precision of area computation

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PRECISION OF AREA COMPUTATION Gerhard Navratil ([email protected]) Institute for Geoinformation / TU Vienna

ABSTRACT Computation of areas is a fundamental task of a geographic information system. The standard methods use coordinates of boundary points for the computation. The trapezoid formula by Gauss is the method used in ArcGIS to solve the problem if the points are connected straight lines. We usually ignore the fact that coordinates have limited accuracy. We use coordinate values as parameters for the area computation and take the result as the value for the area. Unfortunately coordinates result from measurement processes, which are statistical processes. Statistical processes do not produce a single result. We can only specify an interval of results and a probability that the real value (which we do not know) is within that interval. The assumption of having normal distribution allows using two values, the mean value x and the standard deviation s. The probability that the real value is within the interval [x-s, x+s] is 68%. The limited accuracy of the coordinates results in limited accuracy of derived measures, in this case the area. The error propagation law is the theoretical basis for discussion of the effects which limited accuracy of coordinates has on the computation of areas. There are two forms of the error propagation law: The simple form requires uncorrelated data whereas the general form can deal with correlations. I use both forms of the error propagation law and compare the results for a sample data set, which is a cadastral data provided by the Austrian Federal Office of Metrology and Surveying.

1. INTRODUCTION We use areas in many parts of society as a base component. Support for farmers by the European Union or purchase prices for parcels are based on the square measure of the parcel to make the numbers comparable. Area differences, e.g. of forests, are used as indicators for growth. In each case it is necessary to know the area precise enough. But how precise can we be? Computation of areas from coordinates is a basic task in analytical geometry. Restriction to linear connection between boundary points (closed polygons) allows using the formulae of Gauss which can be found in the literature (Kahmen 1993). There are two versions, the trapezoid formula (1) and the triangular formula (2). ArcGIS uses the trapezoid formula for area calculation (ESRI 2003).

∑ ( y − y )( x + x ) = ∑ ( x − x )( y 2F = ∑ y ( x + x ) = ∑ x ( y + y )

2F =

i +1

i

i −1

i

i

i +1

i +1

i +1

i

i −1

i +1

i

i

+ yi +1 )

(1) (2)

From the statistical point of view coordinates are randomized values. Measurements define coordinates. Unfortunately measurements are affected by random influences. Therefore, two measurements of the same quantity result in two different numbers if they are performed independent of each other. Theoretically each measurement has a ‘real’ value. This is the value, which a measured quantity actually has. Our observations of the quantity (the measurements) are influenced by random influences and differ from the real value. Therefore we cannot determine the real value and we only get approximations. The measure for the variation of the observations is the precision of the precision of the measurement. Having a large number of measurements allows the computation of a mean value. The mean value will be close to the real value if the number of measurements is high enough and if there are no systematic influences affecting our measurements. Systematic influences affect all measurements in a similar way resulting in a shift from the real value. The distance between real value and mean value is the accuracy. Accuracy and precision of a measurement form the quality of the measurement. The quality of measurements affects the derived values like coordinates. Surveyors try to eliminate systematic errors and therefore assume that the accuracy can be neglected. Figure 1 shows the connection between real value, accuracy, and precision for one dimension.

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Figure 1 Real value, accuracy, and precision In this paper I discuss the effects of coordinate precision on the computation of areas. I assume that the areas are bound by polygons so I can use the Gaussian formula. Unfortunately the precision of area calculations does not only depend on precision of the coordinates but also on the shape of the area. Therefore, an investigation should use many different areas. The Austrian Federal Office of Metrology and Surveying providing a suitable data set from the Austrian cadastre. The data set consists of 1.238 parcels with a total area of 7 km2. The area includes a village with small, crooked parcels and agricultural areas with large, approximately rectangular parcels. This provides the basis for the investigation. The paper does not check if the computed areas correspond to the real areas of the parcels. The aim of the paper is to investigation the effect of the shape on the precision of area computation. Therefore, the only possible conclusions for the cadastre are about the statistical accuracy under the assumptions used.

2. THEORETICAL ASPECTS OF PRECISION Measurements and values derived from measurements are called randomized values X in statistics. A measurement is an experiment, which can be repeated as often as wanted, in theory even infinite times. A set of values limits the possible result of the experiment. If the experiment is an electronic distance measurement, the set consists of all values a display can show. Each experiment provides a realization x of the randomized value. Infinite repetition of the experiment allows the computation of probabilities for each possible resulting value. The experiment has normal distribution if it is only affected by random influences. Normal distribution is defined by mean value µ and standard deviation σ or variance σ2 (Stoyan 1993). Standard deviation and variance are measures for the precision.

