Hydrological Model Sensitivity to Parameter and Radar Rainfall Estimation Uncertainty

8/2/2019 Hydrological Model Sensitivity to Parameter and Radar Rainfall Estimation Uncertainty

1/28

Hydrological model sensitivity to parameter andradar rainfall estimation uncertainty

Abstract:

Radar estimates of rainfall are being increasingly applied to flood forecastingapplications. Errors are inherent both in the process of estimating rainfall fromradar and in the modelling of the rainfall-runoff transformation. The study aims atbuilding a framework for the assessment of uncertainty that is consistent with thelimitations of the model and data available and that allows a direct quantitativecomparison between model predictions obtained by using radar and raingaugerainfall inputs. The study uses radar data from a mountainous region in northernItaly where complex topography amplifies radar errors due to radar beam occlusionand to variability of precipitation with height. These errors, together with other errorsources, are adjusted for by applying a radar rainfall estimation algorithm. Radar

rainfall estimates, adjusted and not, are used as an input to TOPMODEL for floodsimulation over a medium-sized catchment (116 km2). Hydrological modelparameter uncertainty is explicitly accounted for by use of the GLUE (GeneralisedLikelihood Uncertainty Estimation) framework. Statistics are proposed to evaluateboth the wideness of the uncertainty limits and the percentage of observations thatfall within the uncertainty bounds. Results show the critical importance of properadjustment of radar estimates and of the use of radar estimates as close to groundas possible. Uncertainties affecting runoff predictions from adjusted radar data areclose to those obtained by using a dense raingauge network, at least for the lowestradar observations available.

Keywords: Radar rainfall estimation, runoff simulation uncertainty, parameteruncertainty, sensitivity to errors, TOPMODEL, GLUE.

1. INTRODUCTION

Early warning with the use of radar rainfall observations and hydrologicmodels is crucial for minimizing flood and flash flood related hazards. The potentialbenefit of using radar observations is particularly large in mountainous areas,where the rugged nature of the terrain and altitudinal effects impose significantlimitations to the real-time operation of raingauge networks. Radar rainfallestimation is, though, subject to errors caused by various factors ranging from

instrument issues (e.g., calibration, measurement noise) to the high complexity andvariability in the relationship of the measurement to precipitation parameters(Austin, 1987; Joss and Lee, 1995; Andrieu et al., 1997; Anagnostou andKrajewski, 1999; Borga et al., 2002). The different radar-rainfall error sources canaffect in various ways the accuracy of rainfall-runoff simulations (Borga, 2002).Meaningful hydrological applications of weather radar rainfall estimates requiretherefore the rigorous analysis of the propagation of radar rainfall uncertaintiesthrough rainfall-runoff modelling.


2/28

Many studies have focused on the application of radar-rainfall estimates inflood-forecasting applications (Schell et al., 1992; James et al., 1993;Georgakakos et al., 1996; Bell and Moore, 1998; Vieux and Bedient, 1998;Winchell et al., 1998; Sempere-Torres et al., 1999; Borga et al., 2000; Ogden etal., 2000). Common to these studies is the use of rainfall-runoff models with a

single optimal parameter set. These models were calibrated based on using areference rainfall input (most often based on dense raingauge networks) andcomparisons carried out with runoff simulations obtained by applying radar-basedprecipitation inputs. This allowed to explore issues related to the impact ofuncertainties due to 1) radar-rainfall estimation errors and 2) the spatio-temporalsampling of precipitation fields on runoff simulation.

However, latest research has shown that the concept of the optimumparameter set may be questioned for the case of hydrological modelling (Beven,2001, p. 218). There are many aspects about the hydrological system that areessentially unknowable, especially the nature of flow processes below the ground

in structured soils. One practical alternative strategy to bypass such unknowns inhydrology has been to apply the flow equations at a scale (of space and time) towhich they are appropriate (Woods et al., 1995). But problems still exist to suchalternative techniques of improving model predictability even if the hydrologicsystems are assumed stationary (Weinberg, 1972). Freeze and Harlan (1969) firstproposed the digital blue print on how to model the physical flow processes ofwater in a fashion that would allow runoff prediction. An obvious question asked bythe hydrologic community today is 'what progress has been made since 1969?'Has the physics of subsurface flow in unsaturated and saturated zones been fullyunderstood at all scales to allow reliable hydrologic modelling through the usualdata measurement protocol (e.g. rainfall and stream flow measurement)?

