Site size (km2) Woodland Grassland Arable Urban
16 4.16 9.85 13.97 16.75
Table 8.3 presents the standard deviation as an index of variability for each of the four land-use categories at different site sizes. Two points emerge from this table. First, the standard deviation increases as the site size falls. This is not unexpected as smaller sites are likely to give more variable results as more extreme (larger or smaller) percentages of use in one particular category become more likely. Second, the values of the standard deviations appear to reflect the way these land classes cluster on the ground. Thus urban and arable land cluster more strongly compared with the other two categories. The spatial configuration of the land-use categories thus emerges in this variability (for details see Harrison et al., 1989).
Simulating sampling schemes: size of site
By treating the characteristics of the simulated sampling process (e.g. number of observations (n) and variability (σ)) as ‘population parameters’ it was possible to derive information on estimated sampling distributions for different random sampling schemes.
Assuming a random sampling scheme, for a given sample size and for a particular sample site size, we can calculate the expected standard deviation of the sample mean ( ) using the formula:
There are clearly numerous scenarios which we could investigate with this information. Here we focus on one scenario. We took samples which covered 64km2 of the land surface of Avon in total, about 5% of the land surface (much in line with the MLC sampling scheme used in Avon). Given that the size (area extent) of the sample sites used may vary, this can be done in a number of ways: four 16-km2, eight 8-km2 sites, and so on, down to 128 0.5-km2 sites.
Table 8.4. Random sampling with 64 km
2of cover, standard error of the sample means.
Sample
No. km2 Woodland Grassland Arable Urban
4 16 2.08 4.93 6.99 8.38
8 8 1.80 3.87 5.55 6.22
16 4 1.30 3.09 4.28 4.59
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32 2 1.12 2.50 3.26 3.35 64 1 0.87 2.00 2.53 2.52
128 0.5 0.71 1.60 1.95 1.87
Table 8.4 reports the expected standard errors of the sample means under this experiment for the four land-use categories, based on the standard formula given above.
These results show that, for a given total area of land use, the most efficient sampling approach is to take many small sites. Although this result is perhaps not unexpected, these results do allow us to quantify the expected differences in efficiency of different sampling strategies. Thus using the results in Table 8.4, if we moved from a strategy of 16 4-km2 sites to 64 1-km2 sites we would expect to reduce the standard error of the means by 33% for woodland, 35% for grassland, 41% for arable land, and 45% for urban land. The consistency of the results from this experiment suggests that there are very substantial gains to be had from moving to the use of more smaller sites for sampling land-use cover.
Simulating sampling schemes: random versus systematic
A further important question we can address with these data is the relative efficiency of different sampling strategies. Here we compare random, stratified random (with one observation per stratum) and systematic strategies.
To evaluate systematic sampling, a square grid was superimposed over the data such that the spacing of the grid was an integer multiple r of the side length of the sample sites. Odd values of r were used in the experiments since the central site within each grid square is sampled. For any value of r the origin of the grid may be in one of r2 positions so that an empirical sampling distribution for a systematic scheme may be obtained. The variance of the sample means derived in this way are written Vsys. Here values of r=5, 7, 9 and 11 were used, yielding median sample sizes of 100, 51, 31 and 21, respectively. Due to edge effects, the exact size of the sample varied between the r2 replications.
For stratified random sampling a grid of the same size as for systematic sampling was used, but in this case a site was chosen at random within each grid square. For comparability with systematic sampling the distribution of the sample mean was determined from r2 replications; the variance of this distribution is written as Vst1. The sampling distribution for random sampling was also determined empirically; its variance is denoted by Vran.
The values Vsys, Vst1 and Vran were calculated for the four land-use categories for grid sizes r=5, 7, 9 and 11. Table 8.5 shows the gain in efficiency of moving from random sampling to systematic sampling by recording the ratio Vran/Vsys for each
Problems of sampling the landscape 109
Table 8.5. Comparison (V
ran/V
sys) of the variances of the sample mean from systematic sampling (V
sys) and random sampling (V
ran) for different sampling inten-sities.
r
Land cover 5 7 9 11
Grassland 1.82 1.93 0.98 1.61
Arable 1.39 1.67 2.00 1.87
Urban 3.85 4.21 1.51 1.52
Woodland 2.24 1.11 1.47 0.98
value of r. These experimental results show that, on average, very large gains in efficiency result from moving to systematic sampling from random sampling, with typical values between 67% and 37%. These values are similar to those found in previous studies, which report comparable statistics (Matern, 1960; Payandeh, 1970). The pattern of average values also suggests that larger gains in efficiency tend to occur at greater sampling intensities (smaller values of r).
