Empirical evidence from the humid tropics shows that informal housing can
increase the occurrence of rainfall-triggered landslides. However, informal
housing is rarely accounted for in landslide hazard assessments at community or larger scales. We include informal-housing influences (vegetation removal, slope cutting, house loading, and point water sources) in a slope stability analysis. We extend the mechanistic model CHASM (Combined Hydrology and Stability Model) to include leaking pipes, septic tanks, and roof gutters. We apply this extended model (CHASM

Global and regional landslide records reveal an increase in rainfall- and human-triggered landslides during the last century, mainly in economically developing countries with rapid population growth and urbanisation (Kirschbaum et al., 2015; Froude and Petley, 2018). This increase might be partly due to continuing improvements in landslide recording, but it also indicates the growing impact of climate and urban pressure on landslide occurrence (Larsen, 2008). Understanding the mutual interactions between the natural and urban environment becomes particularly relevant in the humid tropics, where high-intensity and long-duration rainfall events are the main landslide triggers, and urban expansion is poorly regulated (Lumb, 1975; UN-Habitat, 2015). The natural landslide susceptibility of these regions coupled with the lack of urban planning and regulations can increase risk in terms of not only vulnerability and exposure but also hazard.

Potential anthropogenic landslide drivers include slope cutting and filling for house and road construction (Sidle and Ziegler, 2012; Smyth and Royle, 2000), slope degradation with clearance of forested areas (Gerrard and Gardner, 2006; Vanacker et al., 2003), inadequate drainage networks, unplanned redirection of storm run-off, and poorly maintained septic systems (Diaz, 1992; Anderson et al., 2008). In this paper, we use the term “informal housing” to refer to the combination of these urban modifications which influence slope stability by altering its geometry, hydrology, and material strength (Fig. 1).

Examples of informal housing affecting land stability. Panels

However, informal housing is usually neglected or not quantified in landslide hazard assessment at community and larger scales. There are two main reasons for this: lack of reporting and the highly localised scale and heterogeneous nature of human landslide drivers. A landslide is defined as triggered by human activities when there is a direct (and easily recognisable) connection with the failure process (e.g. during mining activities). Landslides of this type are small and often not recorded (Petley, 2012). When considering rainfall-triggered landslides, human landslide drivers are often either not considered or not distinguished from the natural drivers (SafeLand, 2011). Urban construction activities are localised, and even if they contribute to land instability, they remain difficult to observe either in situ (e.g. leaking pipes) or via satellite images. For these reasons, there are numerous site-specific analyses that investigate the influence of urban construction activities for individual slopes with known soil and rainfall trigger characteristics (e.g. Preuth et al., 2010; Zhang et al., 2012), but there are few studies that explore the influence of informal housing more widely for different combinations of human landslide drivers, soils, slope geometry, and rainfall triggers. This limits the transferability of the findings from slope to larger scales, where fewer detailed data are available.

Empirical-statistical and heuristic methods have been used in regional studies to link informal housing to the spatial and temporal occurrence of landslides. For example, precipitation and landslide records have been analysed in relation to lithology and land use change (Meusburger and Alewell, 2008; Gerrard and Gardner, 2006) or in relation to soil type and type of settlement (Smyth and Royle, 2000). Here, most of the recorded landslides were found to be associated with poorly regulated construction techniques, water management, and land degradation. Rainfall thresholds for triggering landslides were observed to depend on the proportion of impervious surfaces (Diaz, 1992). However, these analyses did not enable the differentiation of the relative role of natural and human landslide drivers, precluding the translation of the results into actions at the slope, i.e. engineering, scale (Anderson et al., 2013; Maes et al., 2017).

Mechanistic slope hydrology and stability models can be used to represent the landslide drivers for historical, current, and potential future climate conditions (e.g. Ciabatta et al., 2016; Almeida et al., 2017). If these models included the effect of informal housing, the analysis of different combinations of slope, urban, and climate properties could lead to the assessment of the relative role of natural and urban properties in triggering landslides and to the identification of the conditions at which urban construction activities become most detrimental. This could be useful information for engineers to prioritise slopes that are currently at risk, to identify those at higher risk of being impacted in the future, and to deduce appropriate hazard mitigation or preparedness actions. The inclusion of informal housing in slope stability analysis could also lead to considerations about the reliability of rainfall thresholds for triggering landslides within highly urbanised communities since they might be underestimating the level of the hazard (Mendes et al., 2018).

