The proposed approach was explored using daily mean streamflow data from the reference quality streamgages included in the GAGES-II database within the conterminous USA for the period from 1 October 1980 through 30 September 2013. To allow for the interpolation, rather than extrapolation, of all quantiles considered later, streamgages were screened to ensure that at least 14 complete water years (1 October through 30 September) were available for each record considered; 1168 such streamgages were available. The selected reference streamgages are indicated in Fig. . The streamflow data were obtained directly from the website of the National Water Information System (NWISWeb, http://waterdata.usgs.gov; last access: 20 September 2017). For each streamgage, associated basin characteristics were obtained from the GAGES-II database .

To control for streamflow distributions that vary over orders of magnitude, the simulation and analysis of streamflow at these streamgages is best explored through the applications of logarithms. To avoid the complication of taking the logarithm of a zero, a small value was added to each streamflow observation. The US Geological Survey rounds all mean daily streamflow to two decimal places in units of cubic feet per second (cfs, which can be converted to cubic meters per second using a factor of 0.0283). As a result, any value below 0.005 cfs is rounded to and reported as 0.00 cfs. Because of this rounding procedure, the small additive value applied here was 0.0049 cfs. While there may be some confounding effect produced by the use of an additive adjustment, as long as this value is not subtracted on back transformation, the following assessment of bias and bias correction will remain robust. That is, rather than evaluating bias in streamflow, technically this analysis is evaluating the bias in streamflow plus a correction factor. The conclusions remain valid as the assessment still evaluates the ability of a particular method to remove the bias in the simulation of a particular quantity.

Though the potential for distributional bias applies to any hydrologic simulation , for this study, initial predictions of daily streamflow values for each streamgage were obtained by applying the pooled ordinary kriging approach to each two-digit hydrologic unit (Fig. ) through a leave-one-out cross-validation procedure on the streamgages within the two-digit hydrologic unit. The hydrologic unit system is a common method for delineating watersheds in the USA. As described by , the two-digit hydrologic units, or regions (as seen in Fig. ), roughly align with the major river basins of the USA. This approach considers all pairs of common logarithmically transformed unit streamflow (discharge per unit area) for each day and builds a single, time-invariant semivariogram model of cross-correlation that is then used to estimate ungauged streamflow as a weighted summation of all contemporary observations. A spherical semivariogram was used as the underlying model form. Additional information on the time series simulation procedure is provided by . Note that the choice of pooled ordinary kriging is only made as an example of a streamflow simulation method; it is not implied that the bias observed or methods applied are relevant only to this approach to simulation. Because the novelty of this work is in the application of bias correction, further details on the particular simulation method employed are left for the reader to investigate in the cited works .

Daily period-of-record FDCs were developed independently of the streamflow simulation procedure by following a regionalization procedure similar to that of and . Observed FDCs were obtained by determining the percentiles of the streamflow distribution across complete water years between 1981 and 2013 using the Weibull plotting position . A total of 27 percentiles with exceedance probabilities of 0.02 %, 0.05 %, 0.1 %, 0.2 %, 0.5 %, 1 %, 2 %, 5 %, 10 %, 20 %, 25 %, 30 %, 40 %, 50 %, 60 %, 70 %, 75 %, 80 %, 90 %, 95 %, 98 %, 99 %, 99.5 %, 99.8 %, 99.9 %, 99.95 %, and 99.98 % were considered. The selection of streamgages with at least 14 complete water years ensures that all percentiles can be calculated from the observed data. These percentiles derived from the observed hydrograph represent the “unknowable observation” in an application for prediction in ungauged basins. Therefore, to simulate the truly ungauged case, these same percentiles were estimated using a leave-one-out cross-validation of regional regression.

