1. Introduction

Dynamic hydrological models (also known as rainfall–runoff models) are an essential element of operational streamflow prediction systems that inform water, energy, and hazard management procedures in many parts of the world. At seasonal time scales, the so-called ensemble streamflow prediction (ESP) method described by Day (1985) provides the context for streamflow forecasts, in which the model is run up to the time of forecast using observed precipitation, temperature, and other variables and in the forecast period ensembles taken either from resampling of historic observations or from climate model ensembles. Similar, albeit more deterministic, applications of hydrological models are used in flood forecasting; however, in that case as well the models are run up to the time of forecast with observed forcings.

The typical use of hydrologic models in both seasonal and flood forecasting requires that the models be calibrated and validated during a retrospective period to demonstrate that they are able to reproduce observed streamflow with an acceptable degree of error. Once that criterion is met, the models are used for real-time prediction. A major difficulty in this construct is an inevitable discrepancy between the network of meteorological observations that are available retrospectively for model calibration and validation and the data that are available in real time. This discrepancy is because of the time needed to process and apply quality controls to the observations, which results in lags in availability of many of the observations, which hence are not used in real-time applications. In the United States this lag, in the case of nonreporting Cooperative Observer Program (COOP) stations, is often several months and in some cases longer; however, once these data are pro-cessed, there usually are more stations available for model testing and evaluation than reported in real time. Of equal or perhaps greater importance is that the retrospective observations, which are subjected to quality control, are generally less subject to errors than the real-time data.

In any event, the real-time forcings—temperature and precipitation, and in some cases other variables derived from them such as downward solar radiation and dewpoint—for operational hydrology models are subject to bias relative to which the models are calibrated, and these biases may be propagated to the real-time streamflow simulations. Suppose, for instance, that within the observational network, a significant number of the stations recording the highest precipitation during a given storm or season did not report in real time. In this case, the real-time streamflow simulation would be biased downward relative to the climatology provided by the retrospective calibration and verification data. The challenge of inconsistencies between real-time and retrospective observational networks is also relevant for producers of operational analysis fields—for example, the National Operational Hydrology Remote Sensing Center [for their national snow data assimilation system (SNODAS) snow water equivalent fields] and the National Centers for Environmental Prediction (NCEP) [for the North American Land Data Assimilation System].

The synoptic conditions that govern weather impart spatial coherence to meteorological observations within a neighborhood of stations such that from the perspective of rank, the meteorological observations at these stations behave similarly. For example, the warmest day on record at one station is likely to be accompanied by extreme temperature rankings—if not record warmth—at nearby stations. For precipitation, the spatial correlation is usually related to the spatial structure of rainfall patterns (e.g., Huff and Shipp 1969, 1970; Sharon 1972; Shih 1982; Sivakumar 1990; Bacchi and Kottegoda 1995; Hatfield et al. 1999; Habib et al. 2001), with precipitation from convective storms having shorter decorrelation lengths than from large-scale synoptic systems. Furthermore, spatial decorrelation lengths generally increase as the accumulation interval increases—for example, from daily to monthly totals (Shih 1982; Hatfield et al. 1999).

In calculating hydrologic model forcings, a typical approach is to interpolate observed values to estimate a mean forcing over an area (for spatially lumped models) or at a grid point (for spatially distributed models). This approach is used, for instance, by the National Weather Service to compute mean areal precipitation and mean areal temperature for use in its National Weather Service River Forecast System (NWSRFS). We hypothesize that an alternative approach that interpolates the percentiles of the observations instead, with a subsequent step to retrieve the climatological values matching the percentiles at each target location, not only suffices to estimate the forcing adequately at the target areas or locations but is more robust in the face of missing observations. We propose this technique as a solution to the problem of retrospective versus real-time dataset inconsistencies that complicate the real-time forcing of hydrology models for seasonal streamflow forecasting in the western United States. Because the decorrelation length of precipitation is longer for precipitation resulting from frontal rather than convective storms, the proposed method is expected to be the most appropriate for cold season rainfall. However, from a practical standpoint, most precipitation in the western United States—particularly in mountainous areas that generate the majority of regional runoff—is cold season rainfall (Markham 1970; Mock 1996; Higgins et al. 1999; Leung et al. 2003), hence this should be a minor limitation in practice.

