

PRINT VERSION MODULE











MODULE
OBJECTIVES 





 To understand the type of hydrometeorological
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









UNIT
HYDROGRAPH 





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 1hour 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.









ASSUMPTIONS
IN UNIT HYDROGRAPH THEORY 





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.









DERIVATION
OF UNIT HYDROGRAPH (UH) 





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 multipeaked
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 stepwise.
At the end of this module, it is expected that reader would be able
handle UH assignment independently. 




CASEI
 UNIT HYDROGRAPH FROM A HYDROGRAPH WITH ISOLATED PEAK 





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
runoff 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 nobreak/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  G_{o})^{n}
Where, c, G_{o};
& 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 5075% of T_{c}, time
of concentration. Later part of this chapter describes ways to estimate
T_{c},




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 (ABD) 





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 A^{0.2}
Where, A is the area of the drainage
basin in square kilometers.












Variable Slope Method
(ABCE) 





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





UNIT
HYDROGRAPH FROM COMPLEX FLOOD HYDROGRAPH 





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










UNIT
HYDROGRAPH DETERMINATION BY COLLIN'S METHOD 





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.
Example
The direct runoff hydrograph at siteS
and the effective hydrograph due to a particular storm over the
catchment of riverR are tabulated below. We will use the given
DRH and ERH to derive unit hydrograph because of 1mm effective rainfall.
Catchment area is 8570 km^{2}.









Note
The effective rainfall hyetograph blocks are for 3 hour intervals.
Therefore, the unit duration of the unit hydrograph thus derived will
be of 3hour 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 km^{2 }should be:
Catchment
Area * rainfall depth (1mm) = volume of UH
=
Sum of UH ordinates, SU * time interval
between UH ordinates
Hence,
where A = area of catchment in km^{2},
=8570 km^{2}, & 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 revalidate 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 Table3.
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).








UNIT
HYDROGRAPH DERIVATION BY DECONVOLUTION OF DIRECT RUNOFF HYDROGRAPH 





The discrete convolution
equation allows the computation of direct runoff, Qn given excess
rainfall, P_{m} and the unit hydrograph, U_{nm+1}. 




The reverse process, called deconvolution,
can be utilised to derive a unit hydrograph given data on P_{m}
and Q_{n}. 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 Q_{n}, n = 1,2,….,n,
in terms of (n  m + 1) unknown values of the unit hydrograph.
If Q_{n} and P_{m} are given and U_{nm+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 Q_{1},
assuming total number of rainfall pulse, M = 3
Here, n =1 <
(M =3), hence, m =1, therefore,






Example
An observed hydrograph with rainfall
excess is given as under. The time interval is 3 hours between readings.
Catchment area is 7092 km^{2}.
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, setup
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.





UNIT
HYDROGRAPH BY CLARK MODEL 





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





DETERMINATION
OF TIME CONCENTRATION, Tc 





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 rainfallrunoff
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 HECHMS software).





DETERMINATION
OF BASIN STORAGE COEFFICIENT (K) 





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:





Where;
Q_{1}: 
correspond
to the discharge, after separating base flow, at the point of
inflection on the recession limb of flood hydrograph. 




Q_{2}: 
correspond
to the discharge, after separating base flow, after time t on
the recession limb of flood hydrograph. 




t: 
time interval
between Q_{1} and Q_{2}. 








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





TIMEAREA CONCEPT 





The timearea 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 timearea 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 timearea
concept; and inflow is generated by each timearea 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 (O_{0}=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 O_{i} routed by following equation.
O_{i}
= CI_{i} + (1C)O_{i} 1
Where C = (t/(K+0.5t)),
here, t is routing interval in hr, k is basin storage coefficient;
and I_{i} and O_{i} are the inflow and outflow
at the end of period t_{i}.
In other words, for uppermost
area, outflow is obtained by
O_{1}
= CI_{1} + (1C)* 0
Now, let us consider the
area downstream of first one. Like above, this part generates inflow,
I_{2} and receives O_{1} as outflow from upper area.
Using routing equation, this area will produce O_{2} as
O_{2}
= CI_{2} + (1C)* O_{1}
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,
Q_{i} = 0.5(O_{i}
+ O_{i1})
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 Q_{i} is ordinate
of unit hydrograph of desired duration D = nt.
The computation of a 1hr
unit hydrograph by this method is described ahead with an example.
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.













Solution
Determination of The time of concentration
(T_{c})
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 timearea relationship
of the watersheds. Table 3 presents the resulting TimeArea relationship
for river upto dam site.









Note For HECHMS software savvy
readers, calculation reflected in table 3 is not essential. HECHMS
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 Jul2002, 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 HECHMS and basin is subjected to 10mm=1cm
effective rainfall for 1hr duration. Resulting hydrograph, i.e.
1hr duration UH and its ordinates are available below:



















INSTANTANEOUS
UNIT HYDROGRAPH 

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,
Where,
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:
Where,
M_{DRH1}= First moment arm of Direct Runoff Hydrograph (DRH)
M_{ERH1}= First moment arm of Effective Rainfall Hyetograph (ERH)
M_{DRH2}= Second moment arm of DRH
M_{ERH2}= Second moment arm of ERH
The unit of the ordinates of IUH is per sec
(sec1). 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.
Example
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 nonuniform. 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 onehour 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,
8570
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 semihilly 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 * 10^{6} m^{2},
K = 4 * 60 * 60 sec, (n1)! = (41)! = 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 t^{th} hour ordinates
of the 1hour duration unit hydrograph can be very easily found by simply
taking the average of t^{th} hour and (t1)^{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
1hour duration unit hydrograph.









