Efficient and responsible management of water resources relies on accurate streamflow records. However, many watersheds are ungaged, limiting the ability to assess and understand local hydrology. Several tools have been developed to alleviate this data scarcity, but few provide continuous daily streamflow records at individual streamgages within an entire region. Building on the history of hydrologic mapping, ordinary kriging was extended to predict daily streamflow time series on a regional basis. Pooling parameters to estimate a single, time-invariant characterization of spatial semivariance structure is shown to produce accurate reproduction of streamflow. This approach is contrasted with a time-varying series of variograms, representing the temporal evolution and behavior of the spatial semivariance structure. Furthermore, the ordinary kriging approach is shown to produce more accurate time series than more common, single-index hydrologic transfers. A comparison between topological kriging and ordinary kriging is less definitive, showing the ordinary kriging approach to be significantly inferior in terms of Nash–Sutcliffe model efficiencies while maintaining significantly superior performance measured by root mean squared errors. Given the similarity of performance and the computational efficiency of ordinary kriging, it is concluded that ordinary kriging is useful for first-order approximation of daily streamflow time series in ungaged watersheds.

One of the most fundamental problems confronting the fields of
hydrology and water resources management is the prediction of hydrologic
responses in ungaged basins (PUB)

Techniques for the reproduction of historical records of streamflow largely
fall into two main categories: process-based models and transfer-based,
statistical techniques. This work is concerned with the latter, which rely on
transferring information from an index site or set of index sites to an
ungaged site by the means of a statistical relationship. These techniques
include ungaged applications of record reconstruction techniques like the
drainage-area ratio method (see

The prediction of daily streamflow records in ungaged basins, especially for
statistical transfer methods, has largely been dominated by one-to-one
transfers from an index streamgage to an ungaged site (as in

Geostatistical tools have been used to develop regional maps of measured and
predicted hydrologic and climatic variables for decades. The U.S. Geological
Survey has developed contour or isoline maps of runoff in the United States
as far back as 1894

Geostatistical maps of runoff and other variables are usually based on
kriging, a technique developed in the mining industry (as described by

Despite its wide application for the prediction of streamflow statistics,
kriging, top-kriging, and mapping in general have not widely been used to
predict time series of streamflow and related variables. Despite the need for
sub-monthly predictions of streamflow statistics, the prediction of
sub-monthly variables was originally thought to be computationally
prohibitive

This work explores the potential of ordinary kriging to produce spatially and
temporally continuous predictions of historical daily streamflow in the
southeastern region of the United States. Streamflow is a volumetric quantity
that typically accumulates along a river network; as it is not reasonable to
consider the regionalization of a volumetric quantity, a transformation is
needed. This has been the rationale for the prediction of unit runoff values

Spatial interpolation driven by semivariance – kriging – among daily
streamflows is not new.

Using a data set identical to that used by

Daily streamflow records were obtained from the U.S. Geological Survey
National Water Information System (

Map of the study area showing the locations of the 182 streamgages used for analysis and validation.

Ordinary kriging is a geostatistical tool by which the distance between two points is used to predict the semivariance of some dependent variable. The inter-site semivariances of data from a measured network can be used to create a system of linear equations predicting the semivariance at an unmeasured site to be a linear sum of the semivariance between all observed sites. For an unmonitored site, this allows for the derivation of linear weights between the unmonitored site and all monitored sites in the network. If all the assumptions of ordinary kriging are valid, this tool provides the best linear unbiased estimate.

The single realization of

Ordinary kriging of streamflow time series builds off of previous hydrologic
applications to predict streamflow statistics to produce a method for
handling temporal variation along with spatial variation. Based on initial
exploration by

The model of ordinary kriging presented above assumes a global neighborhood.
That is, all observations are assigned a weight for the prediction of the
ungaged site. In other hydrologic applications

While there are many considerations in the development of a kriging system, this work is mainly focused on kriging time series and the temporal behavior of kriging parameters. As such, the temporal evolution and behavior of variogram parameters was of most interest. As discussed above, there are many considerations in the development of a kriging system. Several were explored, including the binning of empirical variograms, the number of contributing neighbors, and the maximum range of the variogram, but none were found to have only a marginal impact on the resulting estimates. Accordingly, the remainder of this paper considers the unique problems of temporal calibration and prediction.

