Module 3 Precipitation PDF

A Module in Hydrometeorology Prepared By: NATHANIEL R. ALIBUYOG Mariano Marcos State University College of Engineering City of Batac, Ilocos Norte August 2020  Module 3...

A Module in Hydrometeorology Prepared By: NATHANIEL R. ALIBUYOG Mariano Marcos State University College of Engineering City of Batac, Ilocos Norte August 2020  Module 3 Precipitation Introduction Precipitation replenishes surface water bodies, renews soil moisture for plants, and recharges aquifers. Its principal forms are rain and snow. The relative importance of these forms is determined by the climate of the area under consideration. Hydrologic modeling and water resources assessments depend upon a knowledge of the form and amount of precipitation occurring in a region of concern over a time period of interest. Learning Outcomes At the end of the lesson, you should be able to: Compare the different types and forms of precipitation Discuss the variability of precipitation over time and space and the factors affecting them Analyze rainfall data and present results in different type of graphs Compute the mean precipitation using different methods and interpret the results Apply different techniques of adjusting precipitation data Develop and interpret intensity-duration-frequency curves Learning Input 3.1 Formation of Precipitation The conditions for precipitation to take place may be summarized stepwise as follows: 1. Supply of moisture 2. Cooling to below point condensation 3. Condensation 4. Growth of particles The supply of moisture is obtained through evaporation from wet surfaces, transpiration from vegetation or transport from elsewhere. The cooling of moist air may be through contact with a cold earth surface causing dew, white frost, mist or fog, and loss of heat through long wave radiation (fog patches). However, much more important is the lifting of air masses under adiabatic conditions (dynamic cooling) causing a fall of temperature to near its dew point. Five lifting mechanism can be distinguished: 1. Convection, due to vertical instability of the air. The air is said to be unstable if the temperature gradient is larger than the adiabatic lapse rate. Consequently, a parcel moving up obtains a  temperature higher than its immediate surroundings. Since the pressure on both is the same the density of the parcel becomes less than the environment and buoyancy causes the parcel to ascend rapidly. Instability of the atmosphere usually results from the heating of the lower air layers by a hot earth surface and the cooling of the upper layers by outgoing radiation. Convective rainfall is common in tropical regions and it usually appears as a thunderstorm in temperate climates during the summer period. Rainfall intensities of convective storms can be very high locally; the duration, however is generally short. 2. Orographic lifting. When air passes over a mountain it is forced to rise which may cause rainfall of the windward slope. As a result of orographic lifting rainfall amounts are usually highest in the mountainous part of the river basin. 3. Frontal Lifting. The existence of an area with low pressure causes surrounding air to move into the depression, displacing low pressure air upwards, which may then be cooled to dew point. If cold air is replaced by warm air (warm front) the frontal zone is usually large and the rainfall of low intensity and long duration. A cold shows a much steeper slope of the interface of warm and cold air usually resulting in rainfall of shorter duration and higher intensity. 4. Cyclones, tropical depressions or hurricanes. These are active depressions, which gain energy while moving over warm ocean water, and which dissipate energy while moving over land or cold water. They may cause torrential rains and heavy storms. Typical characteristics of these tropical depressions are high intensity rainfall of long duration (several days). 5. Convergence. The Inter-tropical Convergence Zone (ITCZ) is the tropical region where the air masse originating from the Tropic of Cancer and Capricon converge and lift. In the tropics, the position of the ITCZ governs the occurrence of wet and dry seasons. In July, the ITCZ lies to the north of the equator and in January it lies to the south. In the tropics the position of the ITCZ determines the main rain-bringing mechanism which is also called monsoon. Hence, the ITCZ is also called the Monsoon Trough. 