# Debris-flow susceptibility of upland catchments

## Abstract

Over the last three decades, many regional studies in mountain ranges under temperate climate revealed that it is possible to discriminate debris-flow and fluvial fans from morphometric indicators measured at the scale of the catchment and the fan itself. The most commonly used indicators are the Melton index (R), a normalized index of the gravitational energy of the catchment, and the fan slope (S). A wide range of thresholds have been proposed for discriminating purpose, but these are generally based on a small population of catchments and may be highly influenced by ambiguous fans included in the data set. A database of 620 upland catchments from several mountain ranges under temperate climate was compiled from the literature to propose robust discriminant morphometric thresholds for debris-flow versus fluvial responses. Linear discriminant analysis (LDA) and logistic regression (LR) were performed using the whole data set, and a leave-one-out cross-validation was used to evaluate performances of the models. Sensitivity and specificity scores obtained for LDA and LR were 0.96 and 0.73, and 0.95 and 0.75, respectively. It is also shown that the channel slope above which debris-flow is observed decreases with the gravitational energy of the catchment. Limitations of the morphometric discrimination are discussed.

## Keywords

Debris-flows Susceptibility analysis Morphometric controls Linear discriminant analysis Logistic regression## 1 Introduction

Small torrents are prone to extreme run-off events that may have dramatic consequences for exposed people and infrastructures. These events are generally triggered by intense convective rainfalls that occur on steep and eroded terrains. At the catchment outlet, these run-off events can take different forms depending on the sediment concentration of the flowing mixture. As this concentration increases, clear water run-off progressively transforms into a sediment-laden flood and finally into a debris-flow at a threshold around 50 % of sediment concentration. Although extreme floods in small torrents may induce catastrophic damages and loss of lives, it is generally recognized that the most dangerous torrents are those that are prone to debris-flows. A recent study in the Eastern Italian Alps showed that for a same return period, debris-flow volumes are two to three orders of magnitude higher than bedload volumes transported by floods (Mao et al. 2009). It is also recognized that for a same drainage area, debris-flow peak discharges are much higher than those of floods (Hungr et al. 2001). Therefore, in flood hazard assessment in mountains, there is a great concern to determine debris-flow susceptibility. It refers to the spatial probability of occurrence of debris-flow, that is, the likelihood that an event occurs in an area under particular physical conditions, not including the temporal dimension. This kind of analysis is often used as a first step in hazard assessment (Jakob 2005).

Several literature studies focus on morphometric indicators to roughly assess the dominant flow type on alluvial fans and classify them into two categories: alluvial fans dominated by sediment gravity flows named debris-flow fans, and alluvial fans dominated by fluid gravity flows named fluvial fans (Jackson et al. 1987). The pioneer works were initiated in the 1980s by Kostaschuk et al. (1986), Jackson et al. (1987) and Wells and Harvey (1987) who tried to identify the debris-flow hazard on alluvial fans using catchment and fan metrics. Studies conducted in the Canadian Rockies (Kostaschuk et al. 1986; Jackson et al. 1987) showed that the Melton index and the fan slope were the most powerful indicators to predict the dominant process occurring in the few small upland catchments they studied. The Melton index (or Melton’s ruggedness number) is an index of the ruggedness of the catchment (Melton 1965) and is calculated as the ratio of the catchment relief (difference between the maximal elevation of the catchment and the elevation of the fan apex) by the square root of the catchment area measured at the fan apex. It is a normalized index of the gravitational energy of the catchment. Fan slope is conventionally measured as the drop rate between the fan apex and the lowest point along the fan perimeter (Jackson et al. 1987). These early works have been followed up by an increasing number of studies in different upland environments over the last 20 years (e.g. Marchi et al. 1993; Calvache et al. 1997; Sorriso-Valvo et al. 1998; Marchi and Brochot 2000; De Scally and Owens 2004; Rowbotham et al. 2005; Santos Alonso 2011; amongst others).

