A comparison of two procedures to estimate three basic monitoring ...

Environ Monit Assess (2014) 186:4709–4718 DOI 10.1007/s10661-014-3732-7

A comparison of two procedures to estimate three basic monitoring landscape metrics for monitoring Habib Ramezani & Anton Grafström

Received: 20 August 2013 / Accepted: 18 March 2014 / Published online: 11 April 2014 # Springer International Publishing Switzerland 2014

Abstract An interesting alternative to wall-to-wall mapping approaches for the estimation of landscape metrics is to use sampling. Sample-based approaches are costefficient, and measurement errors can be reduced considerably. The previous efforts of sample-based estimation of landscape metrics have mainly been focused on data collection methods, but in this study, we consider two estimation procedures. First, landscape metrics of interest are calculated separately for each sampled image and then the image values are averaged to obtain an estimate of the entire landscape (separated procedure, SP). Second, metric components are calculated in all sampled images and then the aggregated values are inserted into the landscape metric formulas (aggregated procedure, AP). The national land cover map (NLCM) of Sweden, reflecting the status of land cover in the year 2000, was used to provide population information to investigate the statistical performance of the estimation procedures. For this purpose, sampling simulation with a large number of replications was used. For all three landscape metrics, the second procedure (AP) produced a lower relative RMSE and bias than the first one (SP). A smaller sample unit size (50 ha) produced larger bias than a larger one (100 ha), whereas a smaller sample unit size produced a lower variance than a larger sample unit. The efficiency of a metric estimator is highly related to the degree of landscape fragmentation and the selected H. Ramezani (*) : A. Grafström Department of Forest Resource Management, Swedish University of Agricultural Sciences (SLU), SE-901 83, Umeå, Sweden e-mail: [email protected]

procedure. Incorporating information from all of the sampled images into a single one (aggregated procedure, AP) is one way to improve the statistical performance of estimators. Keywords Sampling methods . Landscape pattern analysis . Bias . Root mean square error

Introduction Landscape structure is of primary interest for landscape ecologists since a fundamental assumption is that the structure of landscapes significantly affects many ecological processes, for instance, the dynamics of population and biodiversity (Risser et al. 1984, van Dorp and Opdam 1987, Turner 2005). Haines-Young (2005) stated that landscape structure can be treated as indicators of ecological changes. In order to better understand the pattern-process relations, landscape pattern should first be quantified. For this purpose, a large number of landscape metrics has been developed (McGarigal et al. 1995). These metrics are defined based on measurable attributes such as area, edge length, and the number of patches, where a patch is defined as a relatively homogenous area which differs from its surroundings (Forman 1995). Landscape metrics condense and quantify the complexity of landscape pattern to obtain information that is useful for biodiversity assessment at the landscape level (Benitez-Malvido and Martinez-Ramos 2003). With an appropriate set of metrics, the spatial pattern of

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landscapes can be objectively assessed. Landscape metrics also act as the quantitative linkage between landscape structure and ecological processes as well as species abundance (Krummel et al. 1987, Bunnell 1997). They provide information useful for analyzing fragmentation and connectivity of landscape units (Hargis et al. 1998). These metrics serve as tools in environmental monitoring programs (Hunsaker et al. 1994, NIJOS 2001, Ståhl et al. 2011). Quantification of landscape pattern through landscape metrics is commonly conducted using wall-towall maps from remotely sensed data (e.g., Riitters et al. 2004, Li et al. 2005). This method, however, is not without problems; in some cases, such maps may not be available. In wall-to-wall mapping approaches, all potential patches are delineated either manually or automatically. The approach is time consuming, costly (Corona et al. 2004), and is associated with polygon delineation errors (Carfagna and Gallego 1999). Satellite-based maps with high-resolution satellite images, for example, IKONOS, may be very expensive to produce (Hassett et al. 2011). An interesting approach for assessing landscape pattern through landscape metrics is to make use of sample data, without the use of wall-to-wall mapping (e.g., Hunsaker et al. 1994, Kleinn 2000, Stehman et al. 2003, Corona et al. 2004, Hassett et al. 2011, Kleinn et al. 2011, Ramezani and Holm 2012). The authors of previous studies found that sample-based approaches are very competitive alternatives to the traditional wall-towall approaches both in terms of cost and data quality. In a sample survey, data can be captured with low cost because less time is required for interpreting sampled images and classifying land use or land cover. In addition, measurement errors can be considerably reduced since interpretation is carefully conducted on only a small number of sampling units (e.g., plots; Cochran 1977). The smaller area covered by a sample (relative to wall-to-wall coverage) also requires less data storage, and a sampling approach is likely to be more practical (Stehman et al. 2003). Kleinn (2000) also demonstrated the possibility of deriving some landscape metrics from field-based forest inventories. This allows trend analysis based on existing historical data such as the national forest inventories (e.g., NFIs; Corona et al. 2011). Sample-based approaches are increasingly used in environmental monitoring programs at the national level, for instance, by the US EPA (Hunsaker et al. 1994); the Norwegian monitoring program for agricultural

