Modeling of daily precipitation at multiple ... - Wiley Online Library

WATER RESOURCES RESEARCH, VOL. 45, W12412, doi:10.1029/2008WR007453, 2009

Modeling of daily precipitation at multiple locations using a mixture of distributions to characterize the extremes Yeshewatesfa Hundecha,1,2 Markus Pahlow,1 and Andreas Schumann1 Received 15 September 2008; revised 11 August 2009; accepted 1 September 2009; published 16 December 2009.

[1] A stochastic model for the generation of daily time series of rainfall at multiple

locations in which the amount of daily rainfall is modeled by a mixture of two different probability distribution functions is presented. A mixture model is implemented with the specific objective of characterizing extremes of daily precipitation. The approach is based on the assumption that the extremes within a time series have a different stochastic behavior compared to the normal regime of precipitation. A multivariate autoregressive model is used to model the local probability of occurrence of rainfall and the amount while keeping the intersite covariance structure using a truncated normal distribution. The amount simulated using the truncated normal distribution is further transformed so that it can be regarded as coming from the actual distribution fitted to the daily precipitation at each station using the probability integral transformation. The seasonal cycles of the amount as well as the temporal and spatial correlations of the daily precipitation are incorporated by fitting the model on the monthly basis. Application was made on 122 stations within the Unstrut catchment with an area of 6343 km2 in central eastern Germany. Results show that the model can fairly well reproduce a number of statistical features of daily precipitation including the extreme value distribution of the annual maximum daily and 3 day total precipitation, both at individual stations and at the catchment scale. Citation: Hundecha, Y., M. Pahlow, and A. Schumann (2009), Modeling of daily precipitation at multiple locations using a mixture of distributions to characterize the extremes, Water Resour. Res., 45, W12412, doi:10.1029/2008WR007453.

1. Introduction [2] Hydrological analysis using a sufficiently long series of precipitation and other meteorological variables is required for many practical problems, such as the assessment of water resources of a river basin or the flood risk within or downstream of a basin. Obtaining time series of these meteorological data with the required length and resolution, both in space and time is, however, often difficult. This problem is often circumvented by making use of weather generators, which are stochastic models for the generation of synthetic time series of the weather variables that reproduce the statistical properties of the observed time series. [3] Different stochastic models have been developed in the past for the generation of weather variables using different approaches. Setting up a model for the generation of a precipitation time series at a higher temporal resolution such as daily or hourly is particularly challenging due to the spatial and temporal intermittence that is inherent in precipitation at such scales. Random variables with a mixed distribution of a discrete probability of zero precipitation and a continuous probability of nonzero precipitation are 1 Institute of Hydrology, Water Resources Management and Environmental Engineering, Ruhr-University Bochum, Bochum, Germany. 2 Now at German Research Centre for Geosciences, Helmholtz Center Potsdam, Potsdam, Germany.

Copyright 2009 by the American Geophysical Union. 0043-1397/09/2008WR007453

therefore required to model precipitation. The most common approach of stochastic modeling of precipitation is by implementing separate models for the occurrence of precipitation and the amount of precipitation whenever it occurs [Todorovic and Woolhiser, 1975; Katz, 1977; Stern and Coe, 1984; Woolhiser, 1992]. Richardson [1981] and Richardson and Wright [1984] implemented such an approach in a weather generator that also models daily maximum temperature, minimum temperature, and solar radiation whose statistical characteristics are assumed to be conditional on the occurrence of rainfall. [4] The occurrence of precipitation is commonly modeled by using a two-state Markov process that simulates a chain of dry and wet days [Haan et al., 1976; Richardson, 1981; Roldan and Woolhiser, 1982; Stern and Coe, 1984; Wilks, 1998]. An alternating renewal modeling approach, which is based on the relative frequencies of wet and dry spells, is also used as another approach [Buishand, 1978; Roldan and Woolhiser, 1982]. Since the amount of precipitation generally shows a skewed distribution with a bias toward low amounts, it is usually modeled using random variables that reproduce this property. The gamma distribution has commonly been used to model daily precipitation [Katz, 1977; Richardson and Wright, 1984; Stern and Coe, 1984; Wilks, 1989], although a truncated and power transformed normal distribution [Ba´rdossy and Plate, 1992; Hutchinson, 1995], an exponential and a mixed exponential distribution [Richardson, 1981; Woolhiser and Roldan, 1982; Wilks, 1989, 1999; Wilson et al., 1992] have also been applied in other studies.