A regional regression across the streamgages in each two-digit hydrologic unit of each of the 27 FDC percentiles was developed using best subsets regression. Best subsets regression is a common tool for exhaustive exploration of the space of potential explanatory variables. All models with a given number of explanatory variables are computed, exploring all combinations of variables. The top models for a given number of explanatory variables are then identified by a performance metric like the Akaike information criterion. This is repeated for several model sizes to fully explore the possibilities for variables and regression size. For each regression, the drainage area was required as an explanatory variable. At a minimum, one additional explanatory variable was used. The maximum number of explanatory variables was limited to the smaller of either six explanatory variables or 5 % of the number of streamgages in the region, rounded up to the next larger whole number. The maximum of six arises from what is computationally feasible for the best subsets regression function used, whereas the maximum of 5 % of streamgages was determined from a limited exploration of the optimal number of explanatory variables as a function of the number of streamgages in a region. Explanatory variables were drawn from the GAGES-II database . As documented by and , a subset of the full GAGES-II dataset was chosen to avoid strong correlations. As the focus of this work is not on the estimation of the FDCs, the reader is referred to the works of and to explore the exact procedures.

In order to allow different explanatory variables to be used to explain percentiles at different streamflow regimes, the percentiles were grouped into a maximum of three contiguous streamflow regimes based on the behavior of the unit FDCs (i.e., the FDCs divided by drainage area) in the two-digit hydrologic units. The regimes are contiguous in that only consecutive percentiles from the list above can be included in the same regime; the result is a maximum of three regimes that can be considered “high”, “medium”, and “low” streamflows, though the number of regimes may vary across two-digit hydrologic units. The percentiles in each regime were estimated by the same explanatory variables, allowing only the fitted coefficients to change. The final regression form for each regime was selected by optimizing the average adjusted coefficient of determination, based on censored Gaussian (Tobit) regression to allow for values censored below 0.005 cfs, across all percentiles in the regime. The addition of a small value was used to avoid the presence of zeros and enable a logarithmic transformation, but this does not avoid the problem of censoring. Censoring below the small value added must still be accounted for so that smaller numbers do not unduly affect the regression. This approach to percentile grouping was found to provide reasonable estimates while minimizing the risk of non-monotonic or otherwise concerning behavior. Further details on this methodology can be found in the associated data and model archive and in .

When estimating a complete FDC as realized through a set of discrete points, non-monotonic behavior is likely . If the regression for each percentile were estimated independently, non-monotonicity would be almost unavoidable. By using three regimes and keeping the explanatory variables the same within each, the potential for non-monotonicity is reduced. The greatest risk of non-monotonic behavior occurs at the regime boundaries. If the FDC used to bias-correct is not perfectly monotonic, the effect will be to alter the relative timing of streamflows. While it would be ideal to avoid any risk of non-monotonic behavior, it is a rather difficult task. An alternative might be to consider the FDC as a parametric function, but demonstrate how difficult this can be for daily streamflows. Of course, the use of regional regression is not the only tool for estimating an FDC for reviews, see.

To implement bias correction, the initial predictions of the daily streamflow values using the ordinary kriging approach were converted to streamflow nonexceedance probabilities using the Weibull plotting position . The nonexceedance probabilities were then converted to standard normal quantiles and linearly interpolated along an independently estimated FDC. For the linear interpolation, the independently estimated FDC was represented as the standard normal quantiles of the associated nonexceedance probabilities versus the common logarithmic transformation of the streamflow percentiles. In the case for which the standard normal quantile being estimated from the simulated hydrograph was beyond the extremes of the FDC, the two nearest percentiles were used for linear extrapolation. In this way, the ordinary kriging simulations were bias-corrected, based on the assumption that the simulated volumes are less accurate than the relative ranks of the simulated values, by rescaling the simulated volumes to an independently estimated FDC. By changing the magnitudes of the simulated streamflow distribution, this approach rescales the distribution of the simulated streamflow.

