Stochastic Nonparametric Techniques for Ensemble Streamflow Forecast : Applications to

Truckee/Carson and Thailand Streamflows

Balaji Rajagopalan, Katrina Grantz, Nkrintra Singhrattna

Department of Civil and Environmental Engg.

University of Colorado, Boulder, CO

Edith Zagona

CADSWES / Dept. Of Civil and Env. Engg.

University of Colorado, Boulder, CO

Martyn Clark

CIRES

University of Colorado

GAPP / PI Meeting – Summer 2003

Hydrologic Forecasting

• Conditional Statistics of Future State, given Current State• Current State: Dt : (xt, xt-, xt-2 , …xt-d1, yt, yt- , yt-2, …yt-d2)

• Future State: xt+T

• Forecast: g(xt+T) = f(Dt)– where g(.) is a function of the future state, e.g., mean or pdf– and f(.) is a mapping of the dynamics represented by Dt to g(.)– Challenges

• Composition of Dt

• Identify g(.) given Dt and model structure

– For nonlinear f(.) , Nonparametric function estimation methods used• K-nearest neighbor• Local Regression• Regression Splines• Neural Networks

The Problem

• Ensemble Forecast/Stochastic Simulation/Scenarios generation – all of them are conditional probability density function problems

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

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

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

( , , ,..., )

Parametric Models• Periodic Auto Regressive model (PAR)

– Linear lag(1) model

– Stochastic Analysis, Modeling, and Simulation (SAMS) (Salas, 1992)

• Data must fit a Gaussian distribution• Expected to preserve

– mean, standard deviation, lag(1) correlation– skew dependant on transformation– gaussian probability density function

y ,

1 y 1– 1–– ,+ +=

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’• Outliers have undue influence on the fit• Not Portable across sites

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

• 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.

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

K-NN Local Polynomial

yt*

yt-1

K-NN Algorithm

yt-1

yt*et*

Residual Resampling

yt = yt* + et

*

Applications

Local-Polynimial + K-NN residual bootstrap• Ensemble Streamflow forecasting

Truckee-Carson basin, NV• Ensemble forecast from categorical probabilistic

forecast – Thailand Streamflows

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

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

Outline of Approach

• Climate DiagnosticsTo identify large scale features correlated to Spring flow in the Truckee and Carson Rivers

• Ensemble ForecastStochastic Models conditioned on climate indicators (Parametric and Nonparametric)

• ApplicationDemonstrate utility of improved forecast to water management

Data

– 1949-1999 monthly averages• Streamflow at Ft. Churchill and Farad• Precipitation (regional)• Geopotential Height 500mb (regional)• Sea Surface Temperature (regional)

Annual Cycle of Flows

Fall Climate Correlations

500 mb Geopotential Height Sea Surface Temperature

Carson Spring Flow

500 mb Geopotential Height Sea Surface Temperature

Carson Spring Flow

Winter Climate Correlations

Winter Climate Correlations

500 mb Geopotential Height Sea Surface Temperature

Truckee Spring Flow

Sea Surface Temperature Vector Winds

High-Low Flow

Climate Composites

Precipitation Correlation

Geopotential Height Correlation

SST Correlation

Flow - NINO3 / Geopotential HeightRelationship

The Forecasting Model• Forecast Spring Runoff in Truckee and Carson Rivers

using Winter Precipitation and Climate Data Indices (Geopotential height index and SST index).

• Modified K-NN Method:– Uses Local Polynomial for the mean forecast– Bootstraps the residuals for the ensemble

Wet Years: 1994-1999

• Overprediction w/o Climate (1995, 1996)– Might release water for flood control– stuck in spring with

not enough water

• Underprediction w/o Climate (1998)

Precipitation Precipitation and Climate

1994 1995 1996

1994 1995 1996

1994 1995 1996

1994 1995 1996

1997 1998 1999 1997 1998 1999

1997 1998 19991997 1998 1999

Dry Years: 1987-1992

• Overprediction w/o Climate (1998, 991)– Might not implement necessary drought

precautions in sufficient time

Precipitation Precipitation and Climate

1987 1988 1989

1987 1988 1989

1987 1988 1989

1987 1988 1989

1990 1991 1992 1990 1991 1992

1990 1991 19921990 1991 1992

Fall Prediction w/ Climate

• Fall Climate forecast captures whether season will be above or below average

• Results comparable to winter forecast w/o climate

Wet Years Dry Years

1987 1988 1989

1987 1988 1989

1990 1991 1992

1990 1991 1992

1994 1995 1996

1994 1995 1996

1997 1998 1999

1997 1998 1999

Simple Water Balance

• St-1 is the storage at time ‘t-1’, It is the inflow at time ‘t’

and Rt is the release at time ‘t’.• Method to test the utility of the model• Pass Ensemble forecasts (scenarios) for It • Gives water managers a quick look at how much storage

they will have available at the end of the season – to evluate decision strategies

For this demonstration,• Assume St-1=0, Rt= 1/2(avg. Inflowhistorical)

St = St-1 + It - Rt

Water Balance

1995 K-NN Ensemble

PDFHistorical

PDF

1995 Storage

Future Work

• Stochastic Model for Timing of the RunoffDisaggregate Spring flows to monthly flows.

• Statistical Physical ModelCouple PRMS with stochastic weather generator (conditioned on climate info.)

• Test the utility of these approaches to water management using the USBR operations model in RiverWare

Region / Data6 rainfall stations

- Nakhon Sawan, Suphan Buri, Lop Buri, Kanchana Buri, Bangkok, and Don Muang

3 streamflow stations(Chao Phaya basin)- Nakhon Sawan, Chai Nat, Ang-Thong

5 temperature stations- Nakhon Sawan, Lop Buri, Kanchana Buri, Bangkok, Don Muang

Large Scale Climate Variables

NCEP-NCAR Re-analysis data(http://www.cdc.noaa.gov)

Composite Maps of High rainfallPre 1980 Post 1980

Composite Maps of Low rainfallPre 1980 Post 1980

Example Forecast for 1997

FlowLow

La Nina 0.000Neu 0.320

La Nina

Conditional Probabilitiesfrom historical data

(Categories are at Quantiles)

Categorical ENSO forecast

Conditional flow probabilitesusing Total Probability Theorem

Ensemble Forecast from Categorical Probabilistic forecasts

• If the categorical probabilistic forecasts are P1, P2 and P3 then– Choose a category with the above probabilities– Randomly select an historical observation from the

chosen category– Repeat this a numberof times to generate ensemble

forecasts

Ensemble Forecast of Thailand Streamflows – 1997

Summary• Nonparametric techniques (K-NN framework in

particular) provides a flexible alternative to Parametric methods for

Ensemble forecasting/Downscaling

• Easy to implement, parsimonious extension to multivariate situations. Water managers can utilize the improved forecasts in operations and seasonal planning

• No prior assumption to the functional form is needed. Can capture nonlinear/non-Gaussian features readily.