Bayesreef: A Bayesian inference framework for modelling reef growth in response to environmental change and biological dynamics

doi:10.1016/j.envsoft.2019.104610

Environmental Modelling & Software

Volume 125, March 2020, 104610

https://doi.org/10.1016/j.envsoft.2019.104610 Get rights and content

Highlights

•
Estimating the impact of environmental processes on vertical reef development in geological time is a very challenging task.
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pyReef-Core is a deterministic carbonate stratigraphic forward model designed to simulate the key biological and environmental processes.
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This paper presents Bayesreef for the estimation and uncertainty quantification of parameters in pyReef-Core.
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The results show that Bayesreef accurately estimates and provides uncertainty quantification for parameters in pyReef-Core.
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Bayesreef provides insights in of pyReef-Core and establishes the groundwork for future research.

Abstract

Estimating the impact of environmental processes on vertical reef development in geological time is a very challenging task. pyReef-Core is a deterministic carbonate stratigraphic forward model designed to simulate the key biological and environmental processes that determine vertical reef accretion and assemblage changes in fossil reef drill cores. We present a Bayesian framework called Bayesreef for the estimation and uncertainty quantification of parameters in pyReef-Core that represent environmental conditions affecting the growth of coral assemblages in geological timescales. Weencounter multimodal posterior distributions and investigate the challenges of sampling using Markov chain Monte-Carlo (MCMC) methods, which includes parallel tempering MCMC. We use a synthetic reef-core to investigate fundamental issues and then apply the methodology to a selected reef-core from the Great Barrier Reef in Australia. The results show that Bayesreef accurately estimates and provides uncertainty quantification of the selected parameters that represent environment and ecological conditions in pyReef-Core. Bayesreef provides insights into the complex posterior distributions of the parameters in pyReef-Core, which provides the groundwork for future research in this area.

Introduction

Developing data-driven models of reef evolution is challenging because the complexity of the process exceeds the amount of available data necessary. Reef evolution is determined by the interaction between environmental factors such as water chemistry, light availability, sedimentation and hydrodynamic energy (Veron, 2008). The data for understanding and modelling reef evolution can be extracted from the geological record of reef-drilled cores, which is sparse, and also expensive to obtain; hence, limited work has been done in this area.

pyReef-Core (Salles et al., 2018) is an example of a carbonate stratigraphic forward model (SFM) for reef evolution that captures a number of important ecological dynamics in coral reef systems. pyReef-Core is the first modelling attempt to constrain hydrodynamic energy and sediment input exposure thresholds for coralgal assemblages on a geological timescale. It is a one-dimensional (1-D) model that simulates the vertical (and not lateral, hence 1-D) coralgal growth patterns observed in a reef drill core. pyReef-Core has several parameters representing external environmental factors which impact reef development. Examples of these factors include sea-level changes and the relationship between sediment input and depth. It also has parameters describing the response of coralgal assemblage growth to these environmental factors, such as water flow and parameters for internal population dynamics such as the Malthusian parameter. Fig. 2 shows the workflow of pyReef-Core, which shows vertical accumulation contributed by different coralgal assemblages over a timeframe.

Given limited and sparse data, we need to quantify uncertainty from different factors in the estimation of unknown parameters in stratigraphic forward models such as pyReef-Core. Moreover, geophysical and stratigraphic forward models rarely have a unique solution (Isakov, 1993; Charvin et al., 2009a) which is also known as non-uniqueness (Burgess and Prince, 2015). For example, different combinations of a range of environmental parameters such as water flow, temperature and population dynamics of the coral assemblages in pyReef-Core may give rise to the same simulated reef-core stratigraphy. Stratigraphic forward models produce a set of solutions that represent multiple and competing hypotheses regarding geological system evolution (Cross et al., 1999; Heller et al., 1993). However, the explicit temporal and depth structure simulated by pyReef-Core presents an opportunity to restrict the number of possible solutions and thus reduce uncertainty.