3.2 Amount of Precipitable Water Estimate of the amount of precipitation that might occur over a given region with favorable conditions are often useful. These may be obtained by calculating the amount of water contained in a column of atmosphere extending up from the earth’s surface. This quantity is known as the precipitable water, W, although it cannot all be removed from the atmosphere by natural processes. The amount of precipitable water in the atmosphere is equal to the water mass contained in a column, having a square base of 1 cm, between elevation zero and some height z. In equation form, it is given as " 𝑊 = ∫# 𝜌! 𝑑𝑧 where rw = absolute humidity, W is the depth of precipitable water in cm. 3.3 Formation of Precipitation 1. Convective Precipitation. Convective precipitation is typical of the tropics and is brought about by heating of the air at the interface with the ground. This heated are expands with a resultant reduction in weight. During this period, increasing quantities of water vapor are taken up; the warm moisture-laden air become unstable; and pronounced vertical current developed. Dynamic cooling takes place, causing condensation and precipitation. Convective precipitation may be in the form of light showers or storms of extremely high intensity. 2. Orographic Precipitation. Orographic precipitation results from the mechanical lifting of moist horizontal air currents over natural barriers such as mountain ranges. Factors that are important in this process include land elevation, local slope, orientation of land slope, and distance from the moisture source. 3. Cyclonic Precipitation. Cyclonic precipitation is associated with the movement of air masses from high-pressure regions to low-pressure regions. These pressure differences are created by the annual heating of the earth’s surface. Cyclonic precipitation may be classified as frontal and nonfrontal.  a. Nonfrontal precipitation - is produce when air is lifted through horizontal convergence of the inflow into a low-pressure area. b. Frontal precipitation - results from the lifting of warm air over cold air at the contact zone between air masses having different characteristics. c. Warm front – if the air masses are moving so that warm air replaces colder air d. Cold front - if the cold air displaces warm air e. Stationary front - if the front is not in motion. 4. Thunderstorms. Thunderstorm cells develop from vertical air movements associated with intense surface heating or orographic effects. 3.4 Types of Precipitation 1. Rain - Water drops have a diameter of at least 0.5 mm. It can be classified based on intensity as Light rain – up to 2.5 mm/h Moderate – 2.5 mm/h to 7.5 mm/h Heavy rain – greater than 7.5 mm/h 2. Snow - Precipitation in the form of ice crystals which usually combine to form flakes, with an intensity density of 0.1 g/cm3 3. Drizzle - Rain droplets of size less than 0.5 mm and rain intensity of less than 1 mm/h 4. Glaze - When rain or drizzle touches ground at 0oC, glaze or freezing rain is formed 5. Sleet - It is frozen raindrops of transparent grains which form when rain falls through air at subfreezing temperature 6. Hail - It is a showery precipitation in the form of irregular pellets or lumps of ice of size more than 8 mm 3.4 Measurement of Precipitation The instrument used to collect and measure the precipitation is called rain gauge. Types of rain gauge 1. Non-recording type. The standard gage has a collector of 8-in diameter. Rain passes from the collector into a cylindrical measuring tube inside the overflow can. The measuring tube has  a cross-sectional area one-tent that of the collector so that 0.1 in rainfall in the tube will have a depth of 1 in. 2. Recording type. The instrument records the graphical variation of the rainfall, the collected quantity in a certain time interval and the intensity of the rainfall (mm/hr). It allows continuous measurement of the rainfall. a. Tipping bucket type. These buckets are so balanced that when 0.25 mm of rain falls into one bucket, it tips bringing the other bucket in position. b. Weighing -bucket type. The catch empties into a bucket mounted on a weighing scale. The weight of the bucket and its contents are recorded on a clock work driven chart. The instrument gives a plot of cumulative rainfall against time (mass curve of rainfall). 