Using correlation of daily flows to identify index gauges for ungauged streams

Authors


Corresponding author: L. L. Yuan, Office of Science and Technology, Office of Water, U.S. Environmental Protection Agency, 1200 Pennsylvania Ave. NW, Mail Code 4304T, Washington, DC 20460, USA. (yuan.lester@epa.gov)

Abstract

[1] Predictions of daily flows in ungauged streams frequently rely on index gauges, where the timing of the daily flow at the index gauge is assumed to be similar to that in the ungauged stream. When some limited flow measurements are available at the ungauged sites, the strength of correlation between these flows and candidate index gauges can inform the selection of the index gauges. Here the use of the correlation of daily flows to select index gauges is systematically evaluated using a large flow data set from the Ohio River Valley, USA. Then, a novel method is introduced for predicting the strength with which daily flows at different gauges are correlated with daily flow at a completely ungauged site, using the physical characteristics of the ungauged site. The index gauges can then be selected based on these predicted correlations. The analysis indicates that this new method provides a means of identifying index sites that will yield a desired level of accuracy in flow predictions at ungauged locations. The ungauged sites at which sufficient index gauges are not available are also identified, and flow predictions are not calculated for these sites. Using this new method improves the overall accuracy of predicted flows, relative to existing methods.

1. Introduction

[2] A wide variety of hydrologic applications (e.g., management of dams and flood control planning) require knowledge of daily flows in ungauged stream reaches. Accurate interpretation of many stream ecological and water quality characteristics can also depend on quantifying flow at the time of sampling. For example, stream nutrient concentrations, such as total phosphorus, are strongly dependent on flow magnitude. High-velocity flows observed during periods of bankfull and larger floods include riparian and general overland flows that carry nutrient-laden sediments into the stream channel, which, in turn, greatly increase nutrient concentrations. Conversely, during periods of low flows, flow often originates from subsurface sources that have comparatively lower total phosphorus concentrations [Banner et al., 2009]. Hence, nutrient concentrations observed in individual samples are best interpreted in the context of the flow conditions at the time of water quality sampling. Similarly, the amount of algae accumulation on stream substrates depends strongly on the flow history prior to the time of sampling [Biggs, 2000].

[3] Flow conditions in different streams vary strongly in both space and time because of differences in the location and timing of precipitation events, differences in the hydrological characteristics of the stream network, and differences in physical characteristics of different watersheds. Thus, predicting daily streamflow in ungauged basins has presented a long-standing analytical and conceptual challenge [Sivapalan, 2003]. Predictions of particular flow statistics (e.g., 100 year flood, base flow) in ungauged basins have been based on statistical models that related catchment characteristics to the flow statistic of interest [Santhi et al., 2008; Haddad et al., 2012], and prediction approaches based on spatial statistical analyses have also been proposed [Skøien et al., 2006; Skøien and Blöschl, 2007; Castiglioni et al., 2009]. However, to predict historical daily flows, flow on each day is inherently a different flow statistic, and hence, in theory, one must calibrate a different model for each daily flow one wishes to predict.

[4] Other approaches more efficiently predict flows on many different days by requiring that one select one or more “index gauges” for each ungauged location. Index gauges are gauges at which the timing of different flow events is assumed to be the same as that of the ungauged location. Then, one computes daily flows by combining the timing information with an estimate of the relative magnitude of the flow in the ungauged basin. For example, one commonly used approach for predicting daily flows assumes that the timing of flow in an ungauged basin is the same as that in the gauged basin, and that the ratio of the magnitude of flows in the two locations is equivalent to the ratio of the basin areas [Hirsch, 1979].

[5] Index gauges have historically been selected based on the geographic proximity to the location of interest and by best professional judgment, but different mathematical and statistical approaches for selecting index gauges have been recently proposed. For example, recent studies have demonstrated that the use of multiple index gauges can improve the accuracy of flow predictions [Smakhtin, 1999; Zhang and Kroll, 2007; Shu and Ouarda, 2012]. Further improvements in predictions were also observed when the contributions of different gauges were weighted by the distance between index gauges and the ungauged site or weighted by the degree to which certain preselected physical characteristics of the gauged basin were similar to those of the ungauged basin [Shu and Ouarda, 2012].