The prediction of the dominant sediment transport process at a catchment outlet from morphometric properties of both fans and catchments is based on several assumptions. Alluvial fans are sedimentary deposits resulting from a long and complex history of aggradation and degradation cycles related to the changing nature of environmental controls affecting the sediment and water regimes of the catchment (Harvey 1984; Blair and McPherson 1994). During the long-lasting period of fan formation, it is likely that the dominant sediment transport process shifted from one type to another, as a response to land-use or climate change. Therefore, some of the present features visible at the surface or in the stratigraphic record of the fan may result from processes which are no more active today. The prediction of the dominant sediment transport process acting on the fan from morphometric indicators is based on two basic assumptions that may or may not be always valid. It must be assumed that (i) no change in the dominant transport process has occurred at the timescale considered in the study and that (ii) this dominant process can be inferred from the present morphological signature of the fan. Here, the timescale we considered as valid for the identification of the processes occurring on the fans is multi-decadal or secular.

Recent regional-scale studies compared debris-flows inventories with potential driving factors (topography, geology, land-use). Some of them proposed multivariate models integrating geological controls (D’Agostino and Marchi 2001; Crosta and Frattini 2004). These studies suggest that multivariate analyses are efficient methods to discriminate the processes occurring on the fans, but they are more difficult to implement considering the greater number of indicators needed, and their results are not significantly improved compared to more sparing bivariate models. Moreover, most of them are based on a reduced number of catchments (always below 130, and generally few tenths); thus, the discriminant thresholds proposed are not very robust. To build a broader-scale statistical model for flow type identification in upland catchments, we revisited these studies and compiled data from various alpine environments. Our objective was to test whether a robust statistical model, adapted to different regional conditions, could be obtained to discriminate debris-flow catchments from fluvial catchments by using only the two most widely used morphometric indicators for debris-flow susceptibility (Melton index and fan or channel slope). These two basic indicators present the advantage of being easily automatically extracted from DEM or contour-level maps. We tested two multivariate statistical models: a linear discriminant analysis (LDA) and a logistic regression (LR). The results and the performance of these two models were compared. Here, we present (i) the database compiled from the literature and (ii) the results from the two implemented statistical models.

## 2 Materials and methods

### 2.1 Database compilation

References used for database compilation of debris-flow and fluvial catchments classified by alpine regions

Region | | Debris-flow catchments | Fluvial catchments | References |
---|---|---|---|---|

Austrian Alps | 31 | 16 | 15 | Scheidl and Rickenmann (2010) |

Italian Alps | 198 | 175 | 23 | Ceriani et al. (2000), D’Agostino and Marchi (2001), Mambretti (2009), Marchi et al. (1993), Marchi and Cavalli (2007), Scheidl and Rickenmann (2010), Lenzi (2000) |

Apennines | 31 | 26 | 5 | Sorriso-Valvo et al. (1998) |

French Alps | 134 | 64 | 70 | Malet et al. (2004), Marchi and Brochot (2000), Remaitre (2006), Thénard (2009), Liébault (2003) |

Swiss Alps | 40 | 40 | 0 | Scheidl and Rickenmann (2010) |

Pyrenees | 5 | 3 | 2 | Gomez-Villar and Garcia-Ruiz (2000), Hürlimann et al. (2006) |

Canadian Rockies | 51 | 35 | 16 | Jackson et al. (1987), Jordan (2007), Kostaschuk et al. (1986) |

North Cascade USA | 12 | 12 | 0 | Kovanen and Slaymaker (2008) |

Southern Alps NZ | 118 | 73 | 45 | |

Total | 620 | 444 | 176 |

In one of the compiled studies, the fan slope was not reported, but channel slope was available (Liébault 2003). This concerns 51 fluvial-dominated catchments in the Southern French Prealps where the channel slope was measured in the field along a reach near the catchment outlet with a length equal to between 10 and 20 active channel widths. We considered that fan slope can be substituted by channel slope to characterize local (at-a-point) topographic conditions. In alluvial constructions, both are controlled by the sediment transport regime of the reach and both provide local gravitational conditions allowing conveying a debris-flow downstream.

Most of the studies we used identified the dominant flow processes thanks to field expertise of the fan/channel morphology or air-photographs interpretation, and a very few are purely based on historical archives (maps, past inventories or technical reports) about past activity. In those cases, the authors used the recognition of the actual flow process on the field to validate the historical data. Given the unavailability of the raw data used to classify each catchment, it was not possible to evaluate the classification and to propose our own interpretation of the dominant sediment transport process. Therefore, we supposed that the identification of debris-flow process in the case studies as robust and chose to consider only two groups of catchments: (i) those that are known to produce debris-flows as attested by field surveys and/or historical documents and (ii) those that never produce debris-flow. The fans presenting any of the evidences of debris-flow activity (lobes, boulder levees, isolated boulders on the fan surface with diameters exceeding about 1 m, amongst others (Kostaschuk et al. 1986; Jackson et al. 1987; Wilford et al. 2004)) were identified as debris-flow fans. Fans are classified as fluvial if any sign of debris-flow activity was recorded in historical archives and/or field surveys.

