Regional distribution of intensity–duration–frequency (IDF) relationships in Sultanate of Oman

1 Introduction

Rainfall is a fundamental constituent of the hydrological cycle. Consequently, the determination of rainfall events is important for planning and designing of any hydrologic project including storm and drainage designs, geotechnical and structural projects, water resources systems, and others. However, the development of any hydrological project is highly challenging in the arid region where the rainfall is largely random and erratic both temporally and spatially (Al-Amri and Subyan, 2017). The changes in precipitation as the result of extreme weather events in the water-scarce arid countries is often facing long-term droughts and flash floods; damaging coastal, residential and agricultural areas and natural habitats in the arid region (Cosgrove and Loucks, 2015; Gunawardhana and Al-Rawas, 2016). The recent climate change is considered as one of the major challenges for water supply systems and flood risk analysis works (Ishak et al., 2013; Kourtis and Tsihrintzis, 2022). Thus, the quantification of rainfall is performed using intensity–duration–frequency (IDF) curve (Chow et al., 1988) as a tool, to implicate safe design and cost efficiency to the hydrologic and engineering projects for certain return period (Raiford et al., 2007). The site specific IDF is used to study the relationship between rainfall intensity, duration and frequency (or return period) associated with the site location and amenities (Chow et al., 1988).

The IDF relationship was first presented by Bernard (1932). Since then, different forms of IDF relationships have been established by many researchers in the field of engineering, hydrology and environmental studies in several regions of the world. IDF formula was developed by Bell (1969) and Chen (1983) for few regions of United State. Koutsoyiannis et al. (1998) constructed the mathematical framework of IDF curve using data from both rainfall recording and non-recording stations based on probability distribution. But, the probability distributions are considered as stationary and do not change with time (Vinnarasi and Dhanya, 2022). Whereas, the climate change is likely to alter the climatic extremes over the time, given the stationary IDF curve approach often underestimates the precipitation extremes (Cheng and AghaKouchak, 2014; Shrestha et al., 2017). Several studies using non stationary model using climate indices as covariates were explored in few studies (Li et al., 2015; Bracken et al., 2018; Silva et al., 2021; Vinnarasi and Dhanya, 2022). Raiford et al. (2007) updated the existing IDF curves for the different region of United States; and, acquired those curves at ungauged sites throughout the region using the newly developed rainfall frequency analysis methods. El-Sayed (2011) used iso-pluvial maps in Egypt; Awadallah et al (2011) used regional analysis and satellite data in Angola; Yu et al. (2004) used scaling theory in Taiwan; Ouali and Cannon (2018) used quantile regression technique in Canada for developing the regional IDF at the ungauged sites. Likewise, Noor (2022) proposed method for IDF curve construction with related uncertainty at the ungauged sites using bias correction of satellite rainfall data and its comparison with the observed IDF Curve.

Several researchers have developed the IDF curve for the arid Arabian Peninsula using both empirical formulae and frequency analysis. Various theoretical distribution functions (Generalized Extreme Values (GEV), Gumbel distribution, Generalized Pareto Distribution (GPD), Log Pearson Type III, Log Normal, Exponential distribution and others are normally used in frequency analysis (Sherif et al., 2014; Forestieri et al., 2018; M. Bermúdez et al., 2020). Al Shaikh (1985), and Al Areeq et al. (2021) used Gumbel distribution for development of IDF curve in various region of Saudi Arabia. Elsebaie (2012), Al-Amir and Subyan (2017) used both Gumbel and Log Pearson III distribution; while, AlHassoun (2011), (Al-anazi and El-Sebaie, 2013) used Gumbel, Log Pearson III, and Log Normal distribution for development of IDF curve in several locations of Saudi Arabia. There studies did not show much difference in the rainfall analysis of IDF curve for Gumbel and Log Pearson Type III distribution for the semiarid and flat topographic region.

The development of IDF relationship requires the long-term and continuous historical rainfall data, which is typically not available in most semiarid and arid region countries including Sultanate of Oman. Also, the precipitation events in the region are rare but usually of high intensities in the short duration, resulting in flash floods in an inter-annual scale (Uraba et al., 2019; Aldosari et al., 2020). Precipitation frequency analysis is equally important in nonstructural problems concerning natural risks related with ultimate rainfall events (Maidment, 1993). The data required to compute IDF curves are a record of rainfall depth measurements during fixed intervals of time, normally 5 min intervals (Mays, 2005). Thus, the coarse-resolution precipitation data is converted into the fine time-resolution precipitation using the temporal disaggregation technique (Al-Wagdany, 2021).

Very limited studies in rainfall analysis and climate change projection has been conducted for Oman. Awadallah (2017) designed a storm hyetograph of few stations located at the Northern coastal zone of Oman using Alternating Block Method (ABM)-IDF Curve method. While, Uraba (2019) has developed the IDF curve for Tawi-Atair station in Dhofar region of Southern Oman using two-stage downscaling disaggregation approach. Thus, the IDF relationships are not available for most of the regions in Oman. In the absence of a properly developed IDF relationship, the planning and development of water resources systems such as recharge dams, flood protection structures, storm water collection networks and other projects may result in improper design. Also, the developed IDF curves can be adopted to quantify the rainfall rate and predict the flooding for any region. Therefore, this research aims to develop IDF curves for the Oman, by analyzing rainfall data at multiple meteorological stations situated at different elevations and regions throughout the country using Gumbel distribution. The empirical formulae were also developed to evaluate rainfall intensity for several rainfall durations and return periods. Further, the contour maps for all the observed parameters were drawn to establish the IDF relationships for the ungauged location for future predictions and designs.

