ARID ZONE JOURNAL OF ENGINEERING, TECHNOLOGY & ENVIRONMENT AZOJETE June 2022. Vol. 18(2):281-294 Published by the Faculty of Engineering, University of Maiduguri, Maiduguri, Nigeria. Print ISSN: 1596-2644, Electronic ISSN: 2545-5818 www.azojete.com.ng 281 Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng ORIGINAL RESEARCH ARTICLE BEST-FIT PROBABILITY DISTRIBUTION MODEL FOR ANNUAL MONTHLY MAXIMUM RAINFALL IN CALABAR CITY, NIGERIA G. D. Akpen, I. D. Ijasini and G. N. Ekwo Department of Civil Engineering, Joseph Sarwuan Tarka University, Makurdi, Benue State, Nigeria *Corresponding author’s email address: akpen.gabriel@uam.edu.ng 1.0 Introduction Rainfall is one of the most widely researched climatic variables in terms of its absence or presence, extremity or scarcity, stability or variability, as well as its vulnerability or essentiality (Oruonye, 2012; Gabriel et al., 2018; Hassanyar et al., 2018 and Aho et al., 2019a). Rainfall is a major source of fresh water replenishment for planet earth. However, too much or too little can mean the difference between prosperity and disaster (Oyegoke et al., 2017). As a random hydrological event, the occurrence of rainfall cannot be predicted with certainty because its temporal and spatial distribution is very complex and irregular. This uncertainty is further aggravated by the current global problem of climate change. However, it is possible to use rainfall data spanning over a long period of time to estimate the likelihood of a rainfall of a particular magnitude ARTICLE INFORMATION ABSTRACT The choice of a suitable model that could predict the possibility of occurrence of rainfall event of a specified magnitude depends mainly on the characteristics of rainfall data at a particular site. This study was aimed at determining the best-fit probability distribution model for annual maximum monthly rainfall. Fifty years rainfall data (1969-2018) obtained from Nigerian Meteorological Agency (NIMET), Calabar, were collated to form an annual series ranked in a decreasing order of magnitude. The data were then evaluated with 6 probability distributions namely; Normal, Log-Normal, Pearson Type III, Log-Pearson Type III, Gumbel Extreme Value Type I and Log-Gumbel probability distributions. The best-fit probability distribution model at the study area was selected based on the results of 4 goodness of fit tests namely; Mean Absolute Deviation Index (MADI), Chi- Squared(π‘₯2) test, Root Mean Square Error (RMSE) and Relative Root Mean Square Error (RRMSE). The performance of the 6 distributions were ranked and the total scores of 8, 23, 16, 12, 21 and 4 were allocated to Normal, Log-Normal, Pearson Type III, Log-Pearson Type III, Gumbel Extreme Value Type I and Log- Gumbel probability distributions, respectively. The results indicate that the best-fit probability distribution model was Log-Normal, which was used to predict rainfall values of 566.8, 664.0, 721.6, 788.1, 834.4, 878.4, 920.7, 974.8 and 1014.4 mm for different return periods of 2, 5, 10, 25, 50, 100, 200, 500 and 1000 years, respectively. It is, therefore, recommended that Log-Normal distribution is the preferred model for frequency analysis of rainfall data for the planning and design of hydraulic structures in the study area. This is necessary for effective flood prevention and mitigation interventions. Β© 2022 Faculty of Engineering, University of Maiduguri, Nigeria. All rights reserved. Submitted 11 November, 2021 Revised 4 March, 2022 Accepted 16 March, 2022 Keywords: Rainfall-events Probability distribution model Rainfall frequency analysis Goodness of fit Rainfall prediction Flood mitigation http://www.azojete.com.ng/ mailto:akpen.gabriel@uam.edu.ng mailto:akpen.gabriel@uam.edu.ng Arid Zone Journal of Engineering, Technology and Environment, June, 2022; Vol. 18(2):281-294. ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng 282 occurring within a specified period of time using frequency analysis. According to Izinyon and Ajumuka (2013) and Win and Win (2014), there is no universally accepted frequency distribution model for rainfall frequency analysis. Few probability distribution functions have been tested and adopted locally, while some others are peculiar to hydrologic characteristics of other regions. The ability to predict the possibility of occurrence of a particular rainfall magnitude is one of the preliminary stages of storm water drainage design that can help individuals, authorities and engineers to plan for extreme eventualities such as flood, drought, landslides, and thunderstorms, among others (Oyegoke et al., 2017). Tao et al. (2002) stated that several probability models have been developed to describe the distribution of annual extreme rainfalls at a single site. However, the choice of a suitable model is still one of the major problems in engineering practice since there is no general agreement as to which distribution(s) that should be used for the frequency analysis of extreme rainfalls. Khudri et al. (2013) reported that the generalized extreme value and generalized gamma four parameter (4P) distributions provide the best fit for 50% of the data from rainfall gauging stations studied in Bangladesh. Olofintoye et al. (2009) showed that for peak daily rainfalls of selected cities in Nigeria, the Log-Pearson type III (LP3) distribution performed best by occupying 50% of the total stations, while Pearson type III (P3) performed second best by occupying 40% of the total stations and lastly by Log-Gumbel occupying 10% of the total stations. Aho et al.(2019b) carried out frequency analysis and reported that the best fit probability distribution function (PDF) for annual and partial series rainfall data in Makurdi metropolis is generalized extreme value (GEV) distribution and Generalized Pareto (GPA) distribution respectively. The selection of an appropriate model depends mainly on the characteristics of available rainfall data at a particular site. Hence, it is necessary to evaluate many available distributions in order to find a suitable model that could provide accurate extreme rainfall estimates. More so, recent devastation caused by flood in different parts of the world in addition to the challenges currently posed by uncertainties occasioned by climate change phenomenon has made the reliability in estimation of rainfall events more imperative (Akpen et al., 2019). The value of such studies especially, in this era of failure/collapse of bridges, culverts, dams and other drainage structures leading to loss of means of livelihoods and sometimes, loss of lives cannot be over emphasized. The objective of this study is to determine the best-fit probability distribution model for annual monthly maximum rainfall over Calabar city. This is required for planning and design of effective flood prevention, control and mitigation measures. 2.0 MATERIALS AND METHODS 2.1 Study Area Calabar is the capital of Cross River State, Nigeria. The city is adjacent to the Calabar and Great Kwa rivers and creeks of the Cross River (from its inland delta). It lies between latitudes 050 45’ file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:akpen.gabriel@uam.edu.ng Akpen et al: Best-Fit Probability Distribution Model for Annual Monthly Maximum Rainfall in Calabar City, Nigeria. AZOJETE, 18(2):281- 294. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng 283 30” N and 050 08’ 30” N and longitudes 8011’ 21’’ E and 80 30’ 00” E (Figure 1). Calabar is often described as the tourism capital of Nigeria. Administratively, the city is divided into Calabar Municipal and Calabar South Local Government Areas. It has an area of 406 square kilometers and a population of 371, 022 as at 2006 Census (Falola and Warnock, 2007; Ogarekpe, 2014; Ottong et al., 2010). Calabar has a tropical monsoon climate with a lengthy wet season spanning 10 months and a short dry season covering the remaining 2 months each year. The harmattan which significantly influences weather in West Africa is noticeably less pronounced in the city. Temperatures are relatively constant throughout the year with average high temperatures usually ranging from 25 to 28 degrees Celsius. There is only little variance between day time and night time temperatures, with fewer degrees lower during nights. In some wet years (1969, 1976, 1980, 1995, 1996, 1997, 1999, 2001, 2005, 2007, 2008, 2010 – 2015, 2017 and 2018), annual rainfall depths have been observed to rise over 3000 mm and above (Ogarekpe, 2014; Antigha, 2012). Figure 1: Map of Calabar, Nigeria (Source: Erhabor et al., 2019) 2.2 Data Collection and Preparation The monthly rainfall data for 50 years (January, 1969 to December, 2018) was obtained from the Nigerian meteorological Agency (NIMET), Calabar, Nigeria. The data was sorted according to years and months. The monthly maximum rainfall depths for 50 years of record were collated to form an annual series which were ranked in decreasing order of magnitude with their corresponding years and months as shown in Table 1. The ranked annual monthly maximum series of rainfall values were then analyzed. http://www.azojete.com.ng/ mailto:akpen.gabriel@uam.edu.ng Arid Zone Journal of Engineering, Technology and Environment, June, 2022; Vol. 18(2):281-294. ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng 284 The probability of an event being equaled or exceeded in any one year was calculated using the Equation 1 (Weibull, 1939), while the recurrence interval, which is the inverse of the probability of its occurrence is given by Equation 2. Probability of occurrence; π‘ƒπ‘Ÿπ‘œ = π‘š 𝑁+1 (1) Return period; π‘‡π‘Ÿ = 𝑁+1 π‘š (2) Where m is the rank number and N is the total number of observed data Table 1: Annual Monthly Maximum Rainfall, Probability and Return Period in Calabar (1969 - 2018) Y e ar M o n th x i ( m m ) R an k ( m ) P ro ( % ) 𝑇 π‘Ÿ ( ye ar s) Y e ar M o n th x i ( m m ) R an k ( m ) P ro ( % ) 𝑇 π‘Ÿ (y e ar s) (1) (2) (3) (4) (5) (6) (1) (2) (3) (4) (5) (6) 2012 Aug. 861.3 1 1.96 51.0 2009 July 577.4 25 49.02 2.0 2005 July 828.2 2 3.92 25.5 1975 July 552.6 26 50.98 1.96 1997 July 796.6 3 5.88 17.0 1988 Sept. 538.3 27 52.94 1.9 1980 Aug. 728.7 4 7.84 12.8 1986 July 533.8 28 54.90 1.8 1969 Aug. 726.9 5 9.80 10.2 1976 Aug. 526.8 29 56.86 1.76 1978 Sept. 721.2 6 11.76 8.5 1985 May 520.5 30 58.82 1.7 2014 July 714.4 7 13.73 7.3 1981 July 519.7 31 60.78 1.6 1990 July 702.7 8 15.69 6.4 1972 July 510.8 32 62.75 1.59 2015 June 678.7 9 17.65 5.7 1982 July 505.7 33 64.71 1.55 1971 July 661.4 10 19.61 5.1 1991 Aug. 505.6 34 66.67 1.5 2018 Aug. 650.6 11 21.57 4.6 1983 June 504.6 35 68.63 1.46 2011 July 648.6 12 23.53 4.2 1998 June 504.5 36 70.59 1.4 1995 July 632.4 13 25.49 3.9 2013 May 499.9 37 72.55 1.38 1989 July 626.5 14 27.45 3.6 1999 Aug. 494.5 38 74.51 1.34 2002 Aug. 623.5 15 29.41 3.4 1987 Aug. 493.8 39 76.47 1.3 1996 Sept. 615.2 16 31.37 3.2 2001 May 491.4 40 78.43 1.28 2017 Aug. 611.4 17 33.33 3.0 2006 July 484.9 41 80.39 1.24 2010 June 611.3 18 35.29 2.8 1992 Sept. 