Skip to main content
SpringerLink
Log in

Estimation of river and stream temperature trends under haphazard sampling

  • Original Paper
  • Published:
Statistical Methods & Applications Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Full name: The U.S. Army Corps of Engineers’ Upper Mississippi River Restoration (UMRR) Program Long Term Resource Monitoring (LTRM) element.

  2. Calculated using mean daily discharge at Kingston Mines, Illinois (http://waterdata.usgs.gov/il/nwis/dv?referred_module=sw&site_no=05568500).

  3. Temperature data may be obtained at http://www.umesc.usgs.gov/data_library/water_quality/water_quality_data_page.html.

  4. 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.

References

  • Araujo HA, Cooper AB, Hassan MA, Venditti J (2012) Estimating suspended sediment concentrations in areas with limited hydrological data using a mixed-effects model. Hydrol Process 26(24):3678–3688

    Article  Google Scholar 

  • Bühlmann P, Kalisch M, Meier L (2014) High-dimensional statistics with a view toward applications in biology. Ann Rev Stat Its Appl 1:258–278

    Google Scholar 

  • Burgmer T, Hillebrand H, Pfenninger M (2007) Effects of climate-driven temperature changes on the diversity of freshwater macroinvertebrates. Oecologia 151:93–103

    Article  Google Scholar 

  • Bustillo V, Moatar F, Ducharne A, Thiéry D, Poirel A (2014) A multimodel comparison for assessing water temperatures under changing climate conditions via the equilibrium temperature concept: case study of the Middle Loire River, France. Hydrol Process 28(3):1507–1524

    Article  Google Scholar 

  • Caissie D (2006) The thermal regime of rivers: a review. Freshw Biol 51(8):1389–1406

    Article  Google Scholar 

  • Clark JS, Bell DM, Kwit MK, Powell A, Zhu K (2013) Dynamic inverse prediction and sensitivity analysis with high-dimensional responses: application to climate-change vulnerability of biodiversity. J Agric Biol Environ Stat 18(3):376–404

    Article  MathSciNet  MATH  Google Scholar 

  • Demidenko E (2004) Mixed models: theory and applications. Wiley, Hobooken

    Book  Google Scholar 

  • El-Shaarawi A, Teugels J (2007) Environmental statistics: current and future. Int Stat Rev 73(2):233–236

    Article  Google Scholar 

  • Grant PJ (1977) Water temperatures of the Ngaruroro River at three stations. J Hydrol 16(2):148–157

    Google Scholar 

  • Gray BR (2012) Variance components estimation for continuous and discrete data, with emphasis on cross-classified sampling designs. In: Gitzen RA, Millspaugh JJ, Cooper AB, Licht DS (eds) Design and analysis of long-term ecological monitoring studies. Cambridge University Press, New York, pp 200–227

    Chapter  Google Scholar 

  • Hockey JB, Owens IF, Tapper NJ (1982) Empirical and theoretical models to isolate the effect of discharge on summer water temperatures in the Hurunui River. J Hydrol 21(1):1–12

    Google Scholar 

  • Hox JJ (2010) Multilevel analysis: techniques and applications, 2nd edn. Routledge, New York

    Google Scholar 

  • Isaak DJ, Luce CH, Rieman BE, Nagel DE, Peterson EE, Horan DL, Parkes S, Chandler GL (2010) Effects of climate change and wildfire on stream temperatures and salmonid thermal habitat in a mountain river network. Ecol Appl 20(5):1350–1371

    Article  Google Scholar 

  • James W, Barko J (2004) Diffusive fluxes and equilibrium processes in relation to phosphorus dynamics in the Upper Mississippi River. River Res Appl 20:473–484

    Article  Google Scholar 

  • Johnson BL, Hagerty KH (2008) Status and trends of selected resources of the Upper Mississippi River System. US Geological Survey, Report LTRMP 2008-T002

  • Kasurak A, Kelly R, Brenning A (2009) Mixed-effects regression for snow distribution modeling in the Central Yukon. The 66th Eastern Snow Conference, Niagara-on-the-Lake, Ontario, Canada

