The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al.

The influence of rating curve uncertainty on flood inundation predictions Florian Pappenberger*, Patrick Matgen**, Keith J. Beven*, Jean-Baptiste Henry†, Laurent Pfister**, Paul de Fraipont† * Environmental Sciences, Lancaster University, UK (corresponding author: [email protected]) ** Centre de Recherche Public-Gabriel Lippmann, Luxembourg †

Service Régional de Traitement d'Image et de Télédétection, Strasbourg University, France

Abstract The uncertainty of rating curves is well explored and understood in current literature. However, most estimations and methods are usually accompanied by a warning not to extrapolate the rating curve beyond a certain range. This is very often impossible for flooding events. Nevertheless, the uncertainty in using these rating curves for flood inundation models is usually ignored. In this paper we investigate the effect of uncertainty of rating curves on flood inundation predictions. The rating curve has been interpolated with two different equations, which are commonly used. The first method is based on a polynomial representation and the second method interpolates data points with the help of the Manning equation. A set of rating curves which represent the system equally well has been derived via the Generalized Likelihood Uncertainty Estimation (GLUE) and the Multicomponent Mapping (Mx) methodology. The multiple rating curves have been used as upstream boundary of the one dimensional unsteady flow routing model HEC-RAS. The Manning roughness, as well as the model input, have been considered as uncertain and varied within a Monte Carlo framework. The model has been evaluated on inundation information retrieved from three different remote sensing sources. It has been shown that the boundary condition and thus the rating curve parameters have a significant impact on inundation predictions. The impact of the boundary condition depends on the hydrograph characteristics. Under some circumstances the quantification of the input uncertainty is more important than inundation model parameters.

Introduction All flood inundation models work with some form of discharge, water level or a combination of both as upstream, downstream and/or internal boundaries. Uncertainties in the boundary condition are very often neglected. No study to the knowledge of the authors seems to investigate the uncertainty of discharge as the upstream boundary of a flood inundation model Most modellers seem to assume an upstream discharge hydrograph as a given quantity. This is surprising, as it is known that input can dominate flood inundation behaviour and most discharge records are derived from rating-curves which convert measured water levels to a flow (Herschey, 1995). The uncertainty in these rating curve models is well studied and documented (Schmidt, 2002). Several sources can be identified which influence the accuracy of rating curves (modified from the extensive literature review by Pelletier, 1988): Considerable uncertainty exists in the sampling of the cross-sectional area. It might be very difficult to measure in certain locations of the river. Modifications of the river bed after flood events are common. In many cases velocity is combined with the cross-section characteristics to derive discharge. However, velocity fluctuates in a very complex pattern in time and space. Furthermore, the instrument which is used is subject to uncertainties especially because it has normally been calibrated by towing through a still water tank neglecting important processes such as turbulence and water flow from different angles. The measurements acquired have to be combined, very often including assumptions of the velocity profile, and these may not always be completely valid. However, it is not only the discharge determination that is subject to various uncertainties but also the water level, which is supposed to be representative for the time of the measurement and cross-section. In many cases the relationship between discharge and water level will be derived by multiple gauges, which support the later analysis by e.g. measuring water surface slope. Nevertheless in most cases such a determination will be too expensive. This highlights a further problem, which is in the type of equation (model structure) which is used to relate the two variables. A number of equations exist, of which some are physically based and some empirical. Depending on the data availability it might sometimes be difficult to distinguish between the different model structures for the rating curves within the range of observations (Beven, 2002). These models will often have difficulties in modelling all hydraulic processes such as hysteresis, which will cause different discharge determinations at the same stage and might be difficult to be distinguished in light of all the other uncertainties. Finally, the regression method will affect the accuracy and uncertainty of the curve. 1

