IJERD – International Journal of Environmental and Rural Development (2019) 10-2

Research article
erd

Identifying the Minimum Number of Observed Rainfall
Events Required for Optimal TOPMODEL Parameters in
Mid-sized Equatorial Catchments
EMMANUEL OKIRIA
Graduate School of Agriculture, Tokyo University of Agriculture, Tokyo, Japan

HIROMU OKAZAWA*
Faculty of Regional Environment Science, Tokyo University of Agriculture, Tokyo, Japan
Email: h1okazaw@nodai.ac.jp

YURI YAMAZAKI
Faculty of Regional Environment Science, Tokyo University of Agriculture, Tokyo, JapanYUKIMITSU KOBAYASHI
Graduate School of Agriculture, Tokyo University of Agriculture, Tokyo, JapanSHINJI SUZUKI
Faculty of Regional Environment Science, Tokyo University of Agriculture, Tokyo, Japan
Received 26 January 2019

Accepted 20 October 2019(*Corresponding Author)

Abstract TOPMODEL, a topography-based, semi-distributed hydrological model wasapplied to the 84 km2 Atari catchment in Eastern Uganda. The study sought to identify theminimum number of rainfall events needed to optimally calibrate 5 unknown parameters foryearly hydrological simulation. Model input data was daily averaged precipitation, riverdischarge and evapotranspiration for the year 2015 with the output being simulateddischarge. A rainfall event was defined as consecutive days of effective rainfall - effective
rainfall being a daily rainfall ≥ 5.0 mm. Parameterization was done for Sequentially
Accumulated Rainfall Events (SARE), beginning with 1 event and sequentially progressing
until all 54 observed rainfall events in the year were used. All SARE had similar starting
dates with the end dates being variable. The ‘true’ parameters were those derived from
inputting all observed rainfall events while the other instances of the parameters frompartial SARE were classified as ‘non-true’. Elimination criterion of ‘non-true’ parameters
was set at an error of ±30%. Parameter values varied with the change in number of rainfall
events, showing their dependence on rainfall characteristics. Downslope saturated
transmissivity (Te) and maximum root zone storage deficit (SRmax) were the most and least
variable from their means respectively. Also, exponential decay parameter (m) and delay
time constant (td) needed the least and the greatest number of rainfall events to stabilise
within the ±30% error bounds respectively. Therefore, the minimum number of rainfall
events required to calibrate TOPMODEL and to optimise td in mid-sized equatorial
catchments in Eastern Uganda are equivalent. Consequently, it required at least 49 rainfall
events to calibrate TOPMODEL in 2015.
Keywords calibrate, modelling, rainfall event, SARE, TOPMODELINTRODUCTION
Observed hydro-meteorological data are useful for irrigation and drainage planning. In areas thatlack this data, hydrological models are used to predict stream discharge. One such model isTOPMODEL, a conceptual, semi-distributed hydrological model (Beven and Kirkby, 1979) thathas been used worldwide. In Africa, TOPMODEL was applied in a humid tropical climate inⒸ ISERD117

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Nigeria (Campling et al., 2002) and in an Ethiopian catchment (Gumindoga et al., 2014). To date,
the authors have not found published evidence of the application of TOPMODEL in Uganda.
Generally, Uganda has a scarcity of hydro-meteorological data, and worse still, even theavailable data is of poor quality, e.g., temporal gaps, unreliability, and inaccuracy of rating curves.
Notably, most Ugandan rivers are ungauged, and bleaker still, only about 33% of installed waterlevel gauging stations are currently operational (MWE, 2013).
Given the challenges above, hydrological models could come in handy. But hydrologicalmodels need calibration, a process that requires the input of observed hydro-meteorological data.
For this reason, Atari catchment was chosen as a target site for the study. Unlike many catchmentsin Uganda, it is equipped with modern hydrological and meteorological monitoring facilities. TheTOPMODEL concept was thus applied to the Atari catchment located in Eastern Uganda.
According to Coles et al. (1997); (1) Rainfall intensity influences rainfall-runoff response of acatchment, and (2) Calibrated parameters are only relevant to the rainfall event(s) that was used fortheir calibration. From these findings, it would be ideal to use all possible permutations of rainfall
events to achieve representative parameters suitable for yearly hydrological simulations. However,given the previously mentioned observed data deficiencies in Uganda, it is necessary to haveinsight into the premise that limited observed data may be sufficient to calibrate model parameterswhich can be used for yearly hydrologic simulation. Therefore, the purpose of the study is toidentify the minimum number of rainfall events required for optimal yearly TOPMODEL
parameter calibration.
METHODOLOGY
Study Site
The study area is Atari, a headwater catchment of Mt. Elgon in Eastern Uganda, with a drainage
area of 84 km2 above the stream gauging station and a corresponding channel length of 33 km. Its
topography is comprised of mountainous areas from where the main stream (Atari River) originatesand flows to the relatively flat plains. From ASTER GDEM, the difference in height between thelowest and the highest point is 2,389 m. Of the 84 km2, 35 km2 (42%) is forest, 28 km2 (33%) is
agricultural area and 21 km2 (25%) is rangeland.

