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Accuracy assessment of probabilistic visibilities Kris Nackaerts , Gerard Govers & Jos Van Orshoven Published online: 06 Aug 2010.
To cite this article: Kris Nackaerts , Gerard Govers & Jos Van Orshoven (1999) Accuracy assessment of probabilistic visibilities, International Journal of Geographical Information Science, 13:7, 709-721, DOI: 10.1080/136588199241076 To link to this article: http://dx.doi.org/10.1080/136588199241076
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int. j. geographical information science, 1999, vol. 13, no. 7, 709± 721
Research Article Accuracy assessment of probabilistic visibilities
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KRIS NACKAERTS, GERARD GOVERS Laboratory for Experimental Geomorphology, Physical and Regional Geography, Katholieke Universiteit Leuven, Redingenstraat 16, B-3000 Leuven, Belgium Redingenstraat 16, B-3000 Leuven, Belgium e-mail:
[email protected]
and JOS VAN ORSHOVEN Interdepartmental Research & Consultancy Group `Ground for GIS’, Katholieke Universiteit Leuven, Vital Decosterstraat 102, B-3000 Leuven, Belgium (Received 15 February 1998; accepted 20 January 1999 ) Abstract.
Visual impact studies often make use of Boolean viewshed maps created by standard functions implemented in most raster GIS software. The DEM is used in a deterministic way and possible inaccuracies in the DEM are neglected. Monte Carlo simulations of the errors in the digital elevation data (DEM error) can be used to test the e ect of the DEM error on the calculated viewshed. Both an explorational and a quantitative method based on binomial statistics are proposed to analyse the impact of the number of simulations on the accuracy of the probabilistic visibility of a cell. The probabilistic information is used for the creation of a Boolean viewshed map with a known user’s and/or producer’s accuracy. The importance of this methodology is illustrated.
1.
Introduction
A wide variety of applications using visibility information have been described in literature. These include civil engineering, orientation and navigation (Nagy 1994), visual impact analysis (Kent 1986, Hadrian et al. 1988), siting optimisation (De Floriani et al. 1994) and other intervisibility studies (Wheatly 1995). The result of a classical viewshed operation within a raster GIS is a Boolean visibility map: a cell is either classi® ed as visible or invisible. In this operation, a DEM is used in a deterministic way. The existence of errors in the digital elevation data (DEM error) and their in¯ uence on the spatial characteristics of the viewshed are hereby neglected. However, di erent types of inaccuracies are generated during the DEM construction from isolines or topographic maps (Burrough 1992, Bolstad and Stower 1994, Eklundh and MaÊrtensson 1995). Some examples of these inaccuracies are: Internationa l Journal of Geographica l Information Science
ISSN 1365-8816 print/ISSN 1362-3087 online Ñ 1999 Taylor & Francis Ltd http://www.tandf.co.uk/JNLS/gis.htm http://www.taylorandfrancis.com/JNLS/gis.htm
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inaccuracies in the registration of aerial photographs during the construction of the topographic maps, digitizer table accuracy, operator induced inaccuracies during the digitizing process, generalisation errors during the digitizing process, image distortions due to a scanning process, interpolation errors, data structure related errors, e.g. resolution (raster) and topology, vector-to-raster conversion errors, errors caused by map projection conversions.
Contributions of these errors are inherently present in DEM’s and thus the accuracy suggested by a Boolean visibility map may be deceptive and not justi® ed (Sorensen and Lanter 1993, Fisher 1991). Therefore, alternative methods for viewshed calculation have been proposed. Donnay (1992), Sorensen and Lanter (1993), and Wang et al. (1996) focused only on the implementation of data-structure induced errors in visibility assessment. Fisher (1991, 1992, 1993) presented a more general method, based on Monte Carlo simulation to test the e ect of the DEM error on the calculated viewshed area. However, the approach of Fisher does not consider the e ect of the DEM error on the visibility of individual cells (Fisher 1991). This study focuses primarily on the problem of assessing the visibility of individual cells. First, we present two new methods to evaluate the in¯ uence of the number of simulations on the probabilistic visibility of a cell. The two methods (one exploratory and one quantitative) allow a well considered estimate of the number of simulations necessary for an accurate calculation of the probabilistic visibilities. Secondly, we propose a method for the creation of Boolean documents (with known user’s and producer’s accuracy) from this probabilistic information. These Boolean documents are less complex to use compared with the probabilistic information. They easily allow the rational implementation of visual impact assessment in a decision process. 2.
