Stochastc Generaton of Daly Ranfall Data Srkanthan, R. CRC for Catchment Hydrology, Bureau of Meteorology, Melbourne, Australa, E-Mal: r.srkanthan@bom.gov.au Keywords: Stochastc generaton; daly ranfall; nested model ETENDED ABSTRACT As the hstorcal record provdes a sngle realsaton of the underlyng clmate, stochastcally generated data are used to assess the mpact of clmate varablty on water resources and agrcultural systems. A wdely used approach n other parts of the world to modellng daly ranfall has been a two part model n whch the frst part descrbes the ranfall occurrence (dry-wet process and the second part descrbes the dstrbuton of ranfall amounts on wet days. Even though the model preserves the daly ranfall characterstcs, the monthly and annual characterstcs are not preserved. Recently, a daly monthly mxed algorthm (Wang and Nathan, was proposed to preserve the monthly ranfall characterstcs explctly. However, the model fals to preserve the annual ranfall characterstcs. By nestng the two-part daly model n monthly and annual models, the characterstcs of ranfall at daly, monthly and annual levels can be preserved (Srkanthan 4. The transton probablty matrx (TPM method wth Boughton s adustment has been shown to preserve most of the statcal characterstcs of the ranfall data and s wdely used n Australa. However, t has been found that ths approach slghtly overestmates the mean annual ranfall and fals to preserve the monthly seral correlaton. To overcome these defcences, the TPM model s nested n monthly and annual models so that the characterstcs of ranfall at daly, monthly and annual levels are preserved (Srkanthan 5. As an alternatve to the nested TPM model, a smple adustment s proposed to the TPM model to preserve both the annual mean and standard devaton. The man obectve of the paper s to compare the performance of the above three models usng a wde varety of daly ranfall data. In earler studes (Srkanthan 4, 5, comparsons were made usng only Australan ranfall data. In ths paper, data from North Amerca and South Afrca were used n addton to Australan data. A number of statstcs at the daly, monthly and annual tme scales were used to assess the performance of the models. The results showed that all the models preserved the daly statstcs well whle the nested models also preserved the annual (Fgure and monthly correlatons and the skewness of annual ranfall data..4.3.. -. -. Nested TPM -.3 -.3 -. -....3.4.4.3.. -. -. Australa South Afrca North Amerca : Lne Nested two-part -.3 -.3 -. -....3.4.4.3.. -. -. Australa South Afrca North Amerca : Lne mtpm -.3 -.3 -. -....3.4 Australa South Afrca North Amerca : Lne Fgure. Comparson of annual lag one autocorrelaton coeffcents. 95
. INTRODUCTION Daly ranfall s a maor nput to the desgn of water resources and agrcultural systems. As hstorcal data provdes only one realsaton of the underlyng clmate, stochastcally generated data s used to assess the mpact of clmate varablty on water resources and agrcultural systems. Ranfall data generaton s a well researched area n the hydrologcal and clmatologcal lterature (Bushand 978; Chapman, 997; Sharma and Lall 999; Srkanthan and McMahon 985; Srkanthan and McMahon ; Woolhser 99. A common approach to modellng daly ranfall has been a two part model n whch the frst part descrbes the ranfall occurrence (dry-wet process and the second part descrbes the dstrbuton of ranfall amounts on wet days (Woolhser, 99. Ranfall occurrence s represented n two ways: ether as a Markov process, the assumpton beng that the ranfall state on the next day s related to the state of ranfall on a fnte number of prevous days; or as an alternatng renewal process for dry and wet sequences, the approach beng to stochastcally generate the dry and wet spell lengths. Once a day has been specfed as wet, ranfall amount s then generated usng a Gamma or mxed Exponental dstrbuton. Even though the model preserves the daly ranfall characterstcs, the