Determination of dates of beginning and end of the rainy season in the northern part of
Madagascar from 1979 to 1989.
IZANDJI OWOWA Landry Régis Martial*ˡ, RABEHARISOA Jean Marc*, RAKOTOVAO Niry Arinavalona*,
RAMIHARIJAFY Rodolphe*, RATIARISON A Adolphe*
*University of Antananarivo - Doctoral School of Physics and Applications - Physics of the Globe and Environment Dynamics of the Atmosphere Climate and Oceans.
Abstract
This study aims to determine the dates of beginning and end of the rainy seasons, their durations, for the 1979-1989
period. We used: the method of six degree polynomial, the anomalous accumulation, and the optimal least squares
estimator. The results show that on average, the dates of beginning of the seasons move from East to West. In most
cases, the dates of beginning and end, and durations of seasons of rains, show a strong polynomial trend of sixth
degree.
Keywords: six degree polynomial; anomalous accumulation; optimal least-squares estimator; analysis; background;
observation.
INTRODUCTION
2. Characteristic curve of rain intensities
Water cycle is one of the major components of the
climate, and its implications on rainfall patterns are
important. In tropical and subtropical zone,
particularly in Madagascar, rainfall variability is a
major factor of vulnerability of societies. Indeed, the
economy remains very largely based on products of
the agricultural sector (example: vanilla in the
Northern part, product whose Madagascar is the 1st
world producer). Moreover, the capacity of adaptation
of population to the climatic risks is still limited too
much. This study was undertaken to determine the
dates of beginning and end of seasons of rains by
implementing three methods.
Characteristic curve of rain intensities is obtained,
using an even (“pair”) polynomial function of the sixth
degree by filtering the precipitation values, in the
purpose to determinate the dates of the beginning and
the end of the rainy season
For the daily global averages in the whole Northern
part of Madagascar, ranging between 42° and 54°
East and between 11° and 15° South, over the 33
seasons between 1979 and 2012, we obtained
(following equation):
(1)
1.
Material and methods
In this study we used three methods: the method of
polynomial of the sixth degree, the method of
Liebmann (anomalous accumulation) and the optimal
estimator of least-squares. We used precipitations data
from the European Center of Medium Range Weather
forecasting (ECMWF) with the format “netcdf”.
Calculations were made using MATLAB 2015
software and tables were made under EXCEL.
Figure 1: Average seasonal variations of the daily
rains and the corresponding values, and the
characteristic curve of precipitation intensities
represented by the values filtered by a polynomial of
the sixth degree over 33 seasons in the whole of the
Northern part of Madagascar.
ˡ author : IZANDJI OWOWA Landry Régis Martial*ˡ
Doctorant : e-mail : [email protected],
RABEHARISOA Jean Marc* Maître de Conférence,
RAKOTOVAO Niry Arinavalona* Maître de
Conférence,
RAMIHARIJAFY Rodolphe*, Docteur en Physique
et Application,
RATIARISON A Adolphe* Professeur Titulaire
In theory, the two minima on each side of the
maximum, respectively represent the date of the
beginning (before the maximum), and the completion
date (after maximum) of the season of rain.
1
But, if the minima are not on the positive part of “the
y axis” which represents the values of precipitation,
we will take then as minimum, the point of
intersection of the values filtered curve with “the x
axis”. Ref [1]
3. Method of polynomial of the sixth degree
This method, as its name indicates it, uses an even
(“pair”) polynomial function of the sixth degree for
the determination of the dates of the beginning and the
end of the rainy season. This method was explained to
the preceding section. We have below the general
form of the polynomial of the sixth degree.
(2)
4. Method of Liebmann :
Anomalous Accumulation (AA)
The index called “Anomalous Accumulation” (AA) is
given by :
Figure 3 : Comparison between the methods of
Liebmann and the polynomial of the sixth degree
(3)
There are 5 months and 01 day of difference over the
duration of the season between the two methods,
which is very considerable. Therefore we need to find
a way to make a choice.
The beginning and the end of the rain season then are
respectively determined by the date of the minimum
and the maximum on the curve of “Anomalous
Accumulation” Ref [2].
5. Analysis of “optimal” least-squares: Data
assimilation concepts and methods. Ref [3]
The data assimilation indicates the various methods
making it possible to take into account, in the
initialization of a forecasting digital model, certain
observation data arrived to this center after the model
launched its calculations (forecasts in real time). A
comparison is made then, with regular times intervals,
between values produced by the new observations on
the one hand, and corresponding values predicted by
the model on this term on the other hand. From the
differences between the rough outline (draft) of
evolution provided by the model and the new
observations, it then becomes possible to rectify the
walk of the model, by periodically modifying its
forecasting results at determined moments, and these
results, once modified, become new initial data for the
future forecasts.
Figure 2: Anomalous accumulation of the daily
average of annual precipitations over the period of
1979-2012 on the Northern part of Madagascar
1)
A quick analysis makes it possible to note that indeed,
the method of Liebmann (anomalous accumulation)
has a tendency to reduce the real duration of the
season, as shown in the following figure.
Data assimilation analysis
In the presentation of this method of analysis, we will
refer sometimes at the true model state. It is an
expression referring to the best state possible to
represent by the model. That which we try to
approach.
2
It is necessary to make the distinction between reality
itself (which is more complex than the representation
of the state vector) and the best possible representation
of reality by a state vector than we will call
: “true
state at the time of the analysis”. Another important
value of the state vector is , who is the a priori or a
background estimate (draft) of the true state. The
analysis is indicated by . Therefore, the problem of
the analysis reduces to find a correction
or called
analysis increment such as :
analysis can be reduced to a simple scalar expression,
so that :
Xa  Xb  k ( X o  Xb )
Let us notice that the observations
here by .
are expressed
The variance of the error of the estimate (of the
analysis) is given by :
a2  (1  k )2 b2  k 2o2
Therefore, the optimal analysis is :
, being as close as possible of.
 xb
xo   b2 . o2
xa  
2
 2  2 

