Floods can occur simultaneously at multiple locations; i.e. flood occurrences can be spatially dependent (Fig. ).

Such spatial dependence can be quantified using different types of measures, including pairwise measures, such as the number of co-occurrences at a pair of catchments or the correlation between flood magnitudes at a pair of catchments ; catchment-specific measures, such as the distance over which multiple catchments flood near synchronously i.e. the flood synchrony scale; and the expected proportion of sites in a catchment's vicinity that exceed their xth quantile during an event in which this catchment exceeds its xth quantile conditional spatial dependence;; or event-based metrics, such as flood extent and the percentage of catchments affected by flooding within a certain region (Table ).

Spatial dependence is related to flood magnitude to a certain degree. However, spatial dependence has been shown to increase or decrease with event magnitude when using different dependence measures. have shown that conditional spatial dependence is particularly severe for moderate floods and becomes weaker as events get more extreme. That is, they showed that more extreme events are more localized than moderate floods. In contrast, have shown a positive relationship between flood magnitude and extent when using the flood synchrony scale, i.e. increasing spatial scales with increasing flood magnitude. The strength of spatial dependence also depends on location and is highly variable across catchments. have shown that the distance over which multiple catchments flood near synchronously exceeds the size of individual catchments in Europe and shows strong regional variations, with larger floods occurring in lowlands than in mountain catchments.

Regional floods are shaped by both meteorological and land surface processes; i.e. precipitation spatial dependence alone is not sufficient to explain spatial flood dependence . Regional floods often develop when a storm meets favourable antecedent conditions, such as widespread wet soils, or when multiple catchments experience synchronous snowmelt . Therefore, floods are more likely to be spatially connected with seasonal snowmelt contributions in mountain regions than in lowland catchments, where floods are mainly driven by precipitation . Besides climate, spatial flood dependence is shaped by reservoir regulation, which leads to less spatially connected floods in winter compared to under unregulated conditions .

Regional flood characteristics change over time, but the direction of change is yet unclear. have shown historical increases in the distance over which catchments flood near synchronously for catchments in Europe. In contrast, found decreases in the synchrony of flooding between snowmelt-dominated basins because of decreases in snowmelt using simulations of future streamflow. This finding is in line with results by and , who found stronger spatial connectedness for snowmelt-influenced regions than for rainfall-driven regions. While these studies provide first evidence for future changes of regional floods in a warming climate, the direction and magnitude of these changes need to be quantified using further targeted modelling experiments (see Sect. ). The spatial dependencies between flood occurrences at multiple locations need to be considered in flood hazard assessments in order to avoid risk over- or underestimation . Such consideration can be achieved by e.g.  computing probabilities of regional flooding .

Droughts are often regional phenomena; i.e. drought occurrences at different locations are dependent. Similarly to floods, such spatial drought dependence can be quantified using different types of descriptors. Using a pairwise perspective, drought dependence can be quantified by counting the number of drought co-occurrences or the number of months under concurrent drought . Taking a regional perspective, regional droughts can be described by the number of catchments affected by drought or by the drought extent . The main part of the literature studying regional droughts and their extents focuses on meteorological rather than streamflow droughts . Those studies that have assessed the spatio-temporal variation in hydrological drought extents found substantial temporal variations in the number of catchments jointly affected by drought .

Spatial drought extent is driven by different hydro-meteorological conditions, including soil moisture deficits, precipitation deficits, and positive temperature anomalies. The relative importance of these different drivers varies by event and season. In winter and spring, large-scale droughts often co-occur with soil moisture and precipitation deficits, while they co-occur with positive temperature anomalies in summer . While there exist first indications that the relationships between climatic drivers and drought extent are complex, future studies should focus on the identification of atmospheric drivers of widespread streamflow droughts, similarly to studies that assess the link between atmospheric patterns and/or climate indices and the spatial extent of meteorological droughts e.g..

Streamflow drought spatial extents have increased in the United States over time, mainly because of increases in the extent of small droughts and in temperature . Further investigations are needed to assess whether such changes can also be observed in other climate zones such as tropical, arctic, or alpine regions. The spatial extents of streamflow droughts have not just changed in the past; they are also projected to further increase in future, as demonstrated for Great Britain using climate and hydrological model simulations . How such changes translate to other regions remains to be assessed using modelling experiments which focus on reliably reproducing spatial streamflow drought extents.