Environ Monit Assess (2014) 186:4709–4718

landscapes in Norway (3Q) (Dramstad et al. 2002); the Spanish rural landscape monitoring system (SISPARES) (http://www.sispares.com/eng/); the Alberta Biodiversity and Monitoring Institute (http://www.abmi.ca) in Canada; and the national inventory of landscapes in Sweden, NILS (Ståhl et al. 2011). The previous efforts of sample-based estimation of landscape metrics have mainly focused on sampling design and data collection methods. Ramezani et al. (2010) and Ramezani and Holm (2011) used point and line intersect sampling on aerial photographs for gathering data and estimating landscape metrics. Estimation procedures are of equal importance; however, they have received less attention. Hence, it is of relevance to investigate estimation methods for landscape pattern analysis. In this study, we consider two estimation methods. First, landscape metrics of interest are calculated for each sampled image separately and then the image values are averaged to obtain an estimate of the entire landscape (separated procedure, SP). Second, metric components are calculated in all sampled images and then the aggregated information is inserted into the landscape metric formulas (aggregated procedure, AP). Landscape metrics for forest edge density (FED), Shannon’s diversity (SH), and contagion (C) are considered. These metrics are basic and often used in the quantitative description of landscape pattern. In addition, in landscape pattern analysis, a set of metrics is required since landscape pattern cannot comprehensively be described using a single metric. Population parameters and corresponding estimators of these metrics are presented in Section 2.2. In this study, the main objective is to compare two estimation procedures as described above, for the estimation of selected landscape metrics. In addition, the statistical performance (root mean square error (RMSE) and bias) of the procedures will be investigated for different combinations: six sample sizes, two different sample unit sizes, and two sampling designs. This will be done for two study sites having different degrees of landscape fragmentation.

Material and methods Study area The national land cover map (NLCM) of Sweden from 2000 (Hagner and Reese 2007) was used to provide


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population information to investigate the statistical performance of landscape metrics estimators. The NLCM was raster-based with a pixel size of 25 m. For this study, its classes were reclassified into six classes: urban, agriculture, forest, water, wetland, and other lands. Two study sites were selected with one from the north (study site 1) and one from the south of Sweden (study site 2; see Fig. 1) each having the same extent of 10,000 ha (10 km×10 km). Some spatial characteristics of the two study sites are summarized in Table 1. Site 1 is more fragmented in terms of the total number of patches, mean patch size, and edge density.