To answer some of the above questions, Beven (1989) and Grayson et al.(1992) initiated a meaningful debate to highlight the current limitations ofhydrologic prediction models based on the physical principles (as we know themnow) of the flow processes. It was concluded that unless a fundamental change inhydrologic modelling philosophy arrived along with an improvement inmeasurement techniques, this current scenario of limitations of hydrologic modelsis not expected to improve in the near future. Thus, most hydrologic models are socomplex yet limited that there can be many different sets of parameter valueswithin a given model structure that may be compatible with the data available forcalibration. Certainly, one of those parameter sets will be 'optimum' according tosome measure of goodness of fit, but that optimum may not survive application toa different data set or different measure of goodness of fit. Parameter sets thatgive almost equally good fits may also be scattered throughout the parameterspace. These observations are at the heart of the equifinality concept, introducedby Beven and Binley (1992). The crux of the problem is that what one would like toknow about the internal description of the hydrological system which isunfortunately of a substantially higher order than what can be observed about theexternal description of the system. The primary philosophy behind rejecting the


3/28

concept of an optimal parameter set and accepting the concept of equifinality isthat the results from sensitivity analyses carried out based on an 'optimumparameter set' concept may not be adequate to describe the influence of radarrainfall uncertainties on runoff simulation, since equally acceptable parameter setsmay exhibit different sensitivity to rain inputs with varied uncertainty.

Recent study by Carpenter et al. (2001) attempted to address the issue ofuncertainty in hydrologic models in the context of NEXRAD gridded radar rainfalldata products by asking the important question 'are data and models accurate andreliable enough to allow their use in an operational environment?' However noexplicit accounting of parameter uncertainty and the statistical characterisation oferror in rainfall input were made in their study. The errors in radar rainfall wereactually simulated from a cloud model. In a study by Sharif et al. (2002), a purelysimulation exercise is carried out involving a cloud model to simulate storms andradar simulation to simulate the radar-like observed reflectivity. The studydemonstrated the importance of radar rainfall errors for Hortonian runoff prediction

even though the interaction of radar rainfall errors were not linked to uncertainty inthe hydrologic model parameters. The statistical characterization of runoffuncertainty associated with the input-parameter uncertainty interaction is the keyinformation to understand potential improvements in radar algorithms andinvestigate scenarios of rainfall-runoff models.

This study aims at building a framework for the assessment of uncertainty thatis consistent with the limitations of the model and data available. The frameworkallows for a direct quantitative comparison between model predictions obtained byusing different rainfall inputs (eg. rainguage and radar). To achieve this, we use theGLUE (Generalised Likelihood Uncertainty Estimation; Beven and Binley, 1992)

framework to account for the associated rainfall-runoff modeling uncertainties. Thisis a Bayesian Monte Carlo simulation-based technique, developed as an extensionof the Generalised Sensitivity Analysis (GSA) of Spear and Hornberger (1980).The method was outlined in concept by Beven and Binley (1992) and otherapplications using different types of likelihood measure have been demonstratedby a number of researchers. The uncertainty assessment is carried out herethrough application of radar-estimated precipitation to a lumped rainfall-runoffmodel for a medium-sized watershed located in a mountainous region in northernItaly, where major error sources are represented by radar beam partial blockingand variability of reflectivity with the altitude.

Results form this study are expected to provide insights on the use of radarrainfall estimates for hydrological forecasting and, more specifically, to provide anobjective framework to improve the problem definition for questions such as, Howconfident is the prediction based on radar rainfall estimates?, What are the

principal sources of uncertainty in runoff prediction? and How can theseuncertainties be reduced?


4/28

The paper is organised as follows. Section 2 presents the study region anddata set, section 3 describes the rainfall-runoff simulation study, section 4illustrates the GLUE methodology and its application in this context, and section 5completes the paper with discussion and conclusions.

2. STUDY AREA AND DATA

The region chosen for this study is located in northern Italy, close to Veniceand Padua. Details about the study area, including its terrain characteristics andrain climatology, can be found in Borga et al. (2000). Radar data are from a C-band Doppler radar located on the Monte Grande hill, which is 60 km from thePosina catchment (116 km2) and 476 meter above sea level. The radar providessweeps at 10 different elevation angles and covers a radius of 120 km (see Table1 for technical details of the radar). In this study we used only data from thenorthwest quadrant (Figure 1, right panel) of the radar viewing area and up to aradius of 100 km. This radar sector is selected because it encompasses the most

complex relief with peak elevations reaching 2500m. In the Posina, altitude rangesfrom 2230 to 390 m a.s.l. at the outlet (Figure 1, left panel).

The investigation is performed for five storm events (see Table 2 fordescription of these events) that represent a typical meteorological situationassociated with flooding in southern alpine regions. The meteorological situationassociated with those storm systems is characterized by cyclogenesis in the LionGulf or surrounding regions, which often establish over western Mediterranean inautumn months (Bacchi et al., 1996). Cold fronts generated by this cycloniccirculation bring humid warm air from the south and/or the southwest, developingpre-frontal convective clouds and frontal stratiform clouds that impact the northern

coastline.

