A Stochastic Nonparametric Technique for Space-time

Disaggregation of Streamflows

Balaji Rajagopalan, Jim Prairie and Upmanu Lall

May 27, 20052005 Joint Assembly

Motivation

• Develop realistic streamflow scenarios at several sites on a network simultaneously

• Difficult to model the network from individual gauges

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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

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)

BεAZX

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

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

dXZXf

ZXfZXf

),(

),()(

Joint probability

Marginal probability

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

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 Yi and the 12 values is z’, where z’ = z*/√12

Generate disaggregated flows vector x* from

x* = u*RT

2134.1266389556.439

8721.320349112.7

7071068.07071068.0

7071068.07071068.0

6963.1790

7816.453

0435.12066528.584

0874.2326942.221

RXY

R

ZX

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

Example.

Generate Zsim let us say 735.6541

Then

1860.5202

6541.735' simz

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

1860.5203896.439',* )1,2( simzyu

52238.678

13172.57

1860.5203896.4397071068.07071068.0

7071068.07071068.0*

sim

T

T

sim

x

uRx

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

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)

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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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

• 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.

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

Acknowledgements

BOR Upper Colorado Regional and Boulder Canyon Area (Terry Fulp)

Office for Funding the Study

CADSWES for Logistical Support