The objective of this study is to simulate daily
rainfall sequences for Aduthurai to use as inputs to
crop, hydrologic and water resources models.
Review of
Literature:
Markov chains specify the state of each day as
‘wet’ or ‘dry’ and develop a relation between the state of the current day and
the states of the preceding days. The order of the Markov chain is the number
of preceding days taken into account. Most Markov chain models referred in the
literature are first order 3. Many authors have used Markov chains
to model the daily occurrence of precipitation. Gabriel and Neumann (1962)4
analyzed the occurrence of rain by fitting a two-state, first-order Markov
chain. Carey and Haan (1978) 5 used
multi-state Markov chain models to generate daily rainfall depths. The low
order chains are mostly preferable for two reasons. The number of parameters to
be estimated is kept to be a minimum, so that better estimates are obtained.
Second, the subsequent use of the fitted model to calculate other quantities,
such as the probabilities of long dry spells, is simpler. The distribution of
the amounts of rainfall on wet days is usually modeled by gamma distributions 6.
Model
Structure:
The model consists of (i)
rainfall occurrence model and (ii) rainfall magnitude model.
Rainfall
occurrence model:
A Russian mathematician, Markov, introduced the
concept of a process (later named after him ‘a Markov process’) in which a
sequence or chain of discrete states in time for which the probability of
transition from one state to any given state in the next step in the chain
depends on the condition during the previous step7. Daily rainfall
includes the occurrence of rain and the amount of rain. Markov chain process is used to find out rainfall
occurrence. Once rainfall occurrence has been specified, rainfall amount is
then generated using a Gamma or mixed Exponential distribution 8.
A first
order Markov chain is a stochastic process having the property that the value
of the process at time t, Xt, depends only
on its value at time t-1, Xt-1, and not on the sequence of values
that the process passed through in arriving at Xt-1.
A ‘C state’ Markov chain requires that C(C-1)
transition probabilities be estimated and the remaining C Pij can be determined using the relation
(1)
The C2 transition probabilities are
given by the stochastic matrix P.
P
= (2)
Once
P is known, all that is required to determine the probabilistic behavior of the
Markov Chain is the initial state of the chain. In the following, pj(n
) denotes the probability that the chain is in state j at step or
time n. The 1xC vector p(n) has elements pj(n).
Thus
p(n)
= [ p1(n) , p2(n) , …., pc(n) ] and p(1) = p(0) p (3)
Where,
p(0) is the initial probability
vector.
In
general, p(n+c) = p(c) p(n) (4)
Where,
pn is the n th power of p.
Parameter estimation:
The
parameters for the occurrence model are transition probabilities, pij s, which forms the transition matrix P. The
estimate for pij is given by
where, nij is the number of times the
observed data went from state i to state j.
For
finding the daily transition probabilities, nij
s for each day of the year are counted for the entire period of record
and for monthly pij s, nij s are counted for each month of the year
throughout the entire record length. Eleven years (1985 - 1995) of daily
rainfall data at Aduthurai were used to estimate the
parameters.
Model for rainfall magnitudes on wet days
The
rainfall amounts on wet days are modeled by a two parameter gamma distribution
with the density function is given by
Px(x)
= λη x
η-1 e- λx (6)
Γ (η)
Where, Γ (η)
= (η – 1)! for η = 1,2,3…… (7)
and η
= tη-1 e-1 (8)
in which
η and λ are the shape and scale parameters respectively.
Parameter estimation
The
parameters of the gamma distribution η and λ were estimated using
Greenwood and Durand (1960) 9 method as given below
η*=
for 0 
y 
0.5772 (9)
η* =
for 0.5772 y 17.0 (10 )
ln ū
_____
where, y = ___
(11 )
ln u
in which, u
is the rainfall amount on wet days and ¯¯¯ denotes ‘the arithmetic mean of’.
