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Hydrology and
Earth System
Sciences
Open Access
Hydrol. Earth Syst. Sci. Discuss., 10, 15659–15680, 2013
www.hydrol-earth-syst-sci-discuss.net/10/15659/2013/
doi:10.5194/hessd-10-15659-2013
© Author(s) 2013. CC Attribution 3.0 License.
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Discussion Paper
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Technical Note: A measure of watershed
nonlinearity II: re-introducing an IFP
inverse fractional power transform for
streamflow recession analysis
Correspondence to: J. Y. Ding ([email protected])
Discussion Paper
Published by Copernicus Publications on behalf of the European Geosciences Union.
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J. Y. Ding
John Ding & Co., 134 Heatherside Drive, Toronto, Ontario, M1W 1T9, Canada
Received: 11 November 2013 – Accepted: 2 December 2013 – Published: 20 December 2013
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This note illustrates, in the context of Brutsaert–Nieber (1977) model: −dQ/dt = aQb ,
the utility of a newly rediscovered inverse fractional power (IFP) transform of the flow
rates. This method of streamflow recession analysis dates back a half-century. The IFP
b−1
transform ∆b on an operand Q is defined as ∆b Q = 1/Q . Brutsaert–Nieber model by
IFP transform thus becomes: ∆b Q(t) = ∆b Q(0) + (b−1) at, if b 6= 1. The IFP transformed
recession curve appears as a straight line on a semi-IFP plot. The method has both
the advantage of being independent of the size of computational time step, and the
disadvantage of being depending on the parameter b value. This is used to calibrate
the Brutsaert–Nieber recession flow model in which b is a slope (or shape) parameter,
and a is an intercept (or a scale parameter). It is applied to four observed events on
the Spoon River in Illinois (4237 km2 ). The results show that the IFP transform method
gives a narrower range of parameter b values than the regression method in a recession plot. Theoretically, an IFP transformed recession curve for large watersheds falls
between those performed by the reciprocal of the cubic root (RoCR) transform and
the reciprocal of the square root (RoSR) one. In general, the forgotten IFP transform
method merits a fresh look, especially for hillslopes and zero-order catchments, the
building blocks of a watershed system. In particular, because of its origin in hillslope
hydrology, the 1-parameter RoSR transform need be falsified or verified for application
to headwater catchments.
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1 Introduction
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Sustainable streamflow in a watershed is a prerequisite to the survival and prosperity
of a civilization. Its uses include consumption by human and livestock, dilution or treatment of their wastes, irrigation of arable land, inland navigation, hydropower generation,
fish habitat maintenance, to name a few. Its sustainability is characterized by the variation and persistence of the low flow, the latter effect aptly captured by what Mandelbrot
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and Wallis (1968) called the Joseph effect from their reading of the Genesis that in the
land of ancient Egypt, seven years of harvest will be followed by seven years of famine.
(The Noah flood represents for them the other extreme of the hydrologic regime.)
Prediction of the low flow during a prolonged rainless period, called the drought, dry
weather, groundwater or base flow analysis, has long been based on a linear storagedischarge assumption, and the related semi-log plot for displaying the recession limb
of a hydrograph (e.g. Gray, 1970).
In 1970’s, Brutsaert and Nieber (1977) proposed a drought flow model which relates
b
nonlinearly the rate of change of the flow to the flow itself: dQ/dt = aQ . This is best
known for its dQ/dt vs. Q recession plot. The model eliminates the ambiguity of the
start time of a recession hydrograph. However, the approximation of the infinitesimally
time-step size dt by a finite difference ∆t manifests itself in a scattering of the recession
data points, called the cloud (e.g. Troch et al., 1993a; McMillan et al., 2010; Stoelzle
et al., 2013; Chen and Wang, 2013). The originators recommend the use of earlyand late-time lower envelopes for the data cloud, along both of which the effect of
evaporation on the flow is reasoned on physical grounds to be minimal (e.g. Troch
et al., 1993b; Brutsaert, 2005; Chen and Wang, 2013).
There have been numerous attempts to make use, or to reduce the size, of the data
cloud in a recession plot. A first and obvious alternative is to fit the data cloud by a linear
regression line (e.g. Troch et al., 1993a, b; Malvicini et al., 2005; Lyon and Troch, 2007).
In connection with the first one, Rupp and Selker (2006) devised a scheme of using
a variable time-step size which scales to the magnitude of the drop in flow. The rescaled
recession plot reduces the scatter at very low flows (e.g. Clark et al., 2009; Ajami et al.,
2011; Staudinger et al., 2011).