Figure  provides a simplified representation of this bias correction methodology. Starting in panel a and proceeding clockwise, after simulating the hydrograph with a given methodology (pooled ordinary kriging was used here), the resulting streamflow value on a given day can be converted to appropriate nonexceedance probabilities by proceeding from point A, through point B, and down to point C. Moving then from point C to point D maps the estimated nonexceedance probability onto an independently estimated FDC. Finally, the streamflow value produced at point D is mapped to the original date (point E) to reconstruct a bias-corrected hydrograph. Note that this is a simplified description: as described above a slightly more complex interpolation procedure is used for the FDCs represented by a set of discrete points.

As can be seen in Fig. , this methodology is quite similar to that conceived by and . The novelty of this work lies in its application. That is, both and imagine a case in which the original hydrograph from which nonexceedance values will be drawn (Fig. a) is drawn from an index station of some sort; here the temporal structure could be drawn from any technique for at-site hydrograph simulation. This generalization allows bias correction of any hydrograph simulation.

The hypothesis of this work, that distributional bias in the simulated streamflow can be corrected by applying independently estimated FDCs, was evaluated by considering the performance of these bias-corrected simulations at both tails of the distribution. The differences in the common logarithms of both high and low streamflow were used to understand and quantify the bias (simulation minus observed) and the correction thereof. That is, biass=∑i=1nlog⁡10Q^s,i-log⁡10Qs,in, where s indicates the site of interest, Q^ indicates the predicted streamflow, whether the original simulation or the bias-corrected simulation, Q indicates the observed streamflow, and n indicates the number of values being assessed. This difference can be approximated as a percent by computing 10 to the power of the difference and subtracting 1 from this quantity : biass,%=100⋅10biass-1. The differences in the root mean squared error of the common logarithms of the predicted streamflow were used to quantify improvements in accuracy. The root mean squared error of the common logarithms of streamflow is calculated as rmsels=∑i=1nlog⁡10Q^s,i-log⁡10Qs,i2n. Improvements in accuracy may or may not occur when bias is reduced. The significance of both these quantities, and the effects of bias correction on these quantities, was assessed using a Wilcoxon signed rank test . For assessments of bias, the null hypothesis was that the bias was equivalent to zero. For assessments of the difference in bias or accuracy with respect to the baseline result, the null hypothesis was that this difference was zero.

Distributional bias and improvement of that bias were considered in both the high and low tails of the streamflow distribution. Two methods were used to capture the bias in each tail. One method, referred to herein as an assessment of the observation-dependent tails, considers the observed nonexceedance probabilities to identify the days on which the highest and lowest 5 % of streamflow occurred. For each respective tail, the errors were assessed based on the observations and simulations of those fixed days. The other method, referred to herein as an assessment of the observation-independent tails, compares the ranked top and bottom 5 % of observations with the independently ranked top and bottom 5 % of simulated streamflow. Errors in the observation-dependent tails are an amalgamation of errors in the sequence of nonexceedance probabilities (the temporal structure) and in the magnitude of streamflow, whereas errors in the observation-independent tails only reflect bias in the ranked magnitudes of streamflow. In the same fashion, evaluation of the complete hydrograph can be assessed sequentially (sequential evaluation), retaining the contemporary sequencing of observations and simulations, or distributionally (distributional evaluation), considering the observations and simulations ranked independently. Though the overall accuracy will vary between the sequential and distributional case, overall bias will be identical in both cases.

With an analysis of both observation-dependent and observation-independent tails, it is possible to begin to tease out the effect of temporal structure on distributional bias. The bias in observation-independent tails is not directly tied to the temporal structure, or relative ranking, of simulated streamflow. That is, if the independently estimated FDC is accurate, then even if relative sequencing of streamflow is badly flawed, the bias correction of observation-independent tails will be successful. However, even if the distribution is accurately reproduced after bias correction, the day-to-day performance may still be poor. For observation-dependent tails, the temporal structure plays a vital role on the effect of bias correction. If the temporal structure is inaccurate in the underlying hydrologic simulation, then the bias correction of observation-dependent tails will be less successful.