The variogram can be characterized by three parameters: the nugget value, partial sill, and the range. The nugget value is the semivariance of collocated points or, as it is sometimes interpreted, the measurement error, the partial sill represents the regional semivariance, and the range represents the separation distance beyond which the inter-site semivariance is best approximated by the regional semivariance. In some previous hydrologic applications of kriging, the semivariance, which is modeled by the semivariogram, has been assumed to be temporally constant, and thus only a single variogram model need be fit. This is clearly not the case for the reconstruction of historical time series of streamflow. It is therefore important to consider the temporal evolution, or lack thereof, in the spatial semivariance structure, as characterized by variogram parameters, of daily streamflows.

The initial development by

This work considers the temporal evolution of variogram parameters more formally. The streamflow models based on independent daily variogram models are contrasted with a pooled variogram model. The latter model requires the fitting of only a single variogram, while the former requires the fitting of as many variograms as there are days to be simulated. If the parameters of the semivariogram can be reasonably assumed to be constant, then the computational efficiency of the pooled model is highly advantageous for operational prediction.

For a daily variogram, the semivariances for each site pair are plotted
against distance, binned, and averaged to fit a variogram model; the process
is repeated independently for each day. The pooled variogram is described by

In addition to contrasting temporally independent variograms and pooled
variograms, this paper also contrasts these methods with two standard,
transfer-based statistical tools: the drainage-area ratio (DAR)

Previous work

As is described below, the kriging methods were implemented to predict a logarithmic transformation of streamflow. With the exception of the Nash–Sutcliffe model efficiency of the logarithms themselves, all other performance metrics were computed on back-transformed streamflows. No bias correction factor was developed or applied. The development of a bias correction factor that can be applied to ungauged basins is beyond the scope of this work but is essential to future explorations.

An example of the observed and simulated streamflows for a site and
year selected to represent the median performance. The results are from site
02401390, with a drainage area of 365

Using the same metrics, ordinary kriging was contrasted with an application
of top-kriging similar to that defined by

The implementation of top-kriging presented here is not intended to represent the ultimate implementation of top-kriging for this region. Ordinary kriging is an extreme of top-kriging in that top-kriging allows for a variable spatial support for each observation, while ordinary kriging provides only one regularization point for each observation. With this in mind, this implementation of top-kriging is meant to reflect the improvements achieved by allowing for a further discretized spatial support. Certainly, the improvements of either method may be improved by considering a more robust exploration of the underlying variogram model, the number of contributing neighbors or the level of spatial discretization. However, this was left for future research, allowing this work to focus only on the effects of additional spatial discretization.

Summary statistics of several performance metrics for different
streamflow record prediction techniques. Summary statistics include (a) the
median, (b) the 10th and 90th percentiles, and (c) the Wilcoxon signed-rank
probability of a difference between pooled kriging and each other method
equal to or more extreme than observed (commonly referred to as a

Median cumulative distribution of absolute percent errors in daily estimates for streamflow estimated from both daily and pooled variogram parameter sets.

In a leave-one-out validation procedure, both the daily and pooled parameter
sets reproduce historical daily streamflow records quite well.
Table

In addition to having similar point performance metrics, the daily and pooled
variograms produced nearly identical distributions of absolute percent errors
(Fig.

Figure

Finally, with the use of time-varying and time-invariant variograms, it is
useful to consider how well the temporal structure of the daily streamflows
is reproduced. Figure

Median percent error for each non-exceedance probability, binned by single percentage points, for streamflow estimates from both daily and pooled variogram parameter sets.

Observed autocorrelation of daily streamflow, in gray, with simulated autocorrelations from daily and pooled ordinary kriging daily streamflow time series.

The 31-day moving median of daily variogram parameters and the ratio
of nugget to sill. (NOTE: the vertical axis of the range is scaled by a
factor of

Because the pooled variogram parameters produce results fairly similar to the
daily parameter sets, it is important to understand how the pooled parameters
relate to their daily counterparts and how the daily counterparts evolved
over time. Figure

As mentioned previously, the nugget value can be thought of as the semivariance of nearly co-located points. In the context of basins and daily parameters, the nugget on each day, because the semivariance of co-located points is akin to a variance, is an approximation of the average of all at-site variances for that day. The 31-day moving median of the nugget time series suggests that there is a substantial seasonal trend. The nugget, or regional variability, and the variability thereof, are fairly constant from the beginning of January through May and rise to a peak in September and October. The pooled parameter, which can be thought of as a time-averaged variability of an average site, is closer to the peak of the moving-median nugget than to the lower stable January–May values. The pooled parameter is greater than the median of the daily values. This suggests that, for much of the year, the pooled nugget, being greater than the daily values, introduces more daily variability than would be expected. As measurement uncertainty may fluctuate, the fluctuations in the nugget may be tied to fluctuations in the magnitude of streamflow.