monthly and annual characterstcs are not preserved. Wang and Nathan ( proposed a daly monthly mxed algorthm to preserve the monthly ranfall characterstcs explctly. In ths model, two daly ranfall sequences are generated usng daly and monthly parameters and the daly ranfall sequences generated from the daly parameters are adusted usng the other sequence generated from the monthly parameters. Ths adustment ensures that the monthly characterstcs are preserved n the generated daly ranfall sequences. However, the model fals to preserve the annual ranfall characterstcs. A nested two-part daly ranfall model was developed to preserve the daly, monthly and annual characterstcs (Srkanthan 4. The transton probablty matrx (TPM model (Srkanthan and McMahon, 985 s wdely used n Australa for stochastc generaton of daly ranfall, and t appears to preserve most of the characterstcs of daly, monthly and annual ranfall. Whle t performs better than many alternatve models, t consstently under represents the varances of the observed monthly and annual ranfall. Boughton (999 proposed an emprcal adustment to match the observed annual standard devaton (TPMb. Ths adustment mproves the varablty n the annual ranfall by scalng the ranfall amounts on wet days. However, not all the monthly and annual characterstcs are preserved by ths model (Srkanthan et al 3. In order to preserve the monthly and annual characterstcs, the TPM model s nested n a monthly annual model. The generated daly ranfall data are used to drve the monthly model and the resultng monthly ranfalls are used to drve an annual model (Srkanthan 5. The TPMb model consstently overestmates the mean ranfall (Srkanthan et al. 3. A smple adustment s proposed to correct for the overestmaton of mean ranfall and evaluated n ths paper. The modfed model s referred to as the mtpm. The man obectve of the paper s to compare the performance of the above three models usng a wde varety of daly ranfall data. In earler studes (Srkanthan 4, 5, comparsons were made usng only Australan ranfall data. In ths paper, data from North Amerca and South Amerca were used n addton to Australan data to assess the performance of nested two-part, nested TPM and mtpm models.. DAILY RAINFALL DATA Daly ranfall data from Australan, 4 North Amercan and 6 South Afrcan stes were used. The statons are unformly dstrbuted n each country and represents a wde range of clmates rangng from dry clmate wth annual number of wet days as low as 3 days to wet clmates wth clmate wth annual number of wet days as hgh as 6 days. A bref summary of the daly ranfall data s gven n Table. Table. Summary of daly ranfall data. Country Mean record Mean annual length (years Ranfall (mm Australa 4-5 8-49 North Amerca 74-5 - 79 South Afrca 85-5 4-3. NESTED TWO-PART MODEL In the two-part model, the occurrence of ranfall s determned by usng a frst order Markov chan usng the two transton probabltes: p W D, the condtonal probablty of a wet day gven that the prevous day was dry; p W W, the condtonal probablty of a wet day gven that the prevous day was wet. The uncondtonal probablty of a wet day can be derved as 96
pw D π = ( + p p W D W W For wet days, the ranfall depth s obtaned from a Gamma dstrbuton whose probablty densty functon s gven by ( / β α x exp( x / β f ( x = ( βγ( α where α s the shape parameter and β the scale parameter. The mean and varance of the Gamma dstrbuton are gven by μ(x = α β (3 σ (x = αβ (4 The seasonalty n daly ranfall s taken nto account by consderng each month separately. Once the daly ranfall s generated for a month, the monthly ranfall s obtaned by summng the daly ranfall values. The generated monthly ranfall value,, s modfed by usng the Thomas-Ferng monthly model to preserve the monthly characterstcs μ = ρ, σ ( μ( σ ( ( / + ( ρ, μ' ( σ '( (5 where ρ,- s the correlaton coeffcent between months and -. The theoretcal mean and varance of the ranfall total,, over a month of N days s gven by