2 Study area and data collection

Oman (Sultanate of Oman) lies in the southeastern corner of the Arabian Peninsula with total area of 309,500 km² and a coastline 3165 km long. It extends between latitudes of 16°40′ and 26°20′ N and longitudes of 51°50′ and 59°40′ E. The country is bordered by Arabian Gulf in the North, Sea of Oman in the East, Arabian Sea in the Southwest, United Arab Emirates and Saudi Arabia in the West, and Yemen in the Southwest as shown in Fig. 1 (FMO, 2022). Topographically, it is divided into three areas; the coastal plain (fertile plain) extends from Al Batinah Plain in the North to Salalah Plain in South, mountainous region runs from Musandam in the North to Ras Al-Hadd in the Southeast and in Dhofar province, and the internal region (desert, gravel and sand plain) ranges from the coastal plain to the mountainous region covering 82% area of the country (FAO, 2021).

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Fig. 1

Study Area with governorate and studied stations (FMO, 2022).

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Based on rainfall, Oman experiences the hyper arid (less than100 mm rain), arid (100–250 mm rain) and semiarid (250–500 mm rain) climate at various parts of the country (Kwarteng et al., 2009). The long-term average annual rainfall of the country has been estimated to be 62 mm (MRMWR, 2013). Average annual rainfall in the desert and coastal plain is less than 50 mm; while the rainfall in mountain region is up to 350 mm, and is relatively frequent providing recharge to the aquifers situated at the coastal and interior plains (Al Barwani, 2014). Seasonal summer monsoon is observed from June to September in southern parts of the country, especially in Dhofar Governorate causing change in temperature. Whereas, the rainfall occurs during winter from November to April in the northern and central region of the country (FAO, 2021). In summer, the weather is hot and humid in the coastal region, hot and dry in interior region; while, the weather is moderate and rainy throughout the year in the highlands (FAO, 2021). Rainfall in the country is associated by four principal mechanism; convection rainstorms related with localized strong convection developed mostly in summer, cold front trough from Mediterranean Sea or North Atlantic that brings rainfall to northern part of Oman especially from November to April, tropical cyclones originated from Arabian Sea typically from May to June and October to November, and on-shore southwesterly monsoon current that causes humid environment and brings frequent drizzle, mist, fog, rain in Dhofar region from June to September (Roberts and Wright, 1993; MWR, 1995).

In this study, the rainfall data were obtained from the Ministry of Regional Municipalities and Water Resources (MRMWR). Oman is divided into eleven governates, namely; Musandam, Al-Buraimi, Al-Batinah North, Al Batinah South, Muscat, Adh Dhahirah, Ad-Dakhaliya, Al-Sharqia North, Ash-Sharqia South, Al-Wusta, and Dhofar. Sixty-five monitoring stations are selected to cover the ten governates in the country. Al Wusta govenerate, the desert area is not considered in the study due to lack of enough stations and data for analysis. The selected stations had homogenous and short time interval record for longer period. The record length varies from one station to another, which was because of some missing years of data and initial years of gauge installation at the sites. The record periods were between 20 years to 40 years of rainfall data. The location details and years of record of stations that were used for this study are shown in Table 1 and Fig. 1.

3 Methodology and data analysis

Estimation of IDF curves involved various steps. Initially the rainfall data were analyzed and disaggregated for the shorter period. Collected data from monitoring stations were initially sorted according to the years, rainfall depth and duration. Disaggregation of rainfall data to shorter and regular period was done using the Hydrologic Engineering Center’s (HEC) Data Storage System Visual Utility Engine (HEC-DSSVue) software. Maximum rainfall depth is acquired for each monitored year and various durations. Statistical analysis such as mean and standard deviation of the maximum rainfall depth were also obtained for various durations.

Development of IDF is performed by fitting the probability distribution function to extreme rainfall data for specific durations. Based on measurements and fitted relationship, the rainfall intensity over specific duration and return period are determined for the recorded years. Gumbel distribution is used for frequency analysis of the annual maximum rainfall data for the calculation of rainfall depth for each return period. Design durations of 5, 15, 30, 60, 360, 720, 1440 min, and return periods of 2, 5, 10, 25, 50 and 100 years were used in the present study. Further, the parameters of the IDF relationship proposed by Bernard (1932) are obtained using regression. Finally, the contour maps for all the parameters were drawn using SURFER software for determining the IDF relationships for ungauged locations.

