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INTEGRATED KNOWLEDGE ON DISASTER & ENVIRONMENTAL RISK MANAGEMENT _____________________________________________________________________________________ Assess changes in water resources budget and hydrological regime of Soan river (Pakistan) basin due to climate change as projected by GCMs under the CMIP5 project. JAVED Ali, Student ID: 37-156909, Department of Civil Engineering, The University of Tokyo, Japan ABSTRACT: The Soan river basin lies near the vicinity of the capital city of Pakistan. It has monsoon every year, which is the major contributor of precipitaon in this region, because of which the region has huge variaons in the dry and rainy periods requiring the assessment of changes in water resources budget and hydrological regime. In this paper, the model performance has been evaluated for past (1981-2000) and future (2046-2065) by simulang against a reference data set, by using the internet based DIAS CMIP5 tool. The process includes selecon of models from fiſteen global climate models on the basis of the evaluaon for relave distribuon which is spaal correlaon coefficient (Scorr) and evaluaon of absolute value (RMSE). For the selected models, the bias correcon is applied on the threshold data to get the corrected data by using APHRODITE precipitaon data in DIAS CMIP5 tool. The obtained analysis results for precipitaon are interpolated in ArcGIS tool by using kriging interpolaon technique. The graphs are ploed for rainfall probability, monthly average rainfall analysis, annual total rain fall, extreme rain days and no rain days. The results from five global climate models show gradual increase in precipitaon intensity in future where as two models show a rapid increase in precipitaon intensity in future for Soan river basin. The monthly average rainfall analysis shows some variaon in the past and future values. These variaons are different for all models. The kriging interpolaon for past and future shows that the upper Soan river basin will have more rainfall as compare to lower basin. The rainfall intensity is increasing with me so it is more for 200-year period as compare to 50-year period. The kriging interpolaon results for the rainfall frequency of MIROC-5 model show huge increase in precipitaon in future which is even more than twice as compare to the past values, so proper precauons should be taken against such an increase which may cause floods in future. KEYWORDS: Soan River basin, Precipitaon, Hydrological regime, Global climate models, Interpolaon 1-INTRODUCTION OF BASIN: The Soan river is an important stream of Pothohar region which lies in Pothohar plateau hill torrent area and is located between longitude 71 o 45’ to 73 o 35’ and latude 32 o 45’ to 33 o 55’ and it lies in Rawalpindi, Aock and Jhelum districts. Soan river is one of the leſt bank tributaries of Indus river. Lai Nullah originates from Marghalla hills in Islamabad and ouall in Soan river. The Soan river rises from the Southern slopes of the Shivalik range also known as Solasinghi range in the tract to east of the Beas gap cross the Southern periphery of the Kangra valley. It joins the boundary of Himachal Pradesh and Punjab. It enters the plains near Chirah, up to Chirah it drains an area about 255 km 2 . This varies in elevaon from 670.5 m to 2286 m. In the plains Soan river flows in a southwest direcon to join Indus river at about 16 km up stream of Kalabagh. It is fed by its major tributaries including Ling, Korang, Lai Nulah, Sil river and flows smoothly with a gentle gradient of about 1.14 m km - 1 . The Soan river basin up to Dhok pathan has area 6549 km 2 . The slope of the Soan catchment varies from gentle to steep. In summer season its discharge drops drascally, while during monsoon it is at its peak. [1] The locaon of Soan river basin is shown in figure-1. Soan river is the main source to provide Page | 1

Assess changes in water resources budget and hydrological regime of Soan river (Pakistan) basin due to climate change as projected by GCMs under the CMIP5 project

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INTEGRATED KNOWLEDGE ON DISASTER & ENVIRONMENTAL RISK MANAGEMENT_____________________________________________________________________________________

Assess changes in water resources budget and hydrological regime of Soan river (Pakistan) basin due to climate change as projected by

GCMs under the CMIP5 project.

