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to foreshadow the economic activity of the country. The SWM rainfall value shows strong spatial and
temporal variability unlike other atmospheric parameters. The spatial average of the country as a whole,
known as All India Rainfall (AIRF), has a long term time average (LTA) of about 85 cm with standard
deviation of 8 cm. On the other hand rainfall values in smaller regions show greater variability. For
example, the coefficient of variation (%) of the subdivision comprising of Saurashtra & Kutch is as high as
44%.
In the past literature on Indian rainfall modeling studies, two major types of modeling have been
pursued. The first one is dynamical modeling based general circulation theories of the atmosphere and the
oceans to simulate summer monsoon circulation and associated rainfall pattern. With the concept of
simulation with different initial and boundary conditions, various general circulation models with varying
levels of refinement were developed (Latif et al 1994, Goswami 1998, Wang et al 2005, Sajani et al 2007,
Rajendran et al 2008) to simulate Indian mean monsoon and its variability. These could not show the
required skill to accurately simulate the monsoon circulation, rainfall and its interannual variability (Gadgil
and Sajini 1998, Kang et al 2002, Gadgil et al 2005, Krishna Kumar et al 2005, Kumar et al. 2005 and
Wang et al. 2005). These models are highly sensitive to atmospheric initial conditions. It was observed
that atmospheric general circulation models coupled with an ocean model may simulate realistic sea surface
temperatures and rainfall relationships. In this context the models developed by Achuthavarier and
Krishnamurthy 2010a, 2010b, Yang et al. 2008 and Pattanaik and Kumar 2010 depict the interactive
oceanic–atmospheric processes associated with the precipitation anomalies relatively well at different time
scales. However, the achieved skill is still rather poor to be useful for real forecasting purposes.
For long range forecasting of Indian summer monsoon rainfall, two kinds of empirical models have
been pursued. The first approach is the model based on historical relationship between the rainfall data and
other atmospheric and oceanic parameters. The statistical correlation between rainfall and antecedent
climate parameters, whenever significant is attractive in forecasting. Since 1900 regression models are
developed for long range forecasting based on this kind of relationship. But they were successful only
during normal monsoon years. With the improvements in the selection of the number of parameters
associated with the rainfall, more models ( Gowariker et al 1989 and 1991, Delsole and Shukla 2002,
Thapliyal 1990 and 2001, Sahai et al 2003) were developed. However, such model failed to forecast the
drought of 2002 and 2004 (Gadgil et al, 2005). Further new statistical models with two stage forecasting
system was developed (Rajeevan et al 2005), where the first stage required the precursor predictor data set
was up to March and the second stage up to May with six predictors to improve the official operational
forecasting. Ashok Kumar et al (2012) improved the above with step wise linear regression and nonlinear
ANN techniques for three stage forecasting (April, June and July) of SWM rainfall over India. It is
generally found that ANN methods worked better in comparison with linear methods of forecasting (Eisner
and Tsonis, 1992). Now these models have been used by Indian meteorological department as the present
forecasting system.
Second approach of empirical modeling to the problem is to handle rainfall data as a time series with
its past values and no other climate parameters in either modeling or forecasting. Sahai et al (2000)
proposed ANN techniques to forecast monsoon rainfall using only past data. They considered SWM data
and spatial average of each of the four monsoon months (June, July, August, and September) of all India
time series data for the period 1871-1960. This network with 25 input nodes, 2 hidden layers and 1 output
node used 276 model parameters with the effective sample size of 335. The modeling efficiency obtained
by them in terms of variance explained was found to be 0.8. This is not surprising as the number of
independent parameters is more than half the sample size. Guhathakurta (2008) proposed ANN model for
36 subdivisions of India including 11 to 12 antecedent rainfall values in the input layer with 3 neurons in
the hidden layer and one output. The number of parameters used was 40 to 43, which is more than half the
length of the sample size of 51 in the training period (1941-1991). The modeling efficiency in the training
period of 51 years for all India time series was shown to be 0.7 and for the subdivisions it was found to be
0.8. Pritpal and Bhogeswar (2013) proposed five different three layered (input, hidden and output) ANN
architectures for all India data with 43, 57, 73, 91 and 111 number of parameters. The efficiency of this
ensemble model in the training period of 84 years was found to be 0.65. It is observed that some of the
3. Forecasting Indian Monsoon Rainfall Including Within Year Seasonal Variability
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ANN approaches above are not skilful since by increasing the number of parameters to the sample size a
polynomial function can be made to fit the data series exactly. A new ANN model was developed by the
present authors (Kokila Ramesh and Iyengar 2015 under review) for SWM data of all India and its
homogeneous regions directly on SWM data with 6 input nodes, a hidden layer with 5 nodes and a single
output. The number of parameters was 41 with a modeling skill of 0.89 and the variance explained by the
model was as high as 0.80. The number of parameters in this case is less than half of the sample size of
100. This model includes only year to year variation, where the within year variation is ignored. Therefore
to further improve this model, within year i.e season-to-season variations have been incorporated in terms
of pre-monsoon and northeast monsoon annual cycles to construct a new ANN model for modeling and
forecasting SWM data in the present paper.
