For decades, several hydrological variables as the rainfall depths were measured on a calendar day basis, usually at 09:00 a.m. local time (Morbidelli et al., 2021). In Italy, those daily measurements are published in the Hydrological Yearbooks, a collection of data usually available only in printed format. The availability of shorter duration records was limited by the requirement of continuous monitoring rainfall gauges. After the replacement of the mechanical rain gauges with the automatic ones in the last decades of the 20th century, the reference time for the calendar day rainfall measure was shifted to 00:00 a.m. However, the daily measurement approach was never ended, even if it is widely known that such discretization in time leads to bias when dealing with extreme events (Papalexiou et al., 2016). Considering the vast heritage of measurements and its simplicity, the majority of annual maximum rainfall depths in the world are at the daily scale, i.e. expressed over fixed 24 h (F-maxima).

It is clear, however, that sliding 24 h maxima obtained from continuous measurements (S-maxima) are more useful values. They refer to 24 h periods starting at any instant and cannot be less than the F-maxima. An analysis conducted over the United States proved that the S-maxima statistics like the median, the mean and the standard deviation are higher than those of F-maxima and have a greater dispersion; the difference between the F- and S-maxima statistics increases as the time interval increases and appears to become constant for time scales larger than 36 h (Papalexiou et al., 2016).

The ratio between F- and S-maxima, named Hershfield factor (H), has been suggested to correct the incongruences induced by the different time discretization, allowing to take advantage of the relevant amount of information included in historical records of F-maxima (Hershfield, 1961; Harihara Ayyar and Tripathi, 1973; van Montfort, 1990; Dwyer and Reed, 1995; van Montfort, 1997; Papalexiou et al., 2016; Llabrés-Brustenga et al., 2020).

H acts as a multiplier that aims at correcting the underestimation caused by the daily time sampling. Hershfield itself suggested to use a value equal to 1.13 after an analysis carried out using data covering the United States (Hershfield, 1961). In a recent analysis in the United States, Papalexiou et al. (2016) obtained a value equal to 1.14. An application over India suggested a value equal to 1.15 (Harihara Ayyar and Tripathi, 1973). Over China, van Montfort (1997) obtained values in the range 1.04–1.17. Dwyer and Reed (1995) suggested a mean value of 1.16, while proposing differences according to different regimes: a correction factor of 1.15 for rainfall regimes with more frequent short-duration concentrated events, and 1.17 for sites with rainfall regimes with more frequent longer events, such as those produced by large frontal systems. Llabrés-Brustenga et al. (2020) obtained a mean value of 1.125 over Catalonia (Spain) and highlighted considerable variations within the year: summer is the season in which lower H (1.093) was obtained, justified by the fact that rainfall is often raised by the influence of the diurnal cycle and the S-maxima are not usually split by daily measuring in the morning; in spring, a higher H (1.161) was obtained, possibly determined by the fact that rainfall events are more often split when F-maxima measurements are taken at 08:00 a.m.

All these studies (except Papalexiou et al., 2016 and Llabrés-Brustenga et al., 2020) are based on old records. In our study, we investigated the possibility of using ≈1000 stations with at least 10 years of both F- and S-maxima for the same years, covering the period from early 1900 until today, to enrich the availability of S-maxima rainfall measurements, by selecting the data-rich Po river basin and the Liguria region (North of Italy) as a case study. The objective is to obtain mean values of H for the stations where both F- and S-maxima data are available and make these value available, through spatial interpolation, to F-maxima stations where S-maxima data are unavailable.

For all the stations and all the years where both the F-maxima and S-maxima are available, the Hershfield factor was computed according to its definition: 1H=h24hg, where h24 is the 24 h annual maximum and hg is the daily maximum in a given year.

As a general rule, 1≤H≤2 (van Montfort, 1997). It can be equal to 1 if the S-maxima occurred inside the same calendar day, while it can be equal to 2 if the extreme event is equally divided into two days.

In a first step, we analysed H using only the validated data, whatever the record length, to obtain reliable results that would be compared to those obtained using the entire dataset.

Even in validated data we found cases in which F-maxima were higher than S-maxima. This inconsistency is not unusual, and can be attributed to errors in the digitization, due to the manual daily recording. These occurrences, and the others in which we found H>2, were removed. Figure 5a shows the box plot of the individual H values obtained year by year: the overall median H is 1.112, while the whiskers ends are, respectively, 1 and 1.585. The overall mean value obtained for H is 1.170, somewhat higher than those in the literature.

Having removed only the undeniable anomalies (H<1 or H>2) does not exclude the possibility that other anomalies may have occurred. We then used the limits of the box plot to filter out possible outliers in the process of extending the computation to the entire dataset. Therefore, when computing the H values for all the stations using the entire dataset, made of validated and not validated data, we used series with at least 10 years of data and we used the whiskers ends of the validated data (1 and 1.585) as thresholds to remove the outliers. After this computation, the median H resulted of 1.144 and the mean resulted equal to 1.146, which are in line with the values reported in the literature. By increasing the minimum record length to 20 years we obtained an average of H equal to 1.142, and a median equal to 1.141, suggesting that even a time series of only 10 years could provide reliable estimates.