 2
o   b   o
 b
For a given analysis we use several values of
observation (or data and other information related),
which are contained in an observation vector . To use
its values in the procedure of analysis requires being
able to compare them with the state vector. The key of
the data analysis is the use of “contradictions”
between the observations and the state vector. Those
are given by the vector of departures at the
observation points :




(4)
being as close as possible of the true state
; the
observations are expressed here by
; and
is
the a priori or a background (the first estimate) of the
true state .
The « gain », or the weight of the analysis is given by :
k
 b2
 b2   o2
When they are calculated from the rough outline (first
estimation or background)
, they are called
innovations, and when they are calculated from the
analysis
, they are called analysis residuals.
Studying these “departure vectors” provides important
information about the quality of the procedure of
assimilation.
Where
and
are respectively the error variances
of the background (or first estimate) and of the
observation.
Where , the observation operator, is a linear operator
The error variance analysis is :
-
(5)
the dimension of the model state is
(6)
the dimension of the vector of observations is .
2)
Optimal least-squares analysis equations.

In the limiting case of a very low quality of
observation, we will have
, so
, and then (
) : and the
analysis remains equal to the background.
Because as we saw :
) ).

However, if the observation is of very good
quality (contrary case).
, then
, and the analysis remains equal to the
observation (
)

If both have the same precision :
,
then
, and analysis is simply the
arithmetic average of
and , which
reflects the fact that we trust as much the
observation as the background. We have
then :
, so we make a compromise.
The optimal estimators of least-squares, or analysis
BLUE, are defined by the following interpolation
equations :
-
Optimal analysis : we should seek a
weighted linear average whose form is :
Xa  Xb  K( y  H[ Xb ])
-
Gain or weight of the analysis :
K  BH T (HBH T  R ) 1
Where the linear operator
is called « gain », or
weight matrix of the analysis. The weights are noted
(minuscule). , and are respectively: the operator
of observations, the matrix of covariance of the outline
(background) errors and the matrix of covariance of
the observation errors. The optimal least-squares
3

But in all cases,
which means
that the analysis is a weighted average of the
background and the observation.

