This study presents a first attempt to address the propagation of radar rainfall nowcasting errors to flood forecasts in the context of distributed hydrological simulations over a range of catchment sizes or scales. Rainfall forecasts with high spatiotemporal resolution generated from observed radar fields are used as forcing to a fully distributed hydrologic model to issue flood forecasts in a set of nested subbasins. Radar nowcasting introduces errors into the rainfall field evolution that result from spatial and temporal changes of storm features that are not captured in the forecast algorithm. The accuracy of radar rainfall and flood forecasts relative to observed radar precipitation fields and calibrated flood simulations is assessed. The study quantifies how increases in nowcasting errors with lead time result in higher flood forecast errors at the basin outlet. For small, interior basins, rainfall forecast errors can be simultaneously amplified or dampened in different flood forecast locations depending on the forecast lead time and storm characteristics. Interior differences in error propagation are shown to be effectively averaged out for larger catchment scales.
Flood forecast skill critically depends on the spatial and temporal accuracy of the rainfall forcing that is present at the forecast time. A variety of advanced techniques are now available for improving the quality of rainfall input into hydrological models. These techniques include the coupling of radar rainfall estimates, radar nowcasting algorithms, and meteorological model output with rainfall–runoff forecasting models (e.g., Pereira Fo et al. 1999; Benoit et al. 2000; Yates et al. 2000; Dolciné et al. 2001; Berenguer et al. 2005; Bartholmes and Todini 2005; Sharif et al. 2006; Vivoni et al. 2006). Advances in rainfall forecasting allow an extension of the lead time available for issuing flood warnings and can improve flood forecast accuracy because the assumption of negligible future precipitation is avoided (Collier 1991). Nevertheless, the coupled use of rainfall and flood forecasting models requires assessing and characterizing the propagation of rainfall forecast errors into the hydrological forecasts.
There is a wide range of hydrological models with varying degrees of complexity that can be used to generate forecasts from precipitation products (e.g., Ogden and Julien 1994; Carpenter et al. 2001; Borga 2002). Different models are appropriate for different applications depending on the spatial and temporal scales of interest as well as the forecast variable (i.e., discharge, reservoir level, soil moisture). For each application, the model structure and complexity affect how precipitation errors from estimation or forecast inaccuracies are propagated through the hydrological system. Here, we present a preliminary study using distributed forecasting tools to define error propagation metrics and to assess their spatial and temporal characteristics for two selected storm events.
Rainfall forecasting errors arise because uncertainty exists in the forecast timing, location, and magnitude of storm features in an evolving precipitation field. For short lead times, radar nowcasting is an effective technique for extrapolating precipitation fields observed by weather radar (e.g., Browning and Collier 1989; Dixon and Weiner 1993). As lead time increases, nowcasting skill diminishes because storm morphological changes (e.g., growth and decay of features) are not explicitly simulated. Errors introduced in the forecast rainfall field because of unpredicted storm changes can have significant effects on a flood forecast—in particular, for distributed models that are sensitive to the temporal and spatial distribution of rainfall (e.g., Ogden and Julien 1994; Ivanov et al. 2004a). Further, the correlation of rainfall intensity errors with respect to evolving hydrological properties in a basin, such as the antecedent soil moisture, can have a dramatic impact on flood generation processes, such as runoff production (e.g., Winchell et al. 1998).
Despite its importance, error propagation from radar-based nowcasting algorithms to distributed flood forecasts has not been previously addressed in a quantitative fashion. This differs from prior studies on the propagation of radar estimation errors (e.g., Borga 2002; Sharif et al. 2002, 2004), because radar nowcasting fields are characterized by lead-time-dependent errors based on unpredictable storm changes. Distributed hydrological models provide an opportunity to study radar nowcasting error propagation because they are designed to capture the full information content available from weather radar over a watershed at a high spatial and temporal resolution (e.g., Pessoa et al. 1993; Garrote and Bras 1995; Ivanov et al. 2004b; Vivoni et al. 2006). As a result, distributed models are considered to be appropriate tools to assess and validate rainfall forecast quality and to understand how errors in the rainfall field forecasts may propagate throughout the watershed. Further, the ability to integrate the rainfall forecast into a discharge simulation at multiple locations in a basin allows one to investigate the effects of catchment scale on the propagation of rainfall forecast errors into a flood forecast. To our knowledge, the scale dependency of rainfall forecast error propagation into streamflow predictions has not been previously addressed.
