Derivation of Unit Hydrograph (UH)

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  • To understand the type of hydro-meteorological records needed for UH derivation
  • To familiarize with steps involved in extraction of Direct Runoff Hydrograph (DRH) from flood hydrograph and rainfall excess from rainfall hyetograph
  • To determine UH by different approaches
  • To learn how to convert a UH of given duration to another duration
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  The Unit Hydrograph (UH) is the simplest but at the same time a very powerful tool for hydrological analysis in general and flood forecasting in particular.

The unit hydrograph may be defined as the direct runoff (outflow) hydrograph resulting from one unit of effective rainfall, which is uniformly distributed over the basin at a uniform rate during a specified period of time known as unit time or unit duration. The following paragraphs make this statement still clearer.


Effective rainfall should be uniformly distributed over the basin, i.e. if there are five rain gauges in the basin, which represent the areal distribution of rainfall over the basin, all the five rain gauges should record for almost same amount of rainfall during specified time. A watershed shown on the right here fully marks this stipulation, while converse is true in respect of left one.



In addition, effective rainfall should be at a uniform rate during the unit duration. If the average rainfall over a particular basin during 6 hour is 126mm, a unit hydrograph of 6 hours duration can be derived only if the intensity of rainfall is more or less 21 mm/hour over 6 hours. If the same amount of rainfall is distributed with varied intensity, the unit hydrograph cannot be precisely estimated by simple method.

  The unit quantity of effective rainfall is normally taken as 1mm or 1cm; and the outflow hydrograph is expressed by discharge in cumec. The unit duration may be of 1-hour duration or more, depending upon the size of the catchment, storm characteristics and operational facilities. However, the unit duration cannot be more than the time of concentration or basin lag or period of rise. The concept of time of concentration has been covered in detail later in the chapter.
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  The following are the basic assumption in the unit hydrograph theory:    

The unit hydrograph theory assumes the principle of time invariance. This implies that the direct runoff hydrograph from a given drainage basin due to a given pattern of effective rainfall will be always same irrespective of the time, i.e. even if the basin characteristics change with season etc., the unit hydrograph remains the same.


Unit Hydrograph theory assumes the principle of linearity, superimposition or proportionality. It means that:


  • If the ordinates of a unit hydrograph of say 1 hour duration are 0,1,6,4,3,2,1,0 units respectively, the effective rainfall of 2 units falling in 1 hour will produce a direct runoff hydrographs having ordinates of 0,2,12,8,6,4,2,0 units.

  • Secondly, if the effective rainfall of two units occurs in 2 hours, i.e. 1 unit per hour, the direct runoff hydrograph ordinates will be obtained by summing up the corresponding ordinates of the two unit hydrographs as shown here.
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  Selection of a particular UH derivation techniques primarily governed by three factors:  

The UH is best derived from the observed hydrograph resulting from a storm which fulfils the two basic conditions i.e., the rainfall is more or less uniformly distributed over the basin and has a reasonably uniform intensity. Such a hydrograph will generally form an isolated peak.


In case, such a hydrograph is not available, the UH has to be derived from the analysis of an observed multi-peaked flood hydrograph resulting from several spells of rainfall of varying intensities.


When the observed discharge and rainfall data at short interval are not available, the synthetic UH is derived with the help of basin characteristics. .

  In this module, UH derivation under all three conditions has been illustrated step-wise. At the end of this module, it is expected that reader would be able handle UH assignment independently.  
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  The steps involved in derivation of UH from the analysis of the flood hydrograph with a single peak are as follows;  
  Inspect discharge records at watershed outlet and corresponding rainfall events to identify events exhibiting isolated, well defined and single peak with considerable run-off volume. Pick up as many sets of such records as available. A plot displaying rainfall and corresponding rise in flood hydrograph, such as here, can help selection of records.  
  Note - A no-break/continuous discharge series, as shown in the plot, is developed by transforming hourly river stage (also water level) into discharge with the help of rating equation. Rating equation used for this purpose must be developed for the period to which flood event belongs to.  
  A rating equation/curve is an equation that relates discharge with water level observed at a site, and is mathematically expressed as

Q = c * (G - Go)n

Where, c, Go; & n are constants; and G is water level.

While gathering information as listed above, it is recommended that


 Storms with rainfall should have been active for duration of around 20 to 30 % of basin lag. Various studies estimate basin lag as 50-75% of Tc, time of concentration. Later part of this chapter describes ways to estimate Tc,


Storms should have generated rainfall excess between 1 cm and 4.5 cm.

