPERYEARUNCERTAINTY%STIMATESFORCARBONINVENTORYOFPRIVATETIMBERLANDSFORYEARSANDWERE
BASEDONJOINTCORRELATIONSAMONGFORESTUNITSSETATR(YPOTHETICALAVERAGEANNUALmUX0$&SWERECALCULATEDUSINGCORRELATION
COEFlCIENTSBETWEENYEARSSETATANDPRODUCINGWIDEANDNARROWDISTRIBUTIONSRESPECTIVELY&LUXCALCULATIONSWEREBASED
ONANNUALIZEDDIFFERENCEBETWEENANDDISTRIBUTIONS0OSITIVEmUXINDICATESTHATCARBONISBEINGSEQUESTEREDINTHEFOREST
53$!&OREST3ERVICE'EN4ECH2EP2-23n'42n
#ONSIDERATIONSFOR)NTERPRETING0ROBABILISTIC%STIMATESOF5NCERTAINTYOF&OREST#ARBON
ISINVERSELYPROPORTIONALTOCOVARIANCEBETWEENINVENTO
RIES)FSIMILARINFORMATIONWASUSEDTOESTIMATECARBONIN
EACHOFTHETWOYEARSTHENTHETWODISTRIBUTIONSWOULDBE
HIGHLYCORRELATED4HISWASTHECASEHERETABLEWHERE
AGEANDVOLUMEWERESPECIlEDWITHOUTUNCERTAINTYAND
&/2#!2" MODEL PARAMETERS SIMILARLY APPLIED IN EACH
YEARSESTIMATEWERETHEONLYSOURCESOFUNCERTAINTY
3UMSANDDIFFERENCESOFRELATED0$&SDEPENDONADDI
TION AND SUBTRACTION OF COVARIANCE TERMS RESPECTIVELY
4HESE ARE STRAIGHTFORWARD CALCULATIONS IF COMPLETE VARI
ANCECOVARIANCE TABLES ARE READILY AVAILABLE 3UCH INFOR
MATION MAY BE PROVIDED WITH ORIGINAL DATA SETS OR IT
CANBEEXPLICITLYSIMULATEDWITHINMODELS(OWEVERFULL
KNOWLEDGEOFCOVARIANCESISNOTAVERYREALISTICEXPECTA
TIONWHENFACINGSEPARATELYACQUIREDESTIMATESOFUNCER
TAINTY FROM INDEPENDENT SOURCES .EVERTHELESS EVEN
SIMPLEQUALITATIVEINFORMATIONCANBEUSEFULLYAPPLIEDTO
SORTINGTHROUGHPOSTANALYSIS0$&S&OREXAMPLESIMPLY
KNOWINGTHATSOMEPOSITIVEBUTUNSPECIlEDLEVELOFCOR
RELATION EXISTS BETWEEN A PAIR OF VARIABLES WOULD LEAD
ANANALYSTTOPLACEMORECONlDENCEINSUMMARIESWHERE
VALUES WERE JOINTLY DRAWN FROM SIMILAR REGIONS OF THE
RESPECTIVE 0$&S !NOTHER EXAMPLE OF INFORMATION PRO
VIDEDBYEVENLIMITEDKNOWLEDGEOFCOVARIABILITYISTHE
EFFECT OF UNCERTAINTY IN TWO INVENTORY 0$&S ON UNCER
TAINTY IN ESTIMATED mUX 4HE ASSUMPTION OF INDEPEN
DENCEBETWEENINVENTORIESISACONSERVATIVEASSUMPTION
LEADING TO LARGE UNCERTAINTY IN mUX !NY KNOWLEDGE OF
RELATEDNESSBETWEENTHETWOINVENTORIESWILLREDUCEmUX
UNCERTAINTY EVEN WITHOUT REDUCING UNCERTAINTY OF THE
RESPECTIVEINVENTORY0$&S
)MPLICATIONFORA,ARGER%XTERNAL3YSTEM
$ECISIONMAKERSARESELDOMPROVIDEDPROBABILISTICESTI
MATES OF UNCERTAINTY WITHOUT ANY ACCOMPANYING INFOR
MATIONAPPLICABLETOITSUSEORCONTEXT3IMILARLYTHEYARE
UNLIKELY TO BE FACED WITH SUMMING SEPARATE 0$&S
4HEMODELINGEXAMPLESWEREPROVIDEDHERETOILLUSTRATE
