Stochastic Generation of Spatially Consistent Monthly Rainfall Using SCL | Open Access Journals

ISSN: 2320-2459

Stochastic Generation of Spatially Consistent Monthly Rainfall Using SCL

Rahman Abdul A*

Department of Physics, Faculty of Science, COMSATS Institute of Information Technology, Pakistan

*Corresponding Author:
Abdul AR
Department of Physics, Faculty of Science
COMSATS Institute of Information Technology, Pakistan
Tel: +92 51 9247000
E-mail: andulrahman.aftabafzal@gmail.com

Received Date: 21/01/2018; Accepted Date: 14/03/2018; Published Date: 23/03/2018

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Abstract

Rainfall is the primary and most important component of any hydrological event. There is some amount of uncertainty in the future prediction of rainfall. The purpose of using SCL is the quantification of uncertainty involved in the prediction of rainfall. In the last few decades a considerable variation in the rainfall of the northern Pakistan has occurred. This might be due to the increase in temperature which is going to be threat for the water resources of Pakistan. So there is need to estimate the uncertainties involved in the estimation of the rainfall. In this paper an effort is being carried out to quantify the uncertainties in the average monthly rainfall for UIB using SCL (Stochastic Climate Library). In this paper an effort is being done to quantify the uncertainty involved in the average monthly rainfall data of rain gauge in Astor. Using the SCL the stochastic data is generated based upon the previous average monthly rainfall data of Astor from (1954 to 2000). Several statistics (annual and monthly) are computed to determine the behavior of the model and to compare them with the stochastic data produced.

Keywords

Environment, Regeneration, Agriculture, Hydrology

Introduction

Monthly rainfall data plays a key role in the estimation of runoff generated from large catchment and also to simulate the water resources of any country which is the backbone of the economy. In order to determine how a system will behave against the variation in the climate long data records of stochastically generated data are used.

Precipitation is vital in the climate variations. Srikanthan and Pegram [1] described an increasing trend in the precipitation to 1% from 0.5% in the elevated areas of the northern Pakistan. This increase has been observed in every decade of the 20th century. The statistically insignificance trend in the rainfall all over the world as well as for Himalayas being studied by many researchers in the last century.

Average monthly precipitation data is required to simulate the monthly flows which are main component of runoff produced from large catchments. But these models have a drawback that these models do not work well for the catchments which have considerable number of dry months. McMahon and Srikanthan [1] described the method of fragments to separate the annual rainfall data into its constituents by using the AR model of order 1. The limitation of this method is the inability that it cannot preserve the correlation (monthly) between the 1st month of a year and previous month of last year and also the same patterns for shorter length of data records.

Using stream flows Perera and Maheepala [2] made few changes in the method of fragments that take care of the first limitation of preserves of the monthly correlation. For the second limitation they have used the Thomas-Fiering monthly (stream flow) model. But it becomes troublesome for the sites having considerable number of months having no rainfall.

O’Neill and Sharma [3] used nonparametric method to model the inter annual effects in monthly runoff.

Climate Variability Program in the Cooperative Research Centre (CRC) for Catchment Hydrology has developed and tested many computer programs for generating stochastic climate data at time scales from less than one hour to one year and for point sites to large catchments. The appropriate models are part Stochastic Climate Library. In this research paper the development and testing of annual and monthly data of Astore (Pakistan) station has been studied. Stochastic monthly data can be further used for hydrological modelling and to quantify the uncertainties in the environmental system. A first order Markov chain is used which consist of two parts one is to model the occurrence of rainfall and second is to find the depth of occurrence. A two parameter Gamma distribution is used with correlated random numbers to obtain the rainfall depths. Historical and generated statistic shows that the model preserves the important characteristics of monthly and annual rainfall. Bardossy et al. [4] have studied time series of circulation patterns which are modelled with the help of a semi-Markov field and Rainfall is linked to the circulation patterns using conditional probabilities. Bardossy et al. [4] estimated the parameters method based on the moments of the observed data are developed.

In the last few decades a considerable variation in the rainfall of the northern Pakistan have occurred [5]. This might be due to the increase in temperature which is going to be threat for the water resources of Pakistan. So there is need to estimate the uncertainties involved in the estimation of the rainfall. In this paper an effort is being carried out to quantify the uncertainties in the average monthly rainfall for UIB using SCL (Stochastic Climate Library).

