Normal, Log-normal, Pearson type III, Log-Pearson type III and Gumbel distributions are applied to one-year maximal inundation informations series of 19 rivers across Malaysia. The statistical parametric quantities of adjustments are estimated by method of minute and L-moment. The inundations informations are obtained from Department of Irrigation and Drainage Malaysia ( DID ) .The chief end of this survey is to happen the most suited chance distribution and method of parametric quantity adjustment. Comparison of chance distribution and method of parametric quantity suiting with existent informations will reason which chance distribution and method of parametric quantity adjustments suited for Malaysia. Probability distribution and method of parametric quantity adjustment are developed utilizing excel dispersed sheets to calculates magnitudes of each river. Magnitude or discharge of river for return period of 2, 5, 10, 20 and 50 old ages are determine by utilizing additive insertion method.
Flood appraisal may be carried out by legion agencies. For illustration, ( 1 ) rainfall theoretical account ; ( 2 ) unit hydrograph and losingss theoretical account ; ( 3 ) inundation frequence analysis ; ( 4 ) rational method and so forth. Flood frequence analysis is favorable among other method stated above due to the fact that this method takes existent hydrology informations into history while carried out flood appraisal.
Flood frequence analysis is computation of statistical chance to find the return period for a given magnitude from a given river. One of the primary aims of frequence analysis is to find the return period of a hydrology event of a given magnitude ( Chow et al. , 1988 ) . Return period is the mean clip between happenings of a defined event. Weibull plotting place is employed to find the return period of 2, 5, 10, 20 and 50 old ages for all the chance distributions. Annual maximal inundations informations are employed in this survey and an approximative 600 old ages flood informations are utilize in obtaining the return period and magnitude.
Flood frequence analysis is an attack to find critical design discharge for hydraulic construction that capable of fulfilling both economical and possibly political issues. For illustration, Water Resources Engineering by Larry W.Mays province that “ The consequences of flow inundation frequence analysis can be used for many technology application: ( 1 ) for the design of dikes, Bridgess, culverts, H2O supply systems and inundation control structures ; ( 2 ) find economic value of inundation control undertakings ; ( 3 ) to find the consequence of invasions in inundation field ; ( 4 ) to find reservoir phase for existent estate acquisition and reservoir usage intent ; ( 5 ) to choose of overflow magnitudes for interior drainage, pumping works and local protection undertaking design ; and ( 6 ) for inundation planning zone ( Mays 2001:320-321 ) ” .
There are assorted types of chance distributions employed in gauging flood frequence such as normal, log-normal ( LN2 ) , Pearson type III ( P3 ) , log-Pearson type III ( LP3 ) , Gumbel ( EV1 ) , general extreme value ( GEV ) , Wakeby, kappa, Weibull and log-logistic distribution etc. The statistical parametric quantities suiting for these distributions may possibly gauge by methods of minute ( MOM ) , L-moment ( LM ) , maximum-likely-hood ( ML ) , probability-weighted minute ( PWM ) , entropy, Bobee, mixed-moment and so forth. In this survey, six distributions: normal ; LN2 ; P3 ; LP3 ; EV1 and two methods of statistical parametric quantities suiting: Ma and method of LM will be used.
“ When the measured informations are really positively skewed, the informations are normally log-transformed and the distribution is called log-Pearson type III distributions. These distributions are widely used in hydrology, chiefly because it has been ( officially ) recommended for application to deluge flow by the U.S. Interagency Advisory Committee on Water Data ( 1982 ) . “ ( CHIN 2000:267 )
The chief end for this survey is to find the most suited chance distribution and method of parametric quantity adjustment to be employed in Malaysia. This survey utilizes six distributions and two method of statistical parametric quantity adjustments mentioned above. Taking into consideration an indistinguishable chance distribution is analysed twice by two methods of parametric quantity adjustments, a sum of 12 analyses will be carried out for each river.
Subsequently, comparing between the results of statistical analysis and existent informations will be carried out to place the chance distribution and method of statistical parametric quantity adjustment that yield the best consequences and consistent with the existent information.
Literature Review/Case Studies
2.1Comparison of chance distributions and parametric quantity calculator
Terafuk Haktanir ( 1992 ) carried out trial to happen the suited chance distributions and methods of parametric quantity suiting for 45 unregulated watercourses in Anatolia. The chance distributions are LN2, LN3, P3, LP3, Gumbel, GEV etc. and the method of parametric quantity are MOM, ML and PWM. Additional rating for LP3 utilizing parameter methods of maximal Entropy ( ME ) , Bobee and Method Mixed Moment in determines the best chance distribution and parametric quantity adjustments. Comparison of 100, 1000 and 10000 old ages are computed from a parent population with representative parametric quantities versus those from many man-made series of once more finite lengths are attempted. The consequence shows that LN2 is best for big return period fortunes. LP3 Wakeby and GEV are used as parent distributions in calculating with many different man-made series. LN3 is the best in term of gauging quantile followed by LN2, Gumbel utilizing ML, LP3 utilizing ME and P3 utilizing PWM. For certain instances Gumbel utilizing ML show the best public presentation. Terafuk Haktanir ( 1992 ) conclude that the most suited chance distributions are LN3, Gumbel by utilizing PWM or ML, LP3 by ME or MOM, P3 by PWM and GEV by PWM.