In the literature, other groups may be proposed, such as debris flood (Scheidl and Rickenmann 2010) or mixed fans (Marchi et al. 1993; Marchi and Brochot 2000). We considered that these distinctions may be ambiguous and difficult to determine in the field or by using historical archives, so we decided to adopt a more objective binary classification. Mixed fans were classified as debris-flow fans, since it is recognized that they are susceptible to be affected by debris-flows. Debris flood fans were classified as fluvial fans, in agreement with the commonly used definition of debris flood, a term which refers to a rapid surging flow of water, heavily charged with debris (Hungr et al. 2001). The presence of a free water surface in debris floods clearly shows that this type of flow is closer to water floods than debris-flows. From the 620 catchments in the database, 72 % are classified as debris-flow and 28 % as fluvial. We referred as fluvial processes when no evidences of debris-flow are reported for the site; in this case, we assume the channel is only shaped by bedload transport. The database presents a great prevalence of positive cases (a positive case being defined as an observation of the most dangerous phenomenon, i.e. the debris-flow in our case). This is not very common in natural hazards studies, where negative cases are generally much more prevalent than positive ones since dangerous phenomena are by nature infrequent (Beguería 2006). The prevalence of debris-flow catchments in the compiled database is due to the integration of studies exclusively addressing debris-flow fans and due to the propensity of authors to oversample debris-flow fans in their dataset. Another reason is that the revisited case studies mostly consist of catchments sampled along mountain valleys, where small, rugged catchments building debris-flow fans are much more frequent than large, mild-slope catchments dominated by fluvial processes.

The catchments are located in nine different alpine regions of the world: Apennines, Austrian Alps, Canadian Rockies, French Alps and Prealps, Italian Alps, North Cascade of USA, Pyrenees, Southern Alps of New Zealand, and Swiss Alps. The common physical features of all of these regions are the ruggedness of the relief and the humid-temperate or Mediterranean climates (arid and semi-arid mountains, as well as tropical mountains are not present in the sample, but Mediterranean conditions are: i.e. Apennines and Catalan Pyrenees and to a lesser extent Alps and French Prealps). On the over hand, they cover a very wide spectrum of geological conditions (sedimentary, metamorphic and volcanic rocks), palaeoclimatic heritages (glacial and periglacial conditions during the last glacial period), present-day tectonic activity and hydrological regimes (from rainfall to melt-water dominated run-off).

### 2.2 Statistical analysis

*R*and S were used to satisfy the normality condition of the LDA, but the standardization of the variables did not appeared necessary in our case.

*p*. The LR is based on the hypothesis that the logarithm of the odds ratio (\( \log \left( {\left. {\frac{p}{1 - p}} \right)} \right. \)or logit function) is a linear function of the two control variables:

*n*= 2,

*β*

_{0}is the constant term,

*β*

_{ i }are the coefficients applied to the two predictive variables

*X*

_{ i }which in our case,

*X*

_{1}and

*X*

_{2}, are the log-transformed

*R*and

*S*, respectively. The probability of belonging to the debris-flow response group is given by:

The compiled database of 620 catchments presents much more objects in the debris-flow group than the fluvial group (Table 1). The sample prevalence may have a negative effect on both LDA and LR. This is still under debate in the statistician community (Xue and Titterington 2008; Jiménez-Valverde et al. 2009), and there is no clear theoretical argument to conclude about a significant negative effect. We therefore performed the statistical processing with balanced and unbalanced training sets to test empirically the effect of prevalence on model performance. Balanced training sets containing 90 % of fluvial catchments (158) and an equivalent number of debris-flow catchments (each training set contains 316 objects) were randomly sampled 1,000 times without replacement. Unselected catchments (*n* = 304) were used to characterize the performance of the statistical models (referred as target sets). Unbalanced training sets were also sampled 1,000 times, each containing 90 % of catchments in each category (158 fluvial catchments and 400 debris-flow catchments). Target sets containing unsampled catchments were used to characterize the model performance (18 fluvial catchments and 44 debris-flow catchments). Sampling operation of balanced and unbalanced training and target sets was carried out twice to perform, respectively, 1,000 LDA and 1,000 LR models.

