Introduction
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
. On the other hand, high impervious degrees
may reduce base flows and lead to intermittent flow during dry periods. At
the same time, urbanisation is usually tied to development of urban drainage
infrastructure, associated with artificial flow control as well as higher
peak flows due to increased drainage connectivity. Predicting the degree of
flashiness or base flow reduction associated with urbanisation is not
straightforward, as it depends on the interplay of impervious cover, basin
size and shape, soil properties, basin slope, drainage connectivity, and
control structures such as detention ponds, weirs and pumps
. Traditional analyses
of flow time series tend to focus on specific aspects and flow
characteristics, aiming for example at predicting low flow durations or peak
flow magnitudes. For analysis of change in hydrological response, it may be
beneficial to combine both peak flow and low flow statistics into a single
framework. This applies in particular to the context of urban hydrology where
urbanisation and human intervention alter both high flow and low flow
characteristics of the hydrological response. Combining both aspects in a
single analysis is difficult, as flow distributions are highly skewed and
frequencies of low and high flow values are very different. In this paper, we
show how alternative sampling of flow time series based on inter-amount times
leads to more balanced statistical distributions, better representation of
both high and low flows in a single framework, and more robust behaviour of
statistical distributions across scales.
Statistical analysis of hydrological response
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
e.g..
Traditional statistical analysis techniques tend to focus on either left or
right tail properties of statistical distributions, but not necessarily using
the same statistical framework. Low flow analyses for example are primarily
concerned with the total time the flow stays below a critical threshold
(see for example , for an extensive review). By contrast,
peak flow analysis puts more weight on total accumulated flows at a given
timescale using annual flow maxima or peak-over-threshold values to derive
extreme value statistics and establish flood frequency
curves e.g..
Both approaches are valid and solidly rooted in the context of extreme event
analysis, with numerous applications in drought and flood risk analysis. However,
the statistical frameworks they rely on are not necessarily the same. Low flow
analysis favours “time” as a random variable. Peak flow analysis on the other
hand treats the “flow amount” over a fixed time interval as the main random
quantity. This might seem more intuitive to many but there is no strong
compelling reason to prefer one approach over the other a priori. For example,
one might as well adopt an alternative framework in which the unknown random
variable is the “time” necessary to cumulate a fixed, critical amount of
flow. By doing so, both low flows and peak flows can be analysed using
the same statistical framework. This approach is known as the inter-amount
time (IAT) method and has been previously proposed
to analyse the properties of intermittent rainfall time series. An
important goal of this paper is to derive properties of statistical
distributions obtained by applying the IAT formalism to flow time
series and to compare the results to the ones obtained using the classical fixed-time framework.
Change in hydrological response, basin flashiness
An important characteristic that has been used to analyse change in
hydrological response is basin flashiness, qualitatively described by
as one of the indicators characterising change in
natural flow regimes and how this affects the ecological integrity of river
ecosystems. developed a set of 33 indices, the
Indicators of Hydrological Alteration (IHA), including indicators for
conditions associated with flashiness, such as frequency and duration of high
and low pulses, and rate and frequency of change in flow conditions.
quantified flashiness of 5436 catchments in the
contiguous United States based on peak flows exceeding 1 m3 s-1 km-2
normalised flows (i.e. flows normalised by basin
area). A frequently used index in the literature is the Richards–Baker (R–B)
flashiness index , based on the Richards pathlength
. The R–B index is defined as the sum of absolute
values of changes in flow values divided by the total cumulative flow, and is
usually computed at the daily timescale. Similar to the coefficient of
variation, it measures the relative dispersion of the flow at a given scale.
A downside of the R–B index is that it highly sensitive to the scale of
analysis. argued that for smaller basins (< 50 km2) the use of hourly instead of daily flow data should be
considered to compute the R–B flashiness index, but also found that R–B
flashiness values computed at hourly scale are highly sensitive to diurnal or
other sub-daily low flow fluctuations. An important and still unanswered question
remains how to overcome scale sensitivity of flashiness indicators in
different hydrological basins. This is crucial for establishing how
urbanisation impacts flashiness and how changes relate to basin
characteristics such as size, slope, imperviousness degree, and whether
urbanisation thresholds can be identified, at a value above which basin response is
characteristically urban .
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.
Reference
Method
Sampling
Basins
Time series
Scale break
Value
Value
scale
length
C1
alpha
MFA
day
30 basins in FR
11–30 years
16 days
1–16 days: 0.2 ± 0.1
1–16 days: 1.45 ± 0.25
40–200 km2
16 days
30–4096 days: 0.2 ± 0.1
30–4096 days: 1.45 ± 0.2
SA
Hour
34 basins in FR
16–37 years
8.7 h–7 days
–
–
MA
Hour
12.7–703 km2
16–37 years
10 h-6.25 days∗
–
–
SA
Day
Idem
Idem
12 days
–
–
SA
Day
19 basins USA
9–73 years
8 days
1–8 days: 0.2 ± 0.1
1–8 days: 1.65 ± 0.12,
5–1.8 × 106 km2
9–73 years
8 days
1–8 days: 0.2 ± 0.1
1–8 days: 1.65 ± 0.12
SA
30 min
3 basins in FR
–
1 day
–
–
TMA
30 min
ca. 13 km2
–
16 h
30 min–16 h: 0.22
30 min–16 h: 1.18
> 16 h: 0.35
>16 h: 0.79
∗ only for higher-order moments.
Scaling analysis of hydrological flows
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
and for a review). One
important drawback of multifractal analyses is that scaling of hydrological
flow time series only holds in approximation and only over a limited range of
scales. Many studies report the existence of “scale breaks” at which
scaling parameters change and significant departures from (multi)fractality
can be observed. Table summarises findings from selected
scaling analyses of flow time series in the literature. It shows that the
number and location of the scale breaks, as well as the values of the
multifractal parameters, are sensitive to the method applied to estimate them
and the resolution of the data used to conduct the analysis. For example,
performed spectral analysis and trace moment analysis
for 30 min flow time series and identified different flow regimes with
scale breaks at 1 day for spectral and 16 h for trace moment analysis.
But when they performed the same analysis at daily and at 3 min
resolution, they identified different scaling regimes, with scale breaks at
16 days and 1 h for daily and 3 min resolution, respectively. Similarly,
found different scaling regimes in their scaling
analysis of flows for 34 basins, with scale breaks at 12 days for daily
resolution and scale breaks varying between 8.7 h and 7 days across
basins when using hourly data resolution, based on spectral analysis. When
they applied trace moment analysis for the same time series at hourly
resolution, they found no scale breaks for the lower-order moments and scale
breaks between 10 and 150 h for higher-order moments. This shows that
while most flows exhibit some sort of scaling behaviour, the identified
scaling laws are not very robust or consistent, as they are dependent on
analysis methods and data resolution.