μ( = Nπαβ (6 + pw W pw D σ ( = Nπαβ + α( π (7 pw W + pw D The subscrpt for all the varables n Eq. (6 and (7 s omtted for clarty. The generated daly ranfall data s multpled by the rato /. Once the values for the twelve months of a year (k have been generated, the generated monthly values can be aggregated to obtan the annual value. The aggregated annual value, Z k, s modfed by usng a lag one autoregressve model to preserve the annual characterstcs. Zk μ Z Z = ρ( Z σ ( Z μ( Z + [ ρ ( Z ] σ ( Z ( k / Zk μ' ( Z σ '( Z (8 where ρ s the lag one autocorrelaton coeffcent. If the annual ranfall data exhbts sgnfcant skewness, then the nose term n Eq. (8 s modfed by usng the Wlson-Hlferty transformaton (93. The theoretcal values of the mean and varance of the aggregated annual ranfall are gven by μ ( Z = μ( (9 σ ( Z = + + σ ( + = = = 3 σ ( σ ( σ ( σ ( 3 σ ( ρ ρ,, σ ( ρ ρ,, ρ ρ,, 3 ( The generated monthly ranfall value s multpled by the rato Z k / Zk. Ths wll preserve the annual characterstcs. The modfed monthly ranfall values are used to adust the daly ranfall values. Rather than adustng the daly ranfall values twce, the adustment to the daly ranfall values can be carred out n one step by multplyng the generated ranfall values for each month ( by the rato Z / Z. k 4. NESTED TPM MODEL k The daly ranfall data are frst generated by the TPM model. The seasonalty n occurrence and magntude of daly ranfall s taken nto account by consderng each month separately. The daly ranfall s dvded nto a number of states, up to a maxmum of seven states. State s dry (no ranfall and the other states are wet. The state boundares for ranfall amounts are gven n Table. If the number of states for a month s less than seven, then the upper lmt of the last state s assumed to be nfnte. The shfted Gamma dstrbuton s used to model the ranfall amounts n the hghest state, whle a lnear dstrbuton s used for the ntermedate states. The latter s chosen because daly ranfall usually exhbts a reverse J shape dstrbuton. The parameters of the Gamma dstrbuton are estmated by usng Eq. (3 and (4. 97
Table. State boundares used for the TPM model. State number Upper state boundary lmt (mm..9 3.9 4 6.9 5 4.9 6 3.9 7 The transton probabltes are estmated from f ( k p ( k, =,, C; k =,, ( = C = f ( k where f (k s the hstorcal frequency of transton from state to wthn month k and C the number states. Once the daly ranfall s generated for a month, the monthly ranfall s obtaned by summng the daly ranfall values. As above, the generated monthly ranfall s modfed by usng Eq (5. Expressons for the mean and standard devaton of monthly ranfall obtaned from the TPM model are not avalable. Hence, these are estmated from a number of the generated monthly totals and averaged. Adusted monthly values are then summed to obtan the annual value and adusted usng Eq (8. Fnally, the generated daly ranfall was adusted wth respect to the adusted monthly and annual ranfall values as before. 5. MODIFIED TPM MODEL Boughton (999 appled an adustment to match the standard devaton of the observed annual ranfall. However, t was noted that the resultng sequences over estmated the mean. In the modfed TPM model, the generated daly ranfall values are adusted wth respect to both the mean and standard devaton of annual ranfall. An adustment factor (F s frst obtaned from stdev o F = ( stdev g The standard devaton of the generated annual ranfall s estmated from a number of replcates and averaged. The generated daly ranfall n each year s multpled by the followng rato: Rato { H + ( T G F} = (3 T where G s the generated mean annual ranfall, H the hstorcal mean annual ranfall and T the generated annual ranfall for year. 6. DISCUSSION OF RESULTS One hundred replcates, each of length equal to the hstorc record were generated usng the above four models for all the 5 statons. The number of states for the North Amercan and South Afrcan statons was frst decded based on the mean monthly