Stochastic Nonparametric Framework for Basin Wide Streamflow and Salinity Modeling Application to Colorado River basin Study Progress Meeting James R. Prairie August 17, 2006

Recent progress Stochastic streamflow conditioned on Paleo Flow –Non homogenous Markov Chain with Kernel Smoothing Estimate lag-1 two state transition probabilities for each year using a Kernel Estimator Generate Flow State Conditionally Generate flow magnitude Colorado River Basin Wide flow simulation –Modify the nonparametric space-time disagg approach to generate monthly flows at all the 29 stations simultaneously –Flow simulation using Paleo recontructions

Masters Research Single site Modified K-NN streamflow generator Climate Analysis Nonparametric Natural Salt Model Policy Analysis Impacts of drought Hydrology Water quality Stochastic Nonparametric Technique for Space-Time Disaggregation Basin Wide Natural Salt Model Incorporate Paleoclimate Information Streamflow conditioned on Flow States from Paleo reconstructions

Proposed Methods Generate flow conditionally (K-NN resampling) Generate system state Block bootstrap resampling of Paleo flows Nonhomogeneous Markov model Markov Chain on a 30-yr window or Nonhomogeneous Markov model with smoothing or

Datasets Paleo reconstruction from Woodhouse et al –Water years Observed natural flow from Reclamation –Water years

Addressing previous issues Determined order of the Markov model –used AIC (Gates and Tong, 1976) Indicated order 0 (or 1) - we used order 1 Subjective block length and window for estimating the Markov Chain Transition Probabilities –Nonhomogeneous Markov Chain with Kernel Smoothing alleviates this problem (Rajagopalan et al., 1996)

Nonhomogenous Markov model with Kernel smoothing (Rajagopalan et al., 1996) 2 state, lag 1 model chosen –wet (1) if flow above annual median of observed record; dry (0) otherwise. –AIC used for order selection (order 1 chosen) TP for each year are obtained using the Kernel Estimator

h window = 2h +1 Discrete kernal function

Nonhomogenous Markov model with Kernel smoothing (Rajagopalan et al., 1996) K(x) is a discrete quadratic Kernel (or weight function) –h is the smoothing window obtained objectively using Least Square Cross Validation

TPMs without smoothing

TPMs with smoothing

3 states Window length chosen with LSCV

Simulation Algorithm 1.Determine planning horizon We chose 98yrs (same length as observational record) 2.Select 98 year block at random For example Generate flow states for each year of the resampled block using their respective TPMs estimated earlier NHMC 4.Generate flow magnitudes for each year by resampling observed flow using a conditional K-NN method 1.Repeat steps 2 through 4 to obtain as many required simulations

Advantages over block resampling No need for a subjective window length –i.e., 30 year window was used to estimate the TP Obviates the need for additional sub-lengths within the planning horizon –i.e., earlier 3 30-yr blocks were resampled Fully Objective in estimating the TPMs for each year

No Conditioning ISM 98 simulations 98 year length

No Conditioning ISM 98 simulations 60 year length

Paleo Conditioned NHMC with smoothing 500 simulations 98 year length

Paleo Conditioned NHMC with smoothing 500 simulations 60 year length

Threshold (e.g., mean) Drought Length Surplus Length time Drought Deficit Drought and Surplus Statistics Surplus volume flow

No Conditioning ISM 98 simulations 98 year length

Paleo Conditioned NHMC with smoothing 2 states 500 simulations 98 year length

Paleo Conditioned Markov chain length 31 years 2 states 500 simulations 98 year length

Sequent Peak Algorithm Determine required Storage Capacity (S c ) at various demand levels given specified inflows. Evaluate risk of not meeting the required S c if positive otherwise y = inflow time series (2x) d = demand level S = storage S 0 = 0

No Conditioning ISM 98 simulations 98 year length 60

No Conditioning Traditional KNN 98 simulations 98 year length 60

Paleo Conditioned NHMC with smoothing 500 simulations 98 year length 60

Paleo Conditioned PDF of 16.5 boxplot Red hatch represents risk of not meeting 16.5 demand at a 60 MAF storage capacity

Paleo Conditioned PDF of 16.5 boxplot

Paleo Conditioned NHMC with smoothing 500 simulations 98 year length 60

Paleo Conditioned PDF of 13.5 boxplot Red hatch represents risk of not meeting 13.5 demand at a 60 MAF storage capacity

Paleo Conditioned CDF of 13.5 boxplot

Storage Capacity – Firm Yield function What is the maximum yield (Y) given a specific storage capacity (K) and flow sequence (Q t )? Mathematically this can be answered with optimization Maximize Y Subject to: otherwise if positive otherwise if positive

Paleo Conditioned NHMC with smoothing 500 simulations 98 year length

Basic Statistics Preserved for observed data Note max and min constrained in observed

Conclusions Combines strength of –Reconstructed paleo streamflows: system state –Observed streamflows: flows magnitude Develops a rich variety of streamflow sequences –Generates sequences not in the observed record –More variety: block bootstrap reconstructed streamflows –Most variety: nonhomogeneous Markov chain TPM provide flexibility –Homogenous Markov chains –Nonhomogenous Markov chains –Use TPM to mimic climate signal (e.g., PDO) –Generate drier or wetter than average flows

