Annual Journal of Hydraulic Engineering, JSCE, Vol.53, 2009, February
Annual Journal of Hydraulic Engineering, JSCE, Vol.53, 2009, February

EFFECT OF SPATIAL VARIABILITY OF RAINFALL
ON CATCHMENT RESPONSESIN MESOSCALE MOUNTAINOUS AREA
Giha LEE 1, Yasuto TACHIKAWA2, Takahiro SAYAMA3 and Kaoru TAKARA4
1 Member of JSCE, Dr. Eng., Research Associate, DPRI, Kyoto University (Gokasho, Uji 611-0011, 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)

Detailed information regarding spatial and temporal patterns of rainfall is crucial to adequately reproduce
the dynamics of the natural rainfall-runoff system. Though a remarkable progress has been achieved inmeasuring rainfall using radar and satellite, rainfall data still remain an important source of uncertainty inhydrological modeling. This paper aims to investigate the influence of input uncertainty due to spatial
variability of rainfall data on both global and internal catchment responses in distributed rainfall-runoff
modeling for a small mountainous catchment (211km2). Various rainfall scenarios are generated fromoriginal radar rainfall data. Then, we analyze catchment responses to these scenarios qualitatively andquantitatively. The results show that total amount and temporal variability of rainfall rather than rainfall
spatial patterns influenced runoff simulations more significantly. However, the spatial pattern of plausible
rainfall scenarios led to the large predictive uncertainty in terms of potential streamflow origin traced by acomputational tracer method. The difference of spatial origin of streamflow due to the input heterogeneity
was attenuated when the rainfall over the catchment was routed into the downstream outlet. As a result,temporally-traced runoff components corresponding to specific rainfall durations became similar
regardless of spatial variability of rainfall.
Key Words : input uncertainty, spatial variability of rainfall, computational tracer method, distributed
rainfall-runoff modeling, potential streamflow origin

Though remarkable progress has been achieved inmeasuring rainfall data, even now these data remain
an important source of uncertainty in rainfall-runoff
modeling1).
Initial studies2), 3) regarding rainfall error focusedon how dense network of rain gauges is necessary toestimate the areal rainfall over a watershed. Ball andLuk4) investigated applicability of alternative
surface interpolation models, functions of hydroinformatics tools in order to describe actual spatial
pattern of rainfall. Moreover, a great deal ofresearches devoted to answer the question how thespatial rainfall pattern affects the runoff at acatchment outlet while a few studies5) have been
made to test the influence of rainfall spatial
variability on internal behaviors of hydrologicalvariables such as soil moisture, ground water,
sedimentation and so on.

1. INTRODUCTION
Rainfall-runoff models, regardless of theircomplexity, are simplified representations of thehydrological system in nature. Uncertainty is an
essential element of rainfall-runoff modeling whichchallenges the reliability of model results. For a
responsible dealing with the outcome of simulation
models, it is crucial to identify and assess theuncertainties involved in modeling processes and toquantify the reliability of model prediction results.
In general, prediction uncertainty stems fromdifferent sources such as model parameters, model
structure as well as measurement errors associatedwith the system input and output. In particular,
detailed information about spatial and temporal
patterns of rainfall is crucial to adequately
reproduce the dynamics of hydrological system.

-7-

Fig. 1 Spatial patterns of accumulated rainfall and comparison of areal mean rainfall between original radar data (i.e.,
Scenario 1) and the generated scenarios (i.e., Scenarios 2 ~ 8).

The main objective of this study is to assess theimpact of input uncertainty due to spatial variabilityof rainfall on catchment responses, which are not
only discharge but also potential streamflow origin,in distributed rainfall-runoff modeling. The key
research question that this study addresses is: how
does input uncertainty based on various rainfall
scenarios, which are generated from observed radar
rainfall data, influence model simulations for (a)hydrograph and (b) spatiotemporal origin of
streamflow at the outlet of the study site.

rearranged rainfall field of Scenario 6. Scenario 8changes the origin of the original radar data (i.e.,
Scenario 1) based on X and Y axes in order toclarify the influence of the rainfall spatial pattern
due to relocation of rainfall cells, on catchment
responses.
Fig. 1 shows the spatial distributions of the total
rainfall depths of eight rainfall scenarios and thecomparison of areal mean rainfall between originalradar data of Scenario1 (hereafter, S1) and thegenerated rainfall scenarios (i.e., S2,…,S8).