J.C Smither ( 1994 ) carried out goodness-of-fits trial to analyze chance distributions that are suited to gauge short-duration autumn in South Africa. Parameter calculators employed in this survey is L-moment. A sum of 38 sites with approximative 30 old ages informations were utilized in this trial. The chance distributions are LP3, LN2, LN3, P3, Gumbel, Wakeby, GEV etc. Chi-squared trial, standardised divergence and non-parameter were carried out by utilizing goodness-of-fits to rank chance distributions harmonizing to their public presentation. The consequences show that P3 is reasonable chance distributions to be use but when L-skewness is larger than 0.2, P3 show hapless public presentation. LP3 show hapless public presentation when L-skewness is greater than 0.2 similar to P3 distributions. The goodness-of-fitness trial show that LN3 performed best follow by GEV and P3 and L-EV1 performed the worst follow by LP3. Harmonizing to J.C Smither ( 1994 ) , EV1 is by and large used in South Africa analysis and the consequence show that EV1 is non a good calculator. LN3 and GEV are recommended for the future short-duration autumn analysis.
T.A.McMahon and R.Srikanthan ( 1980 ) carried out analysis on 172 Australia watercourse by utilizing minute ratio diagram for LP3 and 6 others theoretical distributions viz. : normal ; LN2 ; P3 ; EV1 ; Weibull and exponential. T.A.McMahon and R.Srikanthan ( 1980 ) conclude that LP3 distribution is suited for inundation frequence analysis comparison to the others theoretical chance distribution. The theoretical distributions are ill fitted in to the graph harmonizing to T.A.McMahon and R.Srikanthan ( 1980 ) . LP3 and minute diagram with skewness about equal to zero indicate that LP3 is good distribution to be used for inundation frequence analysis because it is non bias.
Joseph D.Countryman et Al. ( 2008 ) found out that LP3 and P3 are undependable and uneffective after conducted a trial to gauge utmost events viz. : 90 % assurance edge of 100years, 200years and 500 old ages flood from 3 rivers in California. The trial show that the 90 % assurance edge was unacceptable because LP3 extensively overestimate the existent 100 old ages, 200years and 500 old ages flood flow informations whereas P3 underestimate the existent 100 old ages, 200years and 500 old ages flood flow informations. , Joseph D.Countryman et Al. ( 2008 ) reference that P3 is more suited for longer period of record informations compare to LP3. For big inundation informations, LP3 is non suited to be employed to do appraisal because it will take to unrealistic design.
Vogel.R.Met Al. ( 1993 ) evaluate LP3, LN2 and LN3, GEV utilizing LM diagram and goodness of fitness process for 10 parts which consists 383 sites of inundation informations in south western of United State. For 100 twelvemonth event, LP3, GEV and LN2 produce exceedances that fall within 95 % of likely interval. As for 1000 old ages event, LP3, GEV and LN3 produce exceedances that fall within 95 % of likely interval. The concluding consequences show that LP3, LN2, LN3 and GEV gave a satisfactory appraisal to the 383 informations observed.
Vogel.R.M and Fennessey.N.M ( 1993 ) performed a instance survey with sample n =5000 of an mean day-to-day watercourse flow in Massachusetts utilizing L-moment diagrams and merchandise minute diagram. The L-moment diagram describe the relationship between L-skewness and L-kurtosis for chance distributions of P3, LN3, GEV, Gumbel, normal, unvarying, exponential, Wakeby and Generalize Pareto. Product minute diagram is comparing the lopsidedness, kurtosis and coefficient of discrepancies with their theoretical opposite number mean regard to discrepancy, kurtosis, lopsidedness. Monte Carlo simulation is carried out by bring forthing watercourse flow hints of length N of 10,20,50,100, 200, 500, 1000 and 5000 from LN2 and Generalize Pareto and populated with coefficient of discrepancies of 1,2,5 and 10. Bias and root mean square mistake of coefficient of discrepancies Cv and kurtosis is computed for LN2. The consequences show that when population and coefficient of discrepancies increase Product minute ratio provides no information on either lopsidedness or coefficient of discrepancies of the samples. Vogel.R.M and Fennessey.N.M ( 1993 ) conclude that L-moment is better due to the fact that the L-skweness, L-kurtosis and L-Cv are close to indifferent. By taking logarithm of ascertained informations, merchandise minute will act similar to L-moment but distributional belongingss of the original informations will be undetermined except for LP3 and LN3.
For the comparing of chance distributions, LP3 and GEV are more favorable compared to P3, Gumbel and LN2. As for the parametric quantity calculator, LM and LM diagram outperformed MOM and minute ratio diagram.