A third and latest alternative is that of Kirchner (2009). He divides the Q scale into
many increments, replaces the dQ/dt values within each of which by an average value,
i.e. “binning” the data, and fits the data cloud by a polynomial, usually of a second
degree (e.g. Teuling et al., 2010).
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Recently, Stoelzle et al. (2013) have surveyed the state of the art of streamflow recession analysis, with a special emphasis on Brutsaert–Nieber model. In their evaluation,
they include the original lower envelopes (Brutsaert and Nieber, 1977), the data binning and polynomial fitting (Kirchner, 2009) as well as the linear regression methods,
but excluding the variable time-step sizing of Rupp and Selker (2006). Their results
suggest serious limitations to the comparability of streamflow recession characteristics
derived with different methods.
During the public comment period of Stoelzle et al. (2013) paper, I brought to their
attention a forgotten solution technique now called the inverse fractional power (IFP)
transform. This was refined and elaborated in a later comment on Chen and Wang
(2013) paper.
This note is a consolidation and extension of these two sets of comment, plus two
additional ones, all on streamflow recession analysis, but some simply correcting errors, mainly mine (c.f. Farlin and Maloszewski, 2013; Troch et al., 2013). It will focus on
the groundwater outflow model originated by Brutsaert and Nieber (1977), and make
use of the recession flow data reported by Chen and Wang (2013). The latter provides
four data segments for the Spoon River, a large watershed in Illinois, USA, and the
centre piece of their nine-watershed study. The purpose is to illustrates, in the context
of Brutsaert–Nieber model, the utility of a newly rediscovered IFP transform method
whose origin traces back a half-century. This is essentially a retrospective piece of
mine forming part two of the headlined HESS paper (Ding, 2011). It shifts the spotlight
of the vIUH (variable instantaneous unit hydrograph) concept, a nonlinear system one,
from floods to droughts. Comparison of the simple IFP transform method with other
more sophisticated or complicated methods as recently reviewed by Stoelzle et al.
(2013) will not be part of this note.
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2.1 Overview
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Brutsaert and Nieber (1977) model is a hydraulic one for outflow from a cross section
of an unconfined riparian aquifer having a horizontal impenetrable bottom layer. Being
a one-dimensional model, it is most applicable to headwater catchments, not affected
or dominated by the presence of streams of significant size. Of special interest are
their derivations of the outflow behavior in term of the recession plot slope b: 3 for
early recession, and 1.5 and 1 for late ones (Brutsaert and Nieber, 1977; Brutsaert,
2005).
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Brutsaert and Nieber model
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The heading of my most recent comment was titled on purpose the “hydrograph recession as a maturing field of mathematical hydrology” (Ding, 2013b). This hopefully
has conveyed my view that in the absence of the rainfall forcing and its effects, the
falling or receding half of a hydrograph can be mathematically modelled starting from
the principles of hydrology. Conceptually, the receding limb is simpler to model than its
rising counterpart. Such a model can be falsified by a few spot measurements of the
volume of water passing through a river section during the rainless period, i.e. by the
dry weather flow. Compared to other field data collection programs, this seems not too
onerously a task on small catchments.
Models, equations or methods representing the flow recession phenomenon for hillslopes and small watersheds described in the latest comment (Ding, 2013b) are recapped and elaborated below. All of these are different faces or expressions of the
same phenomenon. They have a shape (or, in Brutsaert–Nieber model, slope) parameter b, and a scale parameter (or intercept) a.
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2 A brief mathematics of streamflow recession
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−
dQ
= aQb ,
dt
(1)
−1
1/Qb−1 (t) = 1/Qb−1 (0) + (b − 1)at, if b 6= 1
(2a)
log Q(t) = log Q(0) − at, if b = 1.
(2b)
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Equation (2a) expresses a linear relation between an IFP transformed variable
b−1
1/Q (t) and the elapsed time t, and Eq. (2b) the same relation for a log transformed
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Rearranging Eq. (1) as: Q−b dQ = −adt, and integrating from time 0 to t, I show the
solution to be (Ding, 1974, 2013a, b):
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in which Q is the flow rate (mm d ), t is time (d), b is a shape parameter (dimensionless), and a is a scale parameter having a combination of the flow rate and the time
units, mm−(b−1) d−(2−b) . In a log(−dQ/dt) vs. log(Q) plot, parameter b represents the
slope of the recession curve, thus the name of slope parameter; and because when
Q = 1, a = −dQ/dt, parameter a is thus the intercept of the curve at the Q = 1 line.