2. RAINFALL DATA
Four sets of three seasons per year data are considered here for further work. The first is the All India
rainfall value (AIRF) representing the whole country, which is spatial average based on the sub-regions.
The other three data chosen are for the two sub-regions and one subdivision along the west coast with high
variability. In Figure 1, the two regions and a subdivision are marked for clarity (Ref: In Indian institute of
tropical meteorology website). The basic details of the data for the period (1901-2000) are shown in Table
1 for all the three seasons independently. All the data are taken from the data base of the Indian Institute of
Tropical Meteorology (http://www.tropmet.res.in). The data series , = 1, 2 … is nonstationary, the
long term average (LTA) and the long term deviation (LTD) cannot be strictly termed the mean and standard
deviation of the rainfall. It was shown in our previous article submitted that the variation of and
with increasing sample length slowly oscillate over years without converging to a constant value. Since the
data is shown to be non stationary graphically, it may not be important in modeling exercises, but for
forecasting one step ahead will influence. However, for the known sample length LTA can be used as a
standard number to quantify inter annual variability (IAV). This LTA can be cited as a normal value to use
it as a scaling factor to make the data series non-dimensional for further work.
Figure 1 India map with subdivision numbers marked and the homogeneous regions colored for the
present study from Indian Institute of Tropical Meteorology (IITM). The region covered with lines are
hilly regions and are not included in all India data
4. Kokila Ramesh and R.N. Iyengar
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Table 1 Basic Statistics of Rainfall Data of Three Seasons namely Pre-monsoon, SWM and Northeast monsoon
(NEM) (1901-2000)
Name AIRF Core Monsoon NEIND COKNT
Season
Pre-
Mon
SWM NEM
Pre-
Mon
SWM NEM
Pre-
Mon
SWM NEM
Pre-
Mon
SWM NEM
LTA
( cm)
11.90 84.66 12.26 4.53 87.56 6.40 46.91 141.82 18.40 17.62 289.19 24.26
LTD
( cm)
2.36 7.97 3.46 2.13 13.74 3.94 9.34 12.13 7.06 16.59 46.80 11.13
Skewness 0.46 -0.28 0.41 0.74 -0.20 0.89 0.28 0.54 0.43 1.69 0.66 0.48
Kurtosis 2.99 2.48 2.91 3.11 2.44 3.88 2.70 3.96 2.84 5.38 4.92 2.98
AIRF-All India Rainfall, NEIND-Northeast India and COKNT-Coastal Karnataka
3. MODELING
Simulation and forecasting of Indian monsoon rainfall using a suitable model is of considerable interest. In
the present paper, three seasons per year data has been used to model SWM data. These data for about 100
years are normalized using its own mean and standard deviation . In the previous work, only
SWM data was used to model and forecast. Since some type of nonlinear relation can be expected to exist
among the three seasons accounting for inter seasonal variability in the network may help to improve
modeling SWM data more effectively. Hence a new network is developed in the present paper by including
the pre-monsoon and NEM as the inputs in the input layer along with the SWM data. This is symbolically
represented in Figure 2. This network is the improvement of the network used in the earlier work by the
present authors. Four more input nodes consisting of , and , of pre-monsoon and NEM
respectively are added in the input layer of the network used earlier. The network used is an optimal one,
as it is obtained by increasing the number of inputs each time by one and checking for the efficiency.
Beyond this number of inputs, the number of parameters increases and the results obtained may be spurious.