Results of the two models were compared for balanced and unbalanced response groups with two performance indicators not influenced by prevalence (Beguería 2006). The first is the sensitivity score, which was calculated as the ratio of correctly predicted debris-flow catchments (true positives) in the validation set. The second one is the specificity score, which gives the ratio of correctly predicted fluvial catchments in the validation set (true negatives). As the LR models give a probability of debris-flow response, we use probability threshold of 0.5 to establish in which category the response is predicted. It means that catchments with a probability of debris-flow response higher than 0.5 have a predicted debris-flow response, and catchments with a probability to debris-flow response lower than 0.5 have a predicted fluvial response.

After comparing the results of the two designs with the two statistical models, we built the two final models taking into account the entire data set. We compared both final models with receiver operating characteristic (ROC) curves (i.e. a graphical plot of the performance where the x-axis is the false-positive rate (1 specificity) and the y-axis is the true-positive rate (sensitivity) both calculated for different thresholds).

We also evaluated the models by performing a leave-one-out cross-validation (LOOCV) and calculated the 95 % confidence intervals for the LDA and LR coefficients. This validation method consists in *n* repetition of model building (*n* is the number of observed catchments) taking *n*−1 catchments as training data and the remaining catchment as the validation data (i.e. each catchment is used once as validation data).

## 3 Results

### 3.1 ANOVA tests

*p*value <0.0001 (Fig. 1). Mean values of the Melton index are 0.79 and 0.27 for debris-flow and fluvial responses, respectively, and mean values of fan/channel slope were 7.5 and 1.6° for debris-flow and fluvial groups, respectively.

### 3.2 Fan/channel slope versus Melton index

*S*/

*R*scatter plot in the paper) is very instructive (Fig. 2a). At first, it reveals that the fan/channel slope is a power function of the Melton index with an exponent of 1.07 (

*R*² = 0.59,

*p*value <0.0001). Steeper catchments produce steeper fan/channel slope, as already observed in many regional studies. Although a significant relation is obtained, the scatter is very high: for the same Melton index, the fan/channel slope spreads over two orders of magnitude. Secondly, it shows that debris-flow and fluvial catchments are clearly separated on the scatter plot. The fans dominated by debris-flow show higher energy catchments, a greater relief and also smaller catchments (i.e. a higher Melton index). Debris-flow catchments tend to occur for high values of both predictors, and fluvial catchment for low values, but the two groups overlap (Fig. 1). Ninety-five per cent of fan slope observations lie between 2.05 and 22.04° for debris-flow catchments, and between 0.46 and 7.27° for fluvial catchments. Ninety-five per cent of Melton index observations lie between 0.29 and 1.71 for debris-flow catchments, and between 0.10 and 0.90 for fluvial catchments.

No geographic patterns are observed in the scatter plot. Catchments belonging to different mountain ranges have similar ranges of *S*/*R* values, and it is not possible to detect significantly different thresholds between regions (Fig. 2b). The residuals of the linear regression are normally distributed for all the regions except for the Pyrenees and the North Cascade where the residuals are only positive. Catchments in these regions have steeper fan slopes for the same range of Melton values, but this is based on a very limited number of observations (5 for the Pyrenees and 12 for the North Cascade), and it is not possible to conclude about a specific behaviour for these regions. The use of channel slope instead of fan slope does not show any effect on the scattering of the fluvial-dominated catchments (only the 51 catchments in the Southern French Prealps for which fan slope was not available and channel slope was used instead) (Fig. 2c).

### 3.3 Prevalence effect on statistical model performances

When we compare the sensitivity and specificity scores of the LR by taking balanced and unbalanced response groups, the results are similar to those found with LDA models (Fig. 3b). The 95 % confidence interval of sensitivity scores is higher with unbalanced designs (0.89–1.00) compared with balanced designs (0.84–0.92). The median values show the same trend (0.88 for balanced design and 0.96 for unbalanced design). For the specificity scores, the 95 % confidence interval is higher for the balanced designs (0.61–1.00) than for the unbalanced designs (0.56–0.95). Results are equivalent for the median values (0.83 and 0.72 for balanced and unbalanced designs, respectively). Prevalent data sets were also used for the final LR model since unbalanced designs tend to better predict the occurrence of debris-flow catchments.

