that has physically-meaningful parameters, it is possible that measurements could be used to define empirical prior dis-
tributions. Although statistically appealing, this approach could risk over-interpretation of the physical significance
of the model parameters, which are often defined in hydro- logical models at different scales than those at which
suitable physical properties are measured.
3,15
In the case of TOPMODEL, the exponential saturated zone store is
areally integrated, as are flows at the outlet, allowing estimation of the value of the store parameter m directly
from flow recession curve data.
20
However, in the following analysis, m was treated as if its value could not be deter-
mined a priori. It will become apparent that there may still be uncertainty about the m parameter, especially when
simulations of
variables other
than streamflow
are considered.
4.2 Definition of the likelihood measure
In practice, the choice of parameter sampling distribution may not be critical since the prior distributions can be mod-
ified by comparison of simulated data with observations, and this comparison is applied to complete sets of model
parameters. A likelihood measure, L, is used to make the comparison, where, as noted above, the term ‘likelihood’ is
interpreted fairly broadly to indicate goodness-of-fit. The measure chosen for this study was
L ¼ exp ¹
W j
2 e
j
2 o
5 where W is a weighting factor, to be discussed below, j
2 e
is the variance of the simulation errors and j
2 o
the variance of the observed data. The function L would
equal unity if the observed and simulated data were the same
and reduces
towards zero
as the
similarity decreases.
4.3 Rejection of non-behavioral simulations
There may be many parameter sets that lead to acceptable simulations of a given set of observations. Definitions of
acceptability may vary according to context. For example, if a hydrological model is evaluated in terms of some
simple goodness-of-fit function, less good values of this function may be considered acceptable for simulations of
a time period known to pose particular difficulties for the model.
Allowance can be made within GLUE for the explicit definition of acceptability by the rejection of simulations
for which the likelihood measure falls below a chosen threshold. The use of a rejection threshold is similar in
approach to the Generalised Sensitivity Analysis GSA procedure of Spear and Hornberger
28
which divides multi- ple model simulations into a set that are judged to reproduce
the behavior of the system and a set that are rejected as ‘non-behavioral’.
Previous application of TOPMODEL to the Minifelt
19,21
suggested that very different values would be expected for the chosen goodness-of-fit measure eqn 5 when
evaluated for simulations of discharge and water level data, leading to difficulties in the specification of a
consistent rejection threshold. As a response to this, the approach of Binley and Beven
7
has been followed, where model
simulations were
ranked according
to the
likelihood measure and only the best 10 retained as ‘behavioral’.
It is recognised that the choice of a ‘best 10’ rule to reject relatively poor simulations is somewhat arbitrary, and
that the definition of the rejection rule may affect the uncer- tainty bounds computed using the GLUE method. In fact,
tests demonstrated that relaxation of the rejection threshold to define a larger proportion of the total number of simula-
tions as ‘behavioural’ would cause only slight modifications of uncertainty bounds. The reason for this lack of great
sensitivity to the rejection rule is that although the best parameter sets achieved values of L conditioned on flow
data of 0.8, the majority of simulations achieved only extremely small values, particularly when conditioned on
a combination of observed variables. Therefore, even if a larger number of parameter sets were retained as ‘beha-
vioural’, predictions associated with many of the retained parameter sets would fall within the tails of the cumulative
distributions of L and have little effect on the location of uncertainty bounds, computed according to the method
described below.
4.4 Construction of uncertainty bounds
Simulation uncertainty bounds may be estimated from the parameter sets retained as behavioral by ranking the values
of the simulation variable at the ordinate of interest e.g. discharges at a given time step and forming a cumulative
distribution of corresponding values of the likelihood measure. Uncertainty bounds are then drawn at the values
of the simulation variable that correspond to selected per- centiles of the cumulative likelihood distribution, in this
study 5 and 95.
It is worth emphasising that, although the uncertainty bounds are calculated at each ordinate, the likelihood
measure is always evaluated over a whole simulation. The likelihood measure implicitly accounts
14
for any correlation in the marginal distributions of the parameters by taking a
similar value for different parameter combinations that reflect such correlation.