2.1 Radar error sources and adjustment procedures

The major factors affecting radar rainfall estimation in the study area are non-uniform vertical profile of reflectivity (VPR), orographic enhancement ofprecipitation, ground clutter, wavelength attenuation, uncertainty in the reflectivity-to-rainrate (Z-R) conversion and radar calibration stability effects. The radar rainfallalgorithm developed by Dinku et al. (2002) has been implemented here foradjustment of major radar error sources. This is a multi-component radar rainfallestimation algorithm that includes optimum parameter estimation and error

correction schemes associated with radar operation over mountainous terrain.Algorithm pre-processing steps include correction for terrain beam blocking,adjustment for rain attenuation, and interpolation of reflectivity data from polarradar coordinates to a fixed three-dimensional Cartesian grid (hereafter namedConstant Altitude Plan Position Indicator, CAPPI). The error correction schemesinclude also a simple but efficient approach to correct for the vertical variation ofreflectivity at short-medium ranges and a stochastic filtering approach for meanfield radar-rainfall bias (MFB) adjustment (associated with systematic and drift


5/28

errors in the radar calibration and biased Z-R relationship). MFB adjustment isbased on a uniform scaling factor considered representative of the whole regioncovered by the radar. It is defined as the ratio of the true to the radar estimatedmean area rainfall. For VPR adjustment, a correction procedure based on thefollowing steps is devised. First, within a radius of 50 km of a certain Cartesian

radar pixel location, neighboring pixels with rain data and associated blocking levelbelow an upper threshold are identified. Average rainfall is evaluated for theidentified pixel values at a reference CAPPI level (e.g., 1km) and for other CAPPIlevels (2km, 3km, etc.). The VPR correction factor of each upper CAPPI level (i.e.,2km, 3km, etc.) is defined as the ratio of their averages to the lower CAPPI levelaverages. Finally, the rainfall values of the pixels at the 2km and 3km CAPPI levelsare multiplied by the corresponding factors and are used in place of values of lowerCAPPI levels if these are blocked by terrain features. One advantage of thiscorrection scheme is that it is simple to implement. It also takes into account, tosome extent, the spatial variability of VPR since for each pixel the correction factoris computed from neighboring locations. Details about these correction procedures

and assessment of their significance in radar rainfall estimation can be found inDinku et al. (2002). Ground clutter removal was part of the raw radar data qualitycontrol system; hence, it was not part of the pre-processing steps describedherein.

Thirty-nine rain gauge stations within 80 km range from the radar (see Figure1, right panel) are available. Fifteen of those located near the radar (ranges100 km). Hereafter the resultsof the two scenarios will be named as: Scenario One or Two, and Adjusted or Nonadjusted (or Unadjusted) representing whether or not we used the adjustmentprocedures incorporated into the radar rainfall estimation algorithm. Note that the


6/28

unadjusted Scenario is based on radar observations corrected for radar beamocclusion and that the main difference between unadjusted and adjustedScenarios is the adjustment for mean field bias and VPR effects.

Figure 2 compares cumulative hyetographs obtained using raingauge and

radar data for OCT92 flood event. The figure highlights relatively minor radarestimation biases for the Scenario One, while relatively large underestimation(more than 50%)affects Scenario Two unadjusted radar estimates. For ScenarioTwo, adjustment is characterized by 14% overestimation. For a statisticalevaluation of the radar adjustment procedure two criteria are selected:

1) the mean relative error (MRE),

The values of MRE and FSE are reported in Table 3 for each radar rainfallestimation scenario and for each storm event, as well as for the ensemble of thestorm events. Reasonable performances are obtained by applying the adjustmentprocedure, particularly for Scenario One. The overall underestimation which affectsthe unadjusted estimates (particularly for Scenario Two, with 65% overallunderestimation) is greatly reduced for Scenario One (with almost 90% reduction)


7/28


8/28

Figure 2. Cumulative raingauge and radar hyetographs for basin-averagedrainfall for OCT92 flood event.

TOPSIDE OF FIGURE 3: FAISAL HOSSAIN


9/28

Figure 3. Optimal parameter set simulation of hydrograph for the OCT 1992

Storm using various input rainfall scenarios.

3. THE RAINFALL-RUNOFF MODEL

The rainfall runoff model TOPMODEL (Beven and Kirkby, 1979) was chosento simulate the rainfall-runoff processes of the Posina catchment. This modelmakes a number of simplifying assumptions about the runoff generation processesthat are thought to be reasonably valid in this wet, humid catchment with shortslopes and shallow soils. TOPMODEL is a semi distributed watershed model thatcan simulate the variable source area mechanism of storm runoff generation andincorporates the effect of topography on flow paths. The model is premised on thefollowing two assumptions:


10/28

1) that the dynamics of the saturated zone can be approximated bysuccessive steady state representations

2) that the hydraulic gradient of the saturated zone can be approximated bythe local surface topographic slope. As with many other TOPMODEL applications

the topographic index ln is used as an index of hydrological similarity,where a is the area draining through a point, and is the local surface slope.The use of this form of topographic index implies an effective transmissivity profilethat declines exponentially with increasing storage deficits. In this study, thederivation of the topographic index from a 20 m grid size catchment digital terrainmodel utilised the multiple flow direction algorithm by Quinn et al. (1991, 1995).

For the case of unsaturated zone drainage, a simple gravity-controlledapproach is adopted in the TOPMODEL version used here, where a verticaldrainage flux is calculated for each topographic index class using a time delaybased on local storage deficit. The generated runoff is routed to the catchment

outlet by a linear routing mechanism. Beven and Kirkby (1979) proposed that anoverland flow delay function and a channel routing function might be employedwithin the TOPMODEL structure by the use of a distance-related delay. Theoverland flow velocity uses a constant velocity parameter and allows a unique timedelay histogram to be derived on the basis of basin topography for any runoffcontributing area. Channel routing effects are considered using an approach basedon an average flood wave velocity for the channel network.

