Balaji Rajagopalan Katrina Grantz CIVIL, ENVIRONMENTAL AND ARCHITECTURAL ENGINEERING DEPARTMENT

Ensemble forecasts of streamflows using large-scale

climate information : applications to the trcukee-

carson basin

Balaji RajagopalanKatrina Grantz

CIVIL, ENVIRONMENTAL AND ARCHITECTURAL ENGINEERING DEPARTMENT

UNIVERSITY OF COLORADO AT BOULDER

NCAR/ESIG Spring 2004

A Water Resources Management Perspective

Time

Horizon

Inter-decadal

Hours Weather

ClimateDecision Analysis: Risk + Values

Data: Historical, Paleo, Scale, Models

• Facility Planning

– Reservoir, Treatment Plant Size

• Policy + Regulatory Framework

– Flood Frequency, Water Rights, 7Q10 flow

• Operational Analysis

– Reservoir Operation, Flood/Drought Preparation

• Emergency Management

– Flood Warning, Drought Response

INDEPENDENCE

DONNERMARTIS

STAMPEDE

BOCA

PROSSER

TRUCKEERIVER

CARSONRIVER

CARSONLAKE

Truckee

CarsonCity

Tahoe City

Nixon

Fernley

DerbyDam

Fallon

WINNEMUCCALAKE (dry)

LAHONTAN

PYRAMID LAKE

NewlandsProject

Stillwater NWR

Reno/Sparks

NE

VA

DA

CA

LIF

OR

NIA

LAKE TAHOE

Study Area

TRUCKEE CANAL

Farad

Ft Churchill

NEVADA

CALIFORNIA

Carson

Truckee

Study Area

Prosser Creek Dam Lahontan Reservoir

Motivation

• USBR needs good seasonal forecasts on Truckee and Carson Rivers

• Forecasts determine howstorage targets will be met on Lahonton Reservoir to supply Newlands Project

Truckee Canal

Basin Precipitation

NEVADA

CALIFORNIA

Carson

Truckee

Average Annual Precipitation

Data Used

– 1949-2003 monthly data sets:• Natural Streamflow (Farad & Ft. Churchill gaging stations)

(from USBR)

• Snow Water Equivalent (SWE)- basin average

• Large-Scale Climate Variables (SST, Winds, Z500..) from http://www.cdc.noaa.gov

http://www.cdc.noaa.gov/

Basin Climatology

• Streamflow in Spring (April, May, June)

• Precipitation in Winter (November – March)

• Primarily snowmelt dominated basins

Average Monthly Flow Volumes

0

20

40

60

80

100

120

Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep

Month

Vo

lum

e (k

af)

Truckee

Carson

Average Monthly Preciptation

0

0.5

1

1.5

2

2.5

3

3.5

4

Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep

Month

Pre

cip

itat

ion

(in

)

SWE Correlation

• Winter SWE and spring runoff in Truckee and Carson Rivers are highly correlated

• April 1st SWE better than March 1st SWE

0 50 100 150 200 250

0200

400

600

Truckee-Mar 1st SWE

March 1st SWE (% of Normal)

Volu

me (

kaf)

r=0.80

0 50 100 150 200 250

0200

400

600

Truckee-Apr 1st SWE

April 1st SWE (% of Normal)

Volu

me (

kaf)

r=0.93

0 50 100 150 200 250

0200

400

600

Carson-Mar 1st SWE

March 1st SWE (% of Normal)

Volu

me (

kaf)

r=0.81

0 50 100 150 200 250

0200

400

600

Carson-Apr 1st SWE


Volu

me (

kaf)

r=0.90

Hydroclimate in Western US

• Past studies have shown a link between large-scale ocean-atmosphere features (e.g. ENSO and PDO) and western US hydroclimate (precipitation and streamflow)

• E.g., Pizarro and Lall (2001)

Correlations between peak

streamflow and Nino3

The Approach

ClimateDiagnostics

ClimateDiagnostics

ForecastingModel

ForecastingModel

DecisionSupport System

DecisionSupport System

• Forecasting Modelstochastic models for ensemble forecasting - conditioned on climate information

• Climate DiagnosticsTo identify relevant predictors to streamflow / precipitation

• Decision Support System (DSS)Couple forecast with DSS to demonstrate utility of forecast

• Correlations between spring flows and winter (standard) climate indices (NINO3, PNA, PDO etc.) were close to 0!