The
estimate η* was corrected for small-sample bias using Bowman
and Shenton equation,
η
= 
(12)
where, n is
the sample size
the
estimate for λ is
λ
= 
(13)
MODEL CALCULATION
: NOVEMBER 1989 – ONE MONTH
TABLE
1 : RAINFALL DATA AND STATES
|
Rainfall |
States |
|
7.4 0.6 0.0 11.4 0.0 0.0 0.0 0.0 0.0 9.4 4.6 57.2 74.2 17.0 0.8 0.0 0.0 19.4 59.4 16.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.6 34.0 |
3 2 1 3 1 1 1 1 1 3 3 4 4 3 2 1 1 3 4 3 1 1 1 1 1 1 1 1 2 4 |
(a)TRANSITION PROBABILITY MATRIX
NOVEMBER - 1989
STATE - 3
|
RAINFALL |
STATE |
|
0.0 11.4 0.0 9.4 0.0 19.4 |
1 3 1 3 1 3 |
u Range = 40.2
ū Range = 13.4
n =14
__
ln
u = 
=
0.26384
ln ū =
2.59525
ln ū
_____ =
9.83647
y = ___
ln u
η* =
η* = 0.03876
η = 
= 0.034884
λ
= = 0.002603
RESULTS AND
DISCUSSIONS:
Markov
chains indicated that first order Markov chains can adequately represent the
rainfall occurrences in all the months. A number of states are used for
representing rainfall in a wet day as a more good fit can be obtained for the
distribution representing the rainfall amount in each class. The rainfall data are given in Table 1 for
the model calculation. The states and their boundary limits are given in Table
2. The monthly transition probabilities are given in Table 3. The two
parameters, η and λ of the gamma distributions are given in Table
4.
CONCLUSION:
First
order Markov chains with two parameters gamma distributions were found to be
adequate to model daily rainfall sequences at Aduthurai.
The model consists of 9 Markov chains with 144 parameters representing the
rainfall occurrence process in each month and 26 gamma distributions with 52
parameters to model the amount of precipitation corresponding to each state in
each month.
Table 2. States and their boundaries
|
State |
Limits (mm) |
|
1 2 3 4 |
0.0 - 0.0 0.1 - 3.9 4.0 - 27.9 28.0 - ∞ |
Table 3.
Monthly transition probabilities
April
0.9771 0.0000
0.0150 0.0000
0.5000 0.0000 0.5000
0.0000
1.0000 0.0000
0.0000
0.0000
0.5000 0.0000
0.0000
0.5000
May
0.9271 0.0405
0.0324 0.0000
0.0000 0.5000
0.5000
0.0000
0.4000 0.6000
0.0000 0.0000
0.1000 0.0000
0.0000
0.0000
June
0.9320
0.0350 0.0260 0.0000
0.6330
0.1000 0.2660 0.0000
0.5000
0.1250 0.3750 0.0000
0.0000 0.0000 0.5000 0.5000
July
0.8484 0.0617 0.0707
0.0191
0.7500
0.1875 0.0625 0.0000 0.
6250 0.0000 0.3750
0.0000
0.5000 0.5000 0.0000
0.0000
August
|
0.7962 0.0584
0.0520 0.0932 0.3330 0.2272
0.3939 0.0454 0.7045 0.2045
0.0681 0.0227 |
0.5000 0.2500
0.0000 0.2500
September
0.7708 0.1081
0.0721 0.0489
0.6250
0.0000 0.3125 0.0625
0.5890
0.2072 0.2035 0.0000
0.5000
0.0000 0.2500 0.2500
October
|
0.6910
0.1532 0.1178 0.0378 0.2603
0.4114 0.3282 0.0000 0.3259 0.2592 0.3542 0.0606 |
0.8000 0.2000
0.0000 0.0000
November
0.8605 0.0212 0.1181
0.0000
0.3565 0.1885 0.2196
0.2351
0.1514 0.3030 0.2878
0.2576
0.0000 0.0625 0.4374
0.5000
December
0.7617 0.1566
0.0566 0.0250
0.6915 0.2667
0.0416 0.0000
0.2581 0.3167
0.2334 0.1916
0.2221 0.2221 0.2778
0.27