**3.2 Hindcast method: the case of Manfredonia**

This section aims at showing an example of the application of the hindcast method to a POI located in the south of Italy, in the Manfredonia Gulf (41.38°N, 15.55°E). When the wind comes from south or south-east rivers, flow is influenced by the downstream boundary condition, and several areas are flooded. These events cause damage to economic activities, private houses, etc. Moreover, the coastal area on the Gulf suffers from erosion. Although standard protection structures (e.g. [52, 53]) have been rolled out, erosion problems are still unresolved.

In this scenario, the utility of having a tool to perform hazard analysis is clear [3].

As observed, this method relies on the same mathematical hypotheses of the forecast one using a different statistical approach to correct the raw data coming from the dynamical step.

The response functions of the basin have been evaluated in the same way as described in Section 3.1 and extracted in a different point of interest (see **Figure 1**).

Following the description in Section 2.3, to apply the hindcast method, a set of data has been known. In the present case, the wind and atmospheric pressure data are referred to the ERA-Interim database (European Centre for Medium-Range Weather Forecasts (e.g. [54]). The spatial resolution of ERA-Interim data is 0.75°, while the response function of the basin, as described above, has another resolution equal to 3<sup>0</sup> . To overcome this problem, the values of wind and pressure (acting at the centre of each area) have been evaluated performing a linear interpolation.

The tidal data are those collected by means of the tidal gauge station owned by the Apulia Region Meteomarine Network (also referred to as SIMOP, e.g. [55]). It collects wave, wind and tidal data along the Apulian coasts [56]. This station does not gather the measures of atmospheric pressure. Then, in order to compute the term related to pressure gradients, this data have been taken in two locations near Manfredonia: Vieste and Bari (see **Figure 4**) where two tidal gauges of the National Mareographic Network are installed.

### **Figure 4.**

*Sketch of the study area. Circles indicate the point of interest of Manfredonia and the locations mentioned in the paper.*

As for the case of Venice, the raw level can be evaluated using a simplified version of Eq. (4) considering the projection of the wind stress along the principal orientation of the basin (i.e. ≃324°N). The modified equation reads as

$$\eta(t\_k) = \sum\_{i}^{N} \sum\_{j=1}^{j \le M} \mathcal{W}\_{ij} \mathcal{F}\_i^W(t\_k - j\Delta t + \Delta t) + \mathcal{C}\_p \Delta p(t\_k) \tag{9}$$

where *Wij* is the projection of the wind stress impulse along the principal orientation of the basin and Δ*p t*ð Þ*<sup>k</sup>* is the pressure anomaly.

The coefficient *Cp* is the correlation factor between the residual levels estimated by means of the dynamic approach and the related measured pressure anomalies (i.e. pressure and total tide level are mandatory). In this case, as previously declared, the pressure measures are referred to as those acquired in the mareographic stations in Bari and Vieste.

Due to the proximity of the stations to the POI, as might be expected, pressure measurements are in agreement. For a more detailed description, the reader may refer to [39]. It is possible to reach the same conclusion considering the quantiles of the measures in Vieste and Bari (see **Figure 5**) and comparing the quantiles of the pressure extracted from the ERA-Interim database at the centre of area 12 (that is the area the POI belongs to, see **Figure 1**) against the quantiles of the pressure observed at Bari (*PBARI*, left panel) and at Vieste (*PVIESTE*, right panel). **Figure 6** shows the results of the comparison between the quantiles of the ERA-Interim data and observations.

Based on these outcomes, arguing that the field pressure is almost the same in the area between Bari and Vieste, for Manfredonia, a value of *Cp* equal to 0.905 (the average value estimated for Bari and Vieste) has been considered.