Conceptually, Eq. (1) is an expression of the growth or, in the present context, decay phenomena or functions found in nature and explored in science. For example, in
a chemical reaction, the time rate of change in a concentration is proportional to the
concentration raised to some power. The difference between the two is that flow Q is
a time-rate variable, and −dQ/dt, the flow de-acceleration, is a second time-differential
of the flow volume or concentration. Calculation of the de-acceleration requires a high
resolution in time. For small watersheds, this is generally finer than the time resolution of daily or hourly flow measurements. Approximation of (dQ/dt) by the backward
time-differencing one, (Q(t) − Q(t − ∆t))/∆t, is thus the main source of the numerical
instability in parameters characterizing the model. Approximation of Q(t) by the backward time-averaging one, (Q(t) + Q(t − ∆t))/2, is a secondary one.
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This standard model is now commonly written as follows:
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2.1.3 Nonlinear storage–discharge relation
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(3)
in which S is the water storage (mm), N = 1/(2 − b), c = (2 − b)a, and Nc = a. The
physical constraint that S ≥ 0 and Q ≥ 0, imposes numerical limits that if N ≥ 0, then
c ≥ 0, thus b ≤ 2 (e.g. Ding, 2013a; Chen and Wang, 2013). More on this in Sect. 2.3.
Reciprocal of the square root (RoSR) transform
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This is an alternate expression of the Boussinesq model representing an outflow
hydrograph for an unconfined aquifer in a hillslope (e.g. Ding, 1966, 2013a, b). This
classical transform in engineering hydrology dates back a half-century. This was discovered, independently, within a short time span in the 1960’s by Chapman (1964) in
Melbourne, Australia, by Ishihara and Takagi (1965), the latter then a graduate student, in Kyoto, Japan, and by me, then a graduate student at Guelph, Canada (Ding,
1966). In contrast to a hydrodynamic approach (i.e. based on theory of flow in porous
media) followed by them, I derived this from a hydrologic approach (i.e. based on conceptual or lumped storage). Thus the RoSR transform may be considered one of the
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For a parameter b value of 1.5, Eq. (2a) reduces to a reciprocal of the square root
transform of the flow:
q
q
1/ Q(t) = 1/ Q(0) + (a/2)t.
(4)
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Q = cN S N ,
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Defining a vertical strip of the recession hydrograph as an incremental storage, dS, at
time t: dS = Qdt, and again integrating from time t to ∞, I show the storage available
at time t for later release as rearranged below (Ding, 1966):
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one. Linearization of Brutsaert–Nieber model thus removes sources of numerical instability in its parameters.
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(6)
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∆b Q(t) may be abbreviated as ∆(t) with a given b value.
In top half of Eq. (5), its latter expression hints at an intriguing, counter-intuitive transformation of a time series. It weighs the data Q by a self-dependent, thus varying, and
b
counter weight of (1/Q ). But this is compatible to the input-dependent variable IUH
and a logical consequence of this dependency (Ding, 1974).
There are many types, some common or well-known, of IFP transform as tabulated
in Table 1. These depend on values of the shape parameter b. For streamflow recession analysis, only some transforms are considered physically realistic. Their b values
range from a lower limit of one to an upper limit of 2, the latter as indicated following
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Brutsaert–Nieber model by IFP transform (Eqs. 2a and b) thus becomes:
(
∆b Q(0) + (b − 1)at if b 6= 1;
∆b Q(t) =
∆b Q(0) − at
if b = 1.
(5)
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Formally the IFP transform operator ∆b on an operand Q is defined as:
(
1/Qb−1 = Q/Qb if b 6= 1;
∆b Q =
log Q
if b = 1.
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early links between two approaches. For an illustration of a semi-RoSR plot, see Fig. 1
reprinted from Ishihara and Takagi (1965). This happened to be the first known one
recorded in hydrology literature. As noted in Sect. 2.1.1 above, a parameter b value of
1.5 characterizes late-time recession in Brutsaert–Nieber model.
The application of the RoSR transform has rarely been reported in open literature,
and if so, only opaquely, i.e. one had to read between the lines (e.g. Ando et al., 1986).
But Takagi (1977) explicitly applied this transform for his recession analysis of rivers in
Germany and Japan.