2.3Log-Pearson type III
Griffis and Stedinger ( 2007 ) conducted Monte Carlo analysis on parametric quantities appraisal for LP3 distribution: ML, method of mix minute, MOM, method of minute in log infinite without regional skew, method of minute in log infinite with regional skew ( recommended by bulletin 17 ) , and method of minute in existent infinite. Sample size of 25, 50 and 100 were employed for comparing between the parametric quantities appraisal. Mean square mistake ( MSE ) was employed to happen the efficiency of each method. Consequences for 100 old ages event were reported. ML was shown to be underperformed comparison to MOM for the sample size of 25 because MSE for maximal likeliness is 60 % larger than method of minute at.But as the sample size addition, ML outperform MOM. Griffis and Stedinger ( 2007 ) conclude that MOM improved as extremely information regional skew is utilized as suggested in Bulletin 17. Method of mix minute is first-class method non utilizing regional information and can be compared to Bulletin 17. ML turn out non to be every bit efficient as it was originally thought and merely efficient depend on get downing location and any parametric quantity constrain. Method of minute in log infinite with regional skew is better in mean compared to ML.
I.A. Koutrouvelisa and G.C. Canavosb ( 2000 ) evaluated LP3 by comparing 8 methods viz. : direct minute ( MDM ) , method of indirect minute ( IM ) , method of assorted minute ( MMM ) , an adaptative assorted minute method ( AAMMM ) and etc.. MDM is recommended by us Water Resources Council ( 1967 ) whereas MIM is recommended by Bobee ( 1975 ) . Rao ( 1980, 1983 ) is the first to propose MMM but there is restriction on this method. Subsequently, Phien and Hira ( 1983 ) and by Arora and Singh ( 1989 ) proposed simple processs to obtain MMM estimations of parametric quantity which overcome that restriction. AAMMM consists of the application of the assorted minute ‘s method of Koutrouvelis and Canavos ( 1999 ) to the logarithmically transformed informations. Monte Carlo simulation is employed to measure the public presentation of the methods specify above. AAMMM found to hold the best public presentation in term of prejudices follow by MMM and MID. AAMMM, MMM and MID processs maintain smallest absolute value of prejudice in estimate of return period Ta‰?50 and sample size na‰¤50 every bit good as for Ta‰?10, n=any value and high return period and sample size na‰¤50.Standardizedand normalized root mean squared mistake is carried out for assorted return period and sample size. For T=10 and n=25, AAMMM execute the best while generalised direct minute execute the worst. For the instance of Ta‰?50 and na‰¤50, generalised assorted minute performed the best and MID shows the worst public presentation. Generalised assorted minute demo the best public presentation in 98thand 99thpercentile.
In general the merchandise of minute for LP3 is recommended by Griffisand Stedinger ( 2007 ) , A. Koutrouvelisa and G.C. Canavosb ( 2000 ) , Terafuk Haktanir ( 1992 ) and T.A.McMahon and R.Srikanthan ( 1980 ) . Nevertheless, no L-moment is carried out to compare with the merchandise of minute for LP3 distributions. Vogel.R.Met Al. ( 1993 ) conclude that LP3 is suited to be used in south western of America but J.C.Smither ( 1994 ) found out that LP3 is among the worst distributions for short continuance autumn in South Africa. Joseph D.Countryman et Al. ( 2008 ) did non urge LP3 because this distribution overestimates big informations twelvemonth flow informations for 90 % assurance interval.
2.4Pearson type Three
Chen Yuan Fang et.al ( 2002 ) employed Monte-Carlo method to compared P3 with parametric quantity appraisal method of MOM, curve adjustment, PWM and weighted map minute. Evaluation of P3 is carried out by agencies of prejudice, efficiency of the parametric quantity, the quantile and the chance of failure. Root average Square Error and mean of Monte Carlo bring forthing samples show the prejudice, efficiency of the parametric quantity and the chance of failure of parametric quantity estimated method. The comparings show that method of minute is non a good appraisal method because it is biased in quantile appraisal. Comparisons of PWM, curve adjustment and leaden map minute are carried out in two trials which are a simple sample and a simple sample with history informations. For A simple sample, curve adjustment performed the best follow by PWM and weighted map minute. For the latter trial, PWM performed the best follow by curve adjustment and leaden map minute. Chen Yuan Fang et.al ( 2002 ) recommends using PWM and swerve adjustment for parametric quantity gauging as it shows first-class statistical public presentation. However for little size sample, all the parametric quantity methods reference above provide larger expected chance of the quantile estimated than design frequences.