3.5 Presentation of Rainfall Data Hyetograph. It is the plot of rainfall intensity against time. The hyetograph is derived from the mass curve and is usually represented by a bar chart. It is a very convenient way of representing the characteristics of a storm and is particularly important in the development of design storms to predict extreme floods. Mass Curve of Rainfall. It is the plot of accumulated precipitation against time, plotted in chronological order. Mass curves of rainfall are very useful in extracting the information on the duration and magnitude of a storm. Also, intensities at various time intervals in a storm can be obtained by the slope of the curve. Point Rainfall. It is also known as station rainfall. It refers to the rainfall data of a station. Moving average. It is a technique for smoothening out the high frequency fluctuations of a time series and to enable the trend, if any, to be noticed. The basic principle is that a window of time range m years is selected. Starting from the first set of m years of data, the average of the data of m years is calculated and placed in the middle year of the range m. The window is next moved sequentially one time unit (year) at a time and the mean of m terms in the window is determined at each window location. The value of m can be 3 or more years; usually ad odd value.  Hyetograph Mass Curve of Rainfall 3.6 Mean Precipitation Over an Area For most hydrologic analyses, it is important to know the areal distribution of precipitation. The reliability of rainfall measured at one gauge in representing the average depth over a surrounding area is a function of: a. The distance from the gauge to the center of the representative area b. The size of the area c. Topography d. The nature of the rainfall of concern e. Local storm pattern characteristics The following methods are used to measure the average precipitation over an area: 1. Arithmetic mean method 2. Theissen polygon method 3. Isohyetal method 4. Inverse distance weighting Arithmetic Mean Method – Simplest method for determining areal average. In this method, only stations within the watershed boundary are considered. $ 𝑃( = ∑% &'$ 𝑃& % where: Pi = rainfall at the ith raingauge station N = total number of raingauge stations Thiessen Polygon Method - In this method the rainfall recorded at each station is given a weightage on the basis of an area closes to the station. The procedure of determining the weighing area is as follows: Consider a catchment area as shown below containing three raingauge stations. There are three stations outside the catchment but in its neighborhood. The catchment area is drawn to scale and the positions of the six stations marked on it. Stations 1 to 6 are joined to form a network of triangles. Perpendicular bisectors for each of the sides of the triangle are drawn. These bisectors form a polygon around each station. The boundary of the catchment, if it cuts the bisector is taken as the outer limit of the polygon. Thus, for station 1, the bounding polygon is abcd. For station 2, kade is taken as the bounding polygon. These bounding polygons are called Thiessen polygons.  The areas of the polygons are determined either with a planimeter or by using an overlay grid. The average precipitation over the watershed area is computed as: ∑ " )! *! 𝑃( = !