• Need to explore other regions correlation maps of ocean-atmospheric variables?

Winter Climate Correlations

500mb Geopotential Height Sea Surface Temperature

Truckee Spring Flow

Fall Climate Correlations

500 mb Geopotential Height Sea Surface Temperature

Carson Spring Flow

Climate Indices• Use areas of highest correlation to develop

indices to be used as predictors in the forecasting model

• Area averages of geopotential height and SST

500 mb Geopotential Height Sea Surface Temperature

Forecasting Model Predictors

•SWE •Geopotential Height •Sea Surface Temperature

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

020

040

060

0

SST Correlation

Winter SST Anomaly

Tru

ckee

Spr

ing

Vol

ume

(kaf

)

r=0.41

-100 -50 0 50

020

040

060

0

Geopotential Height Correlation

Winter Geopotential Height Anomaly

Tru

ckee

Spr

ing

Vol

ume

(kaf

)

r=-0.59

0 50 100 150 200 250

020

040

060

0

SWE Correlation


Tru

ckee

Spr

ing

Vol

ume

(kaf

)

r=0.93

Persistence of Climate Patterns

• Strongest correlation in Winter (Dec-Feb)

• Correlation statistically significant back to August

Persistence of Correlations between Climate Variables and Spring Flow

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Jul-Sep Aug-Oct Sep-Nov Oct-Dec Nov-Jan Dec-Feb Jan-Mar

Months

Co

rrel

atio

n V

alu

e (a

bs.

)

SST

Geopotential Height

High Streamflow Years Low Streamflow Years

Vector Winds

Climate Composites

High Streamflow Years Low Streamflow Years

Sea Surface Temperature

Climate Composites

Physical Mechanism

L

• Winds rotate counter-clockwise around area of low pressure bringing warm, moist air to mountains in Western US

Climate Diagnostics Summary

• Winter/ Fall geopotential heights and SSTs over Pacific Ocean related to Spring streamflow in Truckee and Carson Rivers

• Physical explanation for this correlation

• Relationships are nonlinear

• Ensemble Forecast/Stochastic Simulation

/Scenarios generation – all of them are conditional probability density function problems

• Estimate conditional PDF and simulate (Monte Carlo, or Bootstrap)

Ensemble Forecast

f yy y y

f y y y y

f y y y y dyt

t t t p

t t t t p

t t t t p t

1 2

1 2

1 2

, , . . . ,( , , , . . . , )

( , , , . . . , )

• Forecast spring streamflow (total volume)

• Capture linear and nonlinear relationships in the data

• Produce ensemble forecasts (to calculate exceedence probabilities)

Forecasting Model Requirements

Parametric Models - Drawbacks

• Model selection / parameter estimation issuesSelect a model (PDFs or Time series models) from candidate modelsEstimate parameters

• Limited ability to reproduce nonlinearity and non-Gaussian features.

All the parametric probability distributions are ‘unimodal’All the parametric time series models are

‘linear’

Parametric Models - Drawbacks

• Models are fit on the entire data setOutliers can inordinately influence parameter estimation(e.g. a few outliers can influence the mean,

variance)

Mean Squared Error sense the models are optimal but locally they can be very poor.

Not flexible

• Not Portable across sites

Nonlinearity Relationship

Geopotential Height Anomaly

Nonparametric Methods

• Any functional (probabiliity density, regression etc.) estimator is nonparametric if:

It is “local” – estimate at a point depends only on a few neighbors around it.

(effect of outliers is removed)

No prior assumption of the underlying functional form – data driven

Nonparametric Methods

• Kernel Estimators

(properties well studied)• Splines• Multivariate Adaptive Regression Splines (MARS)

• K-Nearest Neighbor (K-NN) Bootstrap Estimators • Locally Weighted Polynomials (K-NN Polynomials)

K-NN Philosophy• Find K-nearest neighbors to the desired point x

• Resample the K historical neighbors (with high probability to the nearest neighbor and low probability to the farthest) Ensembles

• Weighted average of the neighbors Mean Forecast

• Fit a polynomial to the neighbors – Weighted Least Squares

– Use the fit to estimate the function at the desired point x (i.e. local regression)

• Number of neighbors K and the order of polynomial p is obtained using GCV (Generalized Cross Validation) – K = N and p = 1 Linear modeling framework.

