As mentioned before, all 1330 considered records report at least damages to buildings (structural and nonstructural parts and installations). Authorities defined the final compensation granted to owners in accordance to ordinance no. 2 of 5 June 2014 and law no. 93 of 26 June 2014, which specifies refund criteria. For instance, considering the total amount of money that authorities had available for the restoration of all kinds of properties, the maximum coverage for each property was set to EUR 85 000 for damages to buildings and EUR 15 000 for damages to contents, setting a fixed amount of money for each room. In addition, owners declarations about the amount of the restoration work of the damaged parts, if higher than EUR 15 000, were verified by authorities by means of expert technical reports. These controls probably reduced the amount of damage claimed by owners, who commonly tend to overestimate their loss and have less competency for estimating damages than professionals have.

Nevertheless, the limited availability of money and the need for a homogeneous criterion for all the affected properties led in many cases to a much higher reduction of the amount of damage refundable to the owners. In fact, refundable assets are only a cut percentage of assets that can be found in a property and, in addition, experienced damages could be higher than the maximum coverage established by authorities. The difference between overall monetary refunded and claimed damages to buildings is equal to about EUR 1.7 million (EUR 15.2 million of declared loss vs. EUR 13.5 million of refunded loss). Given this significant difference, in order to preserve the representativeness and consistency in loss data, we chose to consider observed damages in our study as claimed by citizens in the forms they filled (estimation of the financial need for restoration, without knowing the refund criteria). We are aware that this choice can introduce overestimation of the damages (particularly considering damages below EUR 15 000) for the reason explained before, but we considered this possible error having less influence on loss estimation, both quantitatively and methodologically, relative to the distortions that would be systematically introduced by adopting the result of the compensation phase.

Together with the amount of money requested for compensation, we also extracted from the filled forms the available information on building footprints and structural typology (masonry, reinforced concrete, etc.) because of their potential impact on the damage process and therefore on damage modeling see also previous studies, e.g.,.

In order to evaluate loss in relative terms (as the percentage of suffered damage relative to the total value of the building), we retrieved the economic value of each property from the Italian Revenue Agency reports (Agenzia delle Entrate, AE, 2018). Every 6 months the AE issues the open-market values (EUR m-2) for different assets (e.g., civil houses, offices, stores) in each Italian administrative district (spatial scale of municipality), taking into account different classes of residential and industrial buildings and the overall economic well-being of the region. These values are different for each homogeneous geographical area (OMI zone) and set a minimum and a maximum market value per unit area. Focusing on residential buildings, and in particular on their structural part without including the cost of the land, we defined the buildings' economic value (EUR m-2) as the average of the values provided for each building in the same OMI zone. Only the first floor of each building was considered since the maximum water depth is always lower than or equal to 2.1 m (see Table ). It is important to notice that these economic values do not consider a possible fall in price due to catastrophic events. Also, we are aware that reconstruction costs seem to be more suitable for this kind of analyses, but they are not freely available in Italy or homogeneous at a national level, different from OMI values. Moreover, the use of these economic values at an aggregation level is still informative for future ex ante damage estimation for planning activities and it is in line with previous loss analyses at different scales see, e.g.,.

We also analyze the monetary loss to household un-registered contents (e.g., furniture and household appliances: refrigerator, dishwasher, oven, sink, stove, washer, dryer, TV and personal computers).

Focusing on these data and looking at the refunded loss, because of the stricter criteria for content damage compensation of ordinance no. 2 of 5 June 2014 and law no. 93 of 26 June 2014, the difference between the requested and refunded amount is even more evident. It is equal to about EUR 5.7 million (EUR 10.4 million of overall declared loss to contents vs. EUR 4.7 million of refunded loss) and confirms the choice to consider observed damages as claimed by owners.

Concerning this dataset, it is worth noting that we do not have any specific information for each building on the items recorded under the generic expression “contents”. Therefore, we cannot express these damages in terms of relative loss over the overall movable property value. Also, the damage models to household contents proposed by the scientific literature are fairly rare and isolated some examples are represented by studies performed by. Thus, we investigate the usefulness of an indirect modeling approach, which is based on regressing loss to contents against loss to buildings (see Sect. ), for this type of damage.