#$ ∑* where: Pi = rainfall at the ith raingauge station Ai = area of the polygon enclosing the ith raingauge station N = total number of raingauge stations Note: The Thiessen polygon method is superior to the arithmetic average method as some weightage is given to the various stations on a rational basis. Further, the raingauge stations outside the catchment are also used effectively. Isohyetal Method. An isohyets is a line joining points of equal rainfall magnitude. In the isohyetal method. The catchment area is drawn to scale and the raingauge stations are marked. The recorded values for which areal average 𝑃( is to be determined are then marked on the plot at appropriate stations. Neighboring stations outside the catchment are also considered. The isohyets of various values are then drawn by considering point rainfalls as guide and interpolating between them by the eye. The procedure is similar to the drawing of elevation contours based on spot levels. The area between two isohyets are then determined with a planimeter. If the isohyets go out of the catchment, the catchment boundary is used as the bounding line. The average value of the rainfall indicated by two isohyets is assumed to be acting over the inter-isohyet area. Thus, P1, P2, …, Pn are the values of isohyets and if A1, A2, …, An-1 are the inter-isohyets areas respectively, then the mean precipitation over the catchment of area A is given by % &% % &% % &% *$ + $ ' ,-*' + ' ( ,-⋯-*)*$ + )*$ ) , 𝑃( = ' ' ' * The Isohyet method is superior to other two methods especially when stations are large in number. Inverse DIstance Weighing (IDW) Method. Prediction at a point is more influenced by nearby measurements than that by distant measurements. The prediction at an ungauged point is inversely proportional to the distance to the measurement points. Steps: a. Computed distance (di) from ungauged point to all measurement points 𝑑$/ = +(𝑋$ − 𝑋/ )/ + (𝑌$ − 𝑌/ )/ b. Compute the precipitation at the ungauged point using the formula 𝑃& ∑0&'$ 2 3 𝑑& / 𝑃( = 1 ∑0&'$ 4 / 6 𝑑& where: P = precipitation at station i d12 = distance between point station 1 and 2 n = number of gauged points Example 1. In a catchment area, approximated by a circle of diameter 100 km, four rainfall stations are situated inside the catchment and one station is outside in its neighborhood. The coordinate of the center of the catchment and of the five station are given below. Also given are the annual precipitation recorded by the five stations in 2019 Determine the average annual precipitation by the Thiessen polygon method. Center: (100,100) Diameter: 100 km Distance are in km Station 1 2 3 4 5 Coordinates (30,80) (70,100) (100,140) (130,100) (100,70) Precipitation 85.0 135.2 95.3 146.4 102.2 (cm)  Thiessen polygon Solution: The catchment area is drawn to scale and the stations are marked on it. The stations are joined to form a set of triangles and the perpendicular bisector of each side is then drawn. The Thiessen- polygon area enclosing each station is then identified. It may be noted that Station 1 in this problem does not have any area of influence in the catchment. The areas of various Thiessen polygons are determined either by a planimeter or by placing an overlay grid. Station Boundary of Area (km2) Fraction of Rainfall Weighted P area #Grids total area (cm) (cm) (Col 4 x Col 5) 1 - - - 85.0 - 2 abcd 2141 0.2726 135.2 36.86 3 dce 1609 0.2049 95.3 19.53 4 ecbf 2141 0.2726 146.4 39.91 5 fba 1963 0.2499 102.2 25.54 Total 7854 1.000 121.84 Mean Precipitation = 121.84 cm Example 2. The Isohyets due to a storm in a catchment were drawn as shown below and the area of the catchment bounded by isohyets were tabulated as below. Isohyets (cm) Area (km2) Station – 12.0 30 12.0 – 10.0 140 10.0 – 8.0 80 8.0 – 6.0 180 6.0 – 4.0 20 Estimate the mean precipitation due to the storm. Solution: For the first area consisting of a station surrounded by a closed isohyet, a precipitation value of 12.0 cm is taken. For all other areas, the mean of two bounding isohyets are taken. Isohyets Average Area (km2) Fraction of Weighted P value of P total area (cm) (cm) (Col 3 /450) (Col 2 x Col 4) 1 2 3 4 5 12.0 12.0 30 0.0667 0.800 12.0 – 10.0 11.0 140 0.3111 3.422 10.0 – 8.0 9.0 80 0.1778 1.600 8.0 – 6.0 7.0 180 0.4000 2.800 6.0 – 4.0 5.0 20 0.0444 0.222 Total 450 1.0000 8.844 Mean Precipitation 𝑃(= 8.84 cm 3.6 Adjustments of Precipitation Data Before using the rainfall records of a station, it is necessary to first check the data for continuity and consistency. The continuity of a record may be broken with missing data due to many reasons such as damage or