### 3.4 Statistical models and performances

*VIF*) (O’Brien 2007) in order to characterize the degree of collinearity. In our case, the

*VIF*is only 2.44, which is much less than 4, 5 or 10, the thresholds proposed in the literature above which collinearity is expected to have a strong effect on parameters variance. This shows that some collinearity exists between the predictors but is not excessive. This means that the standard error for the coefficient of Melton is 1.56 times greater than if the collinearity between Melton and the fan slope was almost null.

Collinearity of predictors impacts the precision of the estimated coefficients of the regression, but not their significance (*p* value lower in both cases), and does not induce a statistical bias for the LDA model. We have considered a relation of the inverse form (the response depends primarily on fan slope and then on Melton), and the signs of the coefficients were unchanged (positive). In our case, this collinearity does not change the sign of the coefficients, so does not change the sign of the estimated effect and plays a very negligible role in the statistical analysis.

Performances of the two statistical models were first compared with the sensitivity and the specificity scores calculated on the training data set. The LDA model shows a sensitivity of 0.96 and a specificity of 0.73. The LR model shows a sensitivity of 0.96 and a specificity of 0.75.

Performances were also characterized using a cross-validation with the LOOCV method. For the LDA model, the 95 % confidence interval of the slopes is between −0.87 and −0.83, and for the intercept between 0.21 and 0.25. For the LR models, the 95 % confidence interval of *β* _{0}, *β* _{1} (*R*) and *β* _{2} (*S*) are −0.60 and 0.70, 1.63 and 1.70, and 1.98 and 2.03, respectively. The global sensitivity and specificity for the 620 LOOCV LDA models are 0.96 and 0.73, respectively, and the percentage of correct classification is 0.89. The global sensitivity and specificity for the 620 LOOCV LR models are 0.95 and 0.75, respectively, and the percentage of correct classification is 0.89.

## 4 Discussion and conclusion

Data compilation from 620 upland catchments in various mountain ranges of the world under temperate climate provided the opportunity to test the performance of morphometric indicators for the identification of catchments prone to debris-flow. This study confirms that the Melton index and the fan or channel slope are very good predictors of the dominant sediment transport process of small upland catchments. Debris-flow prone catchments are characterized by steeper catchments and steeper channel slopes than those which only produce bedload transport, as already observed in many regional studies (Kostaschuk et al. 1986; Jackson et al. 1987; Marchi et al. 1993; Marchi and Brochot 2000; De Scally and Owens 2004; De Scally et al. 2010).

Most of the previous studies (Jackson et al. 1987; Marchi and Brochot 2000; De Scally and Owens 2004) provided a unique threshold value for single morphometric predictors above which debris-flow may occur. However, there is no discontinuity between debris-flow and fluvial response in the distribution of both morphometric indicators, and a great proportion of fluvial catchments take values of Melton and fan slope close to the median values of debris-flow catchments (Fig. 1). Instead of using unique thresholds, the combination of the two indicators into the proposed discriminant function (with a slope >0) shows that the fan/channel slope threshold above which debris-flow occurs decreases when the Melton index increases. This is in good agreement with experiments of granular collapses over sloping bed (Hungr 2008; Mangeney et al. 2010) which shows that the slope of the deposit (an analogue to the fan slope) decreases with the increase of the slope of the plane along which the granular flow is routed (an analogue to the catchment gravitational energy). This suggests that the minimum slope at which debris-flows can propagate is controlled by the mean slope of the catchment.

We used LDA and LR to discriminate fluvial and debris-flow processes. These two models show almost identical sensitivity and specificity scores. LDA specificity is a little lower compared to LR model. Thus, false alarms will be more frequent with the LDA. Both statistical models perform well, but better predictions are obtained with LR.