[6] An alternate approach for selecting index gauges is based on the idea that the gauges at which flows are most strongly correlated with the daily flows at the ungauged site should be selected as index gauges. The application of this approach is most easily conceptualized when a discrete or partial flow record is available at the ungauged site. Then, one calculates the correlation between flows at the ungauged site and candidate gauged sites using just the available flow data and selects the index gauge based on the gauge that exhibits the strongest correlation [Reilly and Kroll, 2003; Eng et al., 2011]. One method for predicting the strength of correlation at sites in which no flow data are available has recently been proposed [Archfield and Vogel, 2010]. In this approach, spatial interpolation is used to predict the expected correlation between daily flows at the ungauged site and flows at a candidate index site. This interpolation is repeated for all available candidate index sites, and then a final index site is selected based on the largest predicted correlation coefficient.

[7] Several questions arise that are directly connected with the use of the strength of correlation of daily flows to select index gauges. First, what is the minimum degree with which daily flows in index gauges can be correlated with flows at the ungauged site to ensure that the index gauge provides useful information regarding the flow timing? Second, how many index gauges should one select? Third, and perhaps most important, how can one best select an index gauge for an ungauged site when partial flow data are not available? Here I analyze historical daily flows recorded at gauges in the Ohio River Valley, USA, to answer the first two questions. I then describe a novel method for identifying the reference gauges that uses statistical models to predict the strength with which daily flows at an ungauged site are correlated with the available flow gauges based on the basin physical characteristics and the site location. Based on this prediction, the best index gauges can be selected.

2. Methods

2.1. Data

[8] The daily flows from 702 U.S. Geological Survey flow gauges collected from 1990 to 2010 from the Ohio River Valley were used to develop and demonstrate the method proposed here. The daily flow data were counted, and the gauges that did not have observations from at least 90% of the days during the period of interest were excluded.

[9] Variables characterizing the gauged sites and their upstream catchments that were likely to affect either the flow timing or the magnitude were extracted from the GAGES (Geospatial Attributes of Gages for Evaluating Streamflow) database [Falcone et al., 2010]. These variables characterized catchment land cover, base flow, volume of impounded flow, annual precipitation, catchment area, and geographic location of the gauged site. Methods used to derive these variables are thoroughly documented by Falcone et al. [2010], and so only a brief description of the selected variables is provided here. Catchments were delineated for each gauge and catchment area computed. The proportion of the area of the catchment used for agriculture (CROP) was derived from the National Land Cover Data from 2006 [Xian et al., 2009]. The mean catchment base flow index, the ratio of base flow to total flow (BFI), was estimated for each catchment from mapped data [Wolock, 2003]. The average precipitation in the catchment (PPT) was derived from the mapped climate variables, and the volume of impounded water in the catchment (IMP) was calculated from the data provided by the U.S. Army Corps of Engineers (http://geo.usace.army.mil/pgis/f?p=397:1:0).

[10] Gauges in the data set were screened to exclude the gauged sites that were affected by impoundments because streamflows at these locations are affected by the operation of the impoundment rather than by the precipitation events and the natural movement of flow through the stream network. Hence, the initial list of gauges was screened to only include the gauges at which IMP, scaled by the catchment area, was less than 50 ML/km2. This criterion provides a conservative selection of gauges representing unimpounded flow, as visual inspection of the satellite imagery for catchments with scaled impoundment volumes on the approximate order of 50 ML/km2 indicated that the majority of impoundments in these catchments were small retention basins off of the main channel and therefore unlikely to strongly affect the timing of flow. After this first screen, satellite imagery for remaining gauge sites was examined, and a few sites with main stem impoundments less than 3 km upstream were excluded. This second screen accounted for cases in which the scaled impoundment volume was low, but the impoundments that did exist were directly upstream. After screening for impoundments, data from 257 gauges were retained for further analysis (Figure 1).

Figure 1.

Map of gage locations used in analysis and major stream systems in the Ohio River Valley.

2.2. Statistical Analysis

[11] Two related statistical analyses were conducted to address the questions associated with using the daily flow correlation to identify index gauges. First, the observed (i.e., known) correlations between daily flows at pairs of gauges were used to select index gauges. Using the known correlations provided a means of examining the effects of the strength of correlation between daily flows at ungauged sites and potential index sites without the additional uncertainty associated with predicting correlations. Second, a new method was developed to predict the expected strength of correlation between candidate index gauges and daily flows at ungauged locations.