2. VARIOUS RAINFALL SCENARIOS

3. RAINFALL-RUNOFF MODEL

Radar rainfall data (the Ejiroyama X-band radar)with 1km spatial resolution is available for the study
site, the Kamishiiba catchment (211km2). A
historical flood event by Typhoon No. 9 (15-19
September 1997) is used to generate various spatial
patterns of the original rainfall data. Five grid cells
are selected from the observed radar rainfall field
and then they are assumed as virtual gauge stations
representing the rainfall characteristics ofdownstream (gauge 1), mid-stream (gauge 2), andupstream (gauge 3, 4, and 5) in the study area. Fivesynthetic rainfall fields are generated based on thechosen virtual stations using the nearest
neighborhood interpolation method in the order ofthe number of gauges used in interpolation; from
Scenario 2 to Scenario 6. Rain gauges of Scenario 7
are rearranged randomly to examine the effect of
spatial variability of rain gauges on runoff
simulation. Namely, Scenario 7 indicates the

The Kinematic Wave Method for Subsurface andSurface Runoff (KWMSS)6) is based on onedimensional kinematic wave flow routing and has
inherent capabilities for catchment partitioning,
sub-catchment modeling, sub-catchment linking,and channel network processing. In this model, thedrainage network is represented by sets of slope andchannel elements. The rainfall-runoff transformation
conducted by the KWMSS is based on theassumption that each element is covered with apermeable soil layer, D as shown in Fig. 2. This soil
layer consists of a capillary layer and a
non-capillary layer. In these conceptual soil layers,
water flow are simulated as unsaturated Darcy flow
and saturated Darcy flow, and overland flow occurs
if water depth, h exceeds soil water capacity.
The KWMSS represents these runoff processes by
the stage-discharge relationship7), defined as:-8-

Fig. 2 Schematic model structure and extended stagedischarge relationship of 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)

Fig. 3 Relative errors of total depth of rainfall for eachscenario.

overestimated approximately 25% when comparing
to S1. However, the other spatially-distributed
scenarios lead to comparatively acceptable results interms of total amount of rainfall in this catchment.
All scenarios except for S2 do not exceed therelative error of ±5%. In particular, even though S3
is more spatially-aggregated than S4~S8, therelative error of S3 is smaller than S4~S7 andsimilar with S8.
The relative errors of runoff simulations
corresponding to these rainfall scenarios are
summarized in Fig. 4. Note that REV, REP, and
REPT indicate relative error of total runoff volume,
relative error of peak discharge, and relative error ofpeak time, respectively.
In spite of different spatial variability of rainfall
in the generated scenarios, the runoff simulation
results corresponding to S3~S8 are acceptable since
relative errors of runoff properties are within±10%. The difference of total rainfall depth due torain gauge sampling as shown in Fig. 3 results invery similar error patterns with respect to REV and
REP while REPT is less sensitive to the spatial
patterns of rainfall scenarios used here. Even thepeak time of S2 shows the relative errors of lessthan 10% but the errors of runoff volume and peakflow for this scenario are much deviated from S1.
The results indicate that total amount andtemporal variability of rainfall during the event
period affect runoff simulation more significantly
than the spatial pattern itself of rainfall in the study
catchment.

Flow rate, q of each slope segment is calculated
by Eq.(1) combined with the continuity equation,
Eq.(2), where vc = kc i ; va = kai ; k c = k a / β ;α = i / n ; m=5/3; 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 need to be
optimized in KWMSS. This model is calibrated byonly S1 rainfall data and this data is assumed toprovide the best estimate of input to the model.
Then, the calibrated model is applied to the studysite represented by 500m Digital Elevation Model(DEM).

4. EFFECT OF INPUT UCERTATINTY ON
HYDROGRAPH PROPERTIESFor each rainfall scenario, we compute therelative errors in rainfall and catchment responses.
The relative errors of rainfall and runoff simulation
are computed as:
RS1 − RSi
(3)
× 100 i = 2, LL ,8
RS1
where RS1 is either the accumulated depth of areal
rainfall or the general properties of runoff based on
S1 such as runoff volume, peak flow and timing ofpeak and RSi is the variable of interest in Scenario
i. The relative errors of total amount of rainfall forall scenarios are illustrated in Fig. 3.
This figure demonstrates that the most spatially
uniform rainfall field, S2 fails to estimate accurate
amounts of areal rainfall. The total rainfall of S2 isFig. 4 Effect of rainfall error on runoff properties foreach scenario.