ranfall. It was then adusted f there were not enough tems (> n the largest state. The number of states fnally selected for the North Amercan and South Afrcan statons s not presented due to lack of space and s avalable from the author. The number of states for the Australan statons was selected usng the gudance gven n Srkanthan and McMahon (985 and s avalable n Srkanthan (5. The performance of the models s evaluated usng a number of statstcs at the daly, monthly and annual levels. The daly, monthly and annual statstcs used are lsted n the followng sectons. Due to lack of space, only a few results are presented here. An overall assessment of the results s presented n Table 4. 6.. Daly statstcs The daly statstcs nclude: Mean, standard devaton and coeffcent of skewness daly ranfall mean daly ranfall for dfferent types of wet days; soltary (class, bounded only on one sde by a wet day (class, bounded on both sdes by wet days class 3 correlaton between ranfall depth and duraton of wet spells mean number of wet days maxmum daly ranfall mean, standard devaton and coeffcent of skewness of dry spell length mean, standard devaton and coeffcent of skewness of wet spell length All the models preserved the mean and standard devaton of daly ranfall. None of the models preserved the skewness when t was larger than about 6. The mean daly ranfall for dfferent types of wet days was preserved reasonably well except for class 3 when t was larger than mm. All the models preserved the correlaton between the ranfall depth and duraton. The mean and standard devaton of the dry and wet spells were preserved by all the models. However, the skewness was not preserved by any of the models for wet spell whle 98
for dry spell none of the models could preserve the large skewness values (> 6. Shorter maxmum dry (< and wet (< 5 spell lengths were preserved but the longer ones were not..6.4 Nested TPM 6.. Monthly statstcs The monthly statstcs nclude: mean, standard devaton, coeffcent of skewness and seral correlaton of monthly ranfall maxmum and mnmum monthly ranfall mean number of months of no ranfall. -. -.4 -.4 -...4.6 Australa South Afrca North Amerca Nested two-part (mm (mm Nested TPM 5 5 4 6 8 4 6 8 (mm Australa South Afrca North Amerca Nested two-part 5 5 4 6 8 4 6 8 (mm.6.4. -. -.4 -.4 -...4.6 Australa South Afrca North Amerca TPMm.6.4. -. -.4 -.4 -...4.6 Australa South Afrca North Amerca (mm 5 5 Australa South Afrca North Amerca mtpm 4 6 8 4 6 8 (mm Australa South Afrca North Amerca Fgure. Comparson of standard devaton of monthly ranfall. Fgure 3. Comparson of monthly correlatons. All the models preserved the monthly means well. The two nested models preserved the standard devaton better than the mtpm due to nestng (Fgure. Smaller skewness values are preserved but not the larger ones by all the models. The nested models preserved the correlaton whle the mtpm dd not and resulted n almost zero correlaton (Fgure 3. All the models preserved the number of months of no ranfall. 6.3. Annual statstcs The annual statstcs nclude: mean annual ranfall standard devaton of annual ranfall coeffcent of skewness of annual ranfall 99
lag one auto correlaton maxmum annual ranfall -, 5- and -year low ranfall sums mean annual number of wet days All the models preserved all the annual statstcs except the skewness and lag one autocorrelaton. The two nested models preserved the skewness and lag one autocorrelaton (Fgure whle the mtpm model faled to preserve them. However, mtpm model elmnated the slght overestmaton of the mean whch was a problem wth the TPM model wth the Boughton s correcton (TPMb. 7. CONCLUSIONS The TPM and two-part models were nested n monthly and annual models to preserve the monthly and annual characterstcs. The orgnal TPM model s also modfed to match the annual mean and standard devaton. The two nested and the mtpm models were used to generate daly ranfall data for a number of stes n Australa, South Afrca and