The analysis variance for the optimal
existing differences between methods AA and
POLY6.
is :
6. Choice of the observation and the first
estimate (background) for the calculation of
optimal values of dates of beginning and end
of rainy season
We take as “observations” the dates of beginning
and end observed on the seasonal variations of
precipitations from 1st August until 31st July, which
are provided by the polynomial of the sixth degree.
Indeed, these dates are comparable to “direct
observations”, because they are readable on the graph
of rain variations. We will take as first estimate (or
outline, background), the dates of beginning and end
of season estimated by the method Anomalous
Accumulation. We will take the Root Mean Squared
Error (noted RMSE), as value of
(standard
deviation of observation errors provided by the
polynomial of the sixth degree). The RMSE is
regarded as an estimate of the standard deviation of
calculated answers. Because we do not know really,
which of the two methods (polynomial of the sixth
degree and anomalous accumulation) is closest to the
true state, we will give them the same importance
while imposing as
and consequently that
That is equivalent supposing that the values
resulting from the two methods have the same
probability (“equiprobable”). That suits us, because
the average Root Mean Squared Error of the
anomalous accumulation is much larger (188,28),
compared to that of the polynomial of the sixth degree,
and in this case the analysis will be almost equal to the
observation (polynomial of the sixth degree). Results
of the average duration for the whole of the Northern
part of Madagascar, (average of 33 seasons: from
1979 to 2012) are given in the following table.
Figure 4°: Average dates of beginning and end of
rainy season, obtained by the methods of the
polynomial of sixth degree (POLY6) and anomalous
accumulation (AA), compared with those calculated
by the optimal least squares analysis (ANALYSIS).
In the previous sections, we worked with global
daily averages. Now, to determine year per year the
dates of beginning and end for ten (10) rain seasons,
between 1979 and 1989, we will reduced and
subdivided the Northern part of Madagascar in four
parts.
7.
Reduction and division of the study area
The reduced area is located between 12° and 15° of
South latitude and 47° and 51° of East longitude. We
will divide the area into four parts (following figure).
Table 1 : Duration of the rain season (average of 33
seasons : from 1979 to 2012) in the whole of the
Northern part of Madagascar, and parameters of the
analysis.
DURATIONS OF THE RAIN SEASON
POLY6
AA
optimal ANALYSIS
288 Days :
9 months et 18 days
137 Days :
4 months et 17 days
213 Days :
7 months et 3 days
= 1,2247
Compromise :
= 0,8660
Note: We have named :
- POLY6: Polynomial of sixth degree
- AA: anomalous accumulation
- ANALYSIS: optimal estimator of least-squares
We see the tendency of anomalous accumulation
method to reduce the duration of the season of rains,
and the capacity of least-squares optimal estimator to
optimize the duration of the season by reducing the
Figure 5 : Area subdivided into four parts
4
-
8. Determination of indices of start and end of
the rainy season for ten seasons going to
1979-1980 to 1988-1989, and their duration
Duration of season (in days)
The results we considered were those provided by the
optimal least squares analysis. They are supposed to
be better because they are optimal according to the
optimal least squares analysis. We have summarized
in Figures 6, the average dates of beginning and end of
seasons, and their average earliest and latest dates for
each of the four areas.
Duration of seasons: In zone 3 there was a