The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al. An uncertainty study of a flood inundation model should not only include considerations of the upstream boundary. Several studies have highlighted the problem of determining effective surface roughness (see e.g Aronica et al., 1998; Beven and Carling, 1992; Matgen et al., 2004; Romanowicz and Beven, 2003). In this paper we enhance previous studies on parameter uncertainty with considerations for the input uncertainties for the one dimensional unsteady flow model HEC-RAS (U.S. Army Corps Engineers). The Generalized Likelihood Uncertainty Estimation (GLUE) (Beven and Binley, 1992) method is used on data of the Alzette catchment (Luxembourg), which will supplement the study of Matgen et al .(2004). In the first part of this paper we propose a method to acknowledge rating-curve uncertainty with restricted available measurements. The results of this method are presented thereafter. This is followed by an introduction of the model setup and all available data. Then, the GLUE methodology and the evaluation measured used will be presented. Finally the results will be discussed and conclusions drawn.

Quantification of uncertainty in rating curves This section gives a brief literature review in the treatment of uncertainty in rating curves. It further describes the method applied in this paper. Numerous publications can be found that quantify the uncertainty in stage-discharge rating curves (Herschey, 2002 ). The difficulty of the problem is increased when, during flooding, the rating curve is extrapolated beyond the measurement range (which is very often below overbank conditions). The flow regime changes dramatically as soon as the river is overbank, which is neglected in most extrapolated curves (Knight and Shiono, 1996; Schmidt, 2002). Several models have been published for the rating curve. Amongst them the most widely used are the power law equations (Herschey, 1998).

Q = c(a + h)α Equation 1 Q: h: a,c,α:

low [m3s-1] water level [m] calibration parameters

and the Manning equation:

Q= R(h): h: n: a: S:

1 R(h) a S 0.5 n

Equation 2

Hydraulic Radius Water level Manning Surface roughness exponent of the Hydraulic Radius (measured between ~0.5 and ~0.7 by Manning, 1891) Slope

It has to be mentioned that the effective Manning surface roughness should change for overbank flow (Knight et al., 1989), which is neglected in this representation, because it cannot be supported by available data in this case. One issue of the Manning equation is that additional measurements are required (slope, hydraulic radius), which are not always available. The power law equation and the Manning equation will be used in this study. They will thus also be applied to extrapolate for flood levels, which is similar to the slope area method (Dawdy et al., 2000). In this approach the two equations are used as equally valid representations of this extrapolation, because it will be not possible to distinguish between the different model structures (Beven, 2002; Beven, in press). This is very much in line with the suggestion of DeGagne (DeGagne et al., 1996) which advocates a multiple model approach for extrapolation. This non-identifiability is driven by the poor availability of data. Rantz et al. (1982) argue that ‘the rating should not be extrapolated beyond twice the largest measured discharge except as a last resort’. However, flood inundation is the prediction of extremes, therefore this advice might have to be ignored otherwise no predictions would be made at all. The limited amount of data also forces the use of a simple regression method. For the site under study, only eight data points could be found on records, which have been used to quantify the rating curve. In this example, the largest 2

The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al. measured point on the rating curve is at 1.93 metres and the larges water level measured, for the flood which is investigated here, is at 2.95 meters. The errors reported in literature for the discharge measurements are from 1.8 % to 8.5 % with a large amount around 6% (see literature quoted in Leonard et al., 2000; Schmidt, 2002). The errors which can are quoted for the water level are around 3 to 14 mm, although this seems to be very low values (see literature quoted in Schmidt, 2002). Due to the limited amount of data an ad hoc approach to fit the curves has been seen as the most suitable. The approach is a modified version of the Multicomponent Mapping by Pappenberger and Beven (Pappenberger and Beven, in press) in which uncertainty associated with data points has been expressed by pyramidal frustums. A pyramidal frustrum of a height of one has been assigned to each data point (figure 1).

