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

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

9. Step 4 Steps for Temporal Disagg Step 5 Repeat steps 2 through 5 for additional years Build a vector u* where the first 11 values are first 11 values from Y i and the 12 values is z’, where z’ = z*/√12 Generate disaggregated flows vector x* from x* = u*R T

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

11. Generate Z sim 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 z sim Note the disaggregated components add to z sim = 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

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16. Bluff

17. Lees Ferry

19. Bluff gauge June flows Nonparametric Parametric

20. Lees Ferry Gauge May Flows Nonparametric Parametric

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

22. 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.

23. 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

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