IDENTIFICATION OF UNCERTAINTY OF A PHYSICALLYBASED HYDROLOGICAL MODEL
GUANGHENG NI
Department of Hydraulic Engineering, Tsinghua University, Beijing, 100084, P. R. China
LEI WANG AND HEPING HU
Department of Hydraulic Engineering, Tsinghua University, Beijing, 100084, P. R. China
For the purpose of water resources planning and management, prediction of water
resources for future or designed conditions is often required and physically-based
distributed models are becoming a common tool. However the model predictions should
be associated with estimates of predictive uncertainty. A hydrological similar unit-based
distributed hydrological model is established and applied in the study basin to analyze
the hydrological cycle and its main affecting factors. Model parameters are calibrated
against the observed discharge data of 1988. Uncertainties that main model parameters,
rainfall data input may bring to the simulated discharge are simply identified.
INTRODUCTION
One of the biggest challenges of hydrology research is the prediction of hydrologic
responses, especially which of ungauged basin and of the changing responses resulting
from changes in landscape features, climate. In theory, run based on knowledge of the
basin physical properties, physically based distributed models offer the capacity of
predicting the hydrologic response of basins with more accuracy and reliability. In most
of the current distributed models, partial differential equations are used to describe flow
processes and solved by approximate numerical methods like finite difference or finite
element. For model calculation, basin is discretized into triangular, rectangular, or other
types of elements, or units. Parameters are required for each element. Although attempts
have been being made, there is still no way to directly measure or estimate the parameter
values at each element, and therefore in practice, common approach to applying
physically based models is to perform calibration against observed discharge, soil
moisture, groundwater level etc.
In general, hydrologic prediction is subjected to uncertainties from natural
randomness, data input, model parameter and model structure. Natural uncertainties refer
to the random and temporal and spatial fluctuations inherent in natural processes. Natural
randomness almost always introduces a large amount of uncertainty into the hydrologic
processes. Data uncertainties refer to the inaccuracy, errors and inadequacy in data
measurement, transmission and handling. Model parameter uncertainty refers to the
determination of the proper parameter values. Model structure uncertainty refers to the
imperfect representation of the true hydrological processes by model equations.
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Prediction uncertainty of physically based distributed models has drawn attention from
hydrologists since the development of early distributed models. Beven [1] argued that
the current generation of distributed physically based models are lumped conceptual
models, and demonstrated a significant degree of model prediction uncertainty resulting
from errors in model structure, estimation of parameter values etc. Binley et al. [2]
investigated the determination and usefulness of predictive uncertainty in the Institute of
Hydrology Distributed Mode (IHDM), using the methods of Rosenblueth and Monte
Carlo simulation. Melching et al. [3] presented a framework to estimate the reliability of
runoff modeling by accounting for the uncertainties from different sources. Guo et al. [4]
studied the uncertainty of model parameter and runoff in two basins by using the Monte
Carlo and nonparametric methods. As summarized in detail by Sivapalan et al. [5], the
International Association of Hydrological Sciences (IAHS) new initiative launched the
IAHS Decade (2003-2012) on Predictions in Ungauged Basins, or PUB, with its
scientific program focuses on the estimation of predictive uncertainty, and its subsequent
reduction.
The Qin River is a branch at the middle reach of the Yellow River and has a basin
area of 13,500 square km. The study basin is part of it and takes an area of 9,245 square
km. Observed discharges at a few stations show a serious declining in recent decades, as
well as groundwater level and springs outflow, although no significant changes in land
use and consumptive water uses occurred. It is in an urgent need to identify the main
factors causing this change in river discharges and groundwater level, and predict the
future trend of water resources. For this purpose, here as a preliminary study, the
uncertainty of a physically based hydrological model is analyzed.
MODEL DESCRIPTION
The model used in this study provides a dynamic representation of watershed processes
including interception, depression, unsaturated flow, groundwater flow, overland and
channel flow routing etc. These processes are briefly summarized in the following and
model integration is described. All these processes are integrated into the whole system.
Interception
Only rainfall interception by vegetation canopy is considered. Rainfall is assumed to be
intercepted by vegetation canopy until the vegetation surface storage capacity is filled.