Figure 1: Illustration of pyramidal frustrum at data points The height can be seen as a two-dimensional fuzzy membership function and thus the minimum distance from a line to the middle point gives a membership value. These values have been chosen according to the literature review and communication with the institute responsible for taking these measurements. Especially, the error in the water level measurement is considerable higher than the values found in literature. However, we found the latter to be a realistic value for this river. Parameter combinations have been sampled by Monte Carlo sampling and the membership values at each data point evaluated. A parameter set has been accepted to fit the current data set if a multiplication of all individual memberships gives a value greater than 0. The combined membership value can be seen as a likelihood measure for this particular realisation. The procedure described above is termed the Generalized Likelihood Uncertainty Estimation (GLUE) (Beven and Binley, 1992). It presumes that there may be many parameter sets giving results that are acceptably consistent with the observations (behavioural models). From this concept a likelihood measure can be computed for each parameter set which then can be used to compute uncertainty bounds.

Results of Rating Curve evaluation The uncertainty in the rating curve has been evaluated as discussed previously. Figure 2 and 3 show performance measures fro parameters which have been accepted as behavioural.

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The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al.

Figure 2 Scatter plot of parameters of polynomial equation (see equation 1) As can be seen from these plots, equation 1 is mainly influenced by parameter a and c. Parameter α seems to be insensitive. All parameters show a large range of equifinality.

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The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al.

Figure 3 Scatter plot of parameters of Manning equation (see equation 2) The Manning roughness and the slope have a clear relationship, which could be already seen from equation 2. The Manning equation has been developed for uniform flow and it cannot be expected to hold fully for non-uniform conditions. Therefore, it will be very difficult to quantify an exact value and thus larger uncertainty regarding the Manning roughness remains. All parameters have been estimated on flow conditions, which are lower than the actual maximum discharge level. Hence, parameter sensitivity and uncertainty will most probably not hold at extrapolation (see introduction). However, the latter is frequently done. Figure 5 and 4 show the effects and ranges such an extrapolation has.

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The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al.

Figure 4: 5 and 95 percentiles of rating curve and measured inflow hydrograph for the Manning Equation. In the top right corner is the rating curve for this hydrograph. The time of the evaluation data is marked on the graph. The maximum difference between the two percentiles is 5.6 m3s-1 or ~10%.

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The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al.

Figure 5: 5 and 95 percentiles of rating curve and measured inflow hydrograph for the power law equation. In the top right corner is the rating curve for this hydrograph. The time of the evaluation data is marked on the graph. The maximum difference between the two percentiles is 8.9 m3s-1 or ~7%. Figure 5 and 4 show a considerable range of uncertain inflow. This range is naturally larger at peak discharges than it is within the range of the measurements. Both models have similar peak discharges, however, an uncertainty between 7-10% of the total discharge is introduced in the extrapolated range. The power law equation does show uncertainty bounds which are less wide than the Manning equation. This can have a significant impact apart from the ones investigated in this paper. In many countries flood warning levels are attached to certain water levels or discharges and in such cases an uncertainty of 10% may be considered as high to decision makers. This already indicates the importance of the uncertainty in rating curves for local values and in the following section the impact on a more global scale will be investigated.

Description of the study area and model implementation The study area is the floodplain of the Alzette river (Luxembourg) with an approximate length of 10 km. The comprehensive data set includes pre-flood and flood SAR images, continuous water level measurements upstream and downstream of the river reach, surveyed high water marks and GPS control points of the maximum flood extent. A set of photographs taken during the flooding event is also available. Two radar images, from ERS-2 and ENVISAT SAR instruments, cover ,acquired during the rising limb and at the peak discharge respectively, cover the flooded area at two distinct stages of the event (a further detailed description which includes the pre-processing is given by Matgen et al., 2004) The geometry of the study region has been represented by 74 cross-sections which have been surveyed beforehand. The model has been simplified and only three different roughness values used to represent the left floodplain, the channel and the right floodplain. It would have been desirable to include further information of geometric uncertainty as it can have a major impact on flow paths and inundation extent (Aronica et al., 1998). However it was seen as beyond the scope of this study. 7

The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al.