  2.    A flood hydrograph is a basin (catchment) response driven by occurrence of rainfall event plus contribution of base flow. Had there been no rainfall over the basin, 'Bulge' (rise) in flood hydrograph would have not appeared. Secondly, all water that falls over a catchment does not reach the river/stream because of 'losses'; and only a fraction of it contributes to this 'Bulge'. This bulge is termed as Direct Runoff Hydrograph (DRH). The part that reaches the stream is called as 'Rainfall excess'. Hydrologist seeks to develop a relationship between 'rainfall excess' and DRH. Apparently, the next step is separation of base flow from flood hydrograph to compute volume of DRH. Following are couple of methods outlined for separation of base flow.  
    Fixed base method (A-B-D)

This method suggests the extension of the base flow line along its general trend before the rise of the hydrograph up to a point B directly below the runoff hydrograph peak. From B, a straight line BD is drawn to meet the hydrograph at point D, which is N days away from B in the time scale. 'N' is determined by an empirical relation by Linsley as:

N (in days) = 0.83 A0.2

Where, A is the area of the drainage basin in square kilometers.

    Variable Slope Method (A-B-C-E)

This method requires identification of two additional points on the recession limb of hydrograph - one is inflection point; while the other is point E. At inflection point, curve changes its concavity. This point also indicates end of surface flow to river. This point beyond, discharge is a combination of interflow (also called as sub-surface flow) and base flow. After a while, interflow also ceases; and only base flow remains in the river. The 'E' suggests this stage. Once, these two points are located on the graph, a line from 'E' is drawn backward to meet a vertical line from inflection point. A line A-B-C-E divides the DRH and base flow.

Nevertheless, for flood studies, the base flow component is rather insignificant and hence does not influence the magnitude of peak runoff substantially. Therefore, inaccuracies involved in separation of base flow are not crucial in overall flood studies.


3.    Computation of direct runoff hydrograph ordinates by deducting base flow ordinates from that of the corresponding observed flood hydrograph.

4.    Scanning and analysis of the rainfall data of all rain gauge stations in and around the basin with a view to;


Obtaining areal rainfall over the catchment by appropriate methods, such as Thiessen Polygon or Isohyetal technique, and


Estimating phi-index. Volume of DRH equals the product of catchment area and rainfall excess over the basin. This simple analogy helps us estimate depth of rainfall excess.

Rainfall Excess = Volume of DRH / Catchment Area

A gap between effective rainfall and averaged rainfall points to losses. Here, in the plot, ? - index (also known as loss rate) is drawn in a manner that partitions hyetograph into two parts- lower indicates losses, while upper rainfall excess.


5.    As DRH is a consequence of given rainfall excess, say 'x' unit. Estimation of the ordinates of the UH is obtained by dividing the ordinates of direct runoff hydrograph by 'x' rainfall excess.

  • How to fix duration of UH?

A plot shown here exhibits duration of rainfall as 5-unit, of which only during 3-unit duration rainfall exceeds 'loss rate'. Thus, for this case, UH duration is a 3-unit.

6.    This process is repeated for all records picked up for this purpose.

7.    It is highly probable that UHs derived for more than one record may differ in duration of excess rainfall. They need to be converted to an identical duration before attempting step 8. A discussion on conversion of UH duration has been added toward the end of this module.

8.    All such UHs are eventually averaged. For this, first peaks, Qp, of all UH are averaged to give Qp, followed by time to peak, tp and time base, Tb of UHs. All other ordinates are adjusted in such a way that total run-off volume of UH equals the product of 1cm/mm and catchment area.

More discussions on conversion of UH duration and averaging of UH have been added toward the end of this chapter.

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Flood hydrographs with a single and sharp peak resulting from an intense and uniform rainfall are very uncommon. Often times, observed hydrographs contain multiple peaks of various magnitudes resulting from several spells of rainfall. UH, in such cases, are derived by

- Collins' Method
- Matrix method
- Instantaneous Unit Hydrograph

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This method uses trial and error approximations to compute UH from complex hydrograph. The basic steps involved in this method can be best gathered by an illustrative example. This example is available in excel file of this week schedule, which can be downloaded by participants.


The direct runoff hydrograph at site-S and the effective hydrograph due to a particular storm over the catchment of river-R are tabulated below. We will use the given DRH and ERH to derive unit hydrograph because of 1mm effective rainfall. Catchment area is 8570 km2.


  Note- The effective rainfall hyetograph blocks are for 3 hour intervals. Therefore, the unit duration of the unit hydrograph thus derived will be of 3-hour unit duration.  

The following steps guide the reader to obtain UH by this method.

i) From the observed flood hydrograph and observed rainfall hyetograph DRH and ERH are separated as explained earlier;

ii) The ordinates of the DRH at different time are written under Column 2 of the Table 3;

iii) Trial & error of determining UH ordinates begins with selection of first set of trial values representing ordinates of UH. This we can do by dividing the ordinates of DRH with 11mm effective rainfall, and recording them under column 3 of the Table 3;

iv) The ordinates of the assumed unit hydrograph are summed up as 785.4 cumec. But the sum of ordinates of the unit hydrograph at 3 hour interval, SU for a catchment area of 8570 km2 should be:

Catchment Area * rainfall depth (1mm) = volume of UH

                                          = Sum of UH ordinates, SU * time interval between UH ordinates


where A = area of catchment in km2, =8570 km2, & t = time interval in hour = 3 hours

v) In order to satisfy condition stated above, all the assumed ordinates are multiplied by a factor 1.01 (793/785.4) and the adjusted values are entered in col.4 of the table;

vi) The ordinates on adjusted UH (col.4) are multiplied by 2.4, first burst of rainfall, and are written under col.5. Similarly, the ordinates of adjusted unit hydrograph (col.4) are multiplied by 3.0 mm, third burst of rainfall, and are reflected in col.7 after shifting it by 6 hours. Why this shift is warranted here? - It is so because 3.0 mm rainfall begins after 6hrs from the start of the storm.;

vii) The ordinates in col.5 and 7 are now added together and written under column 8. This gives the DRH resulting from rainfall excess (2.4mm + 3.0mm) excepting the largest one, i.e. 5.6mm;

viii) The DRH ordinates obtained in col.8 are deducted from the ordinates of DRH in col.2, and are noted down in col.6. This is a DRH as a result of 5.6mm of rainfall. First ordinate due to 5.6 mm rainfall is made zero as this represents the beginning of contribution of 5.6mm rainfall;

ix) The values in col.6 are divided by 5.6mm to give the unit hydrograph, and are fed in col.9. Since, this is a UH, it is necessary to re-validate its volume against condition stipulated under Para (iv) above. The sum of ordinates of this UH as written in col. 9 is now 777. These ordinates are, therefore, multiplied by 1.02 (793/777) to readjust its values as reflected in col.10.

x) The weighted average of the two unit hydrographs (the assumed one as in column 4 and calculated one as in col.10) is reached in the following manner:

  xi) The weighted average ordinates are written under column 11 of the Table-3. The ordinates of the unit hydrograph in column 11 and column 4 are now compared for differences. If significant differences are noticed between the two, 2nd iteration begins with unit hydrograph ordinates of column 11; and this will occupy column 4. Thus, the process is iterative and will go on till the differences in the assumed and calculated unit hydrographs reduce to insignificant level.

  The ordinates of the derived unit hydrograph are tabulated below. At times, it is possible that the lower part of the unit hydrograph may not be uniform. In such cases, it is smoothened and redrawn smoothly with least variations in a manner that its volume should fulfil condition of Para (iv).  
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  The discrete convolution equation allows the computation of direct runoff, Qn given excess rainfall, Pm and the unit hydrograph, Un-m+1.  


The reverse process, called de-convolution, can be utilised to derive a unit hydrograph given data on Pm and Qn. Suppose that there are 'M' pulses or burst of rainfall excess and 'n' pulse of direct runoff in the storm considered; then N equation can be written for Qn, n = 1,2,….,n, in terms of (n - m + 1) unknown values of the unit hydrograph.
If Qn and Pm are given and Un-m+1 is required, the set equations is over determined, because there are more equations (N) than unknowns (n - m + 1). The term n < M in the equation restrains the total nos. of P*U terms for Qn. In first case, when n is < M, m = 1, 2, ......n; while in case n is > M, m = 1,2, ….M.

(For more details, reader may refer to Applied Hydrology by Ven Te Chow)

Let us derive term for Q1, assuming total number of rainfall pulse, M = 3

Here, n =1 < (M =3), hence, m =1, therefore,



An observed hydrograph with rainfall excess is given as under. The time interval is 3 hours between readings. Catchment area is 7092 km2.

Table 1: Ordinates of DRH and Effective Rainfall Hyetograph (ERH)


Let us first define number of equations.

There are 3 pulses of rainfall so M = 3. There are 13 pulses of observed direct runoff so n = 13. The number of unit hydrograph ordinates is therefore, n - m +1 = 13 - 3 + 1 = 11 ordinates.

Applying this piece of information, set-up of matrices for Q, P and U appear as below


Where, [Q] is 13 by 1 matrix with discharge ordinates; [P] is a 13 by 11 matrix having three rainfall impulse of given duration; and [U] is unknown matrix of 11 by 1 size whose ordinates are to be determined. In matrix form, information of tabular chart reduces to

[Q]13*1 = [P]13*11 . [U]11*1

To solve this problem for [U] matrix, we choose only 11 equations to obtain 11 unknown UH ordinates using following equation.

[U] = [P]-1 [Q]

Result obtained using MS excel shown next has generated a few negative terms in the falling limb of UH; and this has to be adjusted by the reader in such a way as to total volume of UH must be equal to the volume of runoff emerging form the catchment as a result of 1mm uniform & effective rainfall over it. This example, thus, underlines the likelihood of a few negative and abnormal terms in the calculation and need subsequent readjustment.


Adjusted unit hydrograph is shown below.


At this stage, reader is advised to refer to the excel sheet for familiarizing themselves with calculation part of this example.

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The Clark model uses two parameters, time of concentration, 'Tc', and storage constant, 'K', and a time-area histogram concept. Before, we set out for UH by Clark model, let us first familiarize ourselves with these new terminologies.