2.2.1. Analysis Using Known Daily Flow Correlations

[12] The selection of index sites based on the strength of correlation between daily flows at candidate index gauge sites and ungauged sites was examined with a hold-one-out cross validation procedure, in which each gauged site from the database was sequentially designated as the “ungauged site” and the remaining gauges designated as potential index sites. The Pearson correlation coefficients (r) between the log-transformed daily flows at the designated ungauged site and the potential index gauges were computed, and the index gauges that had the highest correlation coefficient were selected. Prior to log-transforming the daily flows, observations that were zero (i.e., no observed flow) were replaced with half of the minimum observed daily flow from the entire data set to avoid issues with log-transforming values of zero. From 1 to 12 index gauges were selected for each ungauged site, yielding 12 sets of index gauges for each ungauged site. The maximum correlation coefficient associated with the selected index gauges was also recorded for each of these 12 cases. After identifying the index gauges, predictions of flow quantiles at the “ungauged site” for each day in the period of interest were computed as the average of the flow quantiles for that same day at all the selected index gauges.

[13] To predict daily flow, flow quantiles for each day at the ungauged site were combined with an estimate of the flow magnitude for each quantile. To this end, a regional flow duration model [Smakhtin, 1999] was calibrated using daily flows at the potential index sites (i.e., all available gauges except for the specified “ungauged site”). Generalized additive models (GAMs) [Wood, 2006] were used to relate the flow magnitude for 13 different percentiles (1%, 5%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 95%, and 99%) to three predictor variables. These predictor variables (AREA, BFI, and PPT) were selected based on the exploratory data analysis in which the relationship between the flow magnitudes for different percentiles was plotted against the potential explanatory variables. The use of the nonparametric curves, as provided by GAMs, allows for nonlinear relationships between the flow magnitude and each of these variables, increasing the accuracy of the final predictions. These nonparametric curves were estimated using penalized regression splines, which provide a localized, smooth fit to the data [Wood and Augustin, 2002]. After fitting the regional flow duration model, a local flow duration curve for the ungauged site was estimated by applying the model to the known values of AREA, BFI, and PPT at the ungauged site.

[14] Once a local flow duration curve was estimated, a prediction of flow magnitude at any arbitrary flow quantile was estimated using a cubic spline interpolation [Press et al., 2007]. A cubic spline fits a cubic polynomial between pairs of available points subject to the condition that the slope of each segment matches the slope of neighboring segments at available data points. Hence, a cubic spline function is continuous in both its value and its first derivative across all of the available data points. Extrapolated values beyond the range of the data (i.e., quantiles greater than 99% or less than 1%) were computed by assuming that the slope of the interpolated line at the highest and lowest percentiles was constant beyond the range of the data. That is, a straight line was extended from each terminal data point using the interpolated slope at that point (Figure 2). Others have employed log-linear interpolation to estimate flow magnitudes for arbitrary flow quantiles, and the present approach is conceptually identical. Additional refinements to estimating daily flows from flow duration curves, such as using alternate models for the highest and lowest percentiles [Archfield et al., 2010], were not considered but could easily be incorporated.

Figure 2.

Example of use of cubic spline to interpolate between different flow percentiles. Open circles show predicted flow percentiles. Solid line shows the cubic spline interpolation.

[15] So, in summary, to predict historical daily flows at each held-out site, the flow quantile for each day in the period of record was computed as the average of the flow quantiles at the selected index sites. This flow quantile was then converted to a flow magnitude by interpolating between the predicted flow magnitudes at 13 percentiles in the local flow duration curve.

[16] Other approaches for transferring the flow magnitude from the gauged sites to the ungauged locations are also used, such as the weighting flow magnitude by the ratio of the catchment areas of the ungauged and gauged sites. However, recent work has shown that the flow duration method provides the accurate predictions of the flow magnitude [Shu and Ouarda, 2012], and so in the present analysis the flow duration approach is exclusively used.