Statistical analysis of hydrological response based on adaptive sampling using inter-amount times
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
. Increasing the sampling resolutions solves this problem,
but results in oversampling of base flow values with respect to peak flows.
An alternative consists of adopting an adaptive sampling strategy, i.e. one
that adapts the sampling rate to the variability of the signal itself
e.g.. This makes sense for processes that are very
unevenly distributed in time (such as rainfall and hydrological flows), and
means taking more samples during periods of high activity (e.g. peak flows
following storm events) and fewer during lower activity (e.g. periods of
base flow). A well-designed adaptive sampling technique lowers the
probability of missing an interesting feature like peak flow and avoids
oversampling during periods of small flow variations. We examine to what
extent IATs influence the variance, skewness and shape of the sample
distributions and how they can be used to better characterise basin
flashiness and derive more robust scaling laws. Our results show that because
IATs give more weight to rare peak flows compared to common base flows, they
can provide different insights into flow properties and complement
traditional flow time series analyses and metrics. Advantages of IAT
sampling compared to conventional time series analysis are that IAT time
series contain more information about peak flows and evolve in a more
predictable way across ranges of smaller to larger scales. This makes them a
more robust and reliable source of information to make predictions about flow
characteristics at small, unobserved scales, including crucial information
about rapidly evolving peak flows.
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
(km2), imperviousness (%), average 24 h flow (m3), average 24 h flow
normalized by basin area (mm) and length of observation in years.
ID
Name
Area
Imperv.
Dams
Mean flow
Mean norm. flow
N years
825
UBriar
13.3
24.0
22
12 275
0.92
17.4
315
Taggart
13.6
35.0
3
13 559
1.00
17.2
562
Campbell
15.3
28.0
48
13 567
0.89
16.2
175
Steele
17.9
32.0
21
17 838
1.00
17.4
700
McMullen
18.3
21.0
15
20 348
1.11
29.0
255
UMcAlpine
18.9
18.1
100
15 061
0.80
16.3
975
Irvins
21.8
8.0
62
14 821
0.68
16.3
970
Stewart
23.4
33.0
55
38 800
1.66
15.3
348
Coffey
23.8
25.0
72
24 104
1.01
17.0
409
LSugarM
31.7
48.0
2
46 775
1.48
21.0
022
LBriar
48.5
25.0
17
53 246
1.10
19.8
800
SixMile
52.6
15.0
-99
38 914
0.74
8.0
300
UIrwin
78.1
34.0
39
107 119
1.37
29.0
600
MMcAlpine
100.2
20.0
51
105 640
1.05
29.0
507
LSugarA
111.1
32.0
24
199 002
1.79
29.0
530
LSugarP
127.4
26.0
-99
205 202
1.61
18.3
750
LMcAlpine
238.4
19.4
-99
269 534
1.13
29.0
Data and methods
Flow datasets
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 km2 in size. The area is largely covered by low to
high intensity urban development, covering 60 to 100 % of basin areas.
Percentage of impervious cover varies from 8 % in the least developed to 48 % in
the most urbanised basin covering the city centre of Charlotte.
Figure shows a map with the location of the area, boundaries of
hydrological basins and location of stream gauges used in the analysis.
Table summarises the main characteristics of
the 17 basins.
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 summarises the
characteristics of the basins associated with each basin as well as the time
period covered by the data. The temporal scale of observations changed from
15 to 5 min between 2010 and 2014, at different times for each gauge;
overall 20–30 % of the total observation record was covered by 5 min
intervals. Gauges measure water depth using pressure transducers and flow is
derived using stage–discharge curves. These curves were established based on
protocols developed by USGS and include manual flow measurements during site
visits performed by USGS staff. As part of this procedure, stage–discharge
curves are checked and recalibrated during site visits several times per year
(https://waterdata.usgs.gov/nwis/measurements). The percentage of missing
flow data was smaller than 5 % for all gauges included in the analysis;
missing data were treated like zeros. The effect of missing data on IATs is
difficult to predict as this depends on the pattern of missing values and
whether or not they occur during a period of low or peak flow. Sensitivity
studies by have shown that the general effect of
replacing missing values by zeros is that a few sample IATs will be
overestimated. This mostly affects the right tail of the distribution and
tends to have limited impact on peak flow characteristics. Another strategy
would be to replace missing values by mean or median flow value, which may
slightly reduce the overestimation of IATs in case several missing values
occur in row. However, in this paper only the worst-case scenario will be
considered, i.e. missing values were replaced by zeros.
Definition of inter-amount times
In this paper we analyse hydrological flow variability, based on the
distribution of IATs. We use the following definition of
IATs, based on , when Δq > 0 denote a fixed flow amount: the series of IATs τn(Δq)
with respect to Δq is defined as follows.
τn(Δq)=tn(Δq)-tn-1(Δq),
where tn(Δq) denotes the time at which the cumulative flow amount
first exceeded n times (Δq):
tn(Δq)=inf{u:Q(u)≥n⋅Δq},
where Q(u) denotes the cumulated flow at time u, Q(0)=0, and inf stands for
infimum, also known as the greatest lower bound in a set.
A steady flow pattern with constant flow has equal IATs for all values of Δq.
A variable flow pattern, on the other hand, is characterized by a more variable IAT distribution.
Normalized inter-amounts
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 τ‾ (e.g. 24 h) and
determine the inter-amount Δqτ‾ at this timescale:
Δqτ‾=τ‾QNT,
where QN denotes the total cumulative flow amount at the considered
location and T is the length of the studied time period. In other words,
instead of comparing IATs for a fixed accumulation, we choose the mean IAT
τ‾ and compute (Δq)τ‾ such that the series of
IATs {τn(Δqτ‾):n=1,…,N} has mean
τ‾. Two locations with different cumulative flow amounts over a
given period of time, e.g. over a year, therefore have different normalized
inter-amounts.
Sample estimates and minimum inter-amount scale
Inter-amount times can be estimated from a sample flow time series
q1,..,qN with temporal observation scale Δt that may vary in
time. But for simplicity, only the case with fixed temporal resolution
Δt will be considered below. A key step in this procedure is the
determination of the first passage times t1,..,tn in
Eq. (). This is done by considering the sample accumulated flow
amounts Q1<..<QN at times tn=t0+nΔt:
Qn=∑i=1nqin=1,…,N.
The exact first passage times t1,..,tn for a fixed flow amount
Δq>0 are likely to be unknown due to the limited temporal resolution
of the data. But we can approximate them based on linear interpolation:
t^n(nΔq)=ΔtinΔq-QinΔq-nΔqqinΔqn=1,…,N,
where t^n are the estimated passage times and inΔq denotes
the index (in the sample) at which the total cumulated flow first exceeded
n times (Δq):
inΔq=min{i∈N|Qi≥nΔq}n=1,…,N.
The sample IAT estimates are then given by the following:
τ^n(Δq)=t^(nΔq)-t^(nΔq-Δq).