The watershed was discretized into 32 sub-basins and the corresponding rivergeometry identified to run the TOPMODEL at a fine spatial resolution. The outputsare the hourly average and local soil moisture deficits below saturation, and the

hourly discharge, separated into two components (surface runoff on the saturatedarea, and subsurface flow/groundwater discharge). The important parameters are:

the soil hydraulic properties at the surface: (m hr ), the lateral transmissivity

and , (m hr ), the vertical conductivity; the exponential decay rate of theseproperties with depth, m, (m); the maximum storage capacity of the root zone(SRMAX), (m), interpreted here as the soil moisture at field capacity, and TD, (hr),the time delay parameter used to simulate the vertical unsaturated drainage flux;

the runoff propagation parameters: RV, (m hr ), the overland flow velocity

parameter, and CHV, (m hr ), the channel flow velocity parameter. The modelwas run using basin-averaged rainfall input and considering homogeneous soils allover the catchment. Detailed background information of the model and applications

can be found in Beven et al. (1995). The model has been applied in the studyregion by a previous work ofBorga et al (2000).

To initialise the saturated zone, the relationship between the saturated zonestorage and the subsurface flow can be used if an initial discharge is known andcan be assumed to be the result of drainage from the saturated zone only. Thisassumption is used here to derive the initial average subsurface storage deficitfrom the first discharge of each event which is still on a recession curve.


11/28

4. THE GENERALISED LIKELIHOOD UNCERTAINTY ESTIMATION (GLUE)FRAMEWORK

The Generalised Likelihood Uncertainty Estimation (GLUE) framework ofBeven and Binley (1992) was used to assess the resulting uncertainty in thepredictions. The procedure has been described by Romanowicz et al. (1994, p.299) as "in essence a Bayesian approach to uncertainty estimation for nonlinear


12/28

hydrological models that recognises explicitly the equivalence, or nearequivalence, of different parameter sets.in the representation of hydrological

processes". GLUE is based on Monte Carlo simulation: a large number of modelruns are made, each with random parameter values selected from probabilitydistributions for each parameter. The acceptability of each run is assessed by

comparing predicted to observed discharges through some chosen likelihoodmeasure. Runs that achieve a likelihood below a certain threshold may then berejected as 'nonbehavioural'. The likelihoods of these nonbehaviouralparametrizations are set to zero and are thereby removed from the subsequentanalysis. Following the rejection of nonbehavioural runs, the likelihood weights ofthe retained runs are rescaled so that their cumulative total is one.

At each time step the predicted output from the retained runs are likelihoodweighted and ranked to form a cumulative distribution of the output variable fromwhich chosen quantiles can be selected to represent model uncertainty. In such aprocedure the simulations contributing to a particular quantile interval may vary

from time step to time step, reflecting the nonlinearities and varying time delays inmodel responses. In this study estimates of the 5th and 95th percentiles (, respectively, for the ith hour) were adopted to compute

the uncertainty bounds. While GLUE is based on a Bayesian conditioningapproach, the likelihood measure is achieved through a goodness of fit criterion asa substitute for a more traditional likelihood function. The likelihood value isassociated with a particular set of parameter values within a given model structure.The likelihood associated with a particular parameter value may therefore beexpected to vary depending on the values of the other parameters, and there maybe no clear optimum parameter set. The likelihood measure employed in this studyis the classical index of efficiency, E (Nash and Sutcliffe, 1970),

where, is the variance of errors and , the variance of observations. Thislikelihood measure is consistent with the requirements of the GLUE, as it increasesmonotonically as the similarity of behaviour increases.

There is a number of possible ways to define the behavioural threshold usedto refine the likelihood distribution for a set of simulations. If it is possible to make

some binary tests of model predictions against observed behaviour then this canbe used to distinguish behavioural and non-behavioural simulations. It is oftenimpossible to make such a clear-cut decision. In the majority of cases it may benecessary to impose some essentially arbitrary behavioural threshold. This is notnecessarily a drawback of the GLUE methodology. In setting a behaviouralthreshold, the modeller is able to make explicit his/her conditions for acceptance ofa model. Furthermore, changing the behavioural threshold should result in anarrower or wider range of 'acceptable' behaviours, which will in turn change the


13/28

estimated uncertainty which the modeller has about the model. This subjectiveelement is likely to be implicit in any evaluation of a proposed model; the GLUEprocedure helps to make this subjectivity explicit.

In previous studies, a behavioural rejection threshold has been chosen

arbitrarily as a given value of the likelihood function (Freer et al., 1996).Alternatively, a fixed proportion of the simulations can be retained according totheir ranked likelihoods so that the best n solutions are considered behavioural (thebest 50% for Beven and Binley, 1992). This best n% parameter sets can bedefined by virtue of the mean likelihood value of the parameter sets. The morequalitative the parameter sets are in a given best n% range, the higher is thereforethe mean likelihood value of those sets in that range. The advantage of using sucha best n% threshold spanning from 0% to 100% is that it removes the subjectivityassociated with the definition of a behavioural parameter set.

