The common approach to quantifying the precipitation forecast uncertainty is ensemble simulations where a numerical weather prediction (NWP) model is run for a number of cases with slightly different initial conditions. In practice, the spread of ensemble members in terms of flood discharge is used as a measure of forecast uncertainty due to uncertain precipitation forecasts. This study presents the uncertainty propagation of rainfall forecast into hydrological response with catchment scale through distributed rainfallrunoff modeling based on the forecasted ensemble rainfall of NWP model. At first, forecast rainfall error based on the BIAS is compared with flood forecast error to assess the error propagation. Second, the variability of flood forecast uncertainty according to catchment scale is discussed using ensemble spread. Then we also assess the flood forecast uncertainty with catchment scale using an estimation regression equation between ensemble rainfall BIAS and discharge BIAS. Finally, the flood forecast uncertainty with RMSE using specific discharge in catchment scale is discussed. Our study is carried out and verified using the largest flood event by typhoon “Talas” of 2011 over the 33 subcatchments of Shingu river basin (2,360 km^{2}), which is located in the Kii Peninsula, Japan.
Recent advances in weather measurement and forecasting have created opportunities to improve streamflow forecasts. It is possible to combine highresolution numerical weather prediction (NWP) data directly into streamflow forecast systems in order to obtain an extended lead time. The accuracy of weather forecasts has steadily improved over the years, but recent researches represented that direct application of outputs from the NWP model into the hydrological domain can result in considerable bias and uncertainty that are propagated into hydrological domains [
One of the biggest sources of uncertainty in the application of streamflow forecasting comes from forecasted rainfall. The grid size in NWP models is often larger than the subcatchment size in hydrological models, which results in the forecast rainfall data not being at the appropriate resolution required for flood forecasting. In addition, even small errors in the location of weather systems by NWP models may result in forecast rainfall for the catchment concerned being significantly wrong [
It is difficult to understand the full range and interaction of uncertainties in flood forecasting. And the different types of uncertainty will vary with lead time of the forecasts, and with the magnitude of the event and catchment characteristics. Vivoni et al. [
The main objective of this study is to assess the error and uncertainty propagation due to NWP rainfall uncertainty on hydrological response through a distributed hydrologic model depending on catchment scale. In the context of flood forecasts, it is important to assess the forecast rainfall uncertainty in terms of the effect on runoff. And uncertainties based on spatial scale are also important by means of the information for realtime flood forecast and the possible amount of flow to the reservoir and exceeding its capacity to optimize the water volume to be released. Therefore, the coupled use of NWP rainfall output and hydrologic flood forecasting requires an assessment of uncertainty through hydrological response.
The research question is as follows: How does ensemble NWP rainfall error translate into flood forecasting, and how does flood forecast uncertainty propagate as a function of catchment scale dependency? To our knowledge, there exists research about rainfall uncertainty’s direct propagation into the hydrological domain, but the spatial scale dependency of uncertainty propagation of ensemble NWP rainfall into hydrological predictions has not been addressed. First, we compared forecast rainfall error based on the BIAS, which is used to measure error amplification, to flood forecast error driven by ensemble NWP forecast outputs to assess error propagation. Second, we discussed the variability of flood forecast uncertainty according to catchment scale using ensemble spread, which is driven by ensemble NWP rainfall through a distributed hydrologic model. We also assessed flood forecast uncertainty, which is under the condition that ensemble NWP rainfall has not BIAS compared with observed radar rainfall and catchment scale using an estimation regression equation between ensemble NWP rainfall and discharge based on the BIAS. Finally, we assessed flood forecast uncertainty with RMSE using specific discharge in catchment scale. Note that we focused not only on the quantitative error propagation of rainfall forecast into flood forecast but also the variability of flood forecast uncertainty with catchment scale.
This paper has been organized in the following way. After the Introduction, Section
In Japan, an operational oneweek ensemble prediction model from JMA was developed to provide probabilistic information of 51 ensemble members with a horizontal resolution of 60 km, and it used to be applied for hydrological applications (e.g., prior and optimized release discharge for dam operation) [
Both 10 km and 2 km resolution systems used the JMA Nonhydrostatic Model (NHM) as the forecast model [
(a) Forecast domains of 10 km and 2 km horizontal resolution. (b) Schematic of forecast runs with 10 km and 2 km horizontal resolution. The rectangle inside 2 km domain denotes the spatial verification area for Kinki region.