CONSIDERATIONSFORSUMMARIZING0$&SASDESCRIPTIONSOF
UNCERTAINTY 4HE EFFECTS OF TABULAR SUMMARIES AND RELAT
EDNESS ARE ALSO USEFUL WHEN ADDRESSING ISSUES OF MANY
UNCERTAINTIESINACOMPLEXSYSTEM
4HESYSTEMDElNEDBYTHE&/2#!2"MODELISCLEARLY
ASUBSETOFALARGERINTEGRATEDSYSTEM#ONCERNOVERTHE
PROSPECT OF RAPIDLY GROWING UNCERTAINTIES AS MORE ELE
MENTS ARE BROUGHT INTO AN ANALYSIS CANNOT BE QUANTI
TATIVELY ADDRESSED WITHOUT COMPREHENSIVE UNCERTAINTY
ANALYSES (OWEVER THE RESULTS PROVIDED HERE DO ILLUS
TRATE THE EFFECTS OF COVARIABILITY AMONG PARTS AND HOW THE DElNITION OF AN INTERVAL AFFECTS THE PERCEPTION
OFUNCERTAINTY&OREXAMPLETHEINTERVALBETWEENTHETH
ANDTHETHPERCENTILESOFTHECARBONINVENTORY0$&
FORTHE.ORTHEASTERN&OREST)NDUSTRY-APLE"EECH"IRCH
53$!&OREST3ERVICE'EN4ECH2EP2-23n'42n
3MITHAND(EATH
FOREST MANAGEMENT UNIT RESULT NOT SHOWN IS ABOUT PERCENT OF THE MEDIAN 4HE CORRESPONDING INTERVAL FOR
THE NATIONAL TOTAL AFTER ADDING THE ADDITIONAL FOREST
MANAGEMENTUNITSISONLYABOUTPERCENTOFTHEMEDIAN
TABLE 4HE SAME INTERVAL COULD RANGE FROM AND
PERCENT OF THE MEDIAN BY SIMPLY ADOPTING DIFFERENT
ASSUMPTIONS ABOUT COVARIABILITY AMONG FOREST MANAGE
MENTUNITSASILLUSTRATEDINlGURE2ELATIVEUNCERTAINTY
ONE DElNITION OF AN INTERVAL DECREASED WHILE ABSOLUTE
UNCERTAINTY ANOTHER DElNITION OF AN INTERVAL INCREASED
AS FOREST UNITS WERE SUMMED UNDER AN ASSUMPTION OF
INDEPENDENCE4HISWAS BECAUSEBOTHMEDIANANDVARI
ANCETERMSINCREASEDLINEARLYMAKINGTHETHTOTHPER
CENTILE INTERVAL WHICH INCREASED IN PROPORTION TO THE
SQUAREROOTOFTHEVARIANCEANINCREASINGLYSMALLERPRO
PORTIONOFTHEMEDIAN
-ODELSSTRUCTUREDTOSERVEASACCOUNTINGSYSTEMSFOR
EXAMPLETOTALFORESTCARBONINVENTORYCANBENATURALLY
ORGANIZED INTO TWO SEQUENTIAL STEPS &IRST DETERMINE A
PERUNIT VALUE OF THE QUANTITY FOR EXAMPLE CARBON PER
POOL PER HECTARE AND SECOND SUM THESE UNITS ACROSS
ANAPPROPRIATEINDEXFOREXAMPLEFORESTAREA4HISPAT
TERNAPPEARSINMODELS.ILSSONAND3CHOPFHAUSER
(EATHETALANDNATIONALSUMMARIES"IRDSEYAND
(EATH+URZETALASWELLAS)0##RECOMMEN
DATIONS FOR GREENHOUSE GAS INVENTORIES )0##/%#$
)%!B#HOICESANDASSUMPTIONSMADEINTHECOURSE
OFMODELINGAFFECTTHEFORMANDRELATEDNESSOFINTERMEDI
ATE0$&SANDTHESECANAFFECTlNALRESULTS
2ECOMMENDATIONSFORPOOLINGUNCERTAINTIESOFTENCON
TAINIMPLICITBUTNOTCLEARLYSTATEDASSUMPTIONSOFINDEPEN
DENCEFOREXAMPLE6OLUMEP!)0##/%#$)%!