• The residuals within the neighborhood can be resampled for providing uncertainity estimates / ensembles.

0

0.25

0.5

0.75

1

xt

0 25 50 75 100 125

time

•

••

•••

S

DiD2D1D3•

•

1

3

2

Values of xt

A time series from the model

xt+1 = 1 - 4(xt - 0.5)2

Logistic Map Example

State

0

0.25

0.5

0.75

1

xt+1

0 0.25 0.5 0.75 1

xt

A B

1

1

2 3 4

2

3

4

State

x*A x*B

k-nearest neighborhoods A and B for xt=x*A and x*B respectively

4-state Markov Chain discretization

Modified K- Nearest Neighbor(K-NN)

• Uses local polynomial for the mean forecast (nonparametric)

• Bootstraps the residuals for the ensemble (stochastic)

•Produces flows not seen in the historical Produces flows not seen in the historical recordrecord

•Captures any non-linearities in the data, as Captures any non-linearities in the data, as well as linear relationshipswell as linear relationships

•Quantifies the uncertainty in the forecast Quantifies the uncertainty in the forecast (Gaussian and non-Gaussian)(Gaussian and non-Gaussian)

Benefits

K-NN Local Polynomial

Local Regression

-100 -50 0 50

02

00

40

06

00

Spring Flow vs. Winter Geopotential Height


Tru

cke

e S

pri

ng

Vo

lum

e (

kaf)

-100 -50 0 50

02

00

40

06

00

Spring Flow vs. Winter Geopotential Height


Tru

cke

e S

pri

ng

Vo

lum

e (

kaf)

yt* et*

xt*

yt* = f(xt

*) + et*

Residual Resampling

200

220

240

260

280

Truckee Spring Flow 1989

Vol

ume

(kaf

)

Applications to date….

• Monthly Streamflow Simulation Space and time disaggregation of monthly to daily streamflow

• Monte Carlo Sampling of Spatial Random Fields

• Probabilistic Sampling of Soil Stratigraphy from Cores

• Hurricane Track Simulation

•Multivariate, Daily Weather Simulation

• Downscaling of Climate Models

•Ensemble Forecasting of Hydroclimatic Time Series

• Biological and Economic Time Series

• Exploration of Properties of Dynamical Systems

• Extension to Nearest Neighbor Block Bootstrapping -Yao and Tong

Model Validation & Skill Measure

• Cross-validation: drop one year from the model and forecast the “unknown” value

• Compare median of forecasted vs. observed (obtain “r” value)

• Rank Probability Skill Score

ology)RPS(climat

st)RPS(foreca1RPSS

k

j

i

nn

i

nn dP

kdpRPS

1 111

1),(

Model Validation & Skill Measure

• Cross-validation: drop one year from the model and forecast the “unknown” value

• Compare median of forecasted vs. observed (obtain “r” value)

• Rank Probability Skill Score

• Likelihood Skill Score (Rajagopalan et al., 2001)

ology)RPS(climat

st)RPS(foreca1RPSS

k

j

i

nn

i

nn dP

kdpRPS

1 111

1),(

N

N

tc

N

tij

ijP

PL

1

1

1,

,

Forecasting Results

PredictorsPredictors• April 1April 1stst SWESWE• Dec-Feb Dec-Feb geopotential geopotential heightheight

95th

50th

5th

April 1st forecast

95th

50th

5th

climatology

observed

ensembleforecast(P=0.49)

(P=0.92)

valueensemble

observedvalue

climatology(P=0.17)

forecast(P=0.59)

Dry Year Wet Year

Forecasting Results

• Ensemble forecasts provide range of possible streamflow values

• Water manager can calculate exceedence probabilites

0 1- 0 1 3

Forecast Skill Scores

April 1April 1stst forecastforecast

• Median skill scores significantly beat climatology in all year subsets, both Truckee and Carson

• Truckee slightly better than Carson

Use of Climate Index in Forecast

• In general, the boxes are much tighter with geopotential height (greater confidence)• Underprediction w/o climate index– might not be fully prepared with flood control measures

SWE SWE and Geopotential Height

Extremely Wet Years

95th

50th5th

95th

50th5th

95th

50th5th

95th

50th5th

• Tighter prediction interval with geopotential height

• Overprediction w/o climate index (esp. in 1992)– Might not implement necessary drought precautions in sufficient time

SWE SWE and Geopotential Height

Use of Climate Index in ForecastExtremely Dry Years

95th

50th5th

95th

50th5th

95th

5th

95th

50th5th

50th

Truckee RPSS results

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

nov dec jan feb mar apr

Month

Me

dia

n R

PS

S (

all y

ear

s)

GpH & SWE

SWE

Truckee Forecasted vs. Observed Correlation Coeff.