Because of the linear interpolation in Eq. (), each sample IAT estimate,
regardless of its length and the scale of analysis, will be affected by a
small interpolation error εn(Δq)<Δt. This error is
random and has little influence on key statistics as long as IATs remain much
larger than Δt, as is usually the case for large enough values of
Δq and during periods of low to moderate flow. Most of the
interpolation errors happen during peak flows, when large flow amounts are
accumulated over small periods of time. It is therefore important, for any
given gauge, to identify the values of Δq above which reliable IAT
estimates can be derived. To identify the range of scales over which IATs can
be reliably estimated, we consider the worst-case scenario in which all
interpolation errors are equal to ±Δt. In this case, the maximum
relative error affecting IAT estimates is given by the following:
εn(Δq)=Δtτ^n(Δq).
The minimum value of Δq for which IATs can be reliably estimated
depends on how strictly we want to control the estimation errors in Eq. ().
In our analysis, we set the mean of absolute relative errors to
be smaller than 50 %. This is a rather conservative approach as the
estimation errors in Eq. () represent the worst-case scenario and actual
errors are likely to be much smaller than that. This leads to the following
rule for determination of minimum inter-amounts Δq that can be used
for analysis:
Δqmin=min{Δq>0:ε‾Δq<0.5},
where ε‾Δq represents the arithmetic
mean of the maximum relative errors in Eq. ().
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 Δq:
Δqmax=⌊QN100⌋,
where ⌊⌋ denotes the lower integer part and QN is the
total cumulative flow for the considered time series.
It is worth pointing out that the lower bound on the inter-amount in Eq. ()
also provides an indication of the left-tail properties of IATs, and
thus of the degree of flashiness of the hydrological response, i.e. the
smallest scale at which flow variations can be studied given a fixed temporal
observational resolution. We will elaborate on this in Sect. ,
where we discuss this property in relation to basin flashiness. More
generally, the left tail properties of IAT distributions provide a good
indication of what observational resolution is necessary to adequately
capture the most extreme flow variations. For more details on this important
point, the reader is referred to the results section.
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
showed no signs of long-term trends at 7506 gauges in
the contiguous USA in the last 30 years. Indeed, our own analyses revealed no
significant long-term trend in mean IAT or flow variability over the
considered time period.
Distribution of inter-amount times versus flows
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 . We computed the
coefficient of variation (CV), defined as the standard deviation divided by
the mean, as an indicator for relative spread around the mean. Values of
skewness and “medcouple” , a more robust skewness metric
based on ordered statistics instead of statistical moments, were computed to
investigate asymmetry of the histograms and influence of outliers. We
compared coefficient of variation, skewness and medcouple values for IATs
with those for traditional flow time series and investigated relationships of
the three statistics with basin area and imperviousness degree.
Distribution of changes in inter-amount times
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.
Flashiness indicator and minimum observable scale
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 μ
(expressed in hours) at which the 1 % quantile of the IAT distribution equals
the observational scale Δt (15 min in our case). That is, the IAT
flashiness indicates the average time needed to accumulate the amount of flow
that can be accumulated in 15 min or less, 1 % of the time. The larger
the flashiness, the more flow can be accumulated over short amounts of time.
To better interpret results, we compared the IAT flashiness index with the
frequently used R–B flashiness index defined in :
R-Bindex=∑i=1Nqi-qi-1∑i=1Nqi,
where qi denotes the flow at time step i. The R–B flashiness
index is dimensionless and can vary between 0 and 2. It is 0 for constant
flow and 2 for highly variable and continuously changing flow. Its value is
independent of the units chosen to represent flow .
However, index values do depend on the timescale at which they are computed,
as will be discussed later in the results section. In our analysis, we
computed R–B flashiness indices on daily aggregated flow values.
Scaling of inter-amount times
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 q>0 of a stochastic
process Xλ at scale ratio λ are related by a power law:
〈Xλq〉=C(q)λK(q),
where 〈Xλq〉 denote the moments of order
q of X measured at a scale ratio λ, C(q) is a constant (for
each q) and K(q) is called the moment scaling function. Within the
universal multifractal framework, K(q) is characterised with the help of
only three parameters, α, C1 and H
:
K(q)=C1α-1(qα-q)-qHifα≠1C1qln(q)-qHifα=1.
The parameter C1 is referred to as the intermittency and characterises the
clustering of the time series at smaller and smaller scales. C1=0 for a
homogeneous field that fills the embedded space and approaches 1 for an
extremely concentrated field. The parameter α is called the
multifractality index (0<α<2) and it controls how the moments
change when going from one scale to another. Finally, H=-K(1) is called the
Hurst exponent. Note that in the case of IATs, the mean inter-amount time
τ‾ and scaling ratio λ are inversely proportional to
each other (i.e. Δqτ‾∼λ-1). So either
of them can be used here as a measure of scale. The only difference will be
the value of the constant C(q) and the sign of the exponent in Eq. ().
The scaling quality is assessed by noting that if Eq. () is true, the
log moments for fixed values of q should be a linear function of the
log-scale of ln(λ):
ln(〈Xλq〉)=K(q)ln(λ)+ln(C(q)).
The extent to which this equality holds can be assessed by fitting
a linear regression model and computing the R2 values, i.e. the
coefficient of determination of the log moments versus the log-scale for each
value of q. A R2 of 1 indicates perfect scaling. The lower the
coefficient of determination, the larger the deviations from
scale-invariance. The approach was repeated for different values of q and
the mean or minimum value of R2 were chosen as a way to assess the overall
quality of the scaling. Based on recommendations by ,
we refrained from using too low- or high-order moments and only considered
values of q between 0.4 and 2.5, with an equal number of moments above and
below 1 to avoid favouring one tail of the distribution over the other. The
range of IAT scales that was used for the analysis was constrained by the
length of the time series and the minimum and maximum inter-amounts defined
in Eqs. () and (). The corresponding scales varied from 0.1 to
0.6 days up to 28 to 100 days for the longest time series.
Results
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.
Time series and variability analysis of inter-amount times and flow values
Figure shows an example of times series for flows and for IATs
for the gauge at Taggart Creek, a 13.6 km2 basin in the
Charlotte catchment, at 24 h sampling scale. The two graphs bring out
different aspects of flow variability: flow time series have most of their
data points concentrated in the low flow region, with intermittent peak flows
characterising rain events. For IATs, peak flows appear as minima, while
periods of low flow show up as maxima in the time series. The graph
illustrates how IAT samples are more evenly distributed across high and low
values in the time series compared to flows. The mean inter-amount for
Taggart Creek at 24 h sampling scale is 13 559 m3, equivalent to
0.998 mm when normalised by basin area. Hence, in IAT analysis, the time series is
sampled each time 0.998 mm of normalised flow has been accumulated, which
amounts to frequent samples during high flows and fewer samples during low
flow periods. For instance, a high concentration of IAT samples is clearly
visible for the wet year 2003: this year is represented by 802 IAT samples
compared to the 365 samples per year we have on average.