B 3UCH RELATIONSHIPS AMONG UNCERTAINTIES MAY BE
REASONABLE AND ACCURATE BUT COULD EASILY AND INADVER
TENTLYBEHIDDENINASSUMPTIONSASMODELSAREITERATIVELY
ANALYZEDANDREVISED#LEARLYISSUESOFUNCERTAINTYCON
TINUETOCHANGEANDAREUNLIKELYTOBEENTIRELYRESOLVEDnTHE
STATEOFSCIENCEANDTHEQUESTIONSSOCIETYASKSOFSCIENCE
CHANGE CONTINUOUSLY 4HEREFORE A MODEL STRUCTURE THAT
CLEARLYANDASSIMPLYASPOSSIBLESTATESBASICASSUMPTIONS
ISESSENTIALFORSUBSEQUENTUSEOFUNCERTAINTY
$ECISIONS ARE SELDOM MADE ON THE BASIS OF A SINGLE
UNCERTAINTYANALYSISGENERALLYMULTIPLEINmUENCESNEED
TO BE CONSIDERED AND MERGED BY DECISIONMAKERS *OINT
#LIMATE0ROJECT2ECKHOW+LABBERSETAL
0AOLI AND "ASS 0ROBABILISTIC EXPRESSIONS RESULTING
FROMANALYSESAREUSEFULTODECISIONMAKERSFORCONSIDER
INGMULTIPLEINmUENCES(OFFMANAND(AMMONDS
-ORGAN AND $OWLATABADI !SYSTEMS PERSPECTIVE
ISEVENMOREIMPORTANTWHENTHEARRAYOFEXTERNALINmU
ENCES AND ACCOMPANYING UNCERTAINTIES ARE CONSIDERED
'LOBALCHANGEWILLAFFECTFORESTCOMPOSITIONANDGROWTH
AS WELL AS MANAGEMENT PRACTICES AND TIMBER MARKETS
#LIMATE SENSITIVITY AND FOREST SECTOR PROJECTIONS CONTAIN
ADDITIONAL UNCERTAINTIES THAT WE PLAN TO INCORPORATE IN
3MITHAND(EATH
#ONSIDERATIONSFOR)NTERPRETING0ROBABILISTIC%STIMATESOF5NCERTAINTYOF&OREST#ARBON
OURANALYSES7HEREANDHOWTHESEADDEDUNCERTAINTIES
APPROPRIATELY LINK WITH THE EXISTING MODEL CAN STRONGLY
INmUENCERATEOFPROPAGATION
ELING OR USING THE RESULTS FROM MODELING 4HESE ARE NOT
COMPLICATED SETS OF RULES BUT EXAMPLES OF THE NEED FOR
CLEARSTATEMENTSOFDElNITIONSANDASSUMPTIONS
3UMMARY
,ITERATURE#ITED
0ROBABILITIES ARE COMMONLY EMPLOYED TO QUANTIFY
UNCERTAINTY $ISCUSSION IN THIS CHAPTER FOCUSED ON HOW
SUMMARIES OF PROBABILITY DISTRIBUTIONS OR THEIR SUBSE
QUENT USE CAN AFFECT THE INTERPRETATION OF UNCERTAINTY
-ODELRESULTSPRESENTEDHEREREPRESENTPRELIMINARYESTI
MATESOFAPORTIONOFTHEUNCERTAINTYINCARBONBUDGETS
FORPRIVATE53TIMBERLANDS
4RACTABLEUSEOFRESULTSFROMUNCERTAINTYANALYSESOFTEN
REQUIRE TABULAR SUMMARIES OF PROBABILITY DENSITY FUNC
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UNCERTAINTYSHOULDEXCEEDTHELIKELYLOSSOFINFORMATION
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RIES SHOULD STILL REmECT THE ESSENTIAL INFORMATION DESIRED
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STANDINGTHEFORMOFTHESUMMARYCANHELPASSUREANET
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ILLUSTRATESOMEBASICCONSIDERATIONSFORENSURINGTHISLINK
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COVARIABILITYAMONGPARTSTHENBECOMEIMPORTANTCON
SIDERATIONS
0ROBABILISTIC MODELS SUCH AS THE IMPLEMENTATION OF
&/2#!2" REFERENCED HERE EXPLICITLY ACCOUNT FOR SUCH
CHARACTERISTICS OF 0$&S 4HESE GUIDELINES ARE APPLICABLE
WHENEVERUNCERTAINTYISDESCRIBEDINTERMSOFPROBABILI
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CHANGESCIENCEANDPOLICY"OUNDARYORDERINGDEVICESANDAUTHOR
ITY3CIENCE4ECHNOLOGY(UMAN6ALUESn
3MITH*%(EATH,3;ACCEPTEDFORPUBLICATION=#HARACTERIZINGCOM
PONENTSOFINPUTUNCERTAINTYINAFORESTCARBONBUDGETMODEL%NVI
RONMENTAL-ANAGEMENT
3OHNGREN",(AYNES274HEPOTENTIALFORINCREASINGCARBON
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TIONMODELLING*OHN7ILEY3ONS#HICHESTER%NGLANDP
The Impact of Climate Change
on America’s Forests
A Technical Document Supporting the 2000
USDA Forest Service RPA Assessment
U.S. DEPARTMENT OF AGRICULTURE
FOREST SERVICE
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