fault in a raingauge during a period. The missing data can be estimated by using the data of the neighboring stations. In these calculations the Normal Rainfall is used as standard comparison. The normal rainfall is the average value of rainfall at a particular date, month, or year over a specified 30-year period. The 30-year normal are recomputed every decade. Thus, the term Normal annual precipitation at Station A means the average annual precipitation at A based on a specified 30-year of record. Estimation of Missing Data Given the annual precipitation values, P1, P2, P3, …, Pm at neighboring M stations 1, 2, 3, …, M respectively, it is required to find the missing annual precipitation Px at a station X not included in the above M stations. Further, the normal annual precipitations N1, N2, …, Ni at each of the above (M+1) stations including station X are known. If the normal annual precipitations at various stations are within about 10% of the normal annual precipitation at station X, then a simple arithmetic average procedure is followed to estimate Px. Thus $ 𝑃1 = [𝑃$ + 𝑃/ + ⋯ + 𝑃3 ] 2 If the normal precipitation vary considerably, then Px is estimated by weighing the precipitation at the various stations by the ratios of normal annual precipitations. This method, known as the Normal Ratio Method, gives Px as %+ )$ )' ), 𝑃1 = 2 %$ : + %' + ⋯+ %, ; Example 3. The normal annual rainfall at station A, B, C, and D in a basin are 80.97, 67.59, 76.28, and 92.01 cm respectively. In the year 2019, the station D was inoperative and the stations A, B, and C recorded annual precipitation of 91.11, 72.23, and 79.89 cm, respectively. Estimate the annual rainfall at station D in that year. Solution: As the normal rainfall values vary more than 10% , the normal ration method is adopted. Hence, %+ )$ )' ), 𝑃1 = : + + ⋯+ ; 2 %$ %' %, 4/.#$ 4$.$$ 8/./6 84.74 𝑃1 = :7#.84 + + ; 6 98.:4 89./7 𝑃1 = 99.48 𝑐𝑚 Test for Consistency of Record If the conditions relevant to the recording of a raingauge station have undergone a significant change during the period of record, inconsistency would arise in the rainfall data of that station. This inconsistency would be felt from the time the significant change took place. Some of the cause of inconsistence of record are: Shifting of a raingauge station to a new location The neighborhood of the station undergoing a marked change Change in the ecosystem due to calamities, such as forest fires, landslides Occurrence of observational error from a certain date The checking for inconsistency of a record is done by the Double Mass Curve Technique. This technique is based on the principle that when each recorded data comes from the same parent population, they are consistent. Steps: 1. A group of 5 to 10 base stations in the neighborhood of the problem station X is selected. 2. The data of the annual (or monthly or seasonal mean) rainfall of the station X and also the average rainfall of the group of bases stations covering a long period is arranged in the reverse chronological order (i.e., the latest record as the first entry and the oldest record as the last entry in the list) 3. The accumulated precipitation of the station X (i.e., ∑ 𝑃! ) and the accumulated values of the average of the group of base stations (i.e., ∑ 𝑃"#$ ) are calculated starting from the latest record. 4. Values of ∑ 𝑃! are plotted against ∑ 𝑃"#$ for various consecutive time periods. 5. A decided break in the slope of the resulting plot indicates a change in the precipitation regime of station X. 6. The precipitation values at station X beyond the period of change of regime is corrected by using the relation 𝑀; 𝑃;1 = 𝑃1 B D 𝑀< where: Pcx = corrected precipitation at any time period t1 at station X  Px = original recorded precipitation at time period t1 at station X Mc = corrected slope of the double mass curve Ma = original slope of the double mass curve Double Mass Curve Example 4. Annual rainfall data for station M as well as the average annual rainfall values for a group of ten neighboring stations located in a meteorologically homogenous region are given below. Average Average Annual Annual Annual Annual Rainfall of Rainfall of