2 Stochastic generation of monthly and annual precipitation
Generation of monthly and annual rainfall data is important for water resources systems and the estimation of water yield from large catchments. One can suggest to aggregate the generated daily rainfall sequences, detailed in the previous chapter, to monthly and annual rainfall values. However, in doing so the model does not take into consideration the monthly and annual characteristics.
To address this problem, Srikanthan (2005, 2006, 2009) proposed a nested model, an extension of the two-part model for monthly and annual generation for both single and multiple sites, that ensures the monthly and annual covariances are preserved.
2.1 Single site monthly and annual models
2.1.1 Single site monthly model
As previously mentioned, the generated rainfall amounts when aggregated into monthly totals will not, in general, preserve the monthly characteristics. Thus, the daily amounts model was nested and modified by Srikanthan (2005, 2006) in a single site monthly model to improve the monthly at-site characteristics of the generated rainfall data.
Once the daily rainfall amounts at station k are generated for a given month, the monthly rainfall totals, ai(k), are obtained by summing the daily rainfall values. Then the monthly rainfall totals are modified by a lag-one autoregressive monthly model.
Let xi(k) denote the standardized stationary random process of monthly rainfalls that pertains to the kth station on month i having zero mean and unit variance. Therefore, using a lag-one autoregressive process, we have
xi(k) = ci(k)xi−1(k) + di(k)si (2.1)
where ci(k) and di(k) are coefficient numbers defining the temporal correlations between monthly total values, si is a random component that has zero mean and unit variance and is independent of xi−1(k) and n is the number of years.
The coefficients ci(k) and di(k) are defined for each month because each month is considered separately in the model in order to preserve the seasonal characteristics.
If both sides of Equation 2.1 are multiplied by xi−1(k) such us
xi(k)xi−1(k) = ci(k)x2
Applying the expectation leads to
E[xi(k)xi−1(k)] = ci(k)E[x2
(k) + di(k)sixi−1(k) (2.2)
(k)] + di(k)E[sixi−1(k)]
⇐⇒ ci(k) = E[xi(k)xi−1(k)] = ri(k) (2.3) where ri(k) is the lag-one autocorrelation coefficient for station k on month i. Hence the Equation 2.1 becomes
xi(k) = ri(k)xi−1(k) + di(k)si. (2.4)
If both sides of Equation 2.4 are multiplied by xi(k) such us
xi(k)2 = ri(k)xi−1(k)xi(k) + di(k)sixi(k) (2.5) Again, applying the expectation leads to
E[xi(k)2] = ri(k)E[xi−1(k)xi(k)] + di(k)E[sixi(k)]
⇐⇒ 1 = ri(k)E[xi−1(k)xi(k)] + di(k)E[si(ri(k)xi−1(k) + di(k)si)]
⇐⇒ 1 = ri(k)E[xi−1(k)xi(k)] + di(k)ri(k)E[sixi−1(k)] + di (k)E[si ]
⇐⇒ 1 = ri(k)E[xi−1(k)xi(k)] + di (k)
⇐⇒ 1 = ri (k) + di (k)
di(k) = .1 − r2(k) (2.6)
Therefore, the equation 2.1 becomes
xi(k) = ri(k)xi−1(k) + .1 − r2(k) si (2.7)
Srikanthan (2004) proposed to replace the white noise in Equation
2.7 by aˆi(k), the standardized generated monthly totals having zero mean and unit variance. Thus the adjusted value, xi(k), is obtained from
x˜i(k) = rˆi(k)x˜i−1(k) + 1 − rˆ2(k) aˆi(k) (2.8)
i = 2, 3, …, 12n, with x1(k) = aˆ1(k).
where x˜i(k) is a realization of the random process xi(k) and rˆi(k) is the observed lag one autocorrelation coefficient.
2.1.2 Single site annual model
As was the case for monthly rainfall, a similar method is used to nest the generated monthly rainfall into annual sequences.
Having generating the rainfall amounts at station k for a given year, using the nested model, the annual rainfall totals, bj(k), are obtained by aggregating the monthly precipitation values and the annual rainfall totals are also modified using a lag-one autoregressive model.
Denote y˜j(k) the standardized precipitation amounts at station k on year j having zero mean and unit variance. Using a lag-one autoregressive process, we have y˜j(k) = ρˆ(k)y˜j−1(k) + √1 − ρˆ(k)2 ˆbj(k) (2.9)
where ρˆ(k) is the observed lag-one autocorrelation coefficient for station k, ˆbj(k) is the standardized aggregated annual values having zero mean and unit variance and n is the number of years.