Study Area

In this study Astore watershed is selected to determine the uncertainty in the prediction of rainfall [6]. The Astore watershed is located in Pakistan having longitude and latitude 35° 33’ and 74° 42’ respectively, having a catchment area of about 3990 km2 Figure 1. There is only one gauging station installed by Pakistan Metrological Department having an elevation of 2168 m.a.s.l.

pure-and-applied-physics-water-shed

Figure 1: Astore Water Shed.

A store watershed is an elevated area having peak elevation of about 8000 m.a.s.l. in winter it is mostly covered with snow and glaciers. About 15% of the total area is covered with snow [7,8].

Methodology

MMF (Modified Method of Fragments) and AR (1) Model

The historical data of average monthly rainfall is standardized yearly in such a way that sum of average monthly precipitation for any year is one [9,10]. This can be done by dividing the average monthly rainfall by respective annual precipitation. In this way by having the record of k years one can produce the k sets of fragments of average monthly rainfall.

The appropriate number of monthly fragments for a particular year, n, is obtained by taking in account how close the produced yearly rainfall data and the monthly precipitation data for the last month of previous year to the corresponding observed data [2] (Figure 2). It is done by having of the appropriate number of the monthly fragment of a year, z, in the produced monthly data series that will develop a minimum value αz, that is given here:

pure-and-applied-physics-auto-correlation

Figure 2: Historical and Stimulated data of Rainfall in terms of Mean, Standard deviation, skewness, lag one auto-correlation, maximum and minimum.

image

image: Produced annual rainfall for year n

xz : Observed annual rainfall for year n

sx : st. dev of the annual precipitation

image: Observed monthly precipitation for the last month of the year n-1

yz-1: Observed monthly precipitation for the last month of the year z-1

sy= st. dev of the monthly rainfall for the last month of the year

The annual rainfall is generated by Autoregressive model of order 1 (AR (1)).

Monthly Parameters

The average values of the mean, std. dev, skewness, lag one autocorrelation max, min and percentage zero rain for the 100 replicates are compared with the historical values and are shown from Tables 1-7.

Table 1. Comparison between Historical and Generated monthly mean rainfall.

Item Hist. Mean 2.50% 25% 50% 75% 97.50% Tol. Y/N
Jan 43.349 38.871 28.571 35.676 38.536 42.201 47.148 7.50% N
Feb 48.677 43.645 35.759 41.085 43.785 46.771 52.36 7.50% N
Mar 88.632 91.138 72.932 82.783 89.691 98.238 120.49 7.50% Y
Apr 89.609 89.4 74.925 81.03 88.581 95.362 110.684 7.50% Y
May 76.823 81.489 63.354 74.763 81.302 88.013 97.98 7.50% Y
Jun 24.323 24.767 18.859 22.469 24.467 27.081 32.769 7.50% Y
Jul 24.349 25.069 18.98 22.587 24.93 27.418 32.312 7.50% Y
Aug 25.687 27.875 20.833 25.208 27.485 30.169 35.959 7.50% N
Sep 21.106 20.229 14.412 17.354 19.479 22.604 30.058 7.50% Y
Oct 28.731 32.639 21.829 27.996 32.282 36.489 45.568 7.50% N
Nov 21.188 21.14 15.97 18.314 20.876 23.369 27.607 7.50% Y
Dec 29.974 29.917 22.483 27.261 29.433 32.616 37.844 7.50% Y

Table 2. Comparison between Historical and Generated monthly mean rainfall Standard Deviation.

Item Hist. Mean 2.50% 25% 50% 75% 97.50% Tol. Y/N
Jan 30.13 29.49 20.39 26.669 29.208 31.96 38.569 7.50% Y
Feb 29.692 29.881 24.433 27.009 30.067 32.583 36.797 7.50% Y
Mar 57.134 61.978 44.357 55.788 63.38 67.087 83.297 7.50% N
Apr 53.213 53.854 45.017 49.409 53.754 58.348 63.438 7.50% Y
May 59.619 61.62 44.622 57.155 61.56 65.61 77.15 7.50% Y
Jun 19.899 19.774 13.698 15.887 19.245 22.47 29.536 7.50% Y
Jul 22.796 21.107 12.474 18.336 21.516 23.508 30.671 7.50% Y
Aug 22.993 24.207 15.2 21.493 23.942 27.107 32.805 7.50% Y
Sep 26.595 22.449 9.647 13.06 25.13 27.257 40.771 7.50% N
Oct 33.02 38.068 20.373 33.062 38.402 43.957 51.624 7.50% N
Nov 23.678 22.293 15.644 20.169 22.412 25.1 28.712 7.50% Y
Dec 31.051 28.644 21.362 25.299 28.389 31.293 40.631 7.50% N

Table 3. Comparison between Historical and Generated monthly mean rainfall Skewness.

