Marine Geodesy

21 Pages·2017·2.27 MB·English

by  Brian Calder

Marine Geodesy
ISSN: 0149-0419 (Print) 1521-060X (Online) Journal homepage: http://www.tandfonline.com/loi/umgd20
Development of an Uncertainty Propagation
Equation for Scalar Fields
Brian Calder & Paul Elmore
To cite this article: Brian Calder & Paul Elmore (2017) Development of an Uncertainty Propagation
Equation for Scalar Fields, Marine Geodesy, 40:5, 341-360, DOI: 10.1080/01490419.2017.1345811
To link to this article: https://doi.org/10.1080/01490419.2017.1345811
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Jul 2017.
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Download by: [University of New Hampshire] Date: 12 December 2017, At: 11:21
MARINEGEODESY
2017,VOL.40,NO.5,341–360
https://doi.org/10.1080/01490419.2017.1345811
Development of an Uncertainty Propagation Equation
for Scalar Fields
BrianCalderaandPaulElmoreb
aJereA.ChaseOceanEngineeringLaboratory,CenterforCoastal&OceanMapping/JointHydrographicCenter,
UniversityofNewHampshire,Durham,NewHampshire,USA;bMappingChartingandGeodesyBranch,Naval
ResearchLaboratory,JohnC.StennisSpaceCenter,HancockCounty,Mississippi,USA
7
1
0
2 ABSTRACT ARTICLEHISTORY
er Theuncertaintyofascalarfieldisessentialstructuringinformationfor Received20October2016
b
m any estimation problem. Establishing the uncertainty in a dense Accepted14June2017
ece gridded product from sparse or random uncertainty-attributed input KEYWORDS
D data is not, however, routine. This manuscript develops an equation DEM;mapping;hydrographic
2 that propagates the uncertainty of individual observations, arbitrarily survey;oceanmapping;
1 1 distributedinR2,toacommonestimationlocationatwhichtheycan bathymetry;multibeam
1:2 be used to determine the composite uncertainty of the output field. sonar
1 Theequationincludestheeffectofthedistancebetweentheobserva-
at tion and estimation locations, the field and horizontal uncertainty of
e] theobservation,anduser-parameterstocontroltheexpectedvariabil-
hir ityinthefieldasafunctionofdistance.Twocomputationalversionsof
ps the equation, a lower cost conservative approach and a higher cost
m
mean-distance approach, are developed and evaluated for computa-
a
H tionalcostandresultingaccuracyinnumericalexperimentsoversimu-
w latedbathymetricdata.Themean-distanceapproachismoreaccurate,
e
N but more costly; suitable numerical approximations are proposed to
f controlcomputationalcosts.Abenefitoftheworkdescribedisflexibil-
o
y ityandenhancementforapplicationsofthemodel,suchastheCom-
sit binedUncertaintyandBathymetryEstimatoralgorithm,whichisused
r
e asademonstrationofthedifferencebetweenthetwoversionsofthe
v
ni equation.
U
[
y
b
d
e Introduction
d
a
o
nl Considerthefollowingcomputationaltask: estimateoutputinterpolationvaluesfro(cid:2)minp(cid:3)ut
w
o points in R2 space. In polar coordinates, the interpolated value is zbDI fz r;u g.
D i i j j j
The(cid:2) op(cid:3)erator Iif(cid:2)(cid:2)(cid:2)g is an interpolation over the set of input points
fzj rj;uj :j2NðiÞgD½z1ðr1;u1Þ;:::;zJðrJ;uJÞ(cid:3), where NðiÞ is a local neighborhood(cid:2), th(cid:3)at
yields zb. Further, let there be azimuthal symmetry so that we can simplify zbDI fz r g.
i i j j j
The following model of propagation of uncertainty from input data to an interpolated out-
putpointisbasedonthatdevelopedfortheCombinedUncertaintyandBathymetryEstima-
tor(CUBE)algorithm(CalderandMayer2003).
CONTACT BrianCalder [email protected] JereA.ChaseOceanEngineeringLaboratory,CenterforCoastaland
OceanMapping/JointHydrographicCenter,UniversityofNewHampshire,24ColovosRoad,Durham,NH03824,USA.
Colorversionsofoneormoreofthefiguresinthearticlecanbefoundonlineatwww.tandfonline.com/umgd.