The partial sill, a limit on the regional semivariance, shows a much weaker seasonal signal. The 31-day moving median shows a nearly binary structure of two values. The partial sill is small from January through March, transitions quickly in April, remains high through October and then returns towards January values. Again, the pooled parameter plots closer to the higher plateau of the moving median. This means that for parts of the year, the pooled parameter assumes the more distant neighbors hold appreciably less information than they really contain. For a smaller portion of the year, the pooled parameter, being greater than the daily values, assumes the more-distant neighbors hold slightly more information than they really do. However, the pooled partial sill remains within the inter-decile range of the daily parameter values for the majority of the year. As with the nugget, fluctuations in the sill may be tied to fluctuations in the magnitude of streamflow.

The range parameter shows the least complex temporal structure. The 31-day moving median shows that the range varies over an order of magnitude, and year-to-year variability, as shown by the inter-decile range, is consistently large. The year-to-year variability is more pronounced than the season trends. Overall, there is a slight depression in the summer months, which indicates decreased regional homogeneity and more heterogeneity in that the regional semivariance (partial sill) is reached at shorter distances. The pooled parameter is quite similar in magnitude to the median daily value and is almost completely contained by the daily inter-decile ranges.

It is difficult to consider the effects on any one parameter in isolation.
The final row of Fig.

It is clear that there is substantial temporal structure and seasonal variation in the spatial semivariance structure of daily streamflows. Given the strong temporal dependence and seasonality of daily streamflows, this is not surprising. As with streamflow, it is extremely difficult to identify causal factors resulting in these patterns. Though not explicitly explored here, it is probable that the temporal structure is driven by climatic processes. The greater nugget value in the latter half of the year indicates increased streamflow variability, year to year, during the late fall and early winter. The partial sill and range interact strongly with each other, one being the threshold and the other being a sort of “time to threshold”. The decreased summer range suggests that climatic response is more homogeneous in summer months, while the winter and spring rises are emblematic of increased regional heterogeneity (i.e., more localized climatic drivers of streamflow). The partial sill demonstrates an increased regional variability, beyond the range, from late spring through fall; otherwise the sill is smaller, suggesting that, even beyond the range, variability is lower across the region in winter months.

In presenting a new model for daily streamflow reconstructions, it is useful
to contextualize performance by comparing against previous methods. To this
end, two common statistical, transfer-based tools for the prediction of daily
time series are considered: the drainage-area ratio (DAR)

The performance metrics of both DAR and QPPQ are outlined in
Table

The comparison of the pooled ordinary kriging approach and the pooled
top-kriging approach does not provide as definitive a conclusion. The
top-kriging approach provides a significantly greater Nash–Sutcliffe
efficiency at the 5 % significance level. However, the ordinary kriging
approach yielded significantly smaller root mean squared errors. In terms of
bias, top-kriging provides a significantly smaller absolute bias, but the
median signed bias is slightly larger; the average bias is greater, but the
average deviation from unbiasedness is smaller. There is no significant
difference between ordinary kriging and top-kriging with respect to the
correlations between observed and simulated streamflows and the
Nash–Sutcliffe efficiencies of the logarithms of streamflow. The
disagreement on the significance of the difference in correlations between
observed and simulated streamflows and the difference between Nash–Sutcliffe
efficiencies is the result of the interplay of the components of the
Nash–Sutcliffe model efficiency, as discussed by

Based on the varied performance metrics, there is no significant difference between the ordinary kriging and top-kriging approaches. Aside from average performance, the quantiles of the distributions of performance appear improved for top-kriging. For example, 90 % of the ordinary kriging results show a Nash–Sutcliffe model efficiency of the logarithms below 0.91, while 90 % of top-kriging results are below 0.93. It is not immediately apparent why the top-kriging approach might disproportionately accept the extremes of the distribution of performance. However, the pairwise comparison of the Wilcoxon signed-rank test indicates that there is no significant evidence to reject the hypothesis that pooled ordinary kriging and pooled top-kriging produce different performances. If they are not significantly different, the additional discretization of top-kriging does not appear to produce significantly improved performance to warrant the increased complexity. Future research might also consider whether the prediction variances from either method are superior; though not explored here, a more accurate prediction uncertainty may improve the usefulness of simulated streamflows.

Top-kriging was explicitly developed to address both the hierarchical nature
of streamflow and streamflows' aggregate dependency on contributing drainage
areas

The results of this analysis demonstrate that the computationally efficient routine of pooled variogram estimation can be used to fit an ordinary kriging system that produces plausible estimates of daily time series at ungaged sites. The pooled parameter estimation, which ignores temporal variation of the spatial semivariance structure, was able to reproduce observed hydrographs more accurately than other non-kriging methods considered. Both daily and pooled kriging approaches outperformed single-index transfers. It is intriguing that accounting for temporal variation in the variograms resulted in relatively minor changes in the kriging estimates and the performance thereof. Additionally, it is somewhat concerning that the kriging techniques show a general inaccuracy in the tails of the distribution of streamflow. The comparison of ordinary kriging and top-kriging was inconclusive, with some metrics favoring top-kriging, while others favored ordinary kriging, and still others were not significantly different.