For different vegetation, the canopy interception capacity is expressed as,
RI max  kv LAI
(1)
where RI max is the interception capacity of the vegetation (mm); kv is coefficient related
to vegetation type; LAI is the leaf-area-index of the vegetation at different time. For
rainfall R the interception RI is then calculated as,
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R  R Im ax
 R;
RI  
 R Im ax ; R  R Im ax
(2)
Rainfall intercepted will be evaporated when rain ceases, while during rainfall
evaporation is omitted.
Depression
During rainfall after ponding occurs, runoff will first fill depressions in the land surface
and the excess water becomes overland flow. Water in depressions is retained on the
surface and is ultimately infiltrated into the soil or evaporated into the atmosphere.
Depression storage capacity is related with land cover, topography and is determined
from field observation results. In common depression storage capacity is 1.0 mm to 2.5
mm for impervious areas, and 5.0 mm to 15.0 mm for pervious areas.
Flow in unsaturated zone
Flow processes considered in unsaturated zone include infiltration, evapotranspiration,
percolation, interflow and return flow etc. Based on observations, usually soil moisture
changes gradually below two meters from top surface. Therefore unsaturated zone is
divided into two parts, the upper one and the lower one. For the upper part (root zone)
simulation is carried out by using the Richards equation, and lateral flow along slope is
introduced to make the simulation semi-two-dimensional. The mixed form of Richards
equation used is as following.
 ( z , t )  
h

  k ( h)  k ( h)   s ( z , t )  q L
t
z 
z

(3)
where  ( z , t ) is volumetric soil water content; z is the vertical distance from surface
with downward as positive (m); t is time in seconds; h is water head (m); k ( h) is
hydraulic conductivity (m/s); s ( z , t ) is source or sink item as evapotranspiration etc.;
q L is interflow along slope calculated as qL  k (h)  i with i as the surface slope. For
the lower part (transition zone) only vertical flux is considered and a simplified storage
function is adopted to describe the relative slow soil water movement in it [6].
Groundwater flow and interaction between river and groundwater
Flow in saturated zone is described by mass balance and Darcy’s law. Groundwater flow
between simulation elements is calculated from the hydraulic gradient and the hydraulic
property along contact boundary. Other items considered in water balance equation
include recharge from up unsaturated zone, water abstraction, evaporation, leakage to
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deeper aquifer and exchange with river water. The exchange between groundwater and
river flow is estimated as,
kr Ar ( H g  H r ); H g  H r
RGexc  
kr Ar ;
H g  Hr

(4)
where RGexc is the exchange between river and groundwater, with minus value implying
water flows into aquifer; kr is the hydraulic conductivity of river bed materials; Ar is
river wet area in the calculation element; H g and H r are water head for groundwater
and river water respectively.
Overland and channel flow routing
Kinematic wave model is employed to simulate both overland sheet flow and river flow,
in which the momentum equation is given by Manning’s equation. Overland flow reaches
river and becomes lateral river inflow, and river water is then routed downstream to basin
outlet. The Newton’s method is used in the nonlinear kinematic wave solution scheme.
Artificial water uses
Agricultural water use (irrigation), industrial water use and domestic water supply are
considered. For each water use, the sources must be specified as well as the timing and
amount. The sources could be river, aquifer or outside the basin. Irrigation water is
directly added to crop land surface, while a portion of domestic water supply is added to
top soil layer as leakage. Effluent from households and waste water treatment plants is
added to either top soil layer or water bodies according to their discharge locations.
Model integration
For modeling purposes, a basin is partitioned into a number of subbasins. The use of
subbasins in a simulation is particularly beneficial when different areas of the basin are
dominated by land uses or soils dissimilar enough in properties to impact hydrology.
Subbasins in this study are derived as those within a certain range depth to groundwater,
same or similar soil type and within certain range of surface slope. Hydrological
processes within each subbasin are then considered as hydrological similar. Channel
network, watershed boundary, and variable source area within river floodplain or riparian
zones are also taken into account in subbasin partition. Subbasin delineation is often
carried out by taking the advantage of GIS software. Figure 1 shows a subbasin
delineation and the DEM for the study part of the Qin River basin.