To implement the GLUE methodology, each parameter of TOPMODEL was

specified a range of possible values. Table 4 lists the four TOPMODEL parametersused for the GLUE and the ranges assigned to each. Constant (calibrated) valueswere used for three less sensitive parameters. The relative sensitivities of theparameters were identified by the Generalised Sensitivity Analysis of Spear andHornberger (1980). While the possibility of correlation between parameters exists,we have no apriori reason to assume any correlation structure among parameters,so a uniform sampling strategy was adopted.

Following the methodology outlined above, uncertainty associated with modelparameterization was assessed. A sequence of seven flood events, spanning aperiod of 1987-1995 and not including the validation sequence for which radar data

are available, was selected as the conditioning data set. Rainfall input for thisconditioning data set was provided by raingauge data. Model predictions werecarried out, and the model likelihood measure was calculated using the efficiencyindex. From the specified parameter ranges (Table 4), a large number ofsimulations were run that allowed to select 20,000 parameter sets characterised bya simulation efficiency greater than 0.3. The maximum likelihood value (E)achieved on the conditioning data set was 0.81. A number of fixed proportions ofthe simulations were retained according to their ranked likelihoods so that the bestn solutions were considered behavioural. These fixed proportions range from thebest 10% to the best 100% (i.e., all the 20,000 parameter sets with efficiencygreater than 0.3), at a 10% increment.

Uncertainty bounds conditioned on the calibration flood sequence using thebest 10% to the best 100% of the realizations were then propagated to the(validation) flood sequence (see Table 2) using rainfall estimates from theraingauge network and from the two radar rainfall estimation scenarios, both

Adjusted and Non Adjusted (see Table 3).


14/28

Figure 2. Cumulative raingauge and radar hyetographs for basin-averagedrainfall for OCT92 flood event.


15/28

Figure 3. Optimal parameter set simulation of hydrograph for the OCT 1992Storm using various input rainfall scenarios.

rainfall estimation uncertainty


16/28

4.1 Comparing predictive uncertainty given different rainfall inputs

For a better appreciation of how the GLUE procedure allows the modeller togain a better insight of the input-parameter uncertainty implications on runoffsimulation, we first present the conventional but questionable optimal parameterapproach. Assuming that the best parameter set identified from the calibration setof events using rain gauge data were stationary and input-type invariant, theoptimal parameter set validation is shown in Table 5 for the various inputscenarios. In figure 3 a more qualitative comparison is shown in terms ofhydrograph simulation for the OCT 1992 event. While we clearly observe that the

errors in rainfall from radar undermine the runoff simulation accuracy severalquestions remain. What if the optimal parameter set was non-behavioural for theradar rainfall scenarios? The adjusted radar for Scenario Two did not show muchimprovement in efficiency, could it be that it was just a bad realization for thissingle parameter? Are these single parameter set results repeatable under otherconditions? How much improvement in precision and accuracy of modelsimulations do we achieve by adjusting for errors in rainfall input? Answers to most


17/28

of the questions above can be found via the application of the GLUE method asdiscussed below.

The GLUE procedure provides a range of parameter-likelihood-weightedpredictions that may be compared with discharge measurements. It may be found,

as will be shown below, that the observations against which model predictions areto be compared may still fall outside the calculated uncertainty limits. If it isaccepted that a sufficiently wide range of parameter values has been examined,and the deviation of the observations is greater than would be expected frommeasurement error, then this would suggest that the model structure, or theimposed boundary conditions (including the rainfall input), should be consideredinadequate to describe the system under study. Two contrasting issues should beconsidered at this stage: if either the uncertainty limits are too narrow or the wholesimulation envelope is biased, then a comparison with observations suggests thatthe model structure is invalid; if, on the other hand they are too wide, then it maybe concluded that the model structure has little predictive capability. These two

contrasting issues can be considered synonymous with Precision and Accuracy. Anarrower range of uncertainty limits can be considered as a mark of higherprecision while the property of the percentage of observed runoff beingencapsulated by these uncertainty limits can be taken as a measure of accuracy.These observations outline the basic structure for a framework aimed at a directquantitative comparison of predictive uncertainty between model responsesobtained from competing rainfall estimates. In that framework, the wideness of theuncertainty limits and the percentage of observations included in the limits shouldbe evaluated jointly, i.e, precision and accuracy of a model are conjugateproperties. Furthermore, and since any behavioural threshold is arbitrary in somesense, the comparisons should be carried out for a range of behavioural threshold,

thereby essentially eliminating the subjectivity associated with a parameter set'sacceptability.

