Comparison of natural streamflows generated from a parametric and nonparametric stochastic model

James Prairie(1,2), Balaji Rajagopalan(1) and Terry Fulp(2)

1. University of Colorado at Boulder, CADSWES

2. U.S Bureau of Reclamation

Options

Motivation

• Generate future inflow scenarios for decision making models– reservoir operating rules, salinity control

• Estimate uncertainty in model output

• Parametric Techniques– AR, ARMA, PAR, PARMA

• Nonparametric Techniques– K-NN, density estimator, bootstrap

Objective of Study

• Compare nonparametric and parametric techniques for simulation of streamflows – at USGS stream gauge 09180500: Colorado

River near Cisco, UT

Outline of Talk

• Overview of parametric technique

• Explain nonparametric technique

• Compare various distribution attributes– mean

– standard deviation

– lag(1) correlation

– skewness

– marginal probability density function

– bivariate probability

• Conclusions

Parametric

• Periodic Auto Regressive model (PAR)– developed a lag(1) model

– Stochastic Analysis, Modeling, and Simulation (SAMS)

• Data must fit a Gaussian distribution– log and power transformation

– not guaranteed to preserve statistics after back transformation

• Expected to preserve– mean, standard deviation, lag(1) correlation

– skew dependant on transformation

– gaussian probability density function

( )å=

-- S+-F+=p

jjjj yy

1,,,, tnttntttn mm Salas (1992)

season

year

==

tn

Nonparametric

• K- Nearest Neighbor model (K-NN)– lag(1) model

• No prior assumption of data’s distribution– no transformations needed

• Resamples the original data with replacement using locally weighted bootstrapping technique– only recreates values in the original data

• augment using noise function

• alternate nonparametric method

• Expected to preserve– all distributional properties

• (mean, standard deviation, lag(1) correlation and skewness)

– any arbitrary probability density function

Nonparametric (cont’d)

• Markov process for resampling

Lall and Sharma (1996)

Nearest Neighbor Resampling

1. Dt (x t-1) d =1 (feature vector)

2. determine k nearest neighbors among Dt using Euclidean distance

3. define a discrete kernel K(j(i)) for resampling one of the xj(i) as follows

4. using the discrete probability mass function K(j(i)), resample xj(i) and update the feature vector then return to step 2 as needed

5. Various means to obtain k– GCV– Heuristic scheme

( ) ÷÷ø

öççè

æ-= å

=

d

jtjijjit vvwr

1

2/1

Where v tj is the jith component of Dt, and w j are scaling weights.

( )( )å

=

=k

j

j

jijK

1

1

1

Lall and Sharma (1996)Nk =

Annual Water Year Natural FlowUSGS stream gauge 09180500 (Colorado River near Cisco, UT)

0

2000

4000

6000

8000

10000

12000

14000

1906

1909

1912

1915

1918

1921

1924

1927

1930

1933

1936

1939

1942

1945

1948

1951

1954

1957

1960

1963

1966

1969

1972

1975

1978

1981

1984

1987

1990

flo

w (

1000

acr

e-fe

et/y

ear)

Monthly Natural FlowUSGS stream gauge 09180500 (Colorado River near Cisco, UT)

0

500

1000

1500

2000

2500

3000

3500

4000

Oct-05

Oct-08

Oct-11

Oct-14

Oct-17

Oct-20

Oct-23

Oct-26

Oct-29

Oct-32

Oct-35

Oct-38

Oct-41

Oct-44

Oct-47

Oct-50

Oct-53

Oct-56

Oct-59

Oct-62

Oct-65

Oct-68

Oct-71

Oct-74

Oct-77

Oct-80

Oct-83

Oct-86

Oct-89

flo

w (

1000

acr

e-fe

et/m

on

th)

Bivariate Probability Density Function

Conclusions

• Basic statistics are preserved– both models reproduce mean, standard deviation,

lag(1) correlation, skew

• Reproduction of original probability density function– PAR(1) (parametric method) unable to reproduce

non gaussian PDF

– K-NN (nonparametric method) does reproduce PDF

• Reproduction of bivariate probability density function– month to month PDF

– PAR(1) gaussian assumption smoothes the original function

– K-NN recreate the original function well

• Additional research• nonparametric technique allow easy incorporation

of additional influences to flow (i.e., climate)

Exploratory Data

Analysis

Stochastic Flow Model

Salinity Model in

RiverWare

Policy Analysis

GPAT

Comparison of

parametric and

nonparametric model

Natural Salt

Model

Climate Indicators

K-NN and AR(1)

Nonparametric

regression (lowess)

replacement for

CRSM

compare salt mass

pdf

Research Topics Interconnection

Investigation of CRSS

model and data