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The first parameter, time of concentration is the time taken by a water particle from the hydraulically farthest point to the basin outlet. An estimate of this travel time is the time from the end of runoff producing rainfall over the basin to the inflection point on the recession limb of the direct Runoff Hydrograph (DRH). Because of complexities involved in rainfall-runoff process, all such incidences never produce reproducible 'Tc' nor can we ever know true Tc. And therefore, an averaged value of 'Tc' from observed data should be considered for analysis. Alternatively, any empirical equation, most valid for area under study is recommended for use. Some of the empirical equations normally used for estimation of this parameter are as under:


(For more discussion on 'Tc', reader may please refer to Hydrologic Analysis and Design by Richard H. McCuen and Technical Reference Manual of HEC-HMS software).

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This coefficient represents the temporary storage of precipitation excess in the watershed as it drains to the outlet point. K is expressed in terms of time. If observed short interval flood discharges at site are available, then hydraulically, K is estimated by using the formula:



Q1: correspond to the discharge, after separating base flow, at the point of inflection on the recession limb of flood hydrograph.
Q2: correspond to the discharge, after separating base flow, after time t on the recession limb of flood hydrograph.
t: time interval between Q1 and Q2.

Adopting a suitable base flow, the value of K can be computed for different hydrographs and an average value can be worked out.

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The time-area diagram represents the areas that will contribute to the flow at the outlet over successive periods of time. Once the time of concentration is known, lines of equal time interval called isochrones can be drawn with assumption that time of travel is directly proportional to distance from the outlet to isochrones. In the picture shown here, dashed lines in yellow mark isochrones.

The US Army Corps of Engineers recommends following formula to develop time area table/diagram for estimation of inflow from areas bounded between successive isochrones. An example at the end of this discussion explains the use of this equation.


The time-area diagram is considered as the inflow to a hypothetical reservoir (S = KO; this implies that there is absence of wedge storage) and routed through the reservoir to obtain the outflow hydrograph which is the required instantaneous UH for the basin. Before routing, inflow from incremental areas between isochrones is converted into discharge units by following equation;


Where 'ai' is the area in km2 and 't' is the routing period in hours.

Here, we will take a pause to understand as to how this model produces an outflow with time-area concept; and inflow is generated by each time-area zone due to instantaneous 1mm effective rainfall. Let us consider the uppermost part of the catchment. Being uppermost part of the watershed, it does not receive any outflow (O0=0). Instead, it produces I1 runoff which takes 1 hr (if tc for the catchment is 6hrs; and isochrones are separated by 1hr each) to reach at the tip of area just below it with a magnitude of Oi routed by following equation.

Oi = CIi + (1-C)Oi -1

Where C = (t/(K+0.5t)), here, t is routing interval in hr, k is basin storage coefficient; and Ii and Oi are the inflow and outflow at the end of period ti.

In other words, for uppermost area, outflow is obtained by

O1 = CI1 + (1-C)* 0

Now, let us consider the area downstream of first one. Like above, this part generates inflow, I2 and receives O1 as outflow from upper area. Using routing equation, this area will produce O2 as

O2 = CI2 + (1-C)* O1

This process is continued till we reach terminal point of the catchment/study area.

The IUH can be converted to a unit hydrograph of same unit duration as routing interval simply averaging two instantaneous hydrographs lagged by the selected duration that is,

Qi = 0.5(Oi + Oi-1)

To obtain unit hydrograph for durations other than routing interval (provided that it is exact multiple of routing interval t. The following equation is used.

Where Qi is ordinate of unit hydrograph of desired duration D = nt.

The computation of a 1-hr unit hydrograph by this method is described ahead with an example.


Derive UH for a project site using Clark model with physiographic characteristics of the basin given in Table 1. No observed flood hydrograph is available at the project site.



Determination of The time of concentration (Tc)

Since, no observed flood hydrograph is available at project dam site, Tc, time of concentration is determined using the Kirpich, the Kerby and the California formulae as shown in Table 2. The average value as 6.0 hour based on 3 formulae is adopted for project catchment up to project site.


Since the time of concentration adopted is 6 hr, the catchment area has been divided into six isochrones representing 1 hr equal travel time. The equation developed by U.S. Army Corps of Engineers is used to estimate the time-area relationship of the watersheds. Table 3 presents the resulting Time-Area relationship for river upto dam site.


Note- For HEC-HMS software savvy readers, calculation reflected in table 3 is not essential. HEC-HMS software needs two key parameters, i.e. Tc and K to be input appropriately in its environment, and rest of calculation and UH output are handled by software in no time.

From the Table 3, the value of catchment area can be used developing inflow for each computational time interval using excel sheet.

Determination of Basin storage coefficient (K)

Since the site specific observed short interval discharge data is not available for dam site, therefore basin storage coefficient is estimated based on regional values of K available for concerned river.