©2017Taylor&FrancisGroup,LLC
342 B.CALDERANDP.ELMORE
The objective of the interpolation operator is to propagate uncertainty inherent in an
observation from the location of the observation to the interpolation location. This process
allows the inherent uncertainty of the observation to be used for estimating the composite
uncertainty of the field being estimated. Uncertainty at the observation could be estimated
throughaformalpropagationofmeasurementuncertainties,somedefineduncertaintyesti-
mationprotocol,oranempiricalestimationprocess.
Let each measurement m have horizontal uncertainty s and vertical uncertainty s .
j h, j v, j
Calder and Mayer (2003) give a model of propagation of uncertainty, (s , s ), from the
h, j v, j
jthinputdatapointtotheithinterpolatedoutputpointas
(cid:6) (cid:4) (cid:5) (cid:7)
s2Ds2 1C rijCshsh;j a (1)
ij v;j D
g
7
1
0
er 2 whereDgistheoutputgridspacing,andshandaareuser-parametersthatgoverntheconfi-
b dence interval desired and growth rate for uncertainty. For the case of bathymetry with a
m
ce slopedseafloor,Zamboetal.(2015)(q.v.Bourgeoisetal.2016)addanextratermtoaccount
e
D forextrauncertaintywhentheseafloorhasslope’ attheinterpolationpoint
2 i
1
1 (cid:6) (cid:4) (cid:5) (cid:7)
1:2 s 2Ds2 1C rijCshsh;j a Cs2 tan2’i (2)
at 1 ij v;j Dg h;j
]
e
r
hi (usingtheslopeattheinterpolationpointprovidessmootheduncertaintyestimationforthe
s
p
m slopecontribution;see(Zamboetal.2015;Bourgeoisetal.2016)forfurtherdiscussion).
a
H It is important to note that, while (1) and (2) were developed for bathymetry, both are
ew applicabletoscalarfieldsingeneral.For(2),generalizationisachievedbyreplacingthesec-
N
of ondtermwithjrzbij2 sincejrzbijDsHjtan’i(Zamboetal.2015).
y Whatevertheeventualuseofthepredicteduncertainty,themethodbywhichitisgenerated
rsit should, for consistency, follow some general principles. This paper presents these principles,
e
niv provides an analytic basis for (1) and (2), generalizes this propagated uncertainty equation to
U scalarfields,and outlinesanewerapproachthatreducesuncertaintycomparedtotheprevious
[
by generationofuncertaintymodeling(CalderandMayer2003;Zamboetal.2015).
ed Intheremainingpart,“BaseEquation”sectionprovidesprinciplesthatenablefamiliesof
d
a uncertainty propagation equations to be proposed, and examines some properties of these
o
nl functions. “Conservative Approach” section shows how a single equation with three end-
w
Do useradjustableparametersemergesfromthisfamily;(1)comesfromthisfamilywithaspe-
cificchoiceofparameters.
The“Mean-distanceApproach”sectiondevelopsanewversionofthemodelthatreduces
the number of user-specified parameters needed from three to two by considering the
expected distance that an observation will be propagated, given its horizontal uncertainty,
rather thantheworst-casedistance as wasused in previousversions. DuetotheRician dis-
tributionofdistance(underGaussianhorizontaluncertainty),theexpectedvaluehasasolu-
tion through Laguerre functions, discussed in “Analytic Expression Using L (x)” section.
1/2
“The Evaluation of L (x)” and “Chebyshev Series Fit to F ( ¡ 1/2, 1; x)” sections discuss
1/2 1 1
numerical techniques using the confluent hypergeometric function and Chebyshev polyno-
mialstoefficientlycomputetheexpectedpropagationdistance.
MARINEGEODESY 343
“TheExample:EstimatingDepth”sectionprovidesanumericalexampleforestimationof
depth and uncertainty using the models developed in “The Mean-distance Approach” sec-
tion.Thepaperendswith“Discussion”and“Conclusions”sections.
Baseequation
Uncertainty isdefined here tobe standard deviation, denoted s (Taylorand Kuyatt 1994).For
convenience,however, the equationsare writtenintermsofvariance,denoted s2.Methodsfor
predicteduncertaintyofscalarfieldsofinterestinthispaperfollowthesegeneralprinciples:
1. The value of the information in an observation should decay away from its nominal
location;thatis,theuncertaintyshouldincreasewithdistancefromthenominalobser-
vationlocation.