Fig. 3 Instrumentation and land use in Atari catchment
RG. Stn is rain gauge, WLG is water level gauge, Met. Stn is meteorological stationⒸ ISERD

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Under the Project on Irrigation Scheme Development in Central and Eastern Uganda (PISD)(JICA, 2017), hydro-meteorological monitoring equipment were set up in Atari catchment in 2015,
viz., a mid-stream rain-gauge to detect catchment rainfall, a downstream meteorological station tomeasure evapotranspiration parameters and a water level sensor at a control section of Atari River.
TOPMODEL1) Description of TOPMODELTOPMODEL is a conceptual, semi-distributed model suggested by Beven and Kirkby (1979). It
divides the soil layer into root zone, unsaturated zone and saturated zone. The upper soil layers(root zone and unsaturated zone) are analysed at grid-scale as distributed models while the lowerlayer (saturated zone) is computed as a catchment scale lumped model. TOPMODEL evaluates thestate of wetness of the surface layer of a basin from the Topographical Index (TI).
𝒂𝑻𝑰 = 𝒍𝒏 𝐭𝐚𝐧𝒊𝜷(1)𝒊Where ai is upstream contributing area per unit contour length, tan βi is local slope and i isgrid number.
TI is derived from a Digital Elevation Model (DEM) and it spatially evaluates the amount ofsurface flow. Details of TOPMODEL are in Mukae, et al. (2017) and Beven and Kirkby (1979).

Fig. 4 Histogram of TI distribution for Atari catchment
2) Computational Procedure of TOPMODEL
Input data is observed daily precipitation, observed river discharge, actual evapotranspiration (ETa)and a TI that is derived from 30 m grid DEM. Rainfall was measured for each event and otherhydro-meteorological parameters were recorded at 10-minute logging intervals, but the daily
averages were used for computation.
a. The water balance equation of the root zone:
The amount of water that is stored in the root zone is calculated from the water balance of rainfall[L], ETa [L], maximum storage deficit in the root zone (SRmax) [L] and storage deficit in the rootzone (SRZ) [L]. When SRZ < 0, the excess water (EXi) flows to the unsaturated zone (SUZi) [L].
Potential evapotranspiration (ET0) is calculated by the Penman-Monteith method (Allen et al.,1998). ETa is treated as a function of ET0, SRmax and SRZ.
𝑺𝑹𝒁𝒊

𝑬𝑻𝒂 = 𝑬𝑻𝟎 (𝑺𝑹𝒎𝒂𝒙)(2)b. The water balance equation of the saturated zone:
Base flow discharge, (Qsub) [LT-1] is calculated using the parameters m, and Te [L2T-1], catchment
mean TI, (λ) [-] and the mean storage deficit in the watershed 𝑆𝑖 [L]:𝑸𝒔𝒖𝒃 = 𝑻𝒆 𝒆𝝀𝑺𝒊 ⁄𝒎(3)Ⓒ ISERD

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c. The water balance equation of the unsaturated zone
The unsaturated zone is a temporary water storage zone that links the root zone to the saturatedzone. The mean storage deficit in the watershed at the start of the calculation (initial time step),
𝑆𝑖 is obtained from Eq. (3), assuming that the initial discharge is Q0 [LT-1].
𝑸

