Debris-flow risk-to-life: upper-bound preliminary screening

. Where the potential for future debris-flow occurrence is unrecognised, developments can be unknowingly exposed to debris-flow impact, with corresponding risks to lives. Debris-flow modelling is unsuited to routine local office use, so a simple screening procedure is proposed to enable local officials to identify locations where debris flow risk-to-life may be unacceptable, and prioritise where expert modelling and risk analysis are most urgently required for risk-management decision-making . This procedure calculates catchment Melton ratio R from topographic data, uses a linear upper bound of field data relating R to the annual probability of debris-flow occurrence, and matches a model-based debris-flow risk-to-life analysis for Matatā, New Zealand. Our data suggest that any development exposed to debris flows will require a detailed risk assessment to ensure that risk-to-life does not exceed acceptable levels.


Introduction
Debris flows are extreme events triggered in steep, erodible catchments by intense rainfall causing high sediment inputs to streams. Debris flows differ from normal floods in material composition (a dense slurry of fine sediment in water capable of transporting high concentrations of coarse sediment), flow behaviour (multiple discrete surges with bouldery heads and finer, less dense tails) and frequency (decadal or less in many catchments). Thus, while debris-flow risk management is often assumed to be similar to flood risk management, it is fundamentally different [1].
Because debris flows occur infrequently in any given watershed, many catchments susceptible to debris flows have no historic records of their occurrence, so their ability to generate debris flows may be unrecognised. Development of land in such areas means that assets might be vulnerable to debris-flow impact in the future. Future deaths and damage can be avoided only if debris-flow-susceptible areas can be identified easily and reliably. Reconnaissance-level studies [2,3] suggest that significant developments currently occupy areas in New Zealand that may be subject to debris-flow hazards. Recent studies [4] show that debris-flow riskto-life is a growing issue worldwide.
Over recent decades there has been a global move towards management of risk (defined as event probability x event consequence) as the key factor for quantifying the impacts of natural hazards on society. The 2015 Sendai Framework [5] focussed international disaster management on disaster risk reduction, which requires the ability to quantify risk, in turn requiring the ability to identify hazards and quantify their potential for damaging society. * Corresponding author: tim.davies@canterbury.ac.nz To reduce future debris-flow disasters, it is therefore necessary to identify existing and proposed developments located in debris-flow hazard areas and to quantify the corresponding risks. This ability is required by organisations responsible for land-use planning, particularly their in-house and/or external advisers in natural hazard risk, science and engineering. The present work is aimed at such organisations and advisers, particularly in regions where debris-flow risk is presently poorly known, and people with limited skills and time may need to assess large numbers of catchments.
The urgent and widespread need to routinely identify debris-flow hazard zones and realistically constrain their annual probabilities of occurrence is unlikely to be met by debris-flow modelling. This is because such modelling [6,7] requires detailed data acquisition and analysis by a range of specialists, and is thus likely to be both lengthy and expensive. Our purpose herein is to develop an easily-usable methodology to meet this need, readily-available data. Where developments exposed to debris-flows hazards are thus identified, more sophisticated methods can be used to better quantify the risks and develop risk management strategies for those specific locations.

Catchment characteristics
The primary requirement for debris flow occurrence is a large volume of widely-graded sediment available for transport. Thus debris flows usually occur in activelyeroding catchments with steep gradients, and we use a catchment steepness criterion to characterise a catchment's ability to generate debris flows: the Melton ratio (R), defined as the catchment relief divided by the square root of the catchment area. Melton ratio analyses can apply to a whole catchment or to catchment segments bounded at their lowest point by selected points in a stream network [8,9].
The Melton ratio has frequently been used to identify catchments capable of generating debris flows [10 and references therein]. It has the advantage that R can be computed on a regional basis using GIS with digital elevation models of moderate precision [3], but no association between R and debris-flow probability has hitherto been established. Fig. 1 shows data relating R to the presence of evidence of debris-flow occurrence for over 800 catchments in Canada (upper) [11] and over 600 others worldwide (lower) [12]. Clear (but different) relationships can be derived in each case between R and the proportion p of catchments that show evidence of debris flows (Fig. 2). A linear envelope (dashed line) denotes a conservative estimate of the maximum value of p for given R for all the data.

Debris-flow frequency
However, the annual probability of debris-flow occurrence in any given catchment remains unknown,. Using the envelope relationship of Fig. 2, in which p = pe is the maximum probability that a catchment with given R has produced a debris flow within the (unknown) past time te that it takes for debrisflow evidence to be eliminated by fluvial and other processes.