Ding JING and YANG RONGFU ( 1988 ) compute PWM for P3 with the intent of PWM able to use in the instance of being of extraordinary values in the sample. Sample with historical inundation informations is used in this trial. PWM for P3 is compared with ML, MOM and swerve adjustment. For comparing between MOM and PWM, for skew CSa‰¤2.5, PWM outperformed MOM in term of prejudice but have the same performed in term of efficiency. With the presence of extraordinary value, PWM outperformed MOM for both standards. As for CS & gt ; 2.5, PWM have the same public presentation as MOM. For comparing between ML and PWM, for skew CSa‰¤2.5, PWM have the same performed in term of prejudice and underperformed MOM in term of efficiency. As for CS & gt ; 2.5, Ml outperformed PWM for both standard. For comparing between curve adjustment and PWM, PWM outperformed curve adjustments on both standard for CSa‰¤2.5 and CS & gt ; 2.5.DING JING and YANG RONGFU ( 1988 ) conclude that PWM that proposed is reasonable in technology pattern and for CSa‰¤2.5 PWM is a good calculator but frailty versa for CS & A ; gt ; 2.5.
P3 is found to be a good calculator in term of PWM by Chen Yuan Fang et.al ( 2002 ) , DING JING, YANG RONGFU ( 1988 ) , and Terafuk Haktanir ( 1992 ) . However, P3 perform severely in MOM for the ground that P3 is biased as shown by the two surveies above.
2.5 Gumbel ( utmost value 1 )
M. FIORENTINO and S. GABRIELE ( 1984 ) evaluate corrected maximal likeliness for Gumbel by modifying ML appraisal process. Comparison of calculator methods was carried out for sample sizes range from 5 to 300. The calculator methods viz. MOM, ML, corrected maximal likeliness ( CML ) , PWM and Maximum Entropy ( ME ) . Evaluate sample size, mean, discrepancy and average square mistake for each calculator method. Relative efficiency was evaluated by the ratio of calculator methods with regard to ML. PWM and CML provide indifferent estimations of parametric quantity. The other calculator methods provide biased parametric quantity by overrating the parametric quantity. Mean square mistake depends on the discrepancy and biased of calculator which will act upon the result of the efficiency of the calculator regard to the ML. The consequence show that CML and PWM outperformed ML, CML provide the best public presentation. Quantile estimates for the least colored calculator method are PWM follow by CML and MOM.
HUYNH NGOC PHIEN ( 1986 ) estimate the parametric quantity of Gumbel distributions with four methods viz. MOM, ML, ME and PWM. Efficiency of each method was carried out by comparing the discrepancy of the method regard to variance ML. HUYNH NGOC PHIEN ( 1986 ) point out that ML is asymptotically minimal discrepancies which is more efficiency than other calculators. Subsequently, comparing of efficiencies is handily assessed utilizing the discrepancies of the calculators. Monte Carlo Simulation was carried out to gauge parametric quantities and calculation of Root Mean Square Error ( RMSE ) for the parametric quantities. PWM performs ideally in term of prejudice and MOM performs the worst among all the other calculators. In term of RMSE, PWM and MOM underperform the other two calculators and MOM is found to be the least satisfactory calculators. HUYNH NGOC PHIEN ( 1986 ) concludes that MOM is the worst calculator in term of both RMSE and prejudice, PWM is the best in term of prejudice and ML is best for RMSE. By sing both RMSE and bias standards, ME distribution is favour since the public presentation of RMSE for ME and ML are comparatively similar.
For Gumbel distributions PWM performed the best compared to other parametric quantity calculator as suggested by M. FIORENTINO and S. GABRIELE ( 1984 ) , HUYNH NGOC PHIEN ( 1986 ) and Terafuk Haktanir ( 1992 ) . MOM is non a good calculator as stated in the two surveies above.
2.6 Generalized extreme value ( GEV )
J.R.M Hosking et.al ( 1985 ) derives parametric quantity and quantile of GEV by using PWM and analyze the belongings of the GEV. For big sample, the belongings is investigated by utilizing asymptotic theory whereas use computing machine simulation for medium and little samples. The consequence show that big sample for quantile in PWM has higher discrepancy than Maximum-likelihood but smaller prejudices particularly on the higher upper tail distributions. For little sample, the PWM show that standard divergence is smaller than the maximum-likelihood while PWM for medium size is bigger standard divergence compared to maximum-likelihood. Trial on form parametric quantity is carried out to find whether the distributions are Gumbel or GEV. Null hypothesis is carried out by taking form factor as nothing and comparing the statistic Z with the critical value signifier normal distributions. The statistic Z that gives important positive or negative value will be rejected by the void hypothesis. A sample of 35 one-year maximal inundation informations for the river Nidd to Hunsingore, Yorkshire, England is used to suit into utmost value distributions. The trial show that the Z=1.00 and this value is undistinguished therefore this suggested that the river informations assumed to come from Gumbel distribution.