The individual entries in Table 8.5 show great variability both between land cover types and for the same cover type at different sampling intensities. For example, urban land has the largest gains in efficiency at r=5 and 7, but only just above average gains at r=9 and 11. Two possible explanations for these highly variable results may be forwarded. One is that they are due to small sample effects within the simulation exercises. The second is that they reflect actual differences in the spatial configuration of the land-use types.
To investigate the latter hypothesis, spatial autocorrelation functions were calculated for the four land-use types to see if this additional information assisted in the interpretation of Table 8.5. The lag interval used is equivalent to the length of the sample sites (0.71km), and in each case the autocorrelation function was calculated for the east-west and north-south directions, to a maximum of 15 lags. The results are shown in Figure 8.1.
For improved grassland and arable land the autocorrelation functions are close to a geometric decline, the model assumed in much theoretical work. For urban
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Figure 8.1. Autocorrelation functions (ACF) of land-use categories within Avon. (----) North-south direction; (—
—) east-west direction. Lag interval 0.71km.
land the negative correlations at longer lags suggest a very strong degree of association across the study area. Reference to the original data shows this to be the case, since most urban land is concentrated in the large city areas of Bristol and Bath. Finally, the pattern for woodland suggests the possibility of some periodic variation at around lag 10 in the north-south direction.
In some cases these results assist in the examination of Table 8.5: the high degree of correlation for urban land is matched by very large gains in efficiency for that category;
the poor performance of systematic sampling of woodland at r=11 may be due to periodic variation. In other cases the correspondence is less helpful. For example, the poor performance of systematic sampling at r=9 for improved grassland is unexplained. In summary, the autocorrelation functions do assist a fuller interpretation of the results of Table 8.5, but certain aspects remain unclear. There is some suggestion that unusual autocorrelation functions may result from particular spatial configurations, which in turn affect the efficiency of systematic sampling. On a wider note, these empirical results indicate that autocorrelation functions of natural populations may take quite complex
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forms. To assume a geometric form of decline may be appropriate in some cases, but not all.
To complete the analysis of alternate sampling schemes Tables 8.6 and 8.7 present the ratios Vran/Vst1 and Vst1/Vsys, to assess the relative performance of stratified random sampling compared to random and systematic sampling. The gain in efficiency of moving from random sampling to stratified random sampling (Vran/Vst1) shows a similar pattern to the data in Table 8.5: there are, on average, considerable gains in efficiency, although these are less than for systematic sampling. The gain in
Table 8.6. Comparison (V
ran/V
st1) of the variances of the sample mean from random sampling (V
ran) and stratified sampling (V
st1) for different sampling intensities.
Table 8.7. Comparison (V
st1/V
sys) of the variances of the sample mean from stratified sampling (V
st1) and systematic sampling (V
sys) for different sampling intensities.
efficiency of moving from stratified random sampling to systematic (Vst1/Vsys) is generally small, but the average ratio of between 17% and 3% suggests the gain is worth having.
The results reported here illustrate both the advantages of a systematic approach and that certain important unresolved issues exist, many of which relate to the complexity of the two-dimensional case. In accord with previous theoretical and empirical studies (Cochran, 1946; Finney, 1948; Osbourne, 1942; Williams, 1956), systematic sampling consistently outperforms alternative strategies for the natural population studied here. But the gain in efficiency is highly variable, and appears to reflect the complex and varied autocorrelation functions of the data.
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Conclusions
Four broad conclusions arise from these simulation experiments. First, satellite imagery represents a valuable resource for experimental approaches to landscape simulation and for testing the efficiency of different sampling approaches.
Second, moving to using more smaller sites to measure land use should increase the accuracy of measurement compared to an equivalent area of land in larger sites.
Third, in general, systematic sampling provides the most efficient sampling approach.
Given the operational advantage of the design, its use in sample-based land-use surveys has much to recommend it. (However, there may be problems with this approach if periodic variation is present in the landscape in which case a stratified random approach is preferable.)
Fourth, the gains in efficiency are highly variable, being a function of both land-use type and sampling intensity. In part this appears to be due to the complex nature of spatial dependence in the natural populations studied here. Whether such patterns are widespread can only be ascertained by further empirical studies.
Acknowledgement
Part of this work was carried out under contract PECD 7/2/47 placed by the Department of the Environment. The views expressed here are not necessarily those of the Department of the Environment or any other Government department.
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