Disaggregation of highly seasonal monthly rainfall - CiteSeerX

Disaggregation of highly seasonal monthly rainfall Lelys Guenni Centro de Estadstica y Software Matematico Universidad Simon Bolvar Apartado 89.000, Caracas, Venezuela

and Andras Bardossyy

Institute fur Wasserbau Universitat Stuttgart D-70569, Germany

Abstract The need for high resolution rainfall data at temporal scales varying from daily to hourly or even minutes is a very important problem in hydrology. For many locations of the world, rainfall data quality is very poor and reliable measurements are only available at a coarse time resolution such as monthly. The purpose of this work is to apply a mixed method for the disaggregation of monthly precipitation to daily. The method consists of three steps: 1. Assessment of the number of wet days as a random variable conditional to the monthly amount based on the Polya distribution. 2. Division of the monthly sum among the wet days using Aitchinson's relative amount distribution. 3. Restructuring of the series according to various time series statistics (autocorrelation function, scaling properties, seasonality) using a Monte Carlo Markov Chain based algorithm. The method was applied to a data set from a rainfall network of the central plains of Venezuela. A detailed analysis was carried out to study the seasonal and spatial variability of many properties of the daily rainfall as scaling properties and autocorrelation function in order to incorporate the selected statistics and their annual cycle into the objective function. Comparisons between the observed and simulated data suggest the adequacy of the technique in providing rainfall sequences with consistent statistical properties at a daily time scale given the monthly totals. y

e-mail:[email protected] e-mail:[email protected]

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1 Introduction One of the main problems with hydrological times series is that measured values are usually given at a time scale coarser than the one needed. For example, daily rainfall values are usually measured at many rainfall recording stations. But hourly values are usually required for drainage systems design or many other applications in hydrology and environmental sciences. Since a disaggregated series is a realization from the original aggregated time series, stochastic approaches are necessary in order to reproduce the right statistical characteristics of the data at the required time scale. According to Torf (1997), a dissagregation technique must be a statistical simulation technique, since a deterministic approach is impossible in practical terms. The reason for this is that the original process which is normally unknown is reconstructed from the aggregated process. A signi cant amount of contributions on disaggregation methods give support to the importance of this technique to solve this data limitation problem: Valencia and Schaake (1972); Curry and Bras (1978); Brass and Rodriguez-Iturbe (1985); Grygier and Stedinger (1988); Santos and Salas (1992) More recent contributions are presented by Glasbey et al. (1995); Maheepala and Perera (1996); Torfs (1997); Burger (1997); Hershenhorn and Woolhiser (1992); Lebel et al. (1998); Tarboton et al. (1998); Bardossy (1998a). In many of the techniques a distributional assumption about the disaggregated series is required which is usually the normality assumption. Since the most common hydrological variables, rainfall and stream ow, are ussually nonnormal, a normalizing transformation is required (Maheepala and Perera (1996); Burger (1997)). Tarboton et al. (1998), propose a non-parametric approach by which the joint probability density functions are estimated directly from the historical data using kernel density estimates. The required joint probability density function is f (X; Z ), where X is the vector of disaggregate variables (eg. monthly rainfall, daily stream ows) and Z is the aggregate variable (annual rainfall, hourly stream ows). Given the aggregate variable Z the problem can be posed as a sampling from the conditional probability density function:

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f (XjZ ) = f (X; Z )= f (X; Z )dX

(1)

Due to the intermitence characteristics of the rainfall process and the lack of gaussianity, rainfall still poses the most challenging problems in the dissagregation methods. Most of the existing methodologies are used for the disaggregation of rainfall time series at a single location. Recently 2

Figure 1: Precipitation time series for dierente aggregation time periods . Lebel et al. (1998), presented a space-time dissagregation approach for areal storm depth which deals separately the spatial and temporal variabilities. The spatial dissaggregation is based on the turning bands method (TBM) (Matheron (1982)) and the temporal disaggregation is achieve by imposing a standard hyetogram at each location of the spatial disaggregation domain. The methodology used as input data the mean areal rainfall depth as estimated from a dense recording raingauge network for the mesoscale convective complexes occurred in a Niger region. For many parts of the world, rainfall data quality is poor and raingauge networks are very sparse, which poses a limitation in the estimation of spatially continuous rain elds, especially at a short time scale (Guenni et al. (1997)). More reliable data is available, for example, at a monthly time step than at a daily time step. To illustrate the disaggregation problem with rainfall data we present in gure 1 aggregated time series of rainfall data from one location at the central plains of Venezuela in Guarico State. In this gure, the monthly, quarterly and annual aggregated values are shown with the daily values. If the monthly values are available for a particular location, the purpose of the disaggregation procedure is to reproduce a time series that \looks like" the original daily series. 3