Materials
2.1. T he GIS Idrisi for Windows v.1.01 (Eastman 1995) was used for this study. The viewsheds were calculated with the Viewshed program developed by W. Weisgerber (personal communication, 1990) which is more than ten times faster than the Idrisi Viewshed function. The simulation process was executed by means of an Idrisi Macro Language script ® le (Eastman 1995). 2.2. T he DEM The study area is situated around the archaeological site of Sagalassos, approximately 100km to the north of Antalya, in SW-Turkey (Paulissen et al. 1993). The mountainous topography is characterised by two valleys at an average elevation of 900m and an east± west oriented mountain ridge at a height of approximately 2000 m (® gure 1). The available elevation data was derived from topographic maps (scale 1:25 000), more speci® cally, contour lines with a contour interval of 10 m. Due to operational constraints only contour lines with a contour interval of 50 m were manually digitized. Elevation points were added to the contour elevation data to emphasise mountaintops and micro topographic structures in gentle sloping areas. A DEM with a
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Figure 1. Shaded DEM of the study area.
resolution of 50 m by 50 m was linearly interpolated from this dataset using the Intercon procedure (Eastman 1995). This resolution results in acceptable viewshed calculation times. A low-pass ® lter was applied on the DEM resulting in a signi® cant (a= 0.05) reduction of the standard deviation of the DEM error (see below). 2.3. T he DEM error 150 reference points (corresponding to the nodes of a lattice covering the study area) were used to estimate the mean error of the interpolated elevation data in the DEM. For this, the elevations in these points were manually measured on the topographic maps (with a contour interval of 10 m) and compared with the interpolated values. These results show that the DEM error is normally distributed with a mean error of 0 m and a standard deviation of 10 m. Before an error data set can be simulated, the spatial distribution of the error has to be known. A visual inspection of the errors did not reveal any signi® cant patterns. Furthermore, no signi® cant correlations were found when regressing the DEM error against elevation and slope. It was therefore concluded that the DEMerror was of the same magnitude over the whole study area. It was assumed that there was no spatial autocorrelation. However, the proposed methodologies are independent of this assumption. 2.4. V iewpoints The viewpoints used in this study were observation towers located at strategic points to guard the access to the ancient city of Sagalassos. All the viewpoints used in this study were located and described by an archaeologist with a handheld Trimble ENSIGN GPS. Daily measurements on a number of ® xed points indicated an average accuracy of approximately 100m. The co-ordinates of the viewpoints were
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entered in a spreadsheet, geocoded and overlaid on the DEM. Based on ® eld observations, the positional accuracy was maximised for all viewpoints by manually repositioning them to the highest cell in the direct neighbourhood of the geocoded GPS points. Literature and observations by archaeologists led to an estimation of the height of the viewpoints above ground level of 10 m (Loots et al. forthcoming). The relative location and spacing of the observation points as well as the fact that the towers were used to guard the direct access to the city indicate that the main observation area was limited to a fewkm around each tower. Therefore, viewshed calculations were restricted to a bu er area with a radius of 5km around each viewpoint. Downloaded by [KU Leuven University Library] at 08:20 22 April 2015
3.
