Typical Water Resource System - Stochastic Hydrology - Lecture Notes, Study notes of Mathematical Statistics

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STOCHASTIC HYDROLOGY

Course Contents

• Introduction to Random Variables (RVs)
• Probability Distributions - One dimensional RVs
• Higher Dimensional RVs – Joint Distribution; Conditional Distribution; Independence
• Properties of Random Variables
• Parameter Estimation – Maximum Likelihood Method and Method of Moments
• Commonly Used Distributions in Hydrology
• Hydrologic Data Generation
• Introduction to Time Series Analysis
• Purely stochastic Models; Markov Processes

Effluent streamflow Reservoir

Typical Water Resource System

Irrigated Agriculture  Hydro-power River Non Point Source Pollution Groundwater Reservoir Recharge Rainfall Rainfall Base flow Pumping Catchment

Gauge-A Reservoir Streamflow Regulated flow Time (months) Flow (Mm^3 ) Mean flow Observed (historical) flows at Gauge - A Reservoir Design and Operation

• Medium term forecasts for hydropower/ irrigation/water supply,
• Short term forecasts for flood control

Stochastic Hydrology - Applications

History provides a valuable clue to the future

Real-time Flood Forecasting

TOWN A To forecast water levels at A , with sufficient lead time Water level at A: Function of rainfall in the catchment upstream, evaporation, infiltration, storage, vegetation and other catchment characteristics.

Stochastic Hydrology - Applications

Multi-reservoir systems

• Flood forecasting
• Intermediate catchment flows
• Long-term operation of the system

Stochastic Hydrology

• Applications

AET Rainfall Recharge^ GW Pumping Canal Recharge Release D/S Flow Inflow RESERVOIR IRRIGATED AREA AQUIFER Conjunctive use of surface and ground water Net outflow

Stochastic Hydrology - Applications

Water Quality in Streams

Non-point Source Pollution Governed by : Streamflow, Temperature, Hydraulic properties, Effluent discharges, Non-point source pollution, Reaction rates …..

Stochastic Hydrology - Applications

F Joint variation of flows in two or more streams F Urban floods

• Estimates of design rainfall intensity based on probability concepts F Spatial variation in aquifer parameters F Uncertainties introduced by climate change
• Likely changes in frequencies and magnitudes of floods & droughts.
• Likely changes in stream flow, precipitation patterns, recharge to ground water Flow Q 1

Stochastic Hydrology - Applications

Random Variable

Real-valued function defined on the sample space. Element s in the sample space Y(s) Y Sample space of an experiment Possible real values of Y Y is a Random Variable

• Intuitively, a random variable (RV) is a variable whose value cannot be known with certainty, until the RV actually takes on a value; In this sense, most hydrologic variables are random variables

Random Variable

• Any function of a random variable is also a random variable. - For example, if X is a r.v., then Z = g(X) is also r.v.
• Capital letters will be used for denoting r.v.s and small letters for the values they take e.g. X rainfall, x = 30 mm Y stream flow, y = 300 Cu.m.
• We define events on the r.v. e.g. X=30 ; a <Y <b
• We associate probabilities to occurrence of events
  • represented as P[X=30], P[a<Y<b] etc.

Discrete & Continuous R.V.s

• Discrete R.V.: Set of values a random variable can assume is finite (or countably infinite). - No. of rainy days in a month (0,1,2….30) - Time (no of years) between two flood events (1,2….)
• Continuous R.V.: If the set of values a random variable can assume is infinite (the r.v. can take on values on a continuous scale) - Amount of rainfall occurring in a day - Streamflow during a period - Flood peak over threshold - Temperature

Probability Distributions

Cumulative distribution function : discrete RV F(x) =Σ p(x i

xi < x x 1 x 2 x 3 x n x n-

........... . p(x 1 )

p(x 1 )+p(x 2 )

• P[X = x i

] = F(x

i

) – F(x

i-

• The r.v., being discrete, cannot take values other

than x

1

, x

2

, x

3

………. x

n

; P[X = x] = 0 for x ≠ x

1

x

2

, x

3

………. x

n

• Some times, it is advantageous to treat

continuous r.v.s as discrete rvs.

• e.g., we may discretise streamflow at a

location into a finite no. of class intervals and

associate probabilities of the streamflow

belonging to a given class