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

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

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3 16 19 20 1 2 4 5 6 7 8 9 10 11 1213 14 15 17 18

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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

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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)

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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

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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

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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

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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

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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.

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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

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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

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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)

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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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3 16 19 20

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Bluff

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Lees Ferry

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Bluff gauge June flows Nonparametric Parametric

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Lees Ferry Gauge May Flows Nonparametric Parametric

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

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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.

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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

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

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