For estimating the value of storage coefficient 'K', observed complex flood hydrographs from 3rd to 7th September 2001, 22nd to 25th Jul-2002, September 2004 flood hydrographs at one G&D site on the same river have been used. The detail is given below:


Therefore from Table 4, the Storage coefficient K can be taken as 9.0.

Assuming a base flow of 450 m³/s, the value of K has been computed for different hydrographs at another site on the same river as shown in Table 5. The average value of 8.0 hr has been adopted for observed hydrographs at this G&D site, an average value of attenuation constant K equal to 8.0 hr has been worked out based on 4 flood events.


The K is estimated based on the fact that for a given regional value of this parameter at a particular site, the dimensionless parameter (also called as attenuation ratio) approximately remains constant. The value of this parameter for second and first G&D sites is given in the Table 6:


Above value of dimensionless parameter , which is almost equal, suggests in favour of conclusion given by U.S. Army Corps of Engineers that the value of this dimensionless parameter remains more or less constant over a region. Based on this fact, the value of K estimated for given project site is 5.4 hr (for Tc = 6.0hr).

Having determined catchment area, time of concentration, Tc and storage coefficient, K, a model depicting basin is developed in HEC-HMS and basin is subjected to 10mm=1cm effective rainfall for 1-hr duration. Resulting hydrograph, i.e. 1-hr duration UH and its ordinates are available below:

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The instantaneous Unit Hydrograph is defined as unit hydrograph or infinitesimally small duration. In other words, IUH is the direct runoff hydrograph at the outlet of the catchment resulting from 1 unit (1mm) of rainfall falling over the catchment in zero time. Of course, this is only a fictitious situation and a concept to be used in hydrograph analysis.

    Derivation of IUH

There are various methods for the determination of an IUH from the given effective rainfall hyetograph and direct runoff hydrograph. But the most common is the model suggested by Nash in 1957. Nash proposed a conceptual model by considering a drainage basin as 'n' identical linear reservoirs in series. By routing a unit inflow through the reservoirs a mathematical equation for IUH can be derived.

The ordinate of the IUH at time t is given by,


n = no. Of the reservoir; and

K= a reservoir constant, also called as storage coefficient.

The values of K and n in Nash model can be evaluated by the method of moments by using the following relations:

MDRH1= First moment arm of Direct Runoff Hydrograph (DRH)
MERH1= First moment arm of Effective Rainfall Hyetograph (ERH)
MDRH2= Second moment arm of DRH
MERH2= Second moment arm of ERH

The unit of the ordinates of IUH is per sec (sec-1). When the ordinates are multiplied by the total volume of runoff (in cubic meters) resulting from 1mm of rainfall over the catchment area, the unit will be cumecs.

    Derivation of Unit Hydrograph from IUH

For finding the unit hydrograph from IUH, the area under the IUH is plotted with respect to time at the point. The entire area from the start of IUH at different time interval gives points of S Curve. If a unit hydrograph of T hour duration is required, the S Curve so arrived at is shifted by T hour and the difference in the ordinates of the two S Curves is computed and divided by T. The resulting curve forms the unit hydrograph of T hour duration. To illustrate derivation of UH by Nash method, an example is presented ahead.


A storm of mild intensity was experienced in the catchment of river Baitarani during the period from 26.9.75 to 28.9.75. The rainfall was rather non-uniform. The average hourly rainfall over the catchment and the resulting observed discharge at Anandpur site are furnished in Table 1 & 2. The area of catchment of River Baitarani up to Anandpur is 8570 sqkm.

With this set of data pertaining to this storm, an IUH followed by the unit hydrograph of one-hour unit duration have been computed below:

The various steps involved in the procedure are as follows:

1. Separation of base flow to find DRH

  • The anticipated recession curve of the smaller peak just before this flood was continued till the time of the peak, i.e. point B.
  • A suitable point was chosen on the falling limb at a distance of about 2t to 2.5t where 't' is the time from the rise of flood hydrograph to the peak.
  • These points B & C were joined by a straight line. The curve thus separates the base flow from total flood.
  • The base flow thus separated is deducted from the corresponding ordinates of the flood hydrograph to get the direct runoff hydrograph (DRH).


2. Separation of Rainfall Excess from Total Rainfall

The total hourly rainfall (average over the basin) is shown in the table above. Since the storm under consideration is in the month of September and there was a heavy storm in August and yet another storm of smaller intensity in early September, the loss may be considered to take place at a uniform rate.

Let the loss be at the uniform rate of 'X' mm/hour.

Additionally, the total volume of of DRH is worked out to be

Volume of DRH = 4344 cumecs* 3hrs

Where, 4344 is total sum of DRH ordinates and 3hrs is time interval between two adjacent DRH ordinates.

Total direct runoff = 13032x3.6 = 5.475mm,

Where, 8570 is the area of catchment of River Baitarani up to Anandpur in sqkm.