very significant polynomial trend of the sixth
degree for the durations of the seasons.
y = -0,1053x6 + 3,6457x5 - 48,946x4 + 320,08x3 - 1048,6x2 + 1578,9x 633,37
DURATION (in Day)
R² = 0,8808
Linéaire (DURATION (in Day))
y = -5,8242x + 203,73
Poly. (DURATION (in Day))
R² = 0,206
240
230
220
210
200
190
180
170
160
150
140
130
120
1
2
3
4
5
6
7
8
9
10
Rank of the seasons: from 1979 to 1989
Figure 8: Behavior of the durations of the seasons of
rains of 1979 to 1989 in Zone 3 (Northern Part:
South-west)
Remark: We observed in zone 1, a weak linear falling
trend with a coefficient of determination of 0.25
(figure 9 below). We shall see what will give the
linear regression over thirty-three (33) years (i.e. up to
2012). If this tendency is confirmed, it would mean a
reduction in the duration of the seasons of rains.
Figure 6°: Average dates of beginning and end of
season (average of ten seasons studied) for each of
the four zones.
This will be the object of the continuation of our work.
9. Behavior of the dates of beginning and end,
and of the durations of the rain seasons
between 1979 and 1989 (10 seasons)
Duration of season (in days)
y = -0,0275x6 + 0,9138x5 - 11,655x4 + 71,188x3 - 211,61x2 + 273,68x + 40,667
R² = 0,5878
We made this observations: these variables did not
present a significant linear trend for the ten studied
seasons; a significant polynomial regression trend of
the sixth degree was observed, as shown in the
following examples. In the graphs which will follow,
“Linéaire” means linear regression and “Poly” means
polynomial regression.
-
175
170
165
160
155
150
145
140
135
130
125
1
2
3
4
5
6
7
8
9
Figure 9: Behavior of the durations of the seasons of
rains of 1979 to 1989 in Zone 1 (Northern Part:
North-West)
y = -0,0144x6 + 0,3397x5 - 2,4778x4 + 3,8001x3 + 19,107x2 - 49,743x + 130
R² = 0,775
Index of dates (in days)
y = -2,5394x + 159,87
R² = 0,2494
Rank of the seasons: from 1979 to 1989
Dates of beginning: In zone 4, there was a
significant polynomial trend of the sixth
degree.
Index of BEGINNING (in Day)
Linéaire (Index of BEGINNING (in Day))
Poly. (Index of BEGINNING (in Day))
DURATION (in Day)
Linéaire (DURATION (in Day))
Poly. (DURATION (in Day))
y = -1,3636x + 116,8
R² = 0,0442
140
135
130
125
120
115
110
105
100
95
90
85
80
75
Conclusion
1
2
3
4
5
6
7
8
9
In this study, we observed that on average, the rainy
season begin in the East and advances towards the
West. As for the end of the season, it propagate from
West to East in the Northern zone of the Northern Part
of Madagascar, and from East in West in the Southern
zone of this Northern part. In most cases, the dates of
beginning and end, and durations of seasons of rains,
showed a strong polynomial trend of the sixth degree..
10
Rank of the seasons: from 1979 to 1989
Figure 7: Behavior of the dates of beginning of
seasons of rains from 1979 to 1989 in Zone 4
(Northern Part : South-east)
5
10
[2] Apport des données TRMM 3B42 à l’étude des
précipitations au MATO GROSSO. Université Paris 7
- Universidade do Estado do Rio de Janeiro - UERJ
http://lodel.irevues.inist.fr/climatologie/docannexe/file
/746/40_arvor.pdf
Page 56-57
[3] ECMWF, Bouttier and Courtier March 1999 :
Data assimilation : concepts and methods
http://www.ecmwf.int/sites/default/files/Data%20assi
milation%20concepts%20and%20methods.pdf
References :
[1] RAMIHARIJAFY Rodolphe : « Inter –
corrélation entre la pluviométrie et le débit sauvage en
amont de la centrale hydroélectrique du site
d’Andekaleka a Madagascar ».
http://www.recherches.gov.mg/spip.php?page=detail_
article&id_article=3144
Thèse 2014-2015 - Ecole Doctorale Physique et
Application – Université d’Antananarivo.
Page 75-79
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