2.1 Simple Error Propagation Model Precision is usually ignored when applying a function to a randomized value. The function is applied to the mean values only. The result is still a randomized value and applying the mean value only gives the mean value of the result. We must use the error propagation law to compute the standard deviation of the function result (Reißmann 1976). The error propagation law starts with a function f and it’s parameters l1 to ln with the standard deviations σ1 to σn. and defines the standard deviation σ F . F = F (L1 , L2 ,K, Ln ) 2

2

 ∂F   ∂F   ∂F  σ F =  σ 1  +  σ 2  + K +  σ n   ∂L1   ∂L2   ∂Ln 

2

(3)

Computation of the precision for areas starts with formula (1) or (2). The error propagation law uses the partial derivatives of the function. We differentiate formula (2) with respect to the observations to use it with formula (3). The observations in this case are the coordinates of the boundary points. The partial derivatives are

ESRI 2003 – 18. European User Conference/10. Deutschsprachige Anwenderkonferenz ∂F 1 1 = ( yi + yi+1 ) − ( yi−1 + yi ) = ( yi+1 − yi−1 ), ∂xi 2 2 ∂F 1 1 = ( xi − xi+1 ) + (xi−1 − xi ) = (xi−1 − xi+1 ). ∂yi 2 2

3

(4)

We have to assume that the precision of all coordinates is equal if we have no other information. This leads to σx=σy=σ. The combination of (4) with (3) then gives a formula for the standard deviation of the area (Niemeier 2002): σF =

σ 2

∑ (y

i +1

2 2 − yi −1 ) + ( xi −1 − xi +1 ) .

(5)

i

The sum within the square root covers all points of the bounding polygon. The notion (i-1) for the first point of the polygon refers to the last point of the polygon and the notion (i+1) for the last point refers to the first point. Formula (5) shows the impossibility to have a general value for the standard value for area precision. Coordinate differences, which depend on the shape, influence the precision. Therefore it is impossible to stipulate a general value for area precision.

2.2 Correlations between Coordinates The assumption in the simple model was that there are no statistical correlations between the parameters. This might be true for geodetic measurements like distances but it is definitely not true for the coordinates of a point. The determination of coordinates uses formulae which use the same measurements for the computation. Statistical influences therefore affect both coordinates and cause correlation. Thus we have to use the general error propagation law (Reißmann 1976): σ F2 = f T Σ xx f .

(6)

Σ xx is a matrix holding the variances and co-variances of the parameters and the vector f contains the first derivatives of the formula with respect to the parameters. The variances of the parameters are in the principal diagonal and the covariances between a parameter la and lb can be found in row a, column b or in row b, column a. In our case the parameters are the coordinates of the points and the usual order is that coordinates of a point are adjacent. Therefore, variances and covariances for the coordinates of a point form a square with 2 lines and 2 rows. The connection between correlation and covariance is defined as ρ xy =

σ xy σ xσ y

,

(7)

where ρ xy is the correlation between two parameters x and y, σ xy is the co-variance between the parameters, and σ x and σ y are the standard deviations of x and y. The assumption that there are only correlations between the coordinates of a point but no correlations between the coordinates of different points leads to the following structure for Σ xx :

Σ xx

 σ x21   σ x1 y1  = 0  0   M 

σ x1 y1

0

0

σ 2y1

0

0

0

σ x22

σ x2 y 2

0

σ x2 y2

M

M

σ

L  L   L  L   O 

2 y2

M

(8)

The introduction of matrix (8) changes formula (5) to: σ F2 =

1 2

∑(y

i +1

− yi−1 ) σ x2 + ( xi−1 − xi+1 ) σ y2 + 2( yi+1 − yi −1 )( xi −1 − xi +1 )σ xy . 2

2

(9)

i

In general points are correlated, too. Correlations result from using the same fundamental points or combined coordinate computation in an adjustment process. Unfortunately we usually do not know the correlations when using a data set. Ghilani (2000) discusses this case based on some examples.

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3. TEST DATA The test area is the cadastral unit Ebergassing, which is situated 15km southeast of Vienna. The data was provided free of charge by the Austrian Federal Office of Metrology and Surveying for scientific purposes.

