Agricultural Engineering · Land & Water Resources Engineering · also Engineering Hydrology, Hydrology (PRC)
Rain to runoff: measuring precipitation, subtracting losses, estimating runoff and routing floods.
In the exams: GATE AG Section 4 Hydrology (SWCE is the heaviest GATE section, ~16 marks a year 2019–26); ICAR AIEEA-PG Unit VII; ICAR AICE subjects 40 and 43; NCEES PE hydrology; state PSCs
Watershed hydrology is the engineering study of how rain that falls on a piece of land becomes the water that leaves it: how much is caught by leaves, evaporated, soaked into the soil, and how much runs off, how fast, and with what peak. A watershed (also called a catchment or drainage basin) is the area of land that drains to a single outlet; its boundary, the divide, follows the ridge lines. Every soil and water conservation structure an agricultural engineer designs, from a contour bund to a farm pond spillway to a check dam, is sized from a hydrological estimate: a runoff volume, a peak discharge, or a design flood hydrograph.
The subject runs as a chain. Precipitation is measured at points and averaged over an area. Gaps and inconsistencies in the records are filled and corrected. Rainfall of a chosen probability is picked from frequency analysis. Losses (abstractions) such as interception, evaporation and infiltration are subtracted to leave effective rainfall. Effective rainfall is converted to runoff volume (SCS curve number), peak discharge (rational method) or a full hydrograph (unit hydrograph). Streamflow is measured to check and calibrate all of this, and floods are routed through reservoirs and channels to see how they change downstream. At the dry end of the spectrum, drought is classified so that relief and planning can be triggered.
This manual follows that chain in twelve chapters. It is written for students in India, the Philippines, the United States, Nepal, Canada and elsewhere who study the same methods under different course names (Engineering Hydrology, Watershed Hydrology, Hydrology). In India the subject is examined heavily in GATE Agricultural Engineering (the Hydrology part of the Soil and Water Conservation Engineering section, which is the heaviest technical section), in ICAR AIEEA-PG Unit VII, in the ICAR PhD entrance for Soil and Water Conservation and for Irrigation and Drainage Engineering, and in state public service commission papers. In the United States it appears in the NCEES PE Agricultural and Biological Engineering exam under Natural Resources and Environmental Systems (hydrology, curve number, stormwater); in the Philippines under Land and Water Resources Engineering. Most exam questions are calculations, so each chapter gives formulas with units and a worked example. Coefficients quoted are teaching values; they are regional.
The hydrologic cycle is the continuous movement of water between the oceans, the atmosphere and the land, driven by solar energy and gravity. Water evaporates from oceans, lakes and soil, and transpires from plants; the combined loss from a vegetated surface is called evapotranspiration. Vapour cools, condenses and returns as precipitation. On land, part of the precipitation is caught by vegetation (interception), part is held in surface hollows (depression storage), part enters the soil (infiltration) and part flows over the surface as overland flow to channels. Infiltrated water either stays in the root zone as soil moisture, moves laterally just below the surface as interflow (subsurface stormflow), or percolates down to the saturated zone to recharge groundwater. Groundwater discharges slowly to streams as baseflow, which is what keeps perennial rivers flowing between storms.
Most water on Earth is saline ocean water; of the fresh water, the largest share is locked in ice caps and glaciers, and most of the rest is groundwater. Rivers hold a tiny fraction at any moment, yet carry the flows engineers design for.
For any control volume over any time interval, continuity gives: inputs minus outputs equals change in storage. For a catchment this is usually written P − R − G − E − T = ΔS, where P is precipitation, R is surface runoff leaving the outlet, G is net groundwater outflow across the boundary, E is evaporation, T is transpiration and ΔS is the change in stored water. All terms must be in the same units, either depth over the catchment (mm) or volume (m³). To convert, volume = depth × area; 1 mm over 1 km² is 1,000 m³, and 1 mm over 1 ha is 10 m³.
Over a long period, such as a multi-year average, ΔS is close to zero, so long-term evapotranspiration can be estimated as precipitation minus runoff (where groundwater exchange across the divide is small). Over a single storm, E and T are small and the equation reduces to rainfall minus losses equals runoff. A frequent exam slip is mixing a flow rate (m³/s) with a depth (mm): convert a mean annual flow Q to depth by multiplying by the seconds in a year (about 31.54 million) and dividing by the catchment area in m².
Two mechanisms produce most storm runoff. Infiltration-excess overland flow, described by Horton, occurs when rainfall intensity exceeds the soil's infiltration capacity; the excess ponds and runs off even if the soil below is unsaturated. It dominates on crusted, compacted, bare or semi-arid soils and on paved areas. Saturation-excess overland flow, associated with Dunne, occurs when the soil profile fills up from below, so that even gentle rain on saturated ground runs off. It dominates in humid, well-vegetated catchments, where saturated areas form first near streams and in hollows and expand during a storm. This expanding zone is the variable source area. Subsurface stormflow (interflow) and groundwater baseflow add slower components.
The practical terms follow. Direct runoff is the part of streamflow that arrives quickly after a storm (overland flow plus quick interflow). Baseflow is the delayed groundwater contribution. Effective rainfall (rainfall excess) is the part of rainfall that becomes direct runoff; it equals total rainfall minus abstractions. The unit hydrograph, infiltration indices and the curve number method all work with direct runoff and effective rainfall, never with total streamflow.
Precipitation is water falling from the atmosphere in liquid or solid form: drizzle (very fine drops at low intensity), rain, snow, sleet and hail. Precipitation needs moist air lifted and cooled below its dew point. Hydrologists classify precipitation by the lifting mechanism. Convective precipitation results from local heating of air near the ground; the warm air rises rapidly, producing short, intense, localised storms typical of tropical afternoons and summer thunderstorms. Orographic precipitation results when moving air is forced up over a mountain barrier; the windward slopes receive heavy rain and the leeward side lies in a rain shadow. Cyclonic (frontal) precipitation results from the lifting of air in low-pressure systems, where warm air rises over cold air along a front; it covers large areas for long durations.
