creation of conditional dependency matrices based on a stochastic streamflow synthesis techique

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  • 8/12/2019 Creation of Conditional Dependency Matrices Based on a Stochastic Streamflow Synthesis Techique

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    VOL. 9, NO. 2 WATER RESOURCES RESEARCH APRIL 1973

    Creation f Conditionalependencyatrices ased naStochastictreamflowynthesisechnique

    D. M. CLAINOS 1Department o] SystemsEngineering, University o] Arizona, Tucson, Arizona 85721

    T. G. Ro,rsDepartment o] Hydrology and Water Resources,University o] ArizonaTucson,Arizona 85721

    L. DUCKSTEINDepartment o/Systems Engineering, University of Arizona, Tucson,Arizona 85721

    The construction of transition probability matrices by the streamflow synthesis techniquedeveloped by the Hydrologic Engineering Center of the U.S. Army Corps of Engineerswas tried. In certain cases it was found to be necessary to use a transform different fromthe Wilson-Hilferty transform used in the stated routine in order to produce reasonableresults. Other details of the numerical considerations necessary for the definition of discreteconditional probability matrices between monthly flow volumes are described.The work described herein was undertaken

    as a necessarysubtask of a somewhat argerproject. The objective of the overall projectwas to estimate the effect of alternative opera-tion rule determinations n planningdecisions.Specifically, lanningdecisions ould be exam-ined by using simulation outineswith both anoptimal operation rule determinationand anarbitrary (or standard) operation rule deter-mination.

    There are severaloptimization echniqueshatone coulduse to determinean optimal operatingpolicy of a reservoirsystem.Hall and Buras[1961], Hall and Howell [1963], Hall [1967],Hall et al. [1968], Hall and Roefs [1966], andMeier and Beightler [1967] have extensivelystudied complex eservoirsystemsby usingde-terministic streamflows.Young [1966] extendedthe techniqueof usingdeterministic treamflowsby generating several synthetic sequences fstreamflows (each considered to be determin-istic) by means of a Monte Carlo technique.Another technique was offered by Loucks

    Now with the Hewlett Packard Corporation,San Diego, California.Copyright 1973 by the American GeophysicalUnion.

    [1969], who employeda stochasticinear pro-graming model. Obviously,only a few of themany techniqueshave been cited above. Itappears hat a stochastic ynamic programingalgorithm is the most effective way to deter-mine the optimal operating rule for one reser-voir [Hassift, 1968; Roefs, 1968]. Butcher[1971] has shown that some sort of discreteconditionaldependencymatrix (frequently re-ferred to as a transition probability matrix) isrequired as input to a stochasticdynamic pro-graming model.

    Sincea streamflowsynthesis echnique s usedto compare he supposed ptimal and arbitraryoperatingpolicies, t follows hat the assump-tions used in constructing he necessary ondi-tional dependencymatrix must be as congruentto the streamflow synthesis echnique as pos-sible. This paper shows he problems nvolvedin creating such a conditional dependencymatrix and suggests ome techniques o im-prove the quality of the results.SELECT A MODEL

    First a particular stochastic treamflowwn-thesis echniquemustbe selected. he techniqueselected or this study was the model developedby the U.S. Army Corpsof Engineers 1971].481

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    482 CLAINOS ET AL.: BRIEF REPORTThis decision was based not on the fact that itis the best method but on the fact that it is atleast a satisfactorymethod as well as probablythe most used technique. It should be noted,however, that there appear to be some seriousproblems in using logarithmic transforms toobtain 'smoothed' data [Weber and Hawkins,1971]. However, ntil implementableethodsof working in the nontransformeddomain arefound, we have to resort to such methodsas areused here to solve practical reservoir operationplanning problems. If the interstation depen-dencies re disregarded, he steps n the stream-flow synthesismodel of the U.S. Army Corps ofEngineers are:

    1. Add a small increment (1% of the aver-age flow for each month) to each streamflow opreclude negative ogarithms.2. Take the common ogarithm of the result.3. For each monthly logarithmicvector com-pute the mean, standard deviation, and skewfactor; the skew factor is given byNg = N "(X,i=1 -- ')'/(N- 1)(N- 2)S

    where Xi is the result of step 2, S is the standarddeviation, is the computedamplemean,andN is the number of observations.4. The standardized gamma deviate y isdetermined byy = (X,-

    5. The result of step 4 is transformed to anormal deviate K by usingthe transform

    K=L\W+ I -- I -+-6. The correlation coefficient is computedbetween the normal deviates for adjacentmonths.