481.4 42 82.35 1.21 1994 July 609.5 19 37.25 2.7 1993 Aug. 479.0 43 84.31 1.2 1979 June 599.3 20 39.22 2.5 2016 July 454.6 44 86.27 1.16 2008 July 597.7 21 41.18 2.4 1970 July 440.8 45 88.24 1.13 1977 July 597.6 22 43.14 2.3 1984 June 437.3 46 90.20 1.11 2000 July 597.6 22 43.14 2.3 2003 Sept. 399.2 47 92.16 1.09 1974 Sept. 588.4 23 45.10 2.2 2004 Aug. 391.9 48 94.12 1.06 2007 May 583.5 24 47.06 2.1 1973 Aug. 373.3 49 96.08 1.04 KEY: Tr = Return period, Pro= Probability of occurrence, xi = Rainfall depth 2.3 Descriptive Statistics of Annual Monthly Maximum Rainfall Series The Arithmetic Mean (οΏ½Μ…οΏ½), Standard Deviation (𝜎), Coefficient of Variation(𝐢𝑣), The coefficient of skewness (𝐢𝑠) and coefficient of Kurtosis (Ck) were calculated using Equations 3-7 respectively. file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:akpen.gabriel@uam.edu.ng Akpen et al: Best-Fit Probability Distribution Model for Annual Monthly Maximum Rainfall in Calabar City, Nigeria. AZOJETE, 18(2):281- 294. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng 285 οΏ½Μ…οΏ½ = 𝑋1+𝑋2+𝑋3+β‹―+𝑋𝑁 𝑁 = βˆ‘ 𝑋𝑖 𝑁 𝑖=1 𝑁 (3) 𝜎 = √ 1 π‘βˆ’1 βˆ‘ (𝑋𝑖 βˆ’ οΏ½Μ…οΏ½)2𝑁 𝑖=1 (4) 𝐢𝑣 = 𝜎 οΏ½Μ…οΏ½ (5) Cs = π‘š3 𝜎3 π‘œπ‘Ÿ 𝑁 βˆ‘ (π‘‹π‘–βˆ’οΏ½Μ…οΏ½)3𝑁 𝑖=1 (π‘βˆ’1)(π‘βˆ’2)𝜎3 given by Spiegel et al. (2011) (6) Ck = π‘š4 𝜎4 π‘œπ‘Ÿ 𝑁 βˆ‘ (π‘‹π‘–βˆ’οΏ½Μ…οΏ½)4𝑁 𝑖=1 (π‘βˆ’1)(π‘βˆ’2)𝜎4 given by Spiegel et al. (2011) (7) Where, N = total number of observed data, 50 in this case, 𝑋1, 𝑋2, 𝑋3, . . . , 𝑋𝑁 = monthly maximum rainfall data for each year, 𝑋𝑖 = observed monthly maximum rainfall data for a specified year i = specified year (i ranged from 1-50), π‘š3 = Third moment about the mean of the set of data, and π‘š4 = Fourth moment about the mean of the set of data. Table 2: Logarithms of Annual Monthly Maximum Rainfall in Calabar (1969 -2018) Year Month Rainfall Depth (𝑋𝑖) (mm) Rank (m) 𝑦𝑖 = log10 𝑋𝑖 Year Month Rainfall Depth (𝑋𝑖) (mm) Rank (m) 𝑦𝑖 = log10 𝑋𝑖 (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) 2012 Aug. 861.3 1 2.9352 2009 July 577.4 25 2.7615 2005 July 828.2 2 2.9181 1975 July 552.6 26 2.7424 1997 July 796.6 3 2.9012 1988 Sept. 538.3 27 2.7310 1980 Aug. 728.7 4 2.8625 1986 July 533.8 28 2.7274 1969 Aug. 726.9 5 2.8615 1976 Aug. 526.8 29 2.7216 1978 Sept. 721.2 6 2.8581 1985 May 520.5 30 2.7164 2014 July 714.4 7 2.8539 1981 July 519.7 31 2.7158 1990 July 702.7 8 2.8468 1972 July 510.8 32 2.7083 2015 June 678.7 9 2.8317 1982 July 505.7 33 2.7039 1971 July 661.4 10 2.8205 1991 Aug. 505.6 34 2.7038 2018 Aug. 650.6 11 2.8133 1983 June 504.6 35 2.7029 2011 July 648.6 12 2.8120 1998 June 504.5 36 2.7029 1995 July 632.4 13 2.8010 2013 May 499.9 37 2.6989 1989 July 626.5 14 2.7969 1999 Aug. 494.5 38 2.6942 2002 Aug. 623.5 15 2.7948 1987 Aug. 493.8 39 2.6936 1996 Sept. 615.2 16 2.7890 2001 May 491.4 40 2.6914 2017 Aug. 611.4 17 2.7863 2006 July 484.9 41 2.6857 2010 June 611.3 18 2.7863 1992 Sept. 481.4 42 2.6825 1994 July 609.5 19 2.7850 1993 Aug. 479.0 43 2.6803 1979 June 599.3 20 2.7776 2016 July 454.6 44 2.6576 2008 July 597.7 21 2.7765 1970 July 440.8 45 2.6442 http://www.azojete.com.ng/ mailto:akpen.gabriel@uam.edu.ng Arid Zone Journal of Engineering, Technology and Environment, June, 2022; Vol. 18(2):281-294. ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng 286 1977 July 597.6 22 2.7764 1984 June 437.3 46 2.6408 2000 July 597.6 22 2.7764 2003 Sept. 399.2 47 2.6012 1974 Sept. 588.4 23 2.7697 2004 Aug. 391.9 48 2.5932 2007 May 583.5 24 2.7660 1973 Aug. 373.3 49 2.5721 βˆ‘ 𝑋𝑖 28,835.5 βˆ‘ 𝑦𝑖 137.6703 The logarithms of all the annual monthly maximum rainfall depths were calculated using Equation 8 as shown in column (5) of Table 2, thus: 𝑦𝑖 = πΏπ‘œπ‘”π‘₯𝑖 (8) Where, 𝑦𝑖 = logarithmically transformed individual annual monthly maximum value of rainfall. 