Description of the likelihood measures used within the Generalized Likelihood Uncertainty Estimation framework (GLUE) The different inundation data sets can be used for model calibration. However, each of these sets has its own characteristic and is derived in different ways. Therefore, it is necessary to specify the application of each individually. All data sets have in common that they have been used in a fuzzy membership framework. In this way one cannot take only account of the measurement errors but also for the commensurability errors (Beven, 2002) (although the latter might be difficult to quantify). SAR and ERS The use of SAR imaging for the evaluation of flood inundation has been widely accepted (Horritt and Bates, 2003). However, the classification of flooded and non flooded pixels is not always certain. Wind effects, vegetation, urbanisation and wet-lands make an accurate delineation difficult. In this paper a simple threshold approach on the grey scale of the pixels has been used to quantify a fuzzy membership function of flooding along each cross-section which expresses the uncertainty of flooding extent and images geometric resolution (see Matgen et al., 2004). Figure 6 illustrates this methodology which has been also applied in similar ways by Pappenberger et al (in press) and Romanowicz et al. (2003).

Figure 6: Membership function which reflects the probability of pixels being flooded. This is derived via thresholds from the original data. A membership value is computed according to the modelled water level. All individual membership values at each cross-section can be combined to a global performance value by multiplication.

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The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al. Photographs The photographs from three different points in time have been mapped back onto the Digital Terrain Model. Similar to the SAR, a fuzzy membership function has been derived. All performance measures, their derivation and theory have been documented in detail by Matgen et al. (2004).The research presented here concentrates on input uncertainty. The GLUE methodology requires the specification of a value which distinguishes between the behavioural and non behavioural sets of models. In this approach, it has been decided to linearly interpolate the water levels from the upstream to the downstream boundary at the time of the SAR, ERS and photographs. For these interpolated water levels the performance measure has been computed and used to distinguish between behavioural and non behavioural model realisations. All performance measures have been normalized and multiplied to give an overall measure of fit.

Model Results The following figure (figure 7 and 8) shows the scatter (‘dotty’) plots with respect to the total performance measure.

Figure 7: Dotty plots of model with power law equation as input

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The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al.

Figure 8: Dotty plots of model with Manning equation as input In figure 7, the parameters a and c show an identifiability and clearly dominate the overall model performance, whereas all other values indicate model equifinality. This behaviour is as expected from the initial plots (figure 3). In figure 8 the roughness of the channel and parameter a influence the overall model results and all other parameters show a one dimensional equifinality. Note that parameter a would not have been identified as important from the previous analysis. However, the Slope and the Manning roughness are highly correlated and therefore, the main driving source is parameter a. The difference in the sensitivity of the Manning channel roughness rating is explained by the initial uncertainty in the rating curves. Although, only a small percentage difference of uncertainty in the peak flow has been detected, it results in different parameter sensitivities. The Manning roughness of the inundation model does become more important at larger uncertainty in the inflow because it counteracts to achieve acceptable model results. Thus is it not surprising that the channel roughness shows sensitivity to the Manning equation input. The significance of the Manning channel roughness is in line with previous studies (Bates et al., 2003; Pappenberger et al., in press; Romanowicz and Beven, 2003). If the results are broken down to a more local scale a different picture emerges. In the following table the Spearman Rank correlation coefficient between the water level at the time of the three measurements and the first ten, as well as all cross-sections, has been computed. The first ten sections are the initial ten cross-sections below the upstream boundary. Moreover, the relationship to the friction parameters is shown. The table is focused on the Manning equation which shows more sensitivity to the channel friction and is therefore of higher importance in respect to this analysis. Table 1: Spearman Rank Correlation coefficient between the inflow (at the time of the evaluation data and for Manning equation) and the performance of the first ten and all cross sections (summation of membership values). Moreover, the relationship to the surface roughnesses is shown. Eval. data ERS-2 EnviSat Photographs Cross 1-10 All 1-10 all 1-10 All section 10

The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al. Inflow Channel friction Floodplain left friction Floodplain right friction