    7. A uniform (0, 1) random number is gen-erated.

    8. The uniform random number is trans-formed to a standard normal deviate.

    9. The relationshipKt+l = rK, q- (1 --r2)l/2z

    is used to generate he stochasticprocess;and K, are correlated normal deviates at timet + 1 and t, respectively; K is given; K,,where t < 1, is undefined; and Z is the randomnormaI deviate.

    10. The inverseof the transform of step 5y= [(K--)-+-1]-- }is used to produce a standardizedgamma devi-ate.

    11. The standardizedgamma deviate y ismultiplied y S andaddedo , (bothS and werecomputedn step3): X = yS q- '.12. The antilogarithm s taken.13. The small increment is subtracted.Steps 1-6 compose he analysis part of theroutine. Steps 7-13 are the generation part ofthe routineand might,when he routine s usedin a simulationmode, be repeatedseveral hou-sand times. What concerns us at this point,however, is not the use of this routine in asimulation mode but the establishment of a

    conditionaldependencymatrix that is as nearlycongruentspossible.When one actually attempts to create theconditional dependencymatrices, one becomesaware of three important problems: the im-mense amount of computer time required tocompute the discrete conditionalprobabilities,the choice of the size of the flow interval (theeffectsof diseretizationon a continuousproba-bility density function), and highly skewedstreamflows.Each of these problems will bediscussed eparately.

    COMPUTER TIMETo compute he conditionalprobabilities, nemust consider the bivariate normal distribution

    and perform double numerical ntegration sev-eral thousand times. For example, f a 50 X 50dependencymatrix were desired, 2500 condi-tional probabilitieswould have to be computedfor each pair of months. Since there are 12pairs of months, a total of 30,000 conditionalprobabilitieswould have to be computed. f ittook 5 sec for each numerical double integra-tion, one can readily see hat approximately42hours of computer time would be required.Since for most researchbudgets this computercost would be excessive, alternate methodsshould be investigated. Of the three alternatemethods studied by Roefs and Clainos [1971]the midpoint method was found to be theoptimumradeoff betweenrrorandcomputa-tion time. As the name implies, this methodlooks at the flow interval in question, repre-

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    CLAINOS ET AL.'. BRIEF REPORT 483sents one dimension of the interval (the ante-cedent flow) as a point in the middle of thatinterval, and numerically integrates over theother dimensionof the interval (the subsequentflow). The feasibility of this method, of course,directly depends on the size of the particularinterval, which should be sufiqcienfiy mall.

    INTERVAL SIZEThe secondproblem encountered ollows di-rectly from the above discussionabout the sizeof the flow interval. The solution or this prob-lem lies in the application of numericalanalysisprinciples and experimentation.The size of theflow interval is determined by the degree ofresolution desired in the stochastic dynamicprogram. Although it was noted that this prob-lem does ntroduce some error, we do not have

    enough data at this time to determine theactual magnitude of this form of error.HIGHLY SKEWED STREAMFLOWS

    Another problem was encountered whenhighly skewed streamflowswere studied. Sincethe streamflow synthesis technique that wasused required that the gamma deviatesbe trans-formed into normal deviates, a transform toaccomplish his was required. Since the modelof the U.S. Army Corps o Egieers [1971]used the Wilson-Hilferty transform in thestreamflow synthesis technique, we used thistransform. In the study by McGinnis andSammons [1970] the Wilson-Hilferty trans-form was examined. Although at best thistransform is merely a good approximation,results obtained were quite reasonable for lowskew coefiqcients. owever, as the level of skew-ness increased, the amount of error introducedduring the transformation also increased.Whenthe level of skewnessg exceeded2.0, so mucherror was present in the upper and lower partsof the tails of the distribution that the resultswere no longer even. good approximations. Ofthe data that were used n this study, I monthof flows was found to be highly skewed (g2.0). Because of the high level of skewness,computationalproblem was encounteredwhenthe conditionalprobabilitieswere calculated orthe pair of sequentialmonths that included hemonth with highly skewedstreamflows.The problem was assessed, nd two alterna-tives were considered. The first alternative was

    to establish an upper limit of skewness hatthe model would accept. With this method, ifthe skewnesswas actually higher than the estab-lished level, the model would truncate the actuallevel of skewness o the preestablished evel.The advantage to this alternative was that atleast the model would produce seeminglyrea-sonable esults.The disadvantage,however,wasthat one could question the validity of such amethod on the grounds that the model wouldnot be capable of handling streamflows hat arehighly skewed. n other words, f a stream withhighly skewed streamflows were studied, themodel would reduce each highly skewed flow,and the result might be stream characteristicsthat were in fact not characteristic of thatstream at all.