Year Rainfall of Year Rainfall of Station M Station M the Group the Group (mm) (mm) (mm) (mm) 1950 676 780 c 1244 1400 1951 578 660 1966 999 1140 1952 95 110 1967 573 650 1953 462 520 1968 596 646 1954 472 540 1969 375 350 1955 699 800 1970 635 590 1956 479 540 1971 497 490 1957 431 490 1972 386 400 1958 493 560 1973 438 390 1959 503 575 1974 568 570 1960 415 480 1975 356 377 1961 531 600 1976 685 653 1962 504 580 1977 825 787 1963 828 950 1978 426 410 1964 679 770 1979 612 588 Test the consistency of the annual rainfall data of station M and correct the record if there is any discrepancy. Estimate the mean annual precipitation at station M. Solution: a. The data is sorted in descending order of the year, starting from the latest year 1979. b. Cumulative value of station M rainfall (∑ 𝑃3 ) and the ten station average rainfall values (∑ 𝑃 ) are calculated as shown in Table below. c. The data is then plotted with ∑ 𝑃3 on the y-axis and ∑ 𝑃 on the x-axis to obtain a double mass curve plot as shown below d. The value of the year corresponding to the plotted points is also noted on the plot. e. It is that the data plots as two straight lines with a break of grade at the year 1969. This represent a change in the regime of the station M after the year 1968. f. The slope of the best straight line for the period 1979 – 1968 is Mc = 1.029 and the slope of the best straight line for the period 1968 – 1950 is Ma = 0.8779 g. The correction ratio to bring the old record (1950 – 1968) to the current (post 1968) regimes is = Mc/Ma = 1.029/0.8779 = 1.173 h. Each of the pre 1960 annual rainfall value is multiplied by the correction ration of 1.173 to get the adjusted value. i. The adjusted values at station M are shown in Col. 5 of Table. The finalized values of Pm (rounded off to nearest mm) for all the 30 years of record are shown in Col. 7 j. The mean annual precipitation at station M (based on the corrected time series) = (19004/30) = 633.5 mm 1 2 3 4 5 6 7 8 YEA Pm Pave SPm SPave Adjusted Finalize Adjusted R Values of d values Cumulative Pm of Pm value of Pm 1979 612 588 612 588 612 612 1978 426 410 1038 998 426 1038 1977 825 787 1863 1785 825 1863 1976 685 653 2548 2438 685 2548 1975 356 377 2904 2815 356 2904 1974 568 570 3472 3385 568 3472 1973 438 390 3910 3775 438 3910 1972 386 400 4296 4175 386 4296 1971 497 490 4793 4665 497 4793 1970 635 590 5428 5255 635 5428 1969 375 350 5803 5605 375 5803 1968 596 646 6399 6251 698.92 699 6502 1967 573 650 6972 6901 671.95 672 7174 1966 999 114 7971 8041 1171.51 1172 8345 0 1965 1244 140 9215 9441 1458.82 1459 9804 0 1964 679 770 9894 10211 796.25 796 10600 1963 828 950 10722 11161 970.98 971 11571 1962 504 580 11226 11741 591.03 591 12162 1961 531 600 11757 12341 622.70 623 12785 1960 415 480 12172 12821 486.66 487 13272 1959 503 575 12675 13396 589.86 590 13862 1958 493 560 13168 13956 578.13 578 14440 1957 431 490 13599 14446 505.43 505 14945 1956 479 540 14078 14986 561.72 562 15507 1955 699 800 14777 15786 819.71 820 16327 1954 472 540 15249 16326 553.51 554 16880 1953 462 520 15711 16846 541.78 542 17422 1952 95 110 15806 16956 111.41 111 17533 1951 578 660 16384 17616 677.81 678 18211 1950 676 780 17060 18396 792.73 793 19004 Total of Pm = 19004 mm Mean of Pm = 633.5 mm  30000 Cumulative annual rainfall at station M (mm) Old 25000 Adjusted 20000 New Linear (Old) 15000 Linear (New) 10000 y = 0.8779x + 917.93 5000 y = 1.0295x + 12.48 0 0 5000 10000 15000 20000 Cumulative ten station average (mm) 3.7 Frequency of Point Rainfall In many hydraulic engineering applications such as those concerned with floods, the probability of occurrence of a particular extreme rainfall, e.g., 24-h maximum rainfall, will be of importance. Such information is obtained by the frequency analysis of the point-rainfall data. The probability of occurrence of an event of a random variable (e.g., rainfall) whose magnitude is equal to or in excess of a specified magnitude X is denoted by P. The recurrence interval (also known as return period) is defined as T = 1/P This represents the average interval between the occurrence of a rainfall of magnitude equal to or greater than X. Thus, if it is stated that the return