Example of times series for flow (a) and for associated
inter-amount times (b) for the flow gauge at Taggart Creek, a
13.6 km2 basin in the Charlotte catchment.
Figure illustrates the adaptive sampling strategy based on flow
amounts as the sampling unit, instead of fixed time steps.
Figure b shows cumulative flow over a week, where a storm event
occurred on 7 August. In conventional flow time series analysis, flow is
sampled daily (in this example), resulting in one sample representing the
peak period of the event (i.e. on 7 August). In IAT analysis, flow
accumulation determines the sampling frequency, so periods of low flow are
sparsely sampled, while the storm event is represented by eight samples. This
illustrates how, even for 24 h mean inter-amounts, sampling frequency can
be much higher during periods of peak flow.
Illustration of inter-amount data sampling for cumulative
flow over a period of 7 days, for Taggart Creek. (a) Flow data series
at original 15 min observational resolution; (b) cumulative graph for
flows and IATs at the same mean sampling resolution, illustrating how
adaptive sampling based on IATs differs from classical fixed-time sampling.
Histograms of flow time series (a) and time series of inter-amount
times (b) for Taggart Creek and Little Sugar Creek at Archdale (LSugarA), for 24 h scale.
Histograms of flow time series and IATs at daily timescale are plotted in
Fig. , for two basins, Taggart Creek (13.6 km2) and LSugarA (111 km2). The
corresponding inter-amounts are 1 and 1.8 mm of normalised flow (for
Taggart and LSugarA, respectively). Histograms for the other 15 basins are
available in the Supplement to this paper. Figure shows that
both histograms of flows and IATs are positively skewed. In both cases
however, left and right tails represent very different flow characteristics.
The left tail of the flow's histogram essentially features common base flow
values while the right tail captures rare peak flow events. By contrast, the
left tail of IAT distributions, which makes up most of the values,
predominantly features short IAT values associated with periods of high flow.
The rare samples that make up for the right tail represent long waiting times
associated with extended periods of low flow. The low density of the first
bin in the flow histogram for LSugarA reflects the effect of low flow
regulation for this basin. The same effect is reflected in the bi-modal shape
of the IATs histogram. Note that the low density 0–0.5 bin in the flow
histogram for LSugarA corresponds to the >3.5 day bins in the IAT histogram.
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).
name
CV IAT
CV flow
Skew IAT
Skew flow
Mc IAT
Mc flow
Skew dIAT
Skew dflow
Mc dIAT
Mc dflow
UBriar
1.95
3.69
4.91
14.79
0.84
0.41
0.39
-2.81
0.51
-0.30
Taggart
2.11
3.32
4.40
9.13
0.90
0.55
0.00
-0.16
0.57
-0.46
Campbell
2.02
3.25
4.26
10.40
0.84
0.51
-0.02
0.15
0.66
-0.41
Steele
2.24
3.65
4.39
10.21
0.86
0.58
-0.43
-0.41
0.74
-0.46
McMullen
2.22
3.37
5.35
10.10
0.90
0.56
-0.61
-0.40
0.61
-0.35
UMcAlpine
2.04
3.55
5.53
13.51
0.79
0.42
-2.48
2.17
0.63
-0.38
Irvins
2.52
4.32
8.37
11.74
0.89
0.42
-3.84
0.07
0.78
-0.41
Stewart
0.96
2.47
0.84
12.90
0.12
0.37
-0.23
-0.25
0.26
-0.02
Coffey
2.15
2.94
7.34
8.44
0.85
0.54
-1.05
0.21
0.64
-0.41
LSugarM
1.57
2.95
2.06
11.55
0.90
0.55
-0.43
0.73
0.37
-0.31
LBriar
1.74
3.30
3.13
13.77
0.87
0.51
-0.87
1.13
0.56
-0.32
SixMile
2.23
2.59
6.29
6.42
0.82
0.38
-1.34
1.23
0.69
-0.31
UIrwin
1.36
2.70
2.65
14.43
0.69
0.53
-0.32
1.77
0.45
-0.22
MMcAlpine
2.00
3.19
5.42
10.30
0.84
0.50
-1.51
0.66
0.68
-0.38
LSugarA
1.16
2.28
8.62
12.04
0.44
0.51
0.77
0.40
0.33
-0.26
LSugarP
1.04
2.10
1.52
9.20
0.49
0.58
-0.71
-1.04
0.33
-0.33
LMcAlpine
2.16
2.84
6.56
7.65
0.88
0.50
-1.63
0.27
0.50
-0.32
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.
ID
Name
CV IAT
CV flow
Skew IAT
Skew flow
Mc IAT
Mc flow
Skew dIAT
Skew dflow
Mc dIAT
Mc dflow
970
Stewart
0.96
2.47
0.84
12.90
0.12
0.37
-0.23
-0.25
0.26
-0.02
300
UIrwin
1.36
2.70
2.65
14.43
0.69
0.53
-0.32
1.77
0.45
-0.22
825
UBriar
1.95
3.69
4.91
14.79
0.84
0.41
0.39
-2.81
0.51
-0.30
022
LBriar
1.74
3.30
3.13
13.77
0.87
0.51
-0.87
1.13
0.56
-0.32
409
LSugarM
1.57
2.95
2.06
11.55
0.90
0.55
-0.43
0.73
0.37
-0.31
507
LSugarA
1.16
2.28
8.62
12.04
0.44
0.51
0.77
0.40
0.33
-0.26
530
LSugarP
1.04
2.10
1.52
9.20
0.49
0.58
-0.71
-1.04
0.33
-0.33
562
Campbell
2.02
3.25
4.26
10.40
0.84
0.51
-0.02
0.15
0.66
-0.41
255
UMcAlpine
2.04
3.55
5.53
13.51
0.79
0.42
-2.48
2.17
0.63
-0.38
975
Irvins
2.52
4.32
8.37
11.74
0.89
0.42
-3.84
0.07
0.78
-0.41
600
MMcAlpine
2.00
3.19
5.42
10.30
0.84
0.50
-1.51
0.66
0.68
-0.38
750
LMcAlpine
2.16
2.84
6.56
7.65
0.88
0.50
-1.63
0.27
0.50
-0.32
Tables (6th and 7th columns) and summarise statistics of flow and IAT
time series, at 24 h sampling scale. The results show that mean
inter-amounts vary from 12 275 m3 for the smallest to
269 534 m3 for the largest basin in size. Mean normalised
inter-amounts vary from 0.68 mm for Irvins Creek, the least-urbanised basin
(8.2 % imperviousness) to 1.79 mm for Little Sugar Creek at Archdale, one of
the largest basins with a high degree of imperviousness (32 %). Coefficients
of variation at the daily scale are consistently higher for flows than for
IATs (e.g. 1.7 times higher on average), which highlights the more balanced
nature of IAT distributions. Skewness values at the daily timescale are 3.6
times higher for flows than for IATs, on average, and even up to a factor of
15 higher for Stewart Creek. By contrast, medcouple values for flows are
lower than for IATs by a factor of 2.1 on average. This shows that statistical
distributions of flows are strongly influenced by the presence of a few very
large outliers. Most of the weight, however, lies close to the median (low
medcouple). The IAT sampling gives more weight to rare peak flow values and
less to common base flow, therefore producing distributions with lower
skewness and more information about peak flow values. The larger medcouple
values mean that IATs above the median value tend to be much further away
from the median than values below the median. In other words, the right part
of the distribution, which features long waiting times during low flow
conditions, can be very stretched.