2.4 Determination of Best-Fit Probability Distribution Model The ranked data were evaluated with 6 probability distribution models namely: Normal (N), Log- Normal (LN), Pearson Type III (P3), Log-Pearson Type III (P3), Gumbel Extreme Value Type 1 (EV1) and Log-Gumbel (LG) probability distribution models. The best fit model was selected based on the results of goodness of fit tests and the application of a scoring and ranking scheme. Subsequently, the best fit model of the study area was used to predict rainfall depths for return periods between 2 and 1000 years. 2.4.1 Normal Probability Distribution Model The Normal distribution is also called Gaussian distribution or Normal Error Curve. The parameters of the distribution are mean and standard deviation while the Skewness is taken as zero. The estimation of rainfall (XT), for a given return period (Tr) was calculated using the following procedure: The mean (οΏ½Μ…οΏ½) of the observation was calculated using Equation 3, while the standard deviation (𝜎) was found using Equation 4. The standard normal variate (Z), corresponding to an exceedence probability, P (𝑃 = 1 π‘‡π‘Ÿ ) was calculated by finding the value of an intermediate variable W: π‘Š = [𝑙𝑛 ( 1 𝑃2 )] 1 2 (0 < 𝑃 ≀ 0.5) (9) The standard normal variate, z (which is the same as the value of frequency factor KT) that depends on return period for the Normal distribution was then found using Equation 10 (Chow et al., 1988): 𝑍 = w βˆ’ 2.515517+0.802853W+0.010328W2 1+1.432788W+0.189269W2+0.001308W3 (10) When P > 0.5; 1 – P is substituted for P in Equation 9. The estimated rainfall (XT) was calculated from Equation 11 (Chow et al., 1988): 𝑋𝑇 = οΏ½Μ…οΏ½ + π‘πœŽ = οΏ½Μ…οΏ½ + πΎπ‘‡πœŽ (11) file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:akpen.gabriel@uam.edu.ng Akpen et al: Best-Fit Probability Distribution Model for Annual Monthly Maximum Rainfall in Calabar City, Nigeria. AZOJETE, 18(2):281- 294. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng 287 2.4.2 Log-Normal (LN) Probability Distribution Model A random variable has a Log-normal distribution, if the log of the random variable has a normal distribution. The procedure adopted here was similar to that of Normal distribution except that it was applied to the logarithmically transformed observed rainfall depths. 2.4.3 Pearson Type III (P3) Distribution Model The Pearson Type III distribution has 3 parameters and is bound in the left with positive Skewness. The 3 parameters of the distribution are mean, standard deviation and skewness. The frequency factor depends on the return period and the coefficient of skewness, CS. When 𝐢𝑆 = 0, the frequency factor is equal to the standard normal variate Z. When 𝐢𝑆 β‰  0, the frequency factor (𝐾𝑇) was calculated using Equation 12 (Kite, 1977) expressed as: 𝐾𝑇 = 𝑍 + (𝑍2 βˆ’ 1)𝐾 + 1 3 (𝑍3 βˆ’ 6𝑍)π‘˜2 βˆ’ (𝑍2 βˆ’ 1)𝐾3 + 𝑍𝐾4 + 1 3 𝐾3 (12) where, K = 𝐢𝑆 6 . 2.4.4 Log Pearson Type III (LP3) Probability Distribution Model The mean (οΏ½Μ…οΏ½), standard deviation (πœŽβˆ—) and coefficient of skewness (𝐢𝑆 βˆ—) of the logarithmically transformed data were computed and the Pearson frequency factor (𝐾𝑇 ) was found from Equation 12. 2.4.5 Gumbel Extreme Value Type I (EV1) Probability Distribution Model The frequency factor (𝐾𝑇 ) for EV1 was calculated from Equation 13 (Chow, 1953) and the estimated rainfall, 𝑋𝑇 was then calculated using Equation 11. 