Now let the loss rate, X be more than 2 mm/hour.

Substituting the value of X, the effective rainfall at different time is obtained.

For the purpose of analysis, the hourly ordinate will involve a lot of computational work, and therefore, only the three hourly rainfall ordinates have been considered. The hourly rainfalls are assumed to be uniform during three hours and accordingly, the ordinates of DRH and ERH are shown in Table A.1 below

3. Calculation of n and K

Hence, for analysis the values of K and n may be taken as 4 hr and 4 respectively (In case of very sandy soil characteristics of the catchment 'n' may be taken as 5 if its value works out to be 4.47, whereas for hilly or semi-hilly regions of catchment, n should be 4).

4. To estimate the ordinates of IUH

Once n and K are found out, the ordinates of the IUH can be found very easily by using the relation

This will give ordinates in units of Sec-1. To find the ordinates in cumec, it is multiplied by catchment area contribution due to 1mm effective rainfall. Thus for first time unit U(1),

Area = 8570 * 106 m2, K = 4 * 60 * 60 sec, (n-1)! = (4-1)! = 6, t = 1 hr. Pl note that in term (t/K), K would be 4hr as we are considering t in hr. This step is repeated for rest of time units to generate IUH. The ordinates of IUH are given in Table below:

It will be seen that the last ordinate is never zero. This is because of the fact that the recession of IUH is generally defined by an exponential function which has zero value only at infinity. Hence, the recession is terminated at a suitable point and volume adjusted

5. Derivation of 1 Hr. duration Unit Hydrograph from IUH

The tth hour ordinates of the 1-hour duration unit hydrograph can be very easily found by simply taking the average of tth hour and (t-1)th hour ordinates of the IUH. Column 2 of the Table above gives the ordinates of the IUH (rounded off figures) the column 5 gives the ordinates of the 1-hour duration unit hydrograph.

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More often than not, project sites suffer from inadequate length of hydro-meteorological data or even no data to deduce any reliable hydrological inputs or to develop reliable transfer function in the form of a UH. In several cases, constraint of this kind is not uncommon in India, which compels an engineer to resort to develop a synthetic unit hydrograph. A Synthetic Unit Hydrograph (SUH) takes its shape and size according to the physical characteristics of the basin under study. There are few methods of developing SUH. Some of the common methods for derivation of synthetic unit hydrograph for a basin are as follows:

  • Snyder method

    Among several known methods for development of synthetic unit hydrograph, the one suggested by F. F. Snyder (1938) is most commonly used. Snyder analyzed a large number of hydrographs from drainage basins in the Appalachian Mountain region in the United States, ranging in area from 25 sq. km. to 25,000 sq. km.

    To sketch a unit hydrograph, it is necessary to know the time of the peak, the peak flow and the time base. The elements must be determined for every particular or regional location of the drainage basin. Snyder proposed the following empirical formula for the lag time (Hr.) from mid-point of effective rainfall duration tr to peak of a unit hydrograph:

    tp = Ct (L. Lc ) 0.3

    in which tp = the basin lag in hours, from midpoint of effective rainfall duration tr to peak of a unit graph:

    L= the length of the main stream from the outlet to the divide in kms;

    Lc =the distance from the outlet to a point on the stream nearest to the centroid of the basin; and Ct = a coefficient

    The location of the center of area may be determined by cutting the basin outline from cardboard and marking the point of intersection of plumb lines drawn with the map suspended from different corners. The coefficient Ct varies from 1.0 to 2.2 with lower values associated with basins of steeper slopes.

    For the standard duration of effective rainfall tr , Snyder proposed:

  • For the rains of this duration, he found that synthetic unit hydrograph peak Qp in cumecs may be obtained from the equation:

    And in cumecs/sq km by the relation

    Where, A = the drainage area in square kms.

      Cp = coefficient ranging from 4.0 to 5.0

      Qp = peak flood in cumec.

    For the time base T (in days) of the synthetic unit hydrograph U.S. Army Corps of Engineer adopted the following expression


    These equations are sufficient to construct a synthetic unit hydrograph for a storm of duration tr

    The value of Snyder's coefficients Ct and Cp are found to vary considerably depending upon the topography, geology and climate. Snyder indicated that the coefficient Ct is affected by basin slopes S. Linsely, Kohler and Paulhus have suggested and expression for tp in which the basin slope S has been considered.


    Where, N=0.38 and C =1.2 for mountainous drainage areas; 0.72 for foothills; and 0.35 for valley areas.