For engineering, the three properties that matter are depth (mm), intensity (mm/h) and duration. Short convective storms control peak runoff from small watersheds; long frontal or monsoon storms control flood volumes from large ones.
A rain gauge collects precipitation through an opening of known area and reports depth. Non-recording gauges give only the total depth since the last reading, usually read once a day at a fixed hour. India's standard non-recording gauge follows a national standard with collector areas of 100 and 200 cm²; other countries use their own standard gauges.
Recording (self-recording) gauges give a continuous record of cumulative rainfall against time, which is needed for intensity. The tipping-bucket gauge funnels water into a pair of small buckets that tip when a fixed depth has collected; each tip closes an electrical contact, so intensity comes from the count of tips per unit time. It is suited to telemetry but under-records very intense rain, when water is lost during tipping. The weighing-bucket gauge weighs the collected water and records a mass curve; it can record snow. The natural-siphon (float) gauge lifts a float as water collects and siphons automatically when full, drawing a mass curve on a chart.
Siting matters as much as the instrument. The gauge should stand on level, open ground, with no obstruction closer than about twice the obstruction's height, and with the rim at a standard height above ground (about 30 cm in Indian practice; national standards differ). Wind is the largest source of error: it deflects drops over the orifice so that gauges generally under-catch.
How many gauges does a catchment need? Older World Meteorological Organization guidance, still quoted in most textbooks, suggests about one station per 600 to 900 km² in flat regions of temperate, Mediterranean and tropical zones, one per 100 to 250 km² in mountainous regions, and one per 1,500 to 10,000 km² in arid and polar regions, with at least about 10% of the gauges being recording type.
A statistical rule gives the optimum number of gauges for a chosen error in the mean rainfall: N = (Cv/ε)², where Cv is the coefficient of variation of the rainfall values at the existing stations, in percent, and ε is the allowable percentage error in the estimate of mean rainfall (commonly 10%). Cv is 100 × σ/P̄, where σ is the standard deviation computed with (m − 1) in the denominator for m existing stations, and P̄ is their mean. For example, if existing gauges give Cv = 29% and ε = 10%, N = (29/10)² = 8.41, so 9 gauges are needed; always round up. If 5 gauges exist, 4 more are required. Additional gauges should be placed where the existing network is thinnest or where rainfall varies most.
Gauges measure points; runoff calculations need the average depth over the catchment. Three classical methods are used. The arithmetic mean simply averages the gauge readings; it is acceptable when gauges are evenly spread over flat terrain and their readings do not differ much. The Thiessen polygon method weights each gauge by the area closest to it: join adjacent stations by lines, draw perpendicular bisectors of those lines to form polygons, measure each polygon's area inside the catchment, and compute P̄ = Σ(Pᵢ Aᵢ)/ΣAᵢ. Its weights are fixed, but it ignores topography. The isohyetal method draws contours of equal rainfall (isohyets) from the gauge readings measures the area between successive isohyets, assigns each band the mean of its bounding isohyet values, and takes the area-weighted average. It is the most accurate, especially in hills.
Worked example (Thiessen). Four gauges read 42, 55, 61 and 48 mm, with polygon areas of 18, 26, 34 and 22 km² inside a 100 km² catchment. The weighted mean is (42×18 + 55×26 + 61×34 + 48×22)/100 = 53.16 mm, compared with an arithmetic mean of 51.5 mm.
The missing value at station X is estimated from simultaneous readings at three or more surrounding index stations, chosen to be close and evenly spread around X. The normal annual precipitation is the long-term average annual rainfall at a station, conventionally over a standard 30-year period.
If the normal annual precipitation of each index station is within about 10% of that of station X, a simple arithmetic average of the index stations' readings is used: Px = (P₁ + P₂ + … + Pₘ)/m. If any index station's normal differs by more than 10%, the normal ratio method is used, which scales each station's reading by the ratio of normals: Px = (Nx/m) × Σ(Pᵢ/Nᵢ), where Nx and Nᵢ are the normal annual precipitations of X and index station i, and m is the number of index stations.
Worked example. Station X has a normal of 1,080 mm. On a day X was not read, index stations A, B and C recorded 92, 78 and 85 mm, with normals of 1,210, 960 and 1,030 mm. A's normal is 12% above X's, so the simple average is not appropriate. Normal ratio: Px = (1080/3)(92/1210 + 78/960 + 85/1030) = 86.3 mm. Always check the 10% condition first.
Inverse-distance weighting (weights 1/d², d = distance to X) and regression against a correlated neighbour are also used.
A record is consistent if the gauge measured the same way throughout. Relocation, changed exposure (new buildings or trees), a new instrument or new observation practice can shift the record without any real change in climate. Double-mass analysis detects and corrects such shifts.
The procedure: select a group of 5 to 10 surrounding base stations whose records are known to be consistent. For each year, compute the mean annual rainfall of the group. Accumulate both the suspect station's annual rainfall and the group mean, year by year, from the most recent year backwards. Plot cumulative Px (vertical) against cumulative group mean (horizontal). If the record is consistent, the points fall on a straight line. A distinct break in slope marks the year a change occurred.
Correction: the record for the period that is not representative (usually the older one) is multiplied by the ratio of slopes. Pcx = Px × (Mc/Ma), where Mc is the slope of the double-mass line for the period to which records are adjusted and Ma is the slope for the period being corrected. For example, if the recent slope is 1.15 and the older slope is 0.92, an older annual value of 64 cm becomes 64 × 1.15/0.92 = 80.0 cm.
Two cautions. Double-mass analysis tests consistency, not accuracy, and cannot detect errors shared by all stations. Short breaks of a few years are often random and are not corrected without evidence from the station history.
