Annual Journal of Hydraulic Engineering, JSCE, Vol.52, 2008, February
Annual Journal of Hydraulic Engineering, JSCE, Vol.52, 2008, February

INTERNAL RESPONSE OF CATCHMENTTO PLAUSIBLE PARAMETER SETS
UNDER EQUIFINALITY
Giha LEE 1, Yasuto TACHIKAWA2, Takahiro SAYAMA3 and Kaoru TAKARA4
1Student Member, Graduate student, Dept. of Urban and Environmental Eng., Kyoto University(Nishikyo-ku, Kyoto 615-8540, Japan) E-mail:leegiha@flood.dpri.kyoto-u.ac.jp
2Member of JSCE, Dr. Eng., Associate Professor, Dept. of Urban and Environmental Eng., Kyoto University(Nishikyo-ku, Kyoto 615-8540, Japan)
3Member of JSCE, Dr. Eng., Assistant Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)
4Fellow of JSCE, Dr. Eng., Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)

A natural rainfall-runoff process is conceptualized by means of mathematical form reflecting aphysical law and hydrologist’s perception and it is typically calibrated and verified based on streamflow
data, the commonly used catchment response. The streamflow data is obviously required, but is not
sufficient to identify a conceptual parameter of a hydrologic model since numerous parameter
combinations can often result in quite similar numerical values in terms of objective function and evenindistinguishable simulated hydrographs. One of the efficient techniques to resolve this impasse is acombination of the model with streamflow data augmented by other kinds of hydrological information
relevant to the prediction. In this study, internal dynamic response of catchment, spatiotemporally
separated components of hydrograph, to the various mimic parameter sets is presented. It is concluded
that all of simulations based on plausible parameter combinations sampled by using deterministic andstochastic automatic optimization algorithms, are performed equally well in terms of both model
performance measure and visual comparison between observed and simulated hydrograph while internal
behaviors of catchment show totally different aspects. The spatiotemporal information presented here canbe utilized as one of the complementary constraints capable of filtering out non-physical parameter set(s).
Key Words: plausible parameter sets, complementary constraint, internal dynamic response

1. INTRODUCTION
One of significant steps in a rainfall-runoff
modeling is parameter estimation, typically referred
to as calibration. However, despite a remarkable
improvement of automatic optimization algorithms,
uncertainty in the calibrated parameter estimatesstill remains very large since a large number ofdifferent parameter sets can, in many cases, result in
identical measures of model performance and evenindistinguishable streamflow sequences1),2). Bevenand Binley3) used the special term ‘equifinality’ toexplain the possibility of plausible parameter sets
and it has been an interesting major issue incomplicated hydrological modeling.
A rainfall-runoff model is mostly identified
based on streamflow. Kuczera and Franks4) argued
that it is obviously required, but is not sufficient
information to identify the conceptual model
parameter(s). An effective technique to resolve thisinsufficiency is to increase the amount of

information through the use of additional output
variables such as runoff, soil moisture, piezometriclevels measured at different locations within thecatchment, environmental isotopes and so on5) andthen combine the runoff model with streamflow data
augmented by those hydrological variables.
The additional information may be attractive forfurther conditioning or testing of a hydrologic
model, but on the other hand, the utilization of this
auxiliary information often requires additionalmodel complexity in terms of reformulation ofmodel equations due to adding auxiliary assumption
and parameter. It means that this complementary
information is likely to cause over-complexity of themodel and as a result, it precludes falsification of
unsuitable parameter combinations4),6). Therefore, it
is worthwhile developing a way to test models morecomprehensively and incisively, given inherent
limitations of the available streamflow data without
increasing model complexity7).
Experimental methods such as isotope tracers,- 79 -