Table 1 Relative bias in site 2, by aggregated procedure (AP)

Sampling design

16, and 25 were taken for sample unit sizes of 50 ha, while sample sizes of 8, 32, and 50 were taken for sample unit sizes of 100 ha. We used two different sampling designs, where the first design was simple random sampling without replacement (SRSWOR), and implemented by using the routine SRSWOR in the R-package sampling (Tillé and Matei 2011). The second design was a two-dimensional systematic sampling design, which was implemented by combining a

The populations (study sites) were divided into nonoverlapping units, which were labeled from 1 to 100 (for the 1 km×1 km area) and 1 to 200 (for the 500 m× 1,000 m area). To investigate the statistical performance of the estimators of landscape metrics, a sampling simulation with a large number of replications (10,000) was conducted for different combinations of sampling designs (random and systematic), and sample sizes of 4,

Landscape metrics Systematic design Sample size 100 ha

Simple random design Sample size

4

16

25

4

16

25

FED

0.9

−2

−2

−0.9

−1.8

−1.7

SH

−8

−2

−0.5

−9

−2.5

−1.1

C

3

2

0.7

4

1.6

0.9

50 ha

8

32

50

8

32

50

FED

−2

−2

−3

−2

−2

−3

SH

−6

−1.7

−0.8

−7

−1.7

−1.5

C

3

1.4

1.1

4

4

1.4

Fig. 1 The study site 10 km×10 km (10,000 ha) in north and south of Sweden and national land cover map (NLCM) 2000 with six land cover types of study site 1

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systematic sample of rows with a systematic sample of columns. A unit was then selected if both its row and column were selected. The routine UP systematic in the R-package sampling was used to select the samples of rows and columns. Landscape metrics Landscape metrics are defined based on measurable attributes such as the number, area, and edge length of patches. To estimate landscape metrics, the patch attributes (components of metrics) should first be estimated. The ecological significance and estimators of the metrics of interest in this study, namely, forest edge density, Shannon’s diversity, and contagion, are described in the following sections. Forest edge density (FED) Forest edge refers to a border between forest and nonforest lands or between different categories of forest; in a fragmented landscape, forest edge length tends to be large (Lister et al. 2005). Forest edge density is a basic and robust metric and can be employed as a measure of landscape fragmentation (Saura and MartinezMillan 2001). It impacts biodiversity and is thus an important attribute when assessing the pattern of a forest landscape. The population parameter of FED is as follows: n X

Y FED ¼

i¼1 n X

ei ð1Þ ai

i¼1

where ei and ai are forest edge length and forest area of the ith patch, respectively, in the landscape. Using the SP, the estimator of FED for a sample of m images is defined as follows: 1X Yb FED;SP ¼ y m j¼1 j m

nj

ð2Þ

m X nj X

Yb FED;AP ¼

j¼1 m X

i¼1 nj X

j¼1

i¼1

eij ð3Þ aij

where eij and aij are forest edge length and forest area of ith patch, respectively, in sampled image j.

Shannon’s diversity index (SH) The Shannon’s diversity (SH) index refers to both the number of land cover types and their proportions in a landscape (Weaver and Shannon 1949). The index value ranges between 0 and 1. A high value shows that the land cover types present have roughly equal proportion, whereas a low value indicates that the landscape is dominated by one land cover type. The population parameter of SH is defined as follows: n X

Y SH ¼ −

pi lnðpi Þ

i¼1

lnðsÞ

ð4Þ

where pi is the area proportion of the ith land cover type in a landscape, and s is the total number of land cover types in the classification system (here six land cover types). Using SP, the mean of SH for a sample of m images is estimated as follows: 1X Yb SH;SP ¼ Y SH j m j¼1 m

ð5Þ

where Y SHj is the true value for image j, calculated using Eq. 4. Using AP, the component area proportion, pi, and thus Shannon’s diversity index can be calculated for all sampled images as one. In this case, the estimator of SH is as follows: n X

nj

b pi ln b pi

where y j ¼ ∑ ei = ∑ ai and nj is the number of forest

Yb SH;AP ¼ −

patches in sampled image j. Using the AP, the estimator of FED for a sample of m images is as follows:

where b pi is the proportion of land cover type i in all sampled images.

i¼1

i¼1

i¼1

lnðsÞ

ð6Þ


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Contagion (C) Contagion (C) is a measure of clumping of land cover types within a landscape (Li and Reynolds 1993). This metric has been used in many studies for measuring landscape fragmentation (Ricotta et al. 2003). The value of contagion ranges from 0 to 1. A low value shows a highly fragmented landscape, whereas a high value shows an aggregated landscape. The population parameter of C is as follows: s X s X