0.836 0.034

0.834 0.193

0.37 0.008

0.06 0.960

-0.828 -0.002

0.106 -0.012

-0.137

-0.057

0.437

0.039

-0.010

0.008

-0.323

-0.232

-0.764

0.186

-0.099

0.006

The Spearman Rank correlation seems to reveal a stronger relationship for the first ten sections than to all sections for the ENVISAT and Photographs. However, for the early evaluation data (ERS-2) such a pattern cannot be extracted. The first ten sections and all sections give similar results. This is not surprising, considering that the floodplains fill within this period, starting from the upstream end. On the one hand is the system mostly driven by the amount of inflow in this phase; on the other hand the inundation of the upper sections controls the overall performance. Also interesting is the apparent reversion of the relationship between input and performance of the initial cross sections, which is expected at the falling limb of the hydrograph. In this phase the floodplains are emptying rather than filling. The relationship between the surface roughnesses and the inundation performance varies. The channel friction seems to be most important for the overall performance close to the peak, whereas the floodplain roughnesses dominate the initial ten sections. This behaviour will depend heavily on the geometry of the reach and will most probably change from one event to another. The reader should be reminded that the methodology applied does not reveal high dimensional interactions and thus can give only an indication of existing relationships. The analysis presented in this paper did not provide reasons for the rejected of either of the two rating curve models. Therefore, both approaches have to be considered if uncertainty maps of flood information are drawn. Figure 9 does show the probabilities of flood inundation for the time of the ENVISAT. As expected, the uncertainties in the inflow region are very high.

Figure 9: Uncertainty percentiles of the Alzette floodplain for the time of the ENVISAT picture

Conclusions Uncertainty in the input to inundation models is very often neglected. In this paper we introduce a methodology which is based on the Generalized Likelihood Uncertainty estimation framework (Beven and Binley, 1992) and the Multicomponent Mapping (Pappenberger and Beven, in press) methodology to account for input uncertainty. The Manning equation and a power law representation have been used to interpolate and extrapolate the rating curve. Both equations gave a large uncertainty of peak flow between 7-10%. The analysis presented in this paper showed a significant impact of the input uncertainty on local and global scale inundation predictions for all evaluation data. However, any impact on the full study region depends largely on the timing. For the rising hydrograph limb the impact is much larger than at peak flow or on a falling limb. This variance in importancy can be also seen on the Manning surface roughnesses. This parameter does show sensitivity depending on the uncertainty in the input. The calibration of this parameter becomes more important if the uncertainty in the input is lower.

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The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al. The radar images and the photographs have been successfully provided information to extract the differences in behaviour of the Manning Surface roughness of the inundation model and the input uncertainty. In future applications, these images may support a regionalisation of the roughness coefficients (Henry et al., 2004) This analysis stresses further that the boundary conditions are important and influence model results to a large degree. Therefore, this uncertainty should not be neglected. The estimation procedure of this uncertainty should be developed further and a more rigorous mathematical treatment may be required (Smith et al., 2004).

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The influence of rating curve uncertainty on flood inundation predictions, Pappenberger et al.