Hydrological design picks an event of a chosen rarity: a 25-year rainfall for a grassed waterway, a 50- or 100-year flood for a spillway. The return period (recurrence interval) T is the average interval in years between events that equal or exceed a given magnitude. The annual probability of exceedance is p = 1/T. A 50-year event therefore has a 2% chance of being equalled or exceeded in any one year; it does not happen once every 50 years like a clock.
Because years are independent, the probability that the T-year event is not exceeded in n consecutive years is (1 − p)ⁿ. The risk that it is exceeded at least once during a design life of n years is R = 1 − (1 − 1/T)ⁿ. For a structure designed for the 50-year flood with a 20-year life, R = 1 − (0.98)²⁰ = 0.332, about a one-in-three chance. The reliability is 1 − R. Inverting, the return period needed to keep risk at R over n years is T = 1/[1 − (1 − R)^(1/n)]. The probability of exactly k exceedances in n years follows the binomial distribution: C(n,k) pᵏ (1 − p)ⁿ⁻ᵏ.
Given an annual maximum series (the largest value in each year), the simplest frequency analysis ranks the values in descending order and assigns each a plotting position. The Weibull formula is the textbook standard in hydrology: P = m/(N + 1), where m is the rank (1 for the largest) and N the number of years; T = (N + 1)/m. With 30 years of record, the largest value has an estimated return period of 31 years. Values plotted on probability paper are then fitted by eye or by a distribution. Other plotting-position formulas (California, Hazen, Gringorten) give slightly different values; Weibull is expected unless an exam states otherwise.
An annual maximum series takes one value per year; a partial duration series takes every event above a threshold and is preferred for short return periods.
To extrapolate beyond the record, a probability distribution is fitted. Chow's general frequency equation is xT = x̄ + K σ, where x̄ and σ are the mean and standard deviation of the series and K is a frequency factor that depends on T and the distribution.
The Gumbel (extreme value Type I) distribution is widely used for annual maximum rainfall and floods. For a very long record, K = −(√6/π)[0.5772 + ln ln(T/(T − 1))]; for T = 100 this gives K = 3.137. For a finite record of N years, K = (yT − ȳₙ)/Sₙ, where yT = −ln ln[T/(T − 1)] is the reduced variate and ȳₙ and Sₙ are the reduced mean and reduced standard deviation read from tables for the sample size N. Exams state which to use.
The log-Pearson Type III distribution applies the Pearson Type III distribution to the logarithms of the data; K then depends on T and on the skew coefficient of the logs. United States federal flood-frequency guidelines adopted log-Pearson III as the standard for annual peak floods. Other countries' agencies set their own methods; different distributions can differ greatly at long return periods, so report confidence limits.
Runoff methods such as the rational formula need a rainfall intensity for a specific duration and return period. An IDF relationship supplies it. It is built by extracting, for each year, the maximum average intensity for each of several durations (for example 5, 10, 15, 30, 60 minutes, and 2, 6, 12, 24 hours), doing a frequency analysis for each duration, and plotting intensity against duration with one curve per return period. Intensity falls as duration increases, and rises with return period.
IDF curves are commonly summarised by an empirical equation of the form i = K Tˣ/(D + a)ⁿ, where i is intensity (mm/h), T the return period (years), D the duration (minutes or hours) and K, x, a and n are constants fitted for a location. Do not transfer constants between regions.
A design storm is then constructed: for peak flow from a small watershed, the duration equals the time of concentration; for volumes or hydrographs, a storm of set duration is distributed in time with a standard temporal pattern such as the NRCS 24-hour rainfall distributions in United States practice.
Abstractions are the parts of rainfall that do not become direct runoff: interception, depression storage, evaporation, transpiration and infiltration. During a single storm, interception, depression storage and infiltration are the important ones; evaporation matters over days to seasons, especially from reservoirs and farm ponds.
Interception is rainfall caught and held by leaves, stems and litter, which later evaporates without reaching the ground. The rest reaches the ground as throughfall (dripping through or falling between leaves) or as stemflow (running down stems). Interception depends on the vegetation type and density, the season, wind and above all the storm: the canopy fills a limited storage early in a storm, so interception is a large proportion of small, light storms and a small proportion of large storms. Dense forests intercept more than grasses and crops. Depression storage is water trapped in surface hollows, furrows and micro-relief, which later infiltrates or evaporates; tillage and conservation practices that roughen the surface increase it. In runoff methods, interception, depression storage and early infiltration are lumped as the initial abstraction.
Evaporation is the change of liquid water to vapour. It needs energy (the latent heat of vaporisation, supplied mainly by solar radiation) and a vapour-pressure gradient between the water surface and the air. Dalton's law states that the evaporation rate is proportional to the difference between the saturation vapour pressure at the water-surface temperature (ew) and the actual vapour pressure of the air (ea): E = C(ew − ea), where C depends on wind and other factors. Evaporation rises with water temperature, wind and air dryness. Dissolved salts reduce evaporation slightly.
Mass-transfer formulas put Dalton's law into practice. Meyer's formula, common in Indian textbooks, is E = C(ew − ea)(1 + u₉/16) in mm/day, with vapour pressures in mm of mercury, u₉ the monthly mean wind speed at about 9 m height in km/h, and C about 0.36 for large deep waters and 0.50 for small shallow waters. Water-budget, energy-budget and Penman combination methods are alternatives.
The simplest measurement is an evaporation pan: a standard tank of water whose level is read daily, with rainfall added back. Daily pan evaporation = (level at start − level at end) + rainfall in the day, in mm, taking account of any water added. The United States Class A pan is the most widely used internationally: a circular, unpainted galvanised pan about 1.21 m in diameter and 255 mm deep on a wooden frame. India's standard pan, specified by the Bureau of Indian Standards, is of similar size, painted white and covered with a hexagonal wire mesh to keep birds out.
