A classic method to select an evenly spread sample over a two dimensional area is systematic sampling (Bickford et al., 1963; Hazard et al., 1989). To begin, a grid is placed over the study area with l the number of intersection of the grid within the study area. Sampling starts by selecting a random intersection on the grid. Thereafter, every kth intersection is selected such that k = nl. Units including the selected intersections are selected. Alternatively, systematic sampling can be done by selecting a random unit in the study area. Thereafter, every kth unit is selected such that k = Nn. Systematic sampling is illustrated in Figure 3.1. The method described here is the most well known form of systematic sampling. There are several variations of the base design, such as stratified random systematic sampling where a point is selected within each stratum and centric systematic sampling where points are selected at the centre of each stratum.
The efficiency of systematic sampling tends to depend on the spatial correlation of the species distribution. In the case that the species are uniformly distributed, one can expect a similar efficiency with systematic sampling as with SRS. In the abundance of spatial autocorrelation it is known that systematic sampling can have a higher efficiency compared with alternative sampling designs such as SRS or StratRS (Payandeh, 1970; Ripley, 1991; Stevens and Olsen, 2004; Wang et al., 2012, 2013). Because of this, systematic sampling is often implemented when, in the presence of spatial autocorrelation, a precise estimates of the number of individuals of a species is required or if mapping of the species distribution over the study area is wanted. Additionally, systematic sampling can be implemented for logistic reasons. It can for example be easier for field scientist to visit sites at regular intervals.
A potential problem could be that the species density follows a similar pattern as the pattern of the select units. For example, a ploughed field is sampled for seedlings of an invasive weed species. The sample units are placed in the same pattern as the furrows on the field. If you always count seedlings in the base of the furrow the variance will be underestimated. To solve this problem several variations of the classic system sampling algorithm have been proposed in the literature to perturb the purely systematic character of the selection of sampling units Olea (1984).
Although it is shown theoretically that systematic sampling can be more efficient than SRS a particular issue with systematic sampling is that there is no ‘design-based’ unbiased estimator. (Cochran, 1977; Wang et al., 2013). ‘Design based’ in this case means using probability based
sampling theory. In short: Once the initial starting point is selected on the grid, the remaining points of the sample are part of a deterministic sequence. Because of this, the variance estimation can become biased. Furthermore, since the second order inclusion probabilities (which is the probabilities that two units are selected in the same sample) of neighbouring units are often (near) zero. It is therefore difficult to obtain an unbiased variance estimate. Several solutions have been proposed to solve this problem of having an unbiased estimator (Dunn and Harrison, 1993). One classic solution is to post-stratify the area. The area is stratified such that each stratum has (usually) two sampling units per stratum. The systematic sample is then treated as a StratRS sample. If similar sampling units are grouped within the same strata then the
within-strata variance will decrease and hence the overall variance of the estimator will decrease. In the case of anisotrophic spatial autocorrelation this stratification is preferably done in the direction of the spatial autocorrelation, for example long narrow strata following the direction of the spatial autocorrelation McArdle and Blackwell (1989); Ripley (1991). Another method is to treat the systematic sample as a SRS and use the variance estimators for SRS. However this results generally in a conservative variance estimation.
Systemic sampling has a practically attractive design. Potentially, the fact that all the selected intersections on the grid need to be sampled to obtain a spatially balance sample can have its disadvantages. The design can lose its uniform spread when problems like non-response or premature endings arise. By non-response in this case we mean that when a sampling unit cannot be sampled in practise. For example, assume the unit is located on private property or the area is inaccessible for any other reason. Although the non-responses can have an effect on the spatial coverage effects on the precision of the population estimation are generally small. An example of a premature ending is when the sampling sequence starts in one corner but because of time or money related issues the survey is aborted before visiting all units. In that case, some areas end up being unsampled. This affects the spatial balance of the sample. Furthermore, it is difficult to add additional units to the initial sample while maintaining the spatial balance of the sample. Sampling additional units would be considered, for example, when extra funding becomes available to continue the survey. Of course, some of these issues also occur with other sampling designs. Finally, there is no obvious method to select an unequal probability sample with Systematic sampling.(Stevens and Olsen, 2004)