Given that the same conditions are applied throughout GLUE concerning: (1)the sample of a priori parameter set used; (2) the choice of likelihood measure;and (3) the discharge observations used in the calculation of the likelihoodmeasure and for the comparison with the uncertainty limits, the analysis of resultsobtained by conditioning the model with a reference rainfall (derived herein from adense raingauge network) and propagating the uncertainty bounds by usingcompeting rainfall inputs offers a convenient way to quantify the impact of rainfallinput errors on runoff modelling uncertainty. The dichotomous nature of accuracyand precision levels of a model imposed by the quality of input can also beunderstood better. This is carried out by comparing the range of likelihoodweighted predictions, obtained by using alternative competing algorithmicstructures for radar rainfall estimation, with the observations. In accordance with aprevious application of GLUE by Borga (2001), two statistics are defined as followsfor a given behavioural threshold:


18/28

where is the number of times in which measured flow fallsoutside the calculated uncertainty limits, E.R. is the Exceedance Ratio signifyingthe ratio of exceedances for radar input to gage input; and U.R. is the UncertaintyRatio signifying the ratio of the aggregate wideness in runoff simulation uncertainty

for radar input to gage input. The is the total number of time steps forsimulation of storm events, with jpertaining to a given time step, and qsim is thesimulated discharge (or runoff) with its superscript signifying the type of rainfallinput.

It is noted again that the two statistical measures presented herein need to beassessed jointly as individual interpretation can be erroneous due to their inherentcompeting characteristics. Also, if the gage rainfall is assumed to be the ground"truth" of rainfall, then a closer look at the denominator of the E.R and U.R inEquation (4) will reveal that it actually signifies the impact of parameter uncertaintyon runoff prediction. The numerator can then be considered to be representative ofthe combined impact of model parameter uncertainty and rain input error on runoffsimulation uncertainty. Hence the two ratios are essentially a normalization ofcombined input-parameter uncertainty to the corresponding parameter uncertainty.This allows a more direct comparison and analysis with changing either behavioralthresholds or uncertainty levels associated to rainfall estimation scenarios. A ratiovalue nearing one would indicate the runoff predictive uncertainty for the givenradar input mimicking that from a dense raingauge network.

Figures 4 and 5 shows the uncertainty bounds (with use of all the 20,000parameter sets with efficiency greater than 0.3) propagated for the OCT 1992 flood

event by using different rainfall inputs. Examination of the simulations obtained byusing raingauge data (Figure 4) demonstrates that large uncertainties can beassociated with model predictions even when using the reference rainfall.


19/28

However, the uncertainty bounds enclose the observed time series relativelywell, and only a few flow observations can be found outside of the uncertaintyenvelopes, indicating deficiencies in the data and/or model structure. Comparisonof uncertainty bounds for various radar rainfall scenarios show that a widening(narrowing) of the predictive uncertainty is associated to the magnification

(reduction) of the rainfall bias due to radar error adjustment (see Table 3 andFigure 5). This provides evidence that, contrary to conventional wisdom, thatappropriate adjustment of the radar estimates may (rightly) increase predictiverunoff uncertainty, at least when adjustment is associated to the correction of anegative bias in raw radar estimates. For the adjusted case of Scenario One, weobserve that simulation uncertainty is very similar to that for gauge. This providesstrong evidence that use of radar scans close to the ground and adjusted for errorscan characterise the runoff transformation as accurately as a dense gaugenetwork. Note that, for this event (OCT 1992), rainfall estimates from unadjustedScenario Two are affected by more than 50% negative bias (underestimation),while adjustment results in 14% overestimation. Examination of the simulations

obtained by using unadjusted Scenario Two radar estimates show that theuncertainty bounds are considerably lower compared to the use of adjustedestimates, or to the use of Scenario One radar estimates. A global negative biashas been introduced, with the consequence that all discharge observations falloutside of the uncertainty envelopes. Radar error adjustment allows to reduce thisglobal negative bias and results in a widening of the predictive uncertainty. Thiscauses the bounds to bracket the storm hydrograph realistically.

Figure 6 shows the Exceedance Ratios (E.R.) and Uncertainty Ratios (U.R.)evaluated for different behavioural thresholds for all the five storms. Severalfeatures are worth noting in this figure. There is a distinct uncertainty structure

associated with the use of radar rainfall estimates from the two different scenarios.The significance of radar rainfall adjustment difference between adjusted andunadjusted scenarios is clearly higher for Scenario Two. The gradients (hencesensitivity to parameter uncertainty) of E.R and U.R are also higher for ScenarioTwo. This indicates that the input error-parameter uncertainty interaction is higherwhen radar measurements are affected additional error (radar range and complexterrain). Beyond a 40% parameter uncertainty (best 40%), the radar rainfalladjusted for errors at low scans yield very similar runoff simulation uncertainty asgage (precision wise - the U.R gradually converges to 1.0).


20/28


Figure 4. Propagated uncertainty bounds for OCT92 flood event with rainfallinput from raingauge data.


21/28


Figure 5. Propagated uncertainty bounds for OCT92 flood event with rainfallinput from radar rainfall estimates.


22/28


Figure 6. Runoff prediction uncertainty assessment in terms of UncertaintyRatio and of Exceedance Ratio for various behavioural thresholds.