Fig. 1.
Data from [11] (upper) and [12] (lower) relating Melton Ratio R to evidence of debris-flow occurrence. The significance of this relationship is the fact that it is linear. Thus the relative R values of different catchments 1 and 2 are proportional to their relative maximum probabilities of generating debris flows within time te. Further, if we assume that te is similar between catchments, the relative R values also describe their relative maximum likely annual probabilities pa: The quantity of data underlying Fig. 2 (about 1500 catchments in Europe, North America and New Zealand) suggest that the envelope relationship may be universal, at least in these regions. If so, any catchment in these regions in which debris-flow probability has been determined can serve as a calibration catchment.
Only a small number of catchments worldwide have published data on the past occurrence of many debris flows. Table 1 lists some of these catchments (volcano catchments excluded) along with their annual probability of debris-flow occurrence pa and Melton ratio R. It is notable that the instrumented catchments generally have higher frequencies of debris-flow occurrence than the non-instrumented catchments, because they preferentially utilise catchments known to experience frequent debris flows. Significantly, the upper bound of all data for non-instrumented catchments is well-defined by the dashed line in Fig. 3: which is linear and consistent with the expectation deduced from Fig. 2. A much larger database would obviously be useful in better defining this relationship, but such data are rare. This kind of relationship, based on empirical data, appears more realistic than the common strategy of selecting a critical R-value below which it is (arbitrarily) assumed that debris flows never occur [2].

Risk calculation
Calculating risk for a given site (e.g. a small alluvial fan) requires knowledge of the impact a debris flow will have on a development. Since we calculate pamax as the upper bound of the probability vs R relationship, we also use the upper bound impact in calculating risk.
Consider a development with a population of 100 on a fan whose catchment has R = 0.2. In this case the maximum risk-to-life is 0.35*0.2*100 = 7 deaths per year maximum (dpym), assuming the greatest possible impact (i.e anyone affected by a debris flow dies). If the occupancy rate was 70% and occupancy time was 35%, the risk would reduce to about 2 dpym. A further reduction factor might be the proportion of the fan area on which people are present -if this is 50%, then we come down to 1 dpym. Fragility data for debris-flow deaths within dwellings could reduce the dpy by a further order of magnitude [25] to 0.1 dpym. In New Zealand, the acceptable limit is 10 -3 to 10 -4 dpy [26], so further investigation would be indicated in this case.
This example suggests that, for any catchment known to be able to generate debris flows, risk-to-life is likely to be high enough to require detailed investigation of whether mitigation measures will be needed to reduce the risk.

Example: Matatā, New Zealand
At Awatarariki fan, Matatā, North Island, New Zealand, which has R = 0.17, a comprehensive risk analysis was carried out following a destructive debris flow in 2005 [30]. Using the present approach, the upper-bound debris-flow probability would be calculated from (4) as pamax = 0.35*0.17 = 0.06 per year (~ 17-year return period; note that Table 1 gives pa = 0.03 from field data). Applying our value of life-vulnerability in a building on a fan impacted by a debris flow (0.1) to pamax (0.06) gives a maximum individual risk to life (MIRTL) = 6*10 -3 per year for the Awatarariki fan, which is clearly unacceptable. Thus our procedure indicates that debrisflow risk-to-life on the fan could be unacceptable, and detailed risk analysis and management would be needed. The risk analysis for Matatā [27] used a numerical model (RAMMS; [28]) calibrated on the volume, recurrence interval based on rainfall, and deposit extent of the 2005 event. The model generated deposit extents for a range of volumes, with corresponding probabilities estimated from rainfall frequencies. This, together with scenario distributions of occupation and vulnerability, yielded risk-to-life values on Awatarariki fan from 10 -2 to 10 -5 , that led to planned retreat from the area as the only realistic way to reduce the risk to acceptable levels [8]. The risk analysis for Matatā is consistent with the present analysis.

Discussion
The maximum individual risk to life MIRTL of a person in a dwelling in the catchment deposition area is estimated as MIRTL = 0.1*pamax = 0.035R ≈ R/30 If the acceptable risk level is < 10 -3 , then R needs to be < 0.03 in order to indicate no need for detailed investigation. This is conservatively low (the lowest R associated with debris-flow evidence in Fig. 2 is 0.14) as befits a screening analysis, and could be amended by expert personnel in specific cases.