[17] The accuracy of the predicted flow at the held-out site was quantified by calculating the Nash-Sutcliffe model efficiency coefficient [Nash and Sutcliffe, 1970] defined as follows:

display math

in which qi is the log-transformed observed daily flow on day i, inline image is the predicted log-transformed daily flow, qm is the mean of the log-transformed observed daily flows, and n is the number of days in the period of record. Log-transformed flows were used in this computation to emphasize the correct prediction of low flows, but the approach described here can be applied using other transformations of the daily flow values [Oudin et al., 2006].

2.2.2. Selecting Index Gauges Based on Predicted Correlation Strength

[18] The new method described here for identifying index gauges provides a means of predicting the strength with which daily flows at an ungauged site are correlated with daily flows at different candidate index sites. Then, based on this prediction, one selects appropriate index sites. This approach applies existing methods used to predict the composition of biological communities [Wright et al., 1984; Hawkins et al., 2000; Linke et al., 2005] to the problem of predicting flow in ungauged basins.

[19] The first step for the approach is to ordinate the available flow gauges in the study area on the basis of the strength of correlation of their daily flow records. Ordination is a technique used frequently in ecology and other disciplines to “order” different samples in an n-dimensional space based on many different environmental and/or biological variables [Gauch, 1982]. After ordination, sites that are similar to one another with respect to the selected variables are nearer to one another in the n-dimensional space than sites that are dissimilar from one another. This n-dimensional space is also known as “ordination space.” In past work with the biological communities, sites are ordinated based on the similarity between the observed species at each site [Linke et al., 2005]. In the present application, ordination placed sites with strongly correlated daily flow records in close proximity with one another in ordination space, whereas sites with more weakly correlated daily flow records were distant from one another.

[20] After ordinating the available sites, the second step is to develop statistical models that relate the position of each gauged site in ordination space with the physical characteristics of that site and its upstream catchment (e.g., catchment area and geographic location). Third, after these models are fit, they are used to predict the location of an ungauged site in ordination space, given the known physical characteristics of the ungauged site. Because distance in ordination space is related directly to the strength with which the daily flows are correlated, index gauges can be selected as the gauges that are near the predicted location of the ungauged site in ordination space. More specific details for each of the steps of this method are provided in the rest of this section.

2.2.2.1. Ordinating Available Flow Gauges

[21] To ordinate sites, one must first quantify the dissimilarity between each pair of sites. Because we wanted gauged sites with strongly correlated daily flow records to be near one another in ordination space, dissimilarity between pairs of gauges was defined as d = 1−r, where r is the Pearson correlation coefficient between the log-transformed daily flows of a pair of flow gauges. By definition, r values range from −1 to 1, where a value of 1 indicates a perfectly correlated, linearly increasing relationship between the observed daily flow records at two gauges, and a value of 0 indicates no relationship between the daily flows. Hence, 1−r ranges theoretically from 0 to 2, where 0 indicates the identical flow records (i.e., no dissimilarity), and 1 indicates the completely dissimilar flow records. Values greater than 1 were rarely observed, as these would indicate an inverse correlation between the daily flows at a pair of gauges.

[22] The dissimilarity between the flow records at different gauges was ordinated using nonmetric multidimensional scaling (NMS) [Kruskal and Wish, 1978]. An NMS calculation is formulated as an optimization problem, in which the algorithm seeks to minimize the value of an objective function that is defined as the difference between the rank order of distances between the sites in ordination space and the rank order of dissimilarities that are observed in the data. The optimization calculation continues iteratively until further improvement in the objective function is not observed, indicating that the best solution has been achieved. NMS requires that one specifies a priori the number of dimensions used to represent the data, and in general, the more dimensions that are used, the more closely distances in ordination space correspond with observed dissimilarities. To illustrate the method, an NMS ordination using just two dimensions is shown in Figure 3; however, more dimensions are typically required to accurately represent the pairwise dissimilarities in ordination space. To select the number of NMS dimensions used in the final model, I ran a series of NMS ordinations varying the number of dimensions from 3 to 10, and for each ordination I examined the accuracy with which the distance between pairs of gauges in ordination space predicted the observed dissimilarities between the same pair of gauges. This accuracy was quantified by calculating a mean square prediction error (MSPEORD) as follows:

display math

where dORD,i is the Euclidean distance between pair i of gauges; di is the dissimilarity between the daily flows at the same pair of gauges, defined above as 1−r; and Npair is the number of unique pairs of gauges in the data set. The function metaMDS included in the R library, vegan, was used to compute the NMS ordination (Oksanen et al., vegan: Community Ecology Package, 2012, http://cran.r-project.org/web/packages/vegan/index.html).