In this study, we assess the propagation of radar nowcasting errors into distributed flood forecasts at gauged and ungauged locations in an operational basin in Oklahoma. We illustrate how rainfall forecasting errors are transformed by the integrative nature of the hydrological simulations and the control exerted by catchment scale. Our experiments are carried out for two flood events originating from different storm types typical of the regional setting (i.e., a winter airmass storm and a fall squall line). In the following, we briefly describe the radar nowcasting and the flood forecasting models and present our analysis of the propagation of errors in the combined models. We highlight the effect of the rainfall forecast lead time and catchment scale in the error propagation analysis.
2. Rainfall and flood forecasting experiments
We utilize the Storm Tracker Nowcasting Model (STNM; Wolfson et al. 1999) to generate short-term rainfall forecasts over the Arkansas–Red Basin River Forecast Center (ABRFC; ∼500 000 km2). The STNM algorithm is used here to produce high-resolution rainfall forecasts from the Weather Services International (WSI) “NOWrad” product (averaged to 4 km and 15 min) for short lead times (∼0–3 h) (Grassotti et al. 2003). The nowcasting algorithm is distinguished by its ability to separate large-scale storm motion from embedded convection through the use of image filtering. It then derives the cross correlation of successive filtered images to generate large-scale motion vectors, subsequently used to advect the original rainfall field. Forecast rainfall fields from the STNM over a number of storm events have been recently tested over the ABRFC by Van Horne et al. (2006) and used for hydrological forecasting in the Baron Fork (Eldon, Oklahoma) watershed (808 km2) (Vivoni et al. 2006).
We use the STNM forecasts as forcing to the triangular irregular network (TIN)-based Real-time Integrated Basin Simulator (tRIBS) applied over a subbasin of the ABRFC (Ivanov et al. 2004a, b; Vivoni et al. 2005). The tRIBS is a fully distributed hydrologic model that uses TINs to represent topography and its hydrographic features (Vivoni et al. 2004). The distributed model was developed for continuous flood forecasting using precipitation estimates from weather radars, rain gauges, and numerical weather model analyses. Rainfall forecasts can also be used as forcing over a specified forecast period. A key characteristic of the model is its capability to simulate surface–subsurface processes, including the lateral distribution of moisture within a system of interconnected hillslopes and channels. Thus, tRIBS can be used to produce multiple flood forecasts across a range of nested basin scales based on a sequence of estimated and forecast rainfall fields.
The STNM and tRIBS models are combined by introducing rainfall nowcasts to the hydrological model during the storm evolution. Rainfall forcings are observed radar fields for periods of data availability and STNM nowcasts for time steps during which radar data are unavailable. Nowcasting is used to extrapolate rainfall between available radar images spaced at a particular lead time apart. Because this operation fills in missing radar data with a sequence of forecasts, we refer to this method as the interpolation mode [see Vivoni et al. (2006) for details]. As the time between radar observations increases, the evolution of the actual rainfall field that is not captured by the nowcasting scheme leads to rainfall forecast errors as a function of lead time. The sequence of radar observations and STNM forecasts are then fed into the hydrological model to produce a flood forecast. In this study, the lead time between radar observations is varied from 15 min to 3 h to introduce rainfall forecast errors into the flood forecast. Forecast performance metrics are evaluated over the entire forecast interval for each lead time. For example, metrics valid for the 1-h lead time are based on a sequence of nowcasting fields (e.g., 15, 30, 45, 60 min) over intervening periods between available hourly radar rainfall fields.
We apply the STNM and tRIBS models to two flood events in the gauged Baron Fork basin (808 km2). The basin is characterized by complex terrain and a mixture of land use (54% forest and 46% cropland and pasture) and soils (94% silt loam and 6% fine sandy loam) in a rural setting. The hydrological model is applied to a set of nested basins, ranging in area from 0.78 to 808 km2, including two interior gauges and 12 ungauged locations. Figure 1 presents the land use and topographic variability of the nested basins. The WSI radar product over the Baron Fork is composed of 52 radar cells (4 km by 4 km). The rainfall forcing is mapped to the model domain at a temporal resolution of 15 min for each storm event. The selected storms in January and October of 1998 exhibited some degree of frontal organization (e.g., winter airmass storm and autumn convective squall line, respectively) and caused flooding in the nested basins (6.75- and 1.43-yr return periods at Baron Fork at Eldon, respectively). Although flood magnitudes are moderate, each event consisted of different runoff ratios (1.2. and 0.24 for the January and October floods). These rainfall–runoff conditions suggest the occurrence of nonlinear basin responses that present interesting case studies for the error propagation analysis.
