Multiaffine random field model of rainfall - Wiley Online Library

WATER RESOURCES RESEARCH,

Multiarline Merab

random

field model

VOL. 35, NO. 2, PAGES 509-514, FEBRUARY

1999

of rainfall

Menabde

Department of Physics,Universityof Auckland,Auckland, New Zealand

Alan

Seed

CooperativeResearchCentre for CatchmentHydrology,Bureau of Meteorology,Melbourne, Victoria, Australia

Daniel

Harris

and Geoff

Austin

Department of Physics,Universityof Auckland,Auckland, New Zealand

Abstract. A new method for the statisticaldescriptionand simulationof two-dimensional rainfall radar imagesis proposed.It is basedon the theory of multiaffinerandom fields and the boundedlognormalcascademodel. The model has three free parameters,which are shownto providea parsimoniousand robuststatisticaldescriptionof rainfall images. The parametersretrieved from the analysisof the real rainfall data by fitting the one- and two-pointstatisticsare usedfor simulation.The simulatedrainfall fields are in good statisticaland visual agreementwith their real counterparts. 1.

Introduction

Various multiscalingmodelshavebecomeincreasinglypopular for rainfall statisticaldescriptionand simulationin the last decade. For a review see Foufoula-Georgiouand Krajewski [1995],Lovejoyand Schertzer[1995], and referencestherein. These models are based on the empirical evidenceabout scalingpropertiesof rainfall and the analogywith the random multiplicativecascademodels in fully developedturbulence. Cascademodels are purely phenomenological, though there have been some attemptsto justify them by a quite remote analogywith the energycascadeprocessin the fully developed turbulence,the latter havinga semiphenomenological character itself.For rainfall the existenceof a cascade-type processin spaceandtime anditsphysicaloriginare evenlessclear.In our view, at the presentstage,the bestpossiblejustificationfor the use of multiscalingmodelsof rainfall shouldbe basedon the followingproperties:(1) the abilityto providea robuststatistical descriptionof rainfall, incorporatingits scalingproperties with few parameters,and (2) the abilityto simulatea synthetic rainfall field, which reproducesthe observedstatisticalproperties on the basisof the derivedparameters. This paper introducesa new multiscalingmodel of spatial structureof rainfall, asmeasuredby a weather radar, basedon the boundedrandommultiplicativecascade.At presentthere are few various definitions of a multiscalingrandom field, whichmay lead to qualitativelydifferenttypesof behavior.We will define here two different types of multiscalingmodels, multifractal and multiaffine models,which are quite general and are more or less acceptedin the literature. Considera

whereanglebracketsdenotethe ensembleaverageandK(q) is a nonlinear, upward concavefunction of q. The field is called multiaffine [Benziet al., 1993] if its so-calledgeneralizedstruc-

turefunctionGq(l) satisfies thecondition

(Ig(x + 1)- g(x)lq) o•l;(q), (2) whereI = Ill and •(q) is a nonlinear,downwardconcave functionof q. Randomfieldswith scalingpropertiesas in (1) and (2) both have scalingpower spectra,but with different possiblevaluesof the scalingexponent.It is well known [e.g., Menabdeet al., 1997a]that in the two-dimensional (2-D) case a randomfield satisfying(1) hasa powerspectrumof the form

(3) wherek = Ikl, andthe powerspectrumexponent/3 = 2 -

K(2). However,the radar-measuredrainfall fieldsusuallyexhibit/3 > 2. In order to make the multifractalapproachapplicable,it is necessary to transformthe field by powerlaw filtering or a gradienttransformation[e.g.,Menabdee! al., 1997a]. Both these proceduresare not necessarilyunambiguousand robust.It is preferablefrom a theoreticalpoint of view to have a method to describeand simulatethe rainfall field directly. This promptsus to use (2) as a main statisticaldescriptorof rainfall. However,a major problemariseswhen applying(2) directly,as the randomfield satisfying(2) for arbitraryscalel cannotbe stationaryand nonnegative[Venezianoe! al., 1996]. An alternativeis to considerrandom fields satisfying(2) asymptotically,i.e., for I > 1/1o, with/3 = 2 + •(2) (seeappendix).Sucha type is calledmultifractalif its coarse-grained values(i.e., the spa- of power spectrumis typical for rainfall radar images.The tial averagesover a cube of side I or a sphereof radiusl) followingsectionshowsthat it is possibleto constructan assatisfythe condition ymptoticallyscalingmultiaffinerandomfield usinga bounded multiplicativecascademodel.