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HUNDECHA ET AL.: PRECIPITATION MODELING

Figure 1. Schematic representation of the modeling procedure. [5] Many of the models based on the aforementioned approach are developed for the modeling of rainfall series at a single site. There is, however, often a need to generate synthetic series at multiple sites for regional hydrologic analysis and the approach is not as suitable for multiple sites as it is for single site generation due to the need to correctly take into account the inter site correlations of precipitation. There are, nevertheless, extensions of the single site model for multiple locations as well [Wilks, 1998; Thompson et al., 2007]. [6] Generalized linear models [McCullagh and Nelder, 1989] have also been implemented in the stochastic modeling of precipitation [Chandler and Wheater, 2002; Furrer and Katz, 2007]. Using the generalized linear model framework, the transformed means of the probability distributions of both the amounts and occurrence of precipitation can be estimated as linear combinations of covariates such as site effects, precipitation on previous days, month of the year, as well as climate indicators like El Nin˜o –Southern Oscillation or Pacific Decadal Oscillation. Such an approach offers an attractive feature in that different distributions of the same family are fitted for each day so that the temporal variability of precipitation due to the covariates is accounted for. However, the approach does not offer much flexibility in the choice of distributions as it is applicable to a limited family of distributions. Extension to a multisite model is also possible using this framework [Yang et al., 2005]. [7] A nonparametric approach based on resampling of the observed precipitation has also been used in some studies as an alternative approach to model daily series of precipitation both at a single site [Lall and Sharma, 1996; Brandsma and Buishand, 1998; Rajagopalan and Lall, 1999] and multiple sites [Buishand and Brandsma, 2001; Leander et al., 2005]. Since the simulated precipitation is sampled from the historically observed series, this approach does not produce values at sampling time scale that have not been observed. Therefore applicability of this approach for extreme daily events is limited since one cannot extrapolate the extreme daily precipitation. [8] Another approach of modeling daily precipitation series that can also easily be extended to modeling at multiple sites is implementation of a first-order autoregressive process to simulate random numbers that are normally

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distributed. The generated numbers are then censored to define the occurrence of precipitation and power transformed to produce the precipitation amounts. Ba´rdossy and Plate [1992], Hutchinson [1995], and Ailliot et al. [2009] implemented this approach to model the rainfall process at multiple sites. While the power transformation is meant to impart to the distribution the known skewed nature of the distribution of daily precipitation amount, censoring of the simulated values truncates negative values of the distribution so that days with negative simulated values are regarded as dry days, thereby enabling modeling of the precipitation state implicitly. Implementation of a normal distribution makes it easier to account for the intersite correlation structure as it is amenable to the classical multivariate stochastic methods. [9] Despite the common practice of adopting the gamma, exponential, or a mixed exponential distribution to model the amounts of daily precipitation, it is not uncommon to face a problem of properly characterizing the extremes of the daily precipitation using such distributions. Wilks [1999] indicated that while the gamma model generally underestimates the extreme daily precipitation, a mixed exponential distribution offers an improvement only when the extremes are not very high (less than 100 mm). Different approaches have been implemented to improve simulation of extreme precipitation. Semenov [2008] and Qian et al. [2008] tested models in which the amounts of daily precipitation are modeled using a semiempirical distribution, which is a histogram formed by dividing the observation range into a number of bins, in terms of their performance in reproducing the extremes. The results indicate that the models simulate the extremes well, at lease in the range of the observations. Vrac and Naveau [2007] implemented a mixture of gamma and generalized Pareto distributions (GPD) using a dynamic mixing weight and obtained an improvement over a gamma distribution when the precipitation shows a heavy tail behavior. Furrer and Katz [2008] tested a hybrid model of gamma and GPD in which the gamma model is used below a certain threshold while the GPD is used above the threshold. They obtained a substantial improvement over a pure gamma model to simulate the extremes. [10] This paper presents an implementation of a mixture model within a multisite stochastic modeling framework with a specific objective of characterizing the extremes of daily precipitation. A multivariate autoregressive model, which makes use of multivariate normal distributions to model both the amounts and occurrence of daily precipitation, is implemented to generate daily series of precipitation simultaneously at multiple locations. We extend the approach in such a way that any arbitrary distribution can be used for the amounts of daily precipitation. The model is tested by applying it to 122 stations within the Unstrut River Basin (area 6343 km2) in central eastern Germany, where the distribution of daily precipitation shows different tail behavior than the conventionally fitted distribution functions.