3. Results and discussion
The propagation of errors from the radar rainfall nowcasts to the flood forecasts is analyzed in relation to the radar observations and a calibrated model simulation in the Baron Fork and its subbasins for each event. Vivoni et al. (2006) describe the calibration method and model performance relative to discharge observations at the three gauges. Visual and statistical comparisons showed that the model adequately simulated the flood events as compared with streamflow data. Figure 2 presents the simulated discharge and the mean areal precipitation (MAP) for the Baron Fork outlet. Note the differences in rainfall forcing and flood response for the two events with respect to their temporal distribution and peak amounts. It is readily seen that the January event was more pronounced than the October event in its flood response despite similar rainfall accumulations. Differences in the flood response are attributed to spatially varying wetness conditions prior to the arrival of the seasonal storms [see Vivoni et al. (2006) for details].
Figure 2 also presents the rainfall and flood forecasts obtained from the STNM and tRIBS models for each event (gray lines). The basin-averaged rainfall over the Baron Fork (MAP from 52 radar cells over 808 km2) is obtained from the interpolation mode for different values of the forecast lead time (ranging from 15 min to 3 h). Note that the forecast MAP can vary significantly from the observed MAP. Furthermore, the STNM forecast performance over the Baron Fork is considerably better for the January event. This suggests that the rainfall intensities and spatial distribution were more accurately forecast during the winter airmass storm as compared with the autumn squall line. Note the overestimation of basin-mean rainfall from the STNM forecasts during the early part of the October event, which subsequently has an important impact on the flood forecasts.
Comparisons of the tRIBS flood simulation based on the radar observations and the flood forecasts using the STNM nowcasts are also depicted in Fig. 2. The discharge at the Baron Fork outlet represents an integrated measure of the hydrological response to the rainfall forecast. Note that the flood forecasts typically overestimate the hydrograph peak and underestimate the time to peak for both flood events. Flood forecast errors are due to inaccuracies in the precipitation amount and timing in the rainfall forecasts. The flood forecast errors are clearly larger for the October event because of the inaccuracies in the forecast rainfall over the basin. Initial errors in the rainfall forecast lead to an early hydrograph rise that is consistent with the basin lag time (Vivoni et al. 2006). For the January event, the flood forecasts more closely resemble the calibrated simulation. Based on this set of visual comparisons, it is evident that rainfall forecasts’ errors are propagated to the outlet flood forecasts. It is important, however, to quantify error propagation using statistical measures that appropriately capture forecast deviations.
Figure 3 presents a comparison between rainfall and flood forecast errors based on two statistical measures, the bias B and the mean absolute error (MAE) (see the appendix for definitions) for the Baron Fork. In this study, the bias is used to compare the mean conditions in the forecast and observation and to measure error amplification (Grecu and Krajewski 2000). The MAE is used to capture the absolute error between the forecast and observation over the simulation, in units corresponding to the rainfall forcing and flood response (Legates and McCabe 1999). Because the MAE is determined by forecast errors over an entire time series, it is an appropriate metric for statistical analysis of both precipitation and discharge simulations. Note that the statistical metrics of the basin-mean rainfall and discharge are directly compared with one another in Fig. 3 to assess error propagation. Furthermore, the results are classified according to the rainfall forecast lead time (symbol types) and the storm event (open vs closed symbols).
Results from the Baron Fork indicate that rainfall forecast errors are amplified in the flood forecast for each event. Figure 3a illustrates that the discharge bias BD increases more quickly than the rainfall bias BR. This result is reflected in Table 1 in which the slope parameter of a linear regression between the rainfall and discharge bias is greater than 1 for both events. As a result, rainfall forecast errors lead to proportionally higher flood forecast errors. In this analysis, the increasing forecast bias is primarily due to the forecast lead time. Note that the higher biases correspond to lead times of >1–3 h, whereas low biases correspond to the 0–1-h lead time. Comparisons between the two storms indicate that error amplification in terms of bias is greater for the October event (see Fig. 2 for forecast time series). In this event, for the 3-h lead time, a bias of BD = 0.54 is obtained, indicating a large negative deviation of discharge forecasts due to the low forecast rainfall amounts during the storm peak.