(R•) crl-K(q),

(1)

Copyright1999by the American GeophysicalUnion.

2.

Paper number 1998WR900020.

Consider a homogeneousdistributionof a 2-D field, Ro, over a squarewith sideL. On the first stepof the cascadewe

0043-1397/99/WR900020 $09.00 509

Bounded Lognormal Cascade Model

510

MENABDE

ET AL.:

MULTIAFFINE

divide the initial squareinto four equal squaresand assignto

RANDOM

FIELD

RAIN

MODEL

highestresolutionis distributedas a lognormalrandom vari-

00givenby eachof thema valueRi = Row(j), wherew(j) aredifferent ableexp(-0 ø2q- 00X)withthescaleparameter realizationsof a randomvariable W, producedby somegeneratorwiththe probabilitydensity#(w) andwith (W) = 1. On the nextstep,eachsquareis itselfdividedinto four squaresand the procedureis repeated N times, leading to the discrete random field,

RN(j•, j2, '''

, jN) = Row(jOw(j•, j2) ''' w(j•, j2, '''

, j•), (4)

N

002__ Z 00/2 __0002 1--4-" ' i=1

(11)

The propertiesof the two-point statisticsexpressedby the generalizedstructurefunction can be derived analyticallyunder somerestrictionsspecifiedbelow.Consideringtwo points of the field separatedby a distanceI n = L 2 -n, and assuming that the main

where R•v representsthe piecewiseconstantfunctionon the

1 -- 4 -(N+•)n

contribution

to the structure

function

comes

from the field pointshaving(n - 1) commonparentsin the

square of sidel•v= 2-•VL andthesetof indexesj•, J2, '" , J•v cascade,we get indicate4•vdifferentpossible realizations of therandomfield. In contrastto the commonlyusedself-similarcascademodels, Gq(ln)= (IR(x0 - g(x2)lq) we will use the generator#(w), which dependsexplicitlyon : (W•''' Wg_1Wn''' WN - W;... W•vq). (12) the stepnumberin the constructionof the cascadeand hasthe scaleparameter decreasingwith increasingspaceresolution. Usingproperties(9) and (10) of the generator#(w), expresThis type of cascadeis calledboundedrandomcascade[Ca- sion(12) canbe writtenin the form halan et al., 1990;Marshaket al., 1994;Menabdeet al., 1997b]. Gq(ln)= (W•;n-x)(IWn;NW;;NIq), (13) Different typesof generatorscanbe usedto build the cascade, and there are no fundamentaltheoreticalreasonsfor prefer- where ring one of them. From the practicalpoint of view,we need a generatorthat, on the one hand has enoughadjustablefree Wl;n_ 1• exp(- 001;n2 1 q- 001;n _•X), (14) parametersand, on the other hand, is simple enoughto be Wn.• 2 q- 00n;NX) , : exp(-- 00n;N ß (15) treated analytically.A goodcandidatefor thisrole is a lognormal generator,for reasonsthat will be clearlyseenbelow. On The scaleparametersin (14) and (15) are givenby the n th stepof the cascadewe choosea generatorof the form 1 - 4 -nil

Wn= Cn exp (00•Y),

2 : 0002 001;n-1 1 -- 4 -H,

(5)

where Cn is the normalization constant,defined below, 00n decreases with n, andX = Norm(O,1) is a standardnormal random

variable

with zero mean and unit variance.

1 -- 4 -(N-n+•)H

2 __00024-n•1 -- 4 -n 00n;N

We choose

the scaleparameter00nin (5) tO be exponentially decreasing,

00n--000 2-nil,

(6)

and therefore our model has two parameters:000and H. The normalizationof the field to the requiredmean intensityprovidesthe third free parameterfor the model.The momentsof the lognormaldistributionare givenby [e.g.,Devroye,1996]:

(exp (tX)) = exp(t2).

(16) '

(17)

The firstfactorin (13) canbe easilycalculatedby usingthe property(7) of the lognormaldistribution

(W•;n-1) = exp(00•;n-•(q 2 2__q))'

(18)

The secondfactorin (13) canbe simplifiedfor 1 > ko, then for distancesl