Lower trophic level (LTL) ocean ecosystem models are important tools for understanding ocean biogeochemical variability and its role in Earth’s climate system. These models are often replete with parameters that cannot be well constrained by the sparse observational data available. LTL ocean ecosystem model parameter estimation is examined from a probabilistic perspective, using a Bayesian hierarchical model (BHM), in the coastal Gulf of Alaska (CGOA) domain that benefits from ocean station observations obtained in repeated US GLOBEC cruises. Data entering the BHM include daily average SeaWiFS satellite estimates of surface chlorophyll and GLOBEC observations of nutrient and phytoplankton profiles at inner and outer shelf stations on the Seward Line. The final form of the BHM process model component is comprised of a discrete version of the Nutrient-Phytoplankton-Zooplankton-Detritus LTL ecosystem model equations augmented to address iron limitation in the CGOA (i.e., NPZDFe), and including a vertical diffusion term to constrain the timing of the phytoplankton bloom in spring.

Even in the relatively data-rich GLOBEC context, parameter estimation in the BHM requires guidance from a suite of calculations in a coupled physical-biological deterministic model—the Regional Ocean Model System coupled to an NPZDFe component (ROMS-NPZDFe). ROMS-NPZDFe simulations are used to: (1) validate the BHM formulation, (2) separate BHM limitations due to sampling from those due to LTL model approximations, and (3) obtain output distributions for zooplankton grazing rate and phytoplankton nutrient uptake rate using GLOBEC and SeaWiFS data for 2001. Uncertainty is evident from the spreads in output distributions for model parameters in the BHM. Experiments driven by simulated data from ROMS-NPZDFe helped to optimize the utility of GLOBEC observations for LTL ocean ecosystem model parameter estimation, given ever-present uncertainty issues.

The ROMS-NPZDFe simulations are also used to build Bayesian statistical models as surrogates for the deterministic model. Two applications are briefly described. One estimates output distributions for selected ocean ecosystem parameters while accounting for spatial variability across the GLOBEC stations in the CGOA. A second application assimilates SeaWiFS data and simulated data from a ROMS-NPZDFe control run for 2002 to estimate complete fields of surface phytoplankton concentration, with associated spatial and temporal uncertainties.

Milliff, R.F., J. Fiechter, W.B. Leeds, R. Herbei, C.K. Wikle, M.B. Hooten, A.M. Moore, T.M. Powell, and J. Brown. 2013. Uncertainty management in coupled physical-biological lower trophic level ocean ecosystem models. Oceanography 26(4):98–115, https://doi.org/10.5670/oceanog.2013.78.

Berliner, L.M., R.F. Milliff, and C.K. Wikle. 2003. Bayesian hierarchical modeling of air-sea interaction. Journal of Geophysical Research 108, C43104, https://doi.org/10.1029/2002JC001413.

Berliner, L.M., C.K. Wikle, and N. Cressie. 2000. Long-lead prediction of Pacific SSTs via Bayesian Dynamic Modeling. Journal of Climate 13:3,953–3,968, https://doi.org/10.1175/1520-0442(2001)013<3953:LLPOPS>2.0.CO;2.

Brown, J., and J. Fiechter. 2012. Quantifying eddy-chlorophyll covariability in the coastal Gulf of Alaska. Dynamics of Atmospheres and Oceans 55–56:1–21, https://doi.org/10.1016/j.dynatmoce.2012.04.001.

Conti, S., J. Gosling, J. Oakley, and A. O’Hagan. 2009. Gaussian process emulation for dynamic computer codes. Biometrika 96:663–676, https://doi.org/10.1093/biomet/asp028.

Cressie, N., and C.K. Wikle. 2011. Statistics for Spatio-Temporal Data. Wiley Series in Probability and Statistics, John Wiley and Sons Inc., 588 pp.