The number of parameters to be estimated is 61, but this is much less than half the length of the sample size
300. These parameters are found using the back propagation algorithm in MATLAB toolbox. In training
this network using toolbox, 61 random initial weights have to be selected. These initial weights are iterated
till the mean square error (MSE) between the actual data and the simulated converges. Out of such
100 samples the one which produces the least MSE is taken as the best ANN model for . The skill of the
new ANN model is presented in Table 2. Here is the root mean square error between the data and
simulated , is the correlation coefficient between and and is the performance parameter
between and . may not be a very good indicator of the model skill, since even with phase
difference between the data and the model the correlation coefficient may be high which would be spurious.
However is a measure of the variance explained by the model over the complete range of the sample
and goes to unity in the ideal case. Hence for a model to be accepted as useful in forecasting should
be high both in the modeling stage and in an independent verification stage. From Table 3 it is observed
that this new model is capable of explaining 94% of the data variance in all cases. Comparison between the
data and sample simulation is shown in Figure 3 (a-d) for a visual appreciation of the statistical skill of the
ANN model.
5. Forecasting Indian Monsoon Rainfall Including Within Year Seasonal Variability
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Figure 2 ANN with ten input nodes, five neurons in the hidden layer and one output model. Here M-SWM, P-Pre
monsoon N-Northeast-monsoon and the subscript represents year
Figure 3 Comparison between the observed data and the model (a) AIRF, (b) Core Monsoon, (c) NEIND, (d)
COKNT
Mn
Nn-1
Pn
Mn-1
Pn-1
Nn-2
Mn-2
Mn-3
Mn-4
Mn-5
Mn-6
1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
65
75
85
95
105
Year
R(cm)
1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
40
60
80
100
120
Year
R(cm)
1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
115
140
165
185
Year
R(cm)
1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
160
260
360
460
Year
R(cm)
Observed Model
(a)
(b)
(c)
(d)
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4. FORECASTING
The ANN modeling demonstrated above can be thought of as a curve fit with high skill for the rainfall time
series. The monsoon phenomenon recurs annually along with the two other seasons, pre-monsoon and
NEM. It is observed that the inclusion of the intra annual variability in the network through the pre-monsoon
and NEM components explains about 94% of the SWM data variance for the period 94 years (1907:2000).
Hence as with any well defined function, extrapolation to estimate the next year value would be meaningful.
But the parameters obtained in the training period will not be valid for a long time due to its limitation.
Hence the model parameters are updated every year to overcome this problem. The skill of the present
ANN model is verified for a set of observed values for the period (2001-2014) in forecasting SWM rainfall.
This set of observed values is not included in the previous training exercise. The forecast for any year
+ 1 is based only on the parameters computed for the interval 1907, of SWM data retaining the
same ANN architecture as in Figure 2. This way the non-stationarity of data can be addressed effectively.
The above exercise has to be repeated for the next year with the length of the data series increasing by one
year as the data is non-stationary. In Table 3 one year ahead forecast for years (2001 to 2014) as given by
the ANN model is shown along with the observed value. For each year + 1 the standard error of the
point forecast will be equal to the root mean square error between the actual data and the model fit. This is
also shown in the Table 3. The statistical forecast skill of the present approach in terms of , the
correlation between and and the performance parameter is shown in Table 4. It is observed that
even with a small sample length of 14 years the forecast skill of the ANN technique is very promising. A
measure of forecast widely used in India is the rainfall percentage above or below the long term average.
Such a comparison of the actual and forecast percentage deviations above and below normal value (LTA)
in the independent verification stage is presented in Fig. 4(a-d).
Table 4 Performance of the new ANN model in the testing period
5. DISCUSSION
The available time series data of SWM rainfall does not exhibit any significant year to year linear
correlation. But it shows some small correlations with the pre monsoon and the NEM. Even though there
is no much linear relation with the pre-monsoon, but a small correlation of 0.24 can be observed between
NEM and SWM which is above the threshold level of 0.2. It shows a non-linear relationship between them.