There are a number of desirable characteristics that should be preserved from the original time series. As a rst step, the proportion of dry periods within the aggregated period should be reproduced. Serial dependence, simple scaling properties, probability distribution are some of the many characteristics that could be considered. When the series is non-stationary and there is for example, a strong seasonal component, many of the desirable characteristics that should be preserved are also non-stationary and usually seasonal. This is the case of the data set that will be analized in this work. The methodology is based on the approach presented in Bardossy (1998a), where the technique was applied to selected stations in the Ruhr catchment, Germany, to disaggregate daily to hourly time series. The paper is organized as follows: the data set is described in section 2. In section 3 the disaggregation methodology is explained. The results and validation of the methodology are presented in section 4. Finally the discusion and conclusions indicating possible extensions of the methodology are presented.

2 Methodology 2.1 Computational disaggregation procedure Daily rainfall data from fourteen stations of Guarico state, located at the Central Plains of Venezuela were used for parameter estimation. A diagram of the position and elevation of the selected locations is shown in gure 2. These locations were selected among a number of 120 stations available because of their better data quality and larger number of records. Data from 1967 to 1991 were used. Rainfall at these locations is stronly seasonal with a dry season going from November to April; and a rainy season from May to October. A boxplot diagram for each month is shown in gure 3 for two locations. Following Bardossy(1998a) disaggregation of rainfall amounts into higher temporal resolution values consists of the following three steps:

Generate the number of wet sub-periods (e.g. days) within a time period (e.g. months) Generate the sub-period values conditional on the number of wet sub-periods and the total rainfall amount in the period

Re-order iteratively the arbitrary sequence of sub-period values until the new series has a prescribed probability distribution.

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Figure 2: Position and elevation for the 14 locations used . The high seasonality present in the data suggests a dierent probability distribution for each month and this characteristic will be used in the disaggregation procedure. Let n X i (2) Zj = zji (k) j

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is the rainfall amount on month j and year i; zji (k) is the daily amount on day k of month j and year i and nj is the number of days of month j . For each month the relationship between the probability of having a wet day pjw and the amount of rainfall Zji was investigated. A linear model was tted to the probability of having a wet day as a function of the monthly rainfall. A reasonable good linear t was found for all months as it is shown in gure 4. The slope of the linear relationship showed a strong seasonal behaviour and the monthly boxplots for the 14 locations is presented in gure 5. This dependence between the probability of a wet day and the rainfall amount for each month is used as follows:

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1. Generation of the number of wet days within a month 5

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Figure 3: Monthly boxplots of rainfall amounts for the 14 locations . The number of wet days within a month njw it is assumed to follow a Polya distribution (Feller (1971)) with parameters p and a depending on the total monthly rainfall Zj :

1 0 N ? 1 CA p(p + a) : : : [p + (k ? 2)a]q(q + a) : : : [q + (N ? k ? 1)a] (3) p(njw = kjZj ) = B @ 1(1 + a) : : : [1 + (N ? 2)a] k?1

The parameters a and p are estimated from the observations. p is the unconditional probability of getting a wet day and a is the incremental probability of getting a wet day or a dry day after each wet day or dry day occurrence. The unconditional probability of getting a dry day q is 1 ? p. As we are assuming that Zj > 0, at least one day is a rainy day and the Polya distribution is shifted by 1. This explains the N ? 1 and k ? 1 in the previous formula. The number of wet days within a month are then obtained by sampling from the Polya distribution using the tted relationship between the probability of a wet day and the rainfall amount. 2. Generation of the daily precipitation totals Once we have determined the number of wet days within month j (njw ), we need to determine 6

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Figure 4: Relationship between the probability of a wet dat and the monthly rainfall amount for a dry (February) and a wet (August) month . the rainfall amounts at each day of the month (z j (k)). One can de ne the relative daily rainfall amount for day k as: z j (k) (4) R(k) = Z j