Figure 3.

Example of the use of ordination to predict correlation strength. (left) Map of gauges. (right) Gauges in ordination space, calculated using two NMS dimensions. The filled black circle shows the same ungauged site, and the black ring encloses (left) the geographic area or (right) area in ordination space in which daily flows are predicted to be correlated with daily flows at the ungauged site with a Pearson correlation coefficient (r) > 0.95.

2.2.2.2. Modeling Position in Ordination Space

[23] After the ordination was computed, the positions of different gauged sites along each axis in ordination space were modeled as a function of site physical characteristics using GAMs. A different GAM was fit to predict the position along each axis in ordination space. Five predictor variables were identified through exploratory analysis and used in each model: geographic location (LAT and LON, expressed as degrees latitude and degrees longitude), CROP, log-transformed catchment area (AREA), PPT, and BFI. Models for the position along each axis can be expressed as follows:

display math

where xi is the position on the ith axis in ordination space, σ is a normally distributed residual error, and s(·) indicates a penalized regression spline function when applied to a single variable. In the case of latitude and longitude, s(·) represents a thin-plate regression spline, which fits a continuous surface that modeled position on each NMS axis as a function of both latitude and longitude [Wood, 2006].

[24] To estimate the relative contribution of each term in the model, five alternate models were fit, each with one term omitted. Then, the effect of each term was estimated as the difference between the residual deviance of the full model and the residual deviance of the model with the excluded term, divided by the residual deviance of an intercept-only model. The resulting value can be interpreted as the reduction in R2 values one would expect if the selected term was omitted from the predictive equation. However, this value is only an estimate because in the refitted models (with a single excluded variable), the remaining predictor variables can account for some of the variability that was originally explained by the excluded variable. Hence, these values are best interpreted relative to one another. The accuracy of the statistical models in predicting the position along each NMS axis was also quantified as a mean square predictive error, defined as follows:

display math

where N is the number of gauges, xij,obs is the observed position on NMS axis i for sample j, and xij,pred is the predicted position for that same sample.

[25] To demonstrate the application of this method, the results from the simplified model, in which the position on each of the two NMS axes is modeled just as a function of latitude and longitude, are shown in Figure 4. Contour lines showing the predicted positions along the first NMS axis (Figure 4, left) are roughly parallel with the southern half of the Ohio River, indicating that daily flows remain correlated (i.e., located in the same position in ordination space) between pairs of gauges separated by large distances on segments that are parallel with the course of the Ohio River, whereas gauges that are separated by distances along segments that are normal to the course of the river are less strongly correlated. Note that all gauging stations that were directly located on the main stem of the Ohio River were excluded from the data set because of the extensive impoundments that exist along this river (see Figure 1). Hence, the modeled relationships reflect similarities in flow in tributaries to the main stem.

Figure 4.

Example of modeled relationships between geographic location and position in ordination space. Contour lines show predicted position along (left) the first NMS axis and (right) the second NMS axis.

2.2.2.3. Selecting Index Gauges

[26] After fitting the regression models, the best reference gauges for an ungauged location were identified as follows. First, the mean position of the ungauged location in ordination space was predicted using the fitted GAM models and the known physical characteristics of the ungauged site. Second, gauges that were located within a prespecified distance from the predicted location of the ungauged site in ordination space were selected as the index gauges. As discussed earlier, since distance in ordination space provides an estimate for the actual observed dissimilarity, specifying the allowable distance between an index gauge and the predicted location of the ungauged site in ordination space essentially specifies the minimum strength of correlation between daily flows at the ungauged location and any selected index gauge.