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The Spoon River in Illinois (4237 km ) is a large watershed, and not, by any measure,
a small one for which the IFP transform method was originally developed. Chen and
Wang (2013) provide the flow data for four recession events: a base event in Table 2a
numbered herein as Event 0, and in Table 3 for Events 2–4. For the purpose of this
analysis, the whole watershed is considered a lumped storage element or system.
Table 2 is a template for generating IFP transformed hydrographs for May 1994
(Event 0). For b > 1, say 1.5, the statistics shown of the best-fit line are correlation
coefficient (0.99), slope (0.04) and intercept (1.10). Since Eq. (2a) indicates a slope of
(b − 1)a on a semi-IFP plot, thus a = slope/(b − 1) = 0.08.
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Both have a single parameter, b or λ, in the exponent, and that λ = −(b − 1), thus the
opposite directions of their transformations except for the log one. And the former, having one less line of expression, is more compact than the latter. But I shall leave this
cross-discipline subject for others to explore (c.f. Chapter XVI. Transformations, in: Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/).
Project Leader: David M. Lane, Rice University).
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Eq. (3) above. Since the power of variable Q is −(b − 1), thus the name of inverse,
fractional power (IFP) of one for the transform. These plus a special b value of 3 will
be considered for analysis of the four recession events.
As an aside, there is a striking similarity between the IFP transform in hydrology,
and in mathematical statistics the modified Tukey ladder of power transformations. The
latter is defined as follows:
 λ

if λ > 0;
x
y = log x if λ = 0;
(7)


λ
−(x ) if λ < 0.
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Figure 2 shows, for Events 0 and 2, the original data and five or four transformed
recession curves. In contrast to the log transform (b = 1), the IFP transform for b > 1
reverses the trend of the recession curve from downward to upward: the lower the observed flow value, the higher its transformed value; and that the higher the b value, the
greater the transformation. The IFP transform, including the log one, is like a microscope with a varying magnifying power, gradually increasing its magnification towards
the lowest flow value, thus most suitable for low flow analysis.
Table 3 summarizes results of the calculations for four events. Among them, Event
2 has both the lowest observed initial recession flow value and the second longest
recession period, and as turns out, also the highest calibrated a value among four
events, except for the RoCR transform. It shows that the correlation coefficients, R 0 s,
among all transforms for four events are similar, all close to or at 1.0 in absolute number.
As illustrated in Fig. 3, the parameter scatter diagram shows that for an event, as
parameter b value increases, so does parameter a value reflecting the steepness of
a transformed recession curve in Fig. 2; and that starting from b = 1, parameter a
values diverge from the trend lines, especially for Event 2, but not Event 0, the base
event.
On a large river such as the Spoon, the Manning (or Chezy) friction law governs the
nonlinear storage–discharge relation expressed by Eq. (3). I have shown elsewhere in
HESSD (Ding, 2011) that a typical shape parameter N value of 1.67 by Manning (or
1.5 by Chezy) gives a parameter b value of 1.4 (or 1.33) from Eq. (3): b = 2 − 1/N. A b
value of 1.4 by Manning lies between b of 1.33 for the RoCR transform, and that of 1.5
for the RoSR characterizing an unconfined aquifer.
Based on results in Table 3 for four events for the Spoon River, the Brutsaert–Nieber
parameter pair (b, a), calibrated by the IFP transform method, ranges from (1.33, 0.07)
to (1.5. 0.23). These are a compromise between the hydrodynamic (i.e. based on
Manning or Chezy) and the statistical (i.e. based on variance of the scale parameter) approaches.
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4 Discussion and conclusions
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To summarize, the analytical results above show that the recession hydrograph for
a large river can be represented by an IFP (inverse fractional power) transform model.
Its transformed slope is independent of the size of time step, on which, however, depends the accuracy of the Brutsaert–Nieber model parameters when regressed from
data on a recession plot. But it does depend on the shape parameter b value. The
range of b values varies from one to an upper limit of 2, rather than that of 3 adopted
by Brutsaert and Nieber (1977) and others (e.g. Chen and Wang, 2013) for the earlytime lower-envelopes.
Table 4 shows that three of four events have much higher b values than 2, calculated from the −dQ/dt vs. Q plot. It also shows the range of b values determined by
IFP transform and based solely on the highest correlation coefficients for each event.
Among three of four events, the linear regression method yields higher b values than
the IFP transform one, and slightly lower for the fourth event.
