Uncertainties in the DTMs of 2006 and 2019 lead to errors in the calculated DoD (Lane et al., 2003; Bakker and Lane, 2017) and thus also in the calculation of the debris flow volumes. In order to minimize the errors in the DoD, the two lidar datasets had to be coregistered. To optimize this processing step, the study area was divided into several smaller areas of interest so that the algorithms for matching the data were able to work on a more local scale. In these regional patches, areas were identified where no geomorphologic changes were expected in between the lidar data acquisitions. These stable areas were mapped as close to the debris flow depositions as possible, and were selected, if possible, to be of similar steepness. As point cloud data were available only for one of the lidar datasets, we coregistered the two gridded DTMs using the approach of Nuth and Kääb (2011) implemented in the Python package pybob (https://pybob.readthedocs.io, last access: 26 January 2023).

To get a better understanding of the errors, the DoDs within the identified stable areas were analysed regarding the precision (standard deviation) and the accuracy (RMSE – root mean square error), as well as the arithmetic mean and the absolute mean. For a total assessment of the error of the volume of debris flow deposits, the error was calculated following the approach of Anderson (2019), which combines the uncorrelated random error, the spatially correlated random error and the systematic error of the DoD. All debris flow volumes detected from the DoD together with the respective errors are listed in Table A1 in the Appendix.

The volumes of those debris flow deposits that are detectable in the DoD were determined in each case by summing up the values of the DoD in the mapped deposition areas. For those debris flow deposits which are not contained in the DoD (especially for debris flows prior to 2006), only the area of the deposits could be mapped using the respective pair of orthophotos. In order to estimate the volume of different types of mass movements based on the accumulation area, numerous studies derived an empirical relationship between the deposit area (A) and the deposit volume (V) (Guzzetti et al., 2009; Magirl et al., 2010; Larsen et al., 2010). This relationship is expressed by a power law with an exponent γ > 0 and the intercept α: 1V=α×Aγ. The exponent γ in such area–volume relationships depends not only on the analysed process (e.g. rock fall, landslide, debris flow) but also on its subtypes (Larsen et al., 2010; Griswold and Iverson, 2008). Nevertheless, the range of γ usually seems to be within a similar range for several types of mass movements (Hilger, 2017).

The relationship between the volumes and deposition areas is used in order to predict the volumes of depositions for which only the areas are known. Because in this study only slope-type debris flows of the same type are analysed, and no large differences in the debris material are to be expected, the uncertainties here focus on the individual debris flow processes. These include the different content of water or the topography of the deposition area before the debris flow event.

In order to fit Eq. (1) to the empirical data, a variety of different fitting techniques can be used (see for example Guzzetti et al., 2009; Larsen et al., 2010). One simple method includes a least-squares linear fit to the log-transformed data. Another way of fitting a power-law function to the data is by using non-linear regression. In order to be better comparable to other studies calculating such relationships, both methods were applied in the present study by using the statistical software R and the functions lm (linear model) and nls (non-linear least squares) (Baty et al., 2015).

To get a better understanding of the uncertainties involved in the volume calculations, the goodness of fit of the area–volume models has to be described. However for non-linear correlations, the coefficient of determination R2 is not a valid measure (Spiess and Neumeyer, 2010). Instead, we use the 95 % prediction interval of the non-linear regressions. The upper and lower boundary of the prediction interval for each area were used as the maximum and minimum debris flow volume for those events that were not quantified from the DoD. Where the lower limit of the prediction interval is negative (which occurs especially with small deposit areas), it was set to 0 m3; therefore, the lower uncertainty band of the computed volumes is frequently shorter than the upper. For all debris flows included in the DoD, the uncertainty limits were defined by the error assessment in Sect. 3.2.1. For calculating the uncertainty of the total debris flow volume for each of the considered epochs, the uncertainties of each single volume calculation was propagated. Therefore, the total uncertainty of an epoch is the square root of the sum of the squared single uncertainties (Anderson, 2019).