However, the use of data-intensive mechanistic models can be challenging in data-scarce locations, such as in low-income urban settlements. The more complex the model is, the more data will be required to set its parameters and model forcing and the more uncertainties might be introduced into the analysis. Sources of uncertainties can relate to slope and soil properties, urban features, and a limited understanding of physical processes or future scenarios (epistemic uncertainties; see Beven et al., 2018a, for a review of this issue). Many researchers have assessed the impact of uncertainties related to slope properties (e.g. Cho, 2007) and future climate (e.g. Ciabatta et al., 2016) on slope stability at different scales. However, to the best of our knowledge, there are no analyses that consider both sources of uncertainties when modelling informal housing in landslide hazard assessment. Urban construction activities are considered separately (e.g. slope cutting or leaking pipes; e.g. El-Ramly et al., 2006), or the slope properties are varied using discrete conservative values under fixed rainstorm conditions (Anderson et al., 2008; Holcombe et al., 2016). This separation might overlook significant changes in the slope's behaviour for combinations of urban construction activities and/or combinations of slope, soil, and rainfall properties that have not been considered in the analysis but are still likely to occur.

Almeida et al. (2017) demonstrated how mechanistic landslide models can consider uncertainties due to both poorly defined slope properties and to potential future climate changes. The mechanistic model CHASM (Combined Hydrology and Stability Model) was used in a Monte Carlo framework and applied in Saint Lucia, in the eastern Caribbean, where data support is limited, but landslide hazard is particularly high. The uncertainties in slope and soil properties were characterised through probability distributions extrapolated from available data and the literature, while the rainfall properties were varied uniformly across wide ranges also considering rainfall intensity–duration combinations that were not observed in the past but that might occur in the future. A sample of tens of thousands of rainfall events and slopes were stochastically generated from these distributions and simulated in CHASM. By this approach the possible effects of climate change were explored widely instead of focusing on one (or a few) climate projection scenarios (such as those provided by downscaled generalised circulation models) propagated through the modelling chain (Groves and Lempert, 2007; Wilby and Dessai, 2010). This strategy can be extended to include the exploration of both feasible climate and feasible land use futures (Singh et al., 2014). Statistical and data-mining algorithms were then used by Almeida et al. (2017) to quantify the relative role of the input factors (and thus their uncertainties) in the stability of the simulated slopes as well as to identify critical thresholds in slope properties and rainfall drivers likely to lead to slope failure. In this study we extend the work of Almeida et al. (2017) by including informal housing into such a slope stability analysis. We consider the same location of the humid tropics and the same core model, CHASM, but with new functions to represent the mechanistic influences of informal housing (new CHASM+). The core model is a two-dimensional model which has a relatively low data requirement for a mechanistic model even with the inclusion of the new informal-housing functions. In addition to the original ability to represent the mechanical and hydrological effects of vegetation and the effects of slope cutting and loading, we have added the effects of point water sources resulting from leaking septic tanks, water supply pipes, and houses without roof gutters. By varying both the natural and urban factors, we aim to identify the slope and climate conditions under which landslide hazard is significantly increased by the presence of informal housing and how this information can be used for deducing landslide mitigation measures. Thus, for our humid tropical case study scenario, we aim to address the following questions:

How can we identify which informal urban housing characteristics are most detrimental to slope stability?

How is the rainfall threshold for triggering landslides modified when informal housing is considered?

Which landslide mitigation strategies and practices can be deduced from the analysis for current and potential future scenarios of urbanisation and rainfall?

We want to analyse the relative role of informal housing in slope stability
under different natural and climate conditions. The methodology we introduce
here entails the following steps:

Choose a model that represents the main instability mechanisms of the case study area. We are interested in representing the rainfall-triggered landslides and the informal housing of Saint Lucia (Caribbean). We therefore use the mechanistic model CHASM, which represents not only the hydrology-stability routing but also vegetation, slope cutting, and, with the addition of new functions, various forms of water management (creating the extended CHASM

Define the input factors necessary to run the model and their variability space. In our case study, the input factors are the parameters defining the slope soil, geometry, urban characteristics, and rainfall-forcing data. Each input factor is assumed to be a random variable, and its range of variability is determined by a probability distribution. The probability distributions can be defined based on the physical meaning of the input factors, available data, and/or the existing literature. We use information gathered both from fieldwork in Saint Lucia and also from the literature.

