C&ENVENG 4109 / 7109 “Environmental Engineering Design IVB”
Dr Seth Westra
LECTURE 11: STOCHASTIC MODELLING FOR DROUGHT RISK ASSESSMENT
Stochastic: “a process involving a randomly determined sequence of observations each of which is considered as a sample of … a probability distribution.” – www.thefreedictionary.com
In hydrology, stochastic data are random numbers that are modified so that they have the same characteristics (e.g. mean, variance, skew, long-term persistence) as the historical data on which they are based. – www.toolkit.net.au/tools/SCL
WHAT IS A STOCHASTIC MODEL?
We cannot eliminate risk in engineered systems
The challenge is to understand risk, and reduce it to an acceptable level
WHY DO WE NEED STOCHASTIC MODELS?
WHAT IS AN ACCEPTABLE LEVEL OF RISK FOR A RESERVOIR?
Historical data provides only a single “realisation” of the past climate • Produces unreliable estimates of probabilities, particularly
when considering extreme events
We are designing for the future, not the past • Even without anthropogenic climate change, historical
data will not exactly mirror what we can expect in the future
THE CLIMATE AND WEATHER IS INHERENTLY UNCERTAIN
Deterministic Process: Given a set of initial conditions/inputs, the model produces only one possible series of outcomes
Example: Reservoir Storage Volume
WHAT IS A STOCHASTIC MODEL?
1t t t t t t tS S Q P D E O
10 tS C
Inflow
Rainfall Evaporation
Spill Demand
Storage Capacity
Storage Content
Assuming: Historical Inflows, Demand: 10,000 ML pa, no Rain/Evap from Reservoir, Capacity = 70,000 ML
Risk (<50% storage) = 7/68 years ~ 10%
DROUGHT RISK, CORRIN DAM
WHAT IS A STOCHASTIC MODEL?
Stochastic Process: Given a set of initial conditions/inputs there is a random component in the model that means there are many possible series of outcomes
Stochastic processes are a sequence of random variables, known as a stochastic time series
Historical inflows provide only one realisation of past climate => unreliable risk estimates
Stochastic model for inflows, provides multiple realisation of past climate => better estimates of risk
Stochastic Climate data
• Random numbers (stochastic time series models)
• Calibrated to have same statistical characteristics as historical data
Provides multiple time series of climate data
• Each time series is an alternative “realisation” of the climate that is equally likely to occur
WHAT IS STOCHASTIC CLIMATE DATA?
Use as input into models to quantify uncertainty due to climate variability
Hydrological models
Ecological models
Storage yield analysis
• Estimate reservoir size for a given demand and reliability,
• Estimate system reliability (number and levels of water restrictions) for a given storage size and demand
Water resources models (like REALM and IQQM) to estimate system reliability (e.g., water allocation amounts for competing users) for alternative allocation rules and management practice.
WHAT IS STOCHASTIC CLIMATE DATA USED FOR?
STOCHASTIC MODELLING: PROCESS
Input historical data Input historical data
Select, calibrate and evaluate model
Simulate replicates
System Response Model -Water Resource - Hydrological - Ecological Model
A (VERY) SIMPLE STOCHASTIC GENERATOR
Probably the simplest possible daily stochastic generator works as follows: • Simulate rainfall occurrences using a 1st order Markov chain,
requiring two parameters: pdd (dry-dry probability) and pwd (wet-dry probability)
• Simulate amounts on wet days through a one-parameter exponential distribution (𝜆 = 1/𝑥 ) where 𝑥 is the average of wet-day precipitation.
Can you estimate the parameters for the rainfall data on the left?
…
WHAT ARE SOME LIMITATIONS OF THIS MODEL?
For Bartlett-Lewis process, • ‘storm’ arrivals follow
a Poisson process, • ‘cells’ arrives follow a
Poisson process. • the ‘duration’ of
storms are described by an exponential distribution
• cell ‘depth’ and ‘duration’ described by an exponential distribution
Neyman-Scott process is similar – a few different assumptions, parameters.
POISSON CLUSTER MODELS
You will be using the stochastic generator in Source, which is based on that contained in the Stochastic Climate Library (http://www.toolkit.net.au/SCL)
…AND FOR YOUR ASSIGNMENT…
SCL contains numerous models, and theory is complex
As a user, you must evaluate the quality of the stochastic replicates—do they reasonably simulate observed rainfall variability?
Need to select evaluation statistics that are suitable for the application: • Drought Risk: long-term statistics such as annual/monthly mean, standard
deviation, skew, lag-one autocorrelation, min, max, lowest/highest 3-5 year sums
• Flood Risk: short-term statistics such as % dry days, wet/dry period duration, mean, standard deviation of rain days, low exceedance probability events (e.g. 1%ile rain days)
USING SCL
EVALUATING THE REPLICATES
Typically would simulate multiple replicates, calculate the statistic of interest for each replicate, and see if the observed value fits within the generated range.