Forms collected from authorities for the purpose of compensation do not include data on hydraulic variables, such as water depth, water velocity, etc. Since these data are necessary for our analysis, the reconstruction of the flood event is performed by means of TELEMAC-2D, a fully 2-D hydrodynamic model which solves the 2-D shallow water Saint-Venant equations using the finite-element method within a computational mesh of triangular elements seefor details. This computational model complies with the validation protocol by the International Association for Hydro-Environment Engineering and Research (IAHR) and has been successfully applied to case studies around the globe .

Concerning the inundation event, the dynamics of the wetting front were strongly influenced by the presence of topographic discontinuities e.g., road embankments, artificial as well as natural channels belonging to the minor stream network; see. In order to correctly reproduce ground elevation and discontinuities in the model, a detailed lidar DEM with a spatial resolution of 1 m is used and an unstructured triangular finite-element mesh of the study area is generated. The mesh consists of 34 082 nodes connecting 66 596 elements with variable length side from 1 to 200 m in flatter zones, covering a total of 112 km2. This accurate mesh ensures the correct representation of all major linear discontinuities existing in the study area.

The outflowing hydrograph of the levee breach, as reconstructed by the scientific committee that studied the event , is used as a boundary condition, in particular as inflow to the boundary elements representing the levee breach.

The calibration of the 2-D model is performed by varying floodplain roughness coefficients in order to reproduce the real extent of the inundation, at different time steps, as documented by maps and aerial images made available immediately post event by competent authorities and rescuers , and as also confirmed by later studies see, e.g.,. In particular, Manning's coefficient values were differentiated between agricultural areas and urban areas, and resulting coefficients (0.033 and 0.1 m-1/3 s, respectively) are in line with values reported in the scientific literature see, e.g.,.

After the event, local authorities collected information about water depths reached at different points of the inundated area. This information is used for the validation of the model, together with pictures, videos and reports made available on the Internet, as well as through in situ interviews. At about 50 points, uniformly distributed in the study area, simulation outcomes are compared in terms of water depth with the information available. Results show a good agreement between simulated and observed flooding dynamics, with the residuals between observed and simulated water levels always smaller than ±20 cm. In order to avoid errors due to the model uncertainty, we consider the area with simulated water depth greater than 10 cm to be “flooded” see, e.g.,.

The calibrated and validated model is then used to reconstruct the detailed spatiotemporal dynamics of the inundation event and to identify the spatial distribution of the hydraulic variables of interest. In fact, combining 2-D model outcomes and geocoded locations shown in Fig. , it is possible to extract maximum water depth, maximum flow velocity and duration of the inundation at each site (see Table ). Maximum water depth and the maximum flow velocity commonly refer to different time steps of the flood event.

Figure  shows the results of the correlation analysis between relative flood loss to buildings and the available six predictive variables: maximum water depth, maximum water velocity, flood duration, building value per unit area, building area and building structural typology. Since the latter is a categorical variable, it is converted to dummy variable encoding in order to calculate the correlation of continuous and categorical data together. We refer to the Spearman correlation coefficient in order to also take into account nonlinear relationships among variables. Empty boxes represent correlations that are not statistically significant at a 5 % significance level. The variables that are significantly correlated with the relative loss to buildings are maximum water depth, building value per unit area and building structural typology. However, correlation coefficients are low, precisely lower than ±0.18 in all the cases. Similar results were obtained in terms of Pearson's correlation, but the values are not shown for the sake of brevity.

Figure  shows the output of the RF evaluation of the importance of the six predictive variables within the SMV model. This concept is different from the correlation one: in fact, while the Spearman coefficient indicates how well the relationship between two variables can be described using a monotonic function, the RF algorithm evaluates the importance of a variable by assessing the worsening in the performance of the model when that specific variable is not included in the database. In contrast to other studies see, e.g.,, the dataset does not reveal a distinct importance for individual variables; not even water depth stands out. The descriptive capability of water depth is only slightly stronger than water velocity and building area, while the remaining predictors show very little importance.

Figure  shows in the background the observed relative damage to buildings, collected in the three affected municipalities (i.e., Bastiglia, Bomporto and Modena) as a function of maximum water depth (Fig. a), water velocity (Fig. b) and building area (Fig. c). Despite the statistically significant correlation with water depth (see Fig. ), a very large noise can be observed in all diagrams, which implies that one variable alone explains only a very limited part of the damage process. This is confirmed from the outcomes of both the correlation assessment (see Fig. ) and the importance analysis (see Fig. ).