2. Information inherent in an observation with higher horizontal uncertainty should be
7
1 morediffuse,sothattheuncertaintypredictedafterpropagationshouldbehigherthan
0
2
r thatforanobservationwithlowerhorizontaluncertainty.
e
mb 3. Propagationshouldscalewiththeresolutionofanalysissothatthedistanceconsidered
ce significantismeasuredagainstthegridresolution.
e
D 4. Sincetheuncertaintyisbeingtransferredtoapointforconsideration,thefinalpropagated
12 uncertaintyshouldhavenohorizontalcomponent,sincethelocationispreciselydefined.
1
2 Consequently, the equation must be able to combine the effects of positional and field
1:
1 uncertaintyintheobservationintoapurelyfielduncertaintyatthereferencepoint.
at
] Many possible equations could be proposed that follow these principles, but one of the
e
hir simplestis
s
mp s2DaCbda (3)
a
w H where a;b2R>0 areconstants,a2R(cid:2)1,anddistheeffectivedistancebetweentheobserva-
e tionand referencelocations.Itshouldbeclearfromtheformof(3)thatthedistancetermbda
N
f with a (cid:2) 1 satisfies the first general principle, the constant term satisfies the second general
o
y principle,andthewholeformsatisfiesthefourth.Thethirdgeneralprincipleisinducedbycon-
Universit ssi2dfeirsinthgeafipeplrdopvariraiatencbeouonfdthaeryobcosenrdviattioionns,.aTnhdathiesn,cfoe,racoDnssis2tfe.nIncyo,radtedrDtob0,esaebtlse2toDcosn2tfrwolhtehree
[ speed of uncertainty increase with distance, assume a secondary boundary condition such that
y
b s2Dks2 atsomesuitabledistance,suchasthegridsamplespacing,D.Hence
d f g
e
d
oa ks2 D s2CbDa
nl f f g
w k¡1 (4)
Do ) b D s2f Da
g
Substitutingin(3)yields
da
s2 D s2Cs2ðk¡1Þ
f f Da
(cid:6) (cid:6)g (cid:7) (cid:7) (5)
a
d
D s2 1Cðk¡1Þ
f D
g
Eq. (5) forms the base model for which we propose two methods of computation, which
differ in the way they compute parameter d. The distance d used has to reflect the actual
344 B.CALDERANDP.ELMORE
distance traveled by the observation to the interpolation point rather than a nominal dis-
tance, d between themostlikely locationoftheobservationandtheinterpolationlocation.
0
Fortractability,itisassumedthatthehorizontalobservationdistributionisGaussian.
Conservativeapproach
Thesimplestmodeltoincludetheeffectsofs ondistoassumethatdmustbegreaterthan
h
d bysomefactorthatisafunctionofs .The“conservative”approachistocreatethisfunc-
0 h
tion so that uncertainty is maximized. Due to positivity of d and s , if s > 0, this corre-
h h
spondstodDd Cs s sothat(5)becomes
0 h h
(cid:6) (cid:6) (cid:7) (cid:7)
d Cs s a
7 s2Ds2f 1Cðk¡1Þ 0 D h h (6)
1 g
0
2
r
e
b Thisequationhasthedisadvantageofhavingthreeadjustableparameters:a,s ,andk.If
m h
ce k(cid:4)2,(6)canbesimplifiedsincetheuncertaintyintheverticaldoubleswithinoneeffective
e
D samplespacing,irrespectiveofa,sothat
2
1
1 (cid:6) (cid:6) (cid:7) (cid:7)
1:2 s2Ds2 1C d0Cshsh a (7)
1 f D
at g
]
e
r
hi Eq. (7) has the same form as (1), which was developed for the CUBE algorithm (Calder
s
p
m and Mayer 2003). Ithas the benefit of simplicity in that the scale factor s modifies the rate
a h
H atwhichhorizontaluncertaintyisconvertedtofielduncertainty,andthevalueofacontrols
w
e theshapeofthetransfercurve(Figure1).Therateofincreaseofuncertainty(Figure2)can
N
f besignificant,however,particularlyforwhatmightbeconsidered“reasonable”scalefactors
o
y correspondingto,e.g.,the95%CIfortheobservation.
rsit Adjustingthescalefactorkcanadjustthisbehaviorsomewhat,butatthecostofanother
e
niv parameter which interacts with the others to make it more difficult for the algorithm
U designer,oruser,toadjustthetransferbehaviortomatchthespecificproblembeingtackled.