Create synthetic combinations of input factors by stochastically sampling from their probability distributions, and run CHASM

Identify the input factors that most influence slope stability using global sensitivity analysis (Wagener and Pianosi, 2019). In particular, we use a regional sensitivity analysis (RSA) approach (Hornberger and Spear, 1981) to identify which input factors are most influential in leading to slope failure.

Identify parameters' thresholds beyond which the slopes become unstable. The threshold of an input factor over or below which failure is predicted might depend on the value of the other input factors (e.g. slopes with higher slope angles require higher soil strength to maintain stability). Machine learning is a set of methods that computers use to understand trends from data, also considering their mutual interactions. We use CARTs (classification and regression trees) to develop a set of decision rules that predict the combination of soil, geometry, urbanisation, and rainfall input values for which a particular slope is more likely to fail.

Saint Lucia is an eastern Caribbean island with a humid tropical climate. The main landslide trigger is rainfall, and shallow rotational landslides dominate on both steep and shallow slopes (Van Westen, 2016; Anderson and Holcombe, 2013). The geology is almost entirely comprised of volcanic bedrock and deep volcanic deposits. Due to the tropical climate, these volcanic parent materials are subjected to deep weathering, which decreases their strength and increases landslide susceptibility. The strata of a typical slope cross section comprise weathered residual soils overlying decomposed rock and volcanic bedrock. These three types of strata typically correspond, respectively, to the weathering grades V–VI, III–IV, and I–II of the Hong Kong Geotechnical Engineering Office weathering grade classification (GEO, 1988). There is a high variability in terms of engineering soils, but they can broadly be classified as fine-grained soils such as silty clays, clayey silts, and sandy clays (DeGraff, 1985). The combination of tropical climate, steep topography, and volcanic geology renders the region particularly susceptible to rainfall-triggered landslides. Furthermore, landslide risk is increased by informal housing which occupies steep slopes and employs unregulated engineering practices (World Bank, 2012, 226–235). Various sources of information on the slope, soil, rainfall, and urban properties of this region are available from previous studies by government engineers and planners, the local water company, and consultants (e.g. CHARIM, 2015; Mott MacDonald, 2013; Klohn-Crippen, 1995) as well as from community-based projects for the improvement of slope stability with surface water drainage works (Anderson and Holcombe, 2013). In this project, estimates of soil strength properties are based on direct shear tests of local soils (Anderson and Kemp, 1985; DIWI, 2002; Holcombe, 2006) and secondary data sources on similar volcanic tropical residual soils such as those in Hong Kong (Anderson, 1982; Anderson and Howes, 1985). Information about soil type, soil depth, type of house construction, cut slope angles, and the management of surface run-off and waste water on slopes was based on community-based mapping and elicitation of local expert knowledge undertaken by Anderson and Holcombe (2013), who co-developed these datasets with residents, government, and local experts.

CHASM (Combined Hydrology and Stability Model) is a 2-D mechanistic model which analyses dynamic slope hydrology and its effect on slope stability over time. A full description of the model can be found in Anderson and Lloyd (1991) and Wilkinson et al. (2002a, b). Here we briefly describe its hydrology and stability components, whereas the representation of the urban properties is detailed in Sect. 2.3. In CHASM the slope cross section is represented with a regular mesh of columns and cells. Hydrological and geotechnical parameters are specified per cell, while the initial hydrological conditions define the position of the water table and the matric suction of the top cell of each column. The dynamic forcing for CHASM is rainfall specified in terms of intensity and duration. For each computational time step (usually 10–60 s), a forward explicit finite-difference method is used to solve the Richard's (1-D, vertical flow) and Darcy's (2-D flow) equations, which regulate, respectively, the unsaturated and saturated groundwater flow. At the end of each simulation hour, the resulting soil pore water pressures (positive and negative) are used as input for the slope stability analysis, which implements Bishop's simplified circular limit equilibrium method (Bishop, 1955) and uses the coordinates of the slope surface. An automated search algorithm identifies the location of the slip surface with the minimum factor of safety, FoS, which is given as output at the end of each simulated hour. In a validation exercise in Hong Kong, CHASM showed an accuracy of 72.5 % (Anderson, 1990), which is comparable to the performances of other models used for the stability analysis (e.g. Formetta et al., 2014, p. 639). CHASM has been employed in Malaysia, Indonesia, eastern Caribbean, and New Zealand to propose landslide mitigation measures as well as to identify land instability drivers along roads and in urban and rural areas (Brooks et al., 2004; Lloyd et al., 2001). Almeida et al. (2017) used CHASM stochastically in a Monte Carlo framework to account for uncertainties in both slope properties and future climate scenarios.