It was clearly shown that the variogram parameters, characteristic of the spatial semivariance structure, exhibit seasonal and other temporal patterns. However, the averaging that occurs when pooling daily semivariance information actually resulted in a marginal improvement in the accuracy (as measured by several metrics) of resultant streamflow time series. In initial work

Although ignoring the temporal variation in the variogram parameters did not
appreciably degrade performance, it may be possible to gain some improvements
while retaining computational efficiency by preserving some remnants of the
observed temporal variability in variogram parameters. One option might be to
consider a moving-window average of daily parameters, optimizing the
advantages of temporally variable parameters while seeking to smooth out
chaotic daily behavior. Another clear avenue for future research is to
evaluate the possibility of constructing a temporal model of variogram
parameters. One could easily imagine monthly parameter sets or parameter sets
reproduced by an autoregressive integrated moving average (ARIMA)

However, contextualizing ordinary kriging in the context of other hydrologic
applications of geostatistics, a brief comparison of ordinary kriging and
top-kriging was presented here.

The pooling of semivariance to produce a single set of variogram parameters
implicitly assumes that the spatial semivariance structure is constant in
time. While a seasonal fluctuation may be present, that same fluctuation may
occur every year with no systematic change. For the study period, water years
1981 through 2010, the time series of daily variogram parameters were indeed
stationary. Following the procedures of

Pooled variogram estimation and ordinary kriging allow for the efficient and, according to broad metrics, accurate prediction of daily streamflow at ungaged sites. Being able to regionally characterize networks of streamflow may provide additional advantages. Though not explored here, kriging algorithms also allow for the quantification of variances around estimates. This can serve two purposes: (1) it shows where in the network uncertainties are likely to be greatest, which might be a means to identify optimal locations for additional monitoring. (2) It may be able to explicitly provide confidence intervals for estimated daily streamflows. Future studies will explore the accuracy of so-derived intervals. In any case, the theoretically derived structure of the kriging system promises a more “closed-form” interpretation of predictive uncertainty than more traditional single-index hydrologic transfers, which require an ad hoc procedure for uncertainty quantification. While predictive performance was indistinguishable here, more advanced methods like top-kriging may provide significant advantages in their quantification of predictive uncertainty.

One limitation of the kriging approach, as documented here, is the
overestimation of the lower tail of the streamflow distribution and the
underestimation of the upper tail. Similar results were documented by

The estimation of daily streamflow records at ungaged sites is a fundamental problem of water resources management and assessment. Many tools exist to aid in quantifying resources, but this paper discusses a statistical tool that is capable of combining time series at multiple sites for regional prediction. Building on the work of hand-drawn discharge maps, ordinary kriging is proposed as an efficient technique for reproduction of historical streamflow time series at ungaged sites. Using a leave-one-out validation and daily streamflow data from 182 minimally impacted and minimally regulated watersheds, geostatistical techniques are shown to have advantages over other, common statistical approaches.

Ordinary kriging is demonstrated to produce more accurate streamflow time-series estimates than the drainage-area ratio method and nonlinear spatial interpolations using flow duration curves. In addition, using pooled variogram parameters with ordinary kriging produced marginally better performance than using parameters determined at a daily time step. This is surprising, as pooling effectively averages out temporal variation. Though significant improvements are unlikely, it is observed that the variogram parameters, characterizing the spatial semivariance structure, show clear seasonal patterns that may be reproducible in part without requiring the computation of daily variograms. However, in an initial exploration, the advantages of moving towards a more complex kriging system such as that provided by top-kriging are, at best, minimal. Further research may improve the computational parity of top-kriging and continue to elucidate the advantages and disadvantages of ordinary kriging and top-kriging for spatio-temporal hydrologic geostatistics.

This paper represents the evolution of work published as part of the author's PhD dissertation. This research was supported by the Department of the Interior's WaterSMART initiative and the U.S. Geological Survey's National Water Census. Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government. David Wolock and Gregory Koltun, both of the U.S. Geological Survey, provided valuable reviews of the initial manuscript. Edzer Pebesma, Jan Olav Skøien, and Alessio Pugliese provided valuable reviews as part of the public commentary. Edited by: J. Seibert