A pan evaporates more than a lake because it is small, its walls absorb heat, and it has little heat storage. Lake evaporation is estimated as EL = Cp × Epan, where Cp is the pan coefficient. Commonly quoted textbook values are about 0.7 for the Class A pan (range roughly 0.6 to 0.8), about 0.8 for the Indian standard mesh-covered pan, and about 0.78 to 0.8 for sunken and floating pans. For crop water requirements, the FAO method converts pan evaporation to reference evapotranspiration with a different pan coefficient that depends on wind, humidity and fetch.
Example. A 2 km² farm reservoir, Class A pan average 6 mm/day, Cp = 0.7, over 30 days: EL = 0.7 × 6 × 30 = 126 mm, volume = 0.126 × 2 × 10⁶ = 252,000 m³. In semi-arid areas, losses from small ponds are cut by making them deep with a small surface area, by windbreaks, by shading, or by monolayers of cetyl alcohol (hexadecanol), which work in calm conditions but break up in wind.
Infiltration is the entry of water into the soil through its surface. Infiltration capacity (fp) is the maximum rate at which a soil in a given condition can absorb water; the actual infiltration rate equals the rainfall intensity when rain is gentler than the capacity, and equals the capacity when rain is more intense. Capacity is high at the start of a storm, when the soil is dry and the suction gradient is strong, and falls towards a nearly constant final rate as the soil wets up.
Infiltration increases with coarse texture, good structure, organic matter, dense vegetation and mulch, and with a dry initial soil. It decreases with surface crusting (raindrop impact on bare soil), compaction by machinery and livestock, high clay content, swelling clays, water-repellent soils, frozen ground and a high water table.
In the field, the double-ring (concentric-ring) infiltrometer is standard: two rings are driven into the soil, both are kept flooded, and the fall of water level in the inner ring is recorded. The outer ring acts as a buffer, reducing lateral spread of water from the inner ring so that it measures mainly vertical infiltration. Ring infiltrometers overestimate infiltration under rain because there is no drop impact; rainfall simulators (sprinkling-type infiltrometers) avoid that at higher cost.
Horton described the fall in capacity as exponential decay: fp = fc + (f₀ − fc) e^(−kt), where f₀ is the initial infiltration capacity, fc the final (constant) capacity, k a decay constant (per hour) and t time from the start of infiltration. Cumulative infiltration is the integral: F(t) = fc t + [(f₀ − fc)/k](1 − e^(−kt)). The equation fits many measured curves but is empirical: its parameters must be measured for each soil and condition, and it assumes the supply rate always exceeds capacity.
Example. f₀ = 60 mm/h, fc = 10 mm/h and k = 2 per hour. After 1 h, fp = 10 + 50 e⁻² = 16.77 mm/h, and the cumulative infiltration is 10 × 1 + (50/2)(1 − e⁻²) = 31.62 mm.
Philip derived an infiltration equation from the physics of unsaturated flow (Richards' equation). Its two-term form is F = S t^(1/2) + K t, with rate f = ½ S t^(−1/2) + K. S is the sorptivity, which expresses the soil's capacity to absorb water by capillary suction and depends on initial moisture; K here is a transmission term approaching the saturated hydraulic conductivity. At early times the sorptivity term dominates, which is why dry soils absorb water so fast at first; at long times the rate tends to K.
Kostiakov's empirical equation, F = a tᵇ with 0 < b < 1, fits surface irrigation data well, but its rate falls towards zero at long times, so a modified form adds a constant term. The Green-Ampt model treats the wetting front as a sharp boundary moving down through the soil; it uses measurable soil properties (hydraulic conductivity, wetting-front suction and moisture deficit). A student should know which models are empirical (Horton, Kostiakov) and which are physically based (Philip, Green-Ampt).
For runoff calculations on a whole catchment, varying infiltration is often replaced by a constant average loss rate. The φ-index is the average rate of rainfall above which the rainfall volume equals the runoff volume. On a hyetograph, draw a horizontal line at φ; the rainfall above the line is the effective rainfall, and its total must equal the observed direct runoff depth. φ is found by trial: assume which blocks exceed it, solve, check.
Worked example. A 5-hour storm has hourly depths of 6, 14, 22, 10 and 4 mm (total 56 mm), and the direct runoff measured at the outlet is 24 mm. Try assuming only the 14, 22 and 10 mm blocks exceed φ: (14 − φ) + (22 − φ) + (10 − φ) = 24, so 46 − 3φ = 24 and φ = 7.33 mm/h. Check: 6 and 4 are below 7.33 and 14, 22 and 10 are above, so the assumption holds. Total losses are 56 − 24 = 32 mm, but φ is not 32/5, because the low-intensity hours lost less than φ.
The W-index is the average infiltration rate during the time rainfall exceeds infiltration, after removing initial losses: W = (P − R − Ia)/te, where P is total rainfall, R direct runoff, Ia initial losses and te the time during which rainfall intensity exceeds the infiltration rate. Because initial losses are removed, W is less than or equal to φ. Both vary with antecedent moisture.
The rational method estimates the peak discharge from a small catchment: Q = C i A / 3.6, with Q in m³/s, i the rainfall intensity in mm/h, A the area in km², and C the dimensionless runoff coefficient. With A in hectares the form is Q = C i A / 360. In US customary units, Q (ft³/s) = C i (in/h) A (acres). Mixing the km² and ha constants gives an answer 100 times wrong.
The method's logic: if rain of constant intensity falls long enough, the whole catchment eventually contributes to flow at the outlet, and the flow reaches an equilibrium peak. The time for this is the time of concentration (tc), the time for water to travel from the hydraulically most remote point to the outlet. Therefore the design rainfall duration is set equal to tc, and i is read from the IDF curve for that duration and the chosen return period. Its assumptions are that rainfall is uniform in space and time over the duration, that the return period of the peak equals that of the rainfall, and that C is constant through the storm. These hold only for small catchments; agencies set different size limits. It gives a peak only, so it sizes culverts, waterways and drains, not storage.