For generalized extreme value, the calculator methods suggested are PWM and L-moment. Terafuk Haktanir ( 1992 ) show that the GEV is more suited for PWM comparison to MOM. J.C Smither ( 1994 ) and Vogel.R.M et Al. ( 1993 )
The purpose of this survey is to place the most appropriate chance distributions and methods of statistical parametric quantity adjustment in Malaysia context. A sum of six distributions and 2 methods of statistical parametric quantity adjustment are used in this survey. There is no job with method of minute since it is been develop for rather some clip. L-moment is non every bit set up as method of minute and all chance distributions chance denseness map, cumulative distributions map and quantile map is given except for Log Pearson type 3and two parametric quantity log normal ( Hosking and Wallis,1997 ) .The package utilize in patterning the flow informations is excel dispersed sheet.
The one-year extremum informations are obtained from Department of Irrigation and Drainage Malaysia ( DID ) . These informations consists of 19 rivers across peninsular Malaysia and the natural informations are computed by agencies of Annual Maximum Series. These informations are computed based on the highest value obtained yearly. A sum of 600 old ages informations obtained from the 19 rivers with an norm of 30 inundation informations.
3.2 Excel dispersed sheet processs
Excel Spread sheet is employed for the modeling of the discharges. Mean, standard divergence, return period and lopsidedness of the informations are determined.These parametric quantities are indispensable for the chance distributions to find the discharges. MOM and LM have the similar processs except for the order of the information is ranked. For MOM, The information is sorted harmonizing to falling order whereas LM ranks the informations harmonizing to go uping order.
First, kind informations harmonizing to go uping or falling order. Ranks and sample size are obtained after screening of informations. Therefore return period are determined by using Weibull plotting place as given by equation 1 in appendix. Sample size, N is the figure of flow informations which range from 11 to 50 sets of informations. The rank of falling order is given by equation 2. By cognizing the value of N and m, the return period T can be computed. This survey focuses on return period of 2, 5, 10, and 20 and 50 old ages, therefore insertions of discharges are required to obtain discharges which are X2, X5, X10, X20 and X50.
Selected parametric quantity are obtained by utilizing in built map in excel. Mean and standard divergence are obtained by utilizing map in excel dispersed sheet i.e. AVERAGE and STDEVA. Gamma map is obtained by utilizing map of EXP ( GAMMALN ) to calculate the gamma chance of given variables.
3.3 Comparisons of methods of statistical parametric quantity adjustment and chance distributions
3.3.1 Comparison of methods of statistical parametric quantity adjustment
Pearson type 3
Log-Pearson type 3
Generalized utmost value
Method of minute
Table1: Comparisons of method of minute and L-moment
Comparisons of method of minute and L-moment as shown in table 1 to happen the most suited chance distribution and methods of statistical parametric quantity suiting for tropical state for case Malaysia.
Method of minute is a common and comparatively easy parametric quantity method. Parameter is estimated by comparing the minute of chance distributions map with the minute of sample. There are entire of six chance distributions employed in gauging the discharges of the rivers. All the chance distributions except generalised utmost value utilize frequence factor to calculate discharges. Frequency factor which derived from the cumulative denseness map are used in this survey. The magnitude is given in equation 3 in appendix. Frequency factor depends on the empirical expression for chance distribution. The magnitude is tantamount to the discharge of the rivers in this survey.
L-moment is a additive map of chance weighted minute ( PWM ) ( Hosking, 1986 and 1990 ) .The indifferent sample of L-moment is given from equation 24 to equation 27. The L-coefficient of discrepancy t1 and the L-skewness t3 are given in equation 28 and 29 severally. The L-coefficient of discrepancy and L- lopsidedness are of import parametric quantities to find the discharges.
Validation of the dispersed sheet is carried out by utilizing an illustration given in book. The information given in the illustration is input into the excel dispersed sheet and the consequence is compared with the reply in the book. Once the reply is similar, the expression in the dispersed sheet will be validated.
3.3.2 Comparison of chance distributions
Pearson type 3
Log-Pearson type 3
Generalized utmost value
Method of minute
I±1, I±2, KT
I±1, I±2, KT
I±1, I±2, KT
I±1, I±2, KT
I±1, I±2, KT
U, I± , K
I±1, I±2, U
uy, I?y, U
U, I± , I’ , KT
I? , I± , I’ , KT
I? , I±
U, I± , K
Table 2: parametric quantities for chance distributions for method of minute and L-moment
The parametric quantities for chance distributions for methods of statistical parametric quantity adjustment are computed to obtain the discharges. The parametric quantities that show in table 2 are substitute into the quantile estimation equation to calculate the discharges.
220.127.116.11 Normal distributions
In method of minute, the frequence factor for normal distribution is approximated by an empirical relation ( Abramowitz and Stegun, 1965 ) as shown in equation 4. First determine the exceedance chance p= 1/T.The exceedance chance is govern by the return period, therefore a little return period will take the P to transcend 0.5. This is non allowed as shown above by intermediate variable w. To get the better of this affair, P is substituted with 1-p whenever P & gt ; 0.5. Then find intermediate variable tungsten by replacing the exceedance chance P in to the equation 5. Subsequently frequency factor is computed by replacement the value tungsten into the equation 3. The parametric quantity I±1 and I±2 is average and standard divergence severally. An illustration from H2O resources technology ( WRE ) by Chin ( 2000 ) is cardinal into the dispersed sheet to formalize the dispersed sheet. The reply given by WRE and consequence from dispersed sheet is different. This is due to the disagreement in term of the frequence factor and discharge because WRE did non include the mistake of 0.00045 into equation 4. However, the mistake is so little that it is negligible.