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. ,
behaviour of the statistical distributions of flows and
IATs, as well as associated CV, skewness and medcouple values, will be
analysed across a range of sub-daily to seasonal scales.
Statistical distribution properties comparison across different hydrological basins
Subsequently, we compared properties of IAT and flow
distributions across the 17 basins in relation to basin characteristics.
Figure shows scatter plots of mean normalised inter-amounts,
CV, skewness and medcouple values for flows and IATs as a function of basin
area and imperviousness degree. The results show a positive correlation of
24 h mean normalised flows or inter-amounts with basin size (Spearman
correlation 0.55). This is mainly explained by a lower likelihood of low
flows that have a large influence at this scale (24 h). Mean normalised
flows correlate positively with imperviousness degree (Spearman correlation
0.58), which is likely to be explained by a generally growing importance of
flow regulation, resulting in maintenance of higher mean base flows in
urbanised basins.
Scatter plots for mean normalised flow inter-amounts (a, b), coefficient
of variation (c, d) and medcouple values (e, f) for flows and inter-amount times
versus basin area and imperviousness degree. Grey triangle symbols represent
inter-amount times, black circles represent flows.
Looking at CV values across all basins (Fig. c, d), we found
that CV values for both flows and IATs generally decrease with basin size and
with imperviousness degree. CV values are significantly negatively correlated
with basin size for flows (Spearman rank correlation -0.75). This can be
explained by an increased smoothing effect on flow variation, in particular a
lower likelihood of low flow extremes during dry periods for larger basins.
CV values for IAT distributions do not show a significant correlation with
basin size, while they are significantly negatively correlated with
imperviousness (Spearman rank correlation -0.57). Since IAT distributions put
more weight on high flows compared to low flows as a result of their adaptive
sampling strategy, this probably indicates stronger influence of flow
regulation in 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 is clearly visible for the most urbanised
basin, LSugarM, gauge 409). The effect of urbanisation as reflected by
imperviousness degree on IAT statistics appears to be more important than
basin size.
Scatter plots for skewness and medcouple values
(Fig. e, f, g, h) show generally weak correlation with basin
area (Spearman correlations not significant at the 5 % level). Skewness of
IAT distributions is significantly negatively correlated with imperviousness
(Spearman rank correlation -0.63). Similar to CV values, this probably
indicates stronger influence of flow regulation on flows in urbanised basins.
Medcouple values for IATs clearly show three low-value outliers: for Stewart
Creek (970), LSugarP (530) and LSugarA (507). In these basins, active low
flow control is applied 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. Table
summarises CV, skewness and medcouple values for three sets of sub-basins in
the Charlotte catchment. The results show that variability in skewness and
medcouple values is unrelated to inter-basin connections. The same applies
for flow CV values, while CV values for IATs seem to be clustered by group of
sub-basins, indicating that inter-basin correlation plays a role in explaining
IAT second-order variability. The fact that the effect is only
visible for IAT, not for flows, indicates that correlation is mainly
associated with occurrence of peak flows, which receive more weight in IAT
than in flow statistics.
In 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
e.g. and specifically between CV
of flows and basin area (). These authors also refer
to complexities in hydrological response resulting in deviations from this
general relationship. The same applies for the basins in our study, where
basin area only explains part of the flow variability, especially for smaller
basins. Results showed that larger imperviousness is associated with higher
mean flows and significantly lower CV values for IATs, while there was not
significant correlation between CV values for flows and imperviousness. This
is probably explained by urbanisation being mainly associated with stronger
flow regulation by detention and capacity constraints in the drainage system.
Since IATs are relatively more sensitive to high flows, this effect showed up
more clearly in CV values for IATs than for flows. CV and skewness values are
much higher for flows than for IATs, while medcouple values are lower for
flows, indicating strong asymmetry of the flow distributions and low
representation of high flow extremes in the statistical distribution. While
reported a decrease in CV and skewness associated
with urbanization for basins in the UK, we did not find significant
correlations based on CV and skewness indicators for flows. Skewness for IATs
was significantly negatively correlated with imperviousness; as stated
before, this is probably associated with IAT statistics being more sensitive
to variability in high flows than conventional flow statistics.
histograms of first-order differences in flows (a, b) and inter-amount
times (c, d), at 24 h analysis sampling scale, for Irvins Creek and LSugarM Creek.
Distribution of changes in inter-amount times
Figure shows histograms of first-order differences in IATs and
flows at the 24 h analysis scale, for Irvins Creek (the least-urbanised
basin), LSugarM (the most impervious basin), Stewart Creek (a basin with low
flow regulation) and McAlpine (the largest of all studied basins). In the flow
histograms, negative differences are associated with recession, positive
differences with flow rise. Conversely, negative differences in IATs occur
during flow rise, positive differences during flow recession. Most flow
differences are concentrated in the 0 to -0.5 mm bin, associated with slow
flow recession of 0.5 mm day-1. Most IAT differences are concentrated in the 0 to 0.1 or 0.2 day bin, associated with steeper flow recession of
approximately 5 to 10 mm day-1. This reflects the relatively higher
sampling of rapid flow response for IATs compared to conventional flow
sampling. Skewness and medcouple values of the histograms provide indications
of hydrograph shape, in particular of the steepness of the hydrograph
recession limb: higher skewness, and thus more weight of the distribution
concentrated in one of the tails, indicates slow flow recession compared to
relatively rapid flow rise. Figure shows scatter plots for
skewness and medcouple values versus basin size and imperviousness, for all
basins. The three basins with low flow regulation (970, 530, 507) can be
recognised by their low medcouple values for IAT difference indicating near-symmetrical histograms, i.e. flow rise and recession occur at similar rates.
Most IAT difference histograms are negatively skewed, with a longer left
tail than right tail, i.e. IATs generally decrease quicker (flow rise) than
they increase (flow recession). The strongest negative skewness for IAT
differences was found for the least-urbanised basin (Irvins Creek, gauge
975), indicative of steep flow rise occurring in this basin. Significant
positive correlation was found between skewness of IAT difference histograms
and imperviousness (Spearman correlations 0.75), indicating lower probability of
steep flow rise in higher urbanised basins. Negative correlation was found
between medcouple and imperviousness (Spearman correlation -0.55); thus
relatively more symmetrical hydrographs with flow rise and recession at
similar rates occur for urbanised basins. Here, sub-basin correlation appears
to play role: medcouple values are higher overall in the McAlpine sub-basins
than in Little Sugar Creek and Irwin sub-basins (see
Table ). Significant correlations of IAT
difference skewness and medcouple with imperviousness show that urbanisation
is associated with more regulated flows, confirming findings in
Sect. .