  • From gauged to ungauged basin by transposition of unit hydrograph
    • If unit hydrographs are available for several areas adjacent to a basin for which a unit hydrograph is required but for which necessary data are lacking, then transposition of available unit hydrograph will ordinarily give better results than resorting to a synthetic procedure. Sherman originally proposed that the ordinates and abscissas of unit hydrograph for similar basins might be assumed to be proportional to the square roots respective drainage areas. Further details are available in any textbook on applied hydrology.
  • Based on set of equations recommended by Flood Estimation Reports (FER)

    FER for 26 hydro-meteorologically homogenous subzones in India are reports jointly brought by Central Water Commission (CWC); India Meteorological Department (IMD); Research, Design & Standard Organization (RDSO); and Ministry of Shipping & Transport (MoST). In each report, a number of mathematical relationships between physiographic parameters and components of unit hydrograph, derived on multiple regression technique, exist. The step-by-step procedures as to how to develop UH based on FER are illustrated in paragraphs to follow.


Step-1 Physiographic parameters

  1. Location of catchment area to be identified from Survey of India topo-sheet and measure the catchment area (A)
  2. Measure the length of the longest stream in Km. (L)
  3. Length of the longest stream from a point opposite to C.G. of catchment to the point of study in Km. (Lc)
  4. Compute Equivalent Slope in m/Km. (Seq)

To determine equivalent slope, reader may look at following plot which displays longitudinal profile and formula used for the purpose. We will use this formula a little later to calculate Seq.


With advances in information technology in recent years and also with the availability of Digital Elevation Model (DEM), distillation of physiographic parameters is relatively faster and accurate. Illustrated example in later part of this chapter demonstrates the application of GIS technique to deduce these parameters. The DEM data are available for free at sites


Step-2 To construct 1-hour SUH

Availability of physical parameters of catchment enables one to estimate the components of SUH using following SUH equations for a Zone, say X;


Notations/components used in above table are used to construct an SUH using a plot shown here. If the volume of UH so defined deviates from the volume of runoff generated by catchment because of 1cm rainfall excess, falling limb of the hydrograph is suitably modified without altering the points of synthetic parameters such that the volume of UH equals the theoretical value. The SUH ordinates at one hour interval after corrections are taken as the final estimate of SUH. Succeeding example presents stepwise procedure to develop SUH of 1-hr duration.

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Important Note

  1. This example is based on hypothetical equations and is used to demonstrate the steps required to design a synthetic UH. For real case studies, users are requested to refer to Flood Estimation Report relevant to the area under study. Such reports are available with CWC and concerned State Govt. departments.)
  2. Readers ignorant of application of GIS software and Geo-HMS add-on package will find it difficult to follow initial steps of this example.

The particulars of a catchment/project site are as follows:

Name of watershed
Name of Tributary
Lat 250 47'00"
Long 930 04'48"
Moderate Slope

Step-1: Physiographic Parameters

The Survey of India released topo-sheets marked with contours is primary requisites to extract basin physical parameters. Unquestionably, this process is tedious and fraught with possible errors. Alternatively, Windows based GIS presentation Fig.1 software, such as Arc View 3.X or similar packages can be used to extract these parameters through analysis of DEM. To showcase strength of GIS based analysis of basin, two SRTM grids 55_7 and 55_8 are first merged followed by extraction of a part from whole DEM, which is likely to cover outlet point and its contributing area.

This relatively small sized patch of DEM was imported in Geo-HMS window (an extension package that works on Arc View GIS 3.2a) for terrain and hydrological analysis. A series of options on Geo-HMS supported menus lead users to successfully obtain a network of streams and corresponding catchment areas. This process is termed as terrain processing.

Having reached a stage as shown in Fig.2, it is possible in the system to locate project by defining outlet at pre-determined latitude and longitude. In Fig.2, hypothetical project location is displayed by a point in red. A hydrologist is concerned about area shedding water at this outlet; and therefore, only this area is abstracted at this stage for hydrological analysis subsequently.

Fig.3 displays the output of hydrological processing containing delineated watershed for project location; the location of centre of gravity (CG) of the basin; longest flow path. These features of this basin can be exported to HEC-HMS software for developing design flood hydrograph or any other hydrological analysis. A set of information extracted for estimating SUH equations are as below:

606.52 km2
Length of longest flow path, L
56.4 km
Centroidal flow path, Lc
28.2 km

Fig.4 plots the longitudinal profile of the river/basin along longest flow path. This plot and table presented below are used to estimate equivalent slope.


Step-2 : 1 hr Synthetic UH parameters generated by 1cm effective rainfall

SUH parameters as given below are computed by using equations given in step 2 are as following:

An SUH based on the estimated parameters in step -2 is shown below. The discharge ordinates of this graph at 1 hr interval are multiplied by 1 hr and are summed up to ascertain whether it agrees with principle of UH. Raw UH so arrived overestimates the runoff; and therefore its falling limb ordinates are readjusted in a manner to represent basin's true UH.

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The unit duration of the UH derived from various records may not be alike. In order to compare and average them, it is necessary to convert all of them to the same unit duration.