4 Result and discussion

4.1

4.1 Rainfall analysis

Statistical analysis of the annual rainfall (total rainfall) for 65 study location of Oman from year 1977 to 2017 is shown in Table 2. All the monitoring stations have demonstrated highly variable annual rainfall over the study period. Both annual maximum rainfall of 806.29 mm (Wadi Al Koudh in year 2004) and minimum rainfall of 9.39 mm (Ruwi in year 2003) were observed in Muscat governorate. The Skewness coefficient and Kurtosis coefficient measures the asymmetry and peakedness or flatness of the frequency distribution of the data (Sheskin, 2000). Negative kurtosis and positive kurtosis values indicate the distribution is flatter and sharper in its center than the normal distribution, respectively. Kurtosis coefficient is in the range of −1.429 to 8.167 during the study period that shows higher occurrence of probability near the mean than that in the normal distribution. Skewness coefficient in the range of 0.1 to 2.66 was observed for the stations. The observed positive skewed distribution exhibits much less frequency of occurrence of higher intensity rainfalls and the high frequency of occurrence of annual rainfall below the mean value (see Table 2).

Table 2 Statistical information of annual rainfall (mm) at the monitoring stations.

Governorate	Station Name	Maximum (mm)	Minimum (mm)	Average (mm)	Standard Deviation (mm)	Skewness Coefficient	Kurtosis Coefficient
Musandam	Ghamda	536.15	15.79	165.11	125.24	1.14	0.99
	Khasab	426.43	10.11	136.54	109.98	1.07	0.60
	Rhaibah	515.36	17.73	185.22	126.61	0.75	0.15
	Sal Ala	489.34	17.64	123.25	110.69	1.80	3.42
	Sima	438.03	13.20	128.53	99.02	1.38	1.99
Al-Buraimi	Al-Juwayf	344.87	23.23	99.97	80.19	1.88	3.87
	Al-Ubaylah	302.11	10.04	69.81	72.73	2.01	4.51
	Fayyad	270.44	15.21	96.84	63.02	1.20	1.51
	Khatwah	274.45	17.33	48.30	66.00	2.66	7.19
	Mahdah	285.10	18.80	95.37	70.89	0.88	0.34
	Wadi Salmah	224.97	19.43	64.63	48.30	1.47	2.79
	Wadi Sharm	269.08	10.07	80.40	63.86	1.17	1.45
Al Batinah North	Al-Ghuzayfah	326.91	9.66	88.61	74.73	1.99	3.91
	Al-Jizzi	203.23	16.35	74.05	51.45	1.09	0.44
	Aqair Al-Abreein	415.50	13.53	110.78	108.96	1.61	2.35
	Aqbat Al-Risah	319.66	14.16	92.97	84.92	1.45	1.45
	Hayl Al-Najd	486.00	20.03	102.72	106.91	2.63	6.62
	Saham	275.56	11.35	78.74	64.28	1.31	1.88
Al-Batinah South	Al-Miseen	460.37	13.63	127.50	100.06	1.74	4.46
	Al-Wasit	252.78	15.93	76.01	69.18	1.38	0.64
	Ar-Rustaq	297.25	12.36	103.06	79.76	0.94	0.16
	Barka	178.98	13.22	66.79	37.00	1.29	2.23
	Dhabaah	558.76	19.58	155.03	111.35	1.58	3.98
	Salma	374.25	11.73	161.92	111.20	0.23	−1.09
Muscat	Buei	469.21	18.44	124.22	125.00	1.53	1.83
	Hayfadh	672.08	15.02	83.36	150.11	2.37	6.12
	Mazara 3	526.25	13.65	136.92	121.18	1.77	3.22
	Muscat	271.03	10.95	70.67	68.34	1.93	3.16
	Ruwi	346.40	9.39	111.26	79.25	0.96	1.14
	Wadi Al-Jannah	166.60	39.83	62.81	25.60	2.61	7.07
	Wadi Al-Khawd	860.29	17.83	120.66	213.61	2.52	8.17
Adh-Dhahirah	Dakarah	410.22	14.60	108.80	96.80	1.93	4.25
	Dank	331.43	10.81	69.40	74.16	2.18	5.34
	Dhahir	368.31	10.04	92.39	90.38	1.74	3.60
	Kubarah	200.82	16.27	61.64	50.09	1.40	2.23
	Majzi	536.76	18.19	105.32	130.82	2.30	5.88
	Qarn Al-Kabsa	282.05	19.28	74.03	71.26	1.65	2.66
	Tanam 2	196.20	13.56	61.14	50.34	1.14	0.59
Ad-Dakhliyah	Al-Qusaiba	169.64	12.11	38.41	42.72	1.99	4.10
	Jiwar	249.47	18.07	64.37	62.74	1.43	2.39
	MOD	734.87	23.96	180.21	155.67	2.15	6.31
	Musbit	279.87	11.68	78.73	66.20	1.93	3.99
	Najd Al-Musallah	331.48	10.20	104.79	81.14	1.14	1.39
	Subayb	655.50	27.50	307.10	225.47	0.16	−1.43
	Tawi Zahir	497.43	11.99	142.37	109.00	1.85	4.79
Ash-Sharqiya North	Ad-Dariz	334.47	17.08	72.57	81.71	1.97	4.80
	Al-Mudaybi	331.62	12.66	73.02	79.98	2.05	4.66
	Al-Muqayhfah	334.47	16.08	73.07	81.55	1.96	4.81
	Haimah	392.14	11.25	127.87	110.70	1.47	1.28
	Ibra	382.48	10.49	113.17	96.76	1.47	1.49
	Masroon	344.81	12.57	80.05	76.89	2.46	7.21
	Wadi Bani Khalid	514.33	13.24	129.17	126.03	1.82	3.41
Ash-Sharqiya South	Al-Fuljayj	442.46	10.65	84.90	95.27	2.39	6.80
	Fins	306.30	14.86	93.68	78.06	1.22	1.02
	Jaalan Bani	216.87	20.09	64.64	49.94	2.51	5.93
	Jabal Bani Jabir	765.23	36.53	190.45	165.53	1.97	5.07
	Snaf	368.80	20.70	107.10	106.68	1.31	0.75
	Tahwah 3	442.46	12.36	84.69	91.69	2.52	7.57
Dhofar	Aqarhanawt	438.00	19.01	141.98	125.26	1.30	0.83
	Ghadow	676.93	22.05	136.92	159.37	2.01	5.03
	Hagayf	411.15	11.95	108.61	102.48	2.03	3.98
	Mughsayl	173.97	13.52	83.34	45.09	0.10	−0.78
	Sadh	156.41	33.21	66.45	34.20	1.38	1.35
	Sher	394.35	14.20	154.10	126.92	0.76	−0.56
	Zayk 1	340.50	10.46	98.05	73.22	1.09	2.73
All Stations		860.29	7.83	109.21*	92.82*	1.62*	3.08*

