Stochastic Nonparametric Framework for Basin Wide Streamflow and Salinity Modeling

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

2. 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

3. 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

4. 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

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

6. 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)

7. 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

8. window = 2h +1 Discrete kernal function h

9. 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

10. TPMs without smoothing

11. TPMs with smoothing

12. Window length chosen with LSCV 3 states

13. 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

14. 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

15. ISM 98 simulations 98 year length No Conditioning

16. ISM 98 simulations 60 year length No Conditioning

17. Paleo Conditioned • NHMC with smoothing • 500 simulations • 98 year length

18. Paleo Conditioned • NHMC with smoothing • 500 simulations • 60 year length

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

20. No Conditioning • ISM • 98 simulations • 98 year length

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

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

23. 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

24. No Conditioning • ISM • 98 simulations • 98 year length 60

25. No Conditioning • Traditional KNN • 98 simulations • 98 year length 60

26. Paleo Conditioned • NHMC with smoothing • 500 simulations • 98 year length 60

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

28. Paleo Conditioned • PDF of 16.5 boxplot

29. 60 Paleo Conditioned • NHMC with smoothing • 500 simulations • 98 year length

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

31. Paleo Conditioned • CDF of 13.5 boxplot

32. 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:

33. Paleo Conditioned • NHMC with smoothing • 500 simulations • 98 year length

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

35. 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

36. 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

37. 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

38. Nonparametric disagg K-NN years applied

39. 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

40. 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

41. Lees Ferry • intervening

42. Lees Ferry • Total sum of intervening

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

44. Cisco • Total sum of intervening

45. Green River UT • Total sum of intervening

46. San Juan • Total sum of intervening

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

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

49. Abv Imperial Dam • Total sum of intervening

50. Little Colorado • Total sum of intervening