[
by Thismakesthepropagationequationdifficulttoadjustinpractice.
d
e
d
a
nlo Mean-distanceapproach
w
Do Instead of assuming the worst-case transfer distance for all observations, it is possible to com-
pute the expected distance instead. Assuming isotropic Gaussian distribution in the horizontal,
the distance is Rician distributed so that the expected distance can be computed as an analytic
expression via a (generalized) Laguerre function. Implementing this computation is more
expensivethanthe“conservative”function,buthasthebenefitofavoidingthes parameter.
h
Evaluationoftheexpecteddistancemustbeimplementednumericallyusingthefollowing
stages:
1. Analytic expression using the Laguerre function, L (x), with x D ( ¡ d2 /2s 2); a
1/2 0 h
numericalsolutionisrequired.
2. One-time numerical computation of L (x) via the confluent hypergeometric function
n
(CHGF), F (a,b;x);thiscomputationcanbeslowandintensive.
1 1
MARINEGEODESY 345
7
1
0
2
r
e
b
m
e Figure1. Increaseinpropagatedverticaluncertaintyasafunctionofeffectivedistanceandcontrolparame-
ec ter a for (7). Here, s D 1, D D 1.0, and s2 D 1.0 to normalize the graph. The amplification at one grid
D h g f
2 spacingiscontrolledbytheformoftheequation,whilethecontrolparametermanipulatestheshape.
1
1
2
1: 3. Fit of the CHGF using Chebyshev polynomials of the first kind, T (x), for a rapid
1 n
at approximationtotheone-timeCHGFcomputationresult.
e] In-depthdiscussionsofthesefunctionscanbefoundinAbramowiczandStegun(1965);see
r
hi alsoNationalInstituteof StandardsandTechnology(2015), Olveretal.(2010),andArfken
s
p
m etal.(2012,Ch.18).
a
H
w
e
N AnalyticexpressionusingL (x)
1/2
f
o
y Fora circularlysymmetric Gaussiandistribution inR2 withnon-zeromean vector,thedis-
rsit tributionofdistancefromtheorigin(equivalently,here,thedistancetheobservationtravels)
e
niv isgivenbyaRiciandistribution(Rice1945)
U
[
y
b (cid:6) (cid:7) (cid:8) (cid:9)
d (cid:2) (cid:3) d d d d2Cd2
de p d;d ;s2 D I 0 exp ¡ 0 (8)
nloa 0 h s2h 0 s2h 2s2h
w
o
D
whereI (x)isamodifiedBesselfunctionofthefirstkindoforderzero,andd isthenominal
0 0
distance(magnitudeofthemeanvector).Ifd (cid:4)0,thisreducestotheRayleighdistribution,
0
butingeneralthemeanvalueisgivenby
rffiffiffi
(cid:2) (cid:3)
p
d d0;s2h DE½d(cid:3)Dsh 2L1=2ðxÞ (9)
346 B.CALDERANDP.ELMORE
7
1
0
2
r
e
b
m
e
c
e
D
2
1 1 Figure2. Increaseinpropagatedfielduncertaintyasafunctionofnominaldistancebetweenobservation
1:2 and reference location, and the horizontal uncertainty scale factor, sh, for a unit horizontal uncertainty.
1 Increasingthescalefactorchangestherateofincreaseofuncertainty,butalsooffsetsthezero-distance
at uncertainty,oftenbyverysignificantamounts.
]
e
r
hi
mps wherexD¡d20/(2sh2)andLn(x)isageneralizedLaguerrefunction(AbramowiczandStegun
a 1965,Ch.22).Thus,propagatedvarianceis
H
w (cid:2) (cid:3)! !
of Ne s2 Ds2f 1Cðk¡1Þ d dD0;s2h a
rsity 0 0 rg ffipffiffi 1a1 (10)
Unive D s2fBB@1Cðk¡1ÞBB@sh 2DL1=2ðxÞCCA CCA
y [ g
b
d
e
d
a With(10),theuncertaintymodelnowhasonlytwomanuallysetparameters,kanda,easing
o
nl thespecificationofbehaviorforusers.
w
o
D
EvaluationofL (x)
1/2
ComputationofL (x)in(9)and(10)isnon-trivial(Thompson1997;Muller2001;Pearson
1/2
et al. 2014) because of the non-integer order of the Laguerre function. Since the present
problem requires frequent numerical evaluation of the Laguerre function L (x), computa-
1/2
tionalefficiencyisessentialforimplementation.