The bias correction approach was first tested with the observed FDCs. These observed FDCs would be unknowable in the truly ungauged case, but this test allows for an assessment of the potential utility of this approach. This examination is followed by an application with the regionally regressed FDCs described above, demonstrating one realization of this generalizable method. This general approach to bias correction could be used with other methods for estimating the FDC and could also be used with an observed FDC for record extension, though neither of these possibilities are explored here.

There is statistically significant overall bias at the median (-7.1 %; 10-0.0318-1) in the streamflow distribution simulated by the kriging approach applied here (Fig. , box plot A), but more significant bias is apparent in the upper and lower tails of the distribution (Fig. , box plots A, D, G, and J). Both the observation-dependent and observation-independent upper tails of the streamflow distribution demonstrate significant downward bias (Fig. , box plots D and J). At the median, the observation-dependent upper tail is underestimated by approximately 38 % (Table , row 1; Fig. , box plot D), while the observation-independent upper tail is underestimated by approximately 23 % (Table , row 2; Fig. , box plot J). For the lower tail, the observation-dependent tail shows a median overestimation of 36 % (Table , row 1; Fig. , box plot A), while the observation-independent tail is underestimated by less than 1 % (Table , row 2; Fig. , box plot G). The bias is much more variable, producing greater magnitudes of bias more often, in the lower tails than in the upper tails. Generally, biases in the observation-independent tails are less severe, both in the median and in range, than those in the observation-dependent tails. To provide some information on regional performance and incidence of bias, Fig.  shows the spatial distribution of bias in each tail (discussion of this distribution is provided below).

In both observation-dependent and observation-independent cases, downward bias in the upper tail is more probable than upward biases in the lower tail. For the observation-dependent tails, approximately 89% of streamgages show downward bias for the upper tail (Fig. , box plot D), and approximately 61 % of the streamgages upward bias in the lower tail (Fig. , box plot A). For the observation-independent tails, approximately 80 % of streamgages show downward bias in the upper tail (Fig. , box plot J) and approximately 50 % of the streamgages exhibit upward bias in the lower tail (Fig. , box plot G), indicating, as does the small median bias value, that the lower tail biases are relatively well balanced around zero for the observation-independent case for these simulations.

With respect to their central tendencies, these results show upward bias in lower tails and downward bias in upper tails of the distribution of streamflows from the original simulations for both observation-dependent and observation-independent cases. There is, of course, a great degree of variability around this central tendency. With these baseline results, the bias correction method presented here seeks to mitigate these biases.

The results for this idealized case that could not be applied in practice provide clear evidence that distributional bias in simulated streamflow can be reduced by rescaling using independently estimated FDCs. This evidence is apparent in the reduction of the magnitude and variability of overall bias (Fig. , box plot C; Table , rows 5 and 6) and of the bias in the observation-independent tails of the streamflow distribution (Fig. , box plots I and L) when observed FDCs are used for rescaling. Similarly, the overall distributional accuracy is much improved (Fig. , box plot F; Table , rows 5 and 6), as is the accuracy of observation-independent tails (Fig. , box plot I and L). The effect on observation-dependent tails (Fig. , box plots C and F) and overall sequential accuracy (Fig. , box plot C) is less compelling but still substantial.

Whereas the measures of bias and accuracy are summarized in Tables  and , Tables  and  summarize the change in absolute bias and in accuracy, respectively. With the use of observed FDCs, the overall bias is reduced to a tenth of a percent at the median (Table , rows 5 and 6). This represents a significant median reduction of 0.14 common logarithm units in the overall absolute bias (Table , rows 3 and 4). Overall, the distributional accuracy is improved by a median of 0.21 common logarithm units (Table , row 4). Of all streamgages considered, 99 % saw a reduction in the overall absolute bias, and all saw improvements in overall distributional accuracy. These improvements extend to both observation-independent tails of the distributions. The lower observation-independent tails have a median 0.35 common logarithm unit reduction in absolute bias (Table , row 4). For the upper tail, the median reduction in absolute bias is 0.14 common logarithm units (Table , row 4). Nearly all streamgages (99 %) saw reduction in absolute bias of the observation-independent tails. Table  (row 4) shows similar improvements in tail accuracy: -0.37 and -0.15 units in the lower and upper tails, respectively, with nearly all streamgages (excepting the lower tail of a single streamgage, likely the result of the interpolation procedure) showing improved tail accuracy.