This paper outlines an approach—the index station percentile method (ISPM)—for estimating precipitation from a subset (“index stations”) of the full set of stations that reports in real time. We evaluate the ability of the method to provide accurate estimates of the precipitation that would be estimated by the full set of stations, given different choices in the calculation of precipitation percentiles. The study area for the evaluation is the Klamath River basin, where the elevation ranges from sea level to more than 2200 m and precipitation ranges from 250 to 1750 mm year⁻¹ (Fig. 1). The climate of the Klamath basin is characterized by summer drought, with only 3% of the annual precipitation occurring June–August (JJA) and about 55% of the annual precipitation occurring December–February (DJF).

2. Approach

Precipitation is an intermittent variable, so the observed distribution of daily precipitation at most locations contains more zeros than nonzero values. This is why statistical models of precipitation (e.g., Todorovic and Woolhiser 1975; Katz 1977; Foufoula-Georgiou and Lettenmaier 1986, 1987) often assess a probability of nonzero precipitation as well as a conditional estimate of precipitation amount. Zeros in the precipitation distribution are tied values for which a rank or percentile cannot be uniquely estimated. The proportion of zero precipitation events within the distribution diminishes, however, as the time step of the event increases in length—for example, from days to a week or weeks to a month or longer. We use this property to circumvent the difficulty of transforming precipitation observations to and from percentiles by using a three-step procedure for estimating precipitation on a given day, at a given location. In particular, we wish to estimate precipitation at COOP stations that do not report in real time, using the following procedure.

1. Precipitation at each nearby reporting station is summed over a multiday period (“sliding window period”), leading up to and including the target day, and the percentile (nonexceedance probability) of the accumulated precipitation at each station is calculated.

2. The percentiles are interpolated to the location of interest. A precipitation value corresponding to the interpolated percentile is then drawn from the gridded observed climatological dataset at the location of interest for the given sliding window period. The value of precipitation yielded by this second step pertains to the (multiday) sliding window period and must then be disaggregated.

3. The daily amounts from all of the nearby stations are interpolated to the target location and, after rescaling the resulting daily time series to fractions of the period total, the fractions are multiplied by the precipitation estimated in step 2 for the target location. This third step yields daily time series of precipitation in the sliding window period.

The spatial interpolations above use the Synagraphic Mapping System (SYMAP) algorithm of Shepard (1984), as implemented by Maurer et al. (2002), which has been widely adopted for the production of precipitation climatologies (Widmann and Bretherton 2000; New et al. 2000; Schmidli et al. 2001; Hamlet and Lettenmaier 2005). The SYMAP algorithm estimates values at a set of grid nodes by using an inverse distance squared weighting of observations at stations within a prescribed search radius from the grid point.

The ISPM procedure yields a daily time series of precipitation that includes the day of interest (the last day in the sliding window period) along with estimates for every other day in the period. In practice, the sliding window period is the “spin up” period that produces the hydrologic model’s initial condition for hydrologic forecasts (see Wood and Lettenmaier 2006).

The most common approach to real-time hydrologic simulation is to advance the most recent simulated state (soil moisture and/or snow water storage, perhaps from the prior day) forward to the present. The sliding window period in the ISPM approach is ordinarily longer than the interval between simulation updates. The sliding window period advances with each simulation update, providing new estimates of precipitation both prior to and during the update interval; therefore, data from stations that were not available in real time, but have become available with lag less than the sliding window period, can be used to improve the hydrologic model’s initial conditions.