JAVED Ali, Student ID: 37-156909, Department of Civil Engineering, The University of Tokyo, Japan

ABSTRACT: The Soan river basin lies near the vicinity of the capital city of Pakistan. It has monsoon every year, which is the major contributor of precipitation in this region, because of which the region has huge variations in the dry and rainy periods requiring the assessment of changes in water resources budget and hydrological regime. In this paper, the model performance has been evaluated for past (1981-2000) and future (2046-2065) by simulating against a reference data set, by using the internet based DIAS CMIP5 tool. The process includes selection of models from fifteen global climate models on the basis of the evaluation for relative distribution which is spatial correlation coefficient (Scorr) and evaluation of absolute value (RMSE). For the selected models, the bias correction is applied on the threshold data to get the corrected data by using APHRODITE precipitation data in DIAS CMIP5 tool. The obtained analysis results for precipitation are interpolated in ArcGIS tool by using kriging interpolation technique. The graphs are plotted for rainfall probability, monthly average rainfall analysis, annual total rain fall, extreme rain days and no rain days. The results from five global climate models show gradual increase in precipitation intensity in future where as two models show a rapid increase in precipitation intensity in future for Soan river basin. The monthly average rainfall analysis shows some variation in the past and future values. These variations are different for all models. The kriging interpolation for past and future shows that the upper Soan river basin will have more rainfall as compare to lower basin. The rainfall intensity is increasing with time so it is more for 200-year period as compare to 50-year period. The kriging interpolation results for the rainfall frequency of MIROC-5 model show huge increase in precipitation in future which is even more than twice as compare to the past values, so proper precautions should be taken against such an increase which may cause floods in future. KEYWORDS: Soan River basin, Precipitation, Hydrological regime, Global climate models, Interpolation

1-INTRODUCTION OF BASIN: The Soan river is an important stream of Pothohar region which lies in Pothohar plateau hill torrent area and is located between longitude 71 o45’ to 73o35’ and latitude 32o45’ to 33o55’ and it lies in Rawalpindi, Attock and Jhelum districts. Soan river is one of the left bank tributaries of Indus river. Lai Nullah originates from Marghalla hills in Islamabad and outfall in Soan river. The Soan river rises from the Southern slopes of the Shivalik range also known as Solasinghi range in the tract to east of the Beas gap cross the Southern periphery of the Kangra valley. It joins the boundary of Himachal Pradesh and Punjab. It enters the plains near Chirah, up to Chirah it drains an area about 255 km2. This varies in elevation from 670.5 m to 2286 m. In the plains Soan river flows in a southwest direction to join Indus river at about 16 km up stream of Kalabagh. It is fed by its major tributaries including Ling, Korang, Lai Nulah, Sil river and flows smoothly with a gentle gradient of about 1.14 m km -

1. The Soan river basin up to Dhok pathan has area 6549 km2. The slope of the Soan catchment varies from gentle to steep. In summer season its discharge drops drastically, while during monsoon it is at its peak. [1] The location of Soan river basin is shown in figure-1. Soan river is the main source to provide

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INTEGRATED KNOWLEDGE ON DISASTER & ENVIRONMENTAL RISK MANAGEMENT_____________________________________________________________________________________

water to Simly dam which was constructed on it in 1983 to meet the water requirements of Islamabad which is the capital city of Pakistan. [2] As Pakistan has Monson season which is the main contributor of precipitation in the country which is nearly 1000mm in mountains and 150mm in coastal areas. The highly variable intensity and frequency of precipitation produce floods or droughts. [3] So the study of changes in water resources budget and hydrological regime is important to forecast future precipitation. 2-METHODOLOGY: The Soan river basin model performance has been evaluated for future by simulating against a reference data set by using the internet based DIAS CMIP5 tool. DIAS stands for data integration and analysis system. It is a comprehensive tool which can interpolate future data for a particular region by using a past reference data set. DIAS is based on coupled model inter comparison project (CMIP) which is the standard experimental protocol for studying the output of coupled atmosphere-ocean general circulation models. In this paper six meteorological elements have been used to interpolate results from 15 selected models. These elements include precipitation, air temperature, geopotential height, zonal wind, meridional wind and outgoing longwave radiations.After selecting models which perform acceptably well for the Soan river basin the bias correction is carried out by using APHRODITE precipitation data in DIAS CMIP5 tool. APHRODITE is related to Asian precipitation data which is based on highly resolved observational data integration towards evaluation of the water resources. This bias correction includes historical simulation precipitation output and future projection precipitation output. From this simulation result, the climatology of Soan river basin for past and future has been obtained. These include monthly average rainfall, return period intensities, extreme rain days, no rain days and annual total rainfall of all selected models for past and future. These all results have been shown graphically in an excel sheet. In the end, spatial data analysis is carried out by using ArcGIS tool in which kriging interpolation method for precipitation has been adopted. For the analysis in ArcGIS, the results of 12 different points in Soan river basin for 50,100 and 200 year return period intensity have been used so that a good interpolation