This implies that the following relationship holds: n X j

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Since there are only njw wet days within the month we can remove the remaining R(k) values from the previous sum. We can also assume that the R(k) values dierent from zero are the rst njw values. At this stage of the analysis, the order of the rainy days within the month does not play any role since in the following step the obtained daily rainfall values will be re-ordered. Aitchinson (1982) suggested a normal distribution for the transformation: 7

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One can assume that R(k) > 0 if i < nw and R(nw ) R(k) for all k in a given month. The assumption of normality for W (k) allows a simple mechanism for the relative amounts sampling and the daily rainfall amount estimation since the parameters of this distribution can be estimated from the available observations. 3. Re-ordering of the daily rainfall amounts within a month At this stage we have found a sequence of daily rainfall values z (1); z (2); : : : ; z (nj ) which are not following any particular time structure according to the autocorrelation properties of the original time series. There are other charateristics of the daily time series as the probability distributions of dry and wet runs and the simple scaling properties as described by Burlando and Rosso (1996) that will not be preserved by the unstructured series. A computational intensive procedure based on the Metropolis-Hastings algorithm, a powerful Markov chain method, is used to generate a sequence of Markov chains that will converge to a prescribed multivariate probability distribution . This limiting distribution is de ned with the help of an objective function O. This approach uses the "annealing theorem" combined with the Metropolis-Hastings algorithm to produce realizations corresponding to the minima 8

of O as obtained from the limiting distribution: (t) = K (t)exp(?

O(i) ) T

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where O(i) is the objective function at state i and T is a control parameter called the \temperature" which guarantees convergence of the objective function to the global minima. A detailed description of the Metropolis-Hastings algorithm can be found in Chib and Greenberg (1995). Following Bardossy(1998a) the following procedure is followed to reorder the data. Giving an initial value of the temperature control parameter l for l = 0, repeat M times the following steps: 1. Select the precipitation amount for month j , Zj > 0, at random. 2. Select two indices i1 and i2 from the set 1; 2; : : : ; nj at random such that z j (i1 ) and z j (i2 ) are not both equal to zero. 3. Calculate the value of an objective function for the case the values of z j (i1 ) and z j (i2 ) are swaped (Os ) and for the case the values of z j (i1 ) and z j (i2 ) are unswaped (Ou ). 4. If Os Ou the values of z j (i1 ) and z j (i2 ) are swaped since the objective function decreases. If Os > Ou the values of z j (i1 ) and z j (i2 ) are swaped with probability: P = exp

Ou ? O s l

(8)

The process is repeated from step 2. The lower the value of l , the less likely is the probability of performing a swap that does not decrease the objective function. After M iterations, index l is increased by 1 and we set l+1 < l . Steps 2 to 4 are again repeated M times. The whole process is reapeated until j < crit .

2.2 De nition of the objective function The limiting distribution of the disagregated series will converge asymptotically to the distribution of the original series which is described in terms of an objective function that we want to minimize. The objective function can be formulating by including dierent properties of the rainfall sequences that ones want to preserve, as for example, autocorrelation function, scaling properties, wet and dry duration distributions, etc. 9

For example, for the autocorrelation function, an objective function can be formulated as the squared dierence between a prescribed autocorrelation function (k) and a simulated (k): O1 =

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where K is the number of lags used in the calculations. If the serial dependence of the rainfall occurrences wants to be preserved, an objective function could be de ned as: K X O2 = (I (k) ? I (k))2 (10) k=1

I (k) stands for the \indicator autocorrelation function" which is applied to the transformed rainfall

data through the indicator function:

8 > < 1 if z(t) > I (z (t)) = > : 0 if z(t)

(11)

where is a rainfall threshold representing a small rainfall amount. The \wide sense multiple scaling" property described by Gupta and Waymire (1990) usually holds for the raw moments such that: E [z l (h)] = l E [z l (h)]

(12)

where l is the order of the moment, E [:] is the expected value, zh is the accumulated rainfall at duration h and zh is the accumulated rainfall at duration h and l is the exponent of the scaling relatiosnship. An objective function could be formulated in order to preserve the scaling properties of rainfall as follows: J (E [z j (h)] ? E [z j (h)])2 XX (13) O3 = (E [z j (h)])2 2 j =1

whre J is the number of moment orders and is the set of durations considered. All these objectives functions could be combined as: O = 1 O1 + 2 O2 + 3 O3

