A large number of social and natural phenomena follow a power-law distribution (Pisarenko and Rodkin 2010) rather than a classic Gaussian distribution (the standard example here is body height distribution). Depending on the academic discipline power-laws are also known as Zipf’s laws or Pareto distributions (Newman 2005). When examining phenomena that vary over a large dynamic spectrum and do not peak around a typical value (e.g. statistical mean) chances are that the underlying data behave according to a power-law distribution (the classic example here is the sizes of cities). A number of natural hazards satisfy this kind of distribution, including for instance landslides (Guzzetti et al. 2002), forest fires (Malamud et al. 1998), asteroid impacts (Chapman 2004) and volcanic eruptions (Pyle 2000).

Power laws and flood risk

Some studies suggest that flood hazards also follow a power-law distribution (Malamud et al. 1996; Newman 2005; Pandey et al. 1998; Turcotte and Haselton 1996). Most of these studies however, focus on hydrological and meteorological processes (e.g. river discharge, rainfall data) rather than on the social and economic damage that floods may inflict on flood-prone communities. Since estimates of flood hazards and mitigation measures have wide-ranging economic and social consequences, it is important to understand how the frequency of damage sizes is distributed and what kind of adaptation measures need to be put in place to effectively reduce present and future flood risks. Against this background, I provide a power-law driven theoretical perspective on how adaptation measures may impact the distribution of flood damage sizes.

Adaptation scenarios

The following figures represent a reference scenario without adaptation and four adaptation scenarios. All figures include an illustration on the left panel and a conceptual log-log plot on the right panel to indicate how the power law curve changes due to the implementation of the adaptation measure (note: no tick marks or units are given since the chart is purely conceptual; slopes and shapes of curves only reflect theoretical considerations).

Reference scenario (a)


The reference scenario represents a flood-prone settlement with no adaptation measures in place. The frequencies of damage sizes are plotted next to the illustration in Figure (a). This case reflects a simplified non-adaptation scenario in which damage sizes follow a power-law distribution. Put differently, the settlement is subject to a large number of small flood damages and a small number of severe flood damages.

Dam scenario (b.1)


In this scenario, a structural measure protects the settlement from severe flood events. It appears reasonable to assume that the frequency of small damages is lower compared to the reference scenario. The frequency of severe damages, however, is not likely to be reduced since flood protection measures are only effective up to a specified return period. In this scenario the distribution curve shifts to the bottom until a certain threshold is exceeded (e.g. when the dam reaches its maximum protective capacity). At this threshold the adaptation measure fails to protect the at-risk community and the curve parallels the curve of the reference scenario. The area between the two curves is shaded in grey and reflects the adaptive capacity (e.g. damage reduction) of the measure in question.

Dam scenario (b.2)


Over time, the dam leads to a growing sense of security and therefore to a larger amount of economic assets in the floodplain caused by settlement expansion. In this case, the net effect of the adaptation measure is significantly lower (it can also be zero or negative) compared to the first dam scenario. The adaptation measure is effective in protecting the community from small to mid-size events, but at the critical threshold (when the dam reaches its maximum capacity) flood losses increase due to the accumulation of economic assets in the floodplain. Here, adaptive capacity is offset by human behaviour (adaptive capacity = avoided damages – damage increase).

Relocation scenario (c)


In this scenario flood risk policies are in place with the purpose to clear land by relocating flood-prone communities. If the resettlement is successful, only a few (or no) economic assets will remain in the floodplain. In this case, the curve shifts to the left. The adaptive capacity is indicated by the grey area between the reference and adaptation curves.

Upstream measure scenario (e)


In the upstream measure scenario a retention area (buffer zone) is converted into dry land for settlement expansion or any other designated activity. Due to the dam that protects the former retention area, discharge levels increase at the downstream community. In this scenario the adaptation curve shifts to the right, causing negative externalities on the downstream community. This adaptation measure may effectively protect the upstream community, but the infliction of additional risk on the downstream community may not prove effective for society as a whole.

So what does the power law teach us?

As proposed above, the power-law perspective is a powerful analytical tool to holistically examine the efficacy of flood adaptation measures. By comparing flood risk scenarios against each other, the power-law perspective reveals that expected adaptive capacities may be offset by human behaviour or that certain measures are not beneficial for the larger society if they are put in place without coordination among affected parties. In other words, the power law perspective provides a holistic tool to judge which adaptation measures are effective in a given scenario (note that the combination of measures adds additional complexity of course).

When does adaptation pay off?

The potential of an adaptation measure to mitigate flood risks needs to be weighted against the economic and social costs associated with the implementation of the measure. If these costs exceed the expected risk reduction, it might be wise to reconsider the implementation of this measure. This can be done by cost-benefit analyses for instance. Here, issues of uncertainties also need to be considered, since estimating frequencies of future flood events are problematic due to a lack of agreement on appropriate methodologies.

Practical relevance for flood risk policy

The proposed power-law perspective provides a conceptual framework that can be used to guide decision making in the field of flood risk management. Above, I focused on flood risk scenarios that are highly limited in their spatial extent. The same approach, however, can be used on a more regional, national or even international scale (to assess how upstream adaptation measures affect downstream countries for instance).

The power-law perspective also allows risk policy makers to evaluate the frequency of different damage sizes against a risk layer approach (Mechler et al. 2014) by examining which adaptation measures are most effective on any given position on the distribution curve. According to the risk layering approach, risk reduction may prove most effective on the top left end, private insurance in the middle and (inter)national compensation schemes on the bottom right end of the curve. Mapping risk layers against the power-law perspective could provide important insights for flood risk managers on measures that are most effective at a given return period and how responsibilities in flood risk management could be shared among the stakeholders involved.

Directions for future research

Future research is required to put this theoretical perspective to the test. First, sound empirical data needs to be gathered to assess the validity of the proposed damage-size-frequency distribution. Here, the difficulties are related to completeness and reliability of flood damage data. Most flood-related data only capture damages in economic terms and fail to measure losses that refer to personally valuable goods, human suffering, personal convenience, disturbance of social relations, negative effects on the ecosystem and other externalities. Today, most flood damage data are made available by public authorities (e.g. compensation schemes) or private insurances. These data however, only capture a portion of the damages since flood victims are usually not permitted to claim self-provided labour, which leads to a systematic underestimation of flood damages. Second, long-term time-series data are necessary to test if adaptation has effectively resulted in risk reduction and if the damage-size-frequency curve has changed as theorised in the scenarios above.

Once the validity of the framework is established, empirical and model analysis may bring to the fore for which return periods adaptation measures are most effective. Viewing adaptation through a power-law lens might also be fruitful for fields outside of flood risk research. The theoretical approach proposed above can also be used to assess adaptation options related to other natural and human-induced disasters such as alpine hazards (e.g. landslides, avalanches, rock fall), seismic hazards (e.g. volcanic eruptions, earthquakes), weather related events (e.g. hail storms, hurricanes) or wildfires.


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