1. Spatiotemporal stochastic modeling of multisite stream flows - with application to irrigation water management and risk assessment
Ke-Sheng Cheng, Guest Professor
Faculty and Graduate School of Agriculture,
Kyoto University
Dept. of Bioenvironmental Systems Engineering,
National Taiwan University

2. Outline Introduction Study area and data
Characterizing multi-site flow characteristics
Spatiotemporal model building and stochastic simulation of TDP flows
Model performance assessment
Application to irrigation risk management
Conclusions
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3. In Taiwan, paddy irrigation accounts for a significant proportion of the available water.
The early stage of the winter crop falls in the dry season (November to April) and often experiences irrigation water shortage.
Decision on whether mitigation practices should be taken or what measures need to be implemented must be made in the very early stage of a severe drought or even prior to paddy transplanting.
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4. The objectives of this study are two folds:
Irrigation management during droughts requires characterizing the spatiotemporal variation of stream flows at different stations and familiarity of the irrigation network.
The objectives of this study are two folds:
to develop a stochastic model which is capable of characterizing the spatiotemporal variability of multi-site streamflows within an irrigation district,
to demonstrate the capability of the proposed model for risk assessment of potential drought mitigation practices through stochastic simulation.
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5. Illustration of the spatiotemporal data structure of TDP flows
Modeling a 4-D problem.
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6. Analogy of a spatiotemporal process - Hyperspectral image cube
Time domain
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7. Spatial modeling only. Gamma random field simulation
=> Bivariate Gamma Simulation
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9. Study area and data Introduction
Characterizing multi-site flow characteristics
Spatiotemporal model building and stochastic simulation of TDP flows
Model performance assessment
Application to irrigation risk management
Conclusions
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10. Jianan Irrigation District (JID, 758 km2)
The Jia-nan Irrigation Association manages a complex irrigation network which is comprised of reservoirs, irrigation canals, channels, ditches, flow diversion works, check dams, intake structures, etc.
Traditionally, irrigation scheduling and irrigation water supply are operated on a nominal ten-day-period (TDP, 旬) basis.
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11. Significant spatiotemporal variation of TDP flows can be observed.
Twenty-six years ( ) of TDP flow data available at 12 flow stations were collected.
Low flow season accounts for only approximately 13% of the annual total flow volume.
Significant spatiotemporal variation of TDP flows can be observed.
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12. The Jianan Irrigation District and locations of flow stations.
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13. Seasonal variation of mean TDP flows at different flow stations
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15. Characterizing multi-site flow characteristics
Introduction
Study area and data
Characterizing multi-site flow characteristics
Spatiotemporal model building and stochastic simulation of TDP flows
Model performance assessment
Application to irrigation risk management
Conclusions
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16. The very similar seasonal variation patterns among TDP flows of different flow stations suggest that the TDP flows may exhibit significant spatial correlation, even though most flow stations belong to different rivers or sub-tributaries.
In modeling the multi-site TDP flow characteristics, not only the marginal distributions of the TDP flows at individual stations, but also the temporal and spatial correlations of the TDP flows need to be investigated.
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17. Modeling the characteristics of TDP flows
TDP flows were standardized with respect to their long-term averages and standard deviations.
Site-specific standardized TDP flows (STDPF) are considered as random variables with zero expectation and unit standard deviation.
L-moments-based goodness-of-fit tests were conducted for selection of appropriate distributions for TDP-specific STDPF of individual flow stations.
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18. LMRD for goodness test
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19. LMRD for goodness test
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20. Parameters estimation (method of L-moments)
The STDPF at different sites are modeled by a common Pearson type III distribution having a marginal density with zero expectation and unit variance.
Parameters estimation (method of L-moments)
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21. Modeling the spatial and temporal variations of the standardized TDP flows
The spatial and temporal variations of the STDPF were investigated through the semi-variogram analysis.
Both the spatial and temporal semi-variograms were fitted with an asymptotic value (the sill) of 1 since the standardized TDP flows have a unit standard deviation.
