Handbook of geo-statistics in R for fisheries and marine ecology.

Abstract

Fisheries surveys to estimate the abundance of populations have become a pillar in providing fishery-independent data to determine the status of fish stocks and monitor ecosystems. Since the early 1990s, geostatistics has been used for designing sampling at sea and estimating the precision of estimates of global population biom abun- dance (ICES, 1993; Rivoirard et al., 2000). Now, the ecosystem approach to fisheries management calls for methods that deal explicitly with spatial issues. In effect, the spatial management of human activities and/or the conservation of particular habitats require precise distribution maps of resources at various stages in their life cycle. Geo- statistics offers a range of solutions for mappig and characterizing different aspects of spatial distributions. On more ecological grounds, geostatistics is also useful for modeling habitats and understanding the ecology of spatial distributions. The varied range of geostatistical methods is largely based on the theory of random functions and random fields. The cornerstone of the geostatistical approach to applying this statistical framework for mapping lies in the so-called structural analysis, where the spatial (or spatio-temporal) correlation structure in the data is analyzed and modeled by a so-called variogram. Model types e.g. power, exponential, spherical) are chosen based on their underlying physical and mathematical properties relative to the spatial process to be modelled (Matheron, 1989). Once the model type is chosen, it is best fitted to the data using standard statistical fitting procedures. The model is then used for i nterpolating the data on a grid, which results in a map of the variable studied (local and global estimation) and a map of the estimation error (precision of the esti- mation). It is worth noting that being model-based, the estimation variance calculated by geostatistics applies to any sampling design and particular ly to regular designs, in which sample point locations are spatially correlated. This frees the practicioner from using random designs only to compute design-based statistics, as random designs may provide lower precision than regular designs. Further, geostatistics and classical statis- tics correspond to different approaches when using the same statistical framework of random functions (Matheron, 1989). In particular, geostatistics estimates regional quantities (mean value of the process over a domain) while classical statistics focusses on estimating the process mean. In addition, classical statistics computes the variance of the estimate, while geostatistics also develops the variance of the estima tion error (ICES, 1993; Petitgas, 2001). Depending on the spatial model, sampling intensity , and size of the domain, the estimates may or may not differ, which justifies differentiating between the two approaches (Matheron, 1989). The objective of this handbook is to summarize and explain the basic notions on the wide range of geostatistical methods (linear, multivariate, non-linear, simulations) that are useful for mapping in the context of the ecosystem approach and offer to the reader illustrative case studies with code in R language. Global estimation of population abundance (or biomass) with its precision for different survey designs (even systematic design) is a key issue in fisheries science for which geostatistics provides solutions given a variogram model (Petitgas, 2001; Bez, 2002). This is explained in chapters 4 and 5 on variography and variances. This latter chapter discusses the relationship between structure and scale. Further, when the variable to estimate is a non-linear combination of primary parameters that are those sampled, simulations may be required, as is explained in Chapter 9 on simulations. Variation in spatial distributions with population abundance and/or environmental factors is another key issue. The many aspects of spatial distributions can be character- ized by spatial indicators and monitored over time (Bez and Rivoirard, 2001; Woillez et al., 2007 , 2009a). Chapter 3 is dedicated to spatial indicators. Mapping resources and habitats is clearly paramount. The geostatistical solution to mapping is kriging, which constructs local unbiased estimates of minimum variance. For that, one assumes an underlying random function and its variogram model. The various types of kriging and interpolation settings (Chilès and Delfiner, 2012) are presented in Chapter 6. Mapping habitats may be more complex than kriging fish concentrations. One may be interested in thresholding the data to consider the prevalence in species occurrence or hotspots. Or one may be interested in incorporating in the mapping particular relationships with environmental parameters, some of which may be qualitative variables. Thus , multivariate kriging and non-linear approaches using thresholds (Rivoirard, 1994; Chilès and Delfiner, 2012) are developed in chapters 7 and 8. The applications of a wide range of geostatistical tools are expected to increase with the development of the package RGeostats (Renardet et al ., 2016), which is now freely available for the R language environment. This handbook is intended to summarize the principles of geostatistics and provide to the reader the capability to apply the methods using demonstration scripts in the R language. It compiles the materials of the 2013 and 2014 ICES training courses held by the authors in Fontainebleau. The handbook is constructed from lecture notes presenting the theoretical background with illustrative fisheries survey data studies. The annexes detail the practice in applying the methods. The R package RGeostats is presented in Annex 1. Example data sets used throughout the document are presented in Annex 2. Demonstration Rscripts are provided in Annex 3. Each script allows the user to perform a particular geostatistical study on an example dataset. Each script can be copy/pasted in the R environment for demonstration. The examples illustrating the theory are taken from the Rscripts provided in Annex 3