ModifiedBesselfunctionexpansion
Through fractional differentiation (Herrmann 2014) of the defining equation for the
Laguerre function (Andrews et al. 1999), it is possible to show (for example by expanding
MARINEGEODESY 347
both sides as power series, or through a computer-algebra system) that L (x) can be
1/2
expressed in terms of modified Bessel functions of the first kind (Andrews et al. 1999) of
zerothandfirstorder,viz.:
L1=2ðxÞDexpðx=2ÞðI0ðx=2ÞCxI1ðx=2Þ¡xI0ðx=2ÞÞ (11)
where for efficiency, the two nominal evaluations of the modified Bessel function of order zero
could be trivially collapsed if required (i.e., computing (1 ¡ x)I (x/2) C xI (x/2)). While this
0 1
approachhasanumberoftheoreticalbenefits,mostparticularlyiffurthermanipulationofthefor-
mulaewererequired,evaluationofmodifiedBesselfunctionsisitselfasignificantcomputational
cost,makingthisapproachlessattractivewheremanyevaluationsarerequired,ashere.
17 Useof1F1(a,b;x)forLn(x)
0 Common procedures for computation of L ðxÞ;n2= Z begin by rewriting it in terms of
2 n
er Kummer’s CHGF (Abramowicz and Stegun 1965; Arfken et al. 2012; Buchholz 1969;
b
m NationalInstituteofStandardsandTechnology2015)as
e
c
e (cid:11) (cid:12)
D mCn
2 LðmÞðxÞ: D F ð¡n;mC1;xÞ (12)
1 1 n n 1 1
2
1:
1 whereKummer’sCHGFisdefinedbytheseries
at
shire] 1F1ða;b;xÞD1C abxC abððbaCC11ÞÞx22! C abððabCC11ÞÞððbaCC22ÞÞx33! C (cid:2)(cid:2)(cid:2)
p
m X1 ðaÞ xk
Ha D k (13)
w kD0ðbÞkk!
e
N
of and (a)n Da(aC1)(a C2)(cid:2)(cid:2)(cid:2)(a Cn ¡1)is thePochhammer symbol(or rising factorial);
y (a) (cid:4)1.WithnD1/2andmD0,andhenceaD¡1/2,andbD1,(12)becomes
sit 0
ver L1=2ðxÞD1F1ð¡1=2;1;xÞ (14)
ni
U
[ Thus,useof(9)and(14)providesd,
y
d b (cid:2) (cid:3) qffiffiffiffiffiffiffi (cid:6) (cid:7) (cid:6) (cid:7)
e 3 1
oad d d0;s2h D 2s2hG 2 1F1 ¡ 2;1;x (15)
nl
w
o
D whichcanbeevaluatedbydirectsubstitutionas
(cid:6) (cid:7)
1
(cid:6) (cid:7) ¡
X1
1 2 xk
F ¡ ;1;x D k
1 1 2 ð1Þ k!
kD0 k(cid:6) (cid:7) (16)
1
X1 ¡ xk
2
D ð¡1Þk k
ðk!Þ2
kD0
348 B.CALDERANDP.ELMORE
(since(1) Dk!).Subsequenttermsof(16)alternateinsignandanalysisoftheratioofterms
k
(AbramowiczandStegun1965;NationalInstituteofStandardsandTechnology2015)shows
that the sequences converges, but the terms do not reduce in magnitude as k ! 1 and
therefore the rate of convergence depends on the argument. Practically, therefore, direct
evaluationisonlyfeasibleforsmallvaluesofx940–50,evenusingadouble-precisionfloat-
ing-pointrepresentation.
