The Annals of Applied Statistics

Modeling sea-level change using errors-in-variables integrated Gaussian processes

Niamh Cahill, Andrew C. Kemp, Benjamin P. Horton, and Andrew C. Parnell

Full-text: Open access

Abstract

We perform Bayesian inference on historical and late Holocene (last 2000 years) rates of sea-level change. The input data to our model are tide-gauge measurements and proxy reconstructions from cores of coastal sediment. These data are complicated by multiple sources of uncertainty, some of which arise as part of the data collection exercise. Notably, the proxy reconstructions include temporal uncertainty from dating of the sediment core using techniques such as radiocarbon. The model we propose places a Gaussian process prior on the rate of sea-level change, which is then integrated and set in an errors-in-variables framework to take account of age uncertainty. The resulting model captures the continuous and dynamic evolution of sea-level change with full consideration of all sources of uncertainty. We demonstrate the performance of our model using two real (and previously published) example data sets. The global tide-gauge data set indicates that sea-level rise increased from a rate with a posterior mean of 1.13 mm/yr in 1880 AD (0.89 to 1.28 mm/yr 95% credible interval for the posterior mean) to a posterior mean rate of 1.92 mm/yr in 2009 AD (1.84 to 2.03 mm/yr 95% credible interval for the posterior mean). The proxy reconstruction from North Carolina (USA) after correction for land-level change shows the 2000 AD rate of rise to have a posterior mean of 2.44 mm/yr (1.91 to 3.01 mm/yr 95% credible interval). This is unprecedented in at least the last 2000 years.

Article information

Source
Ann. Appl. Stat., Volume 9, Number 2 (2015), 547-571.

Dates
Received: July 2014
Revised: March 2015
First available in Project Euclid: 20 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1437397101

Digital Object Identifier
doi:10.1214/15-AOAS824

Mathematical Reviews number (MathSciNet)
MR3371325

Zentralblatt MATH identifier
06499920

Keywords
Bayesian statistics integrated Gaussian processes errors-in-variables proxy reconstruction tide gauge

Citation

Cahill, Niamh; Kemp, Andrew C.; Horton, Benjamin P.; Parnell, Andrew C. Modeling sea-level change using errors-in-variables integrated Gaussian processes. Ann. Appl. Stat. 9 (2015), no. 2, 547--571. doi:10.1214/15-AOAS824. https://projecteuclid.org/euclid.aoas/1437397101


