Urban catchments are typically characterised by a more flashy
nature of the hydrological response compared to natural catchments.
Predicting flow changes associated with urbanisation is not straightforward,
as they are influenced by interactions between impervious cover, basin size,
drainage connectivity and stormwater management infrastructure. In this
study, we present an alternative approach to statistical analysis of
hydrological response variability and basin flashiness, based on the
distribution of inter-amount times. We analyse inter-amount time
distributions of high-resolution streamflow time series for
17 (semi-)urbanised basins in North Carolina, USA, ranging from 13 to 238 km

Hydrological response in urban catchments tends to be more flashy compared to
natural ones as a result of their higher degree of imperviousness. Increases
in flashiness are typically characterised by shorter response times to
rainfall, higher run-off ratios and higher peak flows

Many authors have investigated methods for characterising hydrological
response and changes therein, including univariate analysis and multivariate
statistics, combining several hydrograph properties such as flood peak, flood
volume and flood duration

An important characteristic that has been used to analyse change in
hydrological response is basin flashiness, qualitatively described by

Summary of results reported on the literature for (multi)fractal analysis of hydrological flows. MA: moment analysis, MFA: multifractal analysis, SA: spectral analysis, TMA: trace moment analysis.

Scaling behaviour of river flows has been investigated by various authors,
aiming to identify characteristic length and timescales and to detect scale
dependence of hydrological response processes. Among the various statistical
methods that have been proposed to investigate scaling, fractals and
multifractals are among the most popular and powerful. Approaches for fractal
analysis include spectral analysis based on second-order
properties and trace moment analysis based on a wider range of statistical
moments, typically between 0.1 and 4. The universal multifractal framework is
based on the identification of scaling exponents summarising the changes in
flow distributions across a given range of scales, (see

In this paper, the IAT formalism is applied to flow time series and
statistical distributions, and scaling properties are compared to the ones
obtained using the classical fixed-time framework. To do this, we use flow
observations collected in 17 hydrological basins in Charlotte, North
Carolina. We aim to investigate what effects an adaptive sampling strategy
such as IAT sampling has on statistical properties of the time series, in
particular on the tails of the statistical distributions associated with peak
flow and low flow extremes. The main problem with a fixed sampling rate, as
in traditional flow time series analysis, is that it can only accurately
represent frequencies of variations at timescales larger than a certain
threshold. When frequencies higher than that exist, errors are introduced as
information about the higher frequency variability is lost

This paper is organised as follows. In Sect. 2 we present the flow datasets and methods used for analysis. We explain the methodology for deriving normalised IATs and introduce metrics we used to compare properties of flows and IAT time series, to characterise hydrological response and compare response across basins. In Sect. 3, results of the analyses are presented and discussed, first based on results obtained using a daily sampling scale, and followed by results obtained a range of sampling scales, from hourly up to seasonal sampling scale. Conclusions and suggestions for future work are summarised in Sect. 4.

Summary of hydrological basins in the Charlotte area: basin area
(km

The data used in the study were collected at 17 USGS stream gauging stations
in Charlotte–Mecklenburg county, North Carolina. Gauging stations are located
at the outlet of hydrological basins that range from 13
to 238 km

Map with the location of the area, boundaries of hydrological basins and location of stream gauges used in the analysis. NC: North Carolina. SC: South Carolina.

Stream gage data were collected at 5 to 15 min intervals over the period
1986–2011. Table

In this paper we analyse hydrological flow variability, based on the
distribution of IATs. We use the following definition of
IATs, based on

A steady flow pattern with constant flow has equal IATs for all values of

Flow magnitudes strongly vary from one gauge to another. To overcome this
scale dependence and compare flow IATs across basins with different sizes and
flow amounts, one needs to normalize IATs with respect to a common timescale.
A possible way to do this is to fix an average IAT

Inter-amount times can be estimated from a sample flow time series

In addition to the lower bound, we also impose an upper bound on the
inter-amounts used in our analysis. This is necessary to ensure IAT time
series are long enough to compute relevant statistical moments. Typically,
there should be at least 100 consecutive IATs, which yields the following
upper bound for inter-amount

It is worth pointing out that the lower bound on the inter-amount in Eq. (

Note also that analyses of IATs were conducted for all gauges over the entire
period of available data, without distinguishing between year, season or hour
of the day. This was necessary as time series would otherwise be too short to
study IATs across different scales. This means we mostly focus on average
characteristics of IAT and flow distributions with respect to area size and
imperviousness degree and potential influence of flow regulation and
stormwater detention facilities, as far as this information is available for
the 17 basins. We refrain from investigating long-term trends, as our time
series are restricted to maximum 30 years and because a recent study by

