A stochastic nonparametric technique for space-time disaggregation of streamflows May 27, Joint Assembly

Agenda Motivation Current Methods Proposed Technique Step Through Solution Application Results Conclusion Next Steps

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

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

Current Methods Parametric –Valencia and Schaake, 1972 Basic form –Mejia and Rousselle, 1976 –Lane, 1979 –Salas et al –Stedinger and Vogel, 1984 Nonparametric –Tarboton et al –Kumar et al. 2000

Drawbacks of Parametric Data must be transformed to a normal distribution –During transformation additivity is lost There are many parameters to estimate –Al least 36 parameters for annual to monthly Inability to capture non-Guassian and non- linear features

Basic Problem Resampling from a conditional PDF 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

Step 1 Transform monthly flow X with rotational matirx R such that XR=Y Steps for Temporal Disagg Step 2 Generate an annual flow vector Z with an appropriate model Step 3 Build a matrix U where the first 11 columns are from Y and the final column is Z’, where Z’ = Z/√12

Step 4 Resample a vector u from the conditional PDF f(U|Z) Steps for Temporal Disagg Step 5 Back transform the resampled u such that uR T = X monthly values X X are the monthly values that add to Z

Gauge 1 Gauge 2Gauge 1 +2 Solve for R with Gram Schmidt orthonormalization Note the last column of R = 1/√d R T = R -1 Observed data at 2 gauges for 2 years

Generate Z sim let us say Then Next we find the K – nearest neighbors to Zsim The neighbors are weighted so closest gets higher weight We pick a neighbor, let us say year 2 Then we generate U from Y and Z’sim U is a matrix of nyears by dstations

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

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

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

Bluff

Lees Ferry

Bluff gauge June flows Nonparametric Parametric

Lees Ferry Gauge May Flows Nonparametric Parametric

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

Conclusions No need for data transformation Limited parameters Preserves basic statistics Preserves summability Preserves arbitrary PDF Preserves cross correlation Simple to implement

Next Steps Simulate policy and discuss results Analysis paleo-streamflows –Data analysis –Transition Probability Matrix Conditioning on climate

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