Note: * represents the average values.

Table 3 Summary of best fit distribution at various return periods for Wadi Al Jannah station.

Distribution	Return Period
	2	5	10	25	50	100
	Coefficient of Determination (R2)
Gumbel	0.998	0.995	0.992	0.992	0.992	0.991
Log-Pearson Type III	0.999	0.997	0.995	0.991	0.988	0.985

The average annual rainfall exhibited for all studied stations from 1977 to 2017 is 109.21 mm with a standard deviation of 92.82 mm, Skewness coefficient of 1.62 and Kurtosis coefficient of 3.08. The observed highest average annual rainfall is 307.1 mm at Subayb, followed by 190.45 mm at Jabal Bani Jabir, 185.22 mm at Rhaibah and, 180.21 at MOD. While the lowest average of 38.41 mm at Al Qusaiba, succeeded by 48.30 mm at Khatwah, 61.15 mm at Tanam, and 61.64 mm at Kubarah. Interestingly, both highest and lowest average annual rainfall was recorded in Ad-Dakhliyah goveronate. Geographically, Subayb, is in the mountainous range at 1345 m elevation, while Al-Qusaiba station is situated in the flat terrain closer to the Al-Wusta desert region at 373 m elevation. Thus, the study shows that the rainfall in the mountainous region is high compared to the desert and the coastal region of the country. Fig. 2 presents the box and whisker plot of annual rainfall at studied governorates during monitoring period. The dataset exhibits the upward or positive skewness in all governorate with average rainfall higher than the median rainfall.

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Fig. 2

Box and Whisker plot showing annual rainfall at different governorate during study period.

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Variation of average annual rainfall (total rainfall) in various governorates of Oman from 1986 to 2016 is shown in Fig. 3. The highest average of 432.74 mm was recorded in 1997 in Musandam governorate located at the Northern Oman. The years 1990 (382.22 Ash-Sharqiya North), 1997 (360.21 at Ad-Dakhliyah), 2007 (361.21 mm at Muscat), and 2010 (347.37 at Ash-Sharqiya South) also recorded the high average rainfall. The lowest average of 20.64 mm was observed in the year 2008 in Ash-Sharqiya North. Similarly, lowest averages were recorded in the years 2001 (20.68 mm at Al-Buraimi), 2008 (21.09 at Ash-Sharqiya North), 2001 (21.28 at Adh-Dhahirah), and 1985 (23.30 mm Muscat). For the study period of 1986 to 2016, a slightly negative trend in average annual rainfall of −1.195 mm/years was observed for overall stations (Fig. 3). Among the studied region, Musandam governorate in northern part of Oman has the highest annual average of 149.81 mm followed by Mountainous area Ad-Dakhaliyah (130.06 mm). The annual average of 113.54 mm, 111.54 mm, 108.02 mm, and 102.54 mm were observed in Dhofar, Al-Batinah South, Ash-Sharqia North and Muscat governorates. While the lowest annual averages were observed at Ash-Sharqiyah North (95.56 mm), Al-Batinah North (91.84 mm), Adh-Dhahirah (83.04 mm) and Al-Buraimi (82.838 mm).

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Fig. 3

Variation of average annual total rainfall in various governorate.

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4.2

4.2 Intensity-Duration-Frequency (IDF) relationships

Gumbel and Log Pearson Type III distribution were mostly used distribution in arid region in IDF calculation. So, initially both distributions were used at Wadi Al Jannah station to check the best distribution using maximum rainfall records. Rainfall intensities for all the stations is estimated for corresponding rainfall duration (5, 15, 30, 60, 360, 720 and 1440 min) and return periods (2, 5, 10, 25, 50, and 100). Fig. 4 shows the observed and modeled intensity of two distribution; while Table 3 presents the summary of best fit result between two distribution using coefficients of determination (R²) using Eq. (17) at various return periods. Both models showed the good correlation with values greater than 0.9 at all return period. As best fit result did not showed any major difference two distribution, Gumbel distribution is further used in all the stations in this study.