Sample histograms of IATs and flows were analysed to investigate what
different insights they provide into characteristics of the flow regimes. We
plotted sample histograms for all gauges; appropriate bin widths were
determined based on Scott's rule

First-order differences of IATs and flows were computed to look into characteristics of the rising and falling limbs of hydrographs. Because IATs are measured on an inverted scale, positive differences are associated with the falling limb of the hydrograph and negative differences with the rising limb of the hydrograph. Narrow ranges of histogram values for IAT differences indicate slowly varying flow; wide range histograms indicate more flashy behaviour. Positively skewed histograms for IAT differences indicate that the distribution is dominated by values on the rising limb and short recession limbs, while negatively skewed histograms indicate a larger part of the flow is associated with flow recession, i.e. long, slowly receding hydrographs, for instance, induced by a strong groundwater flow component. Differences were computed at the 24 h timescale, imposed by the minimum inter-amount scale rule. Similarly to the other histograms, bin widths were chosen based on Scott's rule.

As mentioned earlier, the lower bound on the inter-amount
provides an indication of left-tail properties of IAT distributions (i.e.
short waiting times) and can therefore be used to characterise the degree of
flashiness of the hydrological response. In flashier catchments, the flow can
rise quicker, resulting in lower IATs during times of heavy rain. The minimum
observable inter-amount represents the smallest scale at which flow
variations can be studied with acceptable interpolation errors, given a fixed
temporal observational resolution. By extension, the lower tail of the IAT
distribution provides a good indication of what observational resolution is
necessary to adequately capture the most extreme flow variations. The IAT
flashiness indicator used in this paper is defined as the mean scale

Multifractal analysis techniques were applied to investigate the scaling
behaviour of IAT time series across different inter-amount scales.
Multifractal analyses are based on the assumption of generalised scale
invariance, in which the statistical moments or order

The scaling quality is assessed by noting that if Eq. (

In the following sections we compare statistical properties of flow and IAT time series and highlight differences that result from the different sampling strategies. Analyses are first conducted at the 24 h timescale and associated mean inter-amount sampling scale. In the second part of this section, we analyse how statistical properties of flow and IAT time series vary across scales, and quantify flashiness and scaling behaviour of both time series.

Figure

Example of times series for flow

Figure

Illustration of inter-amount data sampling for cumulative
flow over a period of 7 days, for Taggart Creek.

Histograms of flow time series

Histograms of flow time series and IATs at daily timescale are plotted in
Fig.

Summary statistics of time series for flows and inter-amount times, at 24 h sampling scale: coefficient of variation (CV), skewness (Skew) and medcouple (Mc).

Summary statistics of time series for flows and inter-amount times, at 24 h sampling scale: coefficient of variation (CV), skewness (Skew) and medcouple (Mc), for three sets of connected sub-basins in the Charlotte catchments: Irwin, Little Sugar and McAlpine.

Tables

These results show that adaptive sampling based on inter-amounts leads to
more balanced representation of high and low flows, resulting in lower
coefficients of variation reflecting more stable statistical variance compared to
traditional flow time series sampling. We would like to point out that these
results were obtained at the 24 h sampling scales. In Sect.

Subsequently, we compared properties of IAT and flow
distributions across the 17 basins in relation to basin characteristics.
Figure

Scatter plots for mean normalised flow inter-amounts

Looking at CV values across all basins (Fig.

Scatter plots for skewness and medcouple values
(Fig.

USGS, water year reports

preventing occurrence of low flow extremes and high IAT extremes. The effect shows up more clearly for IAT medcouple values, as a result of the adaptive sampling strategy that gives more weight to peak flows, leading to generally higher medcouple values, but also reflecting more clearly the absence of low flow extremes. Some of the basins in this study are sub-basins of each other, which implies that flows can be correlated. TableIn this section we discussed distributions of IATs and flows at the 24 h
scale. Results showed that larger basins are generally characterised by
stronger smoothing of flows, resulting in higher mean flow, lower CV and
lower skewness of the flow histograms. Flow variability is clearly correlated
with basin size, which is mainly a result of smoothing of low flows, in the
left tail of the flow histogram. This confirms results previously reported in
the literature on scaling between flows and basin area

histograms of first-order differences in flows

Figure

Scatter plots of skewness

In this section we analyse the variability of IATs and
flows across a wide range of sampling scales. We investigate how the
statistical distributions and hydrological response characteristics change
when moving from inter-event (multiple days) to intra-event (sub-daily)
scales. Figure