Methods
3.1. Probabilistic visibility The information describing the DEM error (see above) is used to calculate an error image ([ERROR] i ) (Fisher 1991). Through Monte Carlo simulation (Heuvelink 1998), random values are generated according to the distribution of the DEM error and assigned to each cell of the image. This process results in an error image with the same statistical and spatial characteristics as the measured DEM error: normal distribution, mean error ( = 0m), standard deviation ( = 10 m), no spatial autocorrelation and equally distributed values. When this image is added to the original DEM ([DEM*]), a sample is simulated ([DEM] i ) from a population describing the theoretical DEM. [DEM] i = [DEM*] + [ERROR] i
(1)
For each simulated [DEM] i , a Boolean viewshed is calculated (BV[DEM] i ), with `0’ indicating an invisible cell and `1’ a visible cell. The probable viewshed (PV[DEM*] n ) is now de® ned as described by (Fisher 1992): PV[DEM*] n = sum(BV[DEM] i )/n
(2)
The value of each cell in the probable viewshed (PV[DEM*] n ) expresses the probability a cell lies within the real viewshed, the probabilistic visibility. 3.2. Minimum number of simulations The number of simulations selected should result in a robust analysis. However, there may be practical constraints due to the limited availability of resources. To assess the e ect of di erent numbers of simulations, an exploratory method is used to visualise the evolution of the probabilistic visibilities during the simulation process. The procedure is as follows. Within the 5km bu er zone 100 sample points were randomly selected. After each simulation, the probabilistic visibility is measured at each sample point and plotted against the number of simulations. The graph visualises the evolution of the probabilistic visibility during the simulation process for di erent cells. A second, quantitative estimate of the probabilistic visibility can be made when the simulations are considered as Bernoulli experiments. In a Bernoulli experiment, the outcome of each trial can be 1 (visible) or 0 (invisible). While the Boolean visibility assessment for a cell in simulation (i ) is independent of the visibility in simulation (iÕ 1 ), both conditions for Bernoulli experiments are satis® ed: E
the trials are independent
Accuracy assessments of probalistic visibilities E
713
the probability of success in each cell is constant over di erent independent trials
The probabilistic visibility after an in® nite number of simulations in a cell corresponds to the probability of `success’ in that cell ( p) and is characterised by a binomial distribution. The sample mean and the standard deviation of the sample mean can be calculated (Sachs 1982): n
±
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Y=
±
Y n
s{ Y } =
= i= 0
Yi (3 )
n
S
±
±
Y (1Õ Y ) n
(4)
with n the total number of simulations (i.e. trials) and Y the total number of times ± a cell is classi® ed as visible during the simulation process. Y is an estimator for p, the population mean. ± The standard deviation of the sample is maximal at Y = 0.5 and minimal (i.e. 0) ± ± at Y = 0 or Y = 1 for a given number of simulations. An increase of the number of simulations will result in a decrease in the standard deviation of the sample. Although the standard deviation is a good indicator of the accuracy of the retrieved probabilistic visibilities, a more precise con® dence interval can be calculated for each probabilistic visibility using the binomial distribution function. The probability that in n simulations a cell is exactly x times classi® ed as visible (or n ± x times as invisible) is then given by the relation (Sachs 1982):
AB n
P(X = x | p, n)= Pn, p (x)=
x
p x q nÕ
x
(5)
with q = 1Õ p. The probability that in n simulations a cell is x or less times classi® ed as visible is now given by: P (X <
x | p, n)=
x
=
i 0
AB n i
p i q nÕ
i
(6)
The lower limit ( p*1 ) of the con® dence interval at a signi® cance level of 0.05 is now de® ned as (Conover 1971): P (Y >
a
y | pÕ
p *1 , n)=
yÕ
1 | pÕ
p *1 , n)= 1Õ
P (Y <
y | pÕ
p *2 , n)=
(7 )
2
or P (Y<
a
2
(8 )
The upper limit ( p*2 ) is de® ned as a
2
(9 )
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If n is greater than 20, a normal approximation may be used (Conover 1971): L=
U=
Y n Y n
Õ
z(1
+ z(1
Õ
Õ
a /2)
Ó Y(n Õ
Y)/n 3
(10)
a /2)
Ó Y (n Õ
Y)/n 3
(11)
where
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L = the lower limit of the confidence interval
U = the upper limit of the confidence interval z1 Õ
a /2
is the quantile of a normally distributed random variable.