The new CHASM

We use 30 input factors to characterise our case study area in CHASM

Input factors of CHASM

U: uniform distribution; Ud: discrete uniform; N: normal distribution; Ln: log-normal distribution.

The slope geometric properties consist of the natural slope (and associated slope height) and the material thickness. Slope angles vary between 20 and 45

The height of the water table is defined as an initial hydrological condition. This water table height is varied between 0 % and 90 % of the
slope height (

The model is forced with rainfall events which are specified in terms of
their duration (in hours) and hourly intensity. The aim is to create both
rainfall events that have been observed in the past and rainfall events
that might occur in the future (e.g. with higher intensity and duration than
observed historically). To constrain the rainfall variability space, we use
the intensity–duration–frequency relationships (IDFs) derived from a Gumbel
analysis of 40 years of daily rainfall data from weather stations across the
island by Klohn–Crippen (1995; Fig. 2). From these IDFs we derive a range of rainfall intensities between 0 and 200 mm h

Rainfall intensity–duration–frequency (IDF) curves for Saint Lucia developed by Klohn–Crippen (1995) using Gumbel analysis of 40 years of daily rainfall data from 15 rainfall gauges. The light-grey section includes rainfall events from observed data (below IDF curves); the dark-grey section represents combinations of rainfall intensity–duration not recorded in the past but that might occur in the future (above IDF curves).

Informal housing is represented by four urban properties: slope cutting,
absence of roof gutters, vegetation removal, and leaking pipes and tanks. While the angle of the cut slope is varied according to its probability distribution, the vegetation, roof gutters, and water leakage are defined as present (option 1: yes) or absent (option 0: no; Fig. 3). The cut slope angle is varied between 39 and 89

Urban properties of informal housing included in the slope stability analysis. Each house corresponds to a cut on the slope. Cut slope angle varies according to its probability distribution, defined in Table 1. Vegetation, roof gutters, and leaking tanks and pipes are stochastically inserted or not. The house on the cut slope is always present, and its load is not varied. The height of the cut slope varies relative to the cut slope angle, but it is forced to be maximum 4 m.

The input factors that define the discretisation of the model, such as the
cell size of 1 m

We use Latin hypercube sampling (McKay et al., 1979) to generate 10 000 different combinations of the 30 independently varying input factors shown in Table 1. Figure 4 illustrates one example of a slope defined by a single combination of these input factors. Due to the randomness of the process, checks are undertaken to ensure that realistic combinations of factors are generated; if not, they are discarded (around 70 % of the time) and replaced by another randomly generated, feasible combination. These
“feasibility” checks are reported in the footnote of Table 1 (letters a–f). The stochastically generated simulations are then run in
CHASM

Example of slope generated by stochastically sampling from the ranges of input factors specified in Table 1.

Global sensitivity analysis is a set of statistical techniques that evaluate
how the variations in a model's outputs can be attributed to the variations
in the model's input factors. In this case we want to identify which input
factors – among geometry, soil, hydrology, rainfall, and urban properties –
have the strongest impact on slope stability. Since in our case the model output is binary as simulated slopes are categorised as failed (if FoS

Classification and regression tree (CART) analysis is a supervised machine
learning method which we use to predict critical thresholds in input factors
over or below which a particular slope is more likely to fail (Breiman et al.,
1984). In this analysis, the predictor model takes the form of a binary
tree. Starting from the whole set of simulations, CART finds the best possible input factor (e.g. slope angle rather than rainfall intensity) and
the best possible value of that input factor (e.g. slope angle greater or
less than 30

In this section we analyse the

Percentage of predicted stable and failed slopes per urban property. An urban property influences slope stability if the percentage of the predicted failures changes with the variation in that urban property.