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Fig. 4

Observed and Modeled rainfall intensity at Wadi Al Jannah Station using A) Gumbel and B) Log Pearson Type III distributions at various return periods (2, 5, 10, 25, 50, 100 years).

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Table 4 and Fig. 5 shows the calculated rainfall intensities at Rhibah, Aqbat Al-Risah, Wadi Al-Jannah, Subayb, and Mughsayl stations for return periods of 2, 5, 10, 25 and 100 years using Gumbel distribution. The estimation showed the rainfall intensities increased with the return period, while the intensities decreased with the increase in the rainfall duration at all the stations. Subayb station at Ad-Dakhliah, the site with higher annual rainfall is likely to experience high rainfall with longer duration and return period compared to other stations.

Table 4 Rainfall Intensity (mm/hr) at Rhibah, Aqbat Al Risah, Wadi Al Jannah, Subayb and Mughsayl station at various duration and return period using Gumbel Distribution.

Station (Governorate)	Elevation (m)	Rainfall Duration, d (hr)	Return Period, T (Years)
			2	5	10	25	50	100
			Frequency Factor, K
			−0.1644	0.7198	1.3052	2.0449	2.5936	3.1383
			Rainfall Intensity-I (mm/hr)
Rhibah (Musandam)	704	0.833	36.04	69.90	92.32	120.65	141.67	162.53
		0.25	21.65	41.04	53.88	70.11	82.15	94.10
		0.5	16.58	29.19	37.54	48.09	55.91	63.68
		1	11.92	19.54	24.59	30.97	35.71	40.40
		360	3.58	6.16	7.87	10.03	11.63	13.22
		720	2.50	3.97	4.94	6.17	7.08	7.98
		1440	1.71	2.60	3.19	3.93	4.48	5.02
Aqbat Al-Risah (Al-Batinah North)	516	0.833	44.34	74.18	93.93	118.89	137.41	155.78
		0.25	26.36	43.81	55.37	69.98	80.81	91.57
		0.5	17.51	30.73	39.48	50.54	58.74	66.88
		1	11.14	19.69	25.35	32.50	37.80	43.07
		360	2.32	4.05	5.20	6.66	7.74	8.80
		720	1.38	2.27	2.86	3.60	4.16	4.71
		1440	0.87	1.39	1.72	2.15	2.47	2.78
Wadi Al-Jannah (Muscat)	220	0.833	18.78	39.97	54.00	71.72	84.87	97.92
		0.25	8.87	19.80	27.04	36.19	42.97	49.70
		0.5	5.94	12.07	16.13	21.26	25.06	28.84
		1	3.65	7.12	9.42	12.33	14.48	16.62
		360	0.89	1.63	2.12	2.74	3.20	3.65
		720	0.56	1.00	1.30	1.67	1.95	2.22
		1440	0.39	0.81	1.09	1.45	1.71	1.97
Subayb (Ad- Dakhaliyah)	1345	0.833	116.65	167.11	200.52	242.73	274.05	305.13
		0.25	74.58	114.22	140.47	173.63	198.23	222.64
		0.5	51.15	80.74	100.32	125.07	143.43	161.66
		1	30.39	47.92	59.52	74.18	85.05	95.85
		360	8.06	13.32	16.80	21.19	24.46	27.69
		720	4.43	7.60	9.70	12.35	14.32	16.27
		1440	2.60	4.5	5.76	7.35	8.53	9.70
Mughsayl (Dhofar)	25	0.833	15.29	35.23	48.42	65.10	77.47	89.75
		0.25	8.24	18.91	25.97	34.89	41.51	48.08
		0.5	5.69	12.50	17.01	22.71	26.93	31.13
		1	3.94	7.71	10.21	13.36	15.70	18.02
		360	1.10	2.20	2.93	3.86	4.54	5.22
		720	0.76	1.44	1.90	2.47	2.90	3.32
		1440	0.51	0.92	1.19	1.53	1.78	2.03

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Fig. 5

Rainfall intensity at different durations and return periods at Rh (Rhibah), Aq (Aqbal Al-Risah), Wa (Wadi Al Jannah), Su (Subayb), and Mu (Mughsayl) stations.

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Also, higher intensity rainfall at various return periods were witnessed at the higher elevation stations compared to the lower elevation stations. Among the stations presented in Table 4 and Fig. 5, Subayb (elevation 1345 m) has the highest rainfall while Mughsayl (elevation 25 m) has the lowest rainfall for all the return periods as compared to the other stations. Similar observations were reported by Kotoub (2004), where the rational method was used to evaluate the rainfall intensities at various return period for Plain, Hills and Mountains region of Oman, for flood peak and wadi characteristic studies for road network development. These studies showed among three studied regions of Oman, Mountains have the higher intensities rainfall followed by Hills, and Plains has the lowest intensities rainfall for various return periods. Thus, in the mountainous region the estimated rainfall intensities for various return periods are high as compared to the desert or interior region, and the coastal region of the country (Table 4; Fig. 5).