2. Methodology 2.1. Model Concept [11] The model implemented in this work is based on the concept of the space-time model introduced by Ba´rdossy

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Figure 2. The Unstrut catchment with the precipitation stations used in the present work. and Plate [1992], which makes use of a truncated power transformed normal distribution to model daily precipitation. In their model, Ba´rdossy and Plate [1992] related the random function Z(t, u), describing the amount of daily precipitation at time t and location u, to a normally distributed random function W(t, u) as Zðt; uÞ ¼

0 if W ðt; uÞ 0 : W b ðt; uÞ if W ðt; uÞ > 0

ð1Þ

[12] They introduced a positive coefficient b to account for the skewed nature of the distribution of daily precipitation. The problem of intermittence of occurrence of precipitation is also handled by this relationship, as the negative values of W are declared as dry days. [13] In the present study, we implement a mixture of gamma and GPD to model the daily amount of precipitation at each station. The underlying idea is that while the bulk of the data can be modeled using the gamma distribution, the upper tail of the distribution can be modeled using a distribution that is suitable to handle the extremes so that

extrapolation to higher return values can conveniently be made. A GPD is a model that is used to model the extremes of a random variable above a sufficiently high threshold [Coles, 2001]. [14] The mixture model is ideally implemented by choosing a threshold and utilizing the gamma model for precipitation values less than the threshold and the GPD for those values exceeding the threshold. However, in addition to the problem of identifying a suitable threshold, this approach makes the resulting density function of the precipitation discontinuous at the threshold. Furrer and Katz [2008] fitted a gamma distribution for the entire data set and a GPD for precipitation above a chosen threshold. Then they retained the estimated shape parameter of the GPD and adjusted its scale parameter so that the resulting distribution becomes continuous at the threshold. [15] An alternative approach of mixing a distribution that describes small and moderate values of a random variable with a GPD to describe its extremes was proposed by Frigessi et al. [2002]. They proposed mixing the two distributions over the entire domain of the random variable with a variable mixing weight that gives more emphasis to

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Figure 3. Observed and simulated monthly mean and standard deviations of daily precipitation together with autocorrelations of daily precipitation. Observed values are shown as dots, solid lines indicate median values of the simulated precipitation, and dotted lines indicate the 90% confidence interval of the simulated values. the GPD for higher values. This way, the difficulty of choosing an appropriate threshold is avoided since no threshold is used. They implemented the proposed approach in the modeling of Danish fire loss by making use of a mixture of GPD and Weibull distribution. Vrac and Naveau [2007] implemented a similar approach to model daily precipitation using a mixture of GPD and gamma distribution. A similar approach is followed in this work to mix the two distributions. [16] The mixture model is given by f ðz; uÞ ¼

½1 pðz; uÞhðz; uÞ þ pðz; uÞgðz; uÞ ; KðuÞ

K(u) is a normalizing constant that ensures the area under f(z, u) is unity. h(z, u) and g(z, u) are the pdf’s of the gamma and GPD, respectively, hðz; uÞ ¼