Figure 3b shows the rainfall forecast error propagation to the flood forecasts using the MAE over the Baron Fork. Note that the rainfall MAE is expressed in meters cubed per second for MAP by multiplying by the outlet basin area (A = 808 km2) for direct comparison with the discharge MAE (m3 s−1). It is evident that an increase in rainfall forecast MAE leads to higher flood forecast MAE. As an approximation, Table 1 presents a linear regression of the MAE, indicating a moderate increase of ∼0.1 m3 s−1 in discharge error for a 1 m3 s−1 increase in rainfall error over the entire basin. Further, the forecast lead time has a strong control on the error propagation, with longer times (2–3 h) experiencing larger MAE. A comparison of the two storms indicates a higher discharge MAE for the January event, which results from the higher absolute discharge values.
While results over the Baron Fork capture the rainfall-to-flood error propagation, the distributed model also allows assessment of the effect of catchment size or scale. To perform this analysis, we use flood forecasts generated at 15 basins ranging in area A from 0.78 to 808 km2 (Table 2; Fig. 1b). For small basins (<16 km2), nowcasting errors occur uniformly over the basin (except where boundaries intersect radar pixels). For larger basins, many radar pixels contribute to the rainfall forecast error propagation in the flood forecast. The discharge bias and MAE between the flood forecast and calibrated simulation is computed for each individual subbasin. Further, we compute the rainfall bias and MAE between the radar nowcast and observation MAP over each subbasin. Note that the number of radar cells used in evaluating MAP decreases as the subbasin area is reduced. To capture the error propagation as a function of catchment size, we compute the ratio of the flood forecast and rainfall forecast metrics, the bias ratio BD/BR and the MAE ratio MAED/(A × MAER), where A is used to obtain a dimensionless ratio.
Figure 4 presents the variation of the bias and MAE ratios with basin area for each storm event and for two rainfall forecast lead times (1 and 2 h). Note that a B or MAE ratio greater than unity implies a higher discharge B or MAE as compared with the rainfall B or MAE in a subbasin (and vice versa). As a result, rainfall forecast errors are amplified (diminished) if the B and MAE ratios increase (decrease). Noting the variation of the B and MAE ratios with the catchment area provides an indication of the effect of basin size on forecast error propagation. For comparisons, it is important to indicate that the results for A = 808 km2 for the 1- and 2-h lead times have been previously shown in Fig. 3. Other catchment areas represent subbasins nested within the Baron Fork with outlets located along the main stem river (see Fig. 1b for outlet locations).
It is evident from Fig. 4 that the bias ratio approaches 1 as basin area increases, implying that nowcasting and flood forecast errors are comparable for large watersheds. For small basins, however, there is a large variability around unity in the B ratio. Several small catchments (A ≤ 100 km2) have high B ratio >1 (e.g., January at 1 h), whereas others exhibit low B ratio <1 (e.g., October at 1 h). This result suggests that error amplification and dampening occur at the same time in different basins. Differences in error propagation are attributed to the spatial variation of nowcast errors and to differences in the basin lag time for each catchment. As A increases, internal differences cancel out, leading to bias ratio ∼1. Exceptions to this behavior are observed. For example, the bias ratio increases to 1.2 for the largest area during October at 2 h. As lead time increases, we note that the B ratio is approximately 1 for most basins, indicating more comparable rainfall and flood forecast bias. Further, we observe large differences between the two events that suggest that the effects of scale and lead time on error propagation are storm dependent. In October, for instance, the B ratio increases for all basins as the lead time is extended from 1 to 2 h, whereas in the January event the B ratio remains fairly constant for these two lead times.
The variation of the MAE ratio with basin area in Fig. 4 shows similar properties to the B ratio. In particular, note the large variability in MAE ratio for small basins (A ≤ 100 km2) and an approach toward stable values closer to unity for large basins. The MAE ratio behavior with catchment scale is composed of two general trends: a decrease in rainfall MAE and an increase in discharge MAE with increasing A (not shown). Small basins either strongly amplify or dampen rainfall errors (MAE ratio from 0.1 to 2.6), whereas large basins exhibit similar rainfall and flood forecast errors. Of interest is that the effect of lead time on the error propagation is highly storm dependent. For January, the 2-h lead time results in greater error dampening at large scales as compared with 1 h. For the October event, the lead-time effect is not as significant. These results indicate how the error propagation analysis can depend on the forecast lead time, basin scale, storm properties, and possibly antecedent conditions for which it is applied. Further analysis with a broader range of storm events would be required to identify the individual contributions of each factor to the hydrologic propagation of nowcast errors.