0

0.2

0.4

0.6

0.8

1


Month

Co

rre

lati

on

Co

eff

GpH & SWE

SWE

Truckee Likelihood Results

0

0.5

1

1.5

2

2.5


Month

Me

dia

n L

ike

liho

od

(al

l ye

ars

)

GpH & SWE

SWE

Model Skills in Water Resources Decision Support System

Ensemble Forecasts are passed through a Decision Support (RIVERWARE) System of the Truckee/Carson Basin

Ensembles of the decision variables are compared against the “actual” values

Truckee-Carson RiverWare Model

Truckee RiverWare Model

• Physical Mechanisms– Reservoir releases, diversions, evap, reach

routing

• Policies– Implemented with “rules” (user defined

prioritized logic)

• Water Rights– Implemented in accounting network

Simplified Seasonal Model

• Method to test the utility of the forecasts and the role they play in decision making

• Model implements major policies in lower basin (Newlands Project OCAP)

• Seasonal timestep

Seasonal Model Policies

• Use Carson water first

• Max canal diversions: 164 kaf

• Storage targets on Lahontan Reservoir: 2/3 of projected April-July runoff volume

• Minimum Lahontan storage target: 1/3 of average historical spring runoff

• No minimum fish flows

Decision Variables

• Lahontan Storage Available for Irrigation

• Truckee River Water Available for Fish

• Diversion through the Truckee Canal

Seasonal Model Results: 1992

• Irrigation Water less than typical– decrease crop size or use drought-resistant crops

• Truckee Canal smaller diversion-start the season with small diversions (one way canal)

• Very little Fish Water- releases from Stampede coordinated with Canal diversions

0 100 200 300 400 500 600

0.00

00.

006

Truckee Spring Flow (kaf)

PD

F

0 100 200 300 400 500 600

0.00

00.

006

0.01

2Carson Spring Flow (kaf)

PD

F

0 100 200 300 400 500 600

0.00

00.

006

0.01

2

Lahontan Storage for Irrigation (kaf)

PD

F

0 100 200 300 400 500 600

0.00

00.

010

Truckee Canal Diversion (kaf)

PD

F

0 100 200 300 400 500 600

0.00

00.

010

0.02

0

Water Remaining in Truckee (kaf)

PD

F

Ensemble forecast results

Climatology forecast results

Observed value results

NRCS official forecast results

Seasonal Model Results:1993

• Irrigation Water more than typical– plenty for irrigation and carryover

• Truckee Canal larger diversion-start the season at full diversions (limited capacity canal)

• Plenty Fish Water- FWS may schedule a fish spawning run

0 100 200 300 400 500 600

0.00

00.

010


PD

F

0 100 200 300 400 500 600

0.00

00.

006

0.01

2

Carson Spring Flow (kaf)

PD

F

0 100 200 300 400 500 600

0.00

00.

006

0.01

2


PD

F

0 100 200 300 400 500 600

0.00

0.04


PD

F

0 100 200 300 400 500 600

0.00

00.

010


PD

F





Seasonal Model Results: 2003

• Irrigation Water pretty average: business as usual

• Truckee Canal diversions normal: not full capacity, but don’t hold back too much

• Plenty Fish Water- no releases necessary to augment low flows, may choose a fish spawning run

0 100 200 300 400 500 600

0.00

00.

015


PD

F

0 100 200 300 400 500 600

0.00

00.

008

Carson Spring Flow (kaf)P

DF

0 100 200 300 400 500 600

0.00

00.

010


PD

F

0 100 200 300 400 500 600

0.00

00.