YC ¼ 1 þ

pkl lnðpkl Þ

k¼1 l¼1

2lnðsÞ

ð7Þ

where pkl(=Nkl/N) is the probability that two randomly chosen adjacent pixels belong to land cover types k and l; Nkl is the number of adjacencies (joins) between pixels of land cover types k and l, and N is the total number of joins within the landscape. Using SP, component pkl and thus the contagion metric are calculated for each sampled image separately and the estimated population parameter is an average from all sampled images (using Eq. 8). The mean of contagion for a sample of m images is estimated as follows: 1X Yb C;SP ¼ YCj m j¼1 m

ð8Þ

where Y C j is the true value for image j. Using AP, component, pkl, and thus the contagion metric can be calculated for all sampled images as one. In this case, the estimator of C is as follows: s X s X

Yb C;AP ¼ 1 þ

b pkl pkl ln b

k¼1 l¼1

2lnðsÞ

ð9Þ

RMSE ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi u X10;000 u t r¼1 Yb r −Y 10; 000

ð10Þ

and relative RMSE (RRMSE) was calculated as follows: RRMSE ¼

RMSE 100 Y

ð11Þ

where Yb r is the estimated attribute (here the selected landscape metrics) for rth replication, and Y is the attribute’s true value (reference value). Estimated bias is the difference between the average of the samples and the true value, i.e., Y −Y and relative bias was estimated as follows: RBias ¼

Y¯ −Y 100 Y

ð12Þ

where Y is the average of all the replications. Variance estimation Approximate variance estimators can be derived for the case of simple random sampling. For instance, Ramezani et al.(2010) and Van Nguyen et al. (2013) demonstrated that Shannon’s diversity and standard Horvitz–Thompson (HT) ratio variance (Särndal et al. 1992) can be used for ratio type landscape metrics such as forest edge density. In this study, however, we compare the estimators by performing a Monte Carlo simulation with a large number of replications. In the case of SP, the estimator can be expressed as sample means, which means that the variance can be estimated by the following: b ðY Þ ¼ 1 1− n ⋅s2 V n N where s2 ¼

1 n−1

n

∑

Y i −Y

ð13Þ 2

i¼1

Efficiency evaluation

where Y is the sample mean and Yi is the estimate of image i. In the case of AP, the jackknife estimator is considered to be a straightforward technique, where one image at a time is systematically deleted from the sample (Thompson 2002). The variance estimator is defined as follows:

Root mean square error (RMSE) and bias were used to evaluate the statistical efficiency of the estimators. The RMSE was calculated as follows:

2 Xn b Yb ¼ n−1 V Yb i −Y¯jack n i ¼ 1

where b pkl is the proportion of two randomly chosen adjacent pixels that belong to land cover types k and l in all sampled images, but excluding pixels at image borders.

ð14Þ

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Environ Monit Assess (2014) 186:4709–4718 n

1 n

where Yjack ¼

∑ Yb i i¼1

where Yb is the estimator of metrics and Yb i is the estimator when leaving image i out.

Results In this study, the statistical performance of the estimator for selected landscape metrics was investigated through sampling simulation for different combinations, including two estimation procedures (SP and AP), six sample sizes, two sample unit sizes, and two sampling designs. Some general results are presented below whereas details are provided in the Appendix. All three landscape metric estimators applied in this study were biased in all combinations considered. Shannon’s diversity was underestimated, whereas forest edge density and contagion were overestimated. A comparison was made between two estimation procedures (AP and SP) for selected landscape metrics. The AP yielded smaller RRMSE and relative bias than the SP. This was true for all combinations considered. Figure 2 shows the magnitude of RRMSE and relative bias of the

