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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
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.