HYDRAULIC CONSTRUCTION 1:
HYDROLOGY
ENSTP – Yaoundé, Cameroun
April‐May 2014
RAIN GAGE (1)
Precipitation consists of rain or snow. Rainfall refers to amount of liquid precipitation.
Rain gages record the the amount of precipitation expressed in millimeter [mm].
Non-recording gage is also generally
installed at a recording gage site to
provide a check on the automatic
gage mechanism
Non‐recording
gage
Recording gage
RAIN GAGE (2)
Record gage may be equipped with 7-day
recorder or a strip-chart recorder that is used
up to 6 months.
When the bucket is full, it turns and it makes
a sign on the paper
The amount of the measured precipitation depends on the exposure of the gage to the
wind and also on the height of the surronding objects.
A poor exposure should be avoided
POINT RAINFALL
Point rainfall is measured at a rain gage station.
The point value is assumed to be applicable only for small areas (up to 25 km2).
The time distribution of point precipitation obtained from a recording rain gage is
shown by a hyetograph, that shows the depths during a selected time interval.
MEAN ANNUAL PRECIPITATION
The following basic factor determine the amount of mean annual precipitation at a
station:
1. Latitude
2. Position and size of continental land mass on which the station is located
3. Distance of the station from the coast or other source of moisture
4. Temperature of ocean and coastwise currents
5. Extend and altitude of adjacent mountain ranges
6. Altitude of the station
CAMEROON MEAN ANNUAL PRECIPITATION
Cameroon is divided into 5 areas:
• Exceedingly hot and humid with a
short dry season on the coast
• Alternation between wet and dry
seasons in the south
• mild climate and high rainfall along
the Cameroon range
• Moderate hot temperature with two
rainy season on the Adamawa Platau
• Arid area with sparse rainfall in the
Northern lowland region
RETURN PERIOD (1)
The information about the rain is necessary to properly size hydraulic structure.
We need data collected over a period of 30-35 years to have a statistic
One of the most important concept in the statistical point of view is the return period.
Definition. It is an estimated of the mean period of time between two events of the
same or greater magnitude.
The statistical rigorous definition states that the return period is the inverse probability
that the event will be exceeded in any one year:
return period
probability that the event H
exceeds the value h
RETURN PERIOD (2)
By using the probability properties, we
have:
Example. The flood having Tr=10 years is a flood that may happen, on average, every
10 years (with the same or greater magnitude).
N.B.= For sizing any hydraulic structure a return period Tr must be given. E.g., the
sewer system is sized for events that have Tr=5÷10 years; a dam for Tr=1000 years.
DEPTH AND INTENSITY OF RAIN
To estimate exceptional rainfall events the depth of rain h can be expressed as:
 is the rain duration
Let us define the intensity of rain j:
N.B.= j increases if  decreases and viceversa.
A short event is more intense than a long one having the same Tr
Our aim is to elaborate an expression for h. The expression is called:
• depth-duration-frequency curves (DDF)
• intensity-duration-frequency curves (IDF)
DEPTH‐DURATION‐FREQUENCY CURVES (1)
The DDF curves:
 Are define for a specific place with
- a given Tr
- an assigned 
 Have to give a synthetic information about
- the exceptional depth of rain
- the exceptional intensity
 Have the purpose of elaborating hyetographs that have to be significant for
- design problem
- test problem
DEPTH‐DURATION‐FREQUENCY CURVES (2)
The mathematical expression for the DDF curves can be expressed by:
Let try to study this curves:
• if 
then h
• if 
then j
,
0
DEPTH‐DURATION‐FREQUENCY CURVES (3)
If n1 > n2 we have h1 > h2 although
Tr1 < Tr2
Parameter a takes into account of Tr
ESTIMATION OF DDF CURVE (1)
To estimate DDF curve we need of rainfall data collected for a period of almost 40
years.
Usually this data are organized such that we have the highest depth of rainfall for
different duration and for each year.
ESTIMATION OF DDF CURVE (2)
To elaborate the DDF curve we need to rewrite the data in a decreasing order for each
duration independently from the year.
# of year
=1h
=3h
=6h
 = 12 h
1
2
3
.
.
.
.
40
Row of the first critical case, i.e., Tr = 40 years
Row of the second critical case, i.e., Tr = 20 years
 = 24 h
ESTIMATION OF DDF CURVE (3)
We obtain the estimation of the
parameters a and n for a given Tr, by
a linear regression of the data in bilog plane
In this analysis:
 The results depends from the ensemble of data we are considering
 The results do not give information about the event having Tr greater than the period
of the observations, e.g., Tr = 100, 200 and 1000 years
GUMBEL PROBABILITY DISTRIBUTION (1)
It is commonly accepted that the extreme
rainfall values follow a Gumbel probability
distribution, that reads ( and u are
parameters):
The estimation of the parameters  and u is realized by the comparison with observed
data.
We compare the probability of non-excedance with the frequency (deducted
by the data) of non-excedance
GUMBEL PROBABILITY DISTRIBUTION (2)
Let us follow this method:
• The available data are rewrite in ascending order, defining hi the value in the i-th
position (with i=1, 2, …, N)
• Thus the frequency of non-excedance for a value hi is given by
• We assign the observed frequency f(hi) with the theoretical one (F(hi)=P(hi))
By applying two times the logaritmic the equation above becomes:
ln
ln
1
with i=1, 2, …, N
GUMBEL PROBABILITY DISTRIBUTION (3)
The best estimation of the parameters  and u
is given by the linear regression of the points
(hi, yi):
In particular we want to minimize the distance between the line and the data. We find
GUMBEL PROBABILITY DISTRIBUTION (4)
We observe that the Gumbel probability P(h) is equal to the probability of nonexcedance P(H ≤ h) for a given return period Tr, i.e.
1
1
ln
1
ln 1
exp
1⁄ 1
:
exp
1
Depth of rainfall for different Tr with
fixed duration 
By reading the values along the arrow, we obtain the DDF curve for Tr = 10 years
GUMBEL PROBABILITY DISTRIBUTION (5)
To elaborate the DDF curve for large Tr we have to recostruct Gumbel curve for each
rainfall duration  recorded.
1
ln
ln 1
1