Using observed station precipitation from the years 2000–04, we evaluate a) the effectiveness of the ISPM to estimate the precipitation that one would calculate retrospectively using a larger set of COOP stations; b) the effect of the length of the sliding window period on the effectiveness of the method; and c) the effect of the density of index stations on the effectiveness of the method. To address the second issue, we compare sliding window period lengths termed ISPM-N, where N is the length of the sliding window period, for which N = 5, 10, 30, and 60 days. To address the third issue, we test a reduced station scenario in which none of the index stations inside the Klamath River basin are assumed to report in real time and only real-time index stations outside the basin are used to estimate precipitation in the basin. Our evaluation design focuses on gridded precipitation fields rather than point station precipitation, in part because ISPM is used in the forecasting system described in Wood and Lettenmaier (2006), which uses a spatially distributed hydrologic model; however, the method is equally applicable for station precipitation and for basin-average mean-areal precipitation that is used to force different hydrologic models (e.g., Leavesley et al. 1983; Burnash 1984).

National Climatic Data Center (NCDC) COOP stations at 172 locations within and adjacent to the Klamath basin were used to construct the gridded climatological datasets used in this paper. To form the baseline analysis or climatological dataset used as the “truth,” we adopted the gridding procedure described in Maurer et al. (2002). The precipitation gauge data for 1961–90 were gridded to 1/16° spatial resolution using the SYMAP algorithm as implemented by Maurer et al. (2002). In this procedure, the gridded daily precipitation data are scaled to match the long-term average of the Precipitation-elevation Regressions on Independent Slopes Model (PRISM) precipitation climatology (Daly et al. 1994, 2002). The correction factors are the monthly scaling factors based on comparison of long-term monthly averages from PRISM and uncorrected, gridded precipitation estimates resulting from the SYMAP interpolation. PRISM uses a regression-based model based on point data, a digital elevation model (DEM), a knowledge base, and human expert parameterization to generate repeatable estimates of annual and monthly climatologies. PRISM-based orographic corrections are also used in NCEP’s Climate Prediction Center (CPC) operational precipitation products (e.g., Higgins et al. 2004). The corrections are typically monthly scaling factors based on the comparison of retrospective long-term monthly averages from PRISM and uncorrected, gridded precipitation estimates resulting from the interpolation of the retrospective station dataset. We term the baseline dataset, which used all 172 stations available retrospectively, the “Maurer” dataset.

In operational use, the correction factors are applied to adjust real-time gridded precipitation estimates that are based only on stations available in real time, the interpolated fields of which may have different statistical properties than the data used to construct the factors. For example, approximately 33 stations (19% of the retrospective stations) are available in real time in the vicinity of the Klamath River basin (see Fig. 1). An alternative approach is to develop correction factors between the PRISM long-term averages and the gridded precipitation averages based only on the stations available in real time and to use these corrections alone to generate real-time gridded precipitation fields. We implement both approaches here and generate two gridded precipitation datasets using only the real-time (or “index”) stations. The first approach, which applies correction factors based on all retrospectively available stations, is termed “Maurer_ALL.” The second approach, which uses correction factors based on the index (real time) stations only, is termed “Maurer_RT.” These and the ISPM-N datasets are summarized in Table 1. The ISPM, Maurer_ALL, and Maurer_RT methods are based on the same set of index stations and all are feasible in real-time estimation. The Maurer_ALL and Maurer_RT methods use already developed PRISM-based correction factors in real time. The ISPM methods correct real-time estimation from climatological datasets on which PRISM-based corrections have been previously performed. The ISPM methods also try to increase correlation among sparse index stations by increasing accumulation interval. We compare the performance of these methods with the Maurer et al. (2002) baseline dataset, which used all 172 stations to calculate correction factors and to produce interpolated precipitation.

3. Results and analysis

Figure 2 shows the spatial patterns of the relative errors in winter (DJF) from 2000 to 2004. The relative error is calculated for daily rainfall on the last day of the sliding window with different lengths. The Maurer_ALL method gives large relative errors over much of the study area. The relative errors are generally reduced by ISPM. The estimated precipitation for smaller N values in ISPM—that is, 5 days—typically has larger errors than for larger N values. The effect of N declines, however, when the N is longer than 30 days. The Maurer_RT method gives smaller relative errors than the Maurer_ALL method. However, larger biases are found in the lower portion of the Klamath basin, which may be caused by precipitation events that are not captured by the sparse index station density.

