Abstract
Long-term temporal trends in water temperature in rivers and streams are typically estimated under the assumption of evenly-spaced space-time measurements. However, sampling times and dates associated with historical water temperature datasets and some sampling designs may be haphazard. As a result, trends in temperature may be confounded with trends in time or space of sampling which, in turn, may yield biased trend estimators and thus unreliable conclusions. We address this concern using multilevel (hierarchical) linear models, where time effects are allowed to vary randomly by day and date effects by year. We evaluate the proposed approach by Monte Carlo simulations with imbalance, sparse data and confounding by trend in time and date of sampling. Simulation results indicate unbiased trend estimators while results from a case study of temperature data from the Illinois River, USA conform to river thermal assumptions. We also propose a new nonparametric bootstrap inference on multilevel models that allows for a relatively flexible and distribution-free quantification of uncertainties. The proposed multilevel modeling approach may be elaborated to accommodate nonlinearities within days and years when sampling times or dates typically span temperature extremes.
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Notes
Full name: The U.S. Army Corps of Engineers’ Upper Mississippi River Restoration (UMRR) Program Long Term Resource Monitoring (LTRM) element.
Calculated using mean daily discharge at Kingston Mines, Illinois (http://waterdata.usgs.gov/il/nwis/dv?referred_module=sw&site_no=05568500).
Temperature data may be obtained at http://www.umesc.usgs.gov/data_library/water_quality/water_quality_data_page.html.
Mean discharges for 1994–2002 and 2004–2010 were 358, 204, 783, 235, 416, 233, 522, 215, 172, and 204, 108, 276, 383, 496, 258 and \(478\, \hbox {m}^3/\hbox {s}\), respectively.
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Acknowledgments
This study was partly funded by the U.S. Army Corps of Engineers’ Upper Mississippi River Restoration Program Long Term Resource Monitoring element. The research of Yulia R. Gel was supported in part by the Natural Sciences and Engineering Research Council of Canada. We thank Bruce Webb for helpful discussions.
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Gray, B.R., Lyubchich, V., Gel, Y.R. et al. Estimation of river and stream temperature trends under haphazard sampling. Stat Methods Appl 25, 89–105 (2016). https://doi.org/10.1007/s10260-015-0334-7
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DOI: https://doi.org/10.1007/s10260-015-0334-7