These con® dence intervals re¯ ect the uncertainty on the probabilistic visibility estimates. More simulations reduce the uncertainty in the probable viewshed. 3.3. Construction of the Boolean viewshed If it is necessary to delineate the area of visible land, probabilistic information is more di cult to use than a Boolean image. Therefore, a new methodology is presented to create a Boolean viewshed from a probable viewshed. The proposed method allows the calculation of Boolean viewsheds with known user’s and producer’s accuracy. The user’s accuracy is a measure of how well the calculated Boolean map represents what is really visible and invisible on the ground. The producer’s accuracy is a measure of how well the visibility of an area on the ground can be mapped by a Boolean viewshed map. All probabilistic visibilities are classi® ed into c visibility classes with c = n+ 1 (n = number of simulations). When N i cells have a probabilistic visibility of p i , only N*i p i of these cells are really visible. (1Õ p i )*N i cells with a probabilistic visibility of p i are actually invisible. When pt is used as a threshold value for the generation of a Boolean viewshed, all the cells with a probabilistic visibility greater or equal to pt (or visibility class t ) are classi® ed as visible, the others as invisible. For a given threshold value pt , the contingency table can be calculated (see table 1). The producer’s and user’s accuracy can then be calculated for each threshold value (Stehman 1997):
Table 1. Contingency table of a Boolean viewshed derived from a probable viewshed. Cells in calculated Boolean viewshed:
Classi® ed as visible
Cells in unknown reference viewshed:
Visible Invisible
t
=
i 1
t
=
i 1
Classi® ed as invisible
(1 Õ
p iN i p i )N i
c
i t 1 =+ c
=+
i t 1
(1 Õ
p iN i p i )N i
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User’s accuracy for visible cells = t
i 1 =t
pi N i (12)
=
i 1
Ni
User’s accuracy for invisible cells = c
(1Õ
=+
i t 1
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p i )N i (13 )
c
Ni
=+
i t 1
Producer’s accuracy for visible cells = t
i 1 =c
=
i 1
pi N i (14) pi N i
Producer’s accuracy for invisible cells = c
i t 1 = c+
=
i 1
(1Õ
(1 Õ
p i )N i (15 ) p i )N i
The user’s accuracy is a measure of how well the calculated Boolean map represents what is really visible and invisible on the ground. The producer’s accuracy is a measure of how well the visibility of an area on the ground can be mapped by a Boolean viewshed map. 4.
Results and discussion
The simulation process and the calculation of Boolean viewsheds were executed by a computer program written in the macro language of Idrisi for Windows (Eastman 1995). Figure 2 shows a probable viewshed out of one viewpoint, created with 50 simulations. 4.1. Exploratory statistics Figure 3 was generated by means of 100 randomly spread sample points. The ® gure clearly demonstrates that the probabilistic visibility of a cell is susceptible to high ¯ uctuations during the ® rst simulations. A decision upon the number of simulations necessary for the calculation of a probable viewshed based on ® gure 3 is subjective and open for discussion. Major ¯ uctuations in cell-level probabilistic visibilities reduce rapidly over approximately 30 simulations. The values stabilise between 30 and 60 simulations although variability (in order of 0.1 units) continues to exist. The probabilistic visibilities during the simulation process show a comparable pattern for cells with di erent `® nal’ probabilistic visibilities. The same graphs were generated for viewpoints located on mountain tops, middle of a slope and valley bottoms, all showing a comparable pattern.
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Figure 2. A sample probable viewshed, calculated using 50 simulations. The viewpoint is located at the centre.
Figure 3. Evolution of probabilistic visibilities during the simulations process. Each line represents the evolution of the probabilistic visibility in one cell over di erent simulations. It can be seen that this value stabilises approximately between 30 to 60 simulations. 60 simulations result in a stable probable viewshed.
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4.2. Binomial statistics An example illustrates the interpretation of the con® dence intervals. Figure 4 shows the con® dence intervals for di erent numbers of simulations for a cell with a probabilistic visibility of 0.5, the value for which the con® dence interval is the largest, as can be derived from equations (10) and (11). Figure 4 shows that for a given cell a probabilistic visibility of 0.5 is calculated with 20 simulations (the probability of being visible is as high as the probability of being invisible), the theoretic real (unknown) value for that cell lies between 0.27 (probably invisible) and 0.73 (probably visible). This implies that, when the probabilistic visibility of the same cell is recalculated with 20 new simulations this will lead to a value between 0.27 and 0.73 for the same cell at a signi® cance level of 0.05. Figure 5 clearly illustrates this for the study area. The cumulative relative frequencies of the absolute di erences between two independently calculated probable viewsheds (from the same viewpoint) are calculated using 20, 50 or 100 independent simulations. When 20 simulations are used, only 20% of the cells within the viewshed (with a total number of 13435 cells) have equal probabilistic visibilities, 40% shows absolute di erences higher than 0.1. For 95% of the cells, the mean di erence varies between approximately 0 and 0.2. These results correspond to the calculated con® dence interval (see ® gure 4). Due to the fact that we take all the probabilistic visibilities into account, the experimentally derived con® dence interval is slightly smaller (Ô 0.2; a = 0.05) than the con® dence interval calculated using equations (10) and (11) for a probabilistic visibility of 0.5 (Ô 0.22; a= 0.05). The use of 50 (14 441 cells within the viewshed) or 100 (15026 cells within the viewshed) simulations results in a reduction of these inaccuracies, corresponding to the calculated con® dence intervals.