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References

  • Abramowitz, M. and Stegun, I. (1965). Handbook of Mathematical Functions. Dover, New York.
  • Atwater, B. F. (1987). Evidence for great holocene earthquakes along the outer coast of Washington state. Science 236 942–944.
  • Banerjee, S. and Fuentes, M. (2012). Bayesian modeling for large spatial datasets. Wiley Interdisciplinary Reviews: Computational Statistics 4 59–66.
  • Banerjee, S., Gelfand, A. E., Finley, A. O. and Sang, H. (2008). Gaussian predictive process models for large spatial data sets. J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 825–848.
  • Barnett, T. P. (1984). The estimation of global sea level change: A problem of uniqueness. Journal of Geophysical Research: Oceans 89 7980–7988.
  • Birks, H. J. B. (1995). Quantitative palaeoenvironmental reconstructions. In Technical Guide 5 161–254. Quaternary Research Association, Cambridge.
  • Boon, J. D. (2012). Evidence of sea level acceleration at U.S. and Canadian tide stations, Atlantic Coast, North America. Journal of Coastal Research 28 1437–1445.
  • Cahill, N., Kemp, A. C., Horton, B. P. and Parnell, A. C. (2015). Supplement to “Modeling sea-level change using errors-in-variables integrated Gaussian processes.” DOI:10.1214/15-AOAS824SUPP.
  • Cazenave, A. and Llovel, W. (2010). Contemporary sea level rise. Annual Review of Marine Science 2 145–173.
  • Chaniotis, A. K. and Poulikakos, D. (2004). High order interpolation and differentiation using B-splines. J. Comput. Phys. 197 253–274.
  • Church, J. A. and White, N. J. (2006). A 20th century acceleration in global sea-level rise. Geophysical Research Letters 33.
  • Church, J. A. and White, N. J. (2011). Sea-level rise from the late 19th to the early 21st century. Surveys in Geophysics 32 585–602.
  • Cramér, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. Sample Function Properties and Their Applications. Wiley, New York.
  • Dey, D. K., Ghosh, S. K. and Mallick, B. K., eds. (2000). Generalized Linear Models: A Bayesian Perspective. Biostatistics 5. Dekker, New York.
  • Donnelly, J. P., Cleary, P., Newby, P. and Ettinger, R. (2004). Coupling instrumental and geological records of sea-level change: Evidence from southern New England of an increase in the rate of sea-level rise in the late 19th century. Geophysical Research Letters 31 L05203.
  • Douglas, B. C., Kearney, M. S. and Leatherman, S. P. (2001). Sea-Level Rise: History and Consequences. Academic Press, San Diego, CA.
  • Engelhart, S. E., Horton, B. P., Douglas, B. C., Peltier, W. R. and Tornqvist, T. E. (2009). Spatial variability of late Holocene and 20th century sea level rise along the Atlantic coast of the United States. Geology 37 1115–1118.
  • Gehrels, W. R. and Woodworth, P. L. (2013). When did modern rates of sea-level rise start? Global and Planetary Change 100 263–277.
  • Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statist. Sci. 7 457–472.
  • Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In Bayesian Statistics, 4 (PeñíScola, 1991) 169–193. Oxford Univ. Press, New York.
  • Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc. 102 359–378.
  • Gornitz, V., Lebedeff, S. and Hansen, J. (1982). Global sea level trend in the past century. Science 215 1611–1614.
  • Haslett, J. and Parnell, A. C. (2008). A simple monotone process with application to radiocarbon-dated depth chronologies. J. R. Stat. Soc. Ser. C. Appl. Stat. 57 399–418.
  • Hay, C. C., Morrow, E., Kopp, R. E. and Mitrovica, J. X. (2015). Probabilistic reanalysis of twentieth-century sea-level rise. Nature 517 481–484.
  • Heidelberger, P. and Welch, P. D. (1983). Simulation run length control in the presence of an initial transient. Oper. Res. 31 1109–1144.
  • Holgate, S. J. and Woodworth, P. L. (2004). Evidence for enhanced coastal sea level rise during the 1990s. Geophysical Research Letters 31 L07305.
  • Holsclaw, T., Sansó, B., Lee, H. K. H., Heitmann, K., Habib, S., Higdon, D. and Alam, U. (2013). Gaussian process modeling of derivative curves. Technometrics 55 57–67.
  • Horton, B. P. and Edwards, R. J. (2006). Quantifying Holocene sea-level change using intertidal foraminifera: Lessons from the British Isles. Cushman Foundation for Foraminiferal Research, Special Publication 40 97.
  • Horton, B. P., Edwards, J. M. and Lloyd, R. J. (1999). UK intertidal foraminiferal distributions: Implications for sea-level studies. Marine Micropaleontology 36 205–223.
  • Houston, J. R. and Dean, R. G. (2011). Sea-level acceleration based on U.S. tide gauges and extensions of previous global-gauge analyses. Journal of Coastal Research 27 409–417.
  • IPCC (2013). Climate change 2013: The physical science basis. Contribution of working group I to the fifth assessment report of the intergovernmental panel on climate change. Cambridge Univ. Press, Cambridge.
  • Jevrejeva, S., Grinsted, A., Moore, J. C. and Holgate, S. (2006). Nonlinear trends and multiyear cycles in sea level records. Journal of Geophysical Research: Oceans 111 C09012.
  • Jevrejeva, S., Moore, J. C., Grinsted, A. and Woodworth, P. L. (2008). Recent global sea level acceleration started over 200 years ago? Geophysical Research Letters 35.
  • Jevrejeva, S., Moore, J. C., Grinsted, A., Matthews, A. and Spada, G. (2014). Trends and acceleration in global and regional sea levels since 1807. Global and Planetary Change 113 11–22.