Figure 2 Land use in Ebergassing The village Ebergassing lies in the center of the area. The light green areas outside the village are areas for agricultural use and the dark green areas are forest. A river forms the south eastern boundary of the area. Table 1 provides an overview on the land use in Ebergassing.

Land use

# of areas

Total area [m2]

Average area [m2]

Constructions

913

167.041

183

Construction area with greenering

724

469.898

649

49

6.081

124

7

2.059

294

13

45.347

3.488

1

4.686

4.686

155

317.628

2.049

5

34.101

6.820

182

4.894.238

26.891

1

763

763

124

1.239.321

9.995

Running waters

34

114.939

3.381

Standing waters

2

4.488

2.244

Construction area w. artificial surface Supply and waste disposal Industry area Deposit Streets Railway Agricultural use Meadow Forest

ESRI 2003 – 18. European User Conference/10. Deutschsprachige Anwenderkonferenz Land use

# of areas

Recreational areas

Average area [m2]

15

86.072

5.738

5

5.498

1.100

2230

7.392.160

Other use Σ=

Total area [m2]

5

Table 1 Land use in Ebergassing The different sizes of the areas are clearly visible. More than 75% of the areas are Constructions and construction areas. These areas only have an average area of less than 1000m2, the constructions themselves even less than 200m2. The biggest areas are areas for agricultural use and forest. However, less than 15% of the areas fall in these categories. Parcels consist of one or more areas with specified use. They are the smallest areas with an identifier in the Austrian cadastre. Ebergassing has 1.238 parcels. The areas range from 3,8m2 to 372.249m2. 783 parcels (more than 60% of the parcels) have an area of less than 1.000m2. 141 Parcels (11%) have an area of more than 10.000m2 and 12 of these even have more than 100.000m2. These 12 parcels form 30% of the total area of Ebergassing. Thus Ebergassing consists of a large number of small parcels and a small number of large parcels which is typical for the Austrian cadastre in rural areas.

4. UNCORRELATED COORDINATES A simple VBA-script computes the standard deviation of the areas. The script extracts the coordinates of the boundary points and applies formula (5). Dim Dim Dim Dim Dim

Output As Double NumSegs As Long aLoop As Long pArea As ISegmentCollection x1,y1,x2,y2 As Double

Set pArea = [Shape] NumSegs = pArea.SegmentCount Output = 0 For aLoop = 1 To NumSegs – 2 x1 = pArea.Segment (aLoop - 1).fromPoint.x y1 = pArea.Segment (aLoop - 1).fromPoint.y x2 = pArea.Segment (aLoop + 1).toPoint.x y2 = pArea.Segment (aLoop + 1).toPoint.y Output = Output + (y2 - y1)*(y2 - y1) + (x2 - x1)*(x2 - x1) Next aLoop x1 = pArea.Segment (NumSegs - 1).fromPoint.x y1 = pArea.Segment (numSegs - 1).fromPoint.y x2 = pArea.Segment (1).toPoint.x y2 = pArea.Segment (1).toPoint.y Output = Output + (y2 - y1)*(y2 - y1) + (x2 - x1)*(x2 - x1) x1 = pArea.Segment (NumSegs - 2).fromPoint.x y1 = pArea.Segment (numSegs - 2).fromPoint.y x2 = pArea.Segment (0).toPoint.x y2 = pArea.Segment (0).toPoint.y Output = Output + (y2 - y1)*(y2 - y1) + (x2 - x1)*(x2 - x1) Output = 0.1 / 2 * Sqr ( Output )

The last line of the script defines the standard deviation of the coordinates with 10cm. This value is derived from the Austrian decree for surveying (1994). The decree sets the limit for the standard deviation of boundary points in the cadastre to 15cm in §7. The assumption of the standard deviation of Helmert s H = s x2 + s 2y

(10)

and equal errors in x and y leads to a standard deviation of 10cm. Figure 3 shows the results of the computation. 907 of the 1.238 parcels have a standard deviation of less than 10m2. The other areas have standard deviations of up to 103m2. As expected, bigger areas have bigger standard deviations. The

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distribution does not change if the standard deviation is changed to 50cm but the maximum value for the standard deviation then changes to 513m2.

Figure 3 Standard deviation of the areas with 10cm standard deviation of the coordinates Another way to express area precision is in relation to the size of the area. The quotient of precision and size makes the precision of areas with different size comparable. The result is a relative accuracy measure. Figure 4 shows the relative precision for the parcels of Ebergassing. In comparison with Figure 3 the distribution changed completely. The biggest values for the relative precision occur where the parcels are rather small. Therefore, the image of the whole study area does not reveal much. Figure 5 shows a zoomed view of the village with parcels having poor ratio between precision and size.