Taking the maximum water depth as the only explanatory variable, Fig. a represents the damages to buildings estimated by means of the univariable models developed based on the Secchia dataset (SEMP, with blue dots, and SREG_d, dark red dots). In a similar fashion, Fig. b, c show the relative loss to buildings as a function of maximum water velocity and building area, estimated by means of SREGv and SREGa, respectively (dark red dots in both diagrams).

Results of the application of the multivariable model (SMV), described in Sect. , are shown in Fig. , which highlights the good performance of this model.

Table  quantifies the discrepancy between observed and predicted loss values for local empirical models in terms of four different performance metrics, namely bias, mean absolute error (MAE), RMSE and the difference between estimated and observed overall monetary loss to buildings (ΔLOSS), which are defined as follows: bias=1n∑i=1nPi-Oi,MAE=1n∑i=1nPi-Oi,RMSE=1n∑i=1nPi-Oi2,ΔLOSS=∑i=1nPi⋅BAi⋅BVi-∑i=1nOi⋅BAi⋅BVi∑i=1nOi⋅BAi⋅BVi⋅100, in which Oi and Pi are observed and predicted relative damages at the ith site, respectively; n is the number of sites in the study area; and BAi and BVi are building area and building value per unit area at the ith site, respectively (see Table ).

SMV is associated with the lowest RMSE value (i.e., 0.062), which is less than half the RMSE value of the second-to-best models (i.e., SREGd and SREGv, with an RMSE value of 0.125). SREGa and SEMP provide slightly worse relative loss estimations than the previous models (RMSE equal to 0.129 and 0.130, respectively). Results are similar in terms of bias and MAE, although some differences can be pointed out for SREGx models, which present a bias value that is slightly lower than the one derived from SMV estimation.

Concerning literature models described in Sect.  and illustrated in Fig. , Table  shows that FLEMOps and JRC Czech Republic outperform the others in terms of RMSE (RMSE equal to 0.125 and 0.127, respectively) and are comparable with the models developed based on Secchia's dataset. RMSE values derived from the relative loss estimation with JRC Netherland, JRC Germany, JRC Belgium and Rhine Atlas are between 0.131 and 0.143, while the worst performance in terms of RMSE is associated with JRC Switzerland, JRC other countries, MCM and JRC UK (RMSE values higher than 0.2). These outcomes reflect the fact that all these latter damage curves are in the upper part of the diagram in Fig.  and significantly apart from the rest of the models, which are instead close to each other. We obtained similar results in terms of bias and MAE.

Analogous results can be observed in terms of ΔLOSS, which is reported in the rightmost column of both Tables  and . This indicator, different from MAE and RMSE and similar to bias, highlights the tendency of models to under- or overpredict damages to buildings; yet ΔLOSS focuses on the overall monetary damage in a given area, whereas bias refers to relative damages. Hence, ΔLOSS clearly shows if a model is biased in predicting the overall monetary loss, that is, if the model systematically predicts higher or lower (positive and negative bias, respectively) damages for the entire study area than those observed. This is shown in Fig. , in which most of the predictions provided by SMV, especially for observed relative damages higher than 10 %, lie under the 1:1 line: this means that the model is negatively biased. Predictions obtained with the other models are spread more evenly around the 1:1 line, denoting a smaller bias. In terms of bias and ΔLOSS, SMV seems to have a slightly worse performance than SREGd, SREGv and SREGa (and FLEMOps, regarding these specific outcomes).

The large overestimation of overall losses associated with JRC UK, MCM, JRC other countries, JRC Switzerland and JRC Belgium reported in Table  is expected from the comparison among these models and empirical data presented in Fig. . The overestimation may result from morphologic and socioeconomic contexts for which these models were constructed, as well as criteria adopted for their development, which might differ considerably from our case study and empirical models. For example, due to the diverse study area topographies and land uses, floods can propagate with various dynamics, differently influencing hazard indicators. Also, building characteristics and the overall well-being of an area can differ considerably among regions and countries, therefore compromising the transferability of literature curves.