Bayesian inference provides a rigorous methodology for estimation and uncertainty quantification of unknown parameters in a given model by incorporating information from multiple sources (Robert and Rousseau, 2009). The information from prior research, expert opinion, and knowledge regarding the nature of specific physical processes can be incorporated via a set of prior beliefs, also known as priors. Moreover, information from observed data is used to update these prior beliefs via the likelihood function. In the case of environmental modelling, Bayesian inference for uncertainty quantification has been deployed for a number of problems (O'Hagan, 2012; Hassan et al., 2009; Raje and Krishnan, 2012; Refsgaard et al., 2007).

We present a novel probabilistic framework for the estimation and uncertainty quantification of environmental processes and factors which impact the depth and temporal distribution of communities of corals and coralline algae (coralgal assemblages) found in fossil reef drill cores. The framework is called Bayesreef which employs Bayesian inference using Markov Chain Monte Carlo (MCMC) sampling to estimate the unknown parameters of pyReef-Core Although we chose pyReef-Core to demonstrate the idea, the framework is general, and can be adapted, for other stratigraphic forward models. The goal of Bayesreef is to provide estimation and uncertainty quantification for complex processes with sparse data which presents four significant contributions to the literature.

First, we transform a selected deterministic stratigraphic forward model (pyReef-Core) into a probabilistic framework known as Bayesreef, where the parameters are sampled via MCMC and represented using a probability distribution. Bayesreef employs a multinomial likelihood using the data from drilled reef-core featuring the coralgal assemblages over a timeframe. The expert opinion and the results of previous studies are incorporated into the prior, while knowledge of the physical processes from the pyReef-Core is connected to the observed assemblage in the reef-core via the multinomial likelihood.

Second, we use Bayesreef to constrain the number of solutions that represent the unique palaeo-environmental history of the reef core by incorporating two different types of data representation, where one features the temporal structure and the other the depth structure of the reef-core. By depth structure, we refer to the thickness and type of coralgal assemblage at varying depths. By temporal structure, we refer to the thickness and type of sediment and/or coralgal material laid down at varying points in time.

Third, we demonstrate the effectiveness of Bayesreef with two sampling methods, that includes single-chain MCMC and parallel tempering MCMC, for synthetic and real reef-core drilled from a selected location in the Great Barrier Reef.

Fourth, we make the methodology and its implementation available for other researchers as a software tool ,¹ which can be used to make inference regarding the factors affecting reef evolution.

The rest of the paper is outlined as follows: Section 2 provides background and related work, while Section 3 presents the methodology and techniques used including the multinomial likelihood function. Section 4 presents experiments and results. Section 5 provides discussion, and Section 6 concludes the paper with a discussion of future research.

Section snippets

Coral reef evolution

The ability of corals to vigorously grow and build reef structures is dependent upon favourable environmental conditions (Done, 2011). Three related environmental factors examined in pyReef-Core are vital in influencing coral reef evolution on multi-decadal to centennial timescales. They are water depth (accommodation), hydrodynamic energy, and autochthonous (reef-derived) sediment input.

The accommodation is the vertical space in the water column above the substrate within which corals can grow

Methodology: Bayesreef

While deterministic models of long-term interactions of organisms in a marine ecosystem exist (Clavera-Gispert et al., 2017), and software tools for coral reef evolution are available (Salles et al., 2018; Barrett and Webster, 2017), there are no probabilistic methods which combine these deterministic models with observed data. We note that such models have several or many non-unique solutions; however, even if we manage to constrain them to be unique, there remains uncertainty with the

Experimental design and results

We investigate the performance of Bayesreef using the multinomial likelihood function outlined in the previous section for the time and depth structure and their associated time-based and depth-based likelihoods as the basis of prediction and evaluation of results. We use two different MCMC sampling methods for synthetic reef-core and apply to a selected application from the Great Barrier Reef. The experiments follow the plan below.