Figure 4 Standard deviation of the areas in percent of the area with 10 cm standard deviation of the coordinates Shape and size of parcels depends on the use of parcels. Parcels for construction are usually rectangular as are parcels for agricultural use. The major difference is the size, as shown in Table 1. Streets however are long and thin areas and in some cases a parcel even forms a network of roads as in the western forest. Therefore, parcels for streets have a worse shape than parcels for construction or agricultural use. The mean value for the relative precision of street parcels is 2% with a maximum of 9%. Only 27% of the street parcels have a relative precision of less that 1%. Parcels for buildings, however have a mean relative precision of 0,8% with a maximum of 6%. In this case 77% of the parcels have a relative precision of less that 1%. The situation is even better for parcels for agricultural use and for forest parcels. Here the mean relative precision is 0,6% with a maximum of 5% and 85% of the parcels have a relative precision of less than 1%.

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Figure 5 Detail from Figure 4

5. CORRELATED COORDINATES There are two different types of correlations. The correlation between coordinates of a single point emerges from the combined determination of the coordinates. Measuring one plane coordinate of a point in a way that makes it impossible to compute the other coordinate from the measurement data is rather difficult. Therefore, coordinates of a point usually have a strong correlation. Coordinates between different points are also correlated if they are determined in the same process, e.g., measured from the same position. However, this correlation is smaller than the correlation between the coordinates of one point because it involves different measurement data. Storage of correlations between coordinates requires vast amounts of memory. Assuming that in a data set the coordinates of all points are correlated we get a symmetric matrix of correlations. A number of n coordinates therefore leads to n(n − 1) 2 correlation values. There are two methods to deal with this problem: Theoretical models provide values for correlations without storing these values. The other method is storing the measurement data itself. Correlations between measurements are simple and Wolf (1968, p.520-522) shows how to stipulate them. Correlations between points can be computed from these measurements as needed. Buyong (1992) proposed this method for cadastral systems and Joffe (2003) described how Survey Analyst can be used to build such a system. The problem of the second method is the amount of computations necessary. Therefore, I only use theoretical models in this work.

5.1 Correlations between Coordinates of a Point Coordinates are usually correlated as shown in section 2.2. The computation of area precision for correlated coordinates requires formula (9) if we restrict the correlations to the coordinates of a point. The changes in the VBA script are rather small. These changes reflect the differences between formula (5) and formula (9). …

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For aLoop = 0 To NumSegs – 2 … Output = Output + ((y2 - y1)^2 + (x2 - x1)^2)*0.01 Output = Output + 2 * (y2 - y1) * (x1 - x2) * 0.007 Next aLoop … Output = Output + ((y2 - y1)^2 + (x2 - x1)^2)*0.01 Output = Output + 2 * (y2 - y1) * (x1 - x2) * 0.007 … Output = Output + ((y2 - y1)^2 + (x2 - x1)^2)*0.01 Output = Output + 2 * (y2 - y1) * (x1 - x2) * 0.007 Output = Sqr ( Output ) / 2

Formula (5) is used in 3 places in the code. These 3 places require changes. The first line in each case computes the squares of the coordinate differences and multiplies them with the square of the standard deviation of the coordinates. The second line computes the effects of the correlation between the coordinates of the point. The correlation used here is 0,7 leading to a co-variance between the coordinates of 0,007.

Figure 6 Standard deviation of the areas with 10cm standard deviation of the coordinates and 0,7 correlation between the coordinates of a point Figure 6 shows the result of applying correlations. There are only small changes in the result. In comparison to Figure 3 the maximum value for the standard deviation changes from 103m2 to 99m2. The general distribution, however, does not change. The same is valid for the relative precision.

5.2 Fixed Correlations between Points Usually points are not uncorrelated. As stated in section 2.2 combined determination of point coordinates results in correlations between points. Sometimes correlations are known, above all if all measurements are available. This is usually true for photogrammetric evaluations or construction site maps. It is definitely not true for cadastral data sets. In that case we have to assume values for correlations. The following example shall give an impression how correlations influence the results. Correlation between the coordinates of a point is again 0,7 but there are also correlations between coordinates of different points. The value for this correlation was set to 0,5. The example imitates a photogrammetric evaluation where all points are determined in a unique process. The