4.3

4.3 Intensity-Duration-Frequency (IDF) equation

Estimation of the empirical parameters (C, m, and e) of IDF relationship (Eq. (6) was done using nonlinear regression analysis in Microsoft Excel. Goodness of fit between observed and estimated data was checked using R² values. IDF curve for Khasab, Wadi Salmah, Saham, Dhabaah, Wadi Al-Jannah, Dank, Subayb, Ibra, Tahwa and Mughsayl stations presented in log scale are shown in Fig. 4. Also, the IDF curves are parallel to each other (Fig. 6). Table 5 shows the estimated IDF parameter values, IDF equation with R² achieved by IDF data analysis. The empirical parameter values for C ranged from 417.5 to 8.95, m ranged from 0.645 to 0.196, and e ranged from 0.79 to 0.391 for the studied stations. The obtained results showed good correlation between the observed and estimated rainfall intensities with high R² ranging between 0.994 and 0.851 (Table 5). Therefore, the IDF curved generated at the stations could be further used in the rainfall estimation and in design of water related projects in Oman.

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Fig. 6

IDF curve for Khasab, Wadi Salmah, Saham, Dhabaah, Wadi Al-Jannah, Dank, Subayb, Ibra, Tahwa, and Mughsayl station.

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Table 5 IDF parameter and equation with coefficient of determination for studied stations.