isotopic hydrograph separations and stream water
residence time have been useful to provide
additional evaluative criteria for water quantity andquality modeling. However, as noted above, theseapproaches, to some degree, require the revision ofmodel for producing multiple output variables sothat it again results in declination of parameter
identifiability. To avoid this dilemma, Sayama etal.8) proposed a hydrograph separation method
derived from a spatiotemporal record matrix ofstreamflow and combined this scheme with anonlinear distributed model that takes into accountunsaturated, saturated subsurface and surface flow.
This method can temporally split the hydrographinto the same number of corresponding components
to the selected number of rainfall segments without
any hydrochemical measure. Moreover, it can track
the spatial origin of streamflow at a specific time
step and then visualize the distributed runoff
sources. Consequently, this new technique enables
to analyze the effect of mimic parameter
combinations on internal response to both pre-eventand event in a catchment and then to be utilized as aposteriori evaluative benchmark to reject erroneous
parameter combination(s).
In this study, we aim to exemplify both thepresence of mimic parameter sets and the limitationof streamflow data for model identification and thenwe propose complementary information, internal
behavior of catchment, capable of filtering out
unreliable parameter sets from numerous plausible
ones. This additional constraint is obtained from thespatiotemporal record matrix of streamflow8).

2. APPLIED DISTRIBUTED RAINFALLRUNOFF MODEL, OHDIS-KWMSSOHymos based DIstributed hydrologic modelwith Kinematic Wave Method for Surface andSubsurface runoff (OHDIS-KWMSS) assumes thata permeable soil layer covers the hillslope as
illustrated in Fig. 1. The soil layer consists of acapillary layer in which unsaturated flow occurs anda non-capillary layer where saturated flow occurs.
According to this runoff mechanism, if the depth ofwater, h is higher than the soil depth, D thenoverland flow occurs.
The stage-discharge relationship9) is defined as:

Fig. 1 Schematic model structure and extended stagedischarge relationship of OHDIS-KWMSS.

⎧vc d c (h / d c ) β ,0 ≤ h ≤ dc⎪
q = ⎨vc d c + va (h − d c ),dc ≤ h ≤ ds
⎪v d + v (h − d ) + α (h − d ) m , d ≤ hcss⎩ c c a(1)∂h ∂q= r (t )
+
∂t ∂x

(2)

Flow rate, q of each slope segment is calculated
by above governing equations combined with thecontinuity equation (2), where vc = kc i ; va = kai ;k c = k a / β ; α = i / n ; i is slope gradient, k c ishydraulic conductivity of the capillary soil layer,
k a is hydraulic conductivity of the non-capillary soil
layer, n is roughness coefficient, the water depth
corresponding to the water content is d s and thewater depth corresponding to maximum watercontent in the capillary pore is d c . There are fiveparameters (n, k a , d s , d c and β ), which are assumed
to have homogeneous values spatially, to be
optimized in OHDIS-KWMSS.

3. PLAUSIBLE PARAMETER SETS
In this study, seven different parameter sets areprepared to investigate the influence of those on
overall and internal model performances. First threeparameter sets are estimated by deterministic
automatic
optimization
algorithm,
Shuffled
Complex Evolution (SCE) method1) with threedifferent Objective Functions (OFs), SLS, HMLE
and MIA (refer to Lee et al.10) for more details about
OFs). Secondly, the optimal value tuned bystochastic
optimization
algorithm,
Shuffled
Complex Evolution Metropolis (SCEM) method11) isalso utilized (i.e. OPT, approximately near thehighest densities in each posterior parameter
distribution estimated by SCEM). Finally, otherthree remainders are randomly sampled from theestimated posterior parameter distribution based on
different events (i.e. Sample (1), (2) and (3)). Table
1 summarizes all parameter values used here andFig. 2 presents both the marginal posterior
probability distributions for the each parameter of
OHDIS-KWMSS, obtained using behavioral 6000
parameter sets after the SCEM implementation, andthe optimal values of chosen parameter sets.
Table 1 Selected plausible parameter sets.
n [m-1/3 s]ka [m/s]ds [m]

dc [m]