Scatter plots of skewness (a, b) and medcouple values (c, d) of
histograms for differences in flows and inter-amount times, plotted versus
basin size and imperviousness degree. Grey triangle symbols represent
inter-amount times, black circles represent flows.
Inter-amount-time variability across scales, from sub-daily to seasonal sampling scale
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 shows quantile plots for normalised flows and
IATs at scales between 12 h and 64 days, for Taggart Creek. On the
horizontal axis is the sampling scale, i.e. fixed sampling time for
conventional flow statistics or, equivalently, mean inter-amounts for IAT
statistics. Note that for the IAT analysis, mean inter-amounts are normalised
by basin area size and reported in millimetres to allow easier interpretation of flow
magnitudes and to allow easier comparison between basins. For instance, the
normalised inter-amount Δq for Taggart Creek at the daily scale is
0.998 mm. The vertical axis shows quantiles of normalised flows and
IATs corresponding to the sampling scale in time or Δq. Values on
both x and y axes are plotted on log scales to allow easier visualisation of
quantile values that vary by 2 to 4 orders of magnitude. The bold black line
denotes the mean, and the dotted black line shows median values. The central part of
the quantile plots represents the 25–75 percentile range, upper and lower
whiskers 10–90 percentiles and crosses the 1 and 99 percentiles.
Quantile plots of flows (a) and inter-amount times (b) for Taggart
Creek for a range of scales, from 12 h to 60 days. The bold black line
denotes the mean values. The dotted black line shows median values. The
central part of box plots represents the 25–75 percentile range, upper
and lower whiskers the 10–90 percentile range, crosses the 1–99 percentile range.
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. a
and b we see that the 10–90 percentile range of IATs
gradually increases towards smaller scales. For flows, the 10–90 percentile
range remains approximately constant; however, distance between 90 and 99 percentile values rapidly increases 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 increases progressively towards
smaller scales. By contrast, 10–90 and 1–99 percentile ranges for IATs
increase more or less similarly with scale, for sampling scales ranging from
0.51 mm to approximately 10–16 mm. This indicates that the tails of IAT
distributions are more or less equally sampled, at least up to the 1 and 99 percentiles. The upper 75, 90 and 99 IAT percentiles of IATs, associated with
low flow periods, change approximately log-linearly with scale, showing that
upper tail percentiles of IAT values refer to the same low flow periods
across all scales, up to 8–16 mm scale. Associated low flows are
approximately 0.1 mm day-1. The 1 percentiles for flows are associated with
approximately 0.02 mm day-1, for the 12 h to 4-day scale, showing that the
distribution tail associated with low flows captures lower flow extremes in
conventional sampling than in IAT sampling. This is a result of the
relatively high frequency at which low flows are sampled. Conversely, peak
flows, associated with the right tail of the flow distribution, are sampled
less frequently in conventional flow sampling: the 99 percentiles are
associated with peak flows of 0.78 to 0.38 mm h-1 for 12 h to 4-day scale.
The 1 percentiles of IATs are associated with peak flows of about 20 mm h-1, at
the 0.5 to 4 mm inter-amount scale, associated with mean IATs of 12 h to
4 days. This shows that the IAT distribution captures more extreme peak flow
values than conventional flow sampling, at the same sampling scale.
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
shows coefficients of variation, computed as the ratio of the second- over
the first-order moment, for four gauges, across a range of sub-daily (3 to
12 h) up to bi-monthly (60–68 days) scales. Results show that
coefficients of variation for flows vary non-linearly with scale, while they
approximately follow a power law with scale for IATs. For Irvins Creek, the
most natural basin in this study (8.2 % imperviousness,
Fig. a), CV values of IATs and flows are similar over a range
of 10 to 50 days. 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,
as a result of growing skewness of the statistical distribution, caused by
relative oversampling of low flows, or conversely, undersampling of high
flows. CV values for Upper LSugar Creek, the most urbanised basin are lower
than for Irvins Creek, especially at smaller scales (Fig. b).
This is explained by the influence of flow control measures in this basin, as
flows are constrained by the stormwater drainage system. The difference is
more pronounced for IATs, because IAT variance is more sensitive to peak
flows as a result of the adaptive sampling strategy. Figure c
shows that for LMcAlpine, the largest basin (238.4 km2),
CV values for flow are more or less stable between 3 and 24 h scale, due
to strong smoothing of peak flows at this intra-event scale. In contrast,
CV values for IATs increase over this range, due to scale sensitivity of the
upper tail of the IAT distribution, where long IATs at this small scale (0.1
to 1.1 mm for 3 to 24 h) are broken up more unevenly, creating increased
CV and skewness. This shows that 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.
CV values for Stewart Creek in Fig. d show very low CV values
for IATs that vary little with scale, while CV values for flows are much
higher and strongly sensitive to scale. Stewart Creek is a small,
semi-urbanised basin (33 % imperviousness) where active low flow control is
applied. This results in very low variability in IATs across the entire range
of scales, while CV values for flows are lower than those for similar basins,
but highly sensitive to scale, probably due to unbalanced sampling of peak
flows compared to very stable low flows.
In Sect. we analysed skewness and medcouple values of flow
and IAT distributions at the 24 h scale and found that skewness values
were lower and medcouple higher for IATs than for flows. This was explained
by the sensitivity of flow distributions to rare peak flows compared to
frequently sampled low flows. Initial analyses of skewness and medcouple
values across scales showed that results are highly sensitive to the sampling
scale. While CV values show a stable pattern across scales, results for
skewness and medcouple are much more variable, across scales and across
basins. Explanation of this scale sensitivity of skewness metrics and what
information can be derived from this about the tails of the distributions
requires deeper analysis that will be part of future work.
Flashiness indicators and minimum observable scale
Two flashiness indicators were computed, as explained in Sect. : the classical R–B flashiness index and an IAT flashiness
indicator based on characteristics of the IAT distribution.
Table summarises flashiness values
for all gauges, as well as minimum and maximum observable inter-amounts, as
defined in Eqs. () and (). IAT flashiness indicators
vary between 12.5 and 165 h; higher values are generally associated with
smaller basins. R–B flashiness values vary between 0.8 and 1.3, indicative of
moderately variable flows (R–B flashiness can vary between 0 and 2). Values
are in the same range as those reported by for smaller
basins: they found R–B flashiness values larger than 1 for basins smaller
than 50 km2. R–B flashiness is strongly correlated with
CV values (Fig. c, Spearman correlation 0.77); this confirms
that R–B flashiness is essentially a metric of flow variability.