Sensitivity index for each input factor in the urbanised (full colour) and non-urbanised (pale colour) cases. The bars correspond to the
mean value of sensitivity for each input factor calculated with bootstrapping, while the vertical black lines at the top of the bars represent the confidence interval (number of bootstrap resampling

We then perform RSA on both sets of urbanised and non-urbanised slope simulations, calculating the cumulative marginal distributions of the failed and stable simulations for each input factor. The maximum distance between the two distributions (KS statistic) is computed and used as a sensitivity index. A high value of the sensitivity index suggests that the variation in that input factor significantly influences slope stability. The results are shown in Fig. 6 for both urbanised and non-urbanised slopes. Figure 6 shows that slope stability is insensitive to many input factors and highly sensitive to few, namely effective cohesion and thickness of layer 1 (residual soil), slope angle, and rain intensity and duration. These sensitive input factors represent the main landslide drivers. The sensitivity indices of the urban properties (in orange) are consistent with the findings of Fig. 5, where only the variation in cut slope angle influences slope stability. When looking at the comparison between urbanised and non-urbanised slopes, it appears that the urban presence decreases the sensitivity indices of all the input factors, except for the effective cohesion of layer 1 and the rainfall intensity.

Percentage of slope failures for urbanised and non-urbanised slopes for different categories of input factors. Throughout, urbanised slopes show higher failure rates than non-urbanised slopes. In

We further explore the change in sensitivity caused by urbanisation by plotting the percentage of failed slopes for the main landslide drivers
(Fig. 7). The figure shows how this percentage varies for the urbanised (full colour bars and lines) and non-urbanised cases (pale colour bars and lines). In general, urbanised slopes produce more failures than non-urbanised slopes, though they both display similar trends: an increased percentage of predicted landslides when we would expect the slope to become more susceptible (e.g. when slope angles are higher) or the trigger more severe (when rainfall intensity and duration are larger). For example, in Fig. 7b the percentage of failed slopes in the non-urbanised case linearly increases from

We use the CART analysis to formalise the critical thresholds of input factors above or below which slopes are most likely to be predicted as stable
or failed. Figure 8 represents the two trees for the non-urbanised (Fig. 8a) and urbanised case (Fig. 8b). As expected, the best predictors selected in CART are the same input factors previously identified as most influential (Fig. 6). The boxes with double colour represent the auxiliary variables that combine correlated input factors: the ratio between effective cohesion and thickness of layer 1 to account for their counterbalancing effect on slope stability (i.e. slope with more cohesive soil can be thicker without experiencing failure); the negative ratio between the logarithm of rainfall intensity and rainfall duration, which represent the slope of the rainfall threshold for triggering landslides; and the weighted average of the natural and the cut slope angles to account for the fact that slope susceptibility can significantly increase for low natural slope angles but high cut slopes angles (see Sect. S2 for details about the auxiliary variables and the
change in the model's performance when they are not considered). Using these few
predictors, both trees correctly classify more than 85 % of the simulations as stable or failed (details about the pruning in Sect. S3). Each branch of the tree shows the paths and thresholds of input factors that lead to slopes most likely to fail (black branch) or most likely to not fail (grey branch). At the end of each branch the black and grey bar shows the fraction of failed and stable simulations, while the thickness of the branch is proportional to the number of simulations following that path. For example, in the tree resulting from non-urbanised slopes (left-hand side), the thickest grey line shows that more than 50 % of simulated slopes were stable 91.2 % of the time for cohesion

Classification tree of slope response for non-urbanised slopes

In the trees resulting from non-urbanised slopes (right-hand side), the
black branch leading to the majority of failures is similar to the non-urbanised tree, but it presents higher splitting thresholds: from the
top, the split happens for cohesion

In this analysis, slope cutting is the urban construction activity with the
strongest effect on slope stability's response (Figs. 5 and 6). Figure 7
indicates that when urbanisation is present, more slope failures are
observed, mainly on slopes with relatively low slope angles and with low
values of both soil (layer 1) thickness and cohesion (Fig. 7b and d; also
reflected by higher effective cohesion

Slope cutting is therefore considered in this analysis to be the practice most detrimental to slope stability. This result is consistent with studies carried out in the humid tropics at regional scales, for which slope cutting was identified as one of the major causes of landslides (e.g. Brand et al., 1984; Froude and Petley, 2018; Holcombe et al., 2016). Cuts with slope angles greater than 60

In

We found that when slopes are urbanised, the most significant increase in the percentage of failed slopes occurs for rainstorm events with high intensity (

The higher the intensity and/or the duration of the rainfall event is, the more slope failures will occur in both cases. However, when informal housing is present, more failures are observed for rainfall durations less than 10 h (short events; Larsen and Simon, 1993). This pushes down the intercept of the rainfall threshold, as reflected in the change in the coefficients of the power law equations (reported in each figure). The slope of the threshold line (i.e. the exponent of the power law) is also steeper in the urbanised case, implying the presence of more failures for lower rainfall intensities throughout the duration axis. These results are compatible with the increase in small-scale landslides previously commented (failure depths less than cut slope's height): to reach saturation at shallow depths, relatively low rainfall intensities and durations can be sufficient to initiate slope failure. Figure 9c confirms this assumption: when slopes are urbanised (black dots), failures tend to occur with smaller radius of slip surface and for higher values of intensity

