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A Stochastic Nonparametric Technique for Space-time Disaggregation of Streamflows

A Stochastic Nonparametric Technique for Space-time Disaggregation of Streamflows Balaji Rajagopalan, Jim Prairie and Upmanu Lall May 27, 2005 2005 Joint Assembly. Motivation. Develop realistic streamflow scenarios at several sites on a network simultaneously

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A Stochastic Nonparametric Technique for Space-time Disaggregation of Streamflows

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  1. A Stochastic Nonparametric Technique for Space-time Disaggregation of Streamflows Balaji Rajagopalan, Jim Prairie and Upmanu Lall May 27, 2005 2005 Joint Assembly

  2. Motivation • Develop realistic streamflow scenarios at several sites on a network simultaneously • Difficult to model the network from individual gauges

  3. 9 10 11 13 12 14 15 1 2 8 3 16 7 4 17 6 5 19 18 20

  4. Motivation • Present methods can not capture higher order features • Present methods can be difficult to implement • Can not easily incorporate climate information • Finding the probability of events • Required for long-term basin-wide planning • Develop shortage criteria • Meeting standards for salinity

  5. Current Methods • Parametric • Basic form – Seminal (Valencia and Schaake, 1972) Variations/Improvements (Mejia and Rousselle; 1976, Lane; 1979; Salas et al. 1980; Stedinger and Vogel, 1984) • Nonparametric • Kernel-based ( Tarboton et al. 1998) • Nearest-Neighbor based (Kumar et al. 2000)

  6. Drawbacks of Parametric Framework • Data must be transformed to a normal distribution • During transformation additivity is lost • There are many parameters to estimate • At least 25 parameters for annual to monthly disaggregation • Inability to capture non-Guassian and non- linear features

  7. Proposed Methodology • Resampling from a conditional PDF • With the “additivity” constraint • Where Z is the annual flow X are the monthly flows • Or this can be viewed as a spatial problem • Where Z is the sum of d locations of monthly flows X are the d locations of monthly flow Joint probability Marginal probability

  8. Steps for Temporal Disagg Step 1 Step 2 Step 3 X = monthly flow matrix. Z = annual flow vector. Transform matrix Y = XR Generate an annual flow z* with an appropriate model Identify k historical years to z*. Pick one of the neighbors with k-nearest neighbor. Tarbaton el al, 1998 Prairie, 2002

  9. Steps for Temporal Disagg Step 4 Step 5 Build a vector u* where the first 11 values are first 11 values from Yi and the 12 values is z’, where z’ = z*/√12 Generate disaggregated flows vector x* from x* = u*RT Repeat steps 2 through 5 for additional years

  10. Example. Gauge 1 Gauge 2 Gauge 1 +2 Obtain the rotation matrix R via Gram Schmidt orthonormalization Note the last column of R = 1/√d RT = R-1

  11. Generate Zsim let us say 735.6541 Then Next we find the K – nearest neighbors to z’sim The neighbors are weighted so closest gets higher weight We pick a neighbor, let us say year 2 Then we build u from y and z’sim

  12. Via back rotation we can solve for the disaggregated components of zsim Note the disaggregated components add to zsim = 735.6541 The only key parameter is K which is estimated with a heuristic scheme K=√N

  13. Application • The Upper Colorado River Basin • 4 key gauges • Perform 500 simulations each of 90 years length • Annual Model • a modified K-NN lag-1 model (Prairie, 2002)

  14. Results • Performance Statistics • Lower order: mean, standard deviation, skew, autocorrelation (lag-1) • Higher order: probability density function, drought statistics • We provide some comparison with a parametric disaggregation model

  15. 3 16 19 20

  16. Bluff

  17. Lees Ferry

  18. Bluff gauge June flows Nonparametric Parametric

  19. Lees Ferry Gauge May Flows Nonparametric Parametric

  20. Lees Ferry Gauge Drought Statistics Annual Model Modified K-NN lag-1 Annual Model 18 year block bootstrap

  21. Conclusions • A flexible, simple, framework for space-time disaggregation is presented • Obviates data transformation • Parsimonious • Ability to capture any arbitrary PDF structure • Preserves all the required statistics and additivity. • Easily be conditioned on large-scale climate information.

  22. Future Extensions • Simulate Decision/Policy strategies via passing the simulated flows through Decision Support System • Incorporate paleo streamflow data to simulate space-time flows back in time and water resources system scenarios. • Conditioning on climate

  23. Acknowledgements BOR Upper Colorado Regional and Boulder Canyon Area (Terry Fulp) Office for Funding the Study CADSWES for Logistical Support

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