Quantile plots of flows

We can see that mean values of normalised flows and IATs decrease log-linearly with sampling scale, as indicated by a straight line in the log–log plot, i.e. the sampling mean follows power-law scaling. As histogram analysis at the 24 h scale already showed, statistical distributions of both flows and IATs are highly skewed. Moreover, skewness increases at smaller scales, as indicated by an increasing distance between mean and median values. Median values for flows follow close to log-linear scaling (albeit steeper compared to the mean) but exhibit stronger departures from log-linear scaling for IATs. In particular, the median of IATs shifts from close to log-linear scaling between 16 and 64 mm (associated with about 16 to 64 days) to non-log-linear scaling between 1 and 14 mm scales (1–14 days) and again to near-log-linear scaling below 1 mm. Coincidentally, these transitions correspond to the range of scales over which IATs generally transition from being inter-event to intra-event dominated. Indeed, IATs at coarser scales mostly combine the properties of multiple storms, resulting in a more symmetric distribution. This effect is much stronger in IAT than in flow distributions, because it is mainly associated with changes in sampling of peak flows which are more frequently sampled in the IAT framework than in the conventional fixed time approach.

Comparing the 10–90 and 1–99 percentile ranges in Fig.

Quantile plots of inter-amounts over a range of scales were created for all 17 gauges included in our analysis (results are added as a Supplement to this paper). This allowed us to compare transition ranges between inter-event-dominated and intra-event-dominated IAT distributions for all basins. Results show that for 10 % IAT quantiles, the lower end of the transition range, where intra-event characteristics start to be mixed with inter-event phenomena, lies roughly between 10 and 25 mm mean inter-amounts, being accumulated in about 1 h in most of the basins. Lower values are found for basins with higher urbanisation degree and for basins where low flow control is applied, reflecting the smoothing influence of flow control measures on peak flows. Similarly, one can compare the amount of flow that is being generated in an hour, compared to the mean flow. This can be derived from the IAT quantile plots by looking at the scale at which a given IAT quantile, for instance 10 % or 1 %, equals 1 h. For Taggart Creek, the IAT 1 percentile equals 1 h at sampling scale of 18 mm of mean normalised flow or, equivalently, 18 days of mean IAT. This means there is a 1 % probability of exceeding 18 mm of flow accumulation in 1 h or less, or, in terms of time, it implies that there is a 1 % chance to accumulate the amount of flow measured on average over a period of 18 days in 1 h or less. Thus, higher values of 1 h, 1 percentiles indicate stronger flashiness of basin response. Comparing values across basins, we found that higher values of 1 %, 1 h accumulations were strongly correlated with basin area, while no significant correlation with imperviousness was observed.

Coefficients of variation for flows and inter-amount-time scales across a range of sub-daily (3 to 12 h) up to bi-monthly (60–68 days) scale, for Irvins Creek, LSugarM, Stewart Creek and McAlpine. Grey triangle symbols represent inter-amount times, black circles represent flows.

Subsequently, we investigated scaling behaviour from the perspective of
statistical moments, by looking at coefficients of variation for flows and
IATs across scales. For the purpose of statistical analysis and downscaling
applications, it is important to have a robust scaling model that predicts
how distributions change when going from one scale to another. Scale
invariance means that a distribution can be derived at any scale, especially
small scales, by shifting and scaling the distribution at larger scales. One
way to assess the property of scale invariance is to check if the statistical
moments of distributions follow a power law of scale. Figure

In Sect.

Two flashiness indicators were computed, as explained in Sect.

Scatter plots of flashiness versus basin area and imperviousness, for all gauges. Grey triangle symbols represent inter-amount times, black circles represent flows.

Figure

Minimum and maximum observable scales (in hours), flashiness index
for 15 min observation time (in hours) and fitted multifractal parameters

R–B flashiness indices were computed at the daily scale, to
allow comparison with results obtained by

Example of log–log plots for flows and inter-amount times

Quantile plots of IAT distributions furthermore provide information about the
minimum observable scale at a given observational resolution (15 min, in
the data series used in our analysis), i.e. the degree of flow variability
that occurs at scales smaller than the observation scale. When moving towards
smaller sampling scales, a growing percentage of flow accumulations occurs in
less than 15 min, and hence cannot be analysed at the given observational
resolution. This typically coincides with peak flows and implies that during
peak events, the observational resolution is too low to measure flow
variability. IAT analysis can thus be used to identify a critical resolution
for flow observations, if a given peak flow accumulation is of interest. This
could be associated with, for instance, the capacity of detention ponds or
flooding caused by exceedance of stormwater drainage capacity. For the
example of Taggart Creek (Fig.

