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SYNTHETIC RAINFALL SERIES GENERATION - MDM
Heinz D. Fill, André F. Santana, Miriam Rita Moro Mine
Contacts: [email protected]
1. MONTHLY DISAGGREGATION MODEL - DUM
3. MDM’S VALIDATION
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.
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.
Disaggregation in a monthly
time step
2.1. Annual Precipitation
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(t) is a vector of k (number of sites) cross-correlated random variables,
z(t) is a size k independent random variables vector and B is a coefficients
matrix, obtained from the sites correlation matrix. Variables are attached to a
time index t.
2.2. Monthly Precipitation
The chosen method uses disaggregation coefficients computed from
historical records. It is called Hydrologic Scenarios Method.
For each historical record year, a matrix Dj (j=1, 2, …, m) (m = length of
historical record) with size k x 12 (k = number of sites) is constructed. Its
elements are:
Pim ( j )
d im ( j ) 
Pi ( j )
The model is structured in 2 Modules and performs sequentially the
following steps (Figure 1):
Module 1
Standardize mean annual
precipitation
Compute the disaggregation
matrices (Dj)
Generate k independent
standard normal random
number
It have been generated 1000 series
of 62 year long each one (the same length
as the historical record) and the following
Figure 2: Selected Sites for Validation
statistics have been computed:
•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
1
2
3
4
5
6
7
8
9
Site Name
Monte Carmelo
Monte Alegre
Usina Couro do Cervo
Franca
Fazenda Barreirinho
Tomazina
União da Vitória
Lagoa Vermelha
Caiuá
2500
2000
Compute the correlation
matrix
Compute the coefficient
matrix (B)
Transform standard normal
vector into cross correlated
random vector, using:
1500
1000
500
0
1
2
3
4
5
6
7
8
9
Site
Minimum
Maximum
Average
Observed
Figure 3: Validation - Mean
3500
Cumulative Deficit (mm)
700
Where Pim(j) represents the month m, site i and year j precipitation, while Pi(j) is
the site i and year j annual precipitation. Given an annual precipitations series,
disaggregation proceeds randomly combining each matrix Dj with the annual
amounts.
Compute mean and
variance at each site
3.1. Annual Validation
Standard Deviation (mm)
Annual time step
precipitation generation
Mean (mm)
2. DESCRIPTION OF THE MODEL
600
500
400
300
200
100
0
3000
2500
2000
1500
1000
500
0
1
2
3
4
5
6
7
8
9
1
2
3
Site
Minimum
Maximum
4
5
6
7
8
9
Site
Average
Observed
Figure 4: Validation – Standard Deviation
Minimum
Maximum
Average
Observed
Figure 5: Validation – Cumulative Deficit
3.2. Monthly Validation
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.
x(t )  B.z (t )
4. CONCLUSIONS
Apply the Hydrologic
Scenarios Method to
disaggregate annual in
monthly precipitation
Obtain the length m
cross correlated annual
precipitation series
Module 2
Figure 1: Procedures 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 “Conselho Nacional de
Desenvolvimento Científico e Tecnológico-CNPq” for the financial support.
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. Elements of physical hydrology, John
Opkins, University Press, Baltimore, 1998.
KELMAN. J. Modelos estocásticos no gerenciamento de recursos hídricos. In:______. Modelos
para Gerenciamento de Recursos Hídricos I. São Paulo: Nobel/ABRH. 1987. p. 387 - 388.