Figure 4. Con® dence intervals for a probabilistic visibility of 0.5 for a given number of simulations.
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Figure 5. Cumulative relative frequencies of the absolute di erences between two independently calculated probable viewsheds (out of the same viewpoint), calculated using 20, 50 or 100 independent simulations.
4.3. Boolean viewshed When a Boolean viewshed is derived from a probable viewshed using the procedure described above, the users and producers accuracies can be calculated for di erent threshold values. Figure 6 shows the users and producers accuracies for a given threshold value,
Figure 6. Example of user’s and producer’s accuracy for a Boolean viewshed created using a speci® ed threshold value.
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calculated for a viewshed in the study area. The choice of the threshold value depends now on the application for which the viewshed is used. Figure 7(a) corresponds to the Boolean viewshed calculated using the standard algorithm (see above) implemented in a GIS software package. It is clear that the use of this Boolean viewshed map may cause important over- and under-estimations of unknown extent. If the viewshed is used to site communication systems, the probability a cell classi® ed as visible is actually invisible should be minimised. When, for example, an error of 5% is accepted, the problem can be solved by using the threshold value for which the user’s accuracy for visible cells equals 0.95. In our example (see ® gure 7
Figure 7. Sample Boolean viewsheds, black cells are visible from the viewpoint located at the centre (a) represents the Boolean viewshed calculated using the standard viewshed operation without the simulation of the DEM error; viewsheds (b) and (c) were calculated out of a probable viewshed and have a known user’s accuracy (e.g. 95% for (b)) and producer’s accuracy (e.g. 90% for (c)).
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and ® gure 7(b)), all cells with a probabilistic visibility greater than or equal to 0.9 should be classi® ed as visible, the others as invisible. If, on the other hand, the potential negative visual impact of an element in a landscape is of importance, the probability of a cell classi® ed as invisible, but actually visible should be maximised (a high producer’s accuracy is required). If for example an error of 10% is acceptable, a threshold value of 0.48 should be used. With this information we know that the area classi® ed as visible (in the Boolean viewshed map, see ® gure 7(c)) fails to identify 10% of the total area with a negative visual impact. 4.4. Spatial-autocorrelation The assumption of no spatial-autocorrelation has no e ect on the methods described. The simulation process can still be described as a Bernoulli experiment: the result of one simulation for a given cell is independent of the result of another simulation for the same cell and the probability of success in each cell is constant over di erent independent trials. When spatial-autocorrelation characteristics of the DEM error would be implemented in the simulation model, this would only a ect the spatial patterns of the probabilistic visibility values reducing the pepper and salt e ect in the Boolean viewsheds when no spatial autocorrelation is assumed. 5.
Conclusions
The exploratory approach to the problem of the estimation of the minimum number of simulations provides insights into the stability of the probabilistic visibilities in a probable viewshed map and produces results that are useful to solve the problem. The determination of the minimum number of simulations necessary to estimate accurate probabilistic visibilities is open to interpretation of the operator. The number of 19 simulations proposed by Fisher (1991), necessary for an accurate estimation of the viewshed area seems to be too low if accurate cell-level probabilistic visibilities are of interest. The method has one main drawback: that the probable viewsheds need to be calculated before any estimation is possible. This is solved by the comparison of the simulation process with Bernoulli trials. The use of the binomial distribution function allows a quanti® cation and explanation of the observed pattern of the probabilistic visibilities in ® gure 3. This demonstrates that the range of con® dence intervals is independent of the DEM error, the topography and the location of the viewpoints. Before a probable viewshed is calculated, an optimal con® dence interval can now be selected. This will determine the number of simulations necessary. Each probable viewshed can now be accompanied with information about the accuracy of the calculated probabilistic visibilities with a more justi® ed and rational use of this probabilistic information as a result. If the visibility of one cell in the DEM is questioned, the probability this cell is visible can be read from the viewshed map, together with the con® dence intervals for this value. The proposed procedure allows the combination of the advantages of the ease of interpretation of a Boolean viewshed map and the information on accuracy from the probabilistic approach of the viewshed problem. The proposed procedure is independent of any spatial-autocorrelation of the DEM error. The Boolean viewshed can now be calculated with a pre-de® ned user’s and producer’s accuracy, accommodating to the user’s needs. As such, the proposed technique allows the calculation of visibility using a `soft’ rather than a `hard’ procedure, allowing for more sophisticated and better informed decision making (Eastman 1995).