Figure a shows that IAT-based flashiness and R–B flashiness are
moderately correlated (Spearman rank correlation 0.55), yet there are some
striking differences. The three low-flow-regulated basins have very low R–B
flashiness values, while IAT flashiness values are in line with values for
other basins. This is explained by R–B flashiness being strongly sensitive to
low flow variability, while IAT flashiness is more sensitive to occurrence of
peak flow values. For instance, the McAlpine basin (gauge 255) has a very high
IAT flashiness as a result of high occurrence of peak flows. On the other
hand LSugarM (gauge 409), the most urbanised basin, has low IAT flashiness as
a result of peak values being capped by maximum capacity of pipes in the
drainage network.
Scatter plots of flashiness versus basin area and imperviousness,
for all gauges. Grey triangle symbols represent inter-amount times, black circles represent flows.
Figure b and d show scatter plots of IAT
flashiness (left y axis) and R–B flashiness (right y axis) versus basin area
and imperviousness, for all gauges. They show a clear relationship between
flashiness and basin area (Spearman correlation -0.83 for IAT, -0.71 for R–B
flashiness), with a large range of flashiness values for the smallest basins
(< approx. 30 km2). Here,other processes than
basin size clearly play a role in explaining flashiness. Correlations between R–B and
IAT flashiness versus imperviousness degree are not significant at the 5 %
level. For R–B flashiness, the most pervious and the most impervious basins
(gauges 975 and 409 respectively) are both in the high range of flashiness
values, showing that other influences, such as basin size and presence or
absence of low flow regulation play a more important role than imperviousness
degree. IAT flashiness tends to decrease for a combination of higher
imperviousness and larger basins, basin size playing a stronger role than
urbanisation. The most urbanised basin, LSugarM (gauge 409, 31.7 km2, 48 % imperviousness) has a relatively low flashiness
value of 48.8 h, while the least impervious basin, Irvins Creek (gauge 975, 21.8 km2, 8 % imperviousness) has a high flashiness
value of 102.8 h. As discussed in Sect. , the effect
of urbanisation on flow patterns for the basins in the study area seems to be
mainly determined by increased flow regulation associated with introduction
of dams, stormwater detention basins and stormwater drains with capacity
limitations. While higher imperviousness leads to higher mean run-off flows
(for instance, 1.5 mm for LSugarM versus 0.68 mm for Irvins Creek, at 24 h
scale), rainfall tends to run off relatively more uniformly in impervious
basins, without rapid flow rise or sharp flow peaks, depending on the degree
of flow regulation. The leads to a mixed effect of basin size, imperviousness
and flow regulation on IAT flashiness and peak flows. In this study, IAT
flashiness values were defined as the time that is needed on average to
accumulate the amount of flow that is accumulated in 15 min or less, 1 %
of the time.
Minimum and maximum observable scales (in hours), flashiness index
for 15 min observation time (in hours) and fitted multifractal parameters
α and C1 for inter-amount time flows.
ID
Min. scale
Max. scale
Flash
RB
Min. R2 IAT
Min. R2 flow
Alpha IAT
Alpha flow
C1 IAT
C1 flow
UBriar
13.75
1462
128.75
1.15
0.999
0.994
1.05
1.53
0.21
0.35
Taggart
12.50
1443
118.75
1.22
0.999
0.993
0.88
1.30
0.26
0.36
Campbell
9.25
1360
106.00
1.17
1.000
0.993
1.01
1.45
0.24
0.33
Steele
9.50
1457
57.25
1.21
1.000
0.991
0.86
1.30
0.25
0.36
McMullen
11.00
2420
92.25
1.25
0.999
0.992
0.94
1.32
0.26
0.32
UMcAlpine
10.00
1367
165.00
0.99
1.000
0.990
1.24
1.59
0.19
0.33
Irvins
13.75
1367
102.75
1.14
0.999
0.991
1.25
1.40
0.22
0.35
Stewart
6.25
1284
64.00
0.82
1.000
0.994
0.72
2.06
0.09
0.24
Coffey
4.75
1422
26.25
1.09
0.999
0.997
1.53
1.37
0.21
0.28
LSugarM
7.50
1752
48.75
1.16
1.000
0.996
0.66
1.48
0.20
0.33
LBriar
6.75
1658
61.50
1.12
1.000
0.996
0.88
1.51
0.20
0.31
SixMile
3.00
672
12.50
0.97
0.999
0.995
1.64
1.36
0.21
0.26
UIrwin
5.00
2420
55.25
0.97
1.000
0.995
1.14
1.81
0.14
0.25
MMcAlpine
5.50
2420
30.75
1.09
1.000
0.996
1.28
1.46
0.20
0.28
LSugarA
3.50
2420
30.75
0.85
0.995
0.996
2.89
1.89
0.07
0.22
LSugarP
2.75
1532
18.00
0.83
1.000
0.996
1.37
1.87
0.09
0.20
LMcAlpine
3.00
2420
15.75
0.98
0.999
0.997
1.64
1.32
0.19
0.24
R–B flashiness indices were computed at the daily scale, to
allow comparison with results obtained by . For a fair
comparison, both flashiness indices should be computed at similar scales, as
far as possible, given that definitions used in the two approaches are
different. We aimed to compute both indices at hourly scale, as this is an
appropriate scale in relation to the size of most of the basins in our
analysis and a reasonable compromise between the 15 min and 24 h timescales used for IAT flashiness and R–B flashiness index respectively. Note
that stated that the hourly scale would be more suitable
for smaller basins (<30 km2), but never computed R–B
flashiness values at this scale, only Richard's pathlengths. When we computed
R–B flashiness indices at the hourly scale, using the same definition, we
found lower flashiness than at the daily scale, which is rather
counterintuitive, as one would expect higher flashiness at smaller scales due
to the fact that Richard's pathlengths increase from daily to hourly scales.
However, R–B flashiness is based on absolute differences of flow values, not
gradients (i.e. differences per unit of time). And since flow differences
decrease when moving toward smaller scales, R–B index also decreases.
Alternatively, one could use discharges instead of flow amounts, but then
values could grow much larger than 2. Regardless of the approach used, R–B
flashiness index appears to be rather sensitive to the scale of analysis. By
contrast, the IAT flashiness index proposed in this paper tends to be much
more robust. Additional sensitivity analyses (not shown) revealed almost no
changes in IAT flashiness estimates for 15 min to 3–6 h aggregation
scales. Beyond that, significant underestimation started to occur as the
resolution is not sufficient anymore to correctly capture peak flow
variability. For data aggregated at 24 h resolution (instead of the original
15 min), IAT flashiness values were underestimated by 20–80 %, depending on
the considered gauge.