Cripps, E., D. Nott, W.T.M. Dunsmuir, and C.K. Wikle. 2005. Space-time modelling of Sydney Harbour Wind. Australian and New Zealand Journal of Statistics 47:3–17, https://doi.org/10.1111/j.1467-842X.2005.00368.x.

Dowd, M. 2007. Bayesian statistical data assimilation for ecosystem models using Markov Chain Monte Carlo. Journal of Marine Systems 68:439–456, https://doi.org/10.1016/j.jmarsys.2007.01.007.

Dowd, M. 2011. Estimating parameters for a stochastic dynamic marine ecological system. Environmetrics 22:501–515, https://doi.org/10.1002/env.1083.

Fiechter, J. 2012. Assessing marine ecosystem model properties from ensemble calculations. Ecological Modelling 242:164–179, https://doi.org/10.1016/j.ecolmodel.2012.05.016.

Fiechter, J., G. Broquet, A.M. Moore, and H.G. Arango. 2011. A data-assimilative, coupled physical-biological model for the coastal Gulf of Alaska. Dynamics of Atmospheres and Oceans 52:95–118, https://doi.org/10.1016/j.dynatmoce.2011.01.002.

Fiechter, J., R. Herbei, W. Leeds, J. Brown, R. Milliff, C. Wikle, A. Moore, and T. Powell. 2013. A Bayesian parameter estimation method applied to a marine ecosystem model for the coastal Gulf of Alaska. Ecological Modelling 258:122–133, https://doi.org/10.1016/j.ecolmodel.2013.03.003.

Fiechter, J., and A.M. Moore. 2009. Interannual spring bloom variability and Ekman pumping in the coastal Gulf of Alaska. Journal of Geophysical Research 114, C06004, https://doi.org/10.1029/2008JC005140.

Fiechter, J., and A.M. Moore. 2012. Iron limitation impact on eddy-induced ecosystem variability in the coastal Gulf of Alaska. Journal of Marine Systems 92:1–15, https://doi.org/10.1016/j.jmarsys.2011.09.012.

Fiechter, J., A.M. Moore, C.A. Edwards, K.W. Bruland, E. Di Lorenzo, C.V.W. Lewis, T.M. Powell, E.N. Curchitser, and K. Hedstrom. 2009. Modeling iron limitation of primary production in the coastal Gulf of Alaska. Deep Sea Research Part II 56:2,503–2,519, https://doi.org/10.1016/j.dsr2.2009.02.010.

Fox, N.I., and C.K. Wikle. 2005. A Bayesian quantitative precipitation nowcast scheme. Weather and Forecasting 2:264–275, https://doi.org/10.1175/WAF845.1.

Friedrichs, M.A.M, M.-E. Carr, R.T. Barber, M. Scardi, D. Antoine, R.A. Armstrong, I. Asanuma, M.J. Behrenfeld, E.T. Buitenhuis, F. Chai, and others. 2009. Assessing the uncertainties in model estimates of primary productivity in the tropical Pacific Ocean. Journal of Marine Systems 76:113–133, https://doi.org/10.1016/j.jmarsys.2008.05.010.

Friedrichs, M.A.M., J.A. Dusenberry, L.A. Anderson, R.A. Armstrong, F. Chai, J.R. Christian, S.C. Doney, J. Dunne, M. Fujii, R. Hood, and others. 2007. Assessment of skill and portability in regional marine biogeochemical models: Role of multiple planktonic groups. Journal of Geophysical Research 112, C08001, https://doi.org/10.1029/2006JC003852.

Haidvogel, D.B., H. Arango, W.P. Budgell, B.D. Cornuelle, E. Curchitser, E. Di Lorenzo, K. Fennel, W.R. Geyer, A.J. Hermann, L. Lanerolle, and others. 2008. Ocean forecasting in terrain-following coordinates: Formulation and skill assessment of the Regional Ocean Modeling System. Journal of Computational Physics 227:3,595–3,624, https://doi.org/10.1016/j.jcp.2007.06.016.

