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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

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slide1

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
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

slide3

Masters Research

Single site

Modified K-NN streamflow generator

Climate Analysis

Nonparametric Natural Salt Model

Stochastic Nonparametric Technique for Space-Time Disaggregation

Basin Wide Natural Salt Model

Incorporate Paleoclimate Information

Streamflow conditioned on “Flow States” from Paleo reconstructions

Policy Analysis

Impacts of drought

Hydrology

Water quality

proposed methods

Generate system state

Generate flow conditionally

(K-NN resampling)

Proposed Methods

Block bootstrap resampling of Paleo flows

Nonhomogeneous Markov model

Markov Chain on a 30-yr window

Nonhomogeneous Markov model with smoothing

or

or

datasets
Datasets
  • Paleo reconstruction from Woodhouse et al. 2006
    • Water years 1490-1997
  • Observed natural flow from Reclamation
    • Water years 1906-2003
addressing previous issues
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
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

slide8

window = 2h +1

Discrete kernal function

h

nonhomogenous markov model with kernel smoothing rajagopalan et al 19961
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

simulation algorithm
Simulation Algorithm
  • Determine planning horizon

We chose 98yrs (same length as observational record)

  • Select 98 year block at random

For example 1701-1798

  • Generate flow states for each year of the resampled block using their respective TPMs estimated earlier NHMC
  • Generate flow magnitudes for each year by resampling observed flow using a conditional K-NN method
  • Repeat steps 2 through 4 to obtain as many required simulations
advantages over block resampling
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
no conditioning1
ISM

98 simulations

60 year length

No Conditioning
paleo conditioned
Paleo Conditioned
  • NHMC with smoothing
  • 500 simulations
  • 98 year length
paleo conditioned1
Paleo Conditioned
  • NHMC with smoothing
  • 500 simulations
  • 60 year length
drought and surplus statistics
Drought and Surplus Statistics

Surplus Length

Surplus volume

flow

Drought Length

Threshold

(e.g., mean)

time

Drought Deficit

slide20

No Conditioning

  • ISM
  • 98 simulations
  • 98 year length
paleo conditioned2
Paleo Conditioned
  • NHMC with smoothing
  • 2 states
  • 500 simulations
  • 98 year length
paleo conditioned3
Paleo Conditioned
  • Markov chain length

31 years

  • 2 states
  • 500 simulations
  • 98 year length
sequent peak algorithm
Sequent Peak Algorithm
  • Determine required Storage Capacity (Sc) at various demand levels given specified inflows.
  • Evaluate risk of not meeting the required Sc

y = inflow time series (2x)

d = demand level

S = storage

S’0= 0

if positive

otherwise

slide24

No Conditioning

  • ISM
  • 98 simulations
  • 98 year length

60

slide25

No Conditioning

  • Traditional KNN
  • 98 simulations
  • 98 year length

60

paleo conditioned4
Paleo Conditioned
  • NHMC with smoothing
  • 500 simulations
  • 98 year length

60

paleo conditioned5
Paleo Conditioned
  • PDF of 16.5 boxplot
  • Red hatch represents risk of not meeting 16.5 demand at a 60 MAF storage capacity
paleo conditioned6
Paleo Conditioned
  • PDF of 16.5 boxplot
paleo conditioned7

60

Paleo Conditioned
  • NHMC with smoothing
  • 500 simulations
  • 98 year length
paleo conditioned8
Paleo Conditioned
  • PDF of 13.5 boxplot
  • Red hatch represents risk of not meeting 13.5 demand at a 60 MAF storage capacity
paleo conditioned9
Paleo Conditioned
  • CDF of 13.5 boxplot
storage capacity firm yield function

if positive

if positive

otherwise

otherwise

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:

paleo conditioned10
Paleo Conditioned
  • NHMC with smoothing
  • 500 simulations
  • 98 year length
slide34

Basic Statistics

  • Preserved for observed data
  • Note

max and min constrained in observed

conclusions
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
slide36

Masters Research

Single site

Modified K-NN streamflow generator

Climate Analysis

Nonparametric Natural Salt Model

Stochastic Nonparametric Technique for Space-Time Disaggregation

Basin Wide Natural Salt Model

Incorporate Paleoclimate Information

Streamflow conditioned on Paleo states

Streamflow conditioned with TPM

Policy Analysis

Impacts of drought

Hydrology

Water quality

full basin disaggregation
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
slide38

Nonparametric disagg

K-NN years applied

advantages
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
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
slide41

Lees Ferry

  • intervening
slide42

Lees Ferry

  • Total sum of intervening
slide43

Lees Ferry

  • Total sum of intervening
  • No first month current year with last month previous year weighting
slide44

Cisco

  • Total sum of intervening
slide45

Green River UT

  • Total sum of intervening
slide46

San Juan

  • Total sum of intervening
slide47

San Rafael

  • Total sum of intervening
  • 1.2% of flow above Lees
  • 6% negatives over 500 sims
lower basin
Lower Basin
  • Resample observed months based on K-NN from Upper basin disaggregation
slide49

Abv Imperial Dam

  • Total sum of intervening
slide50

Little Colorado

  • Total sum of intervening
slide51

Cross Correlation

  • Total sum of intervening
slide52

Cross Correlation

  • Total sum of intervening
slide53

Probability Density Function

  • Lees Ferry
  • Total sum of intervening
slide54

Probability Density Function

  • Lees Ferry
  • Total sum of intervening
slide55

Probability Density Function

  • Lees Ferry
  • Intervening
slide56

Drought Statistics

  • Lees Ferry
  • Total sum of intervening
slide57

Drought Statistics

  • Paleo Conditioned
  • Lees Ferry
  • Total sum of intervening
slide58

Drought Statistics

  • Paleo Conditioned
  • Imperial Dam
  • Total sum of intervening
comments
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
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
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

Additional Research Information

http://animas.colorado.edu/~prairie/ResearchHomePage.html

acknowledgements
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
incorporate paleo state information
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
slide66

Paleo Reconstructed Streamflow Data

Natural Streamflow Data

Choose one path

Block Bootstrap Data

(30 year blocks)

Determine TPMs in smoothed window

Categorize natural flow data

Compute state information

Nonhomogeneous Markov model

Use KNN technique to resample natural flow data

consistent with paleo state information

slide68

No Conditioning

  • ISM
  • 98 simulations
  • 98 year length
disaggregation scheme

Colorado River at Glenwood Springs, Colorado

1

Colorado River near Cameo, Colorado

2

3

4

1

2

3

17

18

4

San Juan River near Bluff, Utah

19

Colorado River near Lees Ferry, Arizona

5

20

6

7

8

1

9

2

100

3

4

11

122

  • temporal disaggregation
    • annual to monthly at index gauge

17

18

  • spatial disaggregation
    • monthly index gauge
    • to monthly gauge

19

20

Disaggregation scheme

Index gauge

slide70

No Conditioning

  • ISM
  • 98 simulations
  • 60 year length
paleo conditioned11
Paleo Conditioned
  • Markov chain length

8 yrs - 00 01 02

6 yrs - 10 11 12

7 yrs - 20 21 22

  • 3 states
  • 500 simulations
  • 98 year length
paleo conditioned12
Paleo Conditioned
  • NHMC with smoothing
  • 2 states
  • 500 simulations
  • 60 year length
paleo conditioned13
Paleo Conditioned
  • Markov chain length

31 years

  • 2 states
  • 500 simulations
  • 60 year length
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