  • Lewis J (2006) Fixed and mixed-effects models for multi-watershed experiments. In: Proceedings of the 3rd federal interagency hydrologic modeling conference, 2–6 April 2006, Reno NV

  • Lyons J, Zorn T, Stewart J, Seelbach P, Wehrly K, Wang L (2009) Defining and characterizing coolwater stream fish assemblages in Michigan and Wisconsin. N Am J Fish Manag 29(4):1130–1151

    Article  Google Scholar 

  • Lyubchich V, Gel YR, El-Shaarawi A (2013) On detecting non-monotonic trends in environmental time series: a fusion of local regression and bootstrap. Environmetrics 24(4):209–226

    Article  MathSciNet  Google Scholar 

  • Lyubchich V, Gray BR, Gel YR (2015) Multilevel random slope approach and nonparametric inference for river temperature, under haphazard sampling. In: Lakshmanan V, Gilleland E, McGovern A, Tingley M (eds) Machine learning and data mining approaches to climate science. Springer, Cham, pp 137–145

    Chapter  Google Scholar 

  • McNamee R (2003) Confounding and confounders. Occup Environ Med 60(3):227–234

    Article  MathSciNet  Google Scholar 

  • Meentemeyer RK, Haas SE, Václavík T (2012) Landscape epidemiology of emerging infectious diseases in natural and human-altered ecosystems. Ann Rev Phytopathol 50:379–402

    Article  Google Scholar 

  • Mohammadpour R, Shaharuddin S, Chang CK, Zakaria NA, Ghani AA, Chan NW (2014) Prediction of water quality index in constructed wetlands using support vector machine. Environ Sci Pollut Res 22:1614–7499

    Google Scholar 

  • Mohseni O, Stefan HG, Erickson TR (1998) A nonlinear regression model for weekly stream temperatures. Water Resour Res 34(10):2685–2692

    Article  Google Scholar 

  • Mohseni O, Stefan HG, Eaton JG (2003) Global warming and potential changes in fish habitat in US streams. Clim Change 59(3):389–409

    Article  Google Scholar 

  • Morris JS (2002) The BLUPs are not “best” when it comes to bootstrapping. Stat Probab 56:425–430

    Article  MATH  Google Scholar 

  • O’Donnell D, Rushworth A, Bowman A, Scott E, Hallard M (2014) Flexible regression models over river networks. J R Stat Soc Ser C Appl Stat 63(1):47–63

    Article  MathSciNet  Google Scholar 

  • Orr HG, Des Clers S, Simpson GL, Hughes M, Battarbee RW, Cooper L, Dunbar MJ, Evans R, Hannaford J, Hannah DM et al (2010) Changing water temperatures: a surface water archive for England and Wales. In: Kirby C (ed) International symposium on the role of hydrology in managing consequences of a changing global environment. British Hydrological Society, Newcastle, pp 1–8

    Google Scholar 

  • Paul WL (2011) A causal modelling approach to spatial and temporal confounding in environmental impact studies. Environmetrics 22(5):626–638

    Article  MathSciNet  Google Scholar 

  • Pilgrim JM, Fang X, Stefan HG (1998) Stream temperature correlations with air temperatures in Minnesota: implications for climate warming. J Am Water Resour Assoc 34(5):1109–1121

    Article  Google Scholar 

  • Preud’homme EB, Stefan HG (1992) Errors related to random stream temperature data collection in Upper Mississippi River watershed. J Am Water Resour Assoc 28(6):1077–1082

    Article  Google Scholar 

  • Regier H, Holmes J, Pauly D (1990) Influence of temperature changes on aquatic ecosystems: an interpretation of empirical data. Trans Am Fish Soc 119:374–389

    Article  Google Scholar 

  • Schertz TL, Alexander RB, Ohe DJ (1991) The computer program estimate trend (ESTREND), a system for the detection of trends in water-quality data. US Geological Survey, Water-Resources Investigations Report 91-4040