Many empirical formulas estimate tc. The Kirpich formula, widely used for small agricultural catchments, is tc = 0.01947 L^0.77 S^(−0.385), with tc in minutes, L the maximum length of travel in metres and S the average slope (m/m) along that path, often taken as the fall H divided by L. Longer paths and flatter slopes give longer tc.
The runoff coefficient C is the ratio of peak runoff rate to rainfall intensity. It is higher for impervious surfaces, steep slopes, clayey soils and bare land, and lower for sandy soils, flat slopes, forest and dense grass. Tables of C exist in every national manual; values differ between them, so use the local table. For a mixed catchment, use an area-weighted coefficient: C = ΣCⱼAⱼ/ΣAⱼ.
Worked example. A 50 ha catchment has 30 ha of cultivated land (C = 0.35) and 20 ha of pasture (C = 0.20). The longest flow path is 850 m with an average slope of 0.02. Kirpich gives tc = 0.01947 × 850^0.77 × 0.02^(−0.385) = 15.8 min. Suppose the local IDF curve gives i = 75 mm/h for that duration at the design return period. Weighted C = (0.35×30 + 0.20×20)/50 = 0.29. Q = 0.29 × 75 × 50/360 = 3.021 m³/s.
The curve number (CN) method, developed by the United States Soil Conservation Service (now the Natural Resources Conservation Service, NRCS), estimates the depth of direct runoff from a storm's rainfall depth. It rests on the hypothesis that the ratio of actual retention to potential maximum retention equals the ratio of actual runoff to potential runoff: F/S = Q/(P − Ia), with continuity P = Ia + F + Q. Here P is rainfall, Q direct runoff, Ia initial abstraction, F retention after runoff begins and S potential maximum retention. Solving gives Q = (P − Ia)²/(P − Ia + S) for P > Ia, and Q = 0 otherwise. With the standard assumption Ia = 0.2S, Q = (P − 0.2S)²/(P + 0.8S).
S is linked to the dimensionless curve number, which runs from 0 to 100: in SI units S (mm) = 25400/CN − 254; in inches, S = 1000/CN − 10. CN = 100 means an impervious or water surface where all rain runs off; lower CN means more retention.
Worked example. CN = 78 and P = 90 mm. S = 25400/78 − 254 = 71.64 mm; Ia = 0.2S = 14.33 mm; Q = (90 − 14.33)²/(90 − 14.33 + 71.64) = 38.87 mm. Over a 50 ha catchment this is 38.87 × 10 × 50 = 19436 m³.
The CN is read from NRCS tables by land use, treatment (for example contoured, terraced, straight row), hydrologic condition (good, fair, poor cover) and hydrologic soil group. The four soil groups are: A, deep sands and well-drained soils with high infiltration and low runoff potential; B, moderately fine to moderately coarse soils with moderate infiltration; C, soils with a layer that impedes drainage or moderately fine texture, with slow infiltration; D, clays, soils with a high water table or shallow soils over impervious material, with very slow infiltration and high runoff potential. For mixed catchments, an area-weighted CN is used.
The standard tables give CN for average antecedent moisture condition (AMC II). Classic textbook practice adjusts for dry (AMC I) and wet (AMC III) conditions, judged from the 5-day antecedent rainfall and season, with conversions CN(I) = CN(II)/(2.281 − 0.01281 CN(II)) and CN(III) = CN(II)/(0.427 + 0.00573 CN(II)). NRCS now calls this antecedent runoff condition; exams still use AMC.
Limits: the method gives runoff depth, not timing; it ignores rainfall intensity within the storm; the fixed Ia = 0.2S has been questioned, with research supporting smaller ratios such as 0.05 for many watersheds; and it is least reliable for small storms and low CN, where small changes in CN change Q greatly.
A hydrograph plots discharge at a point against time. A single-peaked storm hydrograph has a rising limb (concentration curve), a crest segment containing the peak, and a falling limb or recession. The rising limb reflects the growing area contributing flow; the recession reflects the draining of storage in channels, soil and groundwater. Key times: the time to peak (tp) is measured from the start of effective rainfall (or, in some definitions, from its centroid) to the peak; the basin lag is from the centroid of effective rainfall to the centroid or peak of the hydrograph; the time base is the total duration of direct runoff.
Shape depends on the catchment and the storm. Fan-shaped (nearly circular) catchments give high, early peaks because tributaries deliver water together; elongated (fern-shaped) catchments give lower, longer hydrographs. Steep slopes, dense drainage networks and impervious cover raise and sharpen the peak; lakes, floodplains, forests and swamps flatten it. A storm moving downstream gives a higher peak than one moving upstream.
The unit hydrograph works with direct runoff, so baseflow must first be separated. The simplest method draws a straight line from the start of the rising limb to the point on the recession where direct runoff ends. A common way to fix that end point is a time N days after the peak, N = 0.83 A^0.2, with A in km²; this is an empirical guide from large catchments. Other methods extend the prior recession under the peak; for large floods the choice matters little.
Recession curves are commonly described by an exponential decay, Qt = Q₀ Krᵗ, where Kr is the recession constant (less than 1) for the chosen time unit. Plotting log Q against t turns the recession into a straight line; its slope gives Kr. Baseflow recession is also used to estimate groundwater storage and low flows for irrigation planning.
Sherman's unit hydrograph (1932) is the hydrograph of direct runoff resulting from one unit depth of effective rainfall (1 cm in Indian texts, 1 mm or 1 inch elsewhere) occurring uniformly over the catchment at a uniform rate for a specified duration D. Two assumptions make it useful. Linearity (proportionality): the direct runoff ordinates from an effective rainfall of r units of the same duration are r times the unit hydrograph ordinates, and the time base is unchanged. Time invariance: the same effective rainfall produces the same response whenever it occurs. With superposition, a storm of several consecutive D-hour blocks produces a hydrograph equal to the sum of scaled unit hydrographs, each lagged by the start of its block.