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INTEGRATED KNOWLEDGE ON DISASTER & ENVIRONMENTAL RISK MANAGEMENT_____________________________________________________________________________________

results can be obtained. The comparison of graphical results and future forecasting has been done in results and discussions. 3-DATA: 3.1-GLOBAL CLIMATE MODELS: There are many global climate models (GCM) which have been generated to simulate future climate and attribute observed climate change to anthropogenic emissions of greenhouse gases. Global Climate Models (GCMs) have evolved from the Atmospheric General Circulation Models (AGCMs) widely used for daily weather prediction. GCMs have been used for a range of applications, including investigating interactions between processes of the climate system, simulating evolution of the climate system, and providing projections of future climate states under scenarios that might alter the evolution of the climate system. The most widely recognized application is the projection of future climate states under various scenarios of increasing atmospheric carbon dioxide. [4] In this paper, initially fifteen global climate models were used in DIAS CMIP5 tool and then seven of them were selected for final analysis and simulations.3.2-METEOROLOGICAL ELEMENTS: The future forecasting of water resources budget and hydrological regime depends on many meteorological elements. The meteorological data can be obtained by direct readings of measuring instruments and by instruments providing a continuous record of the parameter over time. [5] In this paper seven meteorological elements have been considered for the selection of models for future analysis. 3.3-COUPLED MODEL INTER COMPARISON PROJECT (CMIP): The data integration and analysis tool (DIAS) working is based on coupled model inter comparison project (CMIP). It is a standard experimental protocol for studying the output of coupled atmosphere-ocean general circulation models (AOGSMs). CMIP provides a community based infrastructure in support of climate model diagnosis, validation, inter comparison, documentation and data access. It enables a diverse community of scientists to analyze GCMs in a systematic fashion. Currently its working is under phase-5 so it is termed as CMIP5. [6]

3.4-MODEL SELECTION: As there are many global climate models and different model perform with different uncertainty at different parts of the world. Some of the models may not be able to reproduce main climatological features in a given region so they will not give reliable results even after bias correction so these models must be excluded from the analysis. The model selection is carried out on the basis of the evaluation for relative distribution which is spatial correlation coefficient (Scorr) and evaluation of absolute value (RMSE). The RMSE and Scorr are calculated by the DIAS CMIP5 tool for all models which are then averaged over the whole evaluated period and also averaged over the whole set of analyzed models. Each model is then prescribed with the indices with respect to the calculated average. [7] If Scorr and RMSE are more than all GCM averaged value, then they are scored in positive indices and if Scorr and RMSE are equal to all GCM averaged value then they are scored in zero indices otherwise they are scored in negative indices. The procedure of scoring can also be understood through following criteria; Scorrmodel Scorrtotal average Indexscorr = 1

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Element Scale LevelPrecipitation Small SurfaceAir Temperature Large 850hPaSea Surface Temperature Large SurfaceOutgoing Longwave Radiation Large SurfaceGeopotential Height Large 850hPaZonal Wind Large 850hPaMeridional Wind Large 850hPa

Table-1: Meteorological Elements

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Scorrmodel < Scorrtotal average Indexscorr = 0RMSEmodel RMSEtotal average IndexRMSE = 1 RMSEmodel > RMSEtotal average IndexRMSE = 0From above data set;Indexscorr = 1 and IndexRMSE = 1 Indextotal = 1Indexscorr = 1 and IndexRMSE = 0 Indextotal = 0Indexscorr = 0 and IndexRMSE = 1 Indextotal = 0Indexscorr = 0 and IndexRMSE = 0 Indextotal = -1The models with higher total score of indices are selected for analysis. [7] In table-2 the selection of models for Soan river basin has been shown.3.5-BIAS CORRECTION: The bias correction is applied on the threshold data to get the corrected data. It is carried out by using APHRODITE precipitation data in DIAS CMIP5 tool, 1981-2000 for past climate and 2046-2065 for future climate. APHRODITE stands for Asian Precipitation data which is based on Highly Resolved Observational Data Integration Towards Evaluation of the water resources. The APHRODITE project develops state-of-the-art daily precipitation datasets with high resolution grids for Asia. The datasets are created primarily with data obtained from a rain gauge observation network. [8]

