Stochastic Modelling

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Stochastic Modelling

• Previously Deterministic Modelling
• simulated variables are uniquely determined by
  the model parameters and the input variables

• many hydrological phenomena in which the variable
  of interest cannot be uniquely specified as a
  function of known related variables and
  conditions
• e.g maximum discharge in the next 5 years

• This sort of long term prediction is often
  required by civil engineers

• accept that there is a random element to these
  processes and apply statistical techniques

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What causes the random element?

• inherent unexplainable variability of nature
• Weather

• lack of understanding of all the causes and
  effects in a physical system
• Flow through highly fractured rock

• lack of sufficient data
• Flow record but nothing else

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Random variables

• a numerical variable not subject to precise
  prediction
• description of the random variable is then
  accomplished through the concept of probability
  distributions
• determine the possibility of occurrence of
  particular events and determine the likelihood of
  their occurrence

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Intuition
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Probability

• Let A be an event is the result of an experiment
  (e.g. a given streamflow in nature). If the
  experiment is carried out m times (e.g. number of
  observations of flow made) then

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River in the North West of Scotland
Stationarity? Based on this flow
record P(flowgt220 in any given year)
1/14
Assumption in most stochastic models is that
this probability will not change and hence you
can extrapolate into the future gt Prob of the
flow exceeding 220 in 2010 will also be 1/14
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Probability Density Function fitted to first 14
years of data
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Continuous Random Variable

• the probability that a continuous random variable
  X falls between x and xdx is given by fX(x)dx,
  where fX(x) is called the probability density
  function (pdf) of X

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Discrete random variables

• Countable e.g. Number of rainy days
• The function PX(xi), which gives the probability
  of the discrete random variable X taking the
  value xi is called the probability mass function

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Where are stochastic models used most?

• Determining the probability of rare high and low
  flow events.
• Flood estimation handbook
• Micro-lowflows
• Determining the probability of rare rainfall
  events.
• Flood estimation handbook
• Micro-lowflows
• Stochastic rainfall modelling to generate
  synthetic realisations of rainfall (climate
  change, extending rainfall records)

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Flood frequency

• The aim is to describe the probability of a flow
  Q equalling or exceeded some arbitrary flow rate
  q in any year.

boils down to finding a mathematical expression
to describe it cumulative distribution function
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1. Create a series of annual maxima
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flow (cumecs)
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year (starting Jan 1976)
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2. Rank Them
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Software

• Flood estimation handbook

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Stochastic Rainfall Modelling.

• Empirical statistical models in which the
  distribution of, say, daily rainfall is
  represented by an empirical equation (i.e. you
  fit the best distribution you can to historical
  data)
• Stochastic models that use a limited number of
  parameters to represent rainfall process, the
  parameters being intended to relate to the
  underlying physical phenomena. Can generate
  synthetic realisations of rainfall series.

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Why generate synthetic rainfall?

• Many hydrological processes are a function of
  rainfall intensity and persistence. (e.g.
  infiltration)
• If you only have a few years of rainfall data
  with which to calibrate you model and you want to
  make inferences about rare events then it is
  necessary to generate a longer rainfall sequence.
• GCMs are good at long term predictions of climate
  variables but poor at weather
• Can relate the parameters of the rainfall
  generator to GCM output then you can generate
  possible rainfall scenarios for future climates.
  These can be used as inputs to a hydrological
  model for assessing possible impacts of climate
  change on water resources.

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General Circulation Models
Cloud Physics Mass and Energy Fluxes
Hydrology
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Rainfall Spatial Averaged over a large area

• Problem hydrological processes such as
  infiltration saturation excess flow depend on
  short term dynamics (i.e. intensity and duration
  of rainfall)
• Relate CGM output to short term rainfall
  statistics
• Generate a realisation of rainfall for future
  climate based on these short term statistics.

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Two Rainfall Generators

• 2 state Markov Chain
• Bartlett-Lewis Model

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• Characterise the likelihood of it being a wet or
  dry day (discrete random variable)
• Given that it is wet characterise the probability
  of a particular depth of rainfall occurring

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Characterising prob of wet or dry
Probability of tomorrow being wet or dry is given
by
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Random Sampling

• We know what the probability of tomorrow being
  wet is but how do we decide whether to make it a
  wet day or a dry day?
• Cant say for certain have to introduce an
  element of uncertainty. Randomly select a number
  lying between 0 and 1 assuming that we are
  equally likely to select any number.
• If the number is less than Pw then assign the day
  as being wet.

F(x)
1
x
0
w
d
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Procedure

• For existing rainfall record count
• number of days that a wet day follows a wet day
• divide by number of wet days to get pww
• pwd 1- pww
• number of days a dry day follows a dry day
• divide by number of dry days to get pdd
• pdw 1- pdd
• This defines the transition matrix
• Assume day one is either wet or dry
• If wet then (Pw1,Pd0) and prob of wet and dry
  days tomorrow (pww,pwd)_
• If dry then (Pw0,Pd1) and prob of wet and dry
  days tomorrow (pdw,pdd)_
• Prob of wet and dry days tomorrow (pww,pwd)_
• Select a random number, r, between 0 and1
• if r lt pww then make tomorrow wet (Pw1,Pd0)
• if r gt pdw then make tomorrow dry (Pw0,Pd1)

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Now know whether each day is wet or dry but we
still need to assign a rainfall depth

• Amount of rainfall falling on a wet day has been
  shown to be well represented by an exponential
  distribution
• Let R be the rainfall depth

• Need to devise a method for selecting a rainfall
  depth such that if you select often enough youd
  get an exponential pdf

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Cumulative
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Two state Markov Model

• Wet and dry days decided by a markov chain model
• Rainfall amount by an exponential distribution
• Expected to know how you construct the transition
  probability matrix and how you apply it.
• How you fit an exponential distribution to a
  rainfall data set
• How you sample from an exponential distribution

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Spatial realization of raincells
Cold Front
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X raingauge
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Bartlett_Lewis Stochastic Rainfall Model
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Multiple cell types
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Fitted Statistics

Mean

Variance

Autocovariance at a given lag

Probability of an h hour dry period

Dry/dry and wet/wet transition probabilities

Covariance between gauges

Skewness
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Advantages of this type of model

• Capable of differentiating between different rain
  cell types
• Poisson process is much better at simulating the
  persistence of wet and dry spells
• The fact that it can differentiate between storm
  type and the cell type means that it is easier to
  correlate with climate variables. E.g. Wind
  rotating anticlockwise round a low pressure bring
  storms in from the Atlantic. gt you can derive a
  set of rainfall model parameters for each storm
  type
• GCM can predict changing weather patterns i.e.
  increased frequency of low pressures gt allows
  you to change your model parameters sensibly in
  line with GCM predictions for a future climate.