The Spoon River, having a very large drainage area, is not a catchment I would
choose to illustrate the utility of the newly rediscovered IFP transform. But it has been
because the recession flow data available for analysis were published in an open access journal such as HESS by Chen and Wang (2013). As shown in Fig. 3, this case
study has demonstrated glaring differences, in terms of the scatter of calibrated parameters, among several types of IFP transform. But as shown in Table 3, it has demonstrated only marginally the superiority of IFP transform over the conventional log transform method. But it does serve as a cautionary tale for nonlinear modelling exemplified
by Brutsaert and Nieber (1977) that parameters have meaning only within their model
which has its characteristic time-step size, ∆t, or time scale (e.g. Ding, 2011), which
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For the Spoon, on the basis of a minimum variance of the scale parameter values,
case can be made for a log transform model, i.e. a linear storage system, and even an
un-transformed one, i.e. an autoregressive model of order one.
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Acknowledgements. Tomoki Ruo, formerly of Petro Canada, Oakville, Ontario, helped improve
the legibility of the figures as well as of some in the 2011 HESS paper of mine. Michael Stainton,
of Yok University, Toronto, helped locate some references within academic confines.
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is one day for the Spoon River. Results of model calibration to observed data need
be so interpreted and applied accordingly. However, linearization by IFP transform of
Brutsaert–Nieber model makes it time-scale invariant, thus rendering redundant any
time-scale based optimization schemes, such as Rupp and Selker (2006).
To conclude, the forgotten IFP transform method re-introduced in this note merits
a fresh look by the hydrology community for streamflow recession analysis, especially
for headwater catchments including hillslopes, karsts, zero- or first-order streams of all
shapes and sizes. Of it, the RoSR transform is a special case having a shape parameter
b value of 1.5 derived from groundwater flow theory. Because of its root in hillslopes,
the classical RoSR transform of 1960’s, simple and elegant at once, is a model to be
falsified for headwaters, i.e. used until proven otherwise.
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Troch, P. A., Carrillo, G., Sivapalan, M., Wagener, T., and Sawicz, K.: Climate-vegetation-soil
interactions and long-term hydrologic partitioning: signatures of catchment co-evolution, Hydrol. Earth Syst. Sci., 17, 2209–2217, doi:10.5194/hess-17-2209-2013, 2013. 15662
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IFP inverse fractional
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Shape
parameter
b
Q3
Q2
Q
log Q
√
3
1/√Q
1/ Q
1/Q
2
1/Q
3
1/Q
Cube
Quad
–
log
RoCR
RoSR
Recip
RoQ
RoC
Type
Cubic
Quadratic
The original or un-transformed
Logarithmic
Reciprocal of the cubic root
Reciprocal of the square root
Reciprocal
Reciprocal of the quadratic
Reciprocal of the cubic
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Symbol
|
−2
−1
0
1
1.33
1.5
2
3
4
IFP
transform
∆b
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Table 1. Types of IFP (inverse fractional power) transform, ∆b = 1/Qb−1 .
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Discussion Paper
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IFP inverse fractional
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Table 2. Transformed hydrographs for Spoon River, Illinois, May 1994 (Base event, 0). USGS
Gage 05570000.
|
(mm d−1 )
(mm d−1 )
log
RoCR
RoSR
Recip
RoQ
0
1
2
3
4
5
6
7
8
0.40
0
0
0
0
0
0
0
0.81
0.84
0.78
0.71
0.65
0.61
0.57
0.56
0.52
0.50
−0.17
−0.25
−0.34
−0.43
−0.49
−0.56
−0.58
−0.65
−0.69
1.06
1.09
1.12
1.15
1.18
1.20
1.21
1.24
1.26
1.09
1.13
1.19
1.24
1.28
1.32
1.34
1.39
1.41
1.19
1.28
1.41
1.54
1.64
1.75
1.79
1.92
2.00
1.42
1.64
1.98
2.37
2.69
3.08
3.19
3.70
4.00
0
−0.98
−0.05
0.81
0.05
1.00
−0.99
−0.07
−0.20
0.07
1.33
0.99
0.02
1.07
0.08
1.50
0.99
0.04
1.10
0.08
2.00
1.00
0.10
1.20
0.10
3.00
1.00
0.33
1.37
0.16
Source: Chen and Wang (2013).
∗
a = Slope/(b − 1), if b 6= 1.
|
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(d)
|
May
1994
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Q
|
P
Discussion Paper
Time
b (–)
Correlation R
Slope
Intercept
a [1/(d mmb−1 )]∗
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IFP inverse fractional
power transform
b−1
Date
15
16
17
18
19
20
21
22
23
Type of IFP Transform, 1/Q
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Table 3. Summary of Brutsaert–Nieber model parameters for Spoon River, Illinois.