With the use of a perfect, observed FDC for bias correction, one would expect that nearly all bias would disappear, but the results do not show this. The temporal structure of the simulated hydrograph continues to play a role in the bias of observation-dependent tails. The observation-independent tail continues to exhibit a small degree of residual bias, though it is still slightly nonintuitive. This residual bias arises from the effect of representing the FDC as a set of discrete points and interpolating between them. There may be some additional effect from the small value added to avoid zero-valued streamflows or the censoring procedure, but initial exploration found little impact.

The overall sequential performance (Fig. , box plot C) and the performance of observation-dependent tails (Figs.  and , box plots C and F) demonstrate the degree to which errors in the temporal structure result in bias in the observation-dependent case even when observed FDCs are used for bias correction. Both the observation-dependent lower and upper tails exhibit bias: 30 % and -20 %, respectively, at the median (Table , row 5, with transformation using Eq. 2). Absolute bias in both tails shows median reductions; sequential accuracy and observation-dependent tail accuracy are also improved at the median (Tables  and , row 3). Proportionally, 82 % of the observation-dependent lower tails and 86 % of the observation-dependent upper tails showed reduction in absolute bias (Fig. , box plots C and F); 85 % of observation-dependent lower tails and 79 % of observation-dependent upper tails showed improvements in accuracy (Fig. , box plots C and F). Despite improvements in overall bias and accuracy from rescaling with observed FDCs, the residual bias in the observation-dependent lower tail (Fig. , box plot C) is almost always positive (upward bias) and upper tails (Fig. , box plot F) are almost negative (downward bias), a result which arises primarily from errors in the simulated temporal structure.

To understand the effect of errors in the temporal structure further, consider Fig. , which shows the mean error in the nonexceedance probabilities, i.e., the difference in the ranks of the observed and simulated streamflows, of the observation-dependent upper and lower tails. The nonexceedance percentages in the lower tail are overestimated by a median of 3.8 points, with 5th and 95th percentiles of 0.9 and 20.5, while the percentages in the upper tail are underestimated by 2.4 points, with 5th and 95th percentiles of -0.5 and -12.6 points. The distributions of errors in the nonexceedance probabilities closely reflect the distribution of bias in the observation-dependent tails (Fig. , box plots C and F). These results show that the inaccuracy in the nonexceedance probabilities (i.e., errors in temporal structure) will obscure, at least partially, the improvement offered by bias correction when considering the observation-dependent errors, even when an observed FDC is used for bias correction. These errors in temporal structure also almost always result in errors in a particular direction – low for high flow and high for low flows.

When the uncertainty of regionally regressed FDCs is introduced into the bias correction procedure, the potential value of the bias correction procedure is not as convincing. There is a slight, but significant, increase in the overall bias (Table , rows 1 and 2). Whereas the original estimated streamflow displays a median bias of approximately 7.1 %, the median overall bias is approximately 7.6 % after bias correction with estimated FDCs (Table , rows 3 and 4). Though statistically significant, the distribution of bias does not appear to have changed in a meaningful way (Fig. , box plots A and B). The overall accuracy, sequential and distributional, is also degraded (Fig. , box plots B and E; Table , rows 3 and 4), with more than 60 % of streamgages showing degradation in sequential and distributional accuracy.