period of rainfall of 20 cm in 24 hours is 10 years at a certain station A, it implies that on an average rainfall magnitude equal to or greater than 20 cm in 24 hours occur once in 10 years, i.e., in a long period of say 100 years. However, it does not mean that every 10 years one such event is likely, i.e., periodicity is not implied. The probability of a rainfall of 20 cm in 24 hours occurring in anyone year at station A is 1/T = 1/10 = 0.10. If the probability of an event occurring is P, the probability of the event not occurring in a given year is q=1–p The binomial distribution can be used to find the probability of occurrence of the event r times in n successive years. Thus 𝑃?,0 = 0𝐶? 𝑃? 𝑞0A? 𝑛! 𝑃?,0 = 𝑃? 𝑞0A? (𝑛 − 𝑟)! 𝑟! Where: Pr, n = probability of a random hydrologic event (rainfall) of given magnitude and exceedance probability P occurring r times in n successive years. Thus, for example, a) The probability of an event of exceedance probability P occurring 2 times in n successive years is 𝑛! 𝑃?,0 = 𝑃? 𝑞0A? (𝑛 − 𝑟)! 𝑟! 𝑛! 𝑃/,0 = 𝑃/ 𝑞0A/ (𝑛 − 2)! 2! b) The probability of the event not occurring at all in n successive years is 𝑛! 𝑃B,0 = 𝑃# 𝑞0A# (𝑛 − 0)! 0! 𝑃B,0 = 𝑞0 = (1 − 𝑃0 ) c) The probability of the event occurring at least once in n successive years 𝑛! 𝑃?,0 = 𝑃? 𝑞0A? (𝑛 − 𝑟)! 𝑟! 𝑃$ = 1 − 𝑞0 = 1 − (1 − 𝑃0 ) Example 6. Analysis of data on maximum one-day rainfall depth at Ilocos Norte indicates that a depth of 280 mm had a return period of 50 years. Determine the probability of a one-day rainfall depth equal to or greater than 280 mm at Ilocos Norte occurring (a) once in 20 successive years, (b) two times in 15 successive years, and (c) at least once in 20 successive years. Solution: 1 1 𝑃= = = 0.02 𝑇 50 Using equation 𝑛! 𝑃?,0 = 𝑃? 𝑞0A? (𝑛 − 𝑟)! 𝑟! a) Once in 20 successive years: n = 20, r = 1 𝑛! 𝑃?,0 = 𝑃? 𝑞0A? (𝑛 − 𝑟)! 𝑟! /#! 𝑃$,/# = (/#A$)!$! (0.02)$ (1 − 0.02)/#A$ ; q = 1 - P 𝑃$,/# = 0.272 b) Two times in 15 successive years: n = 15, r =2 𝑛! 𝑃?,0 = 𝑃? 𝑞0A? (𝑛 − 𝑟)! 𝑟!  15! 𝑃/,$: = (0.02)/ (1 − 0.02)$:A/ (15 − 2)! 2! 𝑃/,$: = 0.323 c) At least once in 20 successive years, n=20 𝑃$ = 1 − 𝑞0 = 1 − (1 − 𝑃0 ) 𝑃$ = 1 − (1 − 0.02)/# ) 𝑃$ = 0.332 Plotting Position The purpose of the frequency of an annual series is to obtain a relation between the magnitude of the event and its probability of exceedance. The probability analysis may be made either by empirical or by analytical methods. Partial duration series A simple empirical technique is to arrange the given annual extreme series in descending order of magnitude and to assign an order number m. Thus, for the first entry, m = 1, for the second entry, m= 2, and so one, till the last event for which m = N = number of years of record. The probability P of an event equaled or exceeded is given by the Weibull formula 3 𝑃 = P%-$Q The recurrence interval, T $ %-$ 𝑇= ) = 3 Example: N = 10 years Highest value, m = 1 T = 11/1 = 11 years Least value, m = 10 T = 11/10 = 1.1 years Other formulas use are given in the table below: Plotting position formulae Method P California m/N Hazen (m – 0.5)/N Weibull m/(N+1) Chegodayev (m-0.3)/(N+0.4) Blom (m-0.44)/(N+0.12) Gringorten (m-3/8)/(N+1/4) Having calculated P (and hence T) for all events in the series, the variation of the rainfall magnitude is plotted against the corresponding T on a semi-log paper or log-log paper. By suitable extrapolation on this plot, the rainfall magnitude of specific duration for any recurrence interval can be estimated. Return periods of annual rainfall at Station A Example 7. The record of annual rainfall at Station A covering a period of 22 years is given below. (a) Estimate the annual rainfall with return periods of 10 years and 50 years. (b) What would be the probability of an annual rainfall of magnitude equal to or exceeding 100 cm occurring at Station A? (c) What is the 75% dependable annual rainfall at station A? Year Annual rainfall (cm) Year Annual rainfall (cm) 1960 130 1971 90 1961 84 1972 102 1962 76 1973 108 1963 89 1974 60 1964 112 1975 75 1965 96 1976 120 1966 80 1977 160 1967 125 1978 85 1968 143 1979 106 1969 89 1980 83 1970 78 1981 95 Solution: The data are arranged in descending