4.5 Updating likelihood distributions
The likelihood measure L was first applied to evaluate simulations with respect to the flow data observed for the
autumn 1987 period, resulting in a posterior distribution for the TOPMODEL parameters conditioned upon this set of
observations. Further observations were combined within the likelihood measure by application of Bayes’ equation
308 R. Lamb et al.
in the form L Q
i
lY
1 ,
2
ÿ ¼
1 C
L Q
i
lY
2
ÿ L Q
i
lY
1
ÿ 6
where L Q
i
lY
1 ,
2
is the likelihood measure evaluated for a single set, Q
i
, of TOPMODEL parameters conditioned on two sets of observations Y
1
and Y
2
, L Q
i
lY
1
is a prior value, conditioned on the first set of observations Y
1
, L
Q
i
lY
2
is evaluated with the new observations Y
2
, and C ¼
X L
Q
i
lY
1 ,
2
. Eqn 6 was first applied to ‘update’ the likelihoods conditioned on the 1987 flow data with
borehole water level observations from the 1987 period, rather than with a second period of flow data.
Repeated application of eqns 5 and 6 leads to an expression for the likelihood measure conditioned on n
sets of observations, such that L Q
i
lY
1 , …,
n
ÿ ¼
1 C
exp ¹
W
1
j
2 e
, 1
j
2 o
, 1
þ W
2
j
2 e
, 2
j
2 o
, 2
þ …
þ W
n
j
2 e
, n
j
2 o
, n
ð 7Þ
Note that the use of an exponential function, such as eqn 5, in eqn 6 implies additive combination of the individual
error variance terms within the exponent, allowing the weight given to the ith set of observations to be controlled
by choosing a value for W
i
. This is an important feature of eqn 7 if Q
i
successfully reproduces some sets of observa- tions but gives rise to very poor simulations of other data; it
may be undesirable in this case to reject Q
i
, but this would happen if the individual variance ratios were multiplied
directly. In the results to be presented below, values of W
i
were chosen for each period to give equal weight to errors from
the simulation of each individual set of borehole water levels and to share weighting equally between the combined
error in simulations of all four boreholes and the errors in simulation of the flow data. When observations from the
1987 and 1989 periods were combined, equal weight was given to the simulation errors for each period. An exception
to this scheme was made for the 1987 water level data from borehole 7, which were known to be inaccurate and for
which the errors were given a weight of zero when combin- ing different borehole data sets using eqn 7. The values
chosen for W
i
in eqn 7 are given as a proportion of the sum of weights in Table 2. The weighting of the distributed
piezometer data will be discussed later.
5 RESULTS 5.1 Simulation uncertainties conditioned on a single
period of discharge and borehole water level observations
Initially, simulations of the autumn 1987 flow data were compared with observations using the likelihood measure,
L eqn 5, and the best 10 1000 simulations retained. Cumulative distributions of L are shown in Fig. 2 for the five
TOPMODEL parameters. The solid cumulative curves represent the posterior distributions of the parameters
inferred from L, conditional upon the observed flow data. The uniform random sampling strategy, used to generate
parameter values across the ranges shown, is equivalent to assuming the same value of any likelihood measure for
every parameter set, prior to comparison between the associated simulations and the observed data. These prior
distributions would appear in Fig. 2 as straight lines, plotted from lower left to upper right for each parameter. Deviations
from such a line, seen in Fig. 2, indicate that conditioning on the observed flow data and retaining the best 1000 simula-
tions has modified the prior position of ignorance about the TOPMODEL parameters, except for S
R max
and dv. The strongest modification is suggested for the saturated zone
parameters, m and lnT .
These results are consistent with experience of the calibration of TOPMODEL, where simulated flows are gen-
erally most sensitive to the values taken by m and lnT . In
the relatively short period considered here, the unsaturated
Fig. 2. Rescaled distributions of the likelihood measure L for TOPMODEL parameters, conditional on observations from the
autumn 1987 period t
d
in h m
¹ 1
, T in m
2
h
¹ 1
.
Table 2. Proportional weights applied in eqn 7 to errors in evaluating
combined likelihoods
for different
sets of
observations
1987 1989
Flows 12
12 BH 4
16 18
BH 5 16
18 BH 6
16 18
BH 7 18
Total time series 1
1 Piezometers
1
Use of water table observations to constrain model uncertainty 309
Fig. 3. a Uncertainty bounds computed for the first 300 h of the 1987 period likelihood measure distributions conditioned on observations
from the whole 1987 period.