β [-]
19.8

SLS

0.5

0.013

0.893

0.496

HMLE

0.5

0.011

0.485

0.085

2.95

MIA

0.5

0.010

0.468

0.068

2.59

OPT

0.5

0.013

0.865

0.472

18.6

Sample(1)

0.49

0.049

0.510

0.480

2.95

Sample(2)

0.5

0.050

0.610

0.430

4.47

Sample(3)

0.5

0.019

0.619

0.468

6.20

4. CONCEPT OF HYDROGRAPHSEPARARATION METHOD- 80 -

Fig. 2 (a) ~ (e) Marginal posterior parameter probability distributions of the OHDIS-KWMSS and the selected plausible
parameter sets marked in upper horizontal axes and (f) Initial and constrained parameter uncertainty ranges.

The concept of hydrograph separation method8)
based on spatiotemporal record of streamflow ispresented in Fig. 3(a), (b). Fig. 3(a) illustrates the
temporally separated hydrograph in regard of fiverainfall durations (0~4) within a single event.
Herein, 0 hydrograph component in Fig. 3(a)indicates the runoff of water stored in the catchment
before the event while other partitioned constituents(1~4) are generated by the rainfall during the event.
Additionally, how to partitioning hydrographspatially with respect to 6 subcatchments (A~F) isshown in Fig. 3(b). Spatial separation allows thehydrologists to discriminate between dominant andnon-dominant sub-units of catchment at each time
step whence provides a useful information on water
sources and pathways that are often required inorder to examine ecological issues like ahydrological effects of ecosystem disturbance.
The spatiotemporal record matrix as illustrated inFig. 3(c) is employed so as to track where
streamflow comes from when it rains. The
dimensions of matrix, Ri (t ) are given with S
rows, number of sub-units within catchment and T
columns, number of temporal classes where i is asubcatchment and t is time. Fig. 3(c) shows thespatiotemporal matrix at time t at the outlet, forexample, the value belonging to spatial zone C andtemporal class 2 implies that the proportion of
runoff in (C,2) entry to the stremflow observation attime t is 6%. As summarizing whole values
vertically along the columns, we can obtain theinformation, temporal contribution of rainfall to
streamflow. Likewise, the spatial influence ofsubcatchments on streamflow can be assessed byhorizontal summation along the rows. As a result,
we readily distinguish between old water (i.e.
pre-event water at time class 0 possesses 15%) andnew water (i.e. new water components are 30% attime class 1 and 55% at time class 2, respectively)

Fig. 3 Schematic diagram of hydrograph separation basedon (a) temporal record of streamflow, (b) spatial record of
streamflow and (c) spatiotemporal record matrix ofstreamflow (Sayama et al., 2007); S=6, T=5.

at the target time t. Moreover, it is possible to track
the spatially distributed origin for streamflow
generation, for instance, downstream spatial zones(e.g. D, E and F) contribute more than 60% of
streamflow whereas upstream classess (e.g. A, B
and C) less affect streamflow generation at time t.

5. CASE STUDY
The study site is the Kamishiiba catchment,
upstream area of the Kamishiiba dam, which lies
within Kyusu region in Japan and covers area of
211km2 (see Fig. 5(a)). The topography of this area
is hilly with the elevation varying from 400m to
1700m. Most of land use type is forest, thereby canbe regarded as typical Japanese mountainous area.
In the model, catchment is represented by
250m×250m DEM. The observed discharge data
converted from water level of dam inflow having
10min temporal resolution is available. To show theclear performance of old water, occurred during
pre-event, with respect to plausible parameter
combinations, we select a sufficiently long historicalevent (192hours).

- 81 -

Fig. 4 Hydrograph simulation uncertainty associated withthe behavioral parameter sets (i.e. 5400 (=6000×90%)
hydrographs are plotted) derived using the SCEM andsimulated hydrographs associated with plausible
parameter sets.

