Positional error modelling using Monte Carlo simulations

T. Podobnikar
Scientific Research Centre of the
Slovenian Academy of Sciences and Arts
Spatial Information Centre
Gosposka ul. 13
SI-1000 Ljubljana
Slovenia
E-mail: tomaz@alpha.zrc-sazu.si

Keywords: spatial analysis, Monte Carlo methods, GIS

Abstract

The paper shortly presents some possibilities of using Monte Carlo simulation methods for positional error estimation and visualisation it on different kinds of vector coverages (and DEM) available from spatial databases in Slovenia. In the paper are shown error simulations on two small regions. The simulated data were vector points, lines, and polygons data. For the visibility estimation we used raster of DEM.

The results can be used for introducing spatial data of error visualisation to potential users. Further they can be used for teaching about possibilities of errors incidences to the used data and to estimation the error propagation through the spatial modelling process using spatial analyses. Results can be also statistically evaluated.

Introduction

The main objective of the paper is to present some analyses results of different effects of the positional errors on the vector coverages and DEM that are available from particular Slovene databases, using Monte Carlo simulation methods. An empirical error model has been designed for evaluation inherent errors through simulations.

The other aspect is visualisation of the inherent and even operational errors produced during the spatial analyses. This is useful representation of the possible errors of different (different accuracy, different distribution of the errors on the testing area, etc.) spatial data. The results can be interesting for anyone users of digital spatial data. Visualisation of data accuracy can be presented for example in GIS-catalogues (catalogues of spatial data). Even geodetic measurements can be represented in the same way. Visualisation of the errors can be also used for teaching about error influences on the different spatial data.

Monte Carlo simulations have been calculated with programs based on Arc/Info AML (macro language) combined with procedures made in language C. The objective of the paper is not to introduce the algorithms of the Monte Carlo simulations, which are basically known (i.e. Openshaw et al. 1991, Fisher 1996 etc.). The paper also doesn’t handle with analytical models of error estimation (i.e. Burrough 1986, Heuvelink et al. 1989 etc.). Monte Carlo simulation algorithms have been adapted according to evolved empirical error model, which includes many specific details, for example considering different density of houses or morphology of DEM.

Visualisation an errors of vector data

Positional errors caused by incorrectness of nodes can be represented by Epsilon-band, which describes error zones around the objects – points, lines and polygons (Burrough 1986, Giordano and Veregin 1994). Simple bands for a segment of line are shown in Figure 1 (Ivačič 1994). Standard deviation, which is equal in x and y direction of Cartesian co-ordinate system respective represent boundary of the band.

a) b) c)

d)

Figure 1: Simple forms of Epsilon bands around segment of line (a-c). Example of Monte Carlo simulation of error on segment of line with frame around of the triple standard deviation value (d). Number of simulations is 1000.

Much more realistic Epsilon bands could be provided by Monte Carlo simulation methods (Figure 2). In the case of 68 % of probability and standard deviation, which have to be much more smaller than the length of segment of line, is the area of such Epsilon band approximately 0,79 of original in Figure 1, case a).

 

Figure 2: Perspective views of Epsilon surface (rasterised vectors) of the segment of line, produced by Monte Carlo simulation. White line over the surface describes an original position of line.

With Monte Carlo methods can be simply employed estimation of positional errors of point, line and polygon vector coverages. Very interesting is further step, where error of simulated data is researched using operations such as map algebra.

Example of positional errors on vector data

The practical testing data for Monte Carlo simulation of positional error have been chosen for region around town Novo mesto, which is located in the southeaster part of Slovenia (Figure 3). Positional errors of the data from the region have been roughly divided into two groups:

Figure 3: Digital vector data (1,4 x 1,1 km), prepared for Monte Carlo simulations: roads are double black lines, rivers are rastered with grey dots, centre points of houses are dots, agricultural areas and forests are shown in grey.

For the first group of positional errors is considered that contain translation, rotation, scale, screwing, etc. (s 12). And for the second group is considered that the most important is error of measurements and digitalisation, which depends mostly on chosen methods, instruments, operators, generalisation, etc. (s 22). The result of approximation variance includes both types of errors (s 2) (Drummond 1995):

s 2 = s 12 + s 22.

For the analyses were chosen digital data of Surveying and Mapping Authority of the Republic of Slovenia (Figure 3):

In the Figure 3 can be clearly seen that some of the data don’t fit to each other. The greatest reason is that they were collected from different sources. A relevant error model for every type of data has been evolved before the simulations.

a) b)

Figure 4: Monte Carlo positional error simulation (region 1,4 x 1,1 km) for the roads (5 repetition; a) and rasterised simulated area of agricultural areas and forests (resolution is 10 m, 100 repetitions; b).

Result of the simulations is regions with certain reliability to their position, that can be evaluated visually or statistically (Figure 4b). The next step discovers simulated data behaviour during the spatial analyses processing (modelling). When non-homogeneous data are used in modelling, result could be evidently unforeseen. Described simulating model can even implicate visibility results from error simulated DEM.

Demonstrating visibility on DEM (with simulated error)

Errors of DEM are mostly considered as positional errors of highness (not thematically). For the model has been chosen the area with modesty homogenous and smooth surface without extremely steep relief. We considered that distribution of error is similar as relief, i.e. spatial autocorrelated (Fisher 1996).

There are many visualisation ways for error estimation of DEM. We have chosen visibility operation on simulated DEM on the region around town Ljubljana, which is located in the middle of Slovenia. The observation point of the visibility was calculated from a transmitter on the hill Krim (1107 m) located south from Ljubljana (Figure 5).

DEM of Slovenia from Surveying and Mapping Authority of the Republic of Slovenia with resolution 100 m and estimated error of highness (after testing it) of s  = 10 m and Moran’s coefficient of autocorrelation I = 0,8 has been used.

a) b)

 

Figure 5: Monte Carlo DEM error simulation (region 29 x 23 km) of the visibility from the hill Krim. Example a) shows binary visibility from the original DEM. Example b) shows potential visible areas produced on with error simulated DEMs (resolution 10 m, 100 repetitions). Darker areas means more potential probability that they are not visible while are lighter areas probably more visible.

Result of the simulations is regions with certain reliability to their position, which can be evaluated visually or statistically and could be used, in spatial analyses (for error propagation estimation) together with example in previous chapter (Figure 5b).

Conclusion

Monte Carlo methods have been using mostly experimentally in spatial analyses less than ten years. Errors discovering and their presentation could be especially helpful for users of the spatial data what they really could expect from spatial data. Usually only scale of source data and root mean square error is referred as metric accuracy data. Concerning average user is known that he can’t imagine real metrical quality of data. Usually they mix absolute and relative data accuracy and so they think that the accuracy of used data is the same as a number of decimal places.

We think that in the future useful standard GIS’s will include tools for converting data coverages concerning expected error for producing more realistic results of spatial analyses.

References

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