In this study, we introduced the results of ensemble prediction with a 2 km horizontal resolution due to the viewpoints of high resolution and better predictability of weather phenomena and used 4 sets of ensemble prediction outputs with 30 hours forecast time to assess rainfall forecast uncertainty and to understand how uncertainty in the rainfall forecast may propagate throughout the watershed (Table
Four forecast sets with 30 hours’ forecast time and 2 km horizontal resolution used in the study. Each forecast is overlapped with 6 hours.
Forecast period  First forecast  2011/09/01 03:00–09/02 09:00 JST 
Second forecast  2011/09/02 03:00–09/03 09:00 JST  
Third forecast  2011/09/03 03:00–09/04 09:00 JST  
Fourth forecast  2011/09/04 03:00–09/05 09:00 JST 
The Shingu river basin was selected as the target area to assess rainfall forecast uncertainty into streamflow forecast with spatial scale. The Shingu river basin is located in the Kii Peninsula of the Kinki area, Japan, and covers an area of 2,360 km^{2}. The average elevation of the study site is 644.6 m, and the slope is steep; this basin is a mountainous area. The five dams, Futatsuno, Kazeya, Komori, Nanairo, and Ikehara, are located upstream. The left and right sides of the Shing river basin exhibit different characteristics. The left side is the Totsukawa basin, and the right side is the Kitayamakawa basin. Their characteristics are completely different. The elevation of Totsukawa is higher than that of Kitayamakawa. And Kitayamakawa has a lower level in the channel. We divided the Shingu river basin into 33 subcatchments from 54.24 to 2245 km^{2} (Figure
Subcatchment area at gauged and ungauged points.
Catchment  Area (km^{2})  Catchment  Area (km^{2}) 

1  92.2  18  141.56 
2  165.99  19  347.35 
3  279.78  20  429.07 
4  150.56  21  94.23 
5  444.04  22 (Nanairo dam)  529.49 
6  54.24  23 (Komori dam)  633.22 
7  533.73  24  700.49 
8  105.72  25  1090.92 
9 (Kazeya dam)  656.08  26  56.68 
10  65.97  27  65.20 
11  766.19  28  1268.03 
12  65.04  29  783.85 
13  130.74  30  2091.38 
14 (Futatsuno dam)  1012.15  31  110.92 
15  112.13  32  2212.24 
16  72.65  33 (Ouga station)  2245.56 
17 (Ikehara dam)  203.27 
(a) 33 subcatchments including 6 gauged (5 dams and 1 gauge station) and 27 ungauged locations and (b) connections with flow directions.
We used a spatially distributed hydrologic model, based on onedimensional kinematic wave method for subsurface and surface flow (hereafter, KWMSS) with a conceptual stagedischarge relationship [
Conceptualization of spatial flow movement and flow process in hillslope elements; the arrows indicate element models for calculating hydrological variables, such as water flux.
There was no observed discharge data in subcatchments, except in 5 dams and 1 gauge station. For that reason, the parameter optimization of the hydrologic model was conducted using the Ministry of Land, Infrastructure, Transport and Tourism (MLIT) Cband composite radar data, which has high spatialtemporal resolution to capture the spatial variability of rainfall. However, in spite of the highresolution accuracy of radar data, parameterization associated with soil parameters of hydrological model remains uncertain due to impossibility of direct observation and use of the soil parameters (i.e., discordance between soil properties and model parameters). Therefore, we assumed that parameters of hydrologic model in Table
Optimized parameter values from multicalibration using SCEUA optimization method.
Parameter  Description  Optimal values 


Roughness coefficient [m^{−1/3}s]  0.1284 

Depth of the unsaturated soil layer [m]  0.2369 

Depth of the saturated soil layer [m]  0.1442 

Hydraulic conductivity of the saturated soil layer [m/s]  0.0150 

Nonlinear exponent constant for the unsaturated soil layer [−]  3.7898 
Multicalibration using SCE optimization method and minimizing the objective function of 6 observation points.