Governorate	Station Name	IDF Parameters			IDF Formula ( $\mathit{I}=\frac{\mathit{C}{{\mathit{T}}_{\mathit{R}}}^{\mathit{m}}}{{\mathit{d}}^{\mathit{e}}})$	Coefficient of determination (R2)
		C	m	e
Musandam	Ghamda	73.3	0.435	0.641	$I=\frac{73.3{T_R}^{0.435}}{d^{0.641}}$	0.914
	Khasab	84.6	0.377	0.648	$I=\frac{84.6{T_R}^{0.377}}{d^{0.648}}$	0.931
	Rhaibah	71.4	0.405	0.599	$I=\frac{71.4{T_R}^{0.405}}{d^{0.599}}$	0.930
	Sal Ala	24.0	0.645	0.596	$I=\frac{24{T_R}^{0.645}}{d^{0.596}}$	0.861
	Sima	33.6	0.580	0.614	$I=\frac{33.6{T_R}^{0.58}}{d^{0.614}}$	0.870
Al-Buraimi	Al-Juwayf	144.6	0.319	0.713	$I=\frac{144.6{T_R}^{0.319}}{d^{0.713}}$	0.994
	Al-Ubaylah	111.5	0.317	0.713	$I=\frac{111.5{T_R}^{0.317}}{d^{0.713}}$	0.948
	Fayyad	168.0	0.279	0.685	$I=\frac{168{T_R}^{0.279}}{d^{0.685}}$	0.990
	Khatwah	27.8	0.390	0.575	$I=\frac{27.9{T_R}^{0.39}}{d^{0.575}}$	0.926
	Mahdah	65.4	0.403	0.627	$I=\frac{65.4{T_R}^{0.403}}{d^{0.627}}$	0.934
	Wadi Salmah	75.1	0.398	0.672	$I=\frac{75.1{T_R}^{0.398}}{d^{0.672}}$	0.924
	Wadi Sharm	145.5	0.371	0.691	$I=\frac{145.5{T_R}^{0.371}}{d^{0.691}}$	0.932
Al-Batinah North	Al-Ghuzayfah	9.0	0.347	0.413	$I=\frac{9{T_R}^{0.347}}{d^{0.413}}$	0.937
	Al-Jizzi	130.6	0.444	0.746	$I=\frac{130.6{T_R}^{0.444}}{d^{0.746}}$	0.915
	Aqair Al-Abreein	70.9	0.416	0.618	$I=\frac{70.9{T_R}^{0.416}}{d^{0.618}}$	0.918
	Aqbat Al-Risah	166.1	0.325	0.734	$I=\frac{166.1{T_R}^{0.325}}{d^{0.734}}$	0.946
	Hayl Al-Najd	229.8	0.313	0.740	$I=\frac{229.8{T_R}^{0.313}}{d^{0.74}}$	0.851
	Saham	70.9	0.416	0.618	$I=\frac{70.9{T_R}^{0.416}}{d^{0.618}}$	0.918
Al-Batinah South	Al-Miseen	168.9	0.301	0.651	$I=\frac{168.9{T_R}^{0.301}}{d^{0.651}}$	0.953
	Al-Wasit	137.6	0.350	0.683	$I=\frac{137.6{T_R}^{0.35}}{d^{0.683}}$	0.939
	Ar-Rustaq	129.5	0.360	0.664	$I=\frac{129.5{T_R}^{0.36}}{d^{0.664}}$	0.936
	Barka	90.1	0.566	0.742	$I=\frac{90.1{T_R}^{0.566}}{d^{0.742}}$	0.963
	Dhabaah	82.5	0.437	0.643	$I=\frac{82.5{T_R}^{0.437}}{d^{0.643}}$	0.913
	Salma	176.4	0.311	0.652	$I=\frac{176.4{T_R}^{0.311}}{d^{0.652}}$	0.949
Muscat	Buei	157.8	0.324	0.577	$I=\frac{157.8{T_R}^{0.324}}{d^{0.577}}$	0.957
	Hayfadh	74.8	0.216	0.391	$I=\frac{74.8{T_R}^{0.216}}{d^{0.391}}$	0.991
	Mazara	99.0	0.303	0.743	$I=\frac{99{T_R}^{0.303}}{d^{0.743}}$	0.960
	Muscat	42.8	0.452	0.658	$I=\frac{42.8{T_R}^{0.452}}{d^{0.658}}$	0.922
	Ruwi	167.0	0.307	0.644	$I=\frac{167{T_R}^{0.307}}{d^{0.644}}$	0.990
	Wadi Al-Jannah	58.3	0.412	0.726	$I=\frac{58.3{T_R}^{0.412}}{d^{0.726}}$	0.920
	Wadi Al-Khawd	109.2	0.453	0.568	$I=\frac{109.2{T_R}^{0.453}}{d^{0.568}}$	0.949
Adh-Dhahirah	Dakarah	205.1	0.211	0.654	$I=\frac{205.1{T_R}^{0.211}}{d^{0.654}}$	0.977
	Dank	37.4	0.430	0.612	$I=\frac{37.4{T_R}^{0.43}}{d^{0.612}}$	0.914
	Dhahir	170.9	0.266	0.703	$I=\frac{170.9{T_R}^{0.266}}{d^{0.703}}$	0.961
	Kubarah	175.2	0.373	0.775	$I=\frac{175.2{T_R}^{0.373}}{d^{0.775}}$	0.932
	Majzi	193.8	0.304	0.733	$I=\frac{193.8{T_R}^{0.304}}{d^{0.733}}$	0.951
	Qarn Al-Kabsa	97.6	0.458	0.694	$I=\frac{97.6{T_R}^{0.458}}{d^{0.694}}$	0.914
	Tanam	89.7	0.384	0.683	$I=\frac{89.7{T_R}^{0.384}}{d^{0.683}}$	0.939
Ad-Dakhliyah	Al Qusaiba	177.0	0.299	0.708	$I=\frac{177{T_R}^{0.299}}{d^{0.708}}$	0.959
	Jiwar	175.6	0.398	0.790	$I=\frac{175.6{T_R}^{0.398}}{d^{0.790}}$	0.924
	MOD	309.4	0.196	0.681	$I=\frac{309.4{T_R}^{0.196}}{d^{0.681}}$	0.980
	Musbit	381.0	0.248	0.653	$I=\frac{381{T_R}^{0.248}}{d^{0.653}}$	0.973
	Najd Al-Musallah	251.6	0.332	0.753	$I=\frac{251.6{T_R}^{0.332}}{d^{0.753}}$	0.990
	Subayb	417.5	0.249	0.654	$I=\frac{336.3{T_R}^{0.249}}{d^{0.654}}$	0.972
	Tawi Zahir	263.3	0.219	0.685	$I=\frac{26.3{T_R}^{0.219}}{d^{0.685}}$	0.994
Ash Sharqiya North	Ad-Dariz	180.1	0.211	0.606	$I=\frac{180.1{T_R}^{0.211}}{d^{0.606}}$	0.956
	Al-Mudaybi	195.1	0.218	0.676	$I=\frac{195.1{T_R}^{0.218}}{d^{0.676}}$	0.994
	Al-Muqayhfah	179.8	0.294	0.696	$I=\frac{179.8{T_R}^{0.294}}{d^{0.696}}$	0.956
	Haimah	265.8	0.237	0.685	$I=\frac{265.8{T_R}^{0.237}}{d^{0.685}}$	0.969
	Ibra	144.7	0.425	0.740	$I=\frac{144.7{T_R}^{0.425}}{d^{0.74}}$	0.914
	Masroon	114.1	0.407	0.741	$I=\frac{114.1{T_R}^{0.407}}{d^{0.741}}$	0.955
	Wadi Bani Khalid	279.0	0.238	0.670	$I=\frac{279{T_R}^{0.238}}{d^{0.67}}$	0.973
Ash-Sharqiya South	Al-Fujayj	102.1	0.352	0.583	$I=\frac{102.1{T_R}^{0.352}}{d^{0.583}}$	0.940
	Fins	168.1	0.245	0.631	$I=\frac{168.1{T_R}^{0.245}}{d^{0.631}}$	0.969
	Jaalan Bani	45.9	0.242	0.569	$I=\frac{45.9{T_R}^{0.242}}{d^{0.569}}$	0.970
	Jabal Bani Jabir	105.1	0.397	0.580	$I=\frac{105.1{T_R}^{0.397}}{d^{0.580}}$	0.924
	Snaf	148.6	0.214	0.547	$I=\frac{148.6{T_R}^{0.214}}{d^{0.547}}$	0.985
	Tahwah	94.9	0.345	0.576	$I=\frac{94.9{T_R}^{0.345}}{d^{0.576}}$	0.943
Dhofar	Aqarhanawt	37.6	0.584	0.552	$I=\frac{37.6{T_R}^{0.584}}{d^{0.552}}$	0.968
	Ghadow	25.8	0.632	0.566	$I=\frac{25.8{T_R}^{0.632}}{d^{0.566}}$	0.929
	Hagayf	71.4	0.405	0.548	$I=\frac{71.4{T_R}^{0.405}}{d^{0.548}}$	0.912
	Mughsayl	42.8	0.452	0.658	$I=\frac{42.8{T_R}^{0.452}}{d^{0.658}}$	0.908
	Sadh	90.4	0.268	0.651	$I=\frac{90.4{T_R}^{0.268}}{d^{0.651}}$	0.968
	Sher	114.1	0.407	0.645	$I=\frac{114.1{T_R}^{0.407}}{d^{0.645}}$	0.921
	Zayk	52.4	0.507	0.635	$I=\frac{52.4{T_R}^{0.507}}{d^{0.635}}$	0.887