Another feature of the rightmost column of Table  worth noting is that four of the literature models that perform the best in terms of RMSE (JRC Czech Republic, JRC Netherlands, JRC Germany and Rhine Atlas) underestimate the overall monetary loss. This fact can be explained by several reasons, among which an important one is certainly comparing damages claimed by citizens with the four models listed above, which were developed on the basis of expert-based judgment only, or by considering expert knowledge together with empirical data.

An additional important factor that influences the performance of literature models applied to the Secchia case study is the different scale on which these curves are calibrated and applied: some of them are developed to be applied at the microscale (e.g., MCM, FLEMOps), while others are developed to be applied at the mesoscale (e.g., Rhine Atlas, JRC curves). However, among mesoscale models there is a large variability in terms of performance. In several practical applications, identifying the best performing damage model a priori can be an extremely difficult task. This is also complicated by difficulties in obtaining detailed information about original datasets used for developing literature models (including damage data and characteristics of the flood event and of typology of affected buildings). Deeper investigation on model properties and assumptions (e.g., hazard and vulnerability features based on the context for which they have been derived, values used for translating monetary damage into relative damage, level of aggregation of original data) can guide the selection of models; moreover, a variety of them should be used to additionally obtain information on associated uncertainty .

The results reported in Table  refer to calibrations of empirical models based on our entire dataset. We also validate all empirical models by using a split-sample validation procedure. Specifically, two-thirds of the records are randomly selected from the dataset for calibrating each model, which is then applied to the remaining one-third of the data. Bias, MAE and RMSE calculated in this context and reported in Table  are very similar to the ones reported in Table  concerning SREGx and SEMP. Results of the validation of SMV by means of the same approach instead indicate lower performance of this model, when calibrated on a smaller dataset (see Table ). In fact, values of bias, MAE and RMSE are twice as high as values reported in Table . These outcomes highlight the need for extensive datasets for identifying robust and reliable damage models. From the comparison of the different considered models (uni- and multivariable), it is clear that this aspect is more evident for the multivariable model, whose performance is significantly worse when calibrated on a smaller number of observed data. Conversely, univariable models, though simpler than SMV, appear more robust in the case of a smaller number of calibration data, providing better results in the validation.

Based on the output of Sect. , it is worth noting that the application to the Secchia case study of JRC other countries, in which Italy should be included, provides very poor results in terms of building loss. This confirms how challenging the identification of a regional or large-scale model with a general validity could be see also Sect.  and. This section further assesses the transferability of damage models to very similar socioeconomic contexts.

In order to test the transferability of the empirical locally derived models to similar contexts, we identify analogous models (SREGx, since it is the best model among the locally derived ones, and SMV) on the basis of the building loss data collected in a single municipality and then apply these models for predicting flood building loss in a neighboring municipality. In particular, among the three municipalities considered in the study (i.e., Bomporto, Bastiglia and Modena), we consider Bastiglia (887 observed records) and Bomporto (392 observed records) because of the larger number of data available. We calibrate the models on Bomporto's subset (Bo_MV, Bo_REGd, Bo_REGv and Bo_REGa) and we apply them for predicting Bastiglia's flood damages to buildings. Then, we calibrate the same models on the Bastiglia subset (Ba_MV, Ba_REGd, Ba_REGv and Ba_REGa) and apply them to Bomporto.

Figure  shows the results of these split-sampling experiments. Figure a refers to Bastiglia's relative damages to buildings, estimated via Bo_MV and Bo_REGd, while Fig. b indicates Bomporto's damages estimated via Ba_MV and Ba_REGd; in each graph grey dots represent the estimation of relative loss using the multivariable models and red dots indicate relative damages to buildings estimated with square root regression models.

Square root regression models in Fig.  show rather poor performances, capable of capturing only the average loss, while better results seem to be associated with multivariable models in both graphs. Some differences between the two panels are worth noting: grey dots in Fig. a (application of models calibrated in Bomporto with 392 data to Bastiglia) seem to overestimate relative loss to buildings, while in Fig. b (application of models calibrated in Bastiglia with 887 records to Bomporto) they lie closer to the 1:1 line. The studies performed in terms of relative damages to buildings related to maximum water velocity and building area present very similar results and are reported in the Appendix (see Figs.  and ).

