Ensemble forecasts of streamflows using large-scale climate information : applications to the trcukee- carson basin Balaji Rajagopalan Katrina Grantz CIVIL, ENVIRONMENTAL AND ARCHITECTURAL ENGINEERING DEPARTMENT UNIVERSITY OF COLORADO AT BOULDER NCAR/ESIG Spring 2004

A Water Resources Management Perspective Time HorizonTime Horizon Inter-decadal Hours Weather Climate Decision 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

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 how storage targets will be met on Lahonton Reservoir to supply Newlands Project Truckee Canal

Basin Precipitation NEVADA CALIFORNIA Carson Truckee Average Annual Precipitation

Data Used – 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

Basin Climatology Streamflow in Spring (April, May, June) Precipitation in Winter (November – March) Primarily snowmelt dominated basins

SWE Correlation Winter SWE and spring runoff in Truckee and Carson Rivers are highly correlated April 1 st SWE better than March 1 st SWE

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 Climate Diagnostics Climate Diagnostics Forecasting Model Forecasting Model Decision Support System Decision Support System Forecasting Model stochastic models for ensemble forecasting - conditioned on climate information Climate Diagnostics To 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 HeightSea 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 HeightSea Surface Temperature

Forecasting Model Predictors SWE Geopotential Height Sea Surface Temperature

Persistence of Climate Patterns Strongest correlation in Winter (Dec-Feb) Correlation statistically significant back to August

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

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 issues Select a model (PDFs or Time series models) from candidate models Estimate 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 set Outliers 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

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.

Logistic Map Example k-nearest neighborhoods A and B for x t =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 record Produces flows not seen in the historical record Captures any non-linearities in the data, as well as linear relationships Captures any non-linearities in the data, as well as linear relationships Quantifies the uncertainty in the forecast (Gaussian and non-Gaussian) Quantifies the uncertainty in the forecast (Gaussian and non-Gaussian) Benefits

K-NN Local Polynomial

Local Regression

y t * e t * xt*xt* y t * = f(x t *) + e t * Residual Resampling

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

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)

Forecasting Results Predictors April 1 st SWE April 1 st SWE Dec-Feb geopotential height Dec-Feb geopotential height 95th 50th 5th April 1 st forecast 95th 50th 5th

Dry Year Wet Year Forecasting Results Ensemble forecasts provide range of possible streamflow values Water manager can calculate exceedence probabilites

Forecast Skill Scores April 1 st forecast 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 50th 5th 95th 50th 5th 95th 50th 5th 95th 50th 5th

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 Forecast Extremely Dry Years 95th 50th 5th 95th 50th 5th 95th 5th 95th 50th 5th 50th

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 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 Ensemble forecast results Climatology forecast results Observed value results NRCS official forecast results

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 Ensemble forecast results Climatology forecast results Observed value results NRCS official forecast results

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 Funding Technical Support

K-Nearest Neighbor Estimators A k-nearest neighbor density estimate A conditional k-nearest neighbor density estimate f(.) is continuous on R d, locally Lipschitz of order p k(n) =O(n 2p/(d+2p) ) A k-nearest neighbor ( modified Nadaraya Watson) conditional mean estimate

Classical Bootstrap (Efron): Given x 1, x 2, …... x n 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 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

Define the composition of the "feature vector" D t of dimension d. (1) Dependence on two prior values of the same time series. D t : (x t-1, x t-2 ) ; d=2 (2) Dependence on multiple time scales (e.g., monthly+annual) D t : (x t-  1, x t-2  1,.... x t-M1  1 ; x t-  2, x t-2  2,..... x t-M2  2 ) ; d=M1+M2 (3) Dependence on multiple variables and time scales D t : (x1 t-  1,.... x1 t-M1  1 ; x2 t, x2 t-  2,.... x2 t-M2  2 ); d=M1+M2+1 Identify the k nearest neighbors of D t in the data D 1... D n 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 B r (D*) of D*; r  0, as n , is locally Poisson, with rate (D*) ) for the j th nearest neighbor Selection of k: GCV, FPE, Mutual Information, or rule of thumb (k=n 0.5 )