𝐾𝑇 = βˆ’ √6 πœ‹ {0.5772 + 𝑙𝑛[𝑙𝑛( π‘‡π‘Ÿ π‘‡π‘Ÿβˆ’1 )]} (13) 2.4.6 Log-Gumbel (LG) Probability Distribution Model The mean (οΏ½Μ…οΏ½), standard deviation (πœŽβˆ—) and coefficient of skewness (𝐢𝑆 βˆ—) of the logarithmically transformed data were determined and the frequency factor (𝐾𝑇) was computed from Equation 13. For LN, LP3, and LG distributions, the actual estimated rainfall (XT), corresponding to log transformed estimated rainfall (YT), was computed using Equation 14 thus: 𝑋𝑇 = 10π‘Œπ‘‡ (14) 2.5 Goodness of Fit Test In order to check the adequacy of fit of the observed rainfall data to the six probability distribution models; four goodness of fit tests namely; Mean Absolute Deviation Index (MADI), Chi- squared(π‘₯2), Root Mean Square Error (RMSE) and Relative Root Mean Square Error (RRMSE) tests were adopted. http://www.azojete.com.ng/ mailto:akpen.gabriel@uam.edu.ng Arid Zone Journal of Engineering, Technology and Environment, June, 2022; Vol. 18(2):281-294. ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng 288 2.5.1 Mean Absolute Deviation Index (MADI) The mean absolute deviation index is the average distance between each data value and the mean. It was computed using the formula suggested by Agbonaye and Izinyon (2017): 𝑀𝐴𝐷𝐼 = 1 𝑁 βˆ‘ | π‘‹π‘—βˆ’π‘‹π‘‡π‘— 𝑋𝑗 |𝑁 𝑗=1 (15) Where, 𝑋𝑗 = The observed value 𝑋𝑇𝑗 =The value forecasted by the distribution 𝑁 = The number of data points The smaller the value of MADI obtained for a distribution, the more fitted it is to the observed data (Agbonaye and Izinyon, 2017). 2.5.2 Chi-Squared (π’™πŸ) Test This is a measure of the discrepancy existing between the observed and expected frequencies and is given by: π‘₯2 = (𝑋1βˆ’π‘‹π‘‡1)2 𝑋𝑇1 + (𝑋2βˆ’π‘‹π‘‡2)2 𝑋𝑇2 + (π‘‹π‘˜βˆ’π‘‹π‘‡π‘˜ )2 π‘‹π‘‡π‘˜ = βˆ‘ (π‘‹π‘—βˆ’π‘‹π‘‡π‘— )2 𝑋𝑇𝑗 π‘˜ 𝑗=1 (16) Where, 𝑋𝑗 = Observed frequency 𝑋𝑇𝑗 = Expected frequency 𝑗 =Number of observations (1, 2, . . . k) The smaller the value of π’™πŸ for a distribution, the more fitted it is to the observed data. The test was performed at 5% and 1% levels of significance. 2.5.3 Root Mean Square Error (RMSE) The root mean square error (RMSE) also known as the standard error is the sum of squares of the differences between observed and computed values. However, it was used to measure the difference between values predicted by a model and the observed values. These individual differences are called residuals. The RMSE was obtained using the formula suggested by Tao et al. (2002): 𝑅𝑀𝑆𝐸 = √( βˆ‘(π‘‹π‘—βˆ’π‘‹π‘‡π‘— ) 2 (π‘›βˆ’π‘š) ) (23) Where, 𝑋𝑗(𝑗 = 1, 2, … 𝑛) are observed values 𝑋𝑇𝑗 (𝑗 = 1, 2, … 𝑛) are the corresponding values computed from the assumed probability distribution, n is the number of observations and; m is the number of parameters estimated for the distribution. The smaller the value of RMSE obtained for a distribution, the more fitted it is to the observed data (Agbonaye and Izinyon, 2017). file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:akpen.gabriel@uam.edu.ng Akpen et al: Best-Fit Probability Distribution Model for Annual Monthly Maximum Rainfall in Calabar City, Nigeria. AZOJETE, 18(2):281- 294. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng 289 2.5.4 Relative Root Mean Square Error (RRMSE) The relative root mean square error was computed by dividing the root mean square error by the mean of observed data. The relative root mean square error provides a good picture of the overall fit of a distribution. It computes each error in proportion to the size of observation, thereby reducing the effect of outliers which are commonly found in hydrological data. The RRMSE was computed using the formula by Tao et al. (2002): 𝑅𝑅𝑀𝑆𝐸 = βˆšβˆ‘( π‘‹π‘—βˆ’π‘‹π‘‡π‘— 𝑋𝑗 ) 2 (π‘›βˆ’π‘š) (24) Where, 𝑋𝑗(𝑗 = 1,2, … 𝑛) = observed values 𝑋𝑇𝑗 (𝑗 = 1,2, … 𝑛) = the corresponding values computed from the assumed probability distribution, 𝑛 = number of observations, and; π‘š = the number of parameters estimated for the distribution. The smaller the value of RRMSE obtained for a distribution, the more fitted it is to the observed data (Agbonaye and Izinyon, 2017). 