1) Global response of catchment and propagationof parameter uncertainty
Probabilistic results of the hydrograph are
obtained from the ensemble simulation of OHDISKWMSS for 6000 parameter sets sampled from theposterior parameter distribution. Fig. 4 shows how
the parameter uncertainty propagates into estimatesof hydrograph simulation uncertainty. Moreover, thereproduced hydrographs with respect to plausible
parameter sets are included in Fig. 4. In this figure,the light grey dots indicate the observed streamflow
data and the black shaded region is 90% hydrographsimulation uncertainty associated with the posterior
distribution of the parameter estimates.
In spite of considerable parameter uncertainty(i.e. unknown initial parameter uncertainty range isreduced to constrained boundary through SCEM
trials, see Fig. 2(f)), the resultant boundary ofhydrographs are very narrow. It can be interpreted
that complex rainfall-runoff model is likely to be
exposed to equifinality problem so that it makes
difficult to discriminate quantitatively andqualitatively between reliable and unreliable
parameter sets. Moreover, it supports the fact that
only streamflow data is not sufficient to identify
model parameters precisely.
Indeed, as shown in Table 1 and Fig. 4, all ofparameter sets used here lead to considerably
similar outputs in terms of quantity (i.e.
Nash-Sutcliffe Efficiency values of all hydrographs
are more than 0.95) and quality (i.e. visual
goodness-of-fit between the observed and simulatedhydrographs). Therefore, the complementary
constraint is necessary to augment power to selectthe erroneous parameter combination(s) out from alarge number of plausible ones. In manyhydrological questions, it may be much moreinterest to know what happens inside a catchment
than at the outlet. The subsequent subsections
demonstrate the internal dynamic response ofcatchment to these plausible parameter sets.

2) Spatially distributed origins for plausible
parameter sets
The study catchment is represented by 3190
rectangular slopes (i.e. S=3190) and contributions ofeach slope to each streamflow observation atselected time steps, 1, 48, 134 and 182 hours areillustrated in Figure 5(a). At the beginning of
rainfall-runoff process (e.g. 1hour), the adjacentslopes to river channel, referred to as riparian zone
constitute primarily the streamflow while water
stored in other upstream slopes do not reach to theriver yet. As time goes on, contributive slopes
spread gradually from near to far and eventually, allof slopes influence on streamflow generation. The
contribution of each slope to streamflow isrepresented by Relative Ratio of Total Discharge(RRTD) at the specific time step, defined as:
Di (t )
× 100 (%)
RRTDi (t ) = outlet
D
(t )
(3)
S

∑ RRTD (t ) = 100 (%)
ii =1where i is slope number and Di (t ) is discharge at
the outlet from slope i within catchment at time t.
D outlet (t ) is total discharge of the outlet at time t.
Each RRTD for specific four time steps iscategorized into eight classes as shown in Fig. 5(a).
Colorful snapshots for spatially distributed origin
apparently present that even though global
responses of catchment with respect to the plausible
parameter sets are nearly identical, the internal
responses are completely different.
The variations of spatial origin for the different
parameter sets are summarized by two indices. First,
how many slopes are included in each RRTD classis represented by Relative Frequency of Slope(RFS) index:
8
N j (t )(4)
RFS j (t ) =, ∑ RFS j (t ) = 1
S
j =1
where N j (t ) is the number of slopes included in
RRTD class j at time t, S is the total number ofslopes.
Secondly, how much each RRTD classcontributes to generate streamflow observations isquantified by Ratio of Contribution to Streamflow
Generation (RCSG) index:
RCSG j (t ) = ∑ RRTD j (t )i∈ j8

∑ RCSG (t ) = 100 (%)
j =1

(5)

j

All of values for the RFS and RCSG are plottedin Fig. 5(b). The number of slopes in RRTD class 3
for Sample(1) and (2) is larger than others at the
beginning of runoff but the level of theircontribution to streamflow is very similar (see the
RCSG at the 1hr). It supports that even if thelocations of origin are different, the equivalentamount of streamflow can arise from those. At the- 82 -

(a)(b)
Fig. 5 (a) Spatially distributed origins of streamflow for OPT (note that channel network is plotted as black solidline only in the left top snapshot for OPT at 1hour), Sample(1) and Sample(3) at time 1, 48, 134, 182 hours (b)RFS and RCSG for the selected four time steps.