Fig. 3. b Borehole water level uncertainty bounds for the first 600 h of the 1987 period. z
i
is depth to the water table positive downwards at location i.
310 R. Lamb et al.
Fig. 4. a Uncertainty bounds for the main event in the 1989 period, showing the effect of updating ‘prior’ likelihood measure distributions
conditioned on flows from the 1987 period with the 1989 period flow data.
Fig. 4. b Prior and updated uncertainty bounds for borehole water levels for the 1989 period note that the time axis corresponds to that in
a. Likelihoods were conditioned on observations from each individual borehole only.
Use of water table observations to constrain model uncertainty 311
zone parameters S
R max
and t
d
were unlikely to have as great an effect on the simulated flow data. The posterior distribu-
tion for dv is essentially the same as the prior distribution because the value taken by dv has no effect on the simulated
flows.
The distributions of L were revised by applying eqn 7 to incorporate errors from the simulation of water levels
recorded in the four boreholes during the 1987 period. Re-scaled cumulative distributions dashed curves are
shown in Fig. 2, as are a set of distributions conditioned only on the borehole water level observations dotted
curves. The revised distributions, combining discharge and water level data, differ from the distributions condi-
tioned on flow data alone. The main differences are that wider ranges of m and lnT
appear to be able to explain the combined sets of observations, some modification of
the distribution for dv is seen, and the unsaturated zone parameters revert almost to uniform distributions.
These updated distributions suggest that observations of water level data have not helped to constrain the likely
values of the TOPMODEL parameters, but have instead increased the uncertainty about these values. In the case
of dv, it is interesting that there has been relatively little refinement of the uniform prior distribution. This may be
a result of the interaction between parameters; although dv has no effect on simulated flows, m and lnT
do have an effect on simulated water levels.
Fig. 3a shows flow data from part of the autumn 1987 observation period plotted with two sets of computed uncer-
tainty bounds or prediction intervals, the first conditioned on the observed flows alone and the second after the borehole
water levels were brought into the likelihood measure. Both pairs of bounds enclose the observations, which is desirable.
However, both lower bounds indicate very low flows for most of the period, and the simulation uncertainty is wide
for the main storm event. The uncertainty bounds condi- tioned upon flows and water levels are wider than those
conditioned on flows alone, especially for the first signifi- cant peak in the hydrograph. It is important to note that
because the likelihood measure is calculated for an entire simulation, but uncertainty bounds are computed at every
time step, the plotted bounds do not follow any one particular simulation. Thus, although the lower bounds in
Fig. 3a are relatively flat, individual simulations can be much more dynamic.
Simulation uncertainties are shown for the 1987 water level data in Fig. 3b, which includes a dry period not
shown in Fig. 3a. Each set of borehole observations is plotted along with uncertainty bounds conditioned on the
water levels recorded in that borehole labelled ‘conditioned on local data’ and with a pair of bounds computed after
combining the simulation errors from boreholes 4, 5 and 6 in eqn 7 note again that the information from borehole 7 was
given zero weight due to the unreliability of the data. The water level uncertainty bounds are less variable than the
flow bounds of Fig. 3a. Recessions are not well defined by the uncertainty bounds, which fail to enclose the
observed levels during the drier period. The deep lower bounds at the start of the period suggest that for some para-
meter sets, the stores in TOPMODEL were not properly initialised even though an initialisation period of 100 h
was allowed. Borehole uncertainty bounds conditioned on a combination of flow and water level data are not shown,
but do not differ greatly from the bounds conditioned on the combined borehole data.
The observed water levels in each borehole are better reflected by the bounds conditioned on observations in
that borehole alone, in the sense that the these bounds tend to be narrower, indicate more dynamic responses and
enclose the observations for nearly as much of the time as do the bounds computed using data from the combination of
boreholes. This is important because of the implication that knowledge of water levels in several boreholes increases the
uncertainty in simulations at one given location. Inspection of the observed water levels shows behaviour inconsistent
with the simple assumptions of spatial uniformity in TOP- MODEL. The ponding of water on the surface at borehole 4
is notable, whilst there are slight differences in the timing of water level responses. Given the simple representation of
processes in the model, the extent of the uncertainty shown in Fig. 3b seems realistic, as does the increase in uncer-
tainty when observations from several boreholes are considered.
5.2 Updating uncertainty bounds with a second observation period