The calculated magnitudes, as well as the known ages of the debris flows due to the multi-temporal mapping, allowed us to establish a magnitude–frequency relationship. This has been done for various gravitational processes, such as landslides (Bennett et al., 2012; Gao et al., 2018; Tanyaş et al., 2019; Guzzetti et al., 2009), rockfalls (Ravanel and Deline, 2011), channelized debris flows (Gao et al., 2018) and also slope-type debris flows (Hilger, 2017). Using the poweRlaw package within R (Gillespie, 2015), we calculated an empirical cumulative distribution function (CDF) to represent the relationship between debris flow deposit volumes and their frequencies (Bennett et al., 2012; Hilger, 2017). Subsequently, we were able to fit a continuous power law distribution to the CDF. However, this distribution is only valid for volumes exceeding a minimum magnitude xmin⁡ (Bennett et al., 2012), and the calculated exponent of the power law β, which is based on a cumulative distribution, has to be reduced by 1 when compared with non-cumulative exponents (Brunetti et al., 2009; Haas et al., 2012).

For a total of 58 debris flows it was possible to map the deposition area in the DoD from the 2006 and 2019 terrain models with sufficient accuracy to enable a balance of the deposition volumes (Appendix A1). The volumes range from very small (7.55 m3) to large debris flows (7506 m3). With the help of this data, a relationship between the area and volume of debris flow deposits could be established which follows a power law (Fig. 7). The exponent γ in Eq. (1) could be calculated as γ=1.21 for the fitted linear model. This method tries to fit a linear model to the log-transformed area and log-transformed volume data. The log scaling of both of the input data results in a distortion of the residuals, which are used in the fitting process.

In order to reduce this bias, a non-linear model was fitted using the nls() function in R. This approach determines the non-linear least-squares estimates of the parameters of a power-law model (Bates and Watts, 1988) and results in a mathematical best fit with respect to the residuals. The exponent of the fitted non-linear model results to be slightly lower with γ=0.92±0.077 for the 95 % confidence interval. In the non-linear model plotted in Fig. 7, it is shown that the model slightly overestimates volumes for areas < 500 m2. Again, this is due to the log scaling of the axes.

With the help of the regression of volume on area, the debris flow volume could be calculated for all events with a precisely delimited deposit. In total, the deposition volumes could be calculated (based on the DoD) or estimated (area ∼ volume) for 404 debris flows in GT, LT and ZT. The uncertainties of the volume calculations were carried out as described in Sect. 3.2.3.

Figure 6b reports the annual debris flow volume per epoch. Similar to the mapping results in Fig. 6a, the debris flow volumes show periods with high and low deposition rates per year. Most remarkable is the sudden and strong increase in volumes in the period 1990–1997 compared to the previous periods. After years of relatively few debris flows with little deposited material, the many triggered processes between 1990 and 1997 also transported an above-average amount of material.

Comparable to the mapping results, the volume data of the time intervals between 1954–1973 and 1990–2010 reveal increased deposition, but the period between 2015 and 2018 produced less deposited volume than one could have assumed from the very high number of triggered debris flows in that time interval. Thus, it can be stated that although many events occurred, they have deposited relatively little sediment in total.

The calculated magnitude–frequency relationship is shown in Fig. 8. About 70 % of all debris flows in GT, LT and ZT have an accumulated volume above 100 m3, and about 20 % of all debris flows exceed a volume of 1000 m3. Extreme events, which account for less than 1 % of all debris flows, can reach volumes of more than 10 000 m3. The magnitude–frequency relationship can be described by a continuous power-law distribution with an exponent of β=2.9 for volumes of xmin⁡=1025 m3 and above and therefore is especially usable for large debris flow volumes.

The biggest factor for the concentration of debris flow processes in the sub-catchments GT, LT and ZT is probably the presence of catchments in the steep bedrock above large talus cones, which is typical for slope-type debris flows (Rieger, 1999). Rainwater concentrates at the contact zone between these catchment areas and the slope sediments underneath and potentially triggers debris flows. These catchment morphometries are especially pronounced in GT, LT and ZT.

In these very active and north–south-oriented sub-valleys, 68 % of the active debris flow catchments face west, while only 32 % face east. Becht (1995) attributes this difference to the emergence of cirques in the Pleistocene on east-exposed slopes. West-exposed ones do not show these landforms. Therefore, the stepped profiles on east-exposed slopes caused by the cirques prevent the accumulation of high peak discharges during a rainfall event because of a buffering effect of these cirques. Depending on the amount of loose material in the cirques, these buffering effects can cause longer or shorter delays of the runoff and therefore lower the peak discharge. In addition, the slopes beneath the cirques lack rockfall material supply, which otherwise works as suitable material on the slopes for debris flow initiation.