The observation-independent tails, which are not affected by errors in temporal structure, show a divergence in performance between the results obtained using observed FDCs and those obtained using regionally regressed FDCs. With observed FDCs, both tails demonstrated substantial reductions in absolute bias and improvements in accuracy. With regionally regressed FDCs, the upper observation-independent tails continue to show reductions in absolute bias (Table , row 2; Fig. , box plots J and K) and improvements in accuracy (Table , row 2; Fig. , box plots J and K), while the lower observation-independent tails show a significant increase in absolute bias (Table , row 2; Fig. , box plots G and H) and a degradation of accuracy (Table , row 2; Fig. , box plots G and H). After bias correction with regionally regressed FDCs, only 44 % of observation-dependent lower tails showed reductions in absolute bias; 58 % of upper tails showed reductions in absolute bias.

The effects of the rescaling with FDCs estimated with regional regression on overall and observation-independent tail bias and accuracy can be better understood if the properties of the estimated FDCs are considered. Figure  shows the bias (panel a) and accuracy (panel b) of the lower and upper tails of the regionally regressed FDCs. Recall that the estimated FDCs are composed of 27 quantiles, of which the upper and lower tails contain only the eight values with nonexceedance probabilities 95 % and larger and 5 % and smaller, respectively. The upper tails are reproduced through regional regression with an insignificant 2.5 % median downward bias, but the lower tails exhibit a significant negative median bias of 38.35 % (Table , row 7). Because of this bias in the lower tail of the regionally regressed FDCs, the regionally regressed FDCs are unable to correct the bias in the simulated hydrograph, instead turning a small median bias into large one. As there is no temporal uncertainty in the observation-independent tails, the resulting bias arises from the bias of the regionally regressed FDC. Illustrating this fact, the -38 % bias in the lower tail of the regionally regressed FDC approximates the -33 % in the observation-independent lower tail, while the -2.5 % bias in the upper tail of the regionally regressed FDC approximates the -3.7 % bias in the observation-independent upper tail. The introduction of this additional bias, beyond failing to correct any underlying bias in the simulated hydrograph, also markedly increased the variability of both bias and accuracy.

The results are similar for the observation-dependent tails produced after bias correction with regionally regressed FDCs, even when complicated by the addition of temporal uncertainty as discussed in Sect. 3.2 with reference to Fig. . In some cases, the errors in the temporal structure (nonexceedance probability) counteract the additional bias from regionally regressed FDCs. For example, the observation-dependent lower tails have a median bias of 13 %, which possesses a smaller magnitude and different sign than the median -33 % bias seen in the observation-independent lower tail (Table , rows 3 and 4). The addition of temporal uncertainty actually reduced the increase in absolute bias (Table , rows 1 and 2) and reduced the degradation of accuracy in the lower tail (Table , rows 1 and 2). These slight improvements result from an offsetting of the underestimated regionally regressed FDCs by the overestimated nonexceedance probabilities. While interesting, it seems unlikely that this result can be generalized in a simple way: that is, the errors in estimated FDCs cannot be expected to balance out the errors in nonexceedance probabilities without deleterious effects on other properties. To this point, as noted, rescaling by these regionally regressed FDCs with underestimated lower tails results in similarly underestimated observation-independent lower tails.

The introduction of uncertainty from regionally regressed FDCs diminishes the advantages gained by biased correction with observed FDCs. Considering the observation-independent lower tails, 55 % of streamgages show reductions in absolute bias with observed FDCs that were reversed into increases of absolute bias by the introduction of regionally regressed FDCs. Another 43 % of streamgages show smaller reductions in absolute bias when observed FDCs were replaced with regionally regressed FDCs. For the observation-dependent lower tails, 37 % of streamgages have reversals and 31 % show smaller reductions in absolute bias. For the observation-independent upper tails, 41 % show reversals and 56 % yield smaller reductions in absolute bias. For the observation-dependent upper tails, 24 % produce reversals and 40 % provide smaller reductions in absolute bias. Results are similar with respect to accuracy: while many streamgages saw reversals, a large proportion of streamgages continue to demonstrate improvements.