Volume check. The volume under a unit hydrograph equals 1 unit of depth over the catchment. For a 1 cm unit hydrograph with ordinates Q (m³/s) at interval Δt (hours), ΣQ × Δt × 3600 = A × 10⁶ × 0.01, so A (km²) = 0.36 ΣQ Δt.
Real catchments are not perfectly linear and rain is rarely uniform over large basins, so large basins are split into sub-catchments and routed.
From an isolated storm with nearly uniform effective rainfall of duration D: separate baseflow to get the direct runoff hydrograph; compute the runoff volume by summing ordinates and convert it to depth over the catchment; divide each direct runoff ordinate by that depth (in cm) to get the D-hour unit hydrograph. Several storms are analysed and averaged by peak and time to peak, not ordinate by ordinate.
Applying it (convolution). Suppose a 2-hour unit hydrograph has ordinates 0, 20, 50, 30, 10, 0 m³/s at 2-hour intervals, and a storm gives 2 cm and 3 cm of effective rainfall in two successive 2-hour blocks. Multiply the unit hydrograph by 2, multiply it by 3 and shift one interval, then add: the direct runoff ordinates are 0, 40, 160, 210, 110, 30, 0 m³/s. The peak, 210 m³/s, occurs 6 hours after the start. Adding the baseflow gives the flood hydrograph. For design, the effective rainfall blocks come from a design storm after subtracting losses by the φ-index or curve number.
Unit hydrographs are needed for durations other than the one derived. When the new duration is an integer multiple of the old (for example a 6-hour UH from a 3-hour UH), add two 3-hour unit hydrographs lagged by 3 hours; the sum represents 2 cm in 6 hours, so divide the ordinates by 2 to return to 1 cm.
For any duration, including non-multiples, use the S-curve (S-hydrograph). The S-curve is the hydrograph produced by a continuous effective rainfall at the rate of 1 cm per D hours, lasting indefinitely. It is built by adding an infinite series of D-hour unit hydrographs, each lagged by D hours; in practice, add the ordinates of the unit hydrograph to the S-curve ordinate D hours earlier. The S-curve rises to a constant equilibrium discharge Qe, when the whole catchment contributes at the rainfall rate: Qe = 2.778 A/D m³/s, with A in km² and D in hours (for a 1 cm unit depth). For a 250 km² catchment and a 4-hour unit hydrograph, Qe = 2.778 × 250/4 = 173.6 m³/s. Oscillation near its top shows the unit hydrograph or its duration is inconsistent.
To get a T-hour unit hydrograph, shift the S-curve by T hours and subtract: the difference is the hydrograph of 1 cm × T/D of rainfall in T hours, so multiply by D/T: U(T, t) = (D/T)[S(t) − S(t − T)].
As the duration D shrinks to zero while the depth stays 1 unit, the unit hydrograph tends to a limit called the instantaneous unit hydrograph (IUH), u(t): the response to one unit of effective rainfall applied instantaneously. The IUH is a catchment's impulse response function; its ordinates have units of 1/time, and its area is unity. Any storm's direct runoff is the convolution integral of the effective rainfall intensity with u(t).
Conceptual models give the IUH a formula. The Nash model represents the catchment as a cascade of n identical linear reservoirs, each with storage constant k (S = kQ). Its IUH is u(t) = [1/(k Γ(n))](t/k)ⁿ⁻¹ e^(−t/k); the first moment about the origin (the lag) is nk and the second moment about the centroid is nk². The Clark model routes a time-area diagram (the proportion of catchment area contributing at each travel time, from isochrones) through a single linear reservoir; its parameters are tc and the storage coefficient.
Most small agricultural catchments have no streamflow records, so unit hydrographs are built from catchment characteristics: synthetic unit hydrographs.
Snyder's method, developed for the Appalachian region of the United States, relates basin lag to catchment length: tp = Ct (L Lc)^0.3, where L is the main-stream length to the divide, Lc the distance along the stream to the point nearest the catchment centroid, and Ct a regional coefficient (its value depends on units and on the region). The standard duration is tr = tp/5.5; the peak is Qp = 2.78 Cp A/tp in SI form (Qp in m³/s for A in km², tp in hours, 1 cm unit depth), with Cp a regional peaking coefficient. For other durations tR, the lag is adjusted: tpR = tp + (tR − tr)/4. Because Ct and Cp are regional, Snyder's method must be recalibrated before use outside the region of origin.
The NRCS method uses a standard dimensionless unit hydrograph, or its triangular approximation. Lag tL is taken as 0.6 tc; time to peak Tp = D/2 + tL; time base of the triangle is 2.67 Tp; peak discharge qp = 0.208 A Q/Tp, with qp in m³/s, A in km², Q the runoff depth in mm and Tp in hours (0.208 in SI corresponds to 484 in US units). For a 1 mm unit, Q = 1.
In India, the Central Water Commission's regional synthetic unit hydrograph reports for hydro-meteorological subzones are the design reference.
Discharge equals the sum over the cross-section of velocity times area. The section is divided into vertical segments; at each vertical the depth is sounded and the mean velocity measured, commonly with a current meter. Enough verticals are used that no segment carries a large share of the flow (common textbook guidance is no more than about 10% per segment). Standard observations: in shallow water, one reading at 0.6 of the depth measured from the surface gives the mean velocity; in deeper water, the mean of readings at 0.2 and 0.8 of the depth is taken. Surface velocity multiplied by a coefficient (typically about 0.85, range roughly 0.8 to 0.95) gives an approximate mean, which is how float measurements are used in floods.
In the mid-section method each vertical represents a segment extending halfway to its neighbours: segment discharge = v × y × (half the distance to the previous vertical + half the distance to the next). In the mean-section method, each segment lies between two verticals, and its discharge is the average of their velocities times the average of their depths times the width between them.
