Simulations

Abstract

The natural behavior of earth systems is rather complex and their replicates can be generated by various stochastic models, which take into consideration deterministic and uncertainty components temporally, spatially, or spatio-temporally.

Appendices

Appendix

Figure 6.16 presents the output from the above MATLAB programs the observation sequence with the ensemble average generation data. It is visually obvious that the observation and generation time series have more or less similar statistical parameters; i.e., they are statistically indistinguishable.

Fig. 6.16

Observation and generation time series

Seasonal Generation Process

The first-order Markov synthetic time series simulation methodology assumes that the given time series has a stationarity structure, which means that the statistical parameters are constant over the whole time series duration. Examples for such time series are annual records of natural events that are not affected by the astronomical and seasonal effects. However, daily, weekly and monthly records shorter than annual durations each time series has inborn seasonal effects. The best example for such series is monthly records of natural phenomena such as hydro-meteorological events.

In the case of seasonality, the statistical parameters vary according to months as explained in Chap. 2, Sect. 2.8. Their simulations need a stochastic process with monthly arithmetic average and standard deviations as in Table 2.7 (Chap. 2). In the same section, the elimination of arithmetic average and the standard deviation procedure has been explained. In this manner, the monthly time series is converted to weak stationarity, whereby monthly arithmetic averages are equal to zero and the standard deviation is equal to one. However, this time series still have non-seasonality in correlation coefficient between two successive months, which is difficult to eliminate by simple statistical methodologies.

It is necessary to develop a simulation process, which is similar to first-order Markov process, but with variable statistical parameters and successive monthly correlation coefficients. In the following is the monthly counterpart of the stochastic model in Eq. (6.9) as follows.

$$X_{i} = \mu_{i} + \rho_{i,i - 1} \left( {X_{i - 1} - \mu_{i} } \right) + \sigma_{i} \sqrt {1 - \rho_{i,i - 1}^{2} } \epsilon_{i} \quad \left( {i = 1,2, \ldots ,12} \right)$$

(6.19)

where $X_{i}$ is the monthly value, $\mu_{i}$ and $\sigma_{i}$ are the monthly arithmetic average and the standard deviation, respectively, and $\rho_{i,i - 1}$ is the correlation coefficient between successive i and i – 1 months (i = 1, 2, …, 12), and finally, $\epsilon_{i}$ is an independent normal random variable that accords with the normal (Gaussian) PDF with zero mean and unit variance.

The MATLAB program of this monthly simulation stochastic process is given in the following box. In the hydro-meteorology literature, this model is referred to as the Thomas–Fiering stochastic process for monthly variable generations (Thomas and Fiering 1962).

In the following is another MATLAB language written program for generation of seasonal time series for simulation purpose.

On the other hand, Fig. 6.17 shows synthetic sequence generation of monthly rainfall amounts for 12 months, which are also generated by the aforementioned MATLAB programs.

Fig. 6.17

Monthly total rainfall simulation traces

In order to indicate that the monthly statistics (arithmetic averages, standard deviations, and successive monthly correlation coefficients), similarity between observed and generation values Fig. 6.18 is presented.

Fig. 6.18

Observation and generation sequences: a arithmetic averages, b standard deviations, c correlation coefficients