Area Estimation by Point-Counting Techniques

Abstract

A commonly-used method of estimating the area of a region on a map or aerial photograph is the dot-grid method. The method may be described in general as follows. Consider a map of width M and length N. Divide the map by a grid system into mn rectangles each of area kl, to yield m rows of rectangles and n columns. Each rectangle is of width k and length 1, so M= mk and N= nl. To obtain the sample dots, first choose a dot uniformly distributed on the lower left rectangle of the map. Denote this point by (u, v). The sample dots, one in each rectangle, are located at the set of points {(u + rk, v+sl): r=0,...,m-1; s=O,...,n-1}. This sampling method is, in fact, the aligned systematic sampling method of Quenouille (1949) and the systematic sampling method of Das (1950). Figure 1 illustrates the method. This sampling method may be accomplished easily by using a transparency with systematically located dots on it. The dots are at distances k apart in one direction and I apart in the other. The transparency is laid over the map so that in any rectangular region of area kl on the map one of the transparency dots is uniformly distributed on that area. An estimate of the area of the region of interest is provided by the area of the transparency multiplied by the proportion of dots falling in the region. Because of its simplicity of implementation, systematic sampling is the usual method employed. However, systematic sampling is not necessarilyXan efficient method for area estimation, in the sense of minimizing variance. We show that a stratified sampling design, although less convenient to implement than systematic sampling, is usually more efficient. Also there is no unbiased estimate of variance for systematic sampling. We use 'unbiased' in the sense of 'unbiased with respect to the sampling design'. An unbiased estimate of variance can be obtained, with little loss in convenience, by taking a number of independent repetitions of systematic samples. We show that this procedure leads to a loss in efficiency when compared to a single systematic sample with the same overall sample size. The efficiency comparisons are made using a superpopulation model. The finite population variances for stratified sampling, and for systematic sampling with either a single or multiple random starts, are averaged over the model and compared. We will refer to these finite population variances averaged over the superpopulation as 'average variances' and