As explained in Sect.

Plots in Fig.

Using the empirical log moments, we fitted the multifractal parameters

Multifractal parameters

In this study, we introduced an alternative approach for analysis of
hydrological flow time series, using an adaptive sampling framework based on
inter-amount times (IATs). The main difference between flow time series and
time series for IATs is the rate at which low and high flows are sampled; the
unit of analysis for inter-amount times is a fixed flow amount, instead of a
fixed time window. Thus, in IAT analysis, sampling rate is adapted according
to the local variability in flow time series, as opposed to time series
sampling using fixed time steps. We aimed to investigate the effect of
adaptive IAT sampling on flow statistics, especially on the tails of the
statistical distributions associated with peak flow and low flow extremes. We
analysed and compared statistical distributions of flows and IATs across a
wide range of sampling scales to investigate sensitivity of statistical
properties such as distribution quantiles, variance, scaling parameters and
flashiness indicators to the sampling scale. We did this based on streamflow
time series for 17 (semi-)urbanised basins in North Carolina, USA. The
following conclusions were drawn from the analyses:

Adaptive sampling of flow time series based on inter-amounts leads to higher sampling frequency during high flow periods compared to conventional sampling based on fixed time windows. This results in a more balanced representation of low flow and peak flow values in the statistical distribution. While conventional sampling gives a lot of weight to low flows, as these are most ubiquitous in flow time series, IAT sampling gives relatively more weight to high flow periods, when given flow amounts are accumulated in shorter time. As a consequence, IAT sampling gives more information about the tail of the distribution associated with high flows, while conventional sampling gives relatively more information about low flow values.

Statistical analysis of IATs and flows at the 24 h scale showed that coefficient of variation (CV) and skewness values were much higher for flows than for IATs, while medcouple values were lower for flows, indicating strong asymmetry of the flow distributions and low representation of high flow extremes in the statistical distribution. Larger basins were generally characterised by stronger smoothing of flows, resulting in higher mean flow, lower CV values and lower skewness of the histograms. Flow variability decreased with basin size. Larger imperviousness was associated with higher mean flows and lower variability of IATs, while there was not a clear relation with variability of flows.

Comparison of CV across the 17 basins showed that
CV values of flows were significantly negatively correlated with basin size. CV
values of IAT distributions were not significantly correlated with basin size.
This was explained by basin size having a stronger smoothing effect on low flow
variability, strongly represented in conventional flow time series, than on peak
flows that are more frequently represented in IAT time series. By contrast, CV
values of IAT distributions were negatively correlated with imperviousness, while
correlation between CV values for flows and imperviousness was not significant.
Negative correlation between CV values of IATs and imperviousness probably indicates
a stronger influence of flow regulation by detention and capacity constraints of
stormwater drains in more urbanised basins, resulting in more uniform run-off
during rainy periods. IATs during these periods concentrate relatively more
closely to the mean and show fewer extremes. This result is contrary to
findings reported in the literature, where urbanisation tends to be associated
with higher peak flows.

Histograms of first-order differences showed negative skewness for IATs and
positive skewness for flows, for most of the basins, indicating the prevalence of slow flow recession
compared to flow rise. The three basins with low flow regulation could be recognised by their relatively
low medcouple values (

Quantile plots of flows and IATs plotted over a range of sub-daily to seasonal scales showed the influence of the different sampling strategy for IATs compared to conventional flow sampling on median, 25–75, 10–90 and 1–99 percentile ranges of the distributions. The 25–75 and 10–90 percentile ranges for flows remained approximately constant, but the distance between 90 and 99 percentile values rapidly increased towards smaller scales. This reflects the highly skewed nature of flow distributions caused by oversampling of low flows compared to high flows; an effect that increased progressively towards smaller scales. By contrast, 10–90 and 1–99 percentile ranges for IATs increased more or less similarly with scale, for sampling scales ranging from 0.51 mm to approximately 10–16 mm, largely associated with intra-event flow variability. This indicates that the tails of IAT distributions are more or less equally sampled, at least up to the 1 and 99 percentiles.