Item Hist. Mean 2.50% 25% 50% 75% 97.50% Tol. Y/N
Jan -0.139 -0.022 -0.21 -0.1 -0.031 0.058 0.197 ±0.2 Y
Feb -0.024 0.008 -0.194 -0.055 0.008 0.075 0.208 ±0.2 Y
Mar -0.06 -0.057 -0.342 -0.145 -0.057 0.04 0.232 ±0.2 Y
Apr -0.045 -0.072 -0.339 -0.204 -0.083 0.045 0.294 ±0.2 Y
May 0.002 0.104 -0.119 0.007 0.104 0.167 0.364 ±0.2 Y
Jun -0.072 0.011 -0.285 -0.166 -0.001 0.145 0.395 ±0.2 Y
Jul -0.064 -0.059 -0.203 -0.147 -0.091 -0.008 0.23 ±0.2 Y
Aug -0.105 0.021 -0.271 -0.088 0.004 0.113 0.384 ±0.2 Y
Sep 0.068 0.018 -0.201 -0.104 -0.008 0.117 0.362 ±0.2 Y
Oct -0.116 -0.084 -0.275 -0.162 -0.108 -0.036 0.194 ±0.2 Y
Nov 0.07 0.086 -0.129 0.004 0.081 0.143 0.346 ±0.2 Y
Dec -0.029 0.014 -0.241 -0.107 0.002 0.116 0.327 ±0.2 Y

Table 4. Comparison between Historical and Generated monthly mean rainfall lag one autocorrelation.

Item Hist. Mean 2.50% 25% 50% 75% 97.50% Tol. Y/N
Jan 143.1 134.947 77.592 130.665 140.257 144.96 150.074 10% Y
Feb 144.4 126.583 87 91.899 140.859 151.325 157.418 10% N
Mar 239.8 253.6 222.516 245.346 252.145 256.373 299.613 10% Y
Apr 221.2 210.554 168.11 190.149 217.351 228.006 241.5 10% Y
May 248.7 246.539 205.652 233.352 240.523 255.409 311.121 10% Y
Jun 94.1 87.641 47.582 56.994 93.719 102.147 133.051 10% Y
Jul 116.3 101.828 49.557 88.542 107.286 117.702 145.49 10% N
Aug 106.6 104.477 62.156 105.129 107.78 109.351 112.368 10% Y
Sep 172.3 121.793 47.445 55.575 168.877 174.539 180.697 10% N
Oct 170.6 171.305 93.849 172.43 184.684 191.094 198.838 10% Y
Nov 110 99.698 70.548 81.182 109.537 112.169 114.77 10% Y
Dec 168.1 128.26 81.035 96.23 128.854 164.077 171.866 10% N

Table 5. Comparison between Historical and Generated monthly rainfall maximum.

Item Hist. Mean 2.50% 25% 50% 75% 97.50% Tol. Y/N
Jan 2.3 2.635 0.91 1.983 2.196 3.294 6.867 10% N
Feb 4.2 4.103 3.821 3.95 4.078 4.176 5.011 10% Y
Mar 9.7 10.961 2.48 8.059 9.092 11.446 23.959 10% N
Apr 1.3 4.218 1.281 1.299 1.326 2.981 21.431 10% N
May 4.5 6.14 3.195 4.3 4.43 4.611 14.355 10% N
Jun 2 2.119 1.42 1.966 2.036 2.275 2.863 10% Y
Jul 0.3 0.475 0.262 0.272 0.294 0.795 0.999 10% N
Aug 2.2 2.535 1.703 2.107 2.242 3.206 4.355 10% N
Sep 0.8 1.267 0.754 0.778 0.799 2.224 3.108 10% N
Oct 0.3 0.865 0.272 0.284 0.3 1.394 2.56 10% N
Nov 0.3 0.378 0.287 0.293 0.3 0.313 1.315 10% N
Dec 0.8 1.047 0.725 0.782 0.816 1.213 1.867 10% N

Table 6a. Comparison between Historical and Generated monthly rainfall minimum.