Event
Q(0)
−1
(mm d )
0
0.84
Length
b
Statis
(d)
9
Type of IFP transform, 1/Q
b−1
None
log
RoCR
RoSR
Recip
RoQ
b
0
1.0
1.33
1.5
2.0
3.0
a
R
0.05
−0.98
0.07
−0.99
0.08
0.99
0.08
0.99
0.10
1.00
0.16
1.00
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a
|
4
a
R
0.06
−1.00
0.13
−1.00
0.17
1.00
0.19
1.00
0.29
0.99
0.63
0.98
1
0.29
4
a
R
0.01
−1.00
0.05
−1.00
0.07
0.99
0.09
1.00
0.18
1.00
0.66
1.00
2
0.10
8
a
R
0.01
−0.97
0.07
−0.98
0.15
0.99
0.23
0.99
0.84
0.99
10.88
0.99
Events arranged in the descending order of the initial flow value.
b is the shape parameter (–), a the scale parameter [1/(d mmb−1 )], and R the correlation coefficient.
|
b
0.56
Discussion Paper
a
3
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Shape parameter b
Event
regression
Low
High
2.87
1.13∗
3.33∗
3.54∗
2.0
1.33
1.5
1.33
2.0
1.5
2.0
2.0
Source: Chen and Wang (2013).
Discussion Paper
∗
IFP transform
|
0
3
1
2
Linear
Discussion Paper
Table 4. Comparison of Brutsaert–Nieber shape parameter values for Spoon River, Illinois,
calculated by the linear regression and the IFP transform methods.
|
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Paper
| | Discussion
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Paper
|| Discussion
||
Fig. 1. A streamflow recession hydrograph on a semi-RoSR plot for the Kako River at Kunikane, Japan
(Reprinted with permission from F. Takagi).
||
15678
18
Discussion Paper
Paper
Discussion
Fig. 1. A streamflow recession hydrograph on a semi-RoSR plot for the Kako River at Kunikane,
Japan (reprinted with permission from F. Takagi).
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IFP inverse fractional
power transform
J. Y. Ding
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Discussion Paper
Discussion Paper
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Tables
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J
Discussion Paper
J
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Conclusions
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Discussion Paper
Introduction
|
|
Abstract
I
I
|
Fig. 2. The IFP (inverse fractional power) transformed hydrographs for the Spoon River, Illinois:
on the left panel, May 1994 (Event 0), and right panel, June 1988 (Event 2). b = 0 denotes
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the non-transformed hydrograph; and b = 1, the log transformed one. Note the different vertical
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or(inverse
flow scales
between power)
two panels.
On the righthydrographs
panel and outside
the Spoon
plot areaRiver,
is the Illinois:
RoQ
he
fractional
transformed
for the
on the on
2. IFP
The
IFP
(inverse
fractional
power)
transformed
hydrographs
for
the
Spoon
River,
Illinois:
the
transformed hydrograph (b = 3) having transformed values from a low 95.55 to high 256.82.
Discussion Paper
|
Discussion P
Discussion Pa
|
15679
|
, May May
1994 1994
(Event(Event
0), and0),right
June 1988
2). b = 2).
0 denotes
the non-transformed
panel,
andpanel,
right panel,
June(Event
1988 (Event
b = 0 denotes
the non-transformed
Printer-friendly Version
ph;
and
b
=
1,
the
log
transformed
one.
Note
the
different
vertical
or
flow
scales
between
two
ograph; and b = 1, the log transformed one. Note the different vertical or flow scales between
two
Interactive Discussion
On
the
right
panel
and
outside
the
plot
area
is
the
RoQ
transformed
hydrograph
(b
=
3)
having
els. On the right panel and outside the plot area is the RoQ transformed hydrograph (b = 3) having
med values from a low 95.55 to high 256.82.
sformed values from a low 95.55 to high 256.82.
Discussion Paper
Discussion Paper
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IFP inverse fractional
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Discussion Paper
J. Y. Ding
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Fig. 3. A scatter diagram of the Brutsaert-Nieber model parameters, calibrated by the IFP (inverse fractional power) transform method, for the Spoon River, Illinois.
Discussion Paper
Discussion Paper
Fig. 3. A scatter diagram of the Brutsaert–Nieber model parameters, calibrated by the IFP
(inverse fractional power) transform method, for the Spoon River, Illinois.
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