North Amerca. The results showed that all the models preserved the daly statstcs except the skewness of the wet spell. The nested models also preserved all the monthly and annual statstcs whle the mtpm faled to preserve the monthly and annual correlatons and annual skewness. However, mtpm s an mprovement over TPMb n terms of preservng the annual mean. In general, the nested TPM model performed margnally better than the nested two-part model. However, the nested TPM model needs longer data (generally greater than 3 years to estmate the transton probabltes properly. For long hstorcal data, the nested TPM and for short hstorcal data, the nested two-part models are recommended for the stochastc generaton of daly ranfall data. 8. ACKNOWLEDGMENTS The South Afrcan and North Amercan data were provded by Tom Chapman, Unversty of New South Wales. The South Afrcan data were made avalable by Dr M Deny from the records held at the Computng Centre for Water Research, Unversty of Natal. Most of the North Amercan records were provded by Dr Ben Hardng of Hydrosphere Resource Consultants, Boulder, CO. The records for Corvalls and Vancouver, Oregan were gven by Jnfan Duan of Oregan State Unversty, from records mantaned by the US Forest Servce. The data for Montreal were provded by Dr Fred Fabry. 9. REFERENCES Boughton, W. C. (999 A daly ranfall generatng model for water yeld and flood studes. Report 99/9, CRC for Catchment Hydrology, Monash Unversty, Melbourne, pp. Bushand, T.A. (978, Some remarks on the use of daly ranfall models, Journal of Hydrology, 36, pp. 95-38 Chapman, T.G. (997, Stochastc models for daly ranfall n the Western Pacfc, Mathematcs and Computers n Smulaton, 43, pp. 35-358. Sharma, A. and U. Lall (999, A nonparametrc approach to daly ranfall smulaton, Mathematcs and Computers n Smulaton, 48, pp. 367-37. Srkanthan, R. (4, Stochastc generaton of daly ranfall data usng a nested model. 57 th Canadan Water Resources Assocaton Annual Congress, 6-8 June, 4, Montreal, Canada. Srkanthan, R. (5, Stochastc generaton of daly ranfall usng a nested transton probablty model. 9 th Hydrology and Water Resources Symposum, Engneers Australa, 3 February, 5, Canberra Srkanthan, R. and T.A. McMahon. (985, Stochastc generaton of ranfall and evaporaton data. AWRC Techncal Paper No. 84, 3pp. Srkanthan, R. and T.A. McMahon (, Stochastc generaton of annual, monthly and daly clmate data: A revew, Hydrology and Earth System Scences, 5(4, pp. 653-67. Srkanthan R, T.I. Harrold, A. Sharma and T.A.McMahon (3, Comparson of two approaches for generaton of daly ranfall data, MODSIM3, Townsvlle, 4 7 July 3, pp. 6. Wang, Q.J. and R.J. Nathan ( A daly and monthly mxed algorthm for stochastc generaton of ranfall tme seres. Hydrology and Water Resources Symposum, Melbourne, -3 May. Wlson, E.B. & M.M. Hlferty (93, Dstrbuton of Ch-square. Proc. Natonal Academy of Scence, 7, pp. 684-688. Woolhser D.A. (99, Modelng daly precptaton -progress and problems. In: A.T. Walden and P. Guttorp (Edtors, Statstcs n the envronmental and earth scences. Edward Arnold, London, U.K., 36 pp. 9
Table 4. Evaluaton of the three daly ranfall data generaton models. Nested TPM Nested two-part mtpm Daly Aus SA NA Aus SA NA Aus SA NA Mean number of wet days Maxmum daly ranfall - - - Mean daly ranfall Standard devaton Skewness - - - - - Mean on class wet day - - - Mean on class wet day Mean on class 3 wet day - - - Correl b/w depth & duraton - - - Mean dry spell length Standard devaton - - Skew ness - - - Mean wet spell length Standard devaton Skewness Max dry spell length - - - - - Max wet spell length Monthly Mean monthly ranfall Standard devaton Skewness - - - Correlaton Maxmum Mnmum No of no ranfall months Annual Mean annual ranfall Standard devaton Skewness Correlaton Maxmum Mnmum Adusted range -year low ranfall sum 5-year low ranfall sum -year low ranfall sum Average annual # of wet days - - A few ponts may be away from the 45 degree lne. 9