Uncertainties related to radar rainfall estimates can have significant impacts on hydrologic simulations (e.g., Carpenter et al. 2001; Borga 2002; Sharif et al. 2002, 2004). Radar estimation errors are typically related to beam blockage, brightband contamination, range dependency, or inaccurate reflectivity–rainfall relations. Radar rainfall nowcasting, on the other hand, can introduce additional error sources related to the spatial (storm growth or decay) and temporal changes (feature acceleration or deceleration) in the rainfall field that are not properly captured in the forecasting algorithm (see Van Horne et al. 2006). As a result, radar nowcasting errors vary significantly from radar estimation inaccuracies and need to be assessed with respect to their propagation in flood forecasts. Despite the significance of forecast uncertainty, few studies have determined how radar nowcasting errors propagate to distributed hydrological simulations. In particular, the effects of forecast lead time and catchment scale have not been addressed in this context.
In this study, we have analyzed the propagation of rainfall forecast errors to flood forecasts at a number of gauged and ungauged locations within an operational watershed. Based on the analysis of two flood events, we found the following: 1) increasing the forecast lead time (from 0 to 3 h) results in increasing nowcasting errors, which are reflected in the flood forecast at the basin outlet; 2) catchment scale (A = 0.78–808 km2) controls whether rainfall forecast errors are strongly amplified or dampened (in small basins) or effectively comparable to (in large basins) flood forecast errors at a particular lead time (1 or 2 h); and 3) differences in storm characteristics (winter airmass vs autumn squall line) have a strong effect on the influence of forecast lead time and catchment scale in the error propagation.
It is important to point out that these results are a first step toward identifying the rainfall forecast error propagation through a distributed hydrological model. The results are valid for two case studies with different antecedent conditions and different storm and flood properties in the Baron Fork basin. A broader range of events would be required to generalize the study conclusions beyond the current spatial and temporal extent. For example, tests over a large number of storms would enable deriving an averaged flood forecast skill and its uncertainty (or statistical envelope). In doing so, one could potentially parameterize the flood forecast skill as a function of forecast lead time and basin area. The approach outlined in this study is also sufficiently general for application in other regions and with different precipitation products and hydrological models.
Our results are indicative of the error propagation from radar nowcasting to flood forecasting based on the mean areal precipitation and basin discharge across a range of catchment scales. Nevertheless, these hydrological metrics only capture the basin-averaged rainfall forcing and integrated discharge response. An approach that takes advantage of the distributed nature of the forecasting models would enable one to assess the propagation of spatial and temporal rainfall forecast errors in the context of the forecast hydrologic field. For example, a promising research area is the use of spatial forecast metrics, such as the critical success index, to capture the propagation of rainfall forecast errors into the forecast soil moisture pattern and its temporal evolution in a basin.
This research was sponsored by the U.S. Army Research Office (Contract DAAD19-00-C-0114) and NOAA (Contract NA97WH0033). We thank MIT Lincoln Laboratory for the use of the STNM model and Weather Services International for radar data used in this study. We also thank Rafael L. Bras, Christopher Grassotti, Valeriy Y. Ivanov, and Matthew Van Horne for earlier contributions to this project. We also appreciate the comments of three anonymous reviewers that helped to improve the original manuscript.
The Bias and Mean Absolute Error Statistics
Error propagation is assessed by comparing observations O and forecasts F of the MAP and discharge using the bias B and MAE. Bias B is the ratio of the mean forecasts to observations:
(dimensionless), as defined by Grecu and Krajewski (2000). The MAE (discharge: m3 s−1; MAP: mm h−1) describes the absolute difference between the observations and forecasts without emphasizing outliers (Legates and McCabe 1999):
where N is the sample size. We perform the error propagation analysis by comparing STNM and tRIBS forecasts with radar observations and its associated calibrated flood simulation taken as a ground truth. Use of both statistical metrics is complementary and allows comparisons that capture average differences in dimensionless terms and absolute differences in the unit of the measurement.
Corresponding author address: Enrique R. Vivoni, Department of Earth and Environmental Science, New Mexico Institute of Mining and Technology, 801 Leroy Place, MSEC 244, Socorro, NM 87801. Email: email@example.com