Radar rainfall errors and their adjustment exhibit a much more pronounced

effect for Scenario TWO, where the relatively small values of U.R. for unadjustedestimates and the closer-to-one values for adjusted estimates mirror thecharacteristics of the radar rainfall bias (highly negative and close to zero,respectively) and of its propagation through the rainfall-runoff transformation.Borga et al. (2000) have shown that radar biases magnify through the flood eventhydrological modelling, so these effects are not unexpected. Runoff predictionuncertainty is dominated by radar rainfall uncertainty when using Scenario Two


23/28

unadjusted estimates and evenly influenced by radar and parameter uncertaintywhen using Scenario Two adjusted estimates.

Examination of Figure 6 shows also that there is a tendency for U.R. (E.R.) todecrease (increase) with increasing the number of behavioural realizationsretained in the analysis. The effect of varying the behavioural threshold is to modifythe uncertainty bounds; by setting a stricter threshold the uncertainty bounds arenarrower. This effect suggests that the uncertainty term related to radar rainfallerrors reduces its influence with increasing the model parameter predictiveuncertainty. For Scenario One, the larger than one U.R. values for the lowerthresholds (10-40%) indicates that the optimum model parameter set determinedfrom raingauge input is inadequate for adjusted radar estimates.

The increase of E.R. statistics with increasing the number of behaviouralrealizations indicates that relatively less flow observations are falling within thepredictive uncertainty bounds using radar input than by using raingauge input. Thecombined analysis of both U.R. and E.R. statistics is useful to understand thesignificance of Scenarios One and Two adjusted radar estimates. Examination ofE.R. statistics for these cases show that adjustment ensures that the observed flowdata are enclosed by the uncertainty bounds. However, this is obtained at the priceof increasing considerably the wideness of the uncertainty bounds. These areincreased by a factor of 2.1 - 1.8 with respect to the unadjusted counterparts.Consequently the adjustment of radar rainfall errors improves the accuracy ofmodelling at the cost of losing precision.

The global picture emerging from this analysis demonstrates that runoffprediction uncertainty given radar rainfall input cannot be simply partitioned intotwo additive terms: the term related to model parameter uncertainty and the termrelated to the propagation of radar rainfall uncertainty. Rather, radar rainfalluncertainty - particularly when the rainfall bias term is important - act in a highlynon-linear sense on the model parameter uncertainty, by either magnifying orreducing it according to the nature of the rainfall estimation bias. The aboveconsiderations should not imply that unbiased radar rainfall estimates lead tounitary U.R.: random errors still play a role and indeed examination of U.R. statisticfor Scenario One shows (between best 0% to 40% parameter sets) that its value is

greater than one even when using radar estimates affected by slightly negativebias. In terms of precision and accuracy of a model, we may summarize that radarrainfall unadjusted for errors can potentially render the simulations inaccurate nomatter what degree of precision the model is capable of simulation.

6. CONCLUSIONS


24/28

The GLUE procedure has been used in this study as a means of hydrologicalmodel comparison using different rainfall inputs, provided by a dense rain gaugenetworks and by radar estimates according to various processing scenarios. Theproposed analysis framework allows to evaluate both the wideness of theuncertainty limits and the percentage of observations included in the limits, with

varying the behavioural threshold. This helps to assess the impact of radar rainfallerrors on the output of a hydrological model previously conditioned using rainfalldata from a dense raingauge network. The dual issues of precision and accuracycan be both addressed by the GLUE approach. The evaluation is reported in termsof both structural validity and predictive capability of the resulting model output.

Several features are worth summarising here. Runoff predictions obtained byusing the adjusted radar rainfall for Scenario One (with less errors due to variabilityof reflectivity with height) are similar to those obtained based on basin averagedgage rainfall. The impact of the adjustment is in improving the capability of themodel to bracket the flow observations (and thus improve accuracy of the model)

with increasing only slightly the wideness (precision) of the uncertainty bounds.Runoff simulations appear very sensitive to the impact of errors related tovariability of reflectivity with height, which dominate the radar error structure forScenario Two. The runoff model defined by using unadjusted radar estimates isstructurally invalid due to poorly defined input data.

Use of the type of analysis proposed here provides a clear view of the relativeeffects of input and parameter uncertainty upon model output and indeed is avaluable tool in analysing and ranking the sources of predictive uncertainty. It ishoped that, being explicit about the levels of uncertainty, limitations within the radarprocessing algorithm and the hydrological model can be improved upon, or

additional data can be acquired in order to reduce the predictive uncertainty.

Natural extensions of this work include: (i) the consideration of radar errorstructures and adjustment algorithms different from those used in this analysis; (ii)the implementation of the analysis framework for continuous hydrologicalsimulation instead of event-based flood modelling as presented here; (iii)examining the effects of different likelihood functions and parameter samplingranges within the GLUE methodology. These extensions will further improve ourunderstanding of the predictive uncertainty of radar-based runoff prediction.Development of a proper methodology for the assessment of these uncertainties isessential to prevent the end user (who may not possess a full understanding ofradar error structure and have a more 'conventional' optimal parameter approachto model application) from misinterpreting results or believing that modelpredictions are precise and/or accurate. As has been illustrated here, they are not.The truth about hydrologic models is clearly more complex than that.