-9-In other words, spatially less distributed rainfall
data can also be suited for rainfall-runoff
simulations in this small mountainous catchment if
the aggregated rainfall field by a few rain gauges
captures a total volume of rainfall successfully
during the event period. However, this result does
not imply that scenarios with only several rain
gauges optimally present the actual spatial
variability of rainfall.
Within the context of ‘equifinality’8) (i.e.,
indistinguishable runoff simulation results in spiteof poor consideration of rainfall spatial variability),
it may be worthwhile to find answers to thefollowing questions: 1) how do internal responses of
the catchment react to plausible rainfall scenarios(S3 and S8) containing very similar information
with respect to the total amount and temporalvariability?; and 2) if unlike equivalent hydrographs
reproduced from the plausible input data, internal
catchment responses are sensitive to spatial
variability of rainfall, how can we demonstrate theprediction uncertainty due to input uncertainty?
To account for the influence of plausible input
data on the internal catchment response, we employ
the computational tracer method9) capable of tracing
potential streamflow origin within the catchment.
Then, we compare the prediction results between theoriginal input, S1 and the plausible inputs, S3 and
S8.

5. COMPUATIONAL TRACER METHOD
AND RESULTS OF STREAMFLOWORIGIN TO PLUASIBLE INPUTS
The goal of the computational tracer method is totrace where streamflow comes from when it rains
over the catchment. This method can show thepotential streamflow origin in time and space byusing a spatiotemporal record matrix as illustrated inFig. 5. The dimension of matrix Ri (t ) is given
with Ns (number of sub-units within catchment)rows and NT (number of temporal classes) columns
where i is specific slope element; t is time. Fig. 5(c)shows the spatiotemporal matrix at time t at thecatchment outlet. For example, the value belonging
to spatial zone C and temporal class 2 implies thatthe contribution of (C,2) entry to the outlet
streamflow observation at time t is 6%. Whensummarizing the whole values vertically along thecolumns, the temporal contribution of rainfall to
streamflow can be obtained. Likewise, the spatial
contribution of sub-catchments on streamflow is
calculated by horizontal summation along the rows.
As a result, modelers can divide into two runoff
components; old water (i.e., pre-event water at time

Fig. 5 Schematic diagram of the computational tracer
method based on (a) temporal record of streamflow, (b)spatial record of streamflow and (c) spatiotemporal
record matrix of streamflow; Ns=6, NT=5.

class 0 possesses 15%) and new water (i.e., newwater components are 30% at time class 1 and 55%
at time class 2, respectively) at the specific time t.
Moreover, it is possible to track the
spatially-distributed origin for runoff generation
using this matrix with information stored in eachslope element, for instance, downstream spatial
zones (e.g., D, E, and F) contribute more than 60%
of streamflow whereas upstream zones (e.g., A, B,
and C) affect runoff generation less at time t. Moredetails about this conceptual matrix are presented in
Sayama et al.9).
The drainage network of the Kamishiiba
catchment is represented by 860 slope elements (Ns=860) of 500m DEM. Each slope element isregarded as the potential origin of streamflow. The
spatiotemporal matrix is updated every 1 hour time
step during the model simulation. The contributionof each slope element to runoff generation at the
specific time step is represented by Relative Ratio ofTotal Discharge (RRTD) at the outlet, defined as:
Di (t )RRTDi (t ) = outlet
× 100 (%)
D (t )(4)Ns∑ RRTD (t ) = 100 (%)
ii =1where i is slope element number; Ns is total numberof slope elements; Di (t ) is discharge at the outlet
from slope element i within the catchment at time t;
D outlet (t ) is total discharge of the outlet at time t.
Four particular time steps, 1, 12, 36 and 72 hours
are selected in order to visualize the spatial origin of
the runoff for S1 and then the results are comparedwith the results produced from the plausible- 10 -

Fig. 7 Variation of the Contributing Percentage (CP) of
each sub-catchment due to the three rainfall scenarios.