2.6 Scoring and Ranking Scheme Based on the results of the goodness of fit tests, a scoring scheme in which the best performing distribution with respect to a test criterion was assigned a score of 6, the next test was assigned 5 and the worst test assigned 1. The distribution with the highest total score in the study area based on the goodness of fit criteria was adjudged as the best distribution model for prediction of annual monthly maximum rainfall in the study area. 3. Results and Discussion 3.1 Annual Monthly Maximum Rainfall The annual monthly maximum rainfall depths ranged from 373.3 mm to 861.3 mm, indicating a very large range of fluctuation during the study period (Figure 2). The minimum and maximum rainfall values were in the month of August, 1973 and 2012 respectively. Incidentally, incidences of severe flooding were reported in Calabar in the year, 2012, the year with the maximum monthly rainfall. http://www.azojete.com.ng/ mailto:akpen.gabriel@uam.edu.ng Arid Zone Journal of Engineering, Technology and Environment, June, 2022; Vol. 18(2):281-294. ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng 290 Figure 2: Annual Monthly Maximum Rainfall (mm) for Calabar (1969 - 2018) The computed frequency factor (KT) values and the estimated rainfall (XT) values for different return periods of 2, 5, 10, 25, 50, 100, 200, 500 and 1000 years, are presented in Table 3. The results of the goodness of fit tests for the 6 distributions at the station were also presented in Table 4. Ranking the total scores, a summary of performance of the 6 probability distribution models was obtained as shown in Table 5. The Log-Normal distribution model, with a total score of 23 is adjudged the best fit model for the study area. However, Olofintoye et al. (2009) found Log-Pearson Type III probability distribution model as the best-fit model for the peak daily rainfall in Calabar. It follows that the probability distribution model used for monthly maximum rainfall in a given location may not necessary fit daily maximum rainfall for the same location. The Log- Normal model was then used to predict rainfall values shown in Table 2 for selected return periods of 2, 5, 10, 25, 50, 100, 200, and 500 and 1000 years. These rainfall estimates can provide useful guidance for policy makers and designers. 0 200 400 600 800 1000 1968 1978 1988 1998 2008 2018 M a x im u m R a in fa ll ( m m ) Years file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:akpen.gabriel@uam.edu.ng Akpen et al: Best-Fit Probability Distribution Model for Annual Monthly Maximum Rainfall in Calabar City, Nigeria. AZOJETE, 18(2):281-294. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng 291 Table 3: Computed Values of Frequency Factor (𝐾𝑇) and Rainfall Estimates (𝑋𝑇) given by Six (6) Probability Distributions for different Return Periods for Calabar Table 4: Goodness-of-Fit (GoF) Test Results for the Distributions at Calabar Station GoF Test Criteria Probability Distribution Models N LN P3 LP3 EV1 LG Calabar MADI 0.0099 0.0030 0.0040 0.0048 0.0029 0.0214 π‘₯2 57.1018 4.7976 8.7384 12.0788 7.3030 219.2021 RMSE 84.8856 25.5417 37.1754 43.5557 32.3466 201.4082 RRMSE 0.0854 0.0261 0.0376 0.0439 0.0339 0.2001 For the chi-squared (π‘₯2) test: At Ξ± = 0.05, degree of freedom = 8, and the critical value = 15.5 At Ξ± = 0.01, degree of freedom = 8, and the critical value = 20.1 Return Period (years) Normal (N) Log-Normal (LN) Pearson Type III (P3) Log-Pearson Type III (LP3) Gumbel (EV1) Log-Gumbel (EV1) 𝑲𝑻 𝑿𝑻(mm) 𝑲𝑻 𝑿𝑻(mm) 𝑲𝑻 𝑿𝑻(mm) 𝑲𝑻 𝑿𝑻(mm) 𝑲𝑻 𝑿𝑻(mm) 𝑲𝑻 𝑿𝑻(mm) 2 0.0000 576.7 0.0000 566.8 - 0.0878 567.1 0.0162 568.5 - 0.1642 558.8 -0.1642 549.5 5 0.8414 668.4 0.8414 664.0 0.8043 664.4 0.8456 664.7 0.7195 655.2 0.7195 649.1 10 1.2818 716.5 1.2818 721.6 1.3238 721.0 1.2697 719.9 1.3041 718.9 1.3041 724.6 25 1.7511 767.6 1.7511 788.1 1.9193 786.0 1.7161 783.1 2.0421 799.4 2.0421 832.5 50 2.0541 800.7 2.0541 834.4 2.3226 830.0 2.0015 826.2 2.5921 859.3 2.5921 923.4 100 2.3268 830.4 2.3268 878.4 2.7131 872.5 2.2552 866.8 3.1402 919.1 3.1402 1024.0 200 2.5762 857.6 2.5762 920.7 3.0757 912.1 2.4851 905.1 3.6806 978.0 3.6806 1133.7 500 2.8785 890.6 2.8785 974.8 3.5355 962.2 2.7612 953.5 4.3949 1055.9 4.3949 1296.9 1000 3.0905 913.7 3.0905 1014.4 3.8610 997.7 2.9531 988.6 4.9353 1114.8 4.9353 1435.8 http://www.azojete.com.ng/ mailto:akpen.gabriel@uam.edu.ng Arid Zone Journal of Engineering, Technology and Environment, June, 2022; Vol. 18(2):267-280. ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng 292 Table 5: Scoring and Ranking Scheme of the Statistical Tests Results for the Distributions at Calabar STATION PROBABILITY DISTRIBUTION MODELS GOODNESS-OF-FIT TEST CRITERIA TOTAL SCORE RANK MADI X2 RMSE RRMSE CALABAR N 2 2 2 2 8 5TH LN 5 6 6 6 23 1ST P3 4 4 4 4 16 3RD LP3 3 3 3 3 12 4TH EV1 6 5 5 5 21 2ND LG 1 1 1 1 4 6TH Table 6: Estimated Rainfall (XT) Values (mm) for various Return Periods given by the Best Fit Model for Calabar Station Best-Fit Model Return Period (Years) 2 5 10 25 50 100 200 500 1000 Calabar Log- Normal 566.8 664.0 721.6 788.1 834.4 878.4 920.7 974.8 1014.4 3.2 Rainfall Frequency Curve The annual maximum rainfall estimates for the best fit probability distribution model in Table 6, were used to develop the rainfall frequency curve as presented in Figure 3. More so, the yearly wise annual maximum monthly rainfall depths ranging from 373.3 mm to 861.3 mm during the study period are also presented in the Figure 3. The time series graph is important for showing how data changes over time. Figure 3: Rainfall Frequency Curve for Calabar (based on Log-Normal distribution model) 4. Conclusion Accurate extreme rainfall estimates provided by a suitable model are needed by individuals, authorities and engineers in their planning and designing of hydraulic structures. The results of 0 200 400 600 800 1000 1200 1 10 100 1000 M ax im u m R ai n fa ll (m m ) Return Period (Years) file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:akpen.gabriel@uam.edu.ng Akpen et al: Best-Fit Probability Distribution Model for Annual Monthly Maximum Rainfall in Calabar City, Nigeria. AZOJETE, 18(2):281- 294. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: akpen.gabriel@uam.edu.ng 293 this study, reveals that, Log-Normal distribution was the best-fit model for Calabar in the case of annual maximum monthly rainfall data. The rainfall frequency curve (Figure 3) shows the predicted rainfall values that were obtained from the best-fit model, which may serve as a guide in hydrological designs. This study has provided useful engineering design tool for planning and designing of hydraulic structures and other flood control and mitigation measures. It is recommended that the developed frequency curve (based on the best fit model) should be adopted for prediction of rainfall amount for design of hydraulic structures in the study area. References Agbonaye, AI. and Izinyon, OC. 2017. Best–Fit Probability Distribution Model for Rainfall Frequency Analysis of Three Cities in South Eastern Nigeria. 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