For the bias correction, total twelve points were considered from Soan river basin. These points have been shown in figure-1. So bias correction is obtained for twelve points, corresponding to the seven selected global climate models. 3.6-POINT ANALYSIS: The point analysis of bias corrected precipitation for past and future data is carried out for seven selected global climate models at twelve points of Soan river basin. The results are obtained in the form of txt and csv files which include information of monthly average rainfall, return period intensities, extreme rain days, no rain days and annual total rainfall of all selected models for past and future. There is a modeling realm for CanESM-2 and MIROC-5 models. This indicates high level modeling component of particular relevance for the dataset. [9] It shows an atomic dataset which is assigned to a primary realm but also aliased to the other relevant realms. These could be identified by “r1”, “r2” and “r3” etc. To compare the results with other models CanESM-2@r1i1p1 and MIROC-5@r1i1p1 results have been used as it is not logical to use same model results with different realm or which is cross referenced or aliased to the other relevant realms. 3.7-SPATIAL ANALYSIS: From the point analysis results, frequency data for seven selected models of twelve points of Soan river basin are arranged in csv file which includes frequency of rainfall for 50, 100 and 200 years for past and future. This data is further used in ArcGIS for the spatial analysis to forecast future results by an interpolation method. There are many interpolation methods which could be used in ArcGIS tool. Thiessen polygon, Inverse distance weighting (IDW), spline and kriging are common interpolation methods. Thiessen polygon, IDW and spline are all local interpolation methods which only uses data of the nearest sampling points for climatic mapping. Thiessen polygon is a straight forward

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Table-2: Model Selection

INTEGRATED KNOWLEDGE ON DISASTER & ENVIRONMENTAL RISK MANAGEMENT_____________________________________________________________________________________

method for interpolation in which closest observation value is directly assigned to a point. It is a very simple approach which is not appropriate for variables with gradual spatial variation such as temperature and precipitation. IDW obtain value by calculating weighted average of known values within a specific neighborhood. It assumes that the climatic value at un-sampled point is a distance weighted average of the climatic values at the sampling points. Spline is based on a family of continuous, regular and derived functions adapted to a local variations of climatic data. Its algorithms are quite complex but are standard in the current GIS software. [10] 3.7.1-Kriging interpolation method: It is a geostatistical interpolation method which assumes that the spatial variation of a continuous climatic variable is too irregular to be modeled by a mathematical function and its spatial variation could be better predicted by a probabilistic surface. It estimates the value at a point as a linear combination of neighboring available data. In this paper, kriging interpolation method has been used for future forecasting of rainfall frequency. [10]

3.7.2-Advantages of Kriging interpolation method: Following are the advantages of using kriging interpolation method. [11]

It helps to compensate for the effects of data clustering, assigning individual points within a cluster less weight than isolated data points.

It gives estimate of estimation error (kriging variance), along with estimate of the variable itself (but error map is basically a scaled version of a map of distance to nearest data point, so not that unique).

The availability of estimation error provides basis for stochastic simulation of possible realizations of the nearest data point.

4-RESULTS and DISCUSSIONS: From the bias correction and data analysis, following results are obtained.4.1-RAINFALL PROBABILITY (PAST & FUTURE): The rainfall probability for 50 years, 100 years and 200 years for all selected twelve points in Soan river basin is plotted for past and future for seven models. As there are many plots for all twelve points so here as a reference, plots for one point i.e. 32 o45’ Latitude and 72o30’ Longitude of Soan river basin have been shown in figure-2, figure-3 and figure-4.These plots shows that there is not much difference between past and future precipitation for 50, 100 and 200 years but overall precipitation for 200 years is more as compare to 50-years precipitation. From the analysis of seven global climate model, MIROC-5 and MPI-ESM-MR models are showing some variation in past and future precipitation but others five models are giving similar results so their analysis results are preferred.