References Aronica, G., Hankin, B. and Beven, K.J., 1998. Uncertainty and equifinality in calibrating distributed roughness coefficients in a flood propagation model with limited data. Advances in Water Resources, 22(4): 349-365. Bates, P.D., Marks, K.J. and Horritt, M.S., 2003. Optimal use of high-resolution topographic data in flood inundation models. Hydrological Processes, 17(3): 537-557. Beven, K.J., 2002. Towards an alternative blueprint for a physically based digitally simulated hydrologic response modelling system. Hydrological Processes, 16(2): 189-206. Beven, K.J., in press. A Manifesto for the equifinality thesis. Journal of Hydrology, --(--): --. Beven, K.J. and Binley, A., 1992. The Future of Distributed Models: Model Calibration and Uncertainty Prediction. Hydrological Processes, 6: 279-298. Beven, K.J. and Carling, P., 1992. Velocities, roughness and dispersion in the lowland River Severn. In: P. Carling and G. Petts (Editors), Lowland Floodplain Rivers. John Wiley & Sons, New York. Dawdy, D.R., Lucas, W. and Wang, W.C., 2000. Physical Basis of Stage-Discharge Ratings, Eight International Symposium on Stochastic Hydraulics. Balkema, A.A., Beijing, China, pp. 561564. DeGagne, M.P.J., Douglas, G.G., Hudson, H.R. and Simonovic, S.P., 1996. A decision support system for the analysis and use of stage- discharge rating curves. Journal of Hydrology, 184(3-4): 225-241. Henry, J.B., Matgen, P., Tholey, N., Pfister, L., L., H. and de Fraipont, P., 2004. SAR PCA-based segmentation for hydraulic patterns identification, 5th European Conference on Synthetic Aperture Radar EUSAR 2004, Ulm, Germany. Herschey, R.W., 1995. Streamflow measurement. E & FN Spoon, London. Herschey, R.W., 1998. Flow measurement. In: R.W. Herschey (Editor), Hydrometry: principles and practices. Wiley, Chichester, pp. 9-83. Herschey, R.W., 2002. The uncertainty in a current meter measurement. Flow Measurement and Instrumentation, 13: 281-284. Horritt, M.S. and Bates, P.D., 2003. Evaluation of 1D and 2D numerical models for predicting river flood inundation. Journal of Hydrology, 268(1-4): 87-99. Knight, D.W. and Shiono, K., 1996. Channel and Floodplain Hydraulics. In: M.G. Anderson, D.E. Walling and P.D. Bates (Editors), Floodplain Processes. John Wiley & Sons, New York. Knight, D.W., Shiono, K. and Pirt, J., 1989. Prediction of depth mean velocity and discharge in natural rivers with overbank flow. In: R.A. Falconer, P. Goodwin and R.G.S. Matthew (Editors), Proc. Int. Conf. on Hydraulic and Environmental Modelling of Coastal, Estuarine and River Waters. Gower Technical, University of Bradford, pp. 419-428. Leonard, J., Mietton, M., Najib, H. and Gourbesville, P., 2000. Rating curve modelling with Manning's equation to manage instability and improve extrapolation. Hydrological Sciences Journal-Journal Des Sciences Hydrologiques, 45(5): 739-750. Manning, R., 1891. On the flow of water in open channel and pipes. Trans. Institution of Civil Engineers of Ireland, 20: 161-207. Matgen, P., Henry, J.-B.F., Pappenberger, F., Pfister, L., de Fraipont, P. and Hoffmann, L., 2004. Uncertainty in calibrating flood propagation models with flood boundaries derived from sythetic aperture radar imagery, XXth ISPRS Congress. International Society for Photogrammetry and Remote Sensing, Istanbul, Turkey. Pappenberger, F. and Beven, K., in press. Functional Classification and Evaluation of Hydrographs based on Multicomponent Mapping. International Journal of River Basin Management.

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Pappenberger, F., Beven, K.J., Horritt, M. and Blazkova, S., in press. Uncertainty in the calibration of effective roughness parameters in HEC-RAS using inundation and downstream level observations. Journal of Hydrology, --(--): --. Pelletier, P.M., 1988. Uncertainties in the single determination of river discharge: a literature review. Canadian Journal of Civil Engineering News, 15(5): 834-850. Rantz, S.E., and others, 1982. Measurement and computation of streamflow, Volume 1: Measurement of stage and discharge, Volume 2: Computation of Discharge - U.S. Geological Survey, Water Supply Paper 2175.http://water.usgs.gov/pubs/wsp/wsp2175/, Romanowicz, R. and Beven, K.J., 2003. Estimation of flood inundation probabilities as conditioned on event inundation maps. Water Resources Research, 39(3): art. no.-1073. Schmidt, A.R., 2002. Analysis of stage-discharge relations for open-channel flow and their associated uncertainties, Universitz of Ollinois, Urbana, 328 pp. Smith, P., Beven, K.J. and Tawn, J., 2004. Evaluation of Rainfall-Runoff Models:- The effect of random input errors on output likelihood measures, EGU 1st General Assembly, Nice, France. U.S. Army Corps Engineers, HEC-RAS.http://www.hec.usace.army.mi,

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