[27] In Figure 3, this process is illustrated using the simplified model with two NMS dimensions and position in ordination space modeled only as a function of LAT and LON. In this example, a circle of radius 0.05 is shown in ordination space, and so daily flows at gauges located within this circle are predicted to correlate with daily flows at the ungauged site with r > 0.95. When transformed to geographic space using the GAM models, the area enclosing potential index gauges becomes a narrow oval, oriented along the course of the river. Hence, selection of index sites using the predicted correlation strength accounts for the differences in how daily flows between gauges are correlated, preferentially selecting gauges where daily flows are more likely to be strongly correlated with flows at the ungauged site. The full model using more than two NMS dimensions and all predictor variables is too complex to display using scatter plots, but conceptually the approach is identical to that shown in this example.

[28] The results of the analysis using the known daily flow correlations informed the selection of a minimum strength of correlation of r = 0.9 or equivalently a maximum allowable distance in ordination space of 0.1. For similar reasons, only the four gauges that were closest to the predicted location of the ungauged site were selected as index gauges. After identifying index gauges, flow quantiles at the ungauged site for each day in the period of interest were computed as the average of the quantiles for that same day at all selected index gauges.

2.2.2.4. Assessing Model Performance

[29] To quantify the predictive accuracy of this method, I used the same hold-one-out cross validation described above, in which I sequentially held out each gauge from the data set. Then, using the remaining gauges, I identified index gauges using the method described above and fit regional flow duration curves. Daily flow at the held-out site was then computed for each day of the period of record using the methods described above. The accuracy of the predicted flow was quantified by calculating the Nash-Sutcliffe model efficiency coefficient. The entire validation analysis was repeated with NMS ordinations computed for five, seven, and nine dimensions to examine the effects of the number of NMS dimensions on the final model performance.

[30] To compare the performance of the new index gauge selection procedure described here with other approaches for selecting index gauges that are in common use, I also selected the four nearest geographic neighbors as index gauges and weighted their flow quantiles for each day inversely by geographic distance from the ungauged location. This approach was among the methods that yielded the most accurate predictions of daily flow in a recent study [Shu and Ouarda, 2012].

3. Results and Discussion

3.1. Regional Flow Duration Model

[31] The selected variables (AREA, BFI, and PPT) effectively accounted for the vast majority of variability in flow quantiles, with R2 values for all but the lowest two quantiles greater than or equal to 0.9 (Table 1). Others have also observed that the predictive model performance was weakest when accounting for variations in the lowest flow quantiles [see, e.g., Mohamoud, 2008]. Relationships between explanatory variables and the flow magnitude for different flow quantiles generally conformed with expectations (Figure 5). For example, flow magnitude increased with increased catchment area for all flow quantiles. The effects of BFI were strong for low flow quantiles, but the effects of BFI did not differ significantly from zero for high flow quantiles. These observations make sense intuitively, given the importance of base flow in determining the magnitude of low flow quantiles. Effects of mean annual precipitation were generally weak.

Figure 5.

Partial dependence plots showing relationship between predictor variables and different flow percentiles. Each line in each plot shows the modeled relationship for different percentiles (1%, 5%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 95%, and 99%). For clarity, models for 1%, 50%, and 99% are shown in black, and other percentiles are shown in gray and are stacked according to the numerical order of the percentiles.

Table 1. R2 Values for Models Predicting Flow Magnitude for Different Flow Quantiles Expressed in Terms of Nonexceedance Probability
Flow QuantileR2
0.010.79
0.050.86
0.10.90
0.20.94
0.30.96
0.40.97
0.50.98
0.60.98
0.70.98
0.80.99
0.90.99
0.950.99
0.990.98

3.2. Identifying Index Gauges Using Known Correlation Strengths

[32] When index gauges were selected based on the known correlation strength, the average accuracy of predicted daily flows across all gauges varied strongly with the number of selected index gauges (Figure 6). The accuracy of predictions improved up to four selected index gauges and degraded with further increases in the number of selected index gauges. Others have observed that the use of multiple index sites improves predictive accuracy [Zhang and Kroll, 2007]. Indeed, the use of four index sites was associated with the most accurate predictions in another study [Shu and Ouarda, 2012]. Ultimately though, the best number of index sites to use strongly depends on the configuration and density of candidate index gauges in a study area, and conducting a similar exploratory analysis to that shown here would be expected to provide useful information for making this decision.

Figure 6.