𝑺𝒊 = −𝒎 𝐥𝐧 𝑻 𝒆𝟎−𝝀(4)𝒆Like in Fig. 3, Si [L] expresses the storage deficit of each grid and UZi [LT-1] is the amount ofwater supplied from the unsaturated zone to saturated zone, with i being the grid number. Since,
cells with the same value of TI are considered to be hydrologically similar, computation is done foreach TI class (Fig. 2) and not for every individual grid. If Si is ≤ 0, then the TI class is considered tobe saturated, and excess water inflow from the root zone (EXi) becomes surface flow, as shown inFig. 3. If Si > 0, the excess water inflow is temporarily added to SUZi.
𝑺𝑼𝒁𝑼𝒁𝒊 = 𝑺 𝒕 𝒊(5)𝒊 𝒅where td [TL-1] is the delay time constant, a parameter that expresses the period of retention.
TOPMODEL requires the calibration of 5 unknown parameters, namely; exponential decay
parameter (m), mean value of downslope transmissivity when the soil is just saturated (Te), delay
time constant (td), maximum root zone storage deficit (SRmax) and initial root zone storage deficit(SRZinitial) using the Monte-Carlo method. Nush-Sutcliffe efficiency (NS) and Root Mean Square
Error (RMSE) are the evaluation functions adopted to compare agreement between observed andsimulated discharge. Having evaluated the parameters, it is then possible to simulate riverdischarge following a rainfall-runoff event.
∑𝒏 𝑬𝑽

(6)

𝐍𝐒 = 𝟏 − (∑𝒏𝟏 𝑴𝑽)
𝟏

∑𝒏 𝑬𝑽

(7)

𝐑𝐌𝐒𝐄 = √ 𝟏𝒏

Where EV is error variance ((observed value - simulated value)2), MV is mean variance((observed value – mean observed value)2) and n is number of observation days.

Fig. 5 Schematic of TOPMODEL (Source: Mukae, et al., 2017)
Data Requirement
The input data for TOPMODEL is precipitation, river discharge and meteorological data used toestimate ET0. Averaged daily data for 2015 was used. A rainfall event was defined as consecutive
days of effective rainfall, and effective rainfall as that daily rainfall ≥ 5.0 mm (Ali and Mubarak,2017).
From 2015-March-24 to 2015-December-14, 54 rainfall events were observed.
Correspondingly, the total amount of precipitation, evapotranspiration and river discharge wasⒸ ISERD120

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1,646 mm, 1,149 mm and 499 mm from where it was inferred that 30% of the precipitation isdischarged to the downstream via the river.
Description of Study-defined Terminologies and Methods
Table 1 is a representation of the Sequentially Accumulated Rainfall Events (SARE) concept. All
the SARE had the same starting date but the end dates were variable.
The ‘true’ parameters were those derived from inputting all SARE in 2015 while ‘non-true’
parameters were those got by using partial SARE during model calibration. The ‘true’ parameters
were considered to be optimal for full year hydrological simulation because they were assumed torepresent an average characteristic of all observed events in 2015, which is not the case with ‘nontrue’ parameters.
The percentage error of ‘non-true’ parameters was determined by comparison with the ‘true’parameter value – The ‘non-true’ parameters with maximum error of ± 30% were classified as
good and therefore close enough to ‘true’ value. Equation (8) shows computation of error of ‘nontrue’ parameter,.
𝒏𝒕𝒑−𝒕𝒑) × 𝟏𝟎𝟎%𝒕𝒑(8)𝑬𝒓𝒓𝒐𝒓 = (where ntp is ‘non-true’ parameter value and tp is ‘true’ parameter value.
Parameterisation was done for SARE beginning with the 1st event and sequentiallyprogressing until all 54 rainfall events in 2015 were used. In total, model calibration was done 54
times. However, it was impossible to evaluate the 1st event alone since its NS value was infinity.
Results from 1 SARE were thus omitted from further analysis.
Table 1 The SARE Concept
Rainfall event count12
3･･･53
54

Sequentially Accumulated Rainfall Events (SARE)
1st event
1st event
1st event･･･
1st event
1st event

2nd event
2nd event･･･
2nd event
2nd event

3rd event･･･
3rd event
3rd event→→→･･････

53rd event
53rd event

54th event→→

Calculate unknown parameters(m , T e , t d , SRZ initial and SR max)
non-true parameter value of 1 SARE
non-true parameter value of 2 SARE
non-true parameter value of 3 SARE･･･
non-true parameter value of 53 SARE
true parameter value

Yellow indicates the partial target rainfall event(s), which is used to calculate the unknown parameters for each SARE
occurrence. Orange indicates all SARE.