It is noticeable that in certain time intervals some sub-catchments were significantly more affected by debris flows than other sub-catchments (see Fig. 5). This is most obvious in the period between 1990 and 1997, when the two neighbouring valleys GT and LT show a strongly increased debris flow activity, especially in comparison to the other sub-catchments. Other time intervals (2003–2009, 2009–2010) also show that some sub-catchments were obviously more affected than neighbouring ones. We attribute these findings to the fact that debris-flow-triggering heavy rainfall events often occur during intense convective and thus spatially restricted thunderstorms (Underwood et al., 2016; Berti et al., 2020; Stoffel et al., 2005) that often affect only parts of the study area.

The results of the correlation of the debris flow volumes and the parameters of the respective hydrological catchments in Table 3 indicate that the catchment morphometry is affecting the spatial variability of slope-type debris flows in Horlachtal. Only a few debris flow studies implemented analyses regarding catchment parameters, and most of them deal with different debris flow types and scales than this study (Becht and Rieger, 1997; Wilford et al., 2004; Marchi et al., 2019; Li et al., 2015).

However, the results of this study match the findings of De Haas and Densmore (2019), who worked in a roughly comparable setting in the United States and found statistically significant correlations between debris flow lobe volumes and A, L, P, H and M. This fits our data just as well as the lack of correlation with S, C and R. The only differences are with the variables F and E, which show a stronger relationship in Horlachtal compared to the results in De Haas and Densmore (2019). In addition, a positive correlation between slope-type debris flow volumes and A was detected by Rieger (1999) in LT as well.

The starting points of the vast majority of the debris flows in Horlachtal are located at the contact zone between the bedrock catchments and the adjacent talus slopes. Apart from precipitation and the catchment morphometrics, only the slope gradient is of great importance for debris flow initiation here (Becht, 1995). For the present type of debris flow, a slope threshold of about 27∘ at the starting zones can be found in the literature (Rickenmann and Zimmermann, 1993; Dikau et al., 2019). In Horlachtal, these slope gradients range between 22 and 72∘, with 96 % of the starting points exceeding the threshold of 27∘.

Other factors like vegetation cover or soil properties are of minor importance for debris flow initiation in the study area. Most of the starting points are located above the treeline (96.5 % above 2200 m and 92 % above 2300 m). Thus, no trees or higher vegetation grow in the bedrock catchments even at altitudes below the treeline, as the morphodynamics are too high there. In addition, if there is any soil formation in the catchments, it consists only of shallow initial soils because of the same reasons.

With the help of the area–volume relationship we were able to calculate deposit volumes, which are necessary to establish a relationship between frequency and magnitude. The resulting exponent of γ=0.92 of the nls() method connecting volumes and areas is comparable with the result of Hilger (2017), who used a linear regression method for slope-type debris flows in Kaunertal (γ=1.08). Furthermore, it is consistent with other comparable studies (Jaboyedoff et al., 2020; Larsen et al., 2010; Guzzetti et al., 2009). The established relationship enabled detailed frequency–magnitude analyses. There are rarely any other studies that calculate such a magnitude–frequency relationship only for debris flows and especially slope-type debris flows. Other papers often focus on debris flows in general (Hungr et al., 2008; Riley et al., 2013). An exception is Hilger (2017), who performed such a calculation in a similar geological setting using slope-type debris flows at Kaunertal, Tyrol. While the general shape and β are very much comparable to this study, the magnitudes of debris flows in Horlachtal are larger by an order of a magnitude. Thus, the largest debris flows in Kaunertal show a volume of about 1000 m3, whereas in Horlachtal the volumes can reach nearly 10 000 m3 (Fig. 11a). In addition, a slight shift of debris flow magnitudes could be detected in Horlachtal. The debris flows of the second half of the investigated time span (1983–2020) reach very high volumes more often than debris flows of the first half (1947–1983) (Fig. 11a).