  • Juggins, S. and Birks, H. J. B. (2012). Quantiative environmental reconstructions from biological data. In Tracking Environmental Change Using Lake Sediments 5 431–494. Springer, Berlin.
  • Kemp, A. C., Horton, B. P., Culver, S. J., Corbett, D. R., van de Plassche, O., Gehrels, W. R., Douglas, B. C. and Parnell, A. C. (2009). Timing and magnitude of recent accelerated sea-level rise (North Carolina, United States). Geology 37 1035–1038.
  • Kemp, A. C., Horton, B. P., Donnelly, J. P., Mann, M. E., Vermeer, M. and Rahmstorf, S. (2011). Climate related sea-level variations over the past two millennia. Proc. Natl. Acad. Sci. USA 108 11017–11022.
  • Kemp, A. C., Horton, B. P., Vane, C. H., Corbett, D. R., Bernhardt, C. E., Engelhart, S. E., Anisfeld, S. C., Parnell, A. C. and Cahill, N. (2013). Sea-level change during the last 2500 years in New Jersey, USA. Quaternary Science Reviews 81 90–104.
  • Kopp, R. E. (2013). Does the mid-atlantic united states sea level acceleration hot spot reflect ocean dynamic variability? Geophysical Research Letters 40 3981–3985.
  • Liang, H. and Wu, H. (2008). Parameter estimation for differential equation models using a framework of measurement error in regression models. J. Amer. Statist. Assoc. 103 1570–1583.
  • Lindgren, F., Rue, H. and Lindström, J. (2011). An explicit link between Gaussian fields and Gaussian Markov random fields: The stochastic partial differential equation approach. J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 423–498.
  • Long, A. J., Barlow, N. L. M., Gehrels, W. R., Saher, M. H., Woodworth, P. L., Scaife, R. G., Brain, M. J. and Cahill, N. (2014). Contrasting records of sea-level change in the eastern and western North Atlantic during the last 300 years. Earth and Planetary Science Letters 388 110–122.
  • Mann, M. E., Zhang, Z., Hughes, M. K., Bradley, R. S., Miller, S. K., Rutherford, S. and Ni, F. (2008). Proxy-based reconstructions of hemispheric and global surface temperature variations over the past two millennia. Proc. Natl. Acad. Sci. USA 105 13252–13257.
  • Mardia, K. V., Kent, J. T., Goodall, C. R. and Little, J. A. (1996). Kriging and splines with derivative information. Biometrika 83 207–221.
  • Morris, J. T., Sundareshwar, P. V., Nietch, C. T., Kjerfve, B. and Cahoon, D. R. (2002). Response of coastal wetlands to rising sea level. Ecology 83 2869–2877.
  • Nerem, R. S., Chambers, D., Choe, C. and Mitchum, G. T. (2010). Estimating mean sea level change from the TOPEX and Jason altimeter missions. Marine Geodesy 33 435–446.
  • Nicholls, R. J. and Cazenave, A. (2010). Sea-level rise and its impact on coastal zones. Science 328 1517–1520.
  • O’Hagan, A. (1992). Some Bayesian numerical analysis. In Bayesian Statistics, 4 (PeñíScola, 1991) 345–363. Oxford Univ. Press, New York.
  • Parnell, A. C., Buck, C. E. and Doan, T. K. (2011). A review of statistical chronology models for high-resolution, proxy-based Holocene palaeoenvironmental reconstruction. Quaternary Science Reviews 30 2948–2960.
  • Parnell, A. C., Haslett, J., Allen, J. R. M., Buck, C. E. and Huntley, B. (2008). A flexible approach to assessing synchroneity of past events using Bayesian reconstructions of sedimentation history. Quaternary Science Reviews 27 1872–1885.
  • Peltier, W. R. (2004). Global glacial isostasy and the surface of the ice-age Earth: The ICE-5G (VM2) model and GRACE. Annual Review of Earth and Planetary Sciences 32 111–149.
  • Peltier, W. R. and Tushingham, A. M. (1991). Influence of glacial isostatic adjustment on tide gauge measurements of secular sea level change. Journal of Geophysical Research: Solid Earth 96 6779–6796.
  • Plummer, M. (2003). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. In Proceedings of the 3rd International Workshop on Distributed Statistical Computing, TU Wien.
  • Plummer, M. (2014). rjags: Bayesian graphical models using MCMC. R package version 3-14.
  • Plummer, M., Best, N., Cowles, K. and Vines, K. (2006). CODA: Convergence Diagnosis and Output Analysis for MCMC. R News 6 7–11.
  • Rahmstorf, S. (2007). A semi empirical approach to projecting future sea-level rise. Science 315 368–370.
  • Rasmussen, C. E. and Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. MIT Press, Cambridge, MA.
  • Sallenger, A. H., Doran, K. S. and How’d, P. A. (2012). Hotspot of accelerated sea-level rise on the Atlantic coast of North America. Nature Clim. Change 2 884–888.
  • Scott, D. B. and Medioli, F. S. (1978). Vertical zonations of marsh foraminifera as accurate indicators of former sea levels. Nature 272 528–531.
  • Shennan, I. and Horton, B. P. (2002). Holocene land- and sea-level changes in Great Britain. Journal of Quaternary Science 17 511–526.
  • Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.
  • White, N. J., Church, J. A. and Gregory, J. M. (2005). Coastal and global averaged sea level rise for 1950 to 2000. Geophysical Research Letters 32 L01601.
  • Williams, C. K. I. and Rasmussen, C. E. (1996). Gaussian Processes for Regression. MIT Press, Cambridge.
  • Woodworth, P. L. and Player, R. (2003). The permanent service for mean sea level: An update to the 21st century. Journal of Coastal Research 19 287–295.
  • Woodworth, P. L., White, N. J., Jevrejeva, S., Holgate, S. J., Church, J. A. and Gehrels, W. R. (2009). Evidence for the accelerations of sea level on multi-decade and century timescales. International Journal of Climatology 29 777–789.
  • Yaglom, I. M. (2011). Geometric Transformations. I. Mathematical Association of America, Washington, DC.

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