Apresenta

1
SYNTHETIC RAINFALL SERIES GENERATION - MDM Heinz
D. Fill, André F. Santana, Miriam Rita Moro
Mine Contacts mrmine.dhs_at_ufpr.br
1. MONTHLY DISAGGREGATION MODEL - DUM 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.
3. MDMS VALIDATION 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. 3.1. Annual Validation 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
2. DESCRIPTION OF THE MODEL
Annual time step precipitation generation
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)
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.

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 (j1, 2, , m) (m
length of historical record) with size k x 12 (k
number of sites) is constructed. Its elements
are
Table 1 Sites number convention
Convention Site Name
1 Monte Carmelo
2 Monte Alegre
3 Usina Couro do Cervo
4 Franca
5 Fazenda Barreirinho
6 Tomazina
7 União da Vitória
8 Lagoa Vermelha
9 Caiuá
Figure 3 Validation - Mean
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. The model is structured in 2 Modules
and performs sequentially the following steps
(Figure 1)
Figure 4 Validation Standard Deviation
Figure 5 Validation Cumulative Deficit
Compute mean and variance at each site
Standardize mean annual precipitation
Compute the correlation matrix
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.
Compute the coefficient matrix (B)
Compute the disaggregation matrices (Dj)
Module 1
Transform standard normal vector into cross
correlated random vector, using
Generate k independent standard normal random
number
4. CONCLUSIONS 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
records 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.
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 theEuropean
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.
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.