2.2 Multisite monthly and annual models
2.2.1 Multisite monthly model
As was the case with single site monthly precipitation amounts. Having simulating the daily precipitation amounts at all sites, using the multisite two-part model, for a given month, the monthly precipitation totals, ai(k), at each location are obtained by aggregating the daily rainfall values. Then those monthly totals are standardized by their means and standard deviations and collected in vectors aˆi.
Those standardized, aggregated monthly values are modified by using a nested multisite model (Srikanthan and Pegram 2007) to preserve the monthly spatial and serial correlations.
Let Xi be the stationary vector random process of the standardized adjusted monthly rainfall (zero mean and unit variance) at K station on month i, such that
Using a lag-one autoregressive process, we have
Xi = AiXi−1 + Biaˆi with X1 = aˆ1 (2.11)
where Ai and Bi are coefficient matrices defining the temporal and spatial correlations between monthly totals for month i, which are calculated from the lag-zero, M0, and lag-one, M1, cross-correlation of the observed monthly rainfall and the lag-zero, P0, cross correlation of the standardized and aggregated monthly values coefficient matrices Ai and Bi are derived as follows.
The series aˆi may itself be spatially and serially correlated and it can be represented by a lag-one autoregressive process such as
aˆi = Eiaˆi−1 + Fiλi (2.12)
where Ei and Fi are coefficient matrices which preserve the lag-zero
and-lag one cross-correlations in aˆi and λi is a vector of mutually
independent standardized variates.
Equation 2.11 is equivalent to
aˆi−1 = Bi−1 Xi−1 − Bi−1 Ai−1Xi−2. (2.13)
Applying Equations 2.12 and 2.13 successively in Equation 2.11 results in
Xi = AiXi−1 + BiEiBi−1 Xi−1 − BiEiBi−1 Ai−1Xi−2 + BiFiλi (2.14)
Xi = [Ai + BiEiBi−1 ]Xi−1 − BiEiBi−1 Ai−1Xi−2 + BiFiλi (2.15)
Assuming that aˆi is spatially correlated but serially uncorrelated, so that Ei = 0 for every month i, then
aˆi = Fiλi. (2.16)
And Equation 2.15 reduces to
Xi = AiXi−1 + Giλi where Gi = BiFi. (2.17)
If both sides of Equation 2.17 are multiplied by XT
, the transpose
of Xi−1, such that
applying the expectation leads to
E[XiXT ] = AiE[Xi−1XT ] + GiE[λiXT ] (2.19)
⇐⇒ M1 = AiM0 (2.20)
Ai = M1M0−1
where M0 and M1 are, respectively, the lag-zero and lag-one crosscorrelation matrices of the vector random process Xi.
Thus the Equation 2.17 becomes
Xi = M1M0−1Xi−1 + Giλi (2.22)
If both sides of Equation 2.22 are multiplied by XT , such that
= M1M −1Xi−1XT
again, applying the expectation leads to
E[XiXT ] = M1M −1E[Xi−1XT ] + GiE[λiXT ]
⇐⇒ M0 = M1M −1E[Xi−1XT ] + GiE[λi(M1M −1Xi−1 + Giλi)T ]
⇐⇒ M0 = M1M −1E[Xi−1XT ] + GiE[λi(M1M −1Xi−1)T ]
0 i 0
+ GiE[λi(Giλi)T ]
= M M −1E[X
XT ] + G E[λ XT
](M −1)T MT
+ GiE[λiλT ]GT
⇐⇒ M0 = M1M −1MT + GiGT
= M0 − M1M −1MT
Combining Equation 2.16 with 2.17 gives
Xi = AiXi−1 + GiFi−1aˆi (2.25) The coefficient matrix Fi is obtained from
= P0 (2.26)
where P0 is the cross correlation matrix of aˆi. The matrices Fi and Gi can be calculated using singular value decomposition.
2.2.2 Multisite annual model
Once the monthly precipitation time series are generated at each location for a given year, using multisite monthly model, a similar method can be used to nest the generated monthly rainfall into a multisite annual model.
At first, the monthly rainfall amounts at each location are aggregated into annual precipitation totals, bj(k). Then those annual totals are standardized by their means and standard deviations and collected into vectors ˆbj, which will be modified using a nested multisite annual model to preserve the annual spatial and serial correlations.
Denote Yj the vector of the standardized adjusted annual rainfall at K station on year j, such that
Using a lag-one autoregressive model, Yj is obtained from
Yj = CYj−1 + Dˆbj (2.28)
with Y1 = ˆb1
where C and D are coefficient matrices which are calculated from the lag-one and lag-zero cross-correlation of the observed annual rainfall and lag-zero cross correlation of the standardized and aggregated, already generated, annual values.