Stochastic Nonparametric Framework for Basin Wide Streamflow and Salinity Modeling
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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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Stochastic Nonparametric Framework for Basin Wide Streamflow and Salinity Modeling

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Stochastic nonparametric framework for basin wide streamflow and salinity modeling

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


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

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


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

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


Tpms without smoothing

TPMs without smoothing


Tpms with smoothing

TPMs with smoothing


Window length chosen with lscv

Window length chosen with LSCV

3 states


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


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

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


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

No Conditioning

  • ISM

  • 98 simulations

  • 98 year length

60


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

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


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

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


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

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


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

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


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Lees Ferry

  • intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Lees Ferry

  • Total sum of intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Lees Ferry

  • Total sum of intervening

  • No first month current year with last month previous year weighting


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Cisco

  • Total sum of intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Green River UT

  • Total sum of intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • San Juan

  • Total sum of intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • 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


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Abv Imperial Dam

  • Total sum of intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Little Colorado

  • Total sum of intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Cross Correlation

  • Total sum of intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Cross Correlation

  • Total sum of intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Probability Density Function

  • Lees Ferry

  • Total sum of intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Probability Density Function

  • Lees Ferry

  • Total sum of intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Probability Density Function

  • Lees Ferry

  • Intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Drought Statistics

  • Lees Ferry

  • Total sum of intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • Drought Statistics

  • Paleo Conditioned

  • Lees Ferry

  • Total sum of intervening


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

  • 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


Extra slide follow

Extra Slide Follow


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


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

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


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

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


Stochastic nonparametric framework for basin wide streamflow and salinity modeling

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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