Chapter 2 Two dimensional systematic surveys
Two-dimensional systematic surveys are a commonly applied survey design for estimating the abundance of spatial populations (Millar and Olsen 1995; R. M. Fewster 2011). Systematic surveys enforce a uniform sampling pattern over the entire spatial domain. Unlike random sampling, systematic surveys avoid over and under sampling space.
Systematic surveys can have the advantage over random sampling strategies of being more practical to implement, and in simulation studies, the estimated population density has been shown to provide considerably lower variance than random designs under many conditions e.g., McGarvey, Burch, and Matthews (2016). However, the most common systematic survey design consists of just a single primary sampling unit (PSU) and consequently there is no applicable design-based estimator of this variance. This has led to the development and application of several approximations based on variations of design-based variance estimators designed for random sample designs (Wolter 1984; D’Orazio 2003).
Example of systematic survey. Blue rectangles are sampling units
The only randomly selected sampling unit from
Consider a spatial survey region \(\mathcal{D} \subset \mathbb{R}^2\) of area \(A\). For simplicity \(i\) indexes a sampling unit (quadrat) that is spatially referenced by coordinates in \(\mathcal{D}\). A two-dimensional systematic survey that is based on a single PSU requires the specification of the location of the first sampling unit, from which the location of all other sampling units are deterministically specified. The response variable in this setting is either numbers (abundance) or weight (biomass) in each of the \(n\) sampling units denoted by \(\boldsymbol{y} = (y_1,y_2,\dots, y_n)^T\).
The population characteristic of focus is the total population within the survey region, denoted by \(N\), or equivalently the mean density over \(\mathcal{D}\), \(d_\mathcal{D}=N/A\). If \(\alpha\) is the area of each sampling unit then \(y_\mathcal{D}=\alpha d_\mathcal{D}\) is the expected mean value. Then, \(N=\kappa y_\mathcal{D}\) where \(\kappa=A/\alpha\) is the inverse of the sampling fraction. Design-based methods applied to systematic surveys all use the sample mean estimator of \(y_\mathcal{D}\), \(\bar{y}=\sum\limits_{i = 1}^{n} y_i/n\), and hence the estimator \(\widehat{N}=\kappa \bar{y}\) of \(N\), with variance estimator
\[ \widehat{\rm var} \left(\widehat{N}\right) = \kappa^2 \widehat{\rm var}(\bar{y}) \ . \]
In systematic surveys with multiple PSUs the systematic design-based variance estimator (\(\widehat{\rm var}_{sys}(\bar{y})\)) is known and follows, \[\begin{equation*}\label{eq:exact_var} \widehat{\rm var}_{sys}(\bar{y}) = \frac{1}{B}\sum\limits_{b = 1}^{B} \left(\bar{y}_{b} - \bar{y}\right)^2 \end{equation*}\] where, \(b=1,\dots,B\) are the PSUs, each assumed to contain the same number of sampling units, and \(y_b\) is the sample mean within PSU \(b\). However, \(B\) is typically a relatively small number, so this estimated variance generally has few degrees of freedom.
%, For the systematic design of interest in this study there is just a single PSU (the position of the first quadrat) and hence no design-based variance estimator exists. For this reason, as shown below, some authors have explored the use of the simple random sampling (SRS) variance estimator, ad-hoc estimators utilizing stratified random survey variance estimators (Millar and Olsen 1995; Strand 2017) and adjusted SRS variance estimators (Ambrosio Flores, Iglesias Martı́nez, and Marı́n Ferrer 2003; Strand 2017; Brus and Saby 2016; McGarvey, Burch, and Matthews 2016).