The identification of the main instability drivers and their thresholds can
contribute to create objective rules to classify slopes as hazardous in a
region with scarce data availability. For example, in Saint Lucia our analysis suggests that slopes with effective cohesion of layer 1 less than
12 kPa and thickness less than 2.5 m (effective cohesion

All the results presented are subjected to the assumptions made in our study. The large variation in some of the input factors can lead to overestimating the hazard. Almeida et al. (2017), for example, varied the slope angles between 27 and 30

Finally, Fig. 9b shows that when slopes are urbanised, high-intensity, short-duration rainfall events lead to an increased number of small-scale landslides (failure depths less than 4 m; Fig. 7b; and radius of slip failure less than 10 m; Fig. 9c). Future climate change could potentially increase the frequency of intense precipitation events (e.g. O'Gorman and Schneider, 2009) and therefore the occurrence of these types of landslides in informal communities. However, if small-scale failures produced by anthropogenic factors are neglected in the calculation of rainfall thresholds, current rainstorms events could also be excluded as triggering factors (Crozier, 2010; Mendes et al., 2018). Small-scale, high-frequency landslide events might not lead to major disasters, but they are increasingly seen as indicators of risk accumulation, detrimental to disaster resilience and economic development (Bull-Kamanga et al., 2003). For this reason, these types of landslides deserve greater attention from the scientific community.

We include informal housing into slope stability analysis using a newly extended version of the mechanistic model CHASM in a Monte Carlo framework.
In this way, we consider uncertainties due to both poorly known slope properties and potential future changes in urban and climate conditions. We
demonstrate that informal housing increases landslide hazard and that slope
cutting is the most detrimental construction activity when compared to
vegetation removal, lack of roof gutters, and presence of water leaks. The
presence of informal housing also modifies the relative role that natural
slope angle, soil cohesion, and soil thickness have in maintaining stable slopes, with increased hazard occurrence for low values of these three main
landslide drivers. CART analysis identifies the thresholds of input factors
separating stable and unstable slopes. These thresholds can be used as
objective criteria for guiding local engineers in identifying slopes at risk, deducing landslide mitigation actions, and targeting data acquisition to reduce model prediction uncertainty. Moreover, this analysis allows for the estimation of critical rainfall thresholds at which slope failure is predicted to occur. This rainfall threshold is lower when informal housing is present, with an increased number of small-scale landslides (

Future work will seek to vary the properties that were kept constant in this study, such as the degree of urbanisation and house dimensions, to evaluate their significance for slope stability. This might confirm the importance of household water management such as roof guttering and leaking water supply pipes and septic tanks when the number of households is increased. Analysis of slopes where slope cutting is replaced by other possible construction techniques (such as houses suspended on pile foundations) can identify whether the construction of future hillside settlements could be done in a manner less detrimental to slope stability. Different bioengineering techniques to mitigate hazard likelihood could also be modelled and their effectiveness evaluated. Finally, we seek to transfer the thresholds found in our CART analysis into spatial-scale susceptibility maps in order to identify slopes at higher risk within low-income urban settlements. This would confirm whether the areas suggested to be most hazardous correspond to areas where more landslides have been observed.

Datasets can be accessed by reviewing the data sources stated in Sects. 2.1 and 2.4.

The supplement related to this article is available online at:

EB performed background research, computations, and analysis and wrote the paper. EH, FP, and TW supervised the entire study in all stages, discussed the results, and contributed to the final paper.

The authors declare that they have no conflict of interest.

Existing MATLAB codes from Susana Almeida and Rose Hen-Jones have been adapted and extended for this analysis. We thank Dave Petley (University of Sheffield) for giving permission to use one of the pictures for Kalimpong, India (

The first author was supported by an EPSRC DTP studentship (grant no. EP/N509619/1). Partial support to Thorsten Wagener was provided by a Royal Society Wolfson Research Merit Award (WM170042). Francesca Pianosi is partially funded by the Engineering and Physical Sciences Research Council (EPSRC) “Living with Environmental Uncertainty” Fellowship (EP/R007330/1).

This paper was edited by Margreth Keiler and reviewed by two anonymous referees.