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Stochastic Nonparametric Framework for Basin Wide Streamflow and Salinity ModelingApplication to Colorado River basin
Study Progress MeetingJames R. PrairieAugust 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 ResearchSingle site
Modified K-NN streamflow generatorClimate Analysis
Nonparametric Natural Salt Model
Policy AnalysisImpacts of drought
HydrologyWater quality
Stochastic Nonparametric Technique for Space-Time Disaggregation
Basin Wide Natural Salt Model
Incorporate Paleoclimate InformationStreamflow conditioned on “Flow States” from
Paleo reconstructions
Proposed Methods
Generate flow conditionally(K-NN resampling)
),,( 11 tttt xSSxf
Generate system state )( tS
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. 2006– Water years 1490-1997
• Observed natural flow from Reclamation– Water years 1906-2003
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
nd
i dw
d
ndw
i dw
dw
dw
h
ttK
h
ttK
tPi
i
1
1)( )(1)( tPtP dwdd
h
window = 2h +1Discrete kernal
function1for)1(
)41(
3)( 2
2
xx
h
hxK
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
1for)1()41(
3)( 2
2
xx
h
hxK
TPMs without smoothing
TPMs with smoothing
3 states
Window length chosen with LSCV
Simulation Algorithm
1. Determine planning horizonWe chose 98yrs (same length as observational record)
2. Select 98 year block at randomFor example 1701-1798
3. 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
),,( 11 tttt xSSxf
98...,,2,1)( ttS
)( tx
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 volumeflo
w
No Conditioning
• ISM• 98 simulations• 98 year length
Paleo Conditioned
• NHMC with smoothing
• 2 states• 500 simulations• 98 year length
Paleo Conditioned
• Markov chain length31 years
• 2 states• 500 simulations• 98 year length
Sequent Peak Algorithm
• Determine required Storage Capacity (Sc) at various demand levels given specified inflows.
• Evaluate risk of not meeting the required Sc
0
'1' ii
i
ydSS if positive
otherwise
y = inflow time series (2x)
d = demand level
S = storage
S’0 = 0
'' ,...,max Nic SSS
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 (Qt)?
• Mathematically this can be answered with optimization
Maximize Y
Subject to:
Storage Minimum S
0
SSpill-Y-Q
0
SK - Y - Q
1-ttt
1-ttt
tS
Spillotherwiseif positive
otherwiseif 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 ResearchSingle site
Modified K-NN streamflow generatorClimate Analysis
Nonparametric Natural Salt Model
Policy AnalysisImpacts of drought
HydrologyWater quality
Stochastic Nonparametric Technique for Space-Time Disaggregation
Basin Wide Natural Salt Model
Incorporate Paleoclimate InformationStreamflow 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
http://animas.colorado.edu/~prairie/ResearchHomePage.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 Reclamation’s Lower Colorado Regional Office
• Logistical support provided by CADSWES
Extra Slide Follow
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 dataconsistent 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 schemeColorado River at Glenwood Springs, ColoradoColorado River near Cameo, Colorado
San Juan River near Bluff, UtahColorado River near Lees Ferry, Arizona
1
2
3
4
5
6
7
8
9
100
11
122
1
2
3
4
17
18
19
20
1
2
3
4
17
18
19
20
temporal disaggregation
annual to monthly at index gauge
spatial disaggregation
monthly index gaugeto monthly gauge
No Conditioning
• ISM• 98 simulations• 60 year length
Paleo Conditioned
• Markov chain length8 yrs - 00 01 02
6 yrs - 10 11 12
7 yrs - 20 21 22
• 3 states• 500 simulations• 98 year length
Paleo Conditioned
• NHMC with smoothing
• 2 states• 500 simulations• 60 year length
Paleo Conditioned
• Markov chain length31 years
• 2 states• 500 simulations• 60 year length