Figure 3 illustrates the spatial patterns of the relative errors in summer (JJA) from 2000 to 2004. For ISPM with small N—that is, 5 and 10 days—most biases are negative, whereas for the larger N, the biases are mixed. Both the Maurer_ALL and Maurer_RT methods produce positive biases in the upper Klamath basin, and in general, ISPM does not work nearly as well in summer as in winter. This may be in part because of the large number of zero precipitation values in summer, which precludes the unique estimation of percentiles, and in part because the mechanisms that produce infrequent precipitation in summer may not have the same orographic characteristics as in winter. However, the precipitation biases in summer have little effect because of the small contribution of JJA to annual precipitation.

A histogram of the relative errors from 2000 to 2004 is shown in Fig. 4. The inner quartile distance (IQD) for relative errors over DJF is 21%, which is less than that for the Maurer_ALL and Maurer_RT methods. For N = 5, ISPM tends to underestimate precipitation in contrast to overestimates by the Maurer_ALL and Maurer_RT methods. ISPM underestimates precipitation during JJA for small N, but the biases become small for larger N. The Maurer_ALL and Maurer_RT methods tend to overestimate precipitation during the same period. There are differences between the Maurer_ALL and Maurer_RT methods, which may be attributable in part to the much smaller spatial scale of summer, as contrasted with winter, precipitation. In any event, JJA accounts for a small portion of the annual precipitation in this region—and an even smaller portion of annual runoff; therefore, summertime biases have little practical importance.

Table 2 shows the average daily precipitation based on the retrospective analysis and biases relative to the retrospective analysis by elevation band for the Maurer_ALL and Maurer_RT methods, and the ISPM with different sliding window lengths in DJF of the 5-yr period from 2000 to 2004. The relative error for Maurer_ALL ranges from −15% to 23% over the elevation bands. The relative error of Maurer_RT ranges from 1% to 17%. The relative errors for ISPM with large N are all less than 15% in these elevation bands. All biases are positive for Maurer_RT, whereas for ISPM, most biases are positive, with a declining fraction of the highest elevation bands having negative biases with increasing N (for N = 30 and 60; the higher elevation bands are negatively biased, and the magnitude of the biases are small). The ISPM method implicitly uses PRISM-based orographic corrections as a result of retrieving precipitation from gridded climatological datasets rather than explicitly scaling the precipitation to match the PRISM precipitation climatology. This may be the cause of the elevation dependency of the ISPM method. In general, ISPM produces much better results than Maurer_ALL, especially for the larger values of N.

The accumulated retrospective precipitation and precipitation from ISPM for a range of N are summarized for different elevation bands in Fig. 5. Consistent with Table 2, Maurer_RT substantially overpredicts precipitation in all elevation bands (although the overestimation is very small for the lowest elevation band), whereas Maurer_ALL underestimates for the lowest bands but progressively overestimates for the highest bands, even more so than Maurer_RT. Errors for ISPM—especially for N = 30 and 60—generally have smaller absolute errors, albeit with a tendency for overestimation at low elevations transitioning to a slight downward bias at high elevations.

Figure 6 shows the spatial patterns of the relative errors from 2000 to 2004 averaged over DJF for the reduced station scenario (see Table 1, denoted with the suffix “_r” in the various method designators). Both the Maurer_ALL_r and Maurer_RT_r methods give large relative errors over much of the study area. The Maurer_ALL_r and Maurer_RT_r methods result in large errors when the station number is small. This suggests that the Maurer_ALL_r and Maurer_RT_r methods may not work well when real-time stations are sparse, as is often the case in operational real-time systems. For N larger than 5 days, relative errors are between −20% and 20% of truth for only 35% of grid cells for the Maurer_ALL_r method versus 47% of grid cells for the Maurer_RT_r method and more than 70% of grid cells for the ISPM_r method. The Maurer_ALL_r and Maurer_RT_r methods tend to give positive biases in the southern part of the basin. This may be caused by the larger precipitation at stations south of the basin compared with stations inside the basin (see Fig. 1). ISPM_r has negative bias when the N is small in the northwest part of the basin, where the index stations are sparse. The negative bias is reduced as N increases.