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22. Spatiotemporal model building and TDP flows random field simulation
Introduction
Study area and data
Characterizing multi-site flow characteristics
Spatiotemporal model building and TDP flows random field simulation
Model performance assessment
Application to irrigation risk management
Conclusions
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23. By assuming that the spatial variation and temporal variation are mutually independent, a unique spatiotemporal semi-variogram can be expressed by
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24. The spatiotemporal covariance function can also be expressed as
By rescaling variations in the spatial domain using the anisotropic ratio, the multi-site STDPFs can be treated as an isotropic Pearson type III random field with a unique spatiotemporal semi-variogram .
The spatiotemporal covariance function can also be expressed as
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25. Stochastic simulation of a Pearson type III random field
The whole process is composed of three sequential components:
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26. Conversion between the two bivariate covariance matrices can be achieved using the following equation (Cheng et al., 2011)
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27. Realizations of the standard Gaussian random field are the transformed to realizations of the multi-site standardized TDP flows through the following equation
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28. Model performance assessment
Introduction
Study area and data
Characterizing multi-site flow characteristics
Spatiotemporal model building and TDP flows random field simulation
Model performance assessment
Application to irrigation risk management
Conclusions
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29. In this study the proposed spatiotemporal simulation model was implemented for TDP streamflow generation at eight flow stations (stations 1, 3, 4, 7, 8, 9, 10 and 11) with water intake structures. These stations have completely separate tributaries and none station is in the upstream of the other.
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30. A single simulation run yields streamflows of 36 TDPs (in one year) at each individual station. Since historical flow data are available for a period of 26 years, a total of 26 independent simulation runs is considered as a block-simulation run.
A set of parameters including the mean, standard deviation and coefficient of skewness of the site- and TDP-specific streamflows and their spatial and temporal correlation structure (P) can then be estimated from the results of a block-simulation run.
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32. The spatial and temporal correlation structure is represented by a spatiotemporal correlation matrix P of dimension 288x288. The correlation matrix P is defined as
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33. In order to assess the uncertainties in parameters estimation, a total of 1000 block-simulation runs were conducted in our study.
Such assessments indicate that realizations generated by the proposed spatiotemporal simulation approach can preserve the statistical properties of the marginal density.
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35. Map of standard deviations of the spatiotemporal correlation matrix of simulated TDP streamflows
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37. The space-time lag of 48 is equivalent to a time lag of 6 TDPs since Pij represents the correlation matrix of TDP streamflows at eight flow stations. Notwithstanding the lag in space, the spatiotemporal correlation coefficient drops to lower than 0.1 when the time lag between streamflows of any two flow stations becomes greater than 6 TDPs.
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38. With a very low spatiotemporal correlation coefficient (near zero), the asymptotic standard deviation of the sample correlation coefficient can be approximated by the following equations (Hooper 1958; Priestley 1982):
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39. In our study the sample size n equals 26 and the asymptotic standard deviation of the sample correlation coefficient calculated by either of the two equations yields a value of
Such results also suffice to demonstrate the capability of our spatiotemporal simulation model in preserving the spatiotemporal correlation structure of TDP streamflows.
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40. Map of standard deviations of the spatiotemporal correlation matrix of simulated TDP streamflows
 0.196
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41. Assessment of the spatiotemporal correlation matrix
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43. Application to irrigation risk management
Introduction
Study area and data
Characterizing multi-site flow characteristics
Spatiotemporal model building and TDP flows random field simulation
Model performance assessment
Application to irrigation risk management
Conclusions
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44. The irrigation subgroup often experiences irrigation water shortage during the dry season and a mitigation measure using groundwater from a total of 18 groundwater wells has been investigated. The mitigation measure plans to withdraw groundwater at a rate of 0.01 million cubic meters per TDP from each of the 18 groundwater wells. Using the results of each simulation run, the irrigation management model calculated the amounts of irrigation water that can be provided to individual subdivisions under two different scenarios – (1) without groundwater withdraw and (2) with groundwater withdraw.