RESULTS AND DISCUSSION
Parameter Descriptive Statistics
Table 2 ‘True’ and ‘non-true’ parametersmTe (×106)
td
SRZinitial
24762
0.013
0.006
Min
1840
0.011
0.002
Max
53
987
0.020
0.009
‘non-true’
Mean
21
587
0.017
0.006
value
s.d*
5257
0.003
0.001
CoV**
0.25
0.44
0.16
0.24
*
Standard deviation, ** Coefficient of variance = Standard deviation / Mean‘true’ value

SRmax
0.008
0.004
0.009
0.007
0.001
0.16

NS0.57
0.48
0.92
0.68
0.100.15

RMSE
0.73
0.00
0.73
0.53
0.19
0.36

Table 2 shows the descriptive statistics of the calibrated parameters. ‘Non-true’ parameter values
vary from 18 to 53, 40 to 987, 0.011 to 0.020, 0.002 to 0.009 and 0.004 to 0.009 for parameters m,Te, td, SRZinitial and SRmax respectively. In addition, NS and RMSE varies from 0.48 to 0.92 and fromⒸ ISERD

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0.00 to 0.73 correspondingly. Based on the classification criterion of NS values by Foglia et al.
(2009), the goodness of fit varies from ‘good’ to ‘excellent’. The coefficients of variance (CoV) for‘non-true’ parameters are 0.25, 0.44, 0.16, 0.24 and 0.16 for m, Te, td, SRZinitial and SRmax in thatorder. It follows that, Te and SRmax are the most and least variable respectively. It is then clear thatthe 5 parameters are changeable depending on the number of rainfall events.

Fig. 6 Error plots of calibrated parameters
Parameter Trends
Fig. 6 is a graph of the parameter error under different numbers of SARE conditions and Fig. 5shows the trend of evaluation functions with the number of SARE.

Fig. 7 Evaluation function under different SARE
The mean errors for SRmax, SRZinitial, m, Te and td are -8%, -10%, -13%, -23% and 37%
correspondingly. It takes 3 SARE for parameter m to stabilise within an error rage of ±30%. From
1 up to 16 SARE, Te is highly changeable, with error ranging from -95% to -54%. It requires atleast 17 SARE to stabilise parameter Te within ±30% error range. The first instance of lowobserved event precipitation (9.4 mm at 10th event) seemed to cause Te value to increase sharply.
From 1 up to 18 SARE, td shows no observable trend, with error ranging from 4% to 56%.
However, from 19 to 43 SARE, parameter td stabilises within a narrow error range of 52% to 55%.
The parameter value then begins to steadily decrease at 44 SARE, eventually falling within
acceptable error range at 49 SARE. Although SRmax and SRZinitial are the least variable parameterswith respect to their means, there is no discernible trend of their behaviour with the number ofSARE. This might be because of the gross uncertainties in calibrating soil storage deficits, asⒸ ISERD122

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observed by Coles et al. (1997) and Campling et al. (2002). But, different from Te, it is seen that
there is a large reduction in both SRmax (-38%) and SRZinitial (-69%) at 10 SARE.
NS and RMSE decrease and increase respectively with increasing SARE. It seems that as thenumber of SARE increases, the parameters become an average representation of many more varied
rainfall characteristics, but not the best representation for each individual rainfall event.
Parameter m is the least affected by the rainfall events inputted while parameter td is the mostaffected, requiring at least 3 and 49 SARE respectively to stabilise within ±30% error bounds.
Therefore, the minimum number of rainfall events required to calibrate TOPMODEL and tooptimise td in mid-sized equatorial catchments in Eastern Uganda are equivalent. Consequently, it
requires at least 49 rainfall events to calibrate TOPMODEL in 2015.
Summarily, it is seen that rainfall characteristics influence the parameterization ofTOPMODEL in mid-sized equatorial catchments.
CONCLUSION
A hydrological model, TOPMODEL, was applied to a different number of Successively
Accumulated Rainfall Events (SARE) to examine parameter sensitivity to rainfall characteristics. In
this study we deduce the following: (1) Dependence of calibration parameters on rainfall
characteristics was evident, with exponential decay parameter (m) being the least affected anddelay time (td) being the most affected by rainfall event characteristics; Further, (2) in mid-sized
equatorial catchments, td was the determining parameter for the minimum number of SARE needed
for calibrating TOPMODEL for yearly hydrological simulation. To confirm the observations
presented here, more yearly observed hydro-meteorological data and studies on other catchments isneeded. Furthermore, future studies should consider hydrological conditions like base flow, rainfall
intensity and effective discharge in order to better understand the effect of rainfall events.
ACKNOWLEDGEMENTS
The study was done during graduate studies at Tokyo University of Agriculture, Tokyo, Japan,
courtesy of scholarship from Japan International Cooperation Agency (JICA), Japan.
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