Example of log–log plots for flows and inter-amount times (a, b),
for Mc Alpine Creek, illustrating departures from linearity at high-order
moments. Reported R2 values are for the entire range of results,
without scale breaks. Log–log curve for moment q=2.4 illustrating
scale breaks for flows and inter-amount times (c, d).
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. b), the scale at which 1 %
of flow accumulations occurs in less than 15 min is associated with an
inter-amount sampling scale of 4.76 mm. This implies that flows that exceed
4.76 mm in 15 min, i.e. peak flows above 19.0 mm h-1, cannot be observed
1 % of the time. If correct observation of peak flows of this magnitude or
larger 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.
Scaling of inter-amount times across scales: multifractal analysis
As explained in Sect. , log–log plots of statistical moments
versus sampling scale can be used to study scaling behaviour of time series.
In the following, we plotted the moments 〈Xλq〉 of
order q of IATs as a function of mean inter-amount scale Δq
(proportional to the inverse of the scaling ratio λ), on a log–log
scale, for moments of order 0.6 to 2.4. We applied the same procedure for
flow time series over the same range of equivalent scales.
Figure shows examples of log–log plots for flow volumes and
IATs for McAlpine Creek (gauge 750). They show that log-linear fits are
better for IATs than for flows, especially for higher-order moments; minimum
R2 values, that are associated with fits for higher-order moments, are
0.9972 and 0.9993 for flows and IAT respectively.
Plots in Fig. show stronger departures from linearity in the
log–log plots for flows than for IATs, especially for higher-order moments.
Figure c and d illustrate this for log–log
curves of moment q=2.4, where a scale break was detected at 22 h for
flows. Based on a Davies test , two breakpoints were
significant for flows (p value 0.001). For IATs, there was at least one significant breakpoint, but the test
for two breakpoints returned a p value of
0.071. This shows that scaling is slightly better for IATs than for flows.
Similar analyses were conducted for all gauges, Table summarises
minimum R2 values for log-moment fits for flows and IATs. Log moments for
IATs show near perfect fits for all gauges, with minimum R2 values
between 0.995 and 1.000. Quality of log moments is consistently lower for
flows, for all basins; minimum R2 values are between 0.990 and 0.997,
lower quality fits generally occurring for smaller basins. Investigation of
departures from linearity showed that for flows, most gauges exhibited a
scale break between 8 and 20 days. Similar scale breaks, between timescales
of 8 to 16 days, were found in scaling analyses of flow data by other authors
based on flow data at daily resolution
.
and found scale breaks in the range of 16 to 27 h, respectively, for 30 min hourly resolution. We did not detect any
strong departures from linearity in the IAT framework except for the three gauges
where low flow regulation is applied (LSugarA, 507, LSugarP, 530, Stewart
Creek, 750).
Using the empirical log moments, we fitted the multifractal parameters
C1 and α for IATs and flow amounts. Table summarises
C1 and α values for all basins, for flows and for IATs. Results
show that C1-values, characterising intermittency of the time series,
are lower for IATs than for flows. This makes sense and can be explained by
the adaptive sampling strategy of IATs, especially the fact that low flows
are sampled less often than in the classical fixed-time framework. Values of
the multifractality index α are generally lower for IATs, with the
exception of four basins. Two of these basins are characterised by low flow
regulation; one basin has anomalous land-use distribution with a high
concentration of imperviousness in the upper part of the basin. Time series
of the fourth basin is short (8 years), which might influence
outcomes of the scaling analysis. C1 and α values for flows are
in the range of values found by other authors. Figure shows
scatter plots of values for C1 and α for flow and for IATs versus
basin size and imperviousness. C1 values are clearly negatively
correlated with basin area. Rank correlations for IATs are -0.67 and -0.85 for flows. No significant correlation of C1 with imperviousness was
found, but the three basins with low flow control stand out with lower-than-average C1 values. This shows up both in the IAT analyses and in the
classical approach based on flows. The α values for IATs are
positively correlated with area (0.6) and negatively with imperviousness
(-0.56). No significant correlation with area or imperviousness was
detected. For IATs, negative correlation of α with imperviousness
comes from the fact that IATs in highly impervious basins are redistributed
more evenly when moving from large to small scales (due to high
imperviousness).
Multifractal parameters C1 and alpha for scaling analysis of
flows and inter-amount times, as a function of drainage area and
imperviousness degree. Grey triangle symbols represent inter-amount
times, black circles represent flows.
Summary and conclusions
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. e.g..
On the other hand, several studies have found mixed effects of urbanisation on flow peaks
associated with a combination of imperviousness and flood mitigation measures, especially
for basins in the USA where urbanisation has predominantly taken place after implementation
of stormwater legislation to lower peak
discharges e.g.. For the
basins in Charlotte watershed, urbanisation has taken place before as well as after
stormwater legislation, and a combination of flow regulation by detention facilities and
peak flow restrictions induced by capacity constraints results in an overall effect
of peak flow reduction associated with urbanisation.
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 (<0.4) for IAT differences, showing that hydrographs tend towards being
symmetrical in these basins. Significant correlations were found between skewness and medcouple
of IAT differences and imperviousness (Spearman correlations 0.75 and -0.55), showing
that urbanisation is associated with more regulated flows, thus relatively more
symmetrical hydrographs with flow rise and recession at similar rates and lower
frequencies of steep flow rise. Here, sub-basin correlation appears to play a
role: medcouple values were higher overall in the McAlpine sub-basins than in
Little Sugar Creek and Irwin sub-basins. No significant correlations were found for differences in flows.
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-1 compared to 0.1 mm day-1).
Conversely, IAT distributions capture more extreme peak flow values than conventional
flow sampling, at the same sampling scale: the 99 percentiles for flows are
associated with peak flows of 0.38 to 0.78 mm h-1 (sampling scales 12 h
to 4 days), while 1 percentiles of IATs are associated with peak flows of
about 20 mm h-1 (sampling scales 0.5 to 4 mm inter-amounts, associated with IATs of 12 h to 4 days).
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 R2 values above 0.99. IATs systematically scaled better than flows and
showed departures from multifractality for only three basins, subject to low
flow regulation, while flows exhibited departures from multifractality for most
basins. This showed that IATs can help to better predict peak flow characteristics
at small unobservable scales based on coarse-resolution data. Additionally, they
provide new interesting alternatives for the stochastic modelling and downscaling
of flow data.
This study showed that properties of statistical distributions of flow time
series are very sensitive to the scale at which the statistics have been
derived. This influences values of summary statistics that are used to
characterise flow patterns of hydrological basins, like peak flows at given
recurrence intervals and flashiness indices. Adaptive sampling based on
inter-amount times helped to achieve more stable variance across scales, yet the
behaviour of other statistical properties such as skewness or medcouple is less
clear. Further investigations are needed to interpret changes of statistics
across scales. Future work will focus on multiscale analysis, how to
compare results at different scales and what can be learnt from behaviour at
different scales about flow variability in hydrological basins in relation to
basin characteristics.
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.