For each subbasin, HRUs (Hydrological Response Units) are defined by land cover.
Four types of land cover classification are adopted in this study, i.e. forest, crop land,
bare soil, and urban area. Vertically each HRU has two zones as unsaturated zone and
groundwater aquifer, and the unsaturated zone is further divided into a few layers for
numerical solution scheme. Spatial resolution and temporal step are mainly depended on
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the characteristics of the modeled area, the objectives of the simulation, and the available
computational power. Calculation is carried out for each HRU and their results are
aggregated to form the total response of the subbasin it belongs to. Each subbsin is
hydrologically liked with its surrounding ones through surface and subsurface flow.
Figure 1. Subbasin delineation and the DEM for the study part of the Qin River basin
THE QIN RIVER BASIN AND MODEL PARAMETERIZATION
The Qin River is a branch of the Yellow River and has a basin area of 13,500 square km.
The study basin is part of it and takes an area of 9,245 square km. The Qin River basin is
located in the middle reach of the Yellow river, and its mean annual precipitation is
611mm. At the low reach of the Qin River, urbanization is developed to certain extent,
while most of the middle and up stream of the Qin River are mainly covered by natural
and planted forest, grass land, crops etc. For simulation, meteorological data such as
precipitation and evaporation; landscape features such as soils, topography, vegetation
and land use; agricultural water use, industrial water use and domestic water use are
collected. DEM of 100m resolution is used for stream generation, subbasin delineation
and surface slope calculation. Digitized map of 1:100,000 is used for land use
classification. Discharge records at fiver locations along the main Qin River are also
collected for model calibration.
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Among the model parameters, some of them were determined on the basis of basin
physical property and field observations with the aid of high resolution GIS data, while a
few of them need calibration. Parameters to be calibrated are the hydraulic properties of
soils, river bed materials and aquifers. Calibration was carried out by comparing observed
and simulated discharges at different sites. The goodness of fit for calibrations is
expressed as the mean error ( M error ), and the ratio of absolute error to mean ( Rerror )
which is used by WMO, as defined as the following.
M error
1
1 n Qo  Qs i R
; error 
  i
nQo
n i 1 Qo i
n
Q
i 1
oi
 Qs i
(5)
1000
0
Rain
Simulated Discharge
Observed Discharge
Discharge [m3/s]
800
50
100
Rain [mm/day]
where n is the number of discharge data; Qoi and Qsi are observed and simulated
discharge respectively; Qo is the mean discharge over the whole calibration period.
Calibration was carried out for 1988 whole year. The calibration result for basin outlet is
shown in Figure 2, and the errors of M error and Rerror are0.27 and 0.25 respectively.
600
150
400
200
200
250
0
300
88-1 88-2 88-3 88-4 88-5 88-6 88-7 88-8 88-9 88-10 88-11 88-12
Figure 2. Comparison of observed and simulated discharge at the basin outlet
IDENTIFICATION OF MODEL UNCERTAITY
The reliability of simulation results produced by a hydrological model is a function of
uncertainties in nature, data, model parameters, and model structure. To evaluate the
combined effect of the uncertainties on the reliability of outputs of the hydrological
model, a reliability analysis method, such as first-order second-moment techniques or
Monte Carlo simulation, is usually used. As the development of computer technology and
computational expertise, the Monte Carlo method is considered to be more competitive,
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for by this method more details of the distribution of responses can be obtained. Here as a
preliminary study, only the model uncertainties from sensitive model parameters and
rainfall data input are simply identified.
Uncertainty from model parameters
Model parameters chosen for uncertainties analysis are hydraulic conductivity of top soil,
aquifer and river bed material. Parameter values from calibration are considered to be
standard, uncertainties brought by error in the estimation of these parameters are shown
in Table 1. The parameter estimation error is assumed to be one order, i.e. standard
values multiplied by 0.1 or 10.0 are adopted for analysis. Errors are the comparison
between simulated and observed discharges at the basin outlet.