(14)

where the 0i s are positive weights. They can be arbitrary since dierent realizations will be generated such that O 0. Other aspects of the precipitattion series could be incorporated into the objective function as the distribution of the durations of wet and dry periods, dierent distributions for varying atmospheric conditions as El Ni~no or La Ni~na eects or dierent circulation patterns as considered by (Brdossy and Plate, 1992). 10

3 Application of the Methodology The dierent daily rainfall properties used in the disagregation procedure were calculted for each month and each location in order to investigate their seasonal and spatial variability. These results are presented as follows: In gure 6, the estimated monthly mean and variance for the Aitchinson distribution are shown. Mean of Aitchison Distribution

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Figure 6: Mean and Variance of the Aitchinson Probability Distribution . For each month the dots are the estimated values for the 14 locations. A higher dispersion of these estimates are presented for the driest months (January, Febraury and March). This can be expected since the number of data points used in the estimation process is smaller due to the large number of zeroes in the data. There does not seem to be a clear seasonal pattern in these plots which is is suggesting that one parameter set could be used for the the disagregation of all monthly values. The autocorrelation and indicator autocorrelation function for four dierent months and all locations are shown in gure 7 and gure 8. It is evident from the above gures that dierences from location to location and for the dierent months are present. This is physically realistic, for example, since one can expect a stronger serial dependence in the rainfall occurrence process is for a wet month than for a dry month. In fact the lag-one indicator autocorrelation has an average value of 0.4 in August (wet month) and an average 11

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Figure 7: Autocorrelation function for dierent months and all locations . value of 0.2 in February (dry month) (see gure 8). A lot of scatter is observed in boths functions. The scaling exponents of 12 are shown in gure 3 where two characteristics can be observed: Firstly, the exponents vary with time of the year being higher during the rainy season than during the dry season. Secondly, a higher dispersion is observed during the dry season, which is attributed to the large number of zeroes in the dry months. In gure 3 the relationships between dierent aggregation time periods and their corresponding moments in a log-log scale are shown for March and August at station 1. The relationships are presented for the rst three moments and the slopes of the linear regressions are tted from the data. These slopes are the scaling exponents of 12. The previous characteristics of autocorrelation and scaling properties are used as parameters to the disaggregation model and the annealing process is carried out by specifying the objective function described in 14 for each month and each location. For a given temperature value, a maximum of 5,000 swaps of the rainfall amounts are intended in order to decrease the objective function. To save some computational time, each evaluation is only performed for the part of the objective function which is aected by the swaped values. Then the temperature is decreased until a critical value is reached. As an example, in gure 11 it is observed how the objective function decreases with the temperature and in gure 12 how the 12

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Figure 8: Indicator autocorrelation function for dierent months and all locations . number of swaps decrease with temperature when we disaggregate 17 years of March rainfall at location 4. This is an indication of the convergence of the methodology. In gures 13 and 14 a comparison betweeen the observed daily values and the simulated disaggregated values is presented for two locations. In all cases the years with missing values have been eliminated and consequently at each location, a dierent number of yearly replications for each month have been disaggregated. In the tables 1 and 2 dierent statitics for the observed and simulated values are compared for a dry month (March) and a wet month (August). The quartiles and the mean values for the simulated and observed series show a good agreement. However one can observe some discrepancies in the simulated maxima, indicating that the extreme values need some consideration within the proposed methodology. In order to compare an aditional characteristic of the observed precipitation with the simulated values, the cumulative probability distribution of consecutive wet and dry day durations was estimated for location 1 in August. From gure 15 it is observed, for example, than the frequencies of wet and dry sequences of shorter durations tend to be underestimated by the simulated values. A possible solution for this is to include directly this feature into the objective function. Another possibility is to explore a more dicult relationship between the probability of having a rainy day and the rainfall amounts, which 13

Location Min. q1 1 (Sim) 0 0 1 (Obs) 0 0 2 (Sim) 0 0 2 (Obs) 0 0 3 (Sim) 0 0 3 (Obs) 0 0 4 (Sim) 0 0 4 (sim) 0 0

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Mean q3 Max. 0.1238 0.05 6.29 0.1276 0 28 0.2348 0 52.7 0.2367 0 42.5 0.09519 0 18.55 0.08742 0 19.1 0.2752 0 62.17 0.2738 0 85.8