Also it accounts for the intra-annual or inter-seasonal variability. This has led to the inclusion of these
seasons in the network to model SWM data. However, since the SWM data is strongly non-Gaussian, one
cannot conclude that there are no short and long term patterns in the oscillations of the data series. In fact
the EMD analysis of Iyengar and Raghukanth (2004) on SWM data highlights the existence of a dominant
period of 2-3 years accounting for nearly 60-70% of inter annual variability in the rainfall. This situation
indicates that a complex non-linear relationship is required to model and further forecast Indian SWM
rainfall. Since ANN technique has the flexibility of handling such unstructured nonlinear relations, this
approach has been selected here. In the previous work of the same authors, a six-input node model had been
selected, where only SWM data was used for modeling and forecasting. This was done on the IMD data,
which slightly varies from the IITM data quantitatively. This has been improved in the present paper by
using the network shown in Figure 2 for IITM data having ten input nodes. Further, more addition in the
number of input nodes lead to saturation of the statistical skill and also increase in the number of parameters,
Region
Modeling Period (1907:2000)
(cm)
AIRF 2.00 0.97 0.94
Core Monsoon 3.32 0.97 0.94
NEIND 2.61 0.97 0.94
COKNT 11.45 0.97 0.94
8. Kokila Ramesh and R.N. Iyengar
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which is not desirable. After such preliminary exercises the present ANN model of Figure 2 has been arrived
at as the optimal for use in forecasting of SWM rainfall. The correlation coefficient between the data
and the forecast has to be at least 0.52 to be taken as significant in the independent test period of 14 years.
As seen from Table 4 the forecasting skill is well beyond this threshold level. However, a point forecast is
always a probable value and hence comes with a standard error, like the standard deviation from the mean.
This, forecast error varies slightly in time depending on the length of data up to the antecedent year. These
are also computable in the present approach as shown in Table 4. The average forecast errors for the period
2001-2014 for the four data considered here are (2.19, 3.62, 3.42, and 12.92) respectively. A simple
comparison with the LTD value shown in Table 1 brings out the reduction achieved in the error between
climatic mean (LTA) as a forecast versus the present ANN forecast. A pictorial performance of the present
forecast model is shown in Figure 4. It is observed that in the earlier model that 12 of 56 cases were not
able to capture the same sign of, where only SWM data was used to model and forecast. But in the present
model, 6 of 56 cases are not able to capture the same sign. The number of cases has reduced by 50%, hence
the present model shows an improvement in modeling and forecasting. Also these six cases are well within
the error band shown. Since empirical forecast NR
)
for any year N is statistical in nature, any forecast in
Table 3 has to be treated as a random variable with its forecast error as standard deviation. The probability
density function for this variable can be found in terms of the sample distribution of the model error.
Figure 4 Percentage departure from normal (LTA) for the period of 14 years (2001:2014) for (a) AIRF, (b) Core
Monsoon, (c) NEIND and (d) COKNT
2000 2003 2006 2009 2012 2015
-30
-20
-10
0
10
20
Year
%departurefromLTA
2000 2003 2006 2009 2012 2015
-30
-20
-10
0
10
20
30
40
Year
%departurefromLTA
2000 2003 2006 2009 2012 2015
-30
-20
-10
0
10
20
Year
%departurefromLTA
2000 2003 2006 2009 2012 2015
-30
-20
-10
0
10
20
30
Year
%departurefromLTA
Actual Forecast
(a) (b)
(c) (d)
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6. SUMMARY AND CONCLUSION
An improvement in the earlier work of the same authors has been tried in the present paper. A network is
developed, which used the intra seasonal variability to produce the improved results for SWM data. This is
a new approach for modeling and forecasting the total rainfall of the monsoon season in India, which is
presented in this paper. After a brief literature review and the explanation of the network, it is demonstrated
that a simple ANN architecture with six nodes of SWM data and two nodes each of pre-monsoon and the
NEM at the input layer, a hidden layer with five neurons and an output, is capable of explaining about 94%
of the observed inter-annual variability of observed SWM rainfall data. This has been demonstrated on four
sets of data for the period (1901-2000). The model is capable of updating itself as the length of the sample
increases. This property can be effectively used in forecasting, by extrapolating the results of the model by
one step. The statistical skill of the forecasts have been verified on four sets of observed data not used in
the ANN training exercise and shown to be highly significant. The data series considered here is pre-
monsoon, SWM and NEM rainfall on annual basis. Hence the present work has focused on both inter annual
and intra annual variability. However, the model developed is capable of improvement to include month to
month variations using monthly rainfall data. Such generalizations for monthly rainfall time series will be
presented in a future publication.
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