A histogram of the relative errors from 2000 to 2004 over DJF for reduced station scenario is shown in Fig. 7. The relative errors are concentrated between −20% and 20% for most N values in ISPM_r except for N = 5 days. For N = 5, ISPM_r tends to underestimate precipitation in contrast to overestimates by the Maurer_ALL_r and Maurer_RT_r methods. The basin-averaged relative errors are less than 10% for ISPM_r, whereas the basin-averaged relative errors are 26% and 19% for the Maurer_ALL_r and Maurer_RT_r methods, respectively. The IQD for ISPM_r is 21%, which is less than that for the Maurer_ALL_r and Maurer_RT_r methods. The performance of the Maurer_ALL_r and Maurer_RT_r methods were worse than the Maurer_ALL and Maurer_RT methods, for which all index stations were included. The reason may be that the correlations between stations become weaker in the reduced station scenario as the distances between stations become larger. The performance of ISPM, on the other hand, does not suffer much in the reduced station scenario. One possible reason is that the spatial correlations of percentiles (essentially a rank measure) are stronger than the correlations of precipitation rates.

Table 3 summarizes the biases in the methods for the reduced station scenario for DJF. The relative error for Maurer_ALL_r ranges from −21% to 42% over the elevation bands. The relative error for Maurer_RT_r ranges from −1% to 30%. The relative errors for the ISPM_r are between −15% and 18% in these elevation bands. The smaller bias confirms that ISPM_r in general performs considerably better than the Maurer_ALL_r and Maurer_RT_r methods. Comparing with Table 2, the absolute values of the biases become large for most of the elevation bands for Maurer_ALL_r in the reduced station scenario. The absolute biases increase 5%–19% for the elevation bands except for the band of 500–700 m. For Maurer_RT_r, the absolute values of the biases increase 4%–14% for the elevation bands except for the band 0–250 m. The absolute value of ISPM_r biases decrease in higher elevation bands and slightly increase in lower elevation bands. The increases of the absolute value of the ISPM_r biases are less than 5%. The performance of ISPM does not suffer much for the reduced station scenario, whereas biases increase for the Maurer_ALL and Maurer_RT methods. This suggests that ISPM may be a more stable method than traditional interpolation methods as the number of real-time stations decreases.

4. Conclusions and discussion

This paper outlines a new approach—ISPM—for estimating real-time gridded precipitation from index station percentiles for seasonal streamflow forecasting in the western United States. The ISPM methods are evaluated in the Klamath River basin, OR, through comparison with the Maurer_ALL and Maurer_RT methods, which are based on variations of direct interpolation of the same set of index stations. Because the spatial correlation of precipitation is strongest for large-scale synoptic systems, which are responsible for most precipitation and runoff in the western United States, the proposed ISPM method is particularly appropriate for cold season rainfall and seasonal runoff forecasting in the western United States.

In general, ISPM performs better than the direct application of spatial interpolation methods to index stations over the test domain, which is characterized by considerable topographic complexity typical of much of the western United States. The results show that the performance of ISPM generally improves as the sliding window period length becomes longer; however, there is not much difference between sliding window period lengths N = 30 and 60 days. Therefore, we recommend N = 30 days for the best performance of ISPM. ISPM, in general, does not work nearly as well in summer as in winter. However, the absolute biases are small because of the small contribution of summer to the annual precipitation. Most of the index stations in the Klamath basin lay at relatively low elevations (this is also typical of precipitation networks in the western United States), and errors in ISPM depend on elevation. The relative biases of ISPM for N = 30 were all less than 15%, whereas the direct application of spatial interpolation methods tends to overestimate precipitation for most elevation bands. ISPM errors were generally positive (overestimation) at low elevations transitioning to a downward bias at high elevations. The relative performance of ISPM is relatively unaffected by a reduced (real time) station scenario, whereas the performances of methods based on the direct application of spatial interpolation methods suffers.