4.4

4.4 Empirical IDF parameter contours

The spatial distribution maps of the IDF parameters C, m, and e are shown in Fig. 7. The contours show the smooth variation of parameter over the whole country. But, Due to the absence of monitoring points in the Al-Wusta region, contour lines generated in the middle section of the country are based on the interpolation of data from the neighboring regions. Also, the data from adjacent countries Saudi Arabia and United Arab Emirates were included. Thus, the contour lines extending beyond the Oman’s border were based on the interpolation only.

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Fig. 7

Spatial distribution contour map of empirical IDF parameters (a) C, (b) m, and (c) e.

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All parameter values are relatively high and more condensed in the Northern part of the country compared to the Southern part. Particularly, the high values were observed along the Al-Hajar mountain range that runs parallel to the Al Batinah Coast. Values of the parameters are lower at the flat areas than the ones in the higher altitude areas of Al Batinah region. Also, the values were observed declining steadily along the coastal plains. There is no literature found on the generation of IDF curve and its parameters at the various stations covering whole of Oman. Therefore, the generated contour maps could be used to estimate the empirical parameters; construct the IDF formula and curves and estimate the rainfall intensities for various rainfall duration and return periods at ungauged locations. Especially in the arid region where the rainfall is erratic and unpredictable with both space and time local IDF curves development would be valuable. Also, using new IDF curve concentrated in the actual study area rather than using one generalized regional IDF curve will provide appropriate rainfall data for flood, storm water, road-bridge design and other environmental studies.

5 Conclusions

Intensity–duration–frequency (IDF) curves are utilized in the hydrologic and water engineering projects water resource projects in planning and designing of storm drainage, flood protection, bridges and culverts, water impounding facilities, and other water resources systems. In this study the development of IDF curves was done using Bernard’s equation. Gumbel distribution was used to obtain rainfall intensities for various durations and return periods. The historical rainfall data obtained from the Ministry of Regional Municipalities and Water Resources (MRMWR) at 65 gauging stations situated at different elevation and regions throughout Oman were used in the study. IDF curves and empirical formulas were derived for rainfall durations (5, 15, 30, 60, 360, 720, and 1440 min) for various return periods (2, 5, 10, 25, 50, and 100 years).

Rainfall analysis exhibited the average annual rainfall of 109.21 mm with a standard deviation of 92.82 mm, Skewness coefficient of 1.62 and Kurtosis coefficient of 3.08 for all the studied stations from 1977 to 2017. The study also shows that the rainfall in the mountainous region is high compared to the desert and the coastal region of the country. Also, IDF analysis indicated the higher intensity rainfall at various return periods was witnessed at the higher elevation stations compared to the lower elevation stations. IDF empirical parameters estimation using nonlinear regression provided the parameter values for C ranging from 417.5 to 8.95, m ranging from 0.645 to 0.196, and e ranging from 0.79 to 0.391 for the studied stations.

Finally, the contour maps of spatial distribution of the IDF parameters were plotted for whole country. The parameter values were moderately high and more condensed in the northern part of the country along the Al-Hajar mountain range as compared to the southern part and along the flat terrain of the country. The created contour maps may be helpful in estimating the empirical parameters of the IDF formula and then estimate the rainfall intensities for various rainfall durations and return periods at ungauged locations. The outcome of this study will be helpful in planning, designing and decision making of future water resources and urban drainage projects.

In addition to sampling error, errors due to weather and climate change, and model errors from the short length of data also cause uncertainties in design of rainfall estimation. In any hydraulic and hydrologic structure, the design flow is usually considered for 100 years. Therefore, in this study the rainfall intensities for 100 years were considered despite the short length of rainfall data. So it is recommended to use sufficient length of rainfall data and to use uncertainty analysis methods (Bayesian methods, Cross validation approaches, Bootstrapping, and other methods) in designs to increase credibility of any project. The preliminary research in development of IDF curves in Oman is presented in this study. However, development of regional and more comprehensive studies along with detailed orographic factors in in addition to elevation are intended, in collaboration with neighboring GCC and other countries.