In Horlachtal, the multi-temporal mapping of debris flows, as well as the volumetric measurements, resulted in periods with higher activity (1954–1973, 1990–2010 and 2015–2018) and lower activity in between (Fig. 6a). A consistent linear trend is not recognizable, although it seems that in periods with low activity, the number of debris flows has been rising since 1947. This finding might be biased because of two limitations. First, the most recent orthophotos (especially since 2003) are of very good quality, especially in terms of spatial resolution, light conditions and extent of snow cover. These circumstances allow even small events to be detected and mapped. The second reason is the aforementioned difference in time span lengths, which results in the underestimation of detected debris flows in longer periods.

The mapping results, as well as the calculated deposition volumes, indicate no long-term change in debris flow activity in the past 70 years but with some short-term variabilities. This matches other debris flow records in the Alps (Bollschweiler and Stoffel, 2010; Bollschweiler et al., 2008; Jomelli et al., 2007). Kiefer et al. (2021) detected significant changes over substantially longer time periods when reconstructing the debris flow activity of the past 4000 years based on turbidite measurements of a fan delta. But even in this record, no significant change within the last century is recognizable. In Horlachtal, three periods of increased slope-type debris flow activity can be recognized based on the total number of detected processes (see Fig. 6a): between 1954 and 1973, from 1990 to 2009, and from 2015 to 2018. However, the exact datings of the upper and lower boundaries of the periods with enhanced and low debris flow activity described in this study are predetermined by the used method for establishing the process record. This means that the dates of the acquisition of the aerial images predefine and distort the period boundaries to some extent. In order to delineate the “real” active and inactive periods with higher accuracy, further analyses on the dating of single debris flow events, e.g. by using dendrogeomorphology (Stoffel, 2010), might provide more detailed insights.

A particularly large debris flow event was reported for 31 July 1992, mainly affecting GT and LT (Becht, 1995). These statements can be supported by the debris flow mapping of this study, as many events could be detected in the period 1990–1997 especially in GT and LT (see Fig. 5). In the two subsequent periods until 2009, many events can still be registered in GT and LT but with a decreasing tendency. This could indicate that the large debris flow event in 1992 still had some kind of impact on the debris flow activity of the following years due to the disturbance of the system like e.g. destruction of vegetation or channel deepening. Such a kind of impact on subsequent periods could only be detected for the 1992 event. The high debris flow activities between 1954–1973 and 2015–2018 did not show this pattern of aftereffects, at least not for the temporal resolution predetermined by the orthophotos.

An indication of the outstanding significance of the 1992 event in GT and LT compared to 1954–1973 and 2015–2018 can be provided by considering the debris flow magnitudes. Figure 11b shows that the deposited debris flow volumes of the period 1990–1997 (which includes the 1992 event) are significantly higher in comparison with the magnitudes of 1954–1973 and 2015–2018. In addition, Fig. 11b shows a slightly decreasing tendency in deposited debris flow volumes from 1990–2009, which in turn supports the aforementioned impact of the 1992 event on the following years.

In the heatmap of Fig. 12, each row represents one slope-type debris flow starting zone in GT or LT, which was active at least in two different periods between 1947 and 2020. For each individual starting zone (each row of the heatmap), the magnitudes were normalized between 1 (largest event of the starting zone; dark colouring) and 0 (smallest event of the starting zone; light colouring). If no debris flow could be detected at a starting zone in a specific period, the respective colour was set to grey. The heatmap shows that out of the 82 different starting zones in GT and LT which were active at least twice, the largest debris flow event could be detected within the period 1990–1997 on 42 occasions; 73 % of the starting zones produced their maximum deposited volume between 1990 and 2009, however showing a decreasing tendency. Only on 11 different occasions could the largest debris flows be detected in the period 1954–1973, and within 2015–2018 only four starting zones produced their largest event of the total considered timeframe.

The results of the magnitude comparisons of the most active periods indicate a strong influence of the 1990–1997 event, as it produced the largest debris flow volumes in GT and LT by far. This in turn supports the assumption that the debris flows of 1992 (Becht, 1995) affected the debris flow activity in GT and LT for the following years, and the system needed some time to reach the state of before 1990.