48hour, both RFS and RCSG show relatively thebalanced results (i.e. all of slopes contribute tostreamflow while their amounts of contribution are
different according to parameter sets). Furthermore,
the internal response of catchment in terms of RFS
and RCSG obviously show the different patterns as
the catchment becomes sufficiently saturated, forexample, Sample (1) ~ (3) at 134 and 182 hours are
more deviated, especially, in RRTD classes 6~8, ascompared with other cases. Interesting finding isthat the parameter sets for SLS and OPT provide
similar results in spite of marginal difference
between the values of two parameter sets. It meansthat these two plausible sets should be retained formodel prediction unless and until additional
evidence to the contrary becomes apparent becauseeven auxiliary information used here cannot
constrain the model completely.
3) Temporally separated components of stream
flow for plausible parameter sets
The study event is split into six temporal classes(i.e. T=6; pre-event, 0~25hrs, 26~96hrs, 97~109hrs,
110~120hrs and 121~192hrs) and then the simulatedhydrographs are separated into corresponding six

components by using temporal record of the
proposed matrix. In the same manner, temporally
separated constituents with respect to different
parameter values yield different aspects. In Sample(1) ~ (3) cases, old water posses more than 50% of
streamflow while they are approximate 40% in OPTand SLS cases and less than 35% in HMLE andMIA cases, respectively (see Fig. 6).
The corresponding amount of each hydrographcomponent to selected six rainfall segments issummarized in Table 2. As well as overall behavior
in terms of the simulated hydrograph and thevariation of spatial origin of streamflow above
demonstrated, the temporal components with respect
to SLS and OPT parameter sets provide similar
results in spite of nominal difference in terms of
three parameters, d s , d c and β . However, othercomponents vary differently owing to different
conceptual parameter related with subsurface flow.
From the application results, it can be seen that acomplex distributed rainfall-runoff model canpermit multiple alternative spatiotemporal flow
pathways for rainfall over catchment to transform
into runoff and the commonly used streamflow data
to evaluate such model does not contain any inform-- 83 -

Table 2 Summary of temporally-separated componentsof the simulated hydrographs.
TemporalClass
SampleOld Water (%)

SLS
HMLE
MIA
OPT
Sample(1)Sample(2)Sample(3)

40
34
32
40
54
56
59

New Water (%)
1

2

3

4

5

9
12
13
9
11
9
7

13
15
15
13
10
10
8

11
12
13
11
9
7
7

12
12
12
12
8
8
8

15
15
15
15
8
10
11

global and internal dynamic responses of catchmentwith respect to seven plausible parameter sets.
Summarized results from this study are as follows:
1) The behavioral parameter sets provide similar
results in terms of OFs and simulated hydrographs
even though they have different values.
2) It is possible to separate spatiotemporally hydrograph of a mesoscaled catchment and this result
would be comparable with the separated hydrographby using observable hydrochemical measures.
3) Spatiotemporal internal response of catchment
provides new information capable of narrowing
down more physical and reliable range of parameterset(s).
It is concluded that the incorporation of thecomplementary constraint into evaluation procedurecan be a more advanced approach to solve
equifinality problem than the utilization of only
streamflow data.
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Fig. 6 Hydrograph separation based on temporal recordof streamflow for OPT, Sample(1) and Sample(3).

ation on these possible pathways. As a result, if
additional comparable observed data is available,
the proposed information in terms of the spatio
temporal internal behavior of catchment can beutilized as complementary evaluative criterion tofilter out unreliable parameter set(s).

6. CONCLUSION
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streamflow data to identify the most reliable
parameter set and the necessity of complementary
constraint to select the unreliable one out fromnumerous plausible parameter combinations. The
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