Table 1: Comparisons between simulated and observed statistics of daily rainfall (mm) for March at four locations.( q1,q2 and q3 are the quartiles)

Location Min. q1 q2 Mean q3 Max. 1 (Sim) 0 0 0.93 5.906 5.39 91.6 1 (Obs) 0 0 1.15 6.138 7 128.5 2 (Sim) 0 0 0.705 6.485 5.42 116.1 2 (Obs) 0 0 1 6.787 6.9 107.2 3 (Sim) 0 0 0 5.782 2.26 216.1 3 (Obs) 0 0 0.3 5.881 7.8 59.5 4 (Sim) 0 0 0.46 5.786 3.307 80.07 4 (sim) 0 0 0.75 5.87 8.55 55.4 Table 2: Comparisons between simulated and observed statistics of daily rainfall (mm) for August at four locations.( q1,q2 and q3 are the quartiles)

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is the base of the Polya distribution, using for example, a non-linear relationship instead of a linear one.

4 Discussion and Conclusions The proposed methodology has certain advantages which we can enumerate as follows: 1. It does not need to have a description of the probability density function of the disaggregated series in a parametric, empirical or non-parametric form. This an advantage for most hydrological time series which can not be assumed normally distributed. On the other hand the resulting conditional probability distribution of 1 comes from a sampling process which is forced by the "drivers" of the minimization method which are the selected rainfall properties. 2. Dierent rainfall attributes and occurrence rainfall dependence on the aggregated amounts can be incorporated into the process in a parametric or non-parametric form. In the parametric case the distributional characteristics of the rainfall properties could be used in the de nition of the objective function to consider the inherent uncertainty in the estimated values. Another possibility is using a re-sampling procedure by which the rainfall properties being considered in the objective function are estimated from a sub-sample of the observations at each simulation step. 3. A relatively small number of attributes of the disaggregated series need to be used to simulate a time series at a ner time scale. This is an advantage specially when considering the application of this methodology to multiple sites. Although most of the analized properties show an annual cycle, some parameters as the Aitchinson's distribution mean and variance, could be considered fairly constant through the year and for all locations. But instead of using a constant value, the parameter uncertainty can be incorporated with the methodologies mentioned above. It is clear that the scaling exponents show a strong seasonality and this should be included in the simulation process. With a larger number of locations it is possible to study the spatial variability of the selected properties in more detail. The analized region does not present strong contrasts in precipitation regimes and can be considered a fairly homogeneous region. However dierences are observed and they can be modeled. 15

An extension of the methodology for two or more locations is under study. In this case the cross-correlation and many other characteristics which preserve the spatial dependence of the simulated series could be inserted within the objective function with more computational cost.

Acknowledgements We are grateful to Milagros Loreto for the preparation of many of the gures of this paper and the National Council of Scienti c Research for the partial funding of this project under the Research Grant S1-2770.

References Aitchinson, J. (1982) The statistical analysis of compositional data. Journal of the Royal Statistical Society, B, 44, 139{177. Bardossy, A. (1998a) Disaggregation of daily precipitation. J. of Hydrol.; (submitted). Bardossy, A. (1998b) Generating precipitation time series. Water Resour. Res., 34, 1737{1744. Brass, R. L. and Rodriguez-Iturbe, I. (1985) Random functions in hydrology. Reading,Mass.: Addison-Wesley. Burger, Gerd (1997) On the disaggregation of climatological means and anomalies. Climate Res., 8, 183{194. Burlando, P. and Rosso, R. (1996) Scaling and multiscaling models of depth-duration-frequency curves for storm precipitation. J. of Hydrol., 187, 45{64. Chib, S. and Greenberg, E. (1995) Understanding the Metropolis-Hastings algorithm. The American Statistician, 49, 327{335. Curry, K. and Bras, R. L. (1978) Theory and application of the multivariate broken line, disaggregation and monthly autoregressive stream ow generators for the nile river. Technical Report 78{5. Ralph M. Parsons Lab., Mass. Inst. of Technol., Cambridge.

