1. Why Stochastic Hydrology ?
Providing simulated system behavior based on plausible sequences that are statistically equivalent to observed sequences
System design.
Storage yield analysis and reliability
Evaluation of alternative system configurations
Complex system operation
Evaluation of alternative operation rules. What if evaluation of how a facility will perform for representative future hydrologic inputs.
Operational forecasting
Evaluation of alternative operation rules given current state. What if evaluation of alternatives
Missing data
Filling in of missing data based upon statistical relationships to nearby related data

2. The Monte Carlo Simulation Approach
Streamflow and other hydrologic time series inputs are random (resulting from lack of knowledge and unknowability of boundary conditions and inputs)
System behavior is complex
Can be represented by a simulation model
Analytic derivation of probability distribution of system output is intractable
Inputs generated from a Monte Carlo simulation model designed to capture the essential statistical structure of the input time series
Monte Carlo simulations solve the derived distribution problem to allow numerical determination of probability distributions of output variables

3. From Bras, R. L. and I. Rodriguez-Iturbe, (1985), Random Functions and Hydrology, Addison-Wesley, Reading, MA, 559 p.

4. Streamflow Modeling
A stochastic streamflow model should reproduce what are judged to be the fundamental characteristics of the joint distribution of the flows.
What characteristics are important to reproduce?

5. Marginal Distributions
Mean
Std Deviation
Skewness
Extremes
Full distribution
Normalizing transformations
Box-cox
Parametric PDF
Nonparametric PDF
Measures of fit
Kolmogorov smirnoff
Probability plot correlation (Filliben)
Shapiro-Wilks
Likelihood

6. Parametric vs Nonparametric
Assume data from a known distribution (e.g. Gamma, Normal) and estimate parameters
Nonparametric
Assume that the data has a probability density function but not of a specific known form
Let data speak for themselves
Exploratory data analysis

7. Kernel Density Estimate (KDE)
Place “kernels” at each data point
Sum up the kernels
Width of kernel determines level of smoothing
Determining how to choose the width of the kernel is an important topic!
Narrow kernel
Medium kernel
Wide kernel

8. KDE window width sensitivity

9. Normalizing transformation for arbitrary distribution
Arbitrary distribution F(x)
Normal distribution Fn(y) x y
Normalizing transformation
Back transformation

10. Relationships between variables
Correlation
Autocorrelation
Partial autocorrelation
State dependent correlation
Cross correlation
Spectral density

11. Storage related Threshold related Storage-yield Reliability
Range and Hurst coefficient
Duration above/below
Accumulation above/below

12. Lake Powell Capacity 27 MAF

13. General form of a stochastic hydrology model
Qt=F(Qt-1, Qt-2, …, random inputs)
Example
Regression of later values against earlier values

14. General function fitting

15. Time series function fittingx1 x2 . xt

16. Time series autoregressive function fitting – Method of delays
Embedding dimension
x1 x2 x x4
x2 x3 x x5
x3 x4 x x6
………
xt-3 xt-2 xt-1 xt
Samples data vectors constructed using lagged copies of the single time series
ExampleAR1 model
xt =  xt-1 + 
Trajectory matrix

17. Multiple Random Variables and Joint Distributions
The conditional dependence between random variables serves as a foundation for time series analysis.
When multiple random variables are related they are described by their joint distribution and density functions

19. Conditional and joint density functions
Conditional density function
Marginal density function
If X and Y are independent

20. Marginal Distribution

21. Conditional Distribution

22. The theoretical basis for time series models
A random process is a sequence of random variables indexed in time
A random process is fully described by defining the (infinite) joint probability distribution of the random process at all times

23. Random Processes A sequence of random variables indexed in time
Infinite joint probability distribution
xt+1 = g(xt, xt-1, …,) + random innovation
(errors or unknown random inputs)

24. From Sharma et al., 1997

25. Disaggregation
From Loucks et al., 1981

26. Nonparametric disaggregation
From Tarboton et al., 1998

27. Spectral density function
Spectral representation of a stationary random process
Autocorrelation
Autocorrelation function
Time Series
Fourier Transform
Fourier Transform
Spectral density function
Fourier coefficients
Smoothing

28. Spectral analysis gives us
Decomposition of process into dominant frequencies
Diagnosis and detection of periodicities and repeatable patterns
Capability to, through sampling from the spectrum, simulate a process with any S(w) and hence any Cov()
By comparison of input and output spectra infer aspects of the process based on which frequencies are attenuated and which propagate through

29. Limitations of Stochastic Hydrology
Stochastic Hydrology can not create more information than already exists in the available data. What it does do is provide a methodology to help decision makers find the answers to "what if" questions by providing simulated system behavior based on plausible sequences that mimic the records that are already available (Pegram, 1989)

30. Why Stochastic Hydrology
Because although future rainfalls and streamflows will most likely resemble the past in a broad sense (such as mean values, variability and intercorrelation) all future sequences are almost surely going to be different in detail from what has been observed in the past (Pegram, 1989)