SYNTHETIC STREAM FLOW

16.3 SYNTHETIC STREAM FLOW

The probability of occurrence of floods or droughts are more severe than that observed from the available stream flow records has to be known. On the assumption that the streamflow is essentially a random variable, it is possible to develop a synthetic flow record by statistical methods.

It has been found that high flows are likely to follow high flows and the low flows follow low flows, i.e., any event is dependent on the preceding event. This persistence is measured by

HYDROLOGY

a serial lag coefficient. The lag interval may be one or several time units. A simple one lag Markov generating equation for annual flows Q is

...(16.24) where σ = standard deviation of Q

Q i = Q +r i (Q i –1 – Q )+ε i σ 1 −r 2 i

Q = mean of Q

i = 1 year to n year (flows in series) r i = lag-1 Markov coefficient, which is a portion of the departure from the previous flow

from the mean. The Eq. (16.25) yields a normal synthetic flow that preserves the maximum variance,

and first-order-correlation coefficient of the observed record. Statistical streamflow models are assumed stationary, i.e., the mean and variance of the observations (time series) are un- changed with time.

Thomas and Fiering (1962) used the Markov Chain model for generating monthly flows (by serial correlation of monthly flows) by using the following recursion equation.

...(16.25) where q i ,q i +1 = discharges in the i and i + 1 months, respectively q j , q i +1 = mean monthly discharges in the j and j + 1 months of the annual cycle j b = regression coefficient for estimating the discharge in the j + 1 month from

that in the j month ε i = a random normal deviate at time i with a zero mean and unit variance. j +1 σ = standard deviation of discharges in the j + 1 month j = correlation coefficient between the discharges in the j and j + 1 months. r The Eq. (16.25) is called ‘Lag-one single period Markov Chain Model, where the period

may be day, month or year. To reflect different seasonal or monthly means, the multi-period Markov model is used,

which requires a double indexing subscript as (using Q for annual flows)

Q = i, j Q j +b j (Q i –1, j– 1 – Q j −1 )+ε i σ j 1 2 −r j

HG ...(16.26a) σ j − 1 KJ

where b j =r j , since Q j +1 ≠ Q j

j = number of seasonal periods or months in the year and other terms have been defined in Eq. (16.25) in which q = Q for annual flows.

This model has been used extensively in stream flow analysis. The single-and multi- period Markov generation procedures sometimes result in negative flows. These flows must be retained for generating the next flows in sequence and then they may be discarded.

The procedure assumes that the discharges (or their transform) are normally distrib- uted. A synthetic flow record generated like this can be of any desired length and may well include flow sequences more critical than any in the available observed record. Stochastic analysis can be used to generate a number of synthetic-flow traces of length equal to the ex- pected useful life of project under study.

MATHEMATICAL MODELS IN HYDROLOGY

Stochastic methods may be employed to generate a synthetic record of rainfall, which could be transformed to streamflow (by doing a particular operation), which the available streamflow records are found too short for a stochastic generation. The Markov process for generating a sequence of rainfall data is given by the relation

. ..(16.26b) which expresses the conditional probability of ‘transitioning’ from the state i at period t to

P (X t+ 1 = j|X t = i)

state j at (t + 1). Matalas (1967) used a simple matrix representation of the problem, similar to the

Markovian model, as

...(16.27) where X t = n × 1 matrix representing the non-autocorrelated standardised flow at n stations at time t

X t +1 = AX t + Bε t +1

X t +1 = a similar matrix as above at time (t + 1) ε t +1 = n × 1 matrix of non-autocorrelated random numbers with zero mean and unit

variance

A , B = n × n matrices, the elements being so chosen as to preserve time mean q , vari- ance σ 2 , lag 1-serial correlation coefficient r and the cross correlation coefficients r nm between all the data elements being modelled. Data generated by models can be used to design reservoir capacity by using low-flow- frequency mass diagram and other techniques. Several hundred years of records are gener- ated to obtain an adequate number of high-and low-flow sequences.