Transcript Incorporating Climate Information in Long Term Salinity

Comparison of natural streamflows
generated from a parametric and
nonparametric stochastic model
James Prairie(1,2), Balaji Rajagopalan(1)
and Terry Fulp(2)
1. University of Colorado at Boulder, CADSWES
2. U.S Bureau of Reclamation
Motivation
• Generate future inflow scenarios for
decision making models
– reservoir operating rules, salinity control
• Estimate uncertainty in model output
Options
• Parametric Techniques
– AR, ARMA, PAR, PARMA
• Nonparametric Techniques
– K-NN, density estimator, bootstrap
Objective of Study
• Compare nonparametric and parametric
techniques for simulation of streamflows
– at USGS stream gauge 09180500: Colorado
River near Cisco, UT
Outline of Talk
• Overview of parametric technique
• Explain nonparametric technique
• Compare various distribution attributes
–
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mean
standard deviation
lag(1) correlation
skewness
marginal probability density function
bivariate probability
• Conclusions
Parametric
• Periodic Auto Regressive model (PAR)
– developed a lag(1) model
yn ,t = mt + å F j ,t ( yn ,t - j - mt - j ) + Sn ,t
p
Salas (1992)
j =1
n = year
t = season
– Stochastic Analysis, Modeling, and Simulation
(SAMS)
• Data must fit a Gaussian distribution
– log and power transformation
– not guaranteed to preserve statistics after back
transformation
• Expected to preserve
– mean, standard deviation, lag(1) correlation
– skew dependant on transformation
– gaussian probability density function
Nonparametric
• K- Nearest Neighbor model (K-NN)
– lag(1) model
• No prior assumption of data’s distribution
– no transformations needed
• Resamples the original data with
replacement using locally weighted
bootstrapping technique
– only recreates values in the original data
• augment using noise function
• alternate nonparametric method
• Expected to preserve
– all distributional properties
• (mean, standard deviation, lag(1) correlation and
skewness)
– any arbitrary probability density function
Nonparametric (cont’d)
• Markov process for resampling
Lall and Sharma (1996)
Nearest Neighbor Resampling
1. Dt (x t-1) d =1 (feature vector)
2. determine k nearest neighbors among Dt
using Euclidean distance
æ d
1/ 2 ö
rit = çç å w j (vij - vtj ) ÷÷
è j =1
ø
Where v tj is the jith component of Dt,
and w j are scaling weights.
3. define a discrete kernel K(j(i)) for
resampling one of the xj(i) as follows
K ( j (i )) =
1 j
k
å1
j
j =1
4. using the discrete probability mass
function K(j(i)), resample xj(i) and update
the feature vector then return to step 2 as
needed
5. Various means to obtain k
– GCV
– Heuristic scheme k = N
Lall and Sharma (1996)
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9
ct
-8
6
3
ct
-8
ct
-8
0
7
ct
-8
ct
-7
4
1
ct
-7
ct
-7
8
5
ct
-6
ct
-6
2
9
ct
-6
ct
-5
6
3
ct
-5
ct
-5
0
7
ct
-5
ct
-4
4
1
ct
-4
ct
-4
8
5
ct
-3
ct
-3
2
9
ct
-3
ct
-2
6
3
ct
-2
ct
-2
0
7
ct
-2
ct
-1
4
1
ct
-1
ct
-1
8
5
ct
-0
ct
-0
flow (1000 acre-feet/month)
19
06
19
09
19
12
19
15
19
18
19
21
19
24
19
27
19
30
19
33
19
36
19
39
19
42
19
45
19
48
19
51
19
54
19
57
19
60
19
63
19
66
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19
72
19
75
19
78
19
81
19
84
19
87
19
90
flow (1000 acre-feet/year)
USGS stream gauge 09180500 (Colorado River near Cisco, UT)
Annual Water Year Natural Flow
14000
12000
10000
8000
6000
4000
2000
0
USGS stream gauge 09180500 (Colorado River near Cisco, UT)
Monthly Natural Flow
4000
3500
3000
2500
2000
1500
1000
500
0
Bivariate Probability Density Function
Conclusions
• Basic statistics are preserved
– both models reproduce mean, standard
deviation, lag(1) correlation, skew
• Reproduction of original probability
density function
– PAR(1) (parametric method) unable to
reproduce non gaussian PDF
– K-NN (nonparametric method) does reproduce
PDF
• Reproduction of bivariate probability
density function
– month to month PDF
– PAR(1) gaussian assumption smoothes the
original function
– K-NN recreate the original function well
• Additional research
• nonparametric technique allow easy
incorporation of additional influences to flow
(i.e., climate)
Research Topics Interconnection
Comparison of
parametric and
nonparametric model
Exploratory Data
Climate Indicators
Analysis
Stochastic Flow Model
K-NN and AR(1)
Investigation of CRSS
model and data
Natural Salt
Model
Salinity Model in
RiverWare
Policy Analysis
GPAT
Nonparametric
regression (lowess)
replacement for
CRSM
compare salt mass
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