ð3Þ

where g(u) > 0 and b(u) > 0 are the shape and scale parameters, respectively, corresponding to location u; G(.) is the gamma function: 1 1 xðuÞ 1 xðuÞz gðz; uÞ ¼ ; 1þ aðuÞ aðuÞ

ð2Þ

where f(z, u)is the probability density function of the daily precipitation z at location u; p(z, u) is the mixing weight;

ðz=bðuÞÞgðuÞ1 expðz=bðuÞÞ ; bðuÞGðgðuÞÞ

ð4Þ

where x(u) and a(u) > 0 are the shape and scale parameters, respectively. When x(u) > 0, the distribution displays a

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Figure 4. Observed and simulated monthly mean proportion of wet days and spell lengths. Dots indicate observed values, solid lines indicate median of the simulated, and dotted lines indicate the 90% confidence interval of the simulated values.

heavy tail behavior. When x(u) < 0, it is bounded and when x(u) = 0, the distribution reduces to an exponential distribution g(z, u) = 1/a(u)exp(z/a(u)) with a light tail. These different features of the GPD make it a suitable flexible distribution to model the different extreme behaviors of the daily precipitation. [17] The mixing weight p(z, u) is a function that gives more emphasis to the gamma distribution at low values of precipitation and more weight to the GPD at high values while keeping the continuity of the distribution. We adopt a mixing weight proposed by Frigessi et al. [2002]: pðz; uÞ ¼

1 1 z nðuÞ þ arctan ; 2 p tðuÞ

ð5Þ

distribution governed by the GPD for large z. The normalizing constant K(u) is then given by KðuÞ ¼ 1 þ

1 p

Z1

z nðuÞ ½gðz; uÞ hðz; uÞ arctan dz: tðuÞ

ð6Þ

0

[18] In order to account for the seasonal variability of daily precipitation, different models are fitted to each month of the year. [19] The multisite process is modeled by making use of a multivariate autoregressive random process using a multivariate standard normal distribution W(t) = (W(t, u1),. . ., W(t, un)), which is expressed using the following multivariate autoregressive (AR) (1) model:

where n(u) > 0 and t(u) > 0 are the location and steepness parameters, respectively. Note that the limit of p(z, u) for large z values is unity thereby making the tail of the mixture 5 of 15

WðtÞ ¼ rðWðt 1Þ wÞ þ CYðtÞ þ w;

ð7Þ


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Figure 5. Observed frequencies and median of simulated frequencies of durations of wet and dry days over the entire observation and simulation periods. where Y(t) = (y(t, u1),. . ., y(t, un)) is a random vector of independent normally distributed random variables with zero mean and unit variance; w = (w(u1),. . ., w(un)) is the mean vector of W(t, u) whose elements are zero in our case; r is the lag 1 day autocorrelation function; C is a n n matrix that takes the spatial variability of the process into account. [20] In order to estimate the parameters used in the multivariate AR(1) model, the daily precipitation at each station is transformed so that it can be regarded as coming from a truncated standard normal distribution. A transformation into a truncated standard normal variate x is performed using the cumulative distributions of the fitted distribution and that of the truncated portion of the standard normal distribution. The distribution is truncated based on the probability of wet days pw(u): Fðz; uÞ ¼

FðxÞ ð1 pw ðuÞÞ ; pw ðuÞ

where F(z, u) is the cumulative distribution function of the daily nonzero precipitation amounts at location u, and F() is the cumulative distribution function of a standard normal variate. [21] Both the autocorrelation and the spatial covariance matrices are computed from the transformed precipitation x. Since the transformed variable x comes only from a portion of a normal distribution, computation of both the autocorrelation and the spatial covariance is not straight forward. Instead, the required correlations can be estimated from the indicator correlations estimated from the quantiles of W(t, u). This is carried out by first converting x (including the zero precipitation amounts) or equivalently the precipitation series z(t, u) into an indicator series Iq(t, u) corresponding to any quantile q such that 1 pw(u) < q < 1, where pw(u) is the probability of rainfall at location u using