Examinig the box plots in Fig. 5 one can recognize that the variability of the H computed from the entire dataset turns out to be similar to the one of the validated data, despite a higher number of outliers. This evidence confirms the suitability of using also non-validated data to guarantee an higher spatio-temporal coverage of F-maxima.

After the computation and validation of the H values in the so-reached ∼ 1000 stations, we proceeded to a spatial interpolation at 1 km resolution of the mean H. This procedure was realized using the Ordinary Kriging (Fig. 6). By analyzing the spatial distribution of the interpolated Hershfield factor within the study area, we found variability patterns similar to those available in the literature, i.e. the mean (at station) H values turn out to be non uniform over the space. They seem influenced by the geographic position of the stations, and the patterns entail the possibility to identify some distinct areas with a positive or negative anomaly with respect to the spatial average. More specifically, higher values emerge in the Northern part of the examined region, mainly in Piedmont and Lombardy. The Aosta Valley, despite its location in the northern parts, shows different anomalies. This difference could probably be addressed to different prevailing rainfall mechanisms occurring in the Aosta Valley, possibly due to the presence of high peaks at the borders between Piedmont and the Aosta Valley, which act as a barrier. An influence of the geographic position was also pointed out over Catalonia (Spain) by Llabrés-Brustenga et al. (2020), and over China, where van Montfort (1997) analyzed monthly H values in relation to the location of the station, the autocorrelation within its daily measurements and the fraction of wet days.

The exam of the spatial distribution of H has suggested the necessity to perform an additional data control, as some suspicious isolated anomalies clearly emerged (Fig. 7a). This happened especially in areas where official and quality-controlled annual maxima were not available, so that F-maxima were computed from continuous daily records. For these isolated “peaks” of H, the F-maxima time series of the rain gauges have been checked and the incorrect F-maxima data have been replaced using the values reported in the printed version of the Hydrological Yearbooks, as depicted in Fig. 7b. Note that Fig. 6 already report the fully corrected version of the map.

To increase the spatial detail of the extremes of precipitation over the North of Italy, a consistent database including daily precipitation (F-maxima) can be used. The analysis of the ratio between 24 h and daily extremes (Hershfield factor) throughout the investigated area can allow to estimate annual S-maxima (i.e. over 24 h) in the stations that lack this information. In some areas (mountain regions and the lowlands between Piedmont and Lombardy) the increase in detail is noteworthy, and can provide important improvement over the spatial analysis of 24 h extremes.

Before proceeding with the use of these reconstructed data, some care must be exerted about the reliability of these newly-computed maxima. In these terms, it is worth mentioning that Koutsoyiannis and Iliopoulou (2022) pointed out that there is no theoretical basis behind the “inflation” through the Hershfield factor, even though it can represent an acceptable estimate from an engineering point of view. The authors pointed out that all realizations are stochastically equivalent and there is no theoretical basis for their correction. By correcting the series, its stochastic properties can be distorted.

In terms of series properties, indeed, Papalexiou et al. (2016) had previously noted that the ratios between the mean values of F- and S-maxima deviate from the standard deviation ratios: this implies that the application of the H factor cannot appropriately preserve the standard deviation of the F-maxima. These authors have then introduced an approach that simultaneously correct both the mean and the standard deviation errors.

Despite the caveats, however, we cannot ignore that the possibility to include data recorded in decades at the beginning of the past century (see Fig. 4) can produce relevant benefit in the spatial detail of the process. This suggests us to proceed with the application of the H factor to include the F-maxima in the existing 24 h records.

On the other hand, when looking for trends in the data, to protect the reconstructed data from the distortion mentioned by Koutsoyiannis and Iliopoulou (2022) robust methods can be used, as Sen's method, the non-parametric Mann-Kendall test and the Spearman rank correlation test that all exhibit negligible sensitivity to data distortion (Morbidelli et al., 2021).

In this work and in all the other ones cited in this paper the H values were considered as constant over time. Nevertheless, future works could be directed to investigate if the variations observed all over Italy in the 24 h extremes (Mazzoglio et al., 2022) can be detected also in the H values.

In this study we performed a complex data merging operation to recover all the F-maxima recorded starting from the 20th century by 2149 stations. This dataset was then used to evaluate the Hershfield factor, that is a coefficient computed as the ratio between the S-maxima and the F-maxima that can be relevant for the reconstruction of missing sub-daily annual maxima. The interpolation realized, reported with a map in Fig. 6, allows to reconstruct the missing S-maxima in stations with only F-maxima, and improves the knowledge of the spatial variability of sub-daily rainfall extremes.

In view of the extension of this data-recovering procedure to the entire Italy, further investigations are to be directed to the correction of distortion effects noted in the statistical properties of the reconstructed series. In that sense, it is important to consider the trade-off between the increase of uncertainty on the 24 h series properties and the increase in the spatial detail of S-maxima in data-scarce regions.