Adaptive monitoring of coral health at Scott Reef where data exhibit nonlinear and disturbed trends over time

Abstract Time series data are often observed in ecological monitoring. Frequently, such data exhibit nonlinear trends over time potentially due to complex relationships between observed and auxiliary variables, and there may also be sudden declines over time due to major disturbances. This poses substantial challenges for modeling such data and also for adaptive monitoring. To address this, we propose methods for finding adaptive designs for monitoring in such settings. This work is motivated by a monitoring program that has been established at Scott Reef; a coral reef off the Western coast of Australia. Data collected for monitoring the health of Scott Reef are considered, and semiparametric and interrupted time series modeling approaches are adopted to describe how these data vary over time. New methods are then proposed that enable adaptive monitoring designs to be found based on such modeling approaches. These methods are then applied to find future monitoring designs at Scott Reef where it was found that future information gain is expected to be similar across a variety of different sites, suggesting that no particular location needs to be prioritized at Scott Reef for the next monitoring phase. In addition, it was found that omitting some sampling sites/reef locations was possible without substantial loss in expected information gain, depending upon the disturbances that were observed. The resulting adaptive designs are used to form recommendations for future monitoring in this region, and for reefs where changes in the current monitoring practices are being sought. As the methods used and developed throughout this study are generic in nature, this research has the potential to improve ecological monitoring more broadly where complex data are being collected over time.


A.1. Scott Reef surveys
In Scott Reef, there are three nested sites at each reef location and one observation has been collected from each site for a given survey time. The number of samples collected for each survey time at Scott Reef is summarised in

A.2. Distribution of hard coral cover
We visualise the distribution of coral cover at Scott Reef to see how it varies over time (Figure A.1). Accordingly, it exhibits a nonlinear trend with a sudden shift around the 1998 bleaching event. In addition, the distribution of hard coral cover by reef locations is shown in Figure A.2. The years of cyclone events be provided Table A.2 . It can be seen from the individual figures that different reef locations have been impacted differently due to bleaching and cyclone disturbances.

A.3. Posterior estimation
The joint posterior distribution of model parameters and random effects can be expressed in a Bayesian framework as follows: Table A.1: Survey variations over time at Scott Reef. The first column represents survey times used at Scott Reef where the decimal places indicate the survey times within the given year. The next three columns represent three sites at each reef location. The value seven for a particular site index means that the corresponding site has been surveyed at all the seven reef locations. The last column represents the total number of observations collected over the seven reef locations using three sites which are located at each reef location. Starting from 2016.08, only the first site has been surveyed at each reef location which results in the total number of seven observations for the corresponding survey times.

Site index
Survey times 1 2 3 Total number of observations The fitted blue line is the mean curve produced using geom smooth function in ggplot2 package in R using the default smooth method (i.e. "loess"). The shaded red area represents the corresponding 95% confidence interval for the mean. The sudden shift in hard coral cover due to 1998 bleaching event is marked using a light blue colour box.  1994.83 1995.83 1995.83 1996.83 C 1996.83 1997.92 C C 1997.92 1998.90 1998.90 1999.92 1999.92 2001.90 C 2001.90  Bayesian framework using WinBUGS to sample from the posterior distribution for the purpose of model selection. The fitted blue lines are the mean curves produced using geom smooth function in ggplot2 package in R using the default smooth method (i.e. "loess"). The shaded red areas represent the corresponding 95% confidence intervals for the means. Some badly affected reef locations due to 1998 bleaching event are marked using light blue colour boxes. Here, the impact of a bleaching event can last for years after the event so some locations have been highlighted after the year 1998. Table A.3: The individual components of the adopted prior distribution p(θ).

A.4. Model selection
We used the M-closed perspective to determine the most appropriate model to describe the historical data at Scott Reef. The model selection results using DIC criterion for the formulated models is provided in Table   A.4. Furthermore, it should be noted that bleaching is involved in a significant interaction with cyclones in the model. Thus, this disturbance is having a significant effect depending upon the value of the cyclone variable. In terms of random effects, the posterior distribution summary is provided in Table   A.6. Accordingly, all reef random effects are non-significant. This indicates that there is not a lot of variability in hard coral cover trajectories at the reef location level. However, there are some significant site random effects indicating that there are distinguishable hard coral cover trajectories at the site level.

A.6. Goodness-of-fit
To assess the goodness-of-fit of the model defined above, a posterior predictive check was used. This involved comparing the observed data with data

A.7. Extrapolation results
To investigate the appropriateness of the proposed (Graham et al., 2007) Taylor series approximation to the mean response, this approximation was used to predict future but known hard coral cover in our data set (akin to a leave-one-out cross validation approach), we visualised the extended curves for each of the seven reef locations. For instance, Figure A.5 shows simulated values when the considered design included one site per reef location. As can be seen, the approximation appears to be reasonable for the next sampling

A.8. Disturbance scenarios
Representatives from AIMS provided us with two disturbance scenarios (i.e. two values for z) for which we evaluated designs (Table A.7). In Ta

A.9. Utility evaluations
In the case where the prior and the posterior distributions both are Multivariate Normal, then the KLD utility function can be defined as follows: where θ * m 0 , θ * m 1 and Σ * m 0 , Σ * m 1 are the means and variance covariance matrices of the prior and posterior distributions, respectively and q m is the dimension of the parameter space.

A.10. Informative sites at each reef location
We determine the optimal design that comprises of the most informative site at each reef location subject to two disturbance scenarios under Objective (ii). The selected sites from each reef location into the optimal designs and expected utility values under the two scenarios are provided in Table A.8.