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Figure-2: 50-year probability rainfall of Soan river basin (past and future)

Figure-3: 100-year probability rainfall of Soan river basin (past and future)

INTEGRATED KNOWLEDGE ON DISASTER & ENVIRONMENTAL RISK MANAGEMENT_____________________________________________________________________________________

4.2-MONTHLY AVERAGE RAINFALL ANALYSIS (PAST & FUTURE): The monthly average rainfall analysis for twelve point and seven models for past and future has been shown in graphical form. These are also many graphs so as a reference, the monthly average rainfall analysis of one point i.e. 32 o45’ Latitude and 72o30’ Longitude of Soan river basin have been shown in figure-5 to figure-11 respectively of seven global climate models. The analysis results show huge variations in past and future data and in most of the cases the monthly average rainfall is decreasing in future in the starting months of the year while increasing in the end of the year for Soan river basin. One main reason for this is may be the monsoon period which comes in this basin from July to September. For monthly average rainfall, analysis results of all models vary from each other so a single model results cannot be regarded as best results, furthermore in the analysis results by MIROC-5 model, the result for the month May has huge variation as highlighted in figure-10. This variation is not possible so the results obtained by this model can be regarded as invalid results.

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Figure-4: 200-year probability rainfall of Soan river basin (past and future)

Figure-5: Monthly average rainfall (past and future) for CESM1(BCG) model

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Figure-6: Monthly average rainfall (past and future) for CNRM-CMS model

Figure-7: Monthly average rainfall (past and future) for CanESM-2 model

Figure-8: Monthly average rainfall (past and future) for GFDL-CM3 model

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4.3-ANNUAL TOTAL RAINFALL ANALYSIS (PAST & FUTURE): The annual total rainfall analysis for one point (32o45’ Latitude and 72o30’ Longitude) of Soan river basin has been shown in the graphical pattern in figure-12. The analysis results show small variations in past and future total annual rainfall except MPI-ESM-MR model.

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Figure-9: Monthly average rainfall (past and future) for MIROC-5 model

Figure-10: Monthly average rainfall (past and future) for MPI-ESM-MR model

Figure-11: Monthly average rainfall (past and future) for NorESM1 model

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4.4-NUMBER OF EXTREME RAIN DAYS: The number of extreme rain days for 20 years’ period have been shown in figure-13 for one point (32o45’ Latitude and 72o30’ Longitude) of Soan river basin for seven global climate models. The extreme limit of each point is calculated from observation value of annual total rainfall. i.e. for this point, extreme value = (502.755/365) = 1.375mm so numbers of rain days having rainfall greater than 1.375mm have been counted and results are drawn in the form of chart.

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Annual Total RainfallCESM1(BGC)@r1i1p1 CNRM-CM5@r1i1p1 CanESM-2@r1i1p1 GFDL-CM3@r1i1p1 MIROC-5@r1i1p1 MPI-ESM-MR@r1i1p1 Nor-ESM-MR@r1i1p1

observation 502.75535 502.75535 502.75535 502.75535 502.75535 502.75535 502.75535past 479.1161 503.44005 450.24785 503.5123 501.6789 252.34545 467.93615future 487.6883 477.66265 479.10315 591.45255 537.8686 411.10665 531.9325change (mm/year) 8.5722 -25.7774 28.8553 87.94025 36.1897 158.7612 63.99635%change 1.79 -5.12 6.41 17.47 7.21 62.91 13.68

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CESM1(BGC)@r1i1p1 CNRM-CM5@r1i1p1 CanESM-2@r1i1p1 GFDL-CM3@r1i1p1 MIROC-5@r1i1p1 MPI-ESM-MR@r1i1p1 Nor-ESM-MR@r1i1p1

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Figure-12: Annual total rainfall of Soan river basin (past and future)

CESM1(BGC)@r1i1p1 CNRM-CM5@r1i1p1 CanESM-2@r1i1p1 GFDL-CM3@r1i1p1 MIROC-5@r1i1p1 MPI-ESM-MR@r1i1p1 Nor-ESM-MR@r1i1p1Insitu 1544 1544 1544 1544 1544 1544 1544past 1480 1536 1281 1597 1616 423 1323future 1424 1563 1451 2076 1650 434 1426%change -3.78 1.76 13.27 29.99 2.10 2.60 7.79

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Figure-13: Number of extreme rain days for Soan river basin (past and future)

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4.5-NO RAIN DAYS: The number of no rain days for each model have been counted for 20 years past and future data which is shown is graphical form in figure-14 for one point with 32o45’ Latitude and 72o30’ Longitude. The analysis shows that there is no variation in in-situ, past and future values for CNRM-CM5, GFDL-CM3, MIROC-5 and Nor-ESM-MR models but CESM1(BCG) and CanESM-2 has little variations in past and future values whereas MPI-ESM-MR has huge variation in in-situ value and past/future value.