A current meter converts rotation to velocity. Cup-type (vertical axis) and propeller-type meters are calibrated in a towing tank, giving a rating equation v = aNs + b, where Ns is revolutions per second and b reflects the starting velocity. Acoustic Doppler current profilers now measure the whole section from a moving boat.
Dilution (tracer) gauging suits turbulent mountain streams where current meters cannot be used. In the constant-rate injection method, a tracer of concentration C₁ is injected at a steady rate q; after complete mixing downstream, the stream concentration rises from background C₀ to a plateau C₂. Mass balance gives Q = q (C₁ − C₂)/(C₂ − C₀). In the sudden-injection (gulp) method, a known mass is added once and the concentration-time curve is integrated. Common tracers are common salt (measured by conductivity), fluorescent dyes and, historically, radioactive isotopes. The reach must be long enough for full mixing, and tracer must not be lost to adsorption.
Hydraulic structures such as sharp-crested and broad-crested weirs and flumes (for example the Parshall flume) give discharge from a single head measurement through a calibrated equation, and are used on small streams, irrigation channels and research watersheds. The slope-area method estimates peak flood discharge after the event from high-water marks at two sections, the cross-section, and Manning's equation with an assumed roughness.
Stage is the elevation of the water surface above a fixed datum at a gauging station. It is easy to measure continuously, and discharge is then obtained from a rating curve. Instruments include staff gauges, float recorders in a stilling well (which damps waves), bubble gauges and pressure transducers, and non-contact radar sensors. A good station has a stable, straight reach with a control downstream: a section control such as a rock ledge or a weir, or a channel control where the reach's own roughness and geometry set the stage-discharge relation.
The rating curve converts continuous stage into continuous discharge. It is fitted to many paired measurements of stage G and discharge Q, commonly as Q = Cr (G − a)^β, where a is the stage of zero flow and Cr and β are constants. Taking logarithms, log Q = log Cr + β log(G − a) plots as a straight line; a is found by trial or from the channel profile. Ratings must be checked regularly and shifted when the control changes through scour, deposition or weed growth.
In unsteady flow the relation is not single-valued: for the same stage, discharge is greater on the rising limb than on the falling limb, because the water-surface slope is steeper as the flood wave arrives. The plotted rating forms a loop (hysteresis). Backwater from a downstream confluence, dam or tide also disturbs the relation; a second gauge measures the surface slope and a slope correction is applied. To extend a rating beyond the highest measured flow, logarithmic extrapolation, Manning-based or Stevens methods are used, cautiously, because flow may leave the main channel onto the floodplain.
Flood routing predicts how a flood hydrograph changes as it passes through a reservoir or down a river reach. Storage in the reservoir or channel does two things: it attenuates the peak (lowers it) and it translates or lags it (delays it); the flood volume is the same apart from losses. Hydrologic routing uses only the continuity equation with a storage relationship: I − O = dS/dt, where I is inflow, O outflow and S storage. Over a time step Δt, with average values: (I₁ + I₂)/2 × Δt − (O₁ + O₂)/2 × Δt = S₂ − S₁. Hydraulic routing solves both continuity and momentum (the Saint-Venant equations) and is needed where backwater, tidal effects or very flat slopes matter. The Muskingum-Cunge method links Muskingum parameters to the diffusion wave. The time step must be small compared with the time to peak of the inflow hydrograph, often taken as no more than about one-fifth of it.
In a reservoir with an ungated spillway, storage and outflow are both functions of the water level, so storage is a function of outflow alone. The modified Puls (storage-indication) method rearranges continuity with unknowns on the right: (I₁ + I₂)/2 × Δt + (S₁ − O₁Δt/2) = (S₂ + O₂Δt/2). From the elevation-storage curve and the spillway equation (for example Q = C L H^(3/2) for an overflow weir), a table of O against (S + OΔt/2) is prepared beforehand. At each step, the left side is known, which gives S₂ + O₂Δt/2, from which O₂ is interpolated; then S₂ − O₂Δt/2 = (S₂ + O₂Δt/2) − O₂Δt is carried to the next step. Goodrich's method is the same equation multiplied by 2/Δt: (I₁ + I₂) + (2S₁/Δt − O₁) = (2S₂/Δt + O₂).
A useful property: for an uncontrolled reservoir where storage depends only on outflow, the peak outflow occurs at the moment the outflow hydrograph crosses the falling limb of the inflow hydrograph, because at that instant I = O and storage is at its maximum. Routing is used to size spillways, to find the maximum water level for setting freeboard, and to check farm ponds and check dams against design floods.
In a river reach, storage depends on both inflow and outflow, because during a rising flood the water surface is not parallel to the bed: prism plus wedge storage. McCarthy's Muskingum method writes S = K[xI + (1 − x)O], where K is a storage time constant roughly equal to the travel time through the reach, and x is a weighting factor between 0 and 0.5. x = 0 gives a linear reservoir (storage depends only on outflow, maximum attenuation); x = 0.5 gives pure translation without attenuation. For natural channels x commonly lies between about 0.1 and 0.3.
Substituting into continuity gives O₂ = C₀I₂ + C₁I₁ + C₂O₁ with C₀ = (−Kx + 0.5Δt)/(K − Kx + 0.5Δt), C₁ = (Kx + 0.5Δt)/(K − Kx + 0.5Δt), C₂ = (K − Kx − 0.5Δt)/(K − Kx + 0.5Δt), and C₀ + C₁ + C₂ = 1. Example: K = 12 h, x = 0.2, Δt = 6 h gives C₀ = 0.0476, C₁ = 0.4286, C₂ = 0.5238. To keep all coefficients non-negative, choose 2Kx ≤ Δt ≤ 2K(1 − x).