UNIT
HYDROGRAPH FOR UNGUAGED CATCHMENTS 





More often than not, project sites suffer
from inadequate length of hydrometeorological 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
midpoint of effective rainfall duration t_{r}
to peak of a unit hydrograph:
t_{p}
= C_{t} (L. L_{c} ) ^{0.3}
in which t_{p} =
the basin lag in hours, from midpoint of effective rainfall
duration t_{r} to peak of a unit graph:
L= the length of the main stream
from the outlet to the divide in kms;
L_{c} =the distance
from the outlet to a point on the stream nearest to the centroid
of the basin; and C_{t} = 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
C_{t} varies from 1.0 to 2.2 with lower values
associated with basins of steeper slopes.
For the standard duration of effective
rainfall t_{r} , Snyder proposed:
For the rains of this duration, he
found that synthetic unit hydrograph peak Q_{p}
in cumecs may be obtained from the equation:
And in cumecs/sq km by the relation
Where, A = the drainage area in square
kms.
C_{p}
= coefficient ranging from 4.0 to 5.0
Q_{p} = 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
t_{r}
The value of Snyder's coefficients
C_{t} and C_{p} are found to vary
considerably depending upon the topography, geology and climate.
Snyder indicated that the coefficient C_{t} is
affected by basin slopes S. Linsely, Kohler and Paulhus have
suggested and expression for t_{p }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 hydrometeorologically
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 stepbystep procedures as
to how to develop UH based on FER are illustrated in paragraphs
to follow.








Step1 Physiographic parameters
 Location of catchment area to be
identified from Survey of India toposheet and measure the catchment
area (A)
 Measure the length of the longest
stream in Km. (L)
 Length of the longest stream from
a point opposite to C.G. of catchment to the point of study in
Km. (L_{c})
 Compute Equivalent Slope in m/Km.
(S_{eq})
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
http://srtm.csi.cgiar.org/SELECTION/inputCoord.asp
http://www.gdem.aster.ersdac.or.jp/search.jsp
http://glcf.umd.edu/data/landsat/
http://glcfapp.glcf.umd.edu:8080/esdi/index.jsp








Step2 To construct 1hour 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 1hr duration.





ILLUSTRATIVE
EXAMPLE 





Important
Note
 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.)
 Readers
ignorant of application of GIS software and GeoHMS addon package
will find it difficult to follow initial steps of this example.
The particulars of a catchment/project
site are as follows:
(i)

Name of watershed 
:

A 
(ii)

Name of Tributary 
:

X 
(iii)

Location 
:

Lat 25^{0} 47^{'}00^{"}




Long 93^{0} 04^{'}48^{"}

(vi)

Topography 
:

Moderate Slope 









Step1: Physiographic Parameters
The Survey of India released toposheets
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 GeoHMS window (an extension package that works
on Arc View GIS 3.2a) for terrain and hydrological analysis. A series
of options on GeoHMS 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 predetermined 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 HECHMS
software for developing design flood hydrograph or any other hydrological
analysis. A set of information extracted for estimating SUH equations
are as below:
Area



606.52 km^{2}

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.







Step2 : 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.





CHANGING UNIT
HYDROGRAPH DURATIONS 





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;
 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 superimposition.
Let us grasp it with an example. We have an 1 unit UH of 2hr
duration; and it needs to be converted to a 4hr 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.




 Otherwise, the UH of other
duration can be derived by Scurve.
The Scurve is a hydrograph produced
by a continuous effective rainfall at a constant rate for an indefinite
period. The Shydrograph can be constructed by summing up a series
of identical UHs spaced at intervals equal to the unit duration
of the UH. After the Shydrograph is constructed, the UH of a
given duration is derived with following procedures.
Assume that the Shydrograph
derived is due to effective rainfall intensity of 1/t_{0}
mm/hour. Now, advance or offset the position of Shydrographs
for a period equal to the desired duration of UH, say t_{n}
hours; and tabulate the difference between ordinates of original
Shydrograph and offset Shydrograph. This will be the hydrograph
due to (1/t_{0}) * t_{n} = t_{n}/t_{0}
mm of rainfall occurring in tn hours. Divide ordinates of the
hydrograph thus obtained by t_{n}/t_{0}. The resulting
hydrograph will be the UH of tn hour duration.




 HECHMS software uses UH
as transfer function to generate response of a basin following
occurrences of rainfall. In HECHMS, UH of known duration, say
thr is to be fed in the HECHMS keeping its ordinates spaced
at thr 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.





AVERAGING
UNIT HYDROGRAPH


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 unchangeable 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.





REFERENCES 





 Patra, K C, (2001),
Hydrology & Water Resources Engineering, Narosa Publishing
House
 Ven Te Chow, David
R Maidment, Larry W Mays, (International Edition 1988), Applied
Hydrology, McGrawHill Book Company
 Raghunath, H M, (2006),
Hydrology  Principles, Analysis, Design, New Age International
(P) Limited
 Richard H McCuen,
(1989), Hydrologic Analysis and Design, Prentice Hall, New Jersey
 A multitude of elearning
materials available at
https://www.meted.ucar.edu/





CONTRIBUTORS 

Anup K Srivastava,
Director,
National Water Academy