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Feller, W. (1971) An Introduction to Probability Theory and its Applications. New York: John Wiley. Glasbey, C. A., Cooper, G. and McGechan, M. B. (1995) Disaggregation of daily rainfall by conditional simulation. J. Hydrol., 165, 1{9. Grygier, J. C. and Stedinger, J. R. (1988) Condensed disaggregation procedures and conservation corrections for stochastic hydrology. Water Resour. Res., 24, 1574{1584. Guenni, L., Bardossy, A. and Haberlandt, U. (1997) Spatial interpolation of rainfall data at dierent time scale. Technical Report 97{6. Universidad Simon Bolvar, CESMa. Hershenhorn, J. and Woolhiser, D. A. (1992) Disaggregation of daily rainfall. J. Hydrol., 299{322.

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Lebel, T., Braud, I. and Creutin (1998) A space-time rainfall disaggregation model adapted to sahelian mesoscale convective complexes. Water Resour. Res., 34, 1711{1726. Maheepala, S. and Perera, B.J.C. (1996) Monthly hydrologic data generation by disaggregation. J. Hydrol., 178, 277{291. Matheron, G. (1982) The intrinsic random functions and their applications. Adv. Appl. Probab., 5, 1379{1394. Santos, E. G. and Salas, J. D. (1992) Stepwise disaggregation scheme for synthetic hydrology. J. Hydraul. Eng., 118, 765{1784. Tarboton, D. G., Sharma, A. and Lall, U. (1998) Disaggregation procedures for stochastic hydrology based on nonparametric density estimation. Water Resour. Res., 34, 107{119. Torfs, P. (1997) Disaggregation in hydrology. Technical Report. Agricultural University Wageningen. Valencia, D. R. and Schaake, J. L. (1972) A disaggregation model for time series analysis and synthesis. Technical Report 149. Ralph M. Parsons Lab., Mass. Inst. of Technol., Cambridge.

17

0.8

Firts Moment

0.4 0.2

scalling exponent

0.6

•

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• • •• •

•

-0.2

•

• 2

4

6

8

10

12

months

Second Moment • • •

1.5 0.5 0.0

scalling exponent

1.0

• • •

•• • • ••

• •• • •• • •• •••

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•

-1.0

•

• • 2

4

6

8

10

12

months

Third Moment 2

• • •

•

• • • ••

0

scalling exponent

1

•

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-1

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-2

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4

6

8

10

12

months

Figure 9: Scaling exponents for the rst, second and third moments for all locations

18

March, Station1 3 10

3 3

8

3

3

3

2

4

6

2 2

2

1

1

2

2

2

1

1

1 0.0

1

0.5

1.0

1.5

2.0

2.5

log(lambda)

14

August, Station 1 3 3 12

3 3

10

3 3

2

8

2 2 2

6

2 2

4

1 1

1

1

2

1 1 0.0

0.5

1.0

1.5

2.0

2.5

log(lambda)

Figure 10: Log-log relationships between dierent aggregation periods and the rainfall moments for March and August at station 1. The numbers represent the order of the moment

19

2 1

Objective Function

3

4

March, Station 4

0.20

0.15

0.10

0.05

0.0

Temperature

Figure 11: Objective function vs. Temperature for March at station 1 .

No. of Swaps

1000

1500

2000

2500

March, Station 4

0.20

0.15

0.10

0.05

0.0

Temperature

Figure 12: Number of Swaps vs. Temperature for March at station 1 . 20

10 6 4 0

2

Rain(mm)

8

Observed

days

4

6

Simulated

0

2

Rain(mm)

8

10

March, Station 1

Observed 0 20 40 60 80

Rain(mm)

120

days

20 40 60 80 100

Simulated

0

Rain(mm)

days

days

Figure 13: Observed and disaggregated values for March and August at location 1

21

80 60 40

Rain(mm)

20 0

days

80

March, Station 4

40 0

20

Rain(mm)

60

Simulated

Observed

0

Rain(mm)

10 20 30 40 50

days

days

Simulated

30 20 0

10

Rain(mm)

40

50

August, Station 4

days

Figure 14: Observed and disaggregated values for March and August at location 4

22

0.8 0.6

Simulated Observed

0.4

Probability

1.0

Dry duration probabilities

2

4

6

8

10

days

0.8 0.6 0.4

Simulated Observed

0.2

Probability

1.0

Wet duration probabilities

2

4

6

8

10

12

14

days

Figure 15: Wet and dry duration probability distributions for August at location 1 .

23