ð8Þ

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I q ðt; uÞ ¼

8
0

p>1

p>5

p > 10

p > 20

Month

Obs

5%

Med

95%

Obs

5%

Med

95%

Obs

5%

Med

95%

Obs

5%

Med

95%

Obs

5%

Med

95%

1 2 3 4 5 6 7 8 9 10 11 12

67 61 68 67 66 71 63 59 60 55 66 68

59 60 67 61 59 57 55 54 52 55 60 65

63 62 70 64 63 58 58 58 58 58 64 66

68 72 72 71 67 64 60 59 60 62 69 68

55 53 54 54 57 58 56 50 51 52 54 56

49 49 49 47 50 49 50 47 44 47 50 54

51 51 54 50 51 51 52 52 50 50 54 56

57 63 55 51 55 57 53 53 52 54 59 59

25 23 27 25 34 33 33 30 29 27 27 31

24 22 25 25 30 29 30 30 25 25 27 25

26 23 28 27 33 31 32 34 31 27 29 28

29 31 29 28 36 37 33 36 32 34 32 32

14 12 16 18 20 20 20 19 18 19 16 17

12 11 15 15 20 19 18 23 16 16 15 13

13 14 17 16 21 21 22 25 20 18 18 15

15 18 18 17 26 25 24 27 23 23 20 20

7 7 10 11 10 10 10 9 9 8 10 8

4 5 7 7 9 10 10 14 8 9 6 5

5 6 8 9 11 11 13 15 12 10 8 7

8 11 10 9 18 17 15 19 15 13 11 10

a

Obs is observed; 5%, is 5% simulated; Med is median simulated, and 95% is 95% simulated.

when the precipitation at all stations exceeds a given quantile value at each station is computed and is displayed in Figure 11. Stations are grouped in distance ranges of integer multiples of 10 km. All stations with separation distances within halfway between the previous and next range fall in the same group. From Figure 11, it can be seen that the correlations generally agree well at lower quantiles with the median correlations of the simulated precipitation generally being slightly lower. For higher quantiles, the uncertainty in estimating the correlations is very high due to a few number of events for which all the stations have precipitation exceeding a given quantile. For instance, there are only five events in the observed series over the period 1969 – 2001 for which all stations recorded precipitation amounts exceeding the 90% quantile values at each station. The corresponding median number of events in the simulated series of 100 years length is 22.

4. Summary and Conclusions [39] A stochastic model for the generation of daily time series of precipitation at multiple locations with a specific objective of characterizing the extremes of daily precipitation has been developed and applied to a test catchment in central eastern Germany. A mixture of gamma and GPD

was fitted to the daily amount of precipitation at each station with a dynamic mixing weight that gives more emphasis to the GPD at higher precipitation values. A multivariate autoregressive model is implemented to simulate both the occurrence and transformed precipitation amounts simultaneously at multiple locations. This is done by using a standard normal variate at all locations and truncating the amount to zero when the simulated value at each location is less than or equal to a quantile corresponding to the probability of dry days. The simulated amount is then back transformed so that it can be regarded as coming from the fitted distribution. [40] The model generally performs well in mimicking the annual cycle of the monthly mean precipitation and the monthly variability of the daily precipitation. The observed values generally lie within the 90% confidence interval of the simulated values. The wet and dry day frequencies as well as their durations are also well simulated by the model including the distributions of the spell lengths. Also the autocorrelation structure is well captured and the tendency of progressively decreasing autocorrelation is reflected in the model simulations. [41] The mixture model has been demonstrated to simulate the extremes of daily precipitation well as it reproduced fairly well the distribution of the annual maximum daily

Table 3. Observed and Simulated Joint Exceedance Probabilities of Different Precipitation Amounts at Stations Ilmenau and Balgstaedta Both 0