For l-moment, the normal random variable U for the normal distributions is determined by equation 33. The intermediate variable tungsten is given by equation 34.The intermediate variable tungsten is computed by replacement exceedance chance P into the equation 34 supplying the P is lesser than 0.5.If P & gt ; 0.5, so replace 1-P into equation 34.The normal random variable U is computed by replacing the W into the equation 33. The parametric quantities I±1 and I±2 is given by equation 30 and equation 31 severally one. The quantile estimates is given by equation 32 and the discharge is computed by replacement all the relevant parametric quantities into equation. The dispersed sheet is validated by utilizing an illustration from Flood frequence Analysis ( FFA ) ( Rao, 2000 ) .The replies from dispersed sheets and FFA are similar therefore the dispersed sheet is validated.
18.104.22.168 log-normal distributions
For two parametric quantity log normal distributions utilizing method of minute, the frequence factor is obtained from equation 4. The magnitude XT is given by equation 8. YT is obtained from the transmutation of informations. Magnitude YT is the opposite of magnitude from equation XT and is given by equation 7. The information is transformed by taking logarithm of the informations by keeping the return periods. With this new transformed informations, parametric quantities I±1 and I±2 which are average and standard divergence will be computed. Magnitude YT is obtained by replacing all the related parametric quantity into equation 8. Discharges XT is given by YT to the power of 10. Example from applied hydrology ( AP ) ( Chow, 1988 ) is input into excel dispersed sheet to formalize the preparation for two-parameter log normal. The reply from the dispersed sheet and the book is same, therefore the dispersed sheet is validated.
For L-moment, the quantile estimations for log-normal is given by equation 38. First, transform the information is required to obtain the L-coefficient of discrepancy, t. The log-normal distribution map F is given by in appendix. F is computed by replacing L-coefficient of discrepancy T into the equation shown supra. Then the intermediate variable tungsten is computed by following the process from normal distributions. The frequence factor is acquired by utilizing equation 33 and the parametric quantities uy and I?y are computed by utilizing equation 35 and equation 36. Parameters are substituted into equation 37 to obtain the YT. Last, discharge is obtained by exponential of the parametric quantities in equation 38. The parametric quantities for log normal are non in the book by Hosking and Wallis ( 1997 ) and since the L-moment is additive to PWM, the illustration of PWM for two parametric quantity log normal is used to compare with the excel. Therefore the dispersed sheet is validated by utilizing an illustration from FFA. The replies from dispersed sheets and FFA are same therefore the dispersed sheet is validated
22.214.171.124 Pearson type 3
Pearson type 3 besides known as three-parameter gamma distributions, the frequence factor govern by return period and lopsidedness. In method of minute the frequence factor can be approximative utilizing the relation ( Kite 1977 ) given by equation 9. Substitute lopsidedness, gx into the equation 10 to get the value K. XT ‘ is obtained from the frequence factors of normal distribution. Thus the frequence factor is computed by utilizing the needed parametric quantities. Validation of the preparation for Pearson type 3 is carried out by mentioning to the AP and illustration from WRE. There is disagreement between the consequence from the dispersed sheet and the reply from the WRE. The differences between the two replies are comparatively little. The ground for this disagreement is that WRE recommended to happen the XT ‘ by using Z-distributions table where as the AP recommend to utilize equation 4 find the XT ‘ . Using equation to calculate XT ‘ will be more accurate compared to utilizing tabular array. The differences between the two replies are comparatively little.
In l-moment, the quantile estimation for Pearson type 3 is given by equation 44 in appendix. First, look into L-skewness t3 against the equation 39 and 40 to find which thulium and parametric quantity is suited to be used. Once the parametric quantity is computed, the gamma map and can be computed by utilizing map in dispersed sheet. If the is excessively big, the value is unable to be computed by dispersed sheets. Parameter I± , I’ and u are obtained by equation 41, 42 and 43. The frequence factor of the Pearson type 3 is given by Wilson-Hilferty transmutation is given by equation 45. The lopsidedness of the information is determined by utilizing equation 46 and the standard normal random variable U is computed by utilizing equation 33. The frequence factor is computed by replacing the parametric quantities of and u into equation 45. Discharge is obtained by utilizing equation 44. The parametric quantities for Pearson type 3 recommended by Hosking and Wallis ( 1997 ) have slight different in term of and compared to PWM method in FFA even though the method of deducing the parametric quantity is similar. The PWM parametric quantities are given by equation 47 and 48. Parameters recommended Hosking and Wallis ( 1997 ) are used as it is based on L-moment derivation. Since there is no illustration for l-moment to be validated, a separate spread sheet utilizing PWM calculator is created. Surprisingly, the reply from the dispersed sheet and FFA turn out to be the same therefore the spread sheets is validated.