  • Shang J, Cavanaugh JE (2008) An assumption for the development of bootstrap variants of the Akaike information criterion in mixed models. Stat Probab Lett 78(12):1422–1429

    Article  MathSciNet  MATH  Google Scholar 

  • Sheppard L (2003) Insights on bias and information in group-level studies. Biostatistics 4(2):265–278

    Article  MATH  Google Scholar 

  • Snijders TAB, Bosker RJ (2012) Multilevel analysis: an introduction to basic and advanced multilevel modeling, 2nd edn. Sage, London

    Google Scholar 

  • Soballe DM, Fischer JR (2004) Long term resource monitoring program procedures: water quality monitoring. US Geological Survey, Techn Rep LTRMP 2004-T002-1

  • Stefan HG, Preud’homme EB (1993) Stream temperature estimation from air temperature. J Am Water Resour Assoc 29(1):27–45

    Article  Google Scholar 

  • van der Leeden R, Meijer E, Busing FMTA (2008) Resampling multilevel models. In: de Leeuw J, Meijer E (eds) Handbook of multilevel analysis. Springer, New York, pp 401–433

    Chapter  Google Scholar 

  • Ver Hoef JM, Peterson EE (2010) A moving average approach for spatial statistical models of stream networks. J Am Stat Assoc 105(489):6–18

    Article  MathSciNet  Google Scholar 

  • van Vliet MTH, Ludwig F, Zwolsman JJG, Weedon GP, Kabat P (2011) Global river temperatures and sensitivity to atmospheric warming and changes in river flow. Water Resour Res 47, article id W02544. doi:10.1029/2010WR009198

  • Wakefield J (2003) Sensitivity analyses for ecological regression. Biometrics 59(1):9–17

    Article  MathSciNet  MATH  Google Scholar 

  • Wang L, Akritas MG, Van Keilegom I (2008) An ANOVA-type nonparametric diagnostic test for heteroscedastic regression models. J Nonparametr Stat 20(5):365–382

    Article  MathSciNet  MATH  Google Scholar 

  • Webb BW, Nobilis F (1997) Long-term perspective on the nature of the air-water temperature relationship: a case study. Hydrol Process 11(2):137–147

    Article  Google Scholar 

  • Webb BW, Clack PD, Walling DE (2003) Water-air temperature relationships in a Devon river system and the role of flow. Hydrol Process 17(15):3069–3084

    Article  Google Scholar 

  • Wood S (2006) Generalized additive models: an introduction with R, 2nd edn. Chapman and Hall/CRC Press, Boca Raton

    Google Scholar 

  • Xia R, Demissie M (1999) Hydraulic analyses for LaGrange Pool of the Illinois River: a component of the restoration of large river ecosystems project. Watershed Science Section, Illinois State Water Survey, Contract Report 651

Download references

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.

Author information

Authors and Affiliations

  1. Upper Midwest Environmental Sciences Center, US Geological Survey, 2630 Fanta Reed Road, La Crosse, WI, 54603, USA

    Brian R. Gray & James T. Rogala

  2. Chesapeake Biological Laboratory, University of Maryland Center for Environmental Science, P.O. Box 38, Solomons, MD, 20688, USA

    Vyacheslav Lyubchich

  3. Department of Mathematical Sciences, University of Texas at Dallas, 800 W Campbell Road, Richardson, TX, 75080, USA

    Yulia R. Gel

  4. Wisconsin Water Science Center, US Geological Survey, 8505 Research Way, Middleton, WI, 53562, USA

    Dale M. Robertson

  5. 3M Center, Saint Paul, MN, 55144-1000, USA

    Xiaoqiao Wei

Authors
  1. Brian R. Gray
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vyacheslav Lyubchich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yulia R. Gel
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. James T. Rogala
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Dale M. Robertson
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Xiaoqiao Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brian R. Gray.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 269 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10260-015-0334-7

Keywords

Search

Navigation