To evaluate the accuracy of the ensemble forecast in terms of areal rainfall intensity, we calculated two error indexes. The first is the normalized root mean square error (RMSE), which is normalized by the mean value of the observations during the each forecast period (30 hours). The second is the log ratio bias, which a relative error and provides information about the total amount of rainfall. A log ratio bias value of zero indicates a perfect forecast; positive and negative values indicate underestimated and overestimated forecasts, respectively:
For the spatial verification of ensemble NWP rainfall, the rainfall forecasts have been verified spatially against the MLIT Cband composite radar data. The ensemble forecast was expressed as probabilities of exceeding selected rainfall thresholds (1.0 and 5.0 mm/h). A contingency table can be constructed with a spatial comparison, in which each area with more than selected rainfall threshold is defined as “yes,” and other areas are defined as “no” for both forecasted and observed rainfall fields. In this study, two indexes are considered for spatial verification of ensemble forecast in the Kinki region (Figure
Rainfall forecast error of ensemble outputs from the NWP model is compared with the flood forecast error driven by those rainfall forecasts to assess the uncertainty propagation. It is important, however, to quantify uncertainty propagation from rainfall forecast to flood forecast using statistical measures that appropriately capture forecast deviations. For this reason, the BIAS was used to compare the mean conditions in the forecast and observation in terms of rainfall and flood forecast and to measure error amplification. Note that the BIAS of the basinmean rainfall is directly compared with the discharge BIAS, and the BIAS is used for an average value of 30 hours of forecast time of rainfall and flood forecast results. Furthermore, the results are classified according to the forecast period of ensemble rainfall from the NWP model:
For the evaluation of the variability of flood forecast uncertainty according to catchment scale, the mean value of the coefficient of variation (CV), which is a normalized measure of dispersion of a probability distribution or frequency distribution, was used (Equation (
For the purpose of temporal verification of QPF with ensemble NWP rainfall during the Talas event, the areal rainfall intensity of ensemble forecasts is compared with the Automated Meteorological Data Acquisition System (AMeDAS) over the Shingu river basin. For comparison, the observed rainfall of AMeDAS (18 stations, 10 min step) is interpolated using the Thiessen polygon spatial distribution method.
Figure
(a) Ensemble areal rainfall forecast over the Shingu river basin in the form of box plots plotted from 0 to 24 hr forecast time, excluding overlapped forecast time (from 25 to 30 hr) for the overall comparison for the Typhoon Talas. (b) Verification results of areal rainfall with normalized RMSE and log ratio bias for Typhoon Talas. Red circles and black squares mean the indexes of the control run and the mean value of ensemble forecast, respectively. The lower and upper bounds of the black lines correspond to the minimum and maximum values, respectively.
In the index of normalized RMSE, the control run and ensemble mean have similar values from 1st to 3rd forecast period, but the best index of the ensemble spread could provide good value as compared with the deterministic control run. In the 4th forecast period, as mentioned above, the index of the control run and ensemble spread is relatively large, but the best index of the ensemble is estimated at 0.89 (the control run is 3.85). In the index of the log ratio bias, the best index of ensemble spread could cover zero value (perfect forecast), whereas the control run forecast was underestimated for the 1st, 2nd, and 3rd forecasts and overestimated for the 4th forecast period.
Figure
Spatial rainfall verification using CSI and BIAS with threshold values in verification area of Figure
CSI and BIAS with 1 mm/h threshold value
CSI and BIAS with 5 mm/h threshold value
We conducted the ensemble flood forecasts of 33 subcatchments in the Shingu river basin for an assessment of the ensemble flood forecast driven by ensemble NWP rainfall. Simulated discharges from the observed radar rainfall were used as the initial condition for the ensemble flood forecast in each forecast period. Figure
Flood forecast results in over 33 subcatchments. Grey line represents the each forecasted discharge driven by 11 ensemble NWP rainfall. Red curve illustrates the ensemble average results. Black line represents the observed radar discharge of 33 subcatchments.
Figure
Propagation of rainfall forecast errors to flood forecast errors from the first to the fourth forecast periods.
As mentioned above, the Shingu river basin is divided into 33 subcatchments from 54.24 to 2245 km^{2}, including 6 gauged and 27 ungauged locations, for the assessment of uncertainty of ensemble NWP rainfall into flood forecast with catchment scale. The Shingu river basin has 3 types (small, medium, and large catchments) and 2 characteristics (mountainous and flat area) for evaluation of the variability with catchment scale.
Figure
Flood forecast variability expressed by coefficient of variation with catchment scale and characteristic. Red, blue, and gray points represent the small catchment (<200 km^{2}), medium catchment (200~1000 km^{2}), and large catchment (>1000 km^{2}), respectively. And we also divided catchment characteristics into 2 types, mountainous area (>800 m, big point) and flat area (<800 m, small point) considering average elevation (800 m) of the 33 subcatchments.
