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.