015

0.03

0


PD

F

0 100 200 300 400 500 600

0.00

0.02


PD

F





CONCLUSIONS

• Interannual/Interdecadal variability of regional hydrology (precipitation, streamflows) is modulated by large-scale ocean-atmospheric features

• Incorporating Large scale Climate information in regional hydrologic forecasting models (Seasonal streamflows and precipitation) provides significant skill at long lead times

• Nonparametric methods offer an attractive and flexible alternative to traditional methods.

• capability to capture any arbitrary relationship• data-drive• easily portable across sites

• Significant implications to water (resource) management and planning

Future Work

• Couple ensemble forecasts with RiverWare model

• Temporal disaggregation

• Forecast improvements– Joint Truckee/Carson forecast– Objective predictor selection

• Compare results with physically-based runoff model (e.g. MMS)

Summary & Conclusions

• Climate indicators provide skilful long-lead forecast

• Water managers can utilize the improved forecasts in operations and seasonal planning

• Nonparametric methods provide an attractive and flexible alternative to parametric methods.

Future Work

• Couple ensemble forecasts with RiverWare model

• Temporal disaggregation

• Forecast improvements– Joint Truckee/Carson forecast– Objective predictor selection

• Compare results with physically-based runoff model (e.g. MMS)

Acknowledgements

• CIRES Innovative Reseach Project• Tom Scott of USBR Lahontan Basin Area

Office

• CADSWES personnel

FundingFunding

Technical SupportTechnical Support

K-Nearest Neighbor Estimators

)(rc

k/n =

)(Vk/n

= )(fd

kdkNN xx

xA k-nearest neighbor density estimate

fGNN(x) = 1rk

d(x)nK

x xirk(x)

i1

n

A conditional k-nearest neighbor density estimate

n

1i ),(k

r

)i

,i

(),(K

1 )(k

riK1-)n)(

k(r

1-)n),(k

(r

)(/),( )|(GNNf

Dx

DxDx

x

xxx

Dx

DDxDx

n

i

ff

duuKuduuKuuK )(2,0)(,0)(f(.) is continuous on Rd, locally Lipschitz of order p

k(n) =O(n2p/(d+2p))

n

i kr

K

n

i kr

K

m

1 )(i

1 )(i

i)|(ˆ

D*

D*D

D*

D*Dx

D*DxA k-nearest neighbor ( modified Nadaraya Watson) conditional mean estimate

Classical Bootstrap (Efron):

Given x1, x2, …... xn are i.i.d. random variables with a cdf F(x)

Construct the empirical cdf

Draw a random sample with replacement of size n from

n

iniIF

1/)()(ˆ xxx

)(ˆ xF

Moving Block Bootstrap (Kunsch, Hall, Liu & Singh) :

Resample independent blocks of length b<n, and paste them together to form a series of length n

k-Nearest Neighbor Conditional Bootstrap (Lall and Sharma, 1996)

Construct the Conditional Empirical Distribution Function:

Draw a random sample with replacement from

n

ikiK

krBiIiIF

1/)(*))(()(*)|(ˆ DDxxDx

*)|(ˆ DxF

Define the composition of the "feature vector" Dt of dimension d.

(1) Dependence on two prior values of the same time series.Dt : (xt-1, xt-2) ; d=2

(2) Dependence on multiple time scales (e.g., monthly+annual)Dt: (xt-1, xt-21, .... xt-M11; xt-2, xt-22, ..... xt-M22) ; d=M1+M2

(3) Dependence on multiple variables and time scales Dt: (x1t-1, .... x1t-M11; x2t, x2t-2, .... x2t-M22); d=M1+M2+1

Identify the k nearest neighbors of Dt in the data D1 ... Dn

Define the kernel function ( derived by taking expected values of distances to each of k nearest neighbors, assuming the number of observations of D in a neighborhood Br(D*) of D*; r0, as n , is locally Poisson, with rate (D*))

for the jth nearest neighbor

Selection of k: GCV, FPE, Mutual Information, or rule of thumb (k=n0.5)

1...k =j 1/j

1/jK(j)k

1i

Balaji Rajagopalan Katrina Grantz CIVIL, ENVIRONMENTAL AND ARCHITECTURAL ENGINEERING DEPARTMENT

Documents

Transcript of Balaji Rajagopalan Katrina Grantz CIVIL, ENVIRONMENTAL AND ARCHITECTURAL ENGINEERING DEPARTMENT