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46. (a) TDP-specific average shortage ratios with and without implementation of the groundwater-withdraw mitigation measure. (b) and (c) ECDF of the 7-th TDP shortage ratios of subdivisions A and B, respectively.
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47. Conclusions Introduction Study area and data
Characterizing multi-site flow characteristics
Spatiotemporal model building and TDP flows random field simulation
Model performance assessment
Application to irrigation risk management
Conclusions
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48. In this paper we propose a spatiotemporal stochastic simulation model for multisite streamflow simulation.
Through a rigorous evaluation, the model is found capable of preserving not only the marginal distributions but also the spatiotemporal correlation structure of the multisite streamflows.
The proposed multisite spatiotemporal streamflow simulation model can facilitates the needs of risk-based decision making in water resources management.
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49. References
Cheng, K.S., H.C. Yeh, and C.H. Tsai, 2000: An anisotropic spatial modeling approach for remote sensing image rectification. Remote Sensing of Environment, 73(1),
Cheng, K.S., C. Wei, Y.B. Cheng, and H.C. Yeh, 2003: Effect of spatial variation characteristics on contouring of design storm depth. Hydrological Processes, 17, 1755–1769.
Cheng, K. S., J. L. Chiang, and C. W. Hsu, 2007: Simulation of probability distributions commonly used in hydrological frequency analysis. Hydrol Process, 21(1), 51–60, doi: /hyp.6176
Cheng, K.S., J.C. Hou, J.J. Liou, Y.C. Wu, and J.L. Chiang, 2011: Stochastic simulation of bivariate gamma distribution – A frequency-factor based approach. Stochastic Environmental Research and Risk Assessment, 25(2), 107 – 122, doi: /s
Liou, J.J., Y.F. Su, J.L. Chiang, K.S. Cheng, 2011: Gamma random field simulation by a covariance matrix transformation method. Stochastic Environmental Research and Risk Assessment, 25(2), 235 – 251, doi: /s
Wu, Y.C., J.C. Hou, J.J. Liou, Y.F. Su, and K.S. Cheng, 2012: Assessing the impact of climate change on basin-average annual typhoon rainfalls with consideration of multisite correlation. Paddy and Water Environment, 10(2), , doi: /s
Hsieh, H.I., Su, M.D., Cheng, K.S., Multisite Spatiotemporal Streamflow Simulation - With an Application to Irrigation Water Shortage Risk Assessment. Terrestrial, Atmospheric, Oceanic Sciences, 25(2):
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50. Acknowledgements
Development of the spatiotemporal random field simulation model has been partially supported by the Ministry of Science & Technology (Formerly the National Science Council, NSC) and the Council of Agriculture (COA).
NTU colleagues and former members of the RSLAB
Prof. Ming-Daw Su
Dr. Ju-Chen Hou, Dr. Jun-Jih Liou [NCDR]
Dr. Yii-Chen Wu [NTU], Dr. Hsin-I Hsieh [NTU]
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51. Thanks for listening! Your comments and suggestions are most welcome.
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52. General equation for hydrological frequency analysis
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54. The gamma distribution is a special case of the Pearson type III distribution with a zero location parameter. Therefore, it seems plausible to generate random samples of a bivariate gamma distribution based on two jointly distributed frequency factors.
[A]
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55. Assume two gamma random variables X and Y are jointly distributed.
The two random variables are respectively associated with their frequency factors KX and KY .
Equation (A) indicates that the frequency factor KX of a random variable X with gamma density is approximated by a function of the standard normal deviate and the coefficient of skewness of the gamma density.
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56. Flowchart of BVG simulation (1/2)
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57. Flowchart of BVG simulation (2/2)
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59. [B]
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62. Frequency factors KX and KY can be respectively approximated by
where U and V both are random variables with standard normal density and are correlated with correlation coefficient
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63. Correlation coefficient of KX and KY can be derived as follows:
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66. Since KX and KY are distributed with zero means, it follows that
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68. It can also be shown that
Thus,
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