Computationof F (¡1/2,1;x)
1 1
In general, there is no optimal computational procedure for F (a, b; x) (Lopez and Ester
1 1
2014;Pearsonetal.2014).AsdiscussedindepthinPearsonetal.(2014),differingstrategies
existdependingonthevaluesofthetwoparametersandfunctionalargument.Forthepres-
ent case, aD ¡1=2;bD1;x 2R;x(cid:3) 0, and hence from (Pearson et al. 2014, Section 3),
7 forvaluesofx(cid:3)10,summationoftermsin(16)isefficient.Forlargearguments(e.g.,forx
1
20 > 500), Slater’s expansion (Abramowicz and Stegun 1965; National Institute of Standards
r
e andTechnology2015;Slater1953)
b
m
e
c
De ( )
2 F ða;b;zÞ exxa¡b SX¡1ðb¡aÞ ð1¡aÞ
1:21 1 1 1GðbÞ D GðaÞ nD0 nn! nx¡n (17)
1
at
]
e
hir is effective (National Institute of Standards and Technology 2015, Section 13.29(i)). Practi-
s
p cally, however, evaluation of either (16) or (17) would be prohibitively expensive for large
m
a datasets wheretheexpected distancehastobe computedforeachobservationatleastonce.
H
w Consequently,amoreefficientcomputationalschemeisrequired.
e
N
f
o
y Chebyshevseriesfitto F (¡1/2,1;x)
sit 1 1
er AsrecommendedinPearsonetal.(2014),anefficientapproachtocomputing F (¡1/2,1;
v 1 1
ni x) is to perform a one-time computation using (16) for x < 40 and (17) for x (cid:2) 40. Then,
U
y [ approximatetheresultusingasuitablepolynomialseries,
b
d
e
d
a
wnlo F ða;b;xÞ(cid:5)NX¡1C P ðxÞ (18)
o 1 1 n n
D nD0
where P (x) is a polynomial function of degree n and the choice of N < 1 reflects the
n
desiredlevelofapproximation.
Forthecurrentapplication,Chebyshevpolynomialsofthefirstkind,T (x),formaprag-
n
maticcomputationalsolution.T (x)providenumericaladvantagesoveralternatives,suchas
n
a power series,for rapid convergence and compact representation (i.e., thesequence canbe
computedtouserspecifiedaccuracywithfewerterms)inthedomain[¡1,1](Arfkenetal.
2012;Boyd2001;Pressetal.2007).Withinthisdomain,Chebyshevpolynomialsareorthog-
onalandhaverange[¡1,1].ForChebyshevpolynomials,thecoefficientsrequiredfor(18)
MARINEGEODESY 349
are
2XJ¡1 (cid:2) (cid:3) (cid:2) (cid:3)
C D F a;b;x T x (19)
n 1 1 j n j
J
jD0
andx isthesetofdomainpointsusedtonumericallycomputetheCHGF.
j
Withappropriatesubstitutions,then,(10)canberewrittenas,
"pffiffiffi # !
ps NX¡1 a
s2 (cid:5)s2 1Cðk¡1Þ 2 h C T ðxÞ (20)
f D n n
g nD0
7 givingacomputableformforthepropagateduncertainty.
1
0 Directevaluationof(20)ispossible,butsuffersfromsignificantlyhighererrorsasx!0
2
er (Figures3and4).Toavoidthis,theChebyshevpolynomialscanbefittedtothepre-warped
b
m domainx0(g)Dxg,0<g <1,whichstretchesouttheregionclose toxD0,andresults in
e
ec having to evaluate significantly fewer coefficients for any given level of approximation
D
2 (Figures5and6).
1
1
2
1:
1 Comparisonofimplementationmethods
at
e] A full comparison of the implementation efficiency of the two methods proposed is essen-
r
hi tiallyimpossible,sincethatof(11)dependsstronglyontheimplementationofthestandard
s
mp libraryfunctionforthemodifiedBesselfunctionofthefirstkind.Aninformalassessmentof
a
H
w
e
N
f
o
y
sit
r
e
v
ni
U
[
y
b
d
e
d
a
o
nl
w
o
D
Figure 3.Absolute and relative errors in approximating the CHGF with a Chebyshev polynomial (of 26
coefficients,designedtogiveabsoluteaccuracytowithin0.01ofthecomputedCHGFvalue).Theshape
oftheCHGFasx!0leadstobunchingoferrorripplesaroundxD0.