Scores of the local models when used to predict the response of the 620 catchments

Models | Models formulas | Tp | Fp | Fn | Tn | Sensitivity | Specificity |
---|---|---|---|---|---|---|---|

LDA (this study) | Equation (4) | 427 | 48 | 17 | 128 | 0.96 | 0.73 |

LR (this study) | Equation (6) | 421 | 45 | 21 | 131 | 0.96 | 0.75 |

Jackson et al. (1987) (a) | 385 | 101 | 59 | 75 | 0.87 | 0.43 | |

Jackson et al. (1987) (b) | 383 | 76 | 61 | 100 | 0.87 | 0.57 | |

Marchi et al. (1993) | | 433 | 61 | 11 | 115 | 0.97 | 0.65 |

Marchi and Brochot (2000) (a) | 377 | 68 | 67 | 108 | 0.85 | 0.61 | |

Marchi and Brochot (2000) (b) | 370 | 59 | 74 | 117 | 0.83 | 0.66 | |

De Scally and Owens (2004) | 173 | 14 | 271 | 162 | 0.39 | 0.92 |

Scores of the local models and of the two regional models proposed in this study when used to predict the response of the local databases

Local databases | | Sensitivity of the local models | Specificity of the local models | Sensitivity of LDA (Eq. 4) | Specificity of LDA (Eq. 4) | Sensitivity of LR (Eq. 6) | Specificity of LR (Eq. 6) |
---|---|---|---|---|---|---|---|

Jackson et al. (1987) (a) | 42 | 0.93 | 0.64 | 0.93 | 0.79 | 0.86 | 0.79 |

Jackson et al. (1987) (b) | 42 | 0.90 | 0.78 | 0.93 | 0.79 | 0.86 | 0.79 |

Marchi et al. (1993) | 47 | 1 | 1 | 0.98 | 1 | 0.98 | 1 |

Marchi and Brochot (2000) (a) | 39 | 1 | 1 | 1 | 1 | 1 | 1 |

Marchi and Brochot (2000) (b) | 39 | 1 | 1 | 1 | 1 | 1 | 1 |

De Scally and Owens (2004) | 59 | 0.95 | 0.83 | 1 | 0.17 | 1 | 0.28 |

The morphometric approach of debris-flow susceptibility assessment presents some limits. As we previously said, the distribution of both morphometric indicators for debris-flow and fluvial response overlaps, which is critical for the discrimination of the catchment’s response (Fig. 1). The local slope we took into account is not always the fan slope, especially for fluvial fans in the Southern French Prealps. We considered that the channel slope is a good surrogate for fan slope when the channel is not constrained by bedrock.

The morphometric approach is restricted to the characterization of the minimum gravitational energy required for debris-flows to propagate along the stream network (steep catchments are expected to produce debris-flows susceptible to stop at lower slopes and then to travel longer distances). But debris-flow occurrence is only partly explained by gravitational energy. The spatial variability of debris-flow occurrence along a stream network is also controlled by the magnitude and properties of the sediment supply (Theule et al., 2012). These sediment controls are very difficult to constrain at the regional scale, because they are closely linked with the geomorphic and geological variability of the landscape. We can argue that the morphometric approach provides a conservative assessment of debris-flow catchments, since the discriminant thresholds allow identifying all the catchments presenting enough gravitational energy to produce a debris-flow. Some of them will not produce debris-flow because sediment supply is insufficient or because the grain-size distribution of the sediment supply (which has an influence on the rheological properties of debris-flow) induces a lower than expected run-out distance. Another limitation is that the morphometric approach ignores the conditions of sediment transfer in the catchment, which could be interrupted by sediment trap (i.e. glacial lakes) or a reduction of the gravitational energy along the streams from upstream to downstream (Jackson et al. 1987; Marchi et al. 1993).

This gravitational energy can be quantified by morphometric parameters that can be easily constrained at regional scales from low resolution (~10–50 m) digital elevation models. The statistical models derived from this study are very useful for regional scale mapping of debris-flow susceptibility along a stream network. Each segment of the stream network can be classified according to its probability to be travelled through by a debris-flow.

## Notes

### Acknowledgments

This research is a part of the EU Interreg Alpine Space PARAmount project (“imProved Accessibility: Reliability and security of Alpine transport infrastructure related to mountainous hazards in a changing climate”) and CPER PACA Rhytmme project. The authors thank Lise Vaudor for her advices with statistical processing, Aurélie Carlin for her help with database compilation and preliminary analysis on the subject, and the ISIG platform at ENS of Lyon for material support. This paper benefits from comments by three anonymous reviewers.

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