These outcomes are also visible in Table , which presents the results of the split-sampling experiments in terms of the usual bias, MAE and RMSE indexes. While uni- and multivariable models calibrated on Bastiglia's data and applied to Bomporto's subset do not differ much, with slightly better performances for Ba_MV, Bo_MV is associated with much higher prediction errors when applied to Bastiglia. The worse performance of Bo_MV can be explained by the smaller size of the Bomporto subset of data used for its calibration (less than a half of Bastiglia's sample). As already outlined in Sect. , in order to have robust results from multivariable models, a large number of empirical data are required. Furthermore, the inundated area in Bomporto is larger than in Bastiglia (see Fig. ). This explains rather clearly the difference in terms of accuracy of Ba_MV and Bo_MV in Table : the higher the loss data density the more robust the relationship between different predictor variables and loss data and the higher the ability of the model to explain local characteristics of the study area .

The transferability of the models is also hampered by the different distribution of the water depths in the different municipalities: Fig.  shows that water depths in Bastiglia are lower than in Bomporto, despite the quite similar distribution of observed relative damages. This might be due to the fact that, other than being a hazard, different buildings' vulnerability plays an important role in the damage process too and it also explains prediction errors in the analysis. This aspect has to be taken into consideration whenever the loss estimation is performed by using a model calibrated for a different flood event.

Similar to the procedure for assessing damages to buildings, first of all we analyze the Spearman correlation between the observed flood loss to contents and all potential predictive variables, included monetary damages to buildings. Figure  shows the results of this assessment, where full boxes represent a statistically significant correlation coefficient at a 5 % significance level. On the one hand, similar to the analysis for building loss, the maximum water depth and the structural typology are significantly correlated with damages to contents, although their correlation coefficients are low. On the other hand, damages to contents turn out to be significantly correlated with the building footprint (Spearman correlation coefficient equal to 0.27) instead of the building value. A noteworthy feature of Fig.  is the very strong and statistically significant positive correlation between damages to buildings and their contents (Spearman correlation coefficient equal to 0.59).

We therefore explore the possibility of exploiting the relationship between monetary loss to buildings and contents for predicting the latter. We test different types of mathematical relationships (i.e., linear, square root, logarithmic and bi-logarithmic regressions), and the square root regression is the one with the best prediction performance in terms of RMSE, i.e., the one that best relates monetary building loss with damages to contents. In fact, RMSE is equal to EUR 10 569, while it was EUR 10 882, 10 971 and 15 531 for linear, logarithmic and bi-logarithmic relationships, respectively. The identified regression relationship reads Dcontents=116Dbuildings-2311, where Dcontents (EUR) represents economic damages to contents, and Dbuildings (EUR) indicates loss to buildings. Figure  depicts empirical vs. predicted monetary loss to contents with Eq. ().

In the last component of our analysis, we apply Eq. () for estimating damages to contents as a function of the estimates of monetary building loss resulting from the uni- and multivariable damage models that we considered in our study.

Table  lists the performance metrics bias, MAE, RMSE and ΔLOSS obtained while predicting monetary loss to contents as described. The first row in Table  reports, as a reference term, the same performance indexes that can be obtained when Eq. () is applied to observed damages to buildings. In the second row, the first block of Table  shows the performance in estimating monetary content loss, applying Eq. () to monetary damages to buildings, estimated with empirically derived models. The best performance in terms of RMSE is always associated with SMV, followed by SEMP and SREGx, all with comparable RMSE values. The outcomes for literature models (last block of Table ) also reflect the results that we obtained when modeling building loss, presented in Sect. . The ranking of the best-performing literature models in terms of RMSE for an indirect assessment of content loss is JRC Czech Republic, JRC Netherlands, JRC Germany, FLEMOps, Rhine Atlas and JRC Belgium. Evidently, models associated with poor performances in predicting monetary loss to buildings are also not reliable for indirectly predicting loss to building contents by means of Eq. () (see JRC Switzerland, JRC other countries, MCM and JRC UK). The performance of most considered models, with the exception of the last six in Table , show a difference between overall observed and predicted monetary loss to contents that does not exceed EUR ±20 million. Different from the results obtained when predicting damages to buildings, 11 damage models overestimate content loss, while SEMP, JRC Netherlands, JRC Germany and Rhine Atlas underestimate them. Small differences in the models' ranking, compared to Tables  and , are probably due to the fact that the regression curve for content damages is applied to predicted building damages, which are themselves affected by uncertainty.