Table 1. Uncertainties brought by model parameter estimation
Error
Saturated hydraulic
conductivity of top soils
M error
Hydraulic conductivity of
top soils in slope direction
M error
Hydraulic conductivity of
river bed materials
M error
Hydraulic conductivity of
aquifers
M error
Rerror
Rerror
Rerror
Rerror
Item values multiplied by
0.1
1.0
10.0
0.77
0.27
0.55
0.67
0.25
0.58
0.64
0.27
2.25
0.70
0.25
1.35
0.27
0.27
0.68
0.26
0.25
0.48
0.27
0.27
0.32
0.26
0.25
0.28
1200
0
Rain
Ks×0.1
Ks
Ks×10
Simulated Discharge [m3/s]
1000
800
50
100
600
150
400
200
200
250
0
Rain [mm/day]
Item
300
88-7
88-8
88-9
Figure 3 Uncertainty brought by the hydraulic conductivity of top surface soils (the cases
vertical saturated conductivity is one order larger or smaller than the calibrated values)
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As can be seen from Table 1, variations in top soil hydraulic properties resulted in
serious changes of model outputs; the over estimate of the hydraulic conductivities of
river bed materials also led to larger error between the simulated discharges and the
observed ones; while estimate error in the hydraulic conductivities of aquifers brought
relatively little uncertainties to the model outputs. As shown in Figure 3, increase in
saturated conductivity of top soil resulted in a peak flow reduction of around 400 m 3/s,
while increase in base flow is also obvious because more rainfall infiltrated into soil.
Uncertainty from rainfall data input
Spatial variation of rainfall is described by average (  ), standard deviation (  ), and
coefficient of variation ( Cv ). The number and distribution of the rainfall stations over the
basin surely will affect the model output. As can be seen from Table 2, when the number
of rainfall stations decreased from 49 to 25, i. e. from average one station in 188 square
km to one station in 370 square km, no significant changes in model output is found,
although there is a peak flow reduction by around 200 m3/s. While the number of the
station is further reduced to 10, the difference in model output became obvious in both
peak flows and low flow, and therefore the errors became larger.
Table 2. Uncertainties brought by rainfall input
Number of rainfall stations
 (mm)
 (mm)
Cv
M error
49
25
10
707
683
709
132
107
115
0.19
0.16
0.16
0.27
0.27
0.33
Rerror
0.25
0.27
0.35
CONCLUSIONS
Although the model is based on sound physics and has the potential to simulate and
predict hydrological information of basin, the uncertainty bounds are wide even
parameters are restrained by calibrated values, especially the hydraulic conductivity of
top soils. For the simulated cases, it seems that the selected 25 rainfall stations distributed
over the basin have a fairly good representative. Future work will be carried out to fully
analyze the uncertainties of the model by using the Monte Carlo simulation and predict
the water resources variation under climate changes.
REFERENCES
[1] Beven K., “Changing ideas in hydrology – the case of physically-based models”, J.
Hydrol., Vol. 105, (1989), pp157-172.
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[2] Binley A. M., Beven K. J., Calver A. and Wattl L.G., “Changing responses in
hydrology: Assessing the uncertainty in physically based model predictions”, Water
Resour. Res., Vol. 27, No. 6, (1991), pp1253-1261.
[3] Melching C. S., Yen B. C., and Wenzel H. G., “A reliability estimation in modeling
watershed runoff with uncertainties”, Water Resour. Res., Vol. 26, No. 10, (1990),
pp2275-2286.
[4] Guo S., Li L. and Zeng Z., “Uncertainty analysis of impact of climate change on
hydrology and water resources”, Hydrology, Vol. 6, (1995), pp1-6. (in Chinese)
[5] Sivapalan M., Takeuchi K., Franks S.W., Gupta V. K., Karambiri H., Lakshmi V.,
Liang X., McDonnell J. J., Mendiondo E. M., O'Connell P. E., Oki T., Pomeroy J.
W., Schertzer D., Uhlenbrook S. and Zehe E., “IAHS decade on Predictions in
Ungauged Basins (PUB), 2003-2012: Shaping an exciting future for the hydrological
sciences”, Hydrological Science Journal, Vol. 48, No. 6, (2003), pp 857-880.
[6] Ni G., Herath S., and Musiake K., “A distributed catchment model and its
application to simulate urbanization effect”, 9th APD-IAHR, Singapore, (1994), pp
254-261.