Harmon, R., and P. Challenor. 1997. A Markov chain Monte Carlo method for estimation and assimilation into models. Ecological Modelling 101:41–59, https://doi.org/10.1016/S0304-3800(97)01947-9.

Higdon, D., J. Gattiker, B. Williams, and M. Rightly. 2008. Computer model calibration using high-dimensional output. Journal of the American Statistical Association 103:570–583, https://doi.org/10.1198/016214507000000888.

Hooten, M.B., and C.K. Wikle. 2007. Shifts in the spatio-temporal growth dynamics of shortleaf pine. Environmental and Ecological Statistics 14:207–227, https://doi.org/10.1007/s10651-007-0016-1.

Hooten, M.B., and C.K. Wikle. 2008. A hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian Collared-Dove. Environmental and Ecological Statistics 15:59–70, https://doi.org/10.1007/s10651-007-0040-1.

Hooten, M.B., W.B. Leeds, J. Fiechter, and C.K. Wikle. 2011. Assessing first-order emulator inference for physical parameters in nonlinear mechanistic models. Journal of Agricultural, Biological and Environmental Statistics 16:475–494, https://doi.org/10.1007/s13253-011-0073-7.

Hooten, M.B., C.K. Wikle, R.M. Dorazio, and J.A. Royle. 2007. Hierarchical spatio-temporal matrix models for characterizing invasions. Biometrics 63:558–567, https://doi.org/10.1111/j.1541-0420.2006.00725.x.

Kennedy, M., and A. O’Hagan. 2001. Bayesian calibration of computer models. Journal of the Royal Statistical Society Serie B 63:425–464, https://doi.org/10.1111/1467-9868.00294.

Kishi, M., M. Kashiwai, D.M. Ware, B.A. Megrey, D.L. Eslinger, F.E. Werner, M.N. Aita, T. Azumaya, M. Fujii, S. Hashimoto, and others. 2007. NEMURO: A lower trophic level model for the North Pacific marine ecosystem. Ecological Modelling 202:12–25, https://doi.org/10.1016/j.ecolmodel.2006.08.021.

Leeds, W.B, C.K. Wikle, and J. Fiechter. 2012. Emulator-assisted reduced-rank ecological data assimilation for nonlinear multivariate dynamical spatio-temporal processes. Statistical Methodology 17:126–138, https://doi.org/10.1016/j.stamet.2012.11.004.

Leeds, W.B., C.K. Wikle, J. Fiechter, J.L. Brown, and R.F. Milliff. 2013. Modeling 3-D spatio-temporal biogeochemical processes with a forest of 1-D statistical emulators. Environmetrics 24:1–12, https://doi.org/10.1002/env.2187.

Malve, O., M. Laine, H. Haario, T. Kirkkala, and J. Sarvala. 2007. Bayesian modeling of algal mass occurrences—using adaptive MCMC methods with a lake water quality model. Environmental Modelling & Software 22:966–977, https://doi.org/10.1016/j.envsoft.2006.06.016.

Margvelashvili, N., and E.P. Campbell. 2012. Sequential data assimilation in fine-resolution models using error-subspace emulators: Theory and preliminary evaluation. Journal of Marine Systems 90:13–22, https://doi.org/10.1016/j.jmarsys.2011.08.004.

Milliff, R.F., A. Bonazzi, C.K. Wikle, N. Pinardi, and L.M. Berliner. 2011. Ocean ensemble forecasting. Part 1: Ensemble Mediterranean winds from a Bayesian hierarchical model. Quarterly Journal of the Royal Meteorological Society 137:858–878, https://doi.org/10.1002/qj.767.

Moore, A.M., H.G. Arango, G. Broquet, B.S. Powell, J. Zavala-Garay, and A.T. Weaver. 2011. The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems: Part I – System overview and formulation. Progress in Oceanography 91:34–49, https://doi.org/10.1016/j.pocean.2011.05.004.