Item Hist. Mean 2.50% 25% 50% 75% 97.50% Tol. Y/N
Jan 0 0 0 0 0 0 0 ±5 Y
Feb 0 0 0 0 0 0 0 ±5 Y
Mar 0 0 0 0 0 0 0 ±5 Y
Apr 0 0 0 0 0 0 0 ±5 Y
May 0 0 0 0 0 0 0 ±5 Y
Jun 0 0 0 0 0 0 0 ±5 Y
Jul 0 0 0 0 0 0 0 ±5 Y
Aug 0 0 0 0 0 0 0 ±5 Y
Sep 0 0 0 0 0 0 0 ±5 Y
Oct 0 0 0 0 0 0 0 ±5 Y
Nov 0 0 0 0 0 0 0 ±5 Y
Dec 0 0 0 0 0 0 0 ±5 Y

Table 6b. Comparison between Historical and Generated monthly rainfall% no rain.

Item Hist. Mean 2.50% 25% 50% 75% 97.50% Tol. Y/N
Jan 0 0 0 0 0 0 0 ±5 Y
Feb 0 0 0 0 0 0 0 ±5 Y
Mar 0 0 0 0 0 0 0 ±5 Y
Apr 0 0 0 0 0 0 0 ±5 Y
May 0 0 0 0 0 0 0 ±5 Y
Jun 0 0 0 0 0 0 0 ±5 Y
Jul 0 0 0 0 0 0 0 ±5 Y
Aug 0 0 0 0 0 0 0 ±5 Y
Sep 0 0 0 0 0 0 0 ±5 Y
Oct 0 0 0 0 0 0 0 ±5 Y
Nov 0 0 0 0 0 0 0 ±5 Y
Dec 0 0 0 0 0 0 0 ±5 Y

Table 7. Comparison between Historical and Generated monthly rainfall Statistics.

Statistics Hist. Mean 2.50% 25% 50% 75% 97.50% Tol. Y/N
Rainfall mean 522.448 526.179 469.49 506.27 522.248 544.539 596.525 5% Y
Rainfall St.dev 130.623 139.548 93.109 121.293 138.622 154.408 196.575 5% N
Rainfall Skewness 0.378 0.363 -0.539 -0.059 0.284 0.661 1.766 ±0.5 Y
Rainfall lag oneautocorrelation -0.075 -0.069 -0.464 -0.18 -0.058 0.048 0.277 ±0.15 Y
Rainfall max 1.642 1.685 1.362 1.543 1.629 1.755 2.297 10% Y
Rainfall min 0.538 0.467 0.092 0.384 0.494 0.575 0.687 10% N
Two year low rainfall 1.42 1.24 0.582 1.131 1.299 1.393 1.567 10% N
Three year low rainfall 2.322 2.108 1.37 1.986 2.183 2.299 2.514 10% Y
Five year low rainfall 4.157 3.937 2.967 3.775 3.984 4.159 4.493 10% Y
Seven year low rainfall 6.1 5.816 4.822 5.574 5.871 6.062 6.455 10% Y
Ten year low rainfall 8.756 8.707 7.742 8.45 8.784 8.988 9.378 10% Y

Annual Parameters

The mean values of average, standard deviation, skewness, lag one auto correlations, max, min and two, three, five, seven and ten years low rainfall from 100 replicates are shown in the Table 7.

Results and Conclusions

The main concern of the stochastic rainfall generation is rainfall depth and its occurrence. In this study a stochastic rainfall generation model for the average monthly rainfall is made for Astor. The monthly rainfall model can be applicable in the detailed water budget and environmental and agricultural model studies. The average monthly precipitation data from 1954 to 2000 was made for Astor. The model’s monthly and annual parameter is estimated using SCL (stochastic climate library). The Root mean square values calculated for average of the 100 monthly replicates shows fairly good results. The average values of % no rain for all the 100 replicates have zero values which shows that no month from the year 1954 to 2000 have zero rainfall for the Astor, whereas the historical data shows some month from 1954 to 2000 having zero average monthly rainfall. So on the basis of the comparison for the % no rain we can say that model is over estimating the % occurrence of no rain.

References