126.96.36.199 log-Pearson type 3
For method of minute, the log-Pearson type 3 followed the process of two-parameter log normal to find the mean, standard divergence and lopsidedness. The parametric quantities of I±1 and I±2 are average and standard divergence severally. The frequence factor of Log-Pearson type 3 is similar to frequency factor in Pearson type 3. K is obtained utilizing equation 9 and XT ‘ is obtained utilizing equation 4. Frequency factors will be computed by replacement necessary parametric quantities into the equation 9.Validation of preparation for log-Pearson type 3 by utilizing illustration from AP. The reply get are similar for preparation from dispersed sheet and the AP, therefore the dispersed sheet is validated.
For L-moment, the quantile estimations for Log-Pearson type 3 is given by equation 49. First, transform the information by taking logarithm of the information. Determine the new L-coefficient of discrepancy, L-skewness and mean. The parametric quantities I? , I± and I’ are computed by utilizing the equation 39 or 40, equation 47 and equation 48 severally. KT is obtained by utilizing Wilson-Hilferty transmutation which is similar to Pearson type 3 for l-moment. Discharge is computed by replacing the parametric quantities as shown in table 2 into equation 49. Log-Pearson type 3 is similar to two parametric quantity log normal instance because no information provided for Log-Pearson type 3 in the book by Hosking and Wallis ( 1997 ) . FFA suggested transforming the information by logarithm the information and the other stairss follow the process in Pearson type 3. The process for I± recommended in FFA is different from the Pearson type 3 processs. The parametric quantity recommended by Hosking is used to cipher quantile estimations.
188.8.131.52 Gumbel/extreme value 1
In method of minute, frequence factor for Gumbel or utmost value 1 distributions is given by ( Chow, 1953 ) . The frequence factor is given by equation. Frequency factor is computed by replacement the return period into the equation 3.11. The parametric quantity I±1 and I±2 is given by mean and standard divergence severally. Validation of preparation for Gumbel distributions is carried out by utilizing an illustration from the WRC. The reply obtained is similar to the reply in the book therefore the dispersed sheet is validated.
For L-moment, the quantile estimations for Gumbel or Extreme value 1 is given by equation 52 in appendix. First, Parameter I± and I? are obtained by utilizing equation 50 and equation 51. Subsequently, use the relationship of to obtained F. Discharges is computed by utilizing the equation 52. The dispersed sheet is validated by utilizing an illustration from ( FFA ) . The replies obtained are similar for both dispersed sheets and FFA, therefore the dispersed sheet is validated.
184.108.40.206 Generalized extreme value
In method of minute, the quantile calculator for generalised utmost value is given by equation 23. The parametric quantity K is given by equation 15 or 16 or 17 and it depend on the skew Cs. Iteration will be necessary if the kn in equation 18 is non equal to ko. The loop will go on until the kn about tantamount to kn+1. The parametric quantities for I± and U are computed utilizing the equation 21 and 22. Validation of preparation for generalised utmost value distributions is carried out by utilizing an illustration from the FFA. The reply obtained is similar to the reply in the book therefore the dispersed sheet is validated.
For L-moment, the quantile calculator is similar to MOM. The parametric quantities for K, I± and u are computed by utilizing equation 53 to 55 severally. Validation of preparation for generalised utmost value distributions is carried out by utilizing an illustration from the FFA. The L-skewness t3 obtained is different from the reply from the book but surprisingly the L-CV and mean is similar. The t3 suggested by FFA is input to stand out and the replies obtained is similar to the FFA. The L-skewness t3 is computed utilizing equation 29 but the value is different from FFA therefore the quantile estimation is significantly different.
3.5 Data Analysis
Comparisons of discharges/magnitude between consequences by chance distributions and their calculator methods with the existent informations are conducted. Return periods of 2, 5, 10, 20 and 50 old ages are utilized in these comparings. In affair of fact, there is a few informations that can make 50 old ages of return period and most of the informations reach up to return period 20 old ages. The intent of these comparings is to measure the per centum of mistake for each chance distribution against existent informations. The expression of this comparing is given by:
The per centum difference represents the truth of the chance distribution in gauging the discharge. For positive per centum difference, it indicates that the chance distribution overestimate the flow whereas negative per centum difference indicates that the chance distribution underestimate the flow. Absolute difference is carried out to happen the summing up of differences between chance distributions and existent informations for each river.