**Figure 5.**

*Quantiles of the measured pressure at Bari (PBARI) and Vieste (PVIESTE). The dashed line refers to the perfect fit line.*

*Simplified Methods for Storm Surge Forecast and Hindcast in Semi-Enclosed Basins: A Review DOI: http://dx.doi.org/10.5772/intechopen.92171*

**Figure 6.**

*Comparison between quantiles of the ERA-Interim data and observations at the centre of area 12 against the pressure observed at Bari (PBARI, left panel) and Vieste (PVIESTE, right panel). The perfect fitting line is in marked in grey.*

A total of 39 years (from 1979 to 2017) of residual tide levels have been reconstructed by means of Eq. (9). Also in this case, although Eq. (9) considers also the pressure anomalies (i.e. using the term *Cp*Δ*p t*ð Þ*<sup>k</sup>* ), results strongly depend on the reliability of the selected reanalysis data. In the presented application, ERA-Interim data have been used. As underlined by [57], this database tends to underestimates the hindcast time series. Therefore, also in this case, a statistical correction must be made. Considering that the main aim of this method is to build hindcast time series to be used for return level estimation (i.e. correct hindcast of extreme values, see Section 2.3), the calibration coefficient was evaluated considering the population of the random variable *Ccal* given by Eq. (5). The selection has been made by matching the quantiles of the probability density functions of the hindcast and observed extreme values.

The extreme extraction has been performed by means of a peak over threshold (POT) analysis. The obtained data have been used to define the generalized Pareto distribution (GPD) (e.g. [58]). The threshold selection has been made following the standard technique proposed by [58].

Varying the calibration coefficient *Ccal* ranging from 1.0 up to 2.0, the return levels of the hindcast time series and of the observed values (Xr*M*) have been computed. Measured data show a threshold equal to 0.10 m with 162 values exceeding the threshold, while the estimated GPD parameters are *ξ* ¼ 0*:*22 (shape parameter) and *σ* ¼ 0*:*03 (scale parameter). The calibration coefficient has been obtained varying the ratio Xr*H*/Xr*<sup>M</sup>* as a function of the calibration coefficient (*Ccal*). Taking into account a *Ccal* ¼ 1*:*24 � 0*:*04, the fraction Xr*H*/Xr*<sup>M</sup>* approaches to 1. This means that the corrected hindcast time series (by means of *Ccal*) shows equal values to those evaluated on the basis of observed time series.

In order to gain insight on the ECDF of the observed and hindcast extreme values, the Q-Q plots for the uncorrected series (i.e. *Ccal* = 1) and with a correction equal to 1.28 (see **Figure 7**) have been evaluated. **Figure 7** shows the results and exhibits the usefulness of using the calibration coefficient in improving the reliability of the hindcast.

In addition, the root-mean-square error (*RMSE*) the *Bias*, the correlation coefficient (*R*), the index of agreement (*d*) and the Nash-Sutcliffe efficiency coefficient (*NSE*) have been calculated on the sample of the quantiles of the ECDF of the hindcast and observed extreme values. The *RMSE*, the *Bias* and *R* are commonly

**Figure 7.**

*Q-Q plots for the uncorrected series (black triangle) and with a correction equal to 1.28 (grey circles) of the ECDF quantiles of the extracted extreme values.*

used in the literature (e.g. [59–61]), while *d* and *NSE* are less. The index of agreement (e.g. [62]) measures the model error and varies between 0 (no accordance) and 1 (perfect agreement). Instead, the Sutcliffe efficiency coefficient is widely used to assess the goodness of a fit (e.g. [63]), and its values range from �∞ up to 1 (perfect agreement).

These statistical indicators have been calculated considering the corrected (*Ccal* = 1.28) and uncorrected (*Ccal* = 1) hindcast data.

Results show that there is a moderate increase in the reliability of the corrected data (by means of the calibration coefficient). More in details, the *RMSE* ranged from 0.020 to 0.010, the *Bias* has varied from �0.012 to 0.007, *R* varied from 0.987 to 0.991, *d* changed from 0.952 to 0.990, and *NSE* has varied from 0.853 to 0.960. Remembering the aim of the method, however, the same indexes have been evaluated for the quantiles greater than 0.15 m. In this case, the field contains real extreme values, and the importance of the correction is emphasized (*RMSE*: 0.030 ! 0.012, *Bias*: �0.025 ! 0.009, *R*: 0.975 ! 0.982, *d*: 0.849 ! 0.977, *NSE*: 0.518 ! 0.912).