4.6-ArcGIS KRIGING INTERPOLATION RESULTS: The rainfall frequency data obtained from the point analysis were used further to interpolate results for 50, 100 and 200 years past and future precipitation frequency for CanESM-2, CESM1(BCG) and MIROC-5 global climate models. The results have been shown in figure-15, figure-16 and figure -17 by using the kriging interpolation method for precipitation in ArcGIS. All the models have similar results for Soan river basin that the precipitation intensity is increasing with time in future for the whole basin but upper region of the basin has more precipitation in future as compare to lower region but this increase is gradual in CanESM-2 and CESM1(BCG) models where as MIROC-5 model is showing rapid increase in precipitation in future especially in 200-years model.

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CESM1(BGC)@r1i1p1 CNRM-CM5@r1i1p1 CanESM-2@r1i1p1 GFDL-CM3@r1i1p1 MIROC-5@r1i1p1 MPI-ESM-MR@r1i1p1 Nor-ESM-MR@r1i1p1Insitu 3680 3680 3680 3680 3680 3680 3680past 4724 3677 4918 3747 3687 6823 4160future 4871 3586 4704 3255 3743 6814 4128%change 3.11 -2.47 -4.35 -13.13 1.52 -0.13 -0.77

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Figure-14: Number of no rain days for Soan river basin (past and future)

Figure-15: kriging interpolation results

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5-CONCLUSION: From the analysis of all seven global climate models, it can be seen that all models has diverse results so a single model results cannot be trusted to interpolate water resources budget and hydrological regime. If more models are giving similar trend in past and future, then the forecasted results could be trusted. For Soan river basin, the results from five global climate models (CESM1BCG, CNRM-CM5, CanESM2, GFDL-CM3, Nor-ESM-MR) show gradual increase in precipitation intensity in future where as two models (MIROC-5, MPI-ESM-MR) show a rapid increase in precipitation intensity in future. The monthly average rainfall analysis shows some variation in the past and future values. These variations are different for all models. The number of extreme rain days are increasing and number of no rain days are decreasing in future which shows an increase in the total annual rainfall for Soan river basin. The kriging interpolation for past and future shows that the upper Soan river basin will have more rainfall as compare to lower basin while the results for the rainfall frequency of MIROC-5 model show huge increase in precipitation in future which is even more than twice as compare to the past values so proper precautions should be taken against such an increase which may cause floods in future. As a solution, the capacity of Simply dam, which has been constructed on Soan river basin, can be increased by raising the walls of the dam so that the stored water can be utilized in drought period.

6-REFERENCES: [1]: Soil erosion estimation of Soan river catchment using remote sensing and geographical

information system by G. Nabi, M. Latif, M. Ahsan and M. Anwar, 2008 [2]: Morphological and hydrological responses of soan and khad rivers to impervious land use in

district murree, pakistan by Fida Hussain, Abdul Khaliq, M. Irfan Ashraf, Irshad A. Khan, 2009

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Figure-16: kriging interpolation results Figure-15: kriging interpolation results

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[3]: Land Slide Survey. Government of the Punjab. Upland Rehabilitation and Development Project, Islamabad by Anon, 1998

[4]: Global climate models and their limitations by Anthony Lupo (USA), William Kininmonth (Australia)[5]: Meteorological elements and their observation, working paper no. 14 by Dr. T. Darnhofer, 1985[6]: http://cmip-pcmdi.llnl.gov/[7]: Development of statistical bias correction and downscaling scheme for climate change impact

assessment at a basin scale by Nyunt, cho Tanda 2013. [8]: http://www.chikyu.ac.jp/precip/ [9]: CMIP5 data reference syntax (DRS) and controlled vocabularies by Karl E. Taylor, V. Balaji, Steve

Hankin, Martin Juckes, Bryan Lawrence and Stephen Pascoe, 13 June 2012.[10]: Comparative analysis of interpolation methods in the middle Ebro Valley (Spain): application to the

annual precipitation and temperature by Sergio M. Vicente-Serrano, M. Angel Saz-Sanchez, Jose M. Cuadrat, July 28 2003.

[11]: Kriging by Geoff Bohling, 19 October 2005, http://people.ku.edu/~gbohling/cpe940/

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