25/28


26/28

Borga M. 2001. Use of radar rainfall estimates in rainfall-runoff modeling: Anassessment of predictive uncertainty. In Proceedings of Fifth InternationalSymposium on Hydrological Applications of Weather Radar - Radar Hydrology.Kyoto, Japan: 451-456.

Borga M. 2002. Accuracy of radar rainfall estimates for streamflow simulation.Journal of Hydrology267(1/2):26-39.

Borga M, Anagnostou EN and Frank E. 2000. On the use of real-time radarrainfall estimates for flood prediction in mountainous basins. Journal ofGeophysical Research 105(D2): 2269-2280.

Borga M, Tonelli F, Moore RJ and Andireu H. 2002. Long term assessment ofbias adjustment in radar rainfall estimation. Water Resources Research (In press).

Carpenter TM, Georgakakos KP, Sperfslagea JA. 2001. On the parametric

and NEXrad-RADAR sensitivities of a distributed hydrological model suitable foroperational use. Journal of Hydrology253: 169-193.

Dinku T, Anagnostou EN and Borga M. 2002. Improving radar-basedestimation of rainfall over complex terrain. Journal of Applied Meteorology. Inpress.

Freer J, Ambroise B and Beven KJ. 1996. Bayesian estimation of uncertaintyin runoff prediction and the value of data: An application of the GLUE approach.Water Resources Research 32: 2161-2173.

Georgakakos KP, Sperfslage JA and Guetter AK. 1996. Operational GISbased models for NEXRAD radar data in the U.S. Proceedings of the InternationalConference on Water Resources and Environmental Research, 29-31 October,1996, vol. 1, Water Resources and Environmental Research Center, KyotoUniversity, Kyoto, Japan: 603-609.

Grayson RB, Moore ID and McMahon TA. 1992. Physically-based hydrologicmodelling. 2. Is the concept realistic? Water Resources Research 28: 2659.

James WP, Robinson CG and Bell JF. 1993. Radar-assisted real-time floodforecasting. Journal of Water Resources Planning and Management119(1): 32-44.

Joss J and Lee R. 1995. The application of radar-gauge comparisons tooperational precipitation profile corrections. Journal of Applied Meteorology 34:2612 - 2630.

Nash JE and Sutcliffe JV. 1970. River Flow forecasting through conceptualmodels, 1, A discussion of principles. Journal of Hydrology10: 282- 290.


27/28

Ogden FL, Sharif HO, Senarath SUS, Smith JA, Beck ML and Richardson JR.2000. Hydrologic analysis of the Fort Collins, Colorado, flash flood of 1997.Journal of Hydrology228: 82-100.

Quinn PF, Beven KJ, Chevallier P and Planchon O. 1991. The prediction of

Hillslope flow paths for distributed hydrological modelling using digital terrainmodels. Hydrological Processes 5: 59 - 79.

Quinn PF, Beven KJ and Lamb R. 1995. The ln(a/tanb) index: how to calculateit and how to use it in the TOPMODEL framework. Hydrological Processes 9: 161 -182.

Romanowicz R, Beven KJ and Tawn J. 1994. Evaluation of predictiveuncertainty in non-linear hydrological models using Bayesian approach. In BarnettV. and Turkman K. F. (eds) Statistics for the Environment. II. Water RelatedIssues. John Wiley: Chichester; 297 - 317.

Schell GS, Madramootoo CA, Austin GL and Broughton RS. 1992: Use ofradar measured rainfall for hydrologic modelling. Canadian AgriculturalEngineering34(1): 41-48.

Sempere-Torres D, Corral C, Raso J and Malgrat P. 1999. Use of weatherradar for combined sewer overflows monitoring and control. Journal ofEnvironmental Engineering(ASCE). 125(4): 372- 380.

Sharif HO, Ogden FL, Krajewski WF and Xue M. 2002. Numerical simulationsof radar rainfall error propagation. Water Resources Research 38(8).

Spear RC and Hornberger GM. 1980. Eutrophication in peel inlet, II,Identification of critical uncertainties via Generalized Sensitivity Analysis. WaterResources 4: 43 - 49.

Vieux BE and Bedient PB. 1998. Estimation of rainfall for flood prediction fromWSR-88D reflectivity: A case study, 17-18 October 1994. Weather andForecasting13: 407-415.

Winchell M, Gupta HV and Sorooshian S. 1998. On the simulation ofinfiltration- and saturation-excess runoff using radar-based rainfall estimates:

effects of algorithm uncertainty and pixel aggregation. Water Resources Research34(10): 2655-2670.

Woods R, Sivapalan M, and Duncan M. 1995. Investigating the representativeelementary area concept: An approach based on field data. HydrologicalProcesses 9(3/4): 291.

Weinberg AM. 1972. Weinberg, Science and trans-science. Minerva, 10: 209.


28/28

Documents

Hydrological Model Sensitivity to Parameter and Radar Rainfall Estimation Uncertainty