The 2nd comparings of discharges are between method of minute and L-moment. Return period of 2,5,10, 20 and 50 old ages are used in these comparing. The chief end of this comparing is to measure the difference between both calculator methods. Comparison is conducted by comparing the difference of discharges between method of minute and L-moment with regard to the discharges for method of minute:
The per centum of difference shows which estimator method can gauge higher discharges value. The positive per centum of difference show that MOM come close higher discharge compared to L-moment whereas negative per centum of difference indicate that MOM come close lower discharge than L-moment. Absolute difference is carried out to happen the summing up of differences for chance distributions between method of minute and L-moment for each river.
Assurance intervals ( CI ) of 90 % , 95 % and 99 % are conducted to measure public presentation of chance distributions and calculator method. Confidence interval of 90 % , 95 % and 99 % are given by:
Assurance interval can be computed by utilizing map in excel dispersed sheet every bit good. The existent informations A± CI will obtain a lower bound and upper bound to be compared with the discharges computed from chance distributions and calculator methods. The approximated discharge that autumn in between the bounds will be accepted and will be marked as ‘1 ‘ . For approximated discharges that autumn outside the bounds will see rejected and will be marked as ‘0 ‘ . The chance distribution and calculator method that had the most ‘1 ‘ show that it is most appropriate to be employed in gauging discharges in Malaysia.
Chapter 4: Consequences and treatment
This survey compares six chance distributions utilizing two parametric quantity calculators for 19 rivers in peninsular Malaysia. The comparings are divided to three classs which are the method of minute V existent informations, L-moment V existent informations and method of minute V L-moment. The trials for chance distributions are Chi-squared trial, Kolmogorov-Smirnov trial and assurance interval.
4.1 Method of minute V existent informations
Figure 1 illustrates six chance distributions and existent informations are plotted against return period for all the rivers. A sum of 19 rivers with sample size scope from 11 to 49 are employed in the illustration. For sample size n=40 and above and for sample size n=20 and below, the chance distributions on norm overestimated the inundation informations compared to existent informations. For sample size scope from 20 and 40, the chance distributions averagely underestimated the inundation informations compared to existent informations. Presence of exceeding big inundation in river ( i.e. Lui River, Gemencheh River and Bernam River ) that chiefly consists of little inundation informations, the chance distributions will execute severely such that the differences between the discharges of existent informations and the estimated discharges are reasonably big. From figure 1, the rivers mention supra have the same characteristic which is the inundation informations is overestimated ab initio and so underestimated at the terminal compared to existent informations. However, existent information for Kulim River is lower than bulk of the chance distributions for all return periods even with presence of exceeding big inundation informations. For big inundation informations river ( i.e. Berang River ) , the chance distributions overestimated the inundation informations compared to existent informations throughout the return periods.
The per centum differences between the MOM and existent informations for N scope from -184 % to 47 % , LN2 scope from -85 % to 57 % , P3 scope from -143 % to 47 % , LP3 scope from -94 % to 47 % , EV1 scope from -187 % to 42 % and GEV range from -86 % to 42 % as shown in table 2. EV1 is the distribution with highest negative per centum differences following by N, P3, LP3, GEV and LN2. Nonetheless, EV1 and GEV distribution are the distributions that have the lowest positive per centum difference follow by N, P3, LP3 and LN2. GEV is the distribution with the lowest mean per centum difference while N is the distribution with the highest per centum difference The amount of absolute difference between the existent informations and chances distributions for the 19 rivers are N = 27.5, LN2 = 15.8, P3 = 18.4, LP3 = 14.6, EV1 = 22.62 and GEV = 12.7. From the amount of absolute difference, the GEV is the distribution with the lowest value therefore GEV estimates the inundation informations with the highest truth compared to existent informations for method of minute.
4.2 L- minute V existent informations
From figure 2, the log-Pearson type 3 for river Kepis is non shown because the gamma map is so big that the excel dispersed sheet unable to calculate. For sample size n=40 and above, the chance distributions averagely overestimated the inundation informations compared to existent informations and this is similar to the result for method of minute. For sample size scope from 20 and 40, the chance distributions on norm underestimated the inundation informations compared to existent informations which is similar to the result for method of minute as good. Nevertheless, for sample size n=20 and below, the chance distributions for L-moment on norm are reasonably equal in either underestimating or overrating inundation informations compared to existent informations. For illustration, Buloh River and Pelarit River on mean underestimate the inundation informations while Tasoh River and Berang River averagely overestimate the inundation informations compared to existent informations. Similar to method of minute, the river with the presence of exceeding big inundation informations influence the public presentation of the chance distributions. From figure 2, the river with exceeding big inundation informations will act likewise to characteristic that stated in method of minute. Furthermore, Kulim River in L-moment behaves likewise to the river with exceeding big informations unlike Kulim River instance in method of minute.
Figure 2: Comparison of chance distributions utilizing L- minute and existent informations Vs return period
Figure 2: Comparison of chance distributions utilizing L- minute and existent informations Vs return period
Figure 2: Comparison of chance distributions utilizing L- minute and existent informations Vs return period
Table 2: Percentage difference and absolute differences of existent informations vs L- minute