This model is based on a monthly time step, which avoids reproducing zero rainfall sequences what is a rather complicated procedure. By selecting a disaggregation model one take advantage of the fact that in humid regions annual precipitation has an essentially normal distribution. This has the Central
Limit Theorem support and also has been successfully verified by many statistical tests.
Annual time step precipitation generation
Disaggregation in a monthly time step
It has been assumed that total annual precipitation is not serially correlated, but cross correlation among rainfall stations was considered. Also annual precipitation has been assumed to be normally distributed what is supported both by empirical evidence (Homberger et al ., 1998) and also by the
Central Limit Theorem. So, generation of multisite annual precipitation series is reduced to a multivariate normal distributed random numbers generation.
In serially uncorrelated hydrologic variables case, they may be modeled by the equation (Kelman, 1987): x ( t )

B .
z ( t )
Where x( z( t t ) is a vector of
) is a size k k (number of sites) cross-correlated random variables, independent random variables vector and B is a coefficients matrix, obtained from the sites correlation matrix. Variables are attached to a time index t .
The chosen method uses disaggregation coefficients computed from historical records. It is called Hydrologic Scenarios Method.
For each historical record year, a matrix D historical record) with size elements are: j
(j=1, 2, …, m ) ( m = length of k x 12 ( k = number of sites) is constructed. Its d im
( j )

P im
( j )
P i
( j )
Where P im
( j ) represents the month m , site i and year j precipitation the site disaggregation proceeds randomly combining each matrix D amounts.
j
, while P i
( j ) is i and year j annual precipitation. Given an annual precipitations series, with the annual
The model is structured in 2 Modules and performs sequentially the following steps (Figure 1):
Compute mean and variance at each site
Standardize mean annual precipitation
Compute the correlation matrix
Module 1
Compute the disaggregation matrices (D j
)
Generate k independent standard normal random number
Compute the coefficient matrix (B)
Transform standard normal vector into cross correlated random vector, using: x ( t )

B .
z ( t )
Apply the Hydrologic
Scenarios Method to disaggregate annual in monthly precipitation
Module 2
Obtain the length m cross correlated annual precipitation series
Figure 1: P rocedures Sequence in MDM
ACKNOWLEDGEMENTS:
The research leading to these results has received funding from the
European Community's Seventh Framework Programme (FP7/2007-2013) under Grant
Agreement N
212492. Third author also would like to thank
for the financial support.
For MDM validation some synthetic series statistics have been compared to those computed from the historical records on the selected sites within the La
Plata Basin.
Figure 2 shows their geographical location within the study area.
All the algorithms were developed in
Matlab (R13, The Mathworks Inc, 2000, under license) software.
It have been generated 1000 series of 62 year long each one (the same length as the historical record) and the following statistics have been computed:
Figure 2: Selected Sites for Validation
•Mean, Standard Deviation and Skew Coefficient;
•Number of consecutive years below/above mean;
•Each synthetic series correlation matrix;
•Maximum cumulative deficit for 80% of mean.
The last item has an important effect on flow regulation studies because influences significantly hydropower generation in well regulated systems, such as the Brazilian interconnected system. Some of the results are shown in Figures
3, 4 and 5; sites convention numbers are expressed in Table 1.
Table 1: Sites number convention
# Convention Site Name
1
2
3
4
5
6
7
8
9
Monte Carmelo
Monte Alegre
Usina Couro do Cervo
Franca
Fazenda Barreirinho
Tomazina
União da Vitória
Lagoa Vermelha
Caiuá
2500
2000
1500
1000
500
0
1 2 3 4 5
Site
6 7
Minimum Maximum Average Observed
Figure 3: Validation - Mean
8 9
700
600
500
400
300
200
100
0
1 2 3 4 5
Site
6 7 8 9
Minimum Maximum Average Observed
Figure 4: Validation – Standard Deviation
3500
3000
2500
2000
1500
1000
500
0
1 2 3 4 5
Site
6 7 8 9
Minimum Maximum Average Observed
Figure 5: Validation – Cumulative Deficit
In the monthly step mean, standard deviation and autocorrelation seasonal values were computed, for both historical and synthetic values. Besides, analogous procedure of annual validation was followed for synthetic values, with maximum, minimum and average values calculated.
The first results, however, showed a discrepancy between the original and generated series for some of the sites. This fact was attributed to some programming bug, which will be revised and fixed soon.
Regarding the annual scale generation, it is clear that the value computed from historical record is well within the range of the synthetic series values and, in most cases, close to the average from 1000 series computed. This shows that the synthetic series reasonably preserve most of the historical record’s statistics in terms of annual precipitation.
Next task for the MDM conclusion is a debug procedure, in order to find what is wrong with the monthly step generation.
REFERENCES:
HOMBERGER, G. M., RAFFENSBERGER, J. P., WILBERG, P. L.
Opkins, University Press, Baltimore, 1998.
KELMAN. J. Modelos estocásticos no gerenciamento de recursos hídricos. In:______.
. São Paulo: Nobel/ABRH. 1987. p. 387 - 388.
, John