Relationship between the number of index sites chosen and Nash-Sutcliffe efficiency. Index sites were selected by picking the indicated number of sites with the strongest observed correlation of daily flows with the held-out site.

[33] After specifying four index gauges for each held-out gauge, the average predictive accuracy across all gauges was still strongly associated with the maximum correlation coefficient observed in the selected index sites (Figure 7), a trend that has been observed in previous analyses [Archfield and Vogel, 2010]. Hence, to further improve the model performance, a minimum allowable correlation coefficient could be specified, and if the selected index gauges do not satisfy this requirement, then a prediction of flow would not be calculated. That is, by specifying a minimum number of index gauges and a minimum allowable correlation coefficient, we define the conditions under which acceptably similar flow gauges are available for predicting the flow timing at the ungauged site.

Figure 7.

Relationship between minimum r and Nash-Sutcliffe efficiency. Minimum r is the minimum correlation coefficient among four selected index sites.

[34] When selecting these model parameters (minimum number of index sites and minimum allowable correlation strength), one must weigh the relative benefits of the increased accuracy of daily flow predictions versus the proportion of locations in a study area at which the model is applicable. More specifically, if we require a high degree of accuracy in flow predictions, we are likely to find many sites without a sufficiently similar index gauge. For this analysis, I selected a minimum correlation strength of 0.9 among the four selected index gauges. In the data set, 124 out of 257 gauges did not have index gauges that met this criterion, but the predicted flows at the remaining gauges were characterized by an average Nash-Sutcliffe coefficient of 0.881 (Table 3).

3.3. Identifying Index Gauges Using Predicted Correlation Strength

[35] The accuracy with which distances between pairs of gauges in ordination space predicted the observed dissimilarity between gauges increased with increasing numbers of NMS dimensions as expected (Figure 8). MSPEORD decreased sharply with increased numbers of dimensions up to six dimensions. Further increases in the number of dimensions were associated with less dramatic decreases in MSPEORD.

Figure 8.

Mean square prediction error for predicting observed dissimilarity from Euclidean distance in ordination space.

[36] Statistical models predicted position along each NMS axis with relatively high accuracy with values of MSPEpos of approximately 0.02 for all axes (Table 2). As shown earlier (Figure 4), modeled relationships between geographic location and position along a given ordination axis were complex, but some were interpretable in terms of geomorphology. Overall, geographic location consistently accounted for the vast majority of the variation in the position along each of the NMS axes (see the summary statistics of models predicting position along the nine NMS axes in Table 2). This finding was expected, owing to the fact that flow in rivers that are nearer to one another is likely to be influenced by the same precipitation events. Furthermore, the location of a gauge likely acts as a proxy for other physical variables that influence flow timing, such as characteristics of the stream network in the catchment [Naden, 1992] and the differences in the geological setting that influence flow pathways [Brown et al., 1999].

Table 2. Approximate Reduction in R2 for Models Predicting Position Along Each NMS Axis Associated With Excluding the Indicated Variablea
 NMS Axis
123456789
  1. a

    Last row shows the overall mean-square prediction error for models (MSPEpos) for each NMS axis.

Geographic location0.1070.4290.2490.6070.5090.4060.4890.5690.516
PPT0.0050.0040.0050.0020.0090.0010.0290.0230.006
AREA0.0000.0070.0600.0220.0250.0920.0290.0400.035
CROP0.0010.0020.0090.0100.0000.0280.0050.0000.006
BFI0.0020.0030.0180.0080.0210.0080.0180.0170.036
MSPEpos0.0220.0180.0240.0260.0250.0200.0260.0210.021
Table 3. Summary of Predictive Accuracy of Different Modelsa
 Nash-Sutcliffe CoefficientN (No Predict)
 Min25th PercentileMean75th PercentileMax
  1. a

    Column labeled N (no predict) gives the number of sites for which at least four appropriate index gauges were not identified.