    Another alternative was to construct a newtransform capable of transforming highlyskewed gamma deviates to normal deviateswith more accuracy. Since the alternative thatwas studied was related to the X2 distribution,an algorithm relating the X2 distribution to thePearson type 3 distribution should be con-structed. Hatter [1969] has studied this pro-cedureand has constructeda table of percentagepoints of the Pearson ype 3 distribution.Onemethod that Harter has used to compute per-centagepoints of the X distribution,which inturn can be modified to give the percentagepoints of the Pearson ype 3 distribution,canbe described as follows:

    1. Compute he probability ntegralP(X; r)-- I(u, p) of X with r degrees f freedom:

    I(u,= o v e dv/r(p + 1)where u = X2/(2v)/',p = (v/2) -- 1, andI(u, p) is the incompleteF functionratio.2. Obtain the percentage oints of X2 byperforming nverse nterpolation n the tableof the probability ntegral in order to obiainthe percentage ointsof u and finally multiplyby (2v)3. Relate the. X distributionwith v degreesof freedom to the Pearson type 3 distributionwith mean / = v standard deviation r =(2v)/, and skewness/ = (8/v)/*.The result of Harter's study is a table ofpercentage oints of the Pearson ype 3 dis-tribution for a spedfled level of skewness.

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    48'4 CL.I:NOST .L.' BIIEFRPOlTTABLE 1. Conditional Dependency

    September October Flows,Flows,103 acre-feet 0-50 50-150 150-250 250-350 350-450 450-550 550-650 650-750 750-850 850-950 950-1050

    0-5050-150

    150-250250-350350-450450-550550-650650-750

    0.0 0.01765 0.96470 0 00883 0.00882 0.0 0.0 0.0 0.0 0.0 0.00.0 0.01765 0.96470 0 00882 0.00882 0.0 0.0 0.0 0.0 0.0 0.00.0 0.0 0.67094 0 27067 0.04562 0.00882 0.00256 0 00080 0.00023 0.00016 0.000110.0 0.0 0.13746 0 37639 0.24410 0.11146 0.05850 0.02923 0.01146 0.01037 0.009220.0 0.0 0.04120 0 22960 0.25405 0.16191 0.10851 0.06571 0.02959 0.02938 0.029360.0 0.0 0.01021 0 12286 0.19619 0.16363 0.13180 0.09225 0.05551 0.05442 0.054100.0 0.0 0 00483 0 07657 0.15217 0.14738 0.13228 0.10083 0.06306 0.06978 0.069770.0 0 0 0.00210 0.03805 0.09887 0.11515 0.11825 0.10024 0.08630 0.08549 0.08466

    From this point we constructed he following to be followednext by a probability of 0. Inalgorithm' other words there was no visible trend followed1. The gamma deviatesare computed rom by the probabilities, s one might expect,butthe streamflows. merely an erratic sequenceof numbers. These2. Given the level of skewness,he gamm errors were so pronouncedhat the dependencydeviate is related to the appropriate percentage matrix that was generated was considered opoint by using inear interpolation or Langran- possess ore noise han meaningful nformation.gian interpolation if added accuracy is re- However, when the new transform was used, hequired. conditional probabilities (Table 1) were rea-3. The percentage point of the Pearson sonable inasmuch as the columns summed totype 3 distribution is then related to the per- unity and there were no pronounced ncon-centage point of standard normal distribution. sistencies.4. Iterative numerical integration is per-formedntil onvergencewithinpecifiedrror CONCLUSIONSbounds) is achieved. A number of conclusionscan be made in5. The result is a transformednormal devi- creating conditionaldependencymatricesbasedate. on a stochasticstreamflow synthesis echnique.Although the method describedabove is ex- First, not all assumptionsof current stream-pensive (computer time will vary with the flow synthesis echniquesproduce results thataccuracy equired), it seems o produce eason- are hydrologically easonable.Becauseof theable values. An error bound on this method high value of the skew coefficient he transfor-has not been determined at the time of this marion from the gamma space to the normalwriting, but the method looks promising,and space and back to the gamma space s not atthe results thus far are well worth the com- all accurate. Hence, the result is that the con-puter expense.An exampleof the value of this ditiona probabilities that are generated arenew transform is the conditional dependency clearly not reasonable.matrix for September-Octoberof the study. Second,double numerical integration is notThe flows of October are skewed in excessof computationally feasible because of the large2.0. When the Wilson-Hilferty transform was amount of computer ime required.An approxi-used, the computed conditional probabilities marion, such as the midpoint method, is. re-possessedome inconsistencies.he probabi i- quired o solve he problemof doublenumericalties that were generatedappeared o be hydro- integration n a practicalmanner.logically unreasonable, ecause high flow in Third, it may be possible o use transformsSeptembervirtually dictated that a flow two or efficientlywhen streamflows re highly skewed.three times higher would occur in October and Although this new transform was used success-the probability of lower flows was essentially fully in this study, the real value of such anonexistent. n addition, the sequenceof the transform may be realized in studying water-conditional probabilities causeda disturbance, sheds n an arid region where the coefficient fsince a small conditionalprobability would be skew of the original data may be as large asfollowed by a slightly higher probability, only