SP

Site 1

Site 1

AP

40

40

30 Bias %

RRMSE

two estimation procedures for two sample unit sizes in two study sites. Using AP, less information is missed hence the estimates are more precise than SP. A comparison was also made between two sample unit sizes (50 and 100 ha). The bias of the estimator of selected metrics tended to decrease when a larger sample unit size was used. In contrast, however, the RRMSE of the estimators tended to increase with the larger sample unit size (100 ha). The reason is that for equal total area sampled (i.e., a given sampling intensity), the smaller sample unit produces a large sample size which led to lower variance than from the larger sample unit. Figure 3 shows a comparison between the two different sample unit sizes. In most cases, there was not a large difference, in terms of RRMSE and relative bias, between systematic and simple random sampling designs. This was true for all three metrics (see Appendix). The relationship between RRMSE of the estimators and sample size is shown in Fig. 4. RRMSE tended to decrease when sample size increased, and this held for both systematic and simple random sampling designs as well as for both estimation procedures (i.e., separated procedure (SP) and aggregated procedure (AP)).

20 10

20 0 FED

FED

SH

C

C

-40 Landscape metrics

Landscape metrics Site 2 40

40

30

20

Bias %

RRMSE

SH

-20

0

20 10

0 -20

0 FED

SH

Landscape metrics

C

Site 2

FED

SH

C

-40 Landscape emtrics

Fig 2 RRMSE (left) and relative bias (right) of three landscape metrics for estimation procedures (separated procedure, SP, and aggregated procedure, AP), for sample unit size 50 ha, sample size 8, systematic sampling design, and for two study sites


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50 ha

Site 1

Site 1

100 ha

40

Bias %

RRMSE

30 20 10 0 FED SH Landscape metrics

C

FED

SH

C

Landscape metrics

Site 2

40

Site 2

Bias %

30 RRMSE

40 30 20 10 0 -10 -20 -30 -40

20 10 0 FED SH Landscape metrics

C

40 30 20 10 0 -10 -20 -30 -40

FED

SH

C

Landscape metrics

Fig 3 RRMSE (left) and relative bias (right) of selected landscape metrics for different sample unit sizes (50 and 100 ha), for systematic sampling design and two study sites

Discussion and conclusion

35 30 25 20 15 10 5 0

RRMSE

Fig 4 The relationship between RRMSE and sample size for three landscape metrics, for simple random sampling design, sample unit size 100 ha (left) and 50 ha (right) and for aggregated procedure (AP)

RRMSE

Sample-based estimation of landscape metrics is a fairly new field, and as noted earlier, the main objective of this study was to compare two estimation procedures of three basic landscape metrics derived from sampled images. Many currently used landscape metrics have bias estimators due to various reasons. Such reasons might be due to the metric being a ratio type and originally defined for mapped data of an entire landscape rather than sample data (Ramezani and Holm 2013a).

In the case of the forest edge density (FED) metric, bias can be explained by definition of this metric, which is a ratio between two metric components (patch attributes: area and edge length). Ratio-type estimators would generally produce bias even if their components are estimated unbiasedly. Kleinn and Traub (2003) state that the components of a given metric should be unbiasedly estimated if unbiasedness is required, but Ramezani (2010) demonstrated that unbiased estimation of a metric component is necessary but not a sufficient condition. In some cases, the components might be