Parslow, J., N. Cressie, E. Campbell, E. Jones, and L. Murray. 2013. Bayesian learning and predictability in a stochastic nonlinear dynamical model. Ecological Applications 26:679–698, https://doi.org/10.1890/12-0312.1.

Powell, T.M., C.V.W. Lewis, E.N. Curchitser, D.B. Haidvogel, A.J. Hermann, and E.L. Dobbins. 2006. Results from a three-dimensional nested, biological-physical model of the California Current System and comparisons with statistics from satellite imagery. Journal of Geophysical Research 111, C07018, https://doi.org/10.1029/2004JC002506.

Rougier, J. 2008. Efficient emulators for multivariate deterministic functions. Journal of Computational and Graphical Statistics 17:827–843, https://doi.org/10.1198/106186008X384032.

Royle, J.A., L.M. Berliner, C.K. Wikle, and R.F. Milliff. 1998. A hierarchical spatial model for constructing wind fields from scatterometer data in the Labrador Sea. Pp. 367–381 in Case Studies in Bayesian Statistics IV. C. Gatsonis, R.E. Kass, B. Carlin, A. Cariquiry, A. Gelman, I. Verdinelli, and M. West, eds, Springer-Verlag.

Song, Y., C.K. Wikle, C.J. Anderson, and S.A. Lack. 2007. Bayesian estimation of stochastic parameterizations in a numerical weather forecasting model. Monthly Weather Review 135:4,045–4,059, https://doi.org/10.1175/2007MWR1928.1.

Strom, S.L., M. Brady Olson, E.L. Macri, and C.W. Mordy. 2006. Cross-shelf gradient in phytoplankton community structure, nutrient utilization and growth rate in the coastal Gulf of Alaska. Marine Ecology Progress Series 328:75–92, https://doi.org/10.3354/meps328075.

Ward, B.A., M.A.M. Friedrichs, T.R. Anderson, and A. Oschlies. 2010. Parameter optimization techniques and the problem of underdetermination in marine biogeochemical models. Journal of Marine Systems 81:34–43, https://doi.org/10.1016/j.jmarsys.2009.12.005.

Wikle, C.K. 2003a. Hierarchical Bayesian models for predicting the spread of ecological processes. Ecology 84:1,382–1,394, https://doi.org/10.1890/0012-9658(2003)084[1382:HBMFPT]2.0.CO;2.

Wikle, C.K. 2003b. Hierarchical models in environmental science. International Statistical Review 71:181–199, https://doi.org/10.1111/j.1751-5823.2003.tb00192.x.

Wikle, C.K., R.F. Milliff, D. Nychka, and L.M. Berliner. 2001. Spatiotemporal hierarchical Bayesian modeling: Tropical ocean surface winds. Journal of the American Statistical Association 96:383–397, https://doi.org/10.1198/016214501753168109.

Wikle, C.K., and M.B. Hooten. 2006. Hierarchical Bayesian spatio-temporal models for population spread. Pp. 145–169 in Applications of Computational Statistics in the Environmental Sciences: Hierarchical Bayes and MCMC Methods. J.S. Clark and A. Gelfand, eds, Oxford University Press.

Wikle, C.K., and M.B. Hooten. 2010. A general science-based framework for spatio-temporal dynamical models. TEST 19:417–451, https://doi.org/10.1007/s11749-010-0209-z.

Wikle, C.K., R.F. Milliff, R. Herbei, and W.B. Leeds. 2013. Modern statistical methods in oceanography: A hierarchical perspective. Statistical Science 28:466–486, https://doi.org/10.1214/13-STS436.

This is an open access article made available under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution, and reproduction in any medium or format as long as users cite the materials appropriately, provide a link to the Creative Commons license, and indicate the changes that were made to the original content. Images, animations, videos, or other third-party material used in articles are included in the Creative Commons license unless indicated otherwise in a credit line to the material. If the material is not included in the article’s Creative Commons license, users will need to obtain permission directly from the license holder to reproduce the material.