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    CLAINOSET AL.' BRIEF REPORT 485Matrix for September-October103 acre-feet

    1050-1150 1150-1250 1250-1350 1350-1450 1450-1550 1550-1650 1650-1750 1750-1850 1850-1950 1950-20500.0 0.0 0.0 0.0 0.0 0.00.0 0.0 0.0 0.0 0.0 0.00.00004 0.00001 O. 00001 0.0 O. 0 0.00.00518 0.00211 0.00149 0.00064 0 00053 0.000480.01911 0.00869 0.00666 0.00306 0 00263 0.002610.03907 0.01930 0.01572 0.00758 0.00676 0.005990.05333 0.02767 0.02338 0.01160 0.01055 0.009540.06998 0.03859 0.03412 0.01296 0.01636 0.01518

    oo.o oooo 00047o 00226o 00722o O1178o .01931

    0.0 0.0 0.00.0 0.0 0.00.0 0.0 0.00.00035 0.00035 0.000200 00221 0.00207 0.001400.00600 0.00578 0.004600 01008 0.01176 0.008300.1711 0.02085 0.01845

    Acknowledgment. The work described hereinwas conducted under the auspices of the U.S.Office of Water Resources Research, allotmentgrant A-024 Arizona.

    REFERENCESButcher, W. S., Stochasticdynamic programmingfor optimum reservoir operation, Water Resour.Bull., 7 (1), 115-123, 1971.Hall, W. A., Optimum operations or planningof a complex water resourcessystem, Contrib.122, pp. 1-63, Water Resour.Center, Univ. ofCalif., Los Angeles,1967.Hall, W. A., and N. Buras, The dynamic pro-gramming approach o water resources evelop-ment, J. Geophys. Res., 66(2), 517-520, 1961.Hall, W. A., and D. T. Howell, The optimizationof single-purposeeservoir designwith the ap-plication of dynamic programming o synthetichydrology samples, J. Hydrol., 1(2), 355-363,1963.Hall, W. A., and T. G. Roefs, Hydropower projectoutput optimization,J. Power Div. Amer. $oc.Civil Eng., 92(PO1), 67-69, 1966.Hall, W. A., W. S. Butcher, and A. Esogbue,Optimization of the operation of a multiple-purpose reservoir by dynamic programing,Water Resour. Res., 4(3), 471-477, 1968.HarteL H. L., A new table of percentagepointsof the Pearson type 3 distributions, Techno-metrics, 11 (1), 177-187, 1969.Hassitt, A., Solution of the stochasticprogram-ming model of reservoir regulation, Rep. 320-3506, 18 pp., IBM Wash. Sci. Center, Wheaton,Md., May 1968.

    Loucks, D. P., Stochasticmethods for analyzingriver basin systems, O WRR Proj. C-1034, pp.IV-l, IV-21, research project technical com-pletion report, Dep. of Water Resour. Eng.,Cornell Univ., Ithaca, N.Y., Aug. 1969.McGinnis, D. F., Jr., and W. H. Sammons,Dis-cussion of 'Daily streamflow simulation' byK. Payne, W. R. Neuman, and K. D. Kerri,J. Hydraul. Div. A mer. $oc. Civil Eng.,96(HY5), 1201-1206,1970.Meier, W. L., Jr., and C. S. Beightler, An op-timization method for branching multi-stagewater resource systems, Water Resour. Res.,3 (3), 645-652, 1967.Roefs, T. G., Reservoir management: The stateof the art, Rep. 320-3508,85 pp., IBM Wash.Sci. Center, Wheaton, Md., July 1968.Roefs, T. G., and D. M. Clainos, Conditionalstreamflow probability distributions, in Hy-drology and Water Resourcesn Arizona andthe Southwest, vol. 1, pp. 153-170, AmericanWater Resources Association, Tucson, 1971.U.S. Army Corps of Engineers,Monthly stream-flow simulation, HEC-4, pp. 2-5, Hydrol. Eng.Center, Davis, Calif., Feb. 1971.Weber, J. E., and C. A. Hawkins, The estimationof constant elasticities, S. Econ. J., 38(2),185-192, 1971.Young,G. K., Jr., Techniquesor finding eservoiroperation rules, Ph.D. thesis, Harvard WaterProgram, Harvard Univ., Cambridge, Mass.,1966.

    (Received June 16, 1972.)