There may be two types of cases;

  1. When UH of shorter unit duration't' is known and a UG of longer duration T is to be derived where T is a multiple of t, i.e., T=nt such that n = 1, 2, 3, ....n. This can be achieved simply by the principle of super-imposition. Let us grasp it with an example. We have an 1 unit UH of 2-hr duration; and it needs to be converted to a 4-hr duration. Please refer to the table shown here. The 2 hour UH is displaced by two hours; and is added to first UH. The summed up UH is a result of 2 unit rainfall spread over (2hr+2hr) = 4hr duration. To obtain 1 unit 4hr duration of UH, Last column is divided by 2.

  1. Otherwise, the UH of other duration can be derived by S-curve.

    The S-curve is a hydrograph produced by a continuous effective rainfall at a constant rate for an indefinite period. The S-hydrograph can be constructed by summing up a series of identical UHs spaced at intervals equal to the unit duration of the UH. After the S-hydrograph is constructed, the UH of a given duration is derived with following procedures.

    Assume that the S-hydrograph derived is due to effective rainfall intensity of 1/t0 mm/hour. Now, advance or offset the position of S-hydrographs for a period equal to the desired duration of UH, say tn hours; and tabulate the difference between ordinates of original S-hydrograph and offset S-hydrograph. This will be the hydrograph due to (1/t0) * tn = tn/t0 mm of rainfall occurring in tn hours. Divide ordinates of the hydrograph thus obtained by tn/t0. The resulting hydrograph will be the UH of tn hour duration.
  1. HEC-HMS software uses UH as transfer function to generate response of a basin following occurrences of rainfall. In HEC-HMS, UH of known duration, say t-hr is to be fed in the HEC-HMS keeping its ordinates spaced at t-hr apart. For example, if UH of 1mm rainfall derived is of 3hr duration, ordinates of UH must be entered at 3hr interval. Software automatically converts this UH to other duration according to rainfall interval chosen by user. Thus, conversion of a UH from one duration to another is omitted.
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Three distinct unit hydrographs for Anandpur site on river Baitarani, as tabulated below, have been developed by employing different techniques of UH derivation as highlighted in preceding paragraphs. The figures in bold and italic mark the peak value of respective columns. A cursory look at ordinates recorded under three methods reveals that the peaks, time base, and time of occurrence of three unit hydrographs differs from one another. As a matter of fact, if various storms are considered for development of unit graph for the same catchment, a marked variation will be observed especially in the peak as well as the time of occurrence of the peak. Therefore, it had better derive an average unit hydrograph for practical use. If several unit hydrographs are averaged by averaging concurrent ordinates, it is highly probable that the resulting average unit graph has a broader, and a quite possibly a lower peak than any of the individual graphs.

The correct average unit hydrograph should be obtained by locating the average peak and the average time of occurrence of the peak and sketching a menu unit hydrograph having an area equal to 1mm of runoff and resembling the individual graph as much as possible.

In the backdrop of discussion above, pattern of averaged UH is determined as below. To find the average unit hydrograph, the average peak was found to be 141 cumecs and average time of occurrence of peak was estimated to be 15 hours. Similarly the average base length is estimated to be 48 hours.

Now that essential component of UH are estimated, a suitable unit hydrograph is drawn in a manner such that:

The area of the unit hydrograph is equal to 1mm; and

The shape resembles the shape of the three individual hydrographs.

The average unit hydrograph thus obtained is shown by dotted line.

However, the averaging of the unit hydrograph can't be resorted to all cases. It has been observed that for practical purposes the shape of the unit hydrograph is hugely governed by factors, such as amount of effective rainfall, rainfall distribution pattern, and the storm movement etc

This aspect of UH is briefly demonstrated here with the help of a few diagrams. For more on this subject, readers are encouraged to refer to any good book on hydrology. Adjacent figure illustrates how the storm movement influences the shape of unit hydrograph. Yet another picture exhibits how the concentration of localized rainfall activity over the basin can significantly alter limbs and peak of UH. The third picture demonstrates the distinction between impact of a concentrated and heavy effective rainfall and uniform rainfall of same amount over the catchment.

In brief, it is not necessary that similar features will be reflected in all the storms. As a matter of fact, the formation and distribution of the runoff is quite complex process in which large numbers of factors are involved.

Hence for the operational use, the scheme of the unit hydrograph is to be laid down after taking into account the primary influencing factors. It is not enough, and certainly not, to use one un-changeable unit hydrograph for formulation of flood forecast. Different unit hydrographs should be identified for the various conditions which have various influences on formation and time distribution of the runoff. These unit hydrographs may then be judiciously applied under different conditions.

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  1. Patra, K C, (2001), Hydrology & Water Resources Engineering, Narosa Publishing House
  2. Ven Te Chow, David R Maidment, Larry W Mays, (International Edition 1988), Applied Hydrology, McGraw-Hill Book Company
  3. Raghunath, H M, (2006), Hydrology - Principles, Analysis, Design, New Age International (P) Limited
  4. Richard H McCuen, (1989), Hydrologic Analysis and Design, Prentice Hall, New Jersey
  5. A multitude of e-learning materials available at

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Anup K Srivastava,
Director, National Water Academy