0

10 20 Sample size

30

FED

35 30 25 20 15 10 5 0

SH C

0

20

40

Sample size

60

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estimated without bias, but the estimator will still be biased, with FED being a typical example. The Shannon’s diversity (SH) metric is very sensitive to the number of land cover types and the relative proportion of those land cover types in the landscape (Brower and Zar 1977). In a sample survey, one or more land cover types, particularly rare (in size) land cover types, might be missed in sampling units. As a result, this metric can be estimated with bias (underestimated). Ramezani (2010) states that bias is due to non-linear definition of SH despite that its component (i.e., area proportion) is estimated without bias. Our results are in line with those from Hunsaker et al. (1994) and Hassett et al. (2011) where plot sampling methods were applied on land cover maps made from satellite data. However, the magnitude of bias is usually very small when using a large sample size. As the previous estimators (i.e., FED and SH), the estimator of contagion is also biased. The reason is the same as the SH estimator, in that it is based on a nonlinear definition. In this study, this metric has been overestimated which is consistent with Hassett et al. (2011), whereas the metric was underestimated by Hunsaker et al. (1994) and Ramezani and Holm (2013a). This inconsistency can be explained by different definitions and estimators of the contagion metric, so that direct comparison of the results is difficult. The results show that the smaller sample unit (here 50 ha) produces large bias whereas the larger unit (here 100 ha) produces small bias, which is consistent with O’Neill’ et al. (1996). An explanation to this is given by Hassett et al. (2011) who found that a higher total length of sampled image borders are produced by a smaller sample unit. As a result, more patches are likely truncated by sampled image borders. From a statistical viewpoint, both bias and variance are important criteria, hence a combination of both in terms of RMSE pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Bias2 þ Variance is preferable. On the basis of RMSE as a criterion, in this study, the small sample unit of 50 ha, regardless of the estimation procedure applied, appears to be more efficient than the larger sample unit (100 ha). In this study, cost was not considered. However, with a given sampling intensity, a smaller sample unit, leading to a large sample size, would be more effective statistically but more costly than using a larger plot size. In a sample survey with a small sample unit size, more time (cost) is needed to observe and analyze a large number of sample images.


The results show that the estimators have similar behavior, whereas the two sites are different in terms of RMSE and bias of the estimators. This difference can be explained by the degree of landscape fragmentation; as Table 1 shows, site 1 seems to be more fragmented. The efficiency of estimators is highly related to landscape fragmentation so that in a highly fragmented landscape a large sample size is required in order to achieve reasonable precision. Biasedness of landscape metric estimators is a concern in a sample-based landscape monitoring program (Ramezani et al. 2013). Incorporating information of all sampled images as one (i.e., aggregated procedure, AP) can be considered as one way to improve the statistical performance, in terms of bias, of the estimators. However, note that both procedures considered might fail to detect landscape change over time. Hence, further studies are needed to explore which of the two procedures considered (SP and AP) are adequately sensitive to landscape changes. Acknowledgment The authors are grateful to Dr. Heather Reese for improving language and the constructive comments. We are also grateful for the constructive comments by the anonymous reviewer of the paper.

Appendix Statistical properties (RRMSE and relative bias) of the estimators of three landscape metrics, the two study sites, and for two sampling designs (systematic and simple random sampling)

Table 2 A summary of some characteristics of the two study sites Study site No. patch Proportion Mean patch Edge density (%) size (ha) (mh−1) Forest land 1

58

60.3

103.9

59.2

2

23

76.7

333.8

54

1

2

0.5

2.8

0.2

2

2

0.1

9.4

0.5

1

13

0.7

6

2

2

6

0.3

5

1

Urban area

Agriculture


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Table 3 RRMSE in site 1, by aggregated procedure (AP) Landscape metrics Systematic design Sample size 100 ha

References


4

16

25

4

16

25

FED

26

10

9

30

13

10

SH

19

7

8

17

7

6

C

14

5

5

12

5

4

50 ha

50

8

32

50

8

32

FED

25

9

9

23

11

8

SH

12

6

4

15

6

4

C

9

4

3

10

4

3

Table 4 Relative bias in site 1, by aggregated procedure Landscape metrics Systematic design Sample size 100 ha


4

16

25

4

16

25

FED

0.2

−1.8

−1.6

3.5

−1.5

−2.2

SH

−8

−1.1

−1.3

−8

−1.8

−1.3

C

6

1.2

1.2

5.6

1.7

1.3

50 ha

8

32

50

8

32

50

FED

−6

−3

−3

−1

−3

−3

SH

−4

−9

−0.3

−6

−1.5

−0.5

C

3.4

1.3

0.8

4.8

1.6

0.9

Table 5 RRMSE in site 2, by aggregated procedure (AP) Landscape metrics Systematic design Sample size 100 ha


4

16

25

4

16

25

FED

34

13

10

31

14

10

SH

24

12

10

23

11

5

C

11

5

4

11

5

4

50 ha

8

32

50

8

32

50

FED

23

12

4

27

12

10

SH

17

10

7

21

10

9

C

8

4

3

10

5

4

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