A diverse range of tools can be used to quantify the spatial dependence and spatial extents of floods and droughts. These tools include areal coverage, spatial extent, conditional spatial dependence, synchrony scale, length scale, probability of regional extremes, connectedness, severity–area–frequency curves, and severity–area–duration curves (Table ). A first category of descriptors describes the spatial extent of extreme events at an event scale. This category comprises areal coverage, i.e. the percentage of a region or river basin under extreme conditions; spatial extent, i.e. the area under extreme conditions, usually derived from gridded data; and conditional spatial dependence, i.e. the expected proportion of sites in the vicinity of a specific catchment that exceed their pth quantile during an event in which this catchment exceeds its pth quantile. While these descriptors focus on describing individual events, a second group of descriptors summarizes the behaviour of regional extremes at a catchment scale. For example, the synchrony scale measures over which distance around a catchment multiple rivers experience flooding at the same time. A third group of metrics comprises metrics that summarize regional relationships in terms of the occurrence of extremes, e.g. through a semi-variogram or more specifically the length scale (i.e. the range of the semi-variogram) or the probability of regional extremes, i.e. the probability that a certain percentage of catchments within a region is jointly under extreme conditions. A fourth group of metrics includes pairwise measures such as connectedness determined by either the number of co-occurrences at a pair of catchments or the correlation between flood magnitudes at a pair of catchments. A last group of descriptors is frequency or duration curves, e.g. severity–area–frequency curves or severity–area–duration curves. Depending on which metric is chosen to describe regional extremes, the results of an analysis will differ. For example, change assessments may find different changes in regional extremes when looking at pairwise relationships than when focusing on the event scale.

Flood events cluster in time; i.e. flood-rich periods in which floods are more common alternate with flood-poor periods in which floods are rare . In Europe or China, for example, many catchments show temporal clustering for moderate floods at timescales of 1 year to a few years . However, the strength of temporal clustering decreases substantially with timescale and with an increasing flood threshold . The temporal flood-clustering behaviour to some degree also depends on the region. For example, catchments in the Atlantic and continental regions of Europe are more prone to temporal flood clustering than catchments in Scandinavia .

Flood-rich periods with temporally clustered events are related to climate. and have, for example, shown for Europe that historic flood-rich periods occurred under colder-than-normal climate conditions. Similarly, , , and have shown for catchments in Iowa, China, and Australia, respectively, that the flood-clustering behaviour is influenced by large-scale climate indices. The pronounced link between climate and the temporal flood-clustering behaviour suggests that future changes in temperature and oscillation patterns may lead to changes in temporal flood clustering. How the temporal flood-clustering behaviour changes across different climate zones in a warming climate still needs to be investigated using simulation-based studies. Such simulation-based studies require the development of modelling approaches that reliably represent the temporal clustering behaviour of floods.

Drought events can occur successively or cluster in time as highlighted by studies looking at the occurrence of multi-year droughts and studies assessing the temporal clustering behaviour of droughts. A first body of literature provides evidence for the occurrence of multi-year droughts from both a meteorological and a hydrological perspective. The occurrence of multi-year precipitation deficits has, for example, been documented for France , central Europe , and the United States , and the occurrence of multi-year streamflow deficits has been documented for different parts of Europe and Chile . A second body of literature shows that both meteorological and hydrological drought occurrences are highly variable in time, with alternations between drought-rich and drought-poor periods at multi-year , decadal , and multi-decadal timescales . However, some other studies also provide contrasting evidence by showing a lack of cyclicity in precipitation deficits .

have shown that catchments experiencing multi-year droughts are mostly characterized by a rainfall-dominated flow regime, while catchments with melt-dominated flow regimes are generally not affected by multi-year droughts. In addition, have shown that the temporal clustering of hydrological droughts is substantially more pronounced than the clustering of precipitation deficits. That is, climatic drivers are insufficient to explain the temporal clustering of hydrological droughts, suggesting that additional land surface processes, such as snow storage or the absence thereof, seasonal and interannual groundwater level variations, temporal soil moisture variability, or fluctuations in glacier melt contributions, are needed to explain hydrological drought-clustering behaviour. Catchments prone to temporal hydrological drought clustering are often arid and lack substantial snow storage . As a consequence, changes in the number of catchments showing temporal hydrological drought clustering may be expected in a warming climate because of increases in aridity and decreases in snowmelt. Similarly, multi-year droughts may become more frequent in a future climate as flow regimes transition from snow dominated to rainfall dominated . Detailed modelling assessments are needed to show how the probability of occurrence of multi-year droughts and the temporal-clustering behaviour of droughts are going to change in the future. Such assessments require an adequate representation of temporal streamflow dependencies.

The persistence and periodic features of hydrological extreme events have been documented using a range of measures including the Hurst exponent, power spectra derived using the Fourier transform, dry-to-dry transition probabilities, and others (Table ). A very simple measure to characterize consecutive extremes is the number of consecutive events, e.g. the number of successive extreme months or years. Also related to individual events, one can compute extreme-event-transition probabilities, i.e. the probability of observing a subsequent extreme event given that an extreme event has occurred in the previous time unit (e.g. week, month, or year). Instead of focusing on events, the temporal persistence of extremes can be summarized for entire time series of extreme events, for example by the Hurst exponent, which measures the long-term memory of a time series, or the average power spectrum, i.e. the average power over all frequencies after the Fourier transform. In addition, consecutive extreme events can be described by measures that characterize the temporal-clustering behaviour of extreme events, including the dispersion index, which quantifies the departure of an observed process from a homogeneous Poisson process; Ripley's K, which counts the average number of extreme events in the temporal neighbourhood of extreme events; and the kernel estimation, which estimates the time variation of extreme-event counts as a smooth function of time. Another possibility for describing consecutive extremes is to identify flood- or drought-rich and flood- or drought-poor periods using scan statistics. That is, unusual periods in the observations that are inconsistent with the assumption of independent and identically distributed random variables, i.e. periods encompassing very few or very many events, are identified with a moving-window approach. If it is of interest not only to describe consecutive extremes but also to identify their drivers, one can rely on Cox regression models, which examine the dependence of the rate of occurrence of extremes on covariate processes, e.g. different types of teleconnection patterns. The choice of a specific descriptor will depend on the specific research question or application, i.e. on whether one would like to test for clustering significance, in which case Ripley's K or the dispersion index can be used; whether one would like to identify specific periods particularly abundant in extremes occurrence, in which case scan statistics or kernel estimation can be used; or whether one would like to explain temporal dependence, in which case one can rely on Cox regression models.