Quantile plots for IATs showed different scaling at small scales (up to inter-amount scale 8–10 mm) and large scales (roughly exceeding 20 mm inter-amounts), with a transition range in between. At smaller scales, IATs are mostly dominated by intra-event variability, while at large-scales IATs span multiple events. Flows sampled over fixed time intervals did not clearly exhibit this transition, probably because peak flow variability is being poorly sampled by fixed time window sampling. Because IATs adapt the sampling rate depending on the level of activity, they still capture a fair amount of peak flow statistics and intra-event properties, even at coarser scales.

Comparison of the tails of flows and IAT distributions showed that the
distribution tail associated with low flows captures lower flow extremes in
conventional sampling than in IAT sampling (0.02 mm day

Analysis of CV values of flow and IAT distribution across scales showed that at smaller scales, CV values for flows increase more rapidly than for IATs, indicating that IAT variance remains more stable at smaller scales, while variance rapidly increases at small scales for flows. This is as a result of growing skewness of the statistical distribution of flows, caused by relative oversampling of low flows, or conversely, undersampling of high flows. This shows that for analysis of peak flows, IAT analysis offers advantages of the fixed-time sampling framework, as it samples peak flows more frequently and results in more stable variance across scales. For analysis of low flows, especially in basins characterised by strongly smoothed flow variability, IAT analysis offers little advantage and conventional flow statistics are more suitable.

An IAT flashiness indicator was defined as the inter-amount scale at which 1 % of flow accumulations occur in less than 15 min. Comparison between IAT-based flashiness and the commonly applied R–B flashiness index showed that indices were moderately correlated (Spearman rank correlation 0.55), yet there were some striking differences. R–B flashiness was shown to be strongly sensitive to low flow variability, while IAT flashiness was more sensitive to occurrence of peak values. Both flashiness indices showed strong correlation with basin area. R–B flashiness showed no clear relationship with imperviousness. IAT flashiness tends to decrease for a combination of higher imperviousness and larger basin size, basin size playing a stronger role than urbanisation. The effect of urbanisation on flow patterns for the basins in the study area is a mixture of faster run-off flows due to imperviousness and stronger flow regulation by dams and detention basins. This leads to a mixed effect of basin size, imperviousness and flow regulation on IAT flashiness and peak flows.

A minimum observable inter-amount scale was defined as the smallest scale at which flow variations can be studied given a fixed temporal observational resolution. At higher sampling scales, a growing percentage of flow accumulations occurs in less than the given observational resolution, 15 min in this study. This typically coincides with peak flows and implies that during peak events, the observational resolution is too low to measure flow variability. IAT analysis can thus be used to identify a critical resolution for flow observations, if a given peak flow accumulation is of interest. If correct observation of peak flows of a given magnitude is important, flow data need to be collected at a higher than 15 min resolution during times of peak flows. This is typically the case in urban basins, where stormwater drainage systems are often designed for peak flows associated with 10 to 50-year return periods.

Multifractal analysis of IATs and flows was applied over a range of sub-daily to
seasonal scales. Both approaches exhibited relatively good scaling, as indicated
by

Analyses in this study identified minimum observable scales below which flow variability cannot be captured at a given measurement resolution. The combination of being able to identify these minimum observable scales and to downscale flow data based on IATs is an interesting area for future investigation. Results showed that scaling parameters for IAT time series were more reliable than those based on fixed-time sampling because of smaller departures from linearity in log–log plots. Future work will focus on possible ways to use IATs to downscale coarse-resolution flow data with the help of multifractals and multiplicative random cascades, to see if this leads to more robust and reliable results than downscaling based on conventional flow time series.

Another aspect that remains to be investigated is how IATs computed on flow data compare to IATs computed on associated rainfall time series. Because flow is linked to rainfall, the comparison of the two could help better distinguish which aspects of flow variability are due to rainfall and which relate to basin characteristics and stormwater management.

Flow data used in this study are open datasets made publicly available by USGS. They are available at

The authors declare that they have no conflict of interest.

The authors would like to acknowledge USGS for making available the datasets of flow gauges in Charlotte. The first author would like to thank NWO Aspasia and Delft University of Technology for the grant that supported this research collaboration. The second author acknowledges the funding provided by the Swiss National Science Foundation, grant P300P2_158499 (project STORMS). Edited by: K. Arnbjerg-Nielsen Reviewed by: two anonymous referees