A rainfall threshold is a function of some rainfall quantities that provides the conditions beyond which the probability of debris-flow occurrence is considered significant. Many uncertainties may affect the thresholds calibration and, consequently, its robustness. This study aims to assess the uncertainty in the estimate of a rainfall threshold for stony debris flow based on the backward dynamical approach, an innovative method to compute the rainfall duration and averaged intensity strictly related to a measured debris flow. The uncertainty analysis is computed by performing two Monte Carlo cascade simulations: (i) to assess the variability in the event characteristics estimate due to the uncertainty in the backward dynamical approach parameters and data and (ii) to quantify the impact of this variability on the threshold calibration. The application of this procedure to a case study highlights that the variability in the event characteristics can be both low and high. Instead, the threshold coefficients have a low dispersion showing good robustness of the threshold estimate. Moreover, the results suggest that some event features are correlated with the variability of the rainfall event duration and intensity. The proposed method is suitable to analyse the uncertainty of other threshold calibration approaches.

In mountain regions, rainfall-induced natural phenomena, as shallow landslides and debris flows, are relatively frequent events that have a significant impact on the territory in which they occur, causing damages and, in some cases, casualties

The early warning systems for these phenomena are mainly based on rainfall thresholds

In some studies rainfall thresholds concern a wide typology of phenomena

Power-law thresholds can be derived in the following way. Given a historical dataset of rainfall-induced events, the rainfall associated with each event is determined and described in terms of the couple of synthetic quantities employed in the threshold (e.g. rainfall event cumulated–event duration). Classically, these quantities are defined only on the basis of a hyetograph analysis

One of the critical issues of the calibration is the uncertainty related to both data and model parameters

Some studies have already investigated the uncertainty in threshold determination, focusing on some aspects that can affect the hyetograph or the event synthetic quantities used in the threshold. For instance,

This work focuses on the uncertainty deriving from data and parameters inherent to the BDA, leaving out the uncertainty related to the hyetograph, already investigated in the literature. In particular, the aim is to perform an uncertainty analysis on the threshold calibration to check the robustness of the BDA. To reach this goal, among the different strategies and methods available in the literature

The paper structure is the following. A brief description of the BDA method is presented in Sect.

As mentioned in the Introduction, the BDA determines the rainfall event intensity and duration, namely the couple

The BDA starts from the knowledge of the deposited volume

Conceptual Lagrangian volumetric description of the debris-flow dynamic from

The expression of

The rainfall volume pertaining to the debris flow can also be expressed as the product of the rainfall volume per unit area

On the other hand,

An approximation algorithm, able to determine in a univocal way the unknowns, can be introduced: starting from zero and increasing one unit alternatively

Finally, the duration

Once the

For more details on the BDA and the frequentist method, we refer the reader to the above-mentioned references.

The study area and data used in this analysis are the same as those used in

The regional agencies between 2006 and 2016 have reported

The rainfall data related to these events derive from a radar located in a central position with respect to the region, on Mt Macaion at

Probability distributions of the uncertain variables for each event.

Additional data required for the BDA, namely

Starting from these data and setting the non-exceedance probability equal to

From now on, the quantities involved in the calibration performed by

As described in Sect.

Coherently to the estimate procedure, the uncertainty analysis of the BDA-based threshold calibration is divided into three parts (Fig.

Scheme of the uncertainty analysis performed with two cascade MC simulations.

Regarding the uncertainty characterisation, as explained in the Introduction, the focus of this study is on the uncertainty in the physical and morphological parameters and data used in the BDA to describe, in a simplified way, the debris-flow dynamic. Therefore, in this analysis, the variables considered are

The procedure used to assess the propagation of the uncertainty in

Second, the event characteristics are obtained starting from each input sample, resulting in 100

The uncertainty propagation in the threshold estimate is then quantified with a further MC procedure. In this case, a sample is generated selecting randomly one of the possible

As described in Sect.

Coefficients of variation of the event characteristics related to each debris flow expressed as a percentage.

Reference slope

The CV, by definition, is a standardised measure of dispersion

The

The distributions of

In addition, the concentration distributions show a low spread. As

The distributions of the volumes per unit area have

events with zero variability:

events with low variability:

events with high variability:

Regardless of the uncertainty of the variable, the concentration is always equal to

Despite the fact that

Absolute variability in the

The third category includes

Skewness of the distributions of

Regarding the absolute variability, the

Positive correlation between the absolute variability of

Despite the specificity of each considered event, it's possible to identify some event features that are correlated with the

Positive correlation between the non-null absolute variability of

Values

We define

Regarding the intensity, we define

The result of the second MC simulation is

Mean, standard deviation, variation coefficient CV and mean

In addition, to analyse the absolute variability of the

Hence, both the relative and the absolute variability highlight that the effect of the uncertainty in the variables on the threshold estimate is small. This is mainly due to the zero variability in the

Finally, a comparison between the results of the first and second MC simulation and the reference values is carried out. In particular, we compare

the means of the

the mean threshold (i.e. threshold computed with the mean values of

Bias of

As regards

Regarding the threshold, the differences between the MC intensities

Subsequently, the percentage changes, defined as

It can therefore be generally stated that the outcomes of the uncertainty analyses, both

In the calibration of the BDA-based threshold and in the assumptions of the developed method used to assess the uncertainty, it is possible to identify some elements that may introduce further uncertainty, beyond that considered in this analysis, in the calculation of the event characteristics and, consequently, in the estimate of the threshold. Firstly, the variability ranges and the probability distributions of the parameters and data, namely the uncertainty characterisation of the variables, are uncertain. Secondly, the equations of the BDA may be uncertain since they are based on some simplifications and hypothesis. Finally, the radar data may be affected by uncertainty due to other sources of error, beyond the beam shielding one (considered in this analysis), such as signal attenuation in heavy rain or wet radome attenuation

This study has aimed to assess the effects of the uncertainty in the physical and morphological parameters and data on the BDA-based threshold calibration to evaluate the method robustness. To that end, a suitable methodology composed of two MC cascade simulations has been developed and applied to a specific study area and dataset. The first MC simulation has allowed examining the uncertainty propagation in the event characteristics estimate. The results have highlighted that most of the events (i.e

Overall, the results of this analysis can be useful to calibrate a BDA-based threshold for a different study area since the investigation has highlighted the main elements that could undermine the BDA robustness. In particular, given a debris flow and the related rainfall event, it was noted that some event features are correlated with the variability of

Besides, given an event, further elements likely affecting the estimate of event characteristics have been highlighted in this study: (i) the variability ranges and the probability distributions of the parameters and data, (ii) the equations constituting the BDA model, and (iii) radar data. These elements can be affected by uncertainty and impact the event characteristics estimate. The uncertainty analysis performed in this study does not provide quantitative information on these impacts. Further analysis will assess how these three elements affect the

Moreover, the developed method, composed of two cascade MC simulations, can be applied to assess the uncertainty related to other threshold calibration approaches whose event characteristics estimate is based not only on the hyetograph but also on other variables

Finally, it is worth noting that the results of this analysis are not useful to check the forecast capability of the threshold. Indeed, the variability in the threshold estimate due to the uncertainty of the inputs is not related to its forecast effectiveness but only to its robustness. The threshold forecast capability can be proved only by performing a proper validation analysis, which is essential to make this tool operational. Since the calibration method applied to the specific study area is proved to be robust, further analysis will assess the forecast capability of the threshold, developing an appropriate validation method.

The data used in this analysis are available upon request to the regional agencies of the Trentino-Alto Adige/Südtirol region.

MM designed the experiments, performed the analysis and wrote the paper. DZ and GR supervised the study, analysed the results and wrote the paper.

The authors declare that they have no conflict of interest.

This work has been carried out within the project “Progetto WEEZARD: un sistema integrato di modellazione matematica a servizio della sicurezza nei confronti di pericoli idrogeologici in ambiente montano” (CARITRO Foundation – Cassa di Risparmio di Trento e Rovereto). We thank Stefano Siboni for providing valuable suggestions and Ripartizione Opere Idrauliche and Ufficio Idrografico, Provincia Autonoma di Bolzano (Italy), and Servizio Bacini Montani and Ufficio Previsioni e Pianificazione, Provincia Autonoma di Trento (Italy), for supplying radar and debris-flow data.

This paper was edited by Filippo Catani and reviewed by two anonymous referees.