1.
Single-chain MCMC sampling (Algorithm 2) for two free

Discussion

We applied Bayesreef to infer how coralgal community structure changes in relation to prevailing changes in accommodation, hydrodynamic energy and sediment input. The results show that Bayesreef provides a reliable prediction of the synthetic ground truth data in the experiments that consider the depth and time-based likelihood and different combination of selected free parameters. Moreover, the estimated parameters provide an accurate prediction of the reef-core when compared to the synthetic

Conclusion and future work

We presented Bayesreef, which is a comprehensive Bayesian framework that incorporates multiple sources of information, including forward models, priors and empirical data from geological reef-cores. The Bayesreef framework employ single-chain and parallel tempering MCMC sampling to address the challenges of multimodal posteriors distributions given synthetic reef-core and an application from the Great Barrier Reef. The framework addresses geophysical inverse problem posed by unobserved

Data and software

The source code that implements the framework of Bayesreef can be downloaded from the Github repository .²

Supplementary results for selected experiments are given here ³

Acknowledgements

We would like to acknowledge Australian Research Council (ARC - DP120101793 and FT140101266) and Sydney Research Excellence Initiative 2017, The University of Sydney for providing the grants for supporting this work. We also wish to acknowledge Sydney Informatics Hub for providing research engineering support for this project.

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Bayesreef: A Bayesian inference framework for modelling reef growth in response to environmental change and biological dynamics

Highlights

Abstract

Introduction

Section snippets

Coral reef evolution

Methodology: Bayesreef

Experimental design and results

Discussion

Conclusion and future work

Data and software

Acknowledgements

Mar. Pollut. Bull.

Sediment. Geol.

Mar. Geol.

Appl. Numer. Math.

Neurocomputing

J. Differ. Equ.

Mar. Geol.

Quat. Res.

Glob. Planet. Chang.

Mar. Pollut. Bull.

Environ. Model. Softw

Mar. Pet. Geol.

Environ. Model. Softw

Sediment. Geol.

Earth Sci. Rev.

Environ. Model. Softw.

Environ. Model. Softw

Quat. Sci. Rev.

Palaeogeogr. Palaeoclimatol. Palaeoecol.

Effects of sedimentation on settlement of acropora millepora

Coral Reefs

The great barrier reef: the chronological record from a new borehole

J. Sediment. Res.

Tracing the influence of land-use change on water quality and coral reefs using a bayesian model

Sci. Rep.

Non-unique stratal geometries: implications for sequence stratigraphic interpretations

Basin Res.

Postglacial growth history of a French polynesian barrier reef tract, tahiti, central pacific

Sedimentology

Coral reef response to quaternary sea-level and environmental changes: state of the science

Sedimentology

Multi-core Parallel Tempering Bayeslands for Basin and Landscape Evolution

Coral morphology, diversity and reef growth

Nature

A bayesian approach to inverse modelling of stratigraphy, part 1: Method

Basin Res.

A bayesian approach to inverse modelling of stratigraphy, part 1: Method

Basin Res.

Process-based forward numerical ecological modeling for carbonate sedimentary basins

Comput. Geosci.

Water flow modulates the response of coral reef communities to ocean acidification

Sci. Rep.

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Growth fabrics and growth-rates of holocene reefs in the great barrier-reef

BMR J. Aust. Geol. Geophys.

Relationship between reef growth and sea-level in the great barrier reef

Evolution of the Great Barrier Reef over the Last 130 Ka; a Multifaceted Approach, Integrating Palaeo Ecological, Palaeo Environmental and Chronological Data from Cores

“Corals: Environmental Controls on Growth,” Encyclopedia of Modern Coral Reefs: Structure, Form and Process

Temperature and phytoplankton growth in the sea

Fish. Bull.

Mass-transfer limitation of nutrient uptake by a wave-dominated reef flat community

Limnol. Oceanogr.