Both > 0

Both 5

Both > 5

Both 10

Both > 10

Both 20

Both > 20

Month

Obs

Sim

Obs

Sim

Obs

Sim

Obs

Sim

Obs

Sim

Obs

Sim

Obs

Sim

Obs

Sim

1 2 3 4 5 6 7 8 9 10 11 12

38.50 42.12 41.08 43.67 47.31 41.11 49.43 52.76 49.46 49.22 38.28 36.21

38.52 36.16 36.31 39.48 42.19 42.22 45.48 48.70 46.73 47.63 35.49 34.22

37.15 38.58 37.96 35.78 32.80 37.22 31.53 28.82 31.40 33.71 38.82 42.25

38.58 36.99 39.15 33.33 32.64 34.71 28.21 28.78 30.13 30.61 38.50 40.54

82.41 85.73 82.37 84.56 81.72 78.78 79.19 80.02 83.33 85.22 82.26 80.96

83.18 85.08 81.21 84.58 81.59 80.98 80.33 81.34 84.84 84.95 81.37 82.42

2.81 3.65 5.59 4.78 5.27 6.89 6.35 6.35 6.13 4.99 5.27 4.79

3.92 4.02 5.44 4.90 6.77 7.12 7.08 7.34 5.88 5.12 5.42 5.19

93.76 94.29 93.87 94.00 91.51 90.33 90.32 90.32 93.01 93.13 92.69 91.16

93.93 93.89 92.73 94.18 91.27 90.52 90.70 89.94 94.05 92.92 92.09 92.22

0.42 1.14 1.18 1.56 2.15 2.56 2.29 2.50 1.51 2.29 1.83 1.14

0.57 0.90 1.39 1.44 2.47 2.61 2.40 3.35 1.57 1.83 1.70 1.14

98.65 98.52 98.28 98.33 97.85 96.67 97.29 96.15 98.71 97.81 98.28 97.61

98.92 98.40 98.17 98.63 97.98 97.12 97.66 96.71 98.95 97.98 97.71 98.36

0.10 0.00 0.32 0.44 0.32 0.33 0.62 0.94 0.11 0.21 0.11 0.00

0.00 0.07 0.25 0.26 0.32 0.46 0.38 0.70 0.13 0.25 0.26 0.06

a

Obs, observed; Sim, simulated. Values are given in percent.

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Figure 11. Mean cross correlations of precipitation amounts between stations separated by different distance ranges for different exceedance levels of the amounts of precipitation at all stations. Dots are observed values, solid lines are median values of the simulated series, and dotted lines indicate the 90% confidence interval of the simulated series. precipitation. It is also demonstrated that the model leads to a significant improvement in characterizing the tail behavior of the distribution of the daily precipitation over the gamma model. The good match of the distribution of the simulated and observed extremes for the whole catchment also demonstrates how the spatial structure of the rainfall has been satisfactorily represented in the stochastic model. This would enable one to extrapolate the extreme precipitation for higher return periods with the correct spatial pattern, thereby making it possible to assess the flood risk in a spatially consistent way. [42] The stochastic model presented in this study is applicable under the assumption of stationarity. However, the model can be extended to climate impact studies by parameterizing it conditionally on large-scale circulation indices as in the work by Ba´rdossy and Plate [1992], Stehlı´k and Ba´rdossy [2002], and Hundecha and Ba´rdossy [2008]. [43] Acknowledgments. Funding for this work was provided in part by the German Ministry for Education and Research (BMBF grant 02 WH 0588). Access to data was granted by the Ministry of Agriculture, Nature Conservation and the Environment of Thuringia (TMLNU), and the Ministry of Agriculture and Environment of Saxony-Anhalt (MLU LSA) and is gratefully acknowledged.

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Y. Hundecha, German Research Centre for Geosciences, Helmholtz Center Potsdam, Telegrafenberg, D-14473 Potsdam, Germany. (hundey@ gfz-potsdam.de) M. Pahlow and A. Schumann, Institute of Hydrology, Water Resources Management and Environmental Engineering, Ruhr-University Bochum, Universita¨tsstr. 150, D-44801 Bochum, Germany.

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