# Observing Meteorological Tides: Fifteen Years of Statistics in the Port of La Spezia (Italy)
## 3 Results
multiply \(\Delta p\) by \(J_{ph}\) (taking into account that when the pressure goes up the level falls and vice versa); this corresponds to suppose a linear dependence:  
\[\Delta h=J_{ph}*\ \Delta p\, \tag{2}\]  
as represented by the red straight line in Figure 7, whereas the black dots correspond to the measured pairs (\(\Delta p\), \(\Delta h\)) listed in Table 1.  
A better estimate of the relationship between atmospheric pressure and sea level gradients can be obtained by considering a non-linear law \(\Delta h=f(\Delta p)\); using the least squares method applied to the pairs (\(\Delta p\), \(\Delta h\)) the fixed degree polynomial can be extrapolated that best fits the data cloud (the hydrobarometric transfer factor becomes a sequence of coefficients); for example, assuming a cubic dependence of \(\Delta h\) on \(\Delta p\) and imposing the passage to the origin (because \(\Delta p=0\) implies \(\Delta h=0\)), this estimate of \(\Delta h\) is obtained:  
\[\Delta h=J_{ph3}*\ \Delta p^{3}+J_{ph2}*\ \Delta p^{2}+J_{ph1}*\Delta p\, \tag{3}\]  
where: \(J_{ph3}=0.001\ \mathrm{cm}*\ h\mathrm{Pa}^{-3}\), \(J_{ph2}=-0.05\ \mathrm{cm}*\ h\mathrm{Pa}^{-2}\), \(J_{ph1}=2.3\ \mathrm{cm}*\ h\mathrm{Pa}^{-1}\), as plotted in Figure 7 (green line). The couples (\(\Delta p\), \(\Delta h\)) measured from 2006 to 2021 in the port of La Spezia, linear and cubic approximations are shown in Figure 7, together with the comparison with the \"inverted barometer effect\" of the offshore case. Once the change in atmospheric pressure is measured or predicted, the expected sea level variation can be derived from the Equation (3). Obviously, the two different estimates (linear and cubic) lead to two different errors between the measured value and the estimated trends: starting from data in Table 1, an average error on the expected sea level (difference between forecast and measurement) of 11.3% was obtained in the case of linear approximation, while in the case of cubic function the mean error is 9.6% (against a greater computational load), which is satisfactory for our purposes. Therefore, the fundamental role of the hydrobarometric transfer factor consists in converting a variation of atmospheric pressure into a forecasted sea level change (meteoro  
Figure 7: Measured pairs (\(\Delta p\), \(\Delta h\)) from 2006 to 2021 in the port of La Spezia (black dots), linear (red) and cubic (green) trends and comparison with the offshore case (black dashed line). logical component) for a given port; finally, meteorological tides will have to be added (or subtracted) to the astronomical tides (predictable by means of tide charts) to obtain the forecasted sea level in that basin. Contributions to changes in sea level due to other causes such as, for example, seiches, storm surges or wind effects, which are not predominant in the site examined, are not taken into account in this work. It should be emphasized that the hydrobarometric transfer factor must be updated year after year and periodically recalculated, e.g., following events that modify the topography of the basin examined (dredging operations, coastal erosion, bottom subsidence, sediment deposition). Moreover, a multi-decade statistics is necessary to examine any changes in the hydrobarometric transfer factor due to climate change, as well as the variation over the years of occurrence frequency of events observed [46; 47].