K and x are estimated from an observed flood: compute storage by accumulating (I − O)Δt, then plot S against xI + (1 − x)O for several trial values of x; the value giving the narrowest (most nearly single-valued) loop is chosen, and K is the slope of that line.
Drought is a temporary shortage of water relative to normal conditions, which distinguishes it from aridity, a permanent feature of climate. It develops slowly, has no clear start or end, and spreads over large areas, so it is often called a creeping hazard. Four types are standard. Meteorological drought is a shortfall of precipitation below normal over a period. Agricultural drought is a shortage of soil moisture, at a critical time, below what crops need, leading to yield loss; it can occur with near-normal seasonal rain if the rain is badly distributed. Hydrological drought is a shortfall in streamflow, reservoir storage, lake levels or groundwater; it lags behind meteorological drought and can persist after rain returns. Socio-economic drought occurs when the shortage affects the supply and demand of economic goods and people's livelihoods. Some authors add ecological drought.
India's classic textbook definitions follow the India Meteorological Department (IMD). An area is considered to have meteorological drought when the seasonal rainfall received is less than 75% of its long-term normal, that is, a deficiency of more than 25%; drought is termed moderate when the deficiency is 26 to 50% and severe when it exceeds 50%. For reporting seasonal or weekly rainfall, IMD uses departure categories: excess (20% or more above normal), normal (within ±19%), deficient (−20 to −59%), large deficient (−60 to −99%) and no rain (−100%); an additional large-excess category (60% or more) is also used.
Since 2016 the Government of India's Manual for Drought Management has moved drought declaration from rainfall alone to a combination of indicators: rainfall deviation and dry spells, the standardized precipitation index, remote-sensing vegetation indices, soil moisture, hydrological indices, and crop sowing and yield information, with ground truthing. Thresholds in that manual have been revised over time; use the current edition. Other countries use their own systems, such as the United States Drought Monitor's D0 to D4 categories.
The Standardized Precipitation Index (SPI), proposed by McKee and colleagues in 1993 and recommended by the WMO as a common meteorological drought index, fits a probability distribution (usually gamma) to long-term precipitation totals for a chosen time scale (1, 3, 6, 12 months or more), then transforms the cumulative probability to a standard normal variable. SPI of 0 is the median; negative values are drier than normal. The standard classes are: −1.0 to −1.49 moderately dry, −1.5 to −1.99 severely dry, −2.0 and less extremely dry, with matching wet classes. Short time scales reflect soil moisture and agricultural drought; long time scales reflect streamflow, reservoirs and groundwater. SPI ignores evaporative demand; the Standardized Precipitation-Evapotranspiration Index adds it.
The Palmer Drought Severity Index uses a soil water balance and responds slowly. Agricultural indices include the moisture adequacy index (actual over potential evapotranspiration, as a percentage) and crop moisture indices. Remote-sensing indices include the Normalized Difference Vegetation Index (NDVI) and the Vegetation Condition Index, which scales current NDVI between the historical minimum and maximum for that place and time of year.
A student needs this subject as a toolkit of chained calculations. The core tasks are: average rainfall by Thiessen polygons, fill a missing record by the normal ratio method, compute return period and risk, find a φ-index by trial, estimate runoff by the rational and curve number methods, convolve a unit hydrograph, build an S-curve, and route a flood with Muskingum coefficients. Practise each until the units are automatic: mm to m³, km² to ha, cm unit hydrographs, the 0.36 and 2.778 constants. Most GATE and ICAR marks here come from these calculations, and most lost marks come from unit slips and from not checking conditions such as the 10% rule or P > 0.2S.
An engineer five years into soil conservation, watershed development or irrigation work uses this subject weekly. Typical tasks: sizing a grassed waterway, gully control structure or culvert from a rational-method peak; estimating the runoff yield of a catchment feeding a farm pond or check dam with the curve number method and deciding the pond volume; routing a design flood through a pond to set the spillway length and freeboard; checking rain-gauge and streamflow data before using them. The decisions are practical: which return period the agency requires, which C or CN table applies to the local soils, and whether the catchment is too large for the rational method.
A senior engineer or superintending officer with 25 years' experience sets standards and reviews designs. Tasks include approving design-flood estimates for minor irrigation tanks and check dams, choosing regional methods (synthetic unit hydrograph reports, IDF atlases) for a state or district, deciding where to invest in gauging stations and telemetry, and setting drought triggers with district administrations. The key decisions are about risk: which return period and freeboard are acceptable for structures whose failure would endanger villages, how to allow for land-use change and a changing climate in rainfall statistics, and when to insist on a site-specific study instead of a table.
A researcher with 25 years' experience works where the textbook methods break down. Questions include: how well curve numbers and unit hydrographs represent nonlinear, variable-source-area runoff; calibrating and regionalising parameters for ungauged catchments using remote sensing, GIS-derived geomorphology and machine learning; non-stationary frequency analysis under climate change; coupling hydrological models with crop and groundwater models for water accounting; and validating satellite rainfall, soil-moisture and evapotranspiration products. The PhD-level contribution is honest uncertainty: quantifying how errors in rainfall, parameters and model structure propagate into design floods and drought declarations.
Many coefficients here (runoff coefficients, curve numbers, pan coefficients, Snyder coefficients, drought thresholds) are empirical and regional and are given as teaching values; for design, use your national or state agency's values checked against local data. This manual does not yet cover snow hydrology and snowmelt runoff, detailed depth-area-duration analysis, the probable maximum precipitation, full Saint-Venant hydraulic routing, continuous simulation models (for example SWAT or HEC-HMS model set-up), groundwater hydrology (covered in the groundwater, wells and pumps subject) or watershed morphometry and planning (covered in watershed planning and management). Tables of runoff coefficients, curve numbers, Gumbel reduced-variate statistics and regional Snyder and IDF constants are not reproduced, because values differ by source and region; use the tables of your national agency. Indian drought declaration thresholds are summarised in outline only because the national manual is revised from time to time.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions.
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