Masters Research Single site Modified K-NN streamflow generator Climate Analysis Nonparametric Natural Salt Model Policy Analysis Impacts of drought Hydrology Water quality Stochastic Nonparametric Technique for Space-Time Disaggregation Basin Wide Natural Salt Model Incorporate Paleoclimate Information Streamflow conditioned on Paleo states Streamflow conditioned with TPM

Full basin disaggregation Upper basin –20 gauges (all above Lees Ferry, including Lees Ferry) –Annual total flow at Lees Ferry: modeled with modified K-NN –Disaggregate Lees Ferry: nonparametric disaggregation Results in intervening monthly flows at CRSS nodes Store the years resampled during the temporal disagg Lower basin –9 gauges (all gauges below Lees Ferry) –Select the month values for all sites in a given year based on the years stored above

Nonparametric disagg K-NN years applied

Advantages Paleo-conditioned flows for entire basin Upper Basin –Generate both annual and monthly flows not previously observed –Produces 92% of annual flows above Imperial Dam –Faithfully reproduces PDF and CDF for both intervening and total flows Lower Basin –Produces 8% of annual flows above Imperial Dam –Preserves intermittent properties of tributaries –Faithfully reproduces all statistics –Easily incorporate reconstructions at Lees Ferry

Disadvantages Upper Basin –Generates negative flows at rim gauges (7 out of 10 gauges) Average of 1.5% negatives over all simulations (500 sims) Is this important? Two largest contributors only produce 2.2% –Can not capture cross over correlation (i.e. between last month of previous year and first month of the current year) Improved in recent run (added a weighted resampling) –Can not generate large extremes beyond the observed Annual flow model choice Using Paleo flow magnitudes Lower Basin –Can only generate observed flows

Lees Ferry intervening

Lees Ferry Total sum of intervening

Lees Ferry Total sum of intervening No first month current year with last month previous year weighting

Cisco Total sum of intervening

Green River UT Total sum of intervening

San Juan Total sum of intervening

San Rafael Total sum of intervening 1.2% of flow above Lees 6% negatives over 500 sims

Lower Basin Resample observed months based on K-NN from Upper basin disaggregation

Abv Imperial Dam Total sum of intervening

Little Colorado Total sum of intervening

Cross Correlation Total sum of intervening

Cross Correlation Total sum of intervening

Probability Density Function Lees Ferry Total sum of intervening

Probability Density Function Lees Ferry Total sum of intervening

Probability Density Function Lees Ferry Intervening

Drought Statistics Lees Ferry Total sum of intervening

Drought Statistics Paleo Conditioned Lees Ferry Total sum of intervening

Drought Statistics Paleo Conditioned Imperial Dam Total sum of intervening

Comments Handling negatives in total natural flow –Continuing to explore reducing negatives in simulations –Should we address base data (natural flow)? –How does RiverWare handle negatives at rims? Min 10 constraint K-NN implementation in Lower basin –Robust, simple –Handles intermittent streams –Faithfully reproduces statistics

Next steps Incorporate salinity methods in EIS CRSS Generate stochastic data no conditioning –Flow and salt scenarios –Disaggregate data Generate paleo conditioned data for network –Flow and salt scenarios –Disaggregate data Drive decision support system –Perform policy analysis Compare results from at least two hydrologies –Paleo conditioned streamflows –Index Sequential Method (current Reclamation technique) –Possibly stochastic no conditioning

Continued Steps Submitted revisions for WRR paper Finalize and submit Salt Model Paper –Journal of Hydrology Complete Markov Paper –Water Resources Research Complete Policy Analysis Paper –ASCE Journal of Water Resources Planning and Management or Journal of American Water Resources Association Incorporate all into dissertation

Additional Research Information esearchHomePage.html

Acknowledgements To my committee and advisor. Thank you for your guidance and commitment. –Balaji Rajagopalan, Edith Zagona, Kenneth Strzepek, Subhrendu Gangopadhyay, and Terrance Fulp Funding support provided by Reclamations Lower Colorado Regional Office Logistical support provided by CADSWES

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Incorporate paleo state information Magnitudes of Paleo data in question? –Address issue, use observed data to represent magnitude and paleo reconstructed streamflows to represent system state –Generate streamflows from the observed record conditioned on paleo streamflow state information

Block Bootstrap Data (30 year blocks) Compute state information Use KNN technique to resample natural flow data consistent with paleo state information Categorize natural flow data Paleo Reconstructed Streamflow Data Natural Streamflow Data Nonhomogeneous Markov model Determine TPMs in smoothed window Choose one path

No Conditioning ISM 98 simulations 98 year length

Index gauge Disaggregation scheme Colorado River at Glenwood Springs, Colorado Colorado River near Cameo, Colorado San Juan River near Bluff, Utah Colorado River near Lees Ferry, Arizona temporal disaggregation annual to monthly at index gauge spatial disaggregation monthly index gauge to monthly gauge

No Conditioning ISM 98 simulations 60 year length

Paleo Conditioned Markov chain length 8 yrs yrs yrs states 500 simulations 98 year length

Paleo Conditioned NHMC with smoothing 2 states 500 simulations 60 year length

Paleo Conditioned Markov chain length 31 years 2 states 500 simulations 60 year length