Flood forecast uncertainty focuses on the discharge uncertainty with catchment scale and was assessed when rainfall BIAS was 1, using an estimated linear regression equation between each ensemble rainfall BIAS and discharge BIAS of 33 subcatchments. Figure
Comparison of rainfall and discharge BIAS of ensemble members in each subcatchment and linear regression equation.
Figure
Flood forecast variability expressed by BIAS with catchment scale and characteristic. Red, blue, and gray points represent the small catchment (<200 km^{2}), medium catchment (200~1000 km^{2}), and large catchment (>1000 km^{2}), respectively. And we also divided catchment characteristics into 2 types, mountainous area (>800 m, big point) and flat area (<800 m, small point) considering average elevation (800 m) of the 33 subcatchments.
Figure
Flood forecast variability expressed by RMSE with catchment scale and characteristic. Red, blue, and gray points represent the small catchment (<200 km^{2}), medium catchment (200~1000 km^{2}), and large catchment (>1000 km^{2}), respectively. And we also divided catchment characteristics into 2 types, mountainous area (>800 m, big point) and flat area (<800 m, small point) considering average elevation (800 m) of the 33 subcatchments.
Forecast uncertainty of NWP models is usually assumed to represent the largest source of uncertainty on flood forecasts. However, there are in fact many sources of uncertainties in the flood forecasts which could also be significant, for example, the corrections and downscaling mentioned above and spatial and temporal uncertainties as input into the hydrological simulations including data assimilation. And the different types of uncertainty will vary with lead time of the forecasts and with the magnitude of the event and catchment characteristics. Ensemble flood forecasting by ensemble NWP rainfall is specifically designed to capture the uncertainty, by representing a set of possible future states of the atmosphere. This uncertainty can then be cascaded through flood forecasting systems to produce an uncertain or probabilistic prediction of flooding. In many cases, the potential of flood forecasting is described alongside cautious notes regarding variability, uncertainty, communication of ensemble information, need for decision support, and problems of using short time series [
The main objective of this study is to investigate the error and uncertainty propagation due to NWP rainfall uncertainty on hydrological response through a distributed hydrologic model depending on catchment scale. First, we conducted the ensemble flood forecasts of 33 subcatchments in the Shingu river basin for an assessment of the ensemble flood forecast driven by ensemble NWP rainfall and compared forecast rainfall error based on the BIAS, which is used to measure error amplification, to flood forecast error driven by ensemble NWP forecast outputs to assess error propagation. Second, we discussed the variability of flood forecast uncertainty according to catchment scale using ensemble spread by ensemble NWP rainfall through a distributed hydrologic model. Finally, we assessed the flood forecast uncertainty using an estimation regression equation between ensemble NWP rainfall and discharge based on the BIAS and also assessed the flood forecast uncertainty with RMSE using specific discharge in catchment scale.
From the results, the ensemble flood forecast over the 33 subcatchments and flood forecasts driven by ensemble outputs produced suitable results but showed that in general it has a large proportion of under and overpredictions at low lead times and exhibit a negative bias at longer lead times. And this study demonstrates that uncertainty variability occurs sensitively and diversely at the same time in different catchments, and small catchments have sensitive variability of uncertainty. General findings from this study are the fact that smaller catchments demonstrate a larger uncertainty in the flood forecast. Therefore, flood forecasting in small catchment should be careful due to the large variability of uncertainty. On the other hand, in medium and large catchments, there is less uncertainty than in small catchments as would be expected due to the smoothing effects of modeling a larger catchment. The ensemble forecasts are specifically designed to capture the uncertainty in NWPs, by representing a set of possible future states of the atmosphere. This uncertainty can then be cascaded through flood forecasting systems to produce an uncertain or probabilistic prediction of flooding. In order to use ensemble forecasts of NWP model for flood forecasts effectively, it is important to establish methodologies to analyze ensemble flood forecasts. To reduce the uncertainty of rainfall and flood forecasts, the bias correction and/or hybrid products with radarbased prediction are required to achieve more reliable hydrologic predictions; bias correction and blending method for accuracy improvement was addressed in Yu et al. [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This study was partly supported by the subproject of the field 3 on Next Generation Supercomputer Project, “Prediction of Heavy Rainfalls by a CloudResolving NWP System,” and was based on data from the Ministry of Land, Infrastructure, Transport and Tourism (MLIT) and JPOWER Co. Ltd., Japan. The authors are grateful for their support.