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Acknowledgments
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This text presents research results of the Belgian Programme on Interuniversity Poles of Attraction and the GOA Programme initiated by the Belgian State, Prime Minister’s O ce, Science Policy Programming. Scienti® c responsibility is assumed by its authors. The authors would like to thank Professor Marc Waelkens who made this work possible. They would also like to express their sincere gratitude to the Sagalassos Archaeological Research Project and the Belgian Fund for Collective Fundamental Research (FKFO) for their ® nancial support. References Bolstad, P. V., and Stowe, T., 1994, An evaluation of DEM accuracy: elevation, slope, and aspect. Photogrammetric Engineering & Remote Sensing, 60, 1327± 1332. Burrough, P. A., 1992, Principles of Geographical Information Systems for L and Resources AssessmentÐ Monographs on soil and resources survey, No 12 (Oxford: Oxford University Press). Conover, W. J., 1971, Practical Nonparametricstatistics (New York: John Wiley & Sons Inc.). De Floriani, L., and Marzano, P., 1994, Line-of-sight communication on terrain models. International Journal of Geographical Information Systems, 8, 329± 342. Donnay, J-P., 1992, De termination en mode maille du champ d’intervisibiite dans un modeÁle nume rique de terrain. Cartographica , 29, 75± 82. Eastman, J. R., 1995, Idrisi for W indows User’s Guide (Worcester: Clark University). Eklundh, L., and Ma·rtensson, U., 1995, Rapid generation of Digital Elevation Models from topographic maps. International Journal of Geographical Information Systems, 9, 329± 340. Fisher, P., 1991, First experiments in viewshed uncertainty: the accuracy of the viewshed area. Photogrammetric Engineering & Remote Sensing, 57, 1321± 1327. Fisher, P., 1992, First experiments in viewshed uncertainty: simulating fuzzy viewsheds. Photogrammetric Engineering & Remote Sensing, 58, 345± 352. Fisher, P., 1993, Algorithm and implementation uncertainty in viewshed analysis. International Journal of Geographic Information Systems, 7, 331± 347. Hadrian, D. R., Bishop, I. D., and Mitcheltree, R., 1988, Automated mapping of visual impacts in utility corridors. L andscape and Planning, 16, 261± 282. Heuvelink, G. B. M., 1998, Error Propagation in Environmental Modelling W ith GIS (London: Taylor & Francis). Kent, M., 1986, Visibility analysis of mining and waste tipping sitesÐ a review. L andscape and Urban Planning, 13, 101± 110. Loots, L., Nackaerts, K., and Waelkens, M., in press, Fuzzy viewshed analysis of the Hellenistic city defence system at Sagalassos, Turkey. In Proceedings of the 25th Anniversary Conference of Computer Applications and Quantitative Methods in Archaeology (Oxford: Oxford University Press). Nagy, G., 1994, Terrain visibility. Computer & Graphics, 18, 763± 773. Sachs, L., 1982, Applied Statistics. A Handbook of T echniques (New York: Springer-Verlag). Sorensen, P., and Lanter, D., 1993, Two algorithms for determining partial visibility and reducing data structure induced error in viewshed analysis. Photogrammetric Engineering & Remote Sensing, 59, 1149± 1160. Stehman, S. V., 1997, Selecting and interpreting measures of thematic classi® cation accuracy. Remote Sensing of the Environment, 62, 77± 89. Wang, J., Robinson, G. J., and White, K., 1996, A fast solution to local viewshed computation using grid-based digital elevation models. Photogrammetric Engineering & Remote Sensing, 62, 1157± 1164. Wheatley, D., 1995, Cumulative viewshed analysis: a GIS-based method for investigating intervisibility, and its archaeological application. In Archaeology and geographical information systems edited by G. Lock and Z. Stancic (London: Taylor & Francis),
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