Wind Time-Series Synthesis for Lagoon Simulation

Wind is the prime mover in shallow water, such as the Orbetello lagoon on the Tyrrhenian coast of central Italy (see the inset in Figure 3.22). In the development of a comprehensive lagoon water quality model (Giusti and Marsili-Libelli, 2006), the wind modelling was included as an input gen-erator for long-term simulation to understand the role played by the wind in determining the water movement and the dispersal of seeds, the basis for submerged vegetation expansion.

(a)

Isolate the deterministic part of experimental time-series by smoothing

(b)

Fit a Gaussian distribution to the histogram (Lilliefors test: OK)

(c)

Compute the autocorrelation and check for periodic components (in this case 1 h)

Resid. autocorr.

FIGURE 3.20 The four steps in constructing a synthetic time series. First, isolate the deterministic trend (a), then analyse the residuals (b–d) to generate more random patterns which can be summed to the basic deter-ministic trend (a) to produce an endless number of combinations.

0 100 200 300 400 500 600 50

100 150 200 250 300 350 400 450

500 Try to smooth out the joining

point, though noise may mask the effect

Time (h)

Flow (m3/h)

FIGURE 3.21 Two synthetic time-series modules are joined together to produce a longer record of synthetic data. There may be a joining problem that could be avoided by careful matching the initial and final deriva-tives, though the noise may partially mask the discontinuity.

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90

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180 0

270 300

20 40 6080 30

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240 270 90

120 150 330

180 0

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Orbetello lagoon

FIGURE 3.22 Wind statistics observed in the two ponds of the Orbetello lagoon. (Reproduced with permis-sion from Giusti, E. and Marsili-Libelli, S., Ecol. Model., 196, 379, 2006.)

The model was based on three years (2001–2004) of data acquired by two fixed stations placed in the middle of the two ponds that form the Orbetello lagoon. From those data, it was observed that the wind distribution in the two basins is substantially different, given the differing shield-ing provided by the Monte Argentario promontory on the sea side. Therefore, two wind models were set up, one for each pond, with correlated time series. The available data were used to form the reference daily sequence. These data were originally sampled at 10 min intervals, with about 20% of missing data. To fill the gaps, the most frequent value in the corresponding time slot was considered. Then, a synthetic daily value was extracted by considering the prevailing wind of the day as the one blowing for at least three consecutive hours with a constant direction and a speed of at least 0.5 m/s, which was assumed as the minimum wind speed required to produce a water movement. Several synthetic time series were obtained from these reference data by the following procedure:

1. The deterministic trend was obtained by spline smoothing.

2. The autocorrelation of the residuals was computed to determine the order of the AR model producing the stochastic component of the series.

3. The synthetic sequence was then obtained by adding the residuals generated by the AR models to the deterministic trend data.

Further, because a correlation was observed between the same-day wind speeds in the two basins, a correlated noise was used to drive the AR models and conserve this feature in the synthetic data.

The observed wind series were then decomposed as shown in Figure 3.23, with the smoothed part forming the deterministic backbone of the series and the stochastic part being further analysed for autocorrelation and normal probability density function.

In fact, the synthesized stochastic residuals should have the same autocorrelation as their observed counterparts. The order of the stochastic model was assumed equal to the first non-zero residual, that is, 3, as shown in Figure 3.24, which compares the autocorrelation of the original and synthetic random time series, whereas the Gaussian distributions estimated from the wind samples are shown in Figure 3.25.

0 90 180 270 360

0 5 10

Days

wW (m/s)

Original time-series Smoothed time-series

Deterministic pattern

Stochastic part

Smoothed time-series Residuals

Autocorrelation Model order↓

Normality Forcing function↓

Time-series synthesis

Stochastic model

FIGURE 3.23 Extracting information from the observed wind data to construct a synthetic time series.

0 2 4 6 8 10

−0.5 0.0 0.5 1.0

Lag (d)

0 2 4 6 8 10

Lag (d)

0 2 4 6 8 10

Lag (d)

0 2 4 6 8 10

Lag (d) Original time-series

Synthetic time-series

Original time-series Synthetic time-series

Original time-series Synthetic time-series

Original time-series Synthetic time-series

Residual αE autocorr.

−0.5 0.0 0.5 1.0

Residual wE autocorr.

−0.5 0.0 0.5 1.0

Residual wW autocorr.

−0.5 0.0 0.5 1.0

Residual αW autocorr.

FIGURE 3.24 Autocorrelation of the observed and synthetic wind time series. (Reproduced with permission from Giusti, E. and Marsili-Libelli, S., Ecol. Model., 196, 379, 2006.)

−4 −2 0 2 4

−3 −2 −1 0 1 2 3

Residual w_E (m/s)

Residual w_W (m/s)

−100 −50 0 50 100

0.0 0.1 0.2

Rel. freq. (1/d)

0.0 0.1 0.2

Rel. freq. (1/d)

0.0 0.1 0.2

Rel. freq. (1/d)

0.0 0.1 0.2

Rel. freq. (1/d)

Residual α_E (°)

−100 −50 0 50 100

Residual α_W (°)

FIGURE 3.25 Fitting the experimental wind histograms obtained from the smoothed time series. The normal distribution hypothesis passed the Kolmorogov–Smirnov test. (Reproduced with permission from Giusti, E. and Marsili-Libelli, S., Ecol. Model., 196, 379, 2006.)

The correlated AR models for the wind direction and speed in the two basins were estimated from the data using the Yule–Walker method (Friedlander and Porat, 1984), which minimizes the forward least-squares prediction error based on the estimated autocorrelation function. The following four AR models were obtained for wind angle and speed in the two East and West ponds:

The speed time series are driven by two independent normally distributed uncorrelated noise sources e tE

( )

and e tW

( )

with standard deviations σE and σW. To reproduce the observed same-day speed correlation between the two basins, a driving function was constructed combining the two noises with the correlation coefficient v = 0.732. This provided consistently related time series with a correlation very close to the observed value of 0.7. The synthetic time series were obtained by adding the smoothed deterministic part (trend data) to the pertinent stochastic model, to obtain

Angle East

A sample of a synthetic time series derived by the above method is shown in Figure 3.26, where it is compared to the original data.

3.4 WAVELET SIGNAL PROCESSING: ADAPTIVE COMBINATION

In document Marsili-Libelli, Stefano-Environmental Systems Analysis with MATLAB®-CRC Press LLC (2016) (Page 181-185)