Fig. 6 Effect of input uncertainty due to spatial variabilityof rainfall on internal catchment response: spatiallydistributed origins of streamflow for S1, S3, and S8.

rainfall scenarios, S3 and S8. Each RRTD for theselected four time steps is plotted with seven classes
as shown in Fig. 6.
At the beginning of rainfall-runoff process, theadjacent slope elements to river channel, which isreferred to as riparian zone, constitute primarily of
the streamflow while the water stored in upstream
slope elements do not reach the river channel yet.
As time goes on, contributive areas spread gradually
over the catchment and eventually, all slope
elements contribute for the streamflow generation.
An interesting finding is that even though runoff
simulation properties at the outlet for S1, S3, and S8
are identical (see Fig. 4), the distributions ofstreamflow origin are different in space according to
the spatial pattern of rainfall data. To quantify thevariation of spatial distribution of streamflow origin
at sub-catchment scale, a simple index, Contributing
Percentage of the sub-catchment (CP) is proposed asfollows:
CPj (t ) = ∑ RRTD j (t )i∈ j8

∑ CP (t ) = 100 (%)
j =1

j

(5)

where j is the sub-catchment number; the study
catchment is divided into eight sub-catchments asshown in the left-top panel of Fig. 6 and then thecontribution of each sub-catchment to streamflow
generation is calculated. Fig. 7 shows the results of
CPs for the three scenarios.
The results clearly show that the different spatial
patterns of plausible rainfall scenarios from theoriginal rainfall pattern lead to the significant
predictive uncertainty in terms of potential
streamflow origin unlike their equivalent runoff
properties. At the 1 hour time step, the patterns ofCP are too much different while the variations of CP
become gradually similar after the peak time (36hours) of the event. It means that after stopping
rainfall, S3 and S8 also result in nearly identical CP
values regardless of their different spatial patterns.
Furthermore, the historical event is split into six
temporal classes (i.e., NT=6: pre-event, 0~15hrs,
16~22hrs, 23~30hrs, 31~34hrs and 35~84hrs) andthen the simulated hydrographs based on S1, S3,
and S8 are separated into six corresponding runoff
components to each rainfall duration by using the
temporal matrix of streamflow record.
Fig. 8 illustrates the results under the original andplausible scenarios, which are temporally-separatedhydrographs with respect to selected rainfall
durations. In this figure, the volume of all runoff
components including both pre-event water andevent water did not change according to the spatial
pattern of rainfall. It corroborates that even if thespatial patterns of S3 and S8 are different from S1,
the temporal variations of streamflow origin arevery similar because their total rainfall depth overthe catchment are similar during the study event
period and the best-performing parameter set of theoriginal data is applicable for runoff modeling basedon the plausible input scenarios. Also, it can beconcluded that the KWMSS can lead to multiple
alternative flow pathways, yielding very similar- 11 -

hydrographs due to plausible input data. In otherwords, the effect of rainfall spatial pattern on
spatially- distributed streamflow origin is graduallysmoothen or weakened as the rainfall over thecatchment drains into the downstream outlet in formof either surface/subsurface flow or a mixture of
both. The different CP values in Fig. 7 indicate thatthe different distributions of spatial streamflow
origin can exist due to rainfall spatial patterns but
the different CP values do not influence temporal
streamflow observations as shown in Fig. 8. As aresult, temporally-traced runoff components
corresponding to specific rainfall durations became
similar regardless of spatial variability of rainfall.

6. CONCLUSIONSThis paper aimed to investigate the effect of input
uncertainty due to spatial variability of rainfall dataon runoff simulation and potential origin of
streamflow by using the distributed rainfall-runoff
model, KWMSS and the computational tracer
method. The following observations with respect to
input uncertainty to catchment responses were
made:Fig. 8 Temporally-separated runoff components
corresponding to the six rainfall durations for the threerainfall scenarios.

1) even less spatially-distributed rainfall scenarios(e.g., S3, S6 and S7) than the original data could
provide acceptable runoff simulation results. It
backs up that temporal variability and amount of
rainfall were more important than the spatial
pattern itself of rainfall data in rainfall-runoff
modeling for the small mountainous catchment;2) the spatial patterns of plausible rainfall scenarios,
which led to similar runoff properties with theoriginal rainfall data, resulted in significant
predictive uncertainty in terms of spatial
distribution of streamflow origin, traced by thecomputational tracer method;3) even if the spatial variability of rainfall wasdifferent, the aggregated catchment output,
temporally-separated runoff component has nodifference as the rainfall over the study catchment
was routed into the downstream outlet throughthe drainage network that was represented by anumber of slope elements.
In future works, we will clarify the importance of
spatial variability of rainfall by applying the
proposed methodology to various catchments
containing different characteristics.
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(Received September 30, 2008)- 12 -