Floods are characterized by multiple characteristics, including peak discharge, volume, and duration (Fig. a), which are interdependent . For example, flood duration and volume or flood volume and flood peak show strong correlations (Fig. ); i.e. they show bivariate dependence. These variable relationships vary with the flood generation process; e.g. flash floods, short-rain floods, long-rain floods, and rain-on-snow floods show different forms and strengths of variable dependence . Because of such variations in variable dependence with flood generation processes, variable dependence also varies between low- and high-elevation catchments . For Austrian catchments, found weaker variable dependence in alpine than in lowland catchments because of a mix of flood generation processes. In addition to elevation, variable dependence has also been shown to vary with catchment size. Using a global dataset, have shown that the strength of variable dependence increases with the catchment area. However, overall, variable dependence seems to be more strongly related to climatic factors than to physiographic factors . Because of the link between climatic flood drivers and variable dependence, the strength of variable dependence is changing in a warming climate. For example, found an increase in the dependence between flood volume and peak discharge for the Rhine River, and detected decreases and increases in such dependence for two catchments in Québec. These temporal-change patterns of variable dependence are spatially heterogeneous and cannot be explained by one hydro-meteorological driver alone. Instead, changes in variable dependence are the result of an interplay between changes in precipitation, snowmelt, and soil moisture, resulting in dependence increases in some regions and dependence decreases in other regions . The interdependencies between different flood variables and their potential future changes need to be considered in multivariate hazard and climate impact assessments. That is, flood frequency analyses need to consider variable dependencies if multiple variables are of interest for the application. For example, the dependence between peak and volume should be considered when deriving flood estimates for hydraulic design.

Similarly to floods, droughts can be described by different characteristics, including drought intensity, deficit, and duration (Fig. b), which are also interdependent . Such bivariate interdependence is found for e.g. drought deficit and duration or drought deficit and intensity (Fig. a and b). The strength of dependence varies with climate . Drought deficit increases most strongly with duration in cold seasonal climates because snow accumulation during winter prevents the recovery from summer drought and in monsoonal, Savannah, and Mediterranean climate zones where summer droughts continue into the winter . This relationship between drought variable dependence and climate suggests that the variable interdependence may change in a warming climate. How climate change specifically affects the dependence between different pairs of variables needs to be assessed using targeted modelling experiments focusing on an accurate representation of variable dependencies in hydrological models.

The interdependencies between multiple characteristics of hydrological-extreme events can be assessed using various dependence measures, including different correlation and tail dependence measures focusing on bivariate variable relationships (Table ). Linear relationships can be quantified using Pearson's correlation coefficient, while nonlinear relationships can be described using Spearman's or Kendall's rank correlation coefficients. If the focus is not on the bulk of the distribution but on its tails, one can use the extremal dependence coefficient, which describes the probability of one variable being extreme given that the other one is extreme.

Studies looking at transitions from dry to wet periods mainly focus on transitions in meteorological states, i.e. transitions from negative to positive precipitation or moisture anomalies . These meteorological studies indicate large spatial variability in dry-to-wet-period transition times ranging from a few months to multiple years . In contrast, little is known about consecutive hydrological drought–flood events, i.e. transitions between extremes in streamflow data. For the Amazonas River, studied the abrupt transition from an extreme drought in September 2010 to very high discharge in April 2011, and studied drought termination for river basins in the UK. Still, little is known about the atmospheric and land surface conditions that lead to rapid drought-to-flood transitions and about how transition times and characteristics vary in space and time. Further research is needed in order to better understand the variations of transition times across hydro-climates and the hydro-climatic drivers of rapid drought–flood transitions. Studies looking at future changes in transitions between dry and wet meteorological states suggest more frequent and rapid transitions between wet and dry extremes . Hydrological simulation experiments are needed to assess how these changes in transitions from dry to wet states translate into changes in transitions from hydrological droughts to floods. The possibility of rapid drought–flood transitions under both current and future climate conditions needs to be integrated in disaster risk reduction strategies .

The transitions between dry and wet periods have been described using transition times and transition frequencies, as summarized in Table . The transition time describes the time that elapses between dry and wet periods, while the transition frequency describes the frequency of transitions between dry and wet periods.