order and the rank number assigned to the recorded events. The probability of P of the event being equaled to or exceed ed is calculated by using Weibull formula. It may be noted that when two or more events have the same magnitude (as for m=13 and 14) the probability P is calculated for the largest m value of the set. The return period T is calculated as T= 1/P.  250 200 Annual rainfall (cm) 150 100 50 0 1 10 100 Retrun period T in years Annual Probability Return Period Year rainfall m P= m/(N+1) T = 1/P (years) (cm) 1977 160 1 0.043 23.000 1968 143 2 0.087 11.500 1960 130 3 0.130 7.667 1967 125 4 0.174 5.750 1976 120 5 0.217 4.600 1964 112 6 0.261 3.833 1973 108 7 0.304 3.286 1979 106 8 0.348 2.875 1972 102 9 0.391 2.556 1965 96 10 0.435 2.300 1981 95 11 0.478 2.091 1971 90 12 0.522 1.917 1963 89 13 0.565 1969 89 14 0.609 1.643 1978 85 15 0.652 1.533 1961 84 16 0.696 1.438 1980 83 17 0.739 1.353 1966 80 18 0.783 1.278 1970 78 19 0.826 1.211 1962 76 20 0.870 1.150 1975 75 21 0.913 1.095 1974 60 22 0.957 1.045  a. For T = 10 years, the corresponding rainfall magnitude is obtained by interpolation between two appropriate successive values in the Table, viz, those having T = 11.5 and 7.667 years respectively, as 137.9 cm For T = 50 years, the corresponding rainfall magnitude, by extrapolation of the best fit straight line, is 187 cm. b. Return period of an annual rainfall of magnitude equal to or exceeding 100 cm, by interpolation, is 2.4 years. As such, the exceedance probability 𝑃 = 1/2.4=0.417 c. 75% dependable annual rainfall at Station A = annual rainfall with probability P = 0.75, i.e., T = 1/0.75 = 1.333 years. By interpolation between two successive values in the Table having T = 1.28 and 1.35 respectively, the 75% dependable annual rainfall at Station A is 83.27 cm 83 cm - 1.353 R - 1.333 80 - 1.278 FA76 $.666A$.6:6 = 76A76 $.6:6A$./87 ($.666A$.6:6) 𝑅= (82 − 83) + 83 ($.6:6A$./87) 𝑅 = 83.27 𝑐𝑚 3.8 Depth-Area-Duration Relationship The total amount of rain falling at a point is the usual basic precipitation figure available. For many purposes, however, this is not adequate and information may be required which include the intensity, duration frequency and areal extent. Intensity – This is a measure of the quantity of rain falling in a given time; for example, mm per hour. Duration – This the period of time during which rain falls. Frequency – This refers to the expectation that a given depth of rainfall will fall in a given time. Such an amount may be equaled or exceeded in a given number of days or years. Areal extent – This concerns the area over which a point’s rainfall can be held to apply. Maximum Intensity – Duration Relationship In any storm, the actual intensity as reflected by the slope of the mass curve of rainfall varies over a wide range during the course of the rainfall. If the mass curve is considered divided into N segments of time interval ∆t such that the total duration of the storm is D = N ∆t Then, the intensity of the storm for various sub-duration tj are:  Tj = (1,∆t), (2,∆t), (3,∆t), …, (j,∆t)… and (N,∆t) could be calculated. It will be found that for each duration (say, tj), the intensity will have a maximum value and this could be analyzed to obtain a relationship for the variation of the maximum intensity with duration for the storm. This process is basic to the development of maximum intensity duration frequency relationship for the station. The procedure for analysis of mass curve of rainfall for developing maximum intensity-duration relationship of the storm is as follows: 1. Select a convenient time step ∆t such that duration of the storm D = N ∆t 2. For each duration (say tj = j ∆t) the mass curve of rainfall is considered to be divided into consecutive segments of duration tj. For each segment the incremental rainfall dj in duration tj is noted and intensity Ij = dj/tj obtained. 3. Maximum value of the intensity (Imj) for the chosen tj is noted. 4. The procedure is repeated for all values of j = 1 to N to obtain a data set of Imj as a function of duration tj. 5. Plot the maximum intensity Imj as function of duration t. 6. It is common to express the variation of Im with t as ; 𝐼3 = (G-