Known correlations0.6490.8500.8810.9200.974124
Four nearest neighbors−0.5290.7430.7810.8790.9530
Predicted Correlations
Five NMS dimensions−0.5200.7610.7900.8780.9718
Seven NMS dimensions−0.5200.7850.8110.8930.97156
Nine NMS dimensions−0.5440.7970.8230.8970.96583

[37] After geographic location, catchment area accounted for the largest proportion of variance in the positions along NMS axes, accounting for 6% and 9% of the observed variance for NMS axes 3 and 6, respectively (Table 2). Catchment area is the dominant factor in explaining the variations in flow magnitude, but these results provide support for the notion that catchment area also exerts substantial influence on the timing of flow events, perhaps because potential storage within a catchment increases with area or because routing times of flow are similar to catchments of similar size [Robinson and Sivapalan, 1997]. Other predictor variables did not account for substantial proportions of the observed variability.

[38] On average, daily flows calculated using the predicted correlation strength were more accurate than daily flows calculated using the four nearest neighboring gauges (and weighted by the inverse of the distance). The improvement over the four nearest neighbors was small with the model using five dimensions in the NMS (mean NASH = 0.781 for four nearest neighbors and mean NASH = 0.790 for five NMS dimensions), but prediction accuracy improved with increased numbers of NMS dimensions up to a mean NASH = 0.823 for nine NMS dimensions (Table 3). Much of this improvement can be attributed to increasing numbers of sites for which appropriate index gauges could not be identified: only 8 sites were excluded when five NMS dimensions were used in the model, whereas 83 sites were excluded when nine NMS dimensions were used. Indeed, if the same sites that were excluded by the model using nine NMS dimensions were excluded from the simple four nearest neighbor model, the average Nash-Sutcliffe efficiency associated with the four nearest neighbor model increased to 0.815, a value that is comparable to the nine NMS dimension model. This observation suggests that using the four nearest geographic neighbors as index sites provides accurate predictions, as long as those four index sites satisfy minimal requirements in terms of the correlation of daily flows to the ungauged sites. The model predicting correlation strength presented here provides the means for assessing this similarity.

[39] The average accuracy of the best model (NMS9) was still not comparable to the ideal “best” case, represented by predictions using known correlations, and this difference suggests a potential for further improvement in predictive accuracy is possible. In particular, prediction errors were consistently approximately 0.02 for predictions of position along each NMS axis (Table 2). Hence, the position of each ungauged site in ordination space is predicted with an average error of ±0.02 units, which is 20% of the allowable separation distance between ungauged sites and index sites. These predictive errors highlight the need to identify and quantify other catchment physical characteristics that are associated with flow timing. In particular, characteristics of the flow network [Sivapalan, 2003] and catchment geology [Tague and Grant, 2004] are potentially relevant to flow timing.

4. Conclusions

[40] Predicting flow at ungauged sites is critical to managing both the quality and the quantity of water in stream and rivers, and many different approaches are available to make these predictions. The present analysis focuses on methods for predicting flow that rely on the use of index gauges for predicting the timing of different flow events and has confirmed that the use of the correlation of daily flows provides a valuable approach for selecting index gauges. To further extend the applicability of this approach to more situations, I have proposed a new method for predicting the strength with which daily flows at index gauges correlate with the ungauged locations. These predicted correlations can then be used to select appropriate index gauges for the ungauged location.

[41] The proposed method for identifying index gauges has several advantages over existing techniques. First, the method provides a means of directly quantifying and accounting for the relative importance of different natural factors that influence whether flow at a particular gauge is similar to or different from another site. Other studies have incorporated catchment physical characteristics when identifying index gauges [Shu and Ouarda, 2012], but these characteristics have been selected by a priori assumptions of their relevance. The present approach allows one to directly quantify (and test) the effects of different physical characteristics on whether flows at a particular index site are similar to the ungauged site. Second, the method identifies sites that do not have appropriate index gauges for inferring the flow timing. As discussed earlier, selection of the number of index gauges and the minimum correlation coefficient not only determine the accuracy of daily flow predictions but also determine whether appropriate index gauges are available for different ungauged locations in a study area. This new approach for identifying index gauges clarifies these tradeoffs and provides the means for analysts to make informed decisions regarding the desired accuracy of flow predictions. When available index gauges are not sufficient to achieve the desired accuracy, approaches other than the use of index gauges should be considered.

Acknowledgments

[42] The author thanks J. Oliver, B. Walsh, D. Thomas, and several anonymous referees for reviewing an earlier draft of this manuscript. The views expressed in this paper are those of author and do not reflect the official policy of the U.S. Environmental Protection Agency.

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