The highly active period 1990–2009 with the highest debris flow magnitudes might have affected the debris flow system for even longer. The discrepancy between the high number of detected debris flows from 2015 to 2018 and the relatively small deposited volumes in the same period possibly points to recharge time effects of debris flow channels, as mentioned in Pelfini and Santilli (2008) and demonstrated in Jakob et al. (2005, 2020) and Berger et al. (2011). During the highly active period (1990–2009) the rockfall storages in the bedrock catchments were depleted in some cases. Thus, the debris flows triggered afterwards showed below-average magnitudes. This in turn indicates a very short-term change from transport-limited to supply-limited systems for some of the debris flow channels.

Rainfall events triggering slope-type debris flows can occur very locally (see Sect. 5.1). The precipitation data from the Horlachalm meteorological station can therefore not be related to debris flow processes in the entire Horlachtal. As a consequence, the calculation of triggering thresholds using intensity–duration relationships or other methods (Berti et al., 2020; Segoni et al., 2018) would thus be rather inaccurate. Due to the location of the meteorological station, the data are only set in relation to debris flows in ZT. The threshold value for debris flow triggering of 20 mm per 30 min according to Becht and Rieger (1997) has only been exceeded on 2 d since 1989, namely on 31 July 2002 and 2 July 2009. Nevertheless, the mapping of debris flow processes in ZT since 1947 shows that debris flows have also occurred during periods in which the threshold value of 20 mm per 30 min was not exceeded at the Horlachalm meteorological station. This in turn indicates that a precipitation event < 20 mm per 30 min in the study area can be sufficient to trigger debris flows.

In Fig. 13, all high-intensity rainfall events recorded at the Horlachalm station with precipitation values exceeding 5 mm per 30 min are shown together with the mapped number of debris flows per year in ZT and the calculated deposition volume per year in ZT. The acquisition dates of the orthophotos, which were used for mapping the debris flows, are marked as grey vertical lines in the figure.

Only very few debris flows were mapped in the period between 1990 and 1997 in ZT. The precipitation data show that between the times of the aerial image acquisition in 1990 and 1997, precipitation events of over 10 mm per 30 min were indeed recorded only rarely, and these few precipitation peaks always remained below 15 mm per 30 min. During that period, the precipitation value of 10 mm per 30 min was exceeded a total of 5 times on four different days. The maximum value is 12.6 mm per 30 min on 28 July 1997.

Between the aerial image acquisition in 1997 and 2003, however, considerably more numbers of debris flows and higher debris flow volumes per year were detected in ZT. The precipitation records also show more extreme rainfall events during this period. Precipitation exceeded the value of 10 mm per 30 min a total of 11 times on six different days. The maximum values are also far above the level of the last period, with a maximum value of 20.2 mm per 30 min on 31 July 2002. Even more debris flow processes per year were mapped in the period between 2003 and 2009.

On 4 d between the acquisition of the aerial images of 2010 and 2015, the value of 10 mm per 30 min was exceeded 6 times. However, the maximum of 13.9 mm per 30 min on 7 July 2015 is again relatively low. During this period, comparatively few debris flows per year were mapped in ZT. In the most recent time step between 2015 and 2018, the threshold value was only reached once at the Horlachalm meteorological station on 10 July 2017. At exactly 10.0 mm per 30 min, the maximum value for this period is relatively low. However, many debris flows per year could be determined in ZT during this time step. The corresponding debris-flow-triggering event cannot be traced in the precipitation data. It is therefore probable that this must have been a very local rainfall event that strongly affected ZT but could only be measured at a lower level at the Horlachalm meteorological station.

In general, a correlation between debris flow activity in ZT and precipitation data at Horlachalm station is recognizable. The highest precipitation intensities per 30 min were recorded in the time steps 1997–2003 and 2003–2009, and many debris flow processes could also be mapped during this epoch in ZT. During the periods 1990–1997 and 2010–2015 the maximum precipitation per 30 min was significantly lower, which is also reflected in the lower number of mapped debris flows per year. Only in the time steps 2015–2018 do the datasets not seem to match. This contrast is interpreted as a further indication for very local rainfall events as triggers of debris flows.

The evaluation of high-intensity rainfall events in combination with the mapped slope-type debris flows has shown that the threshold of 20 mm per 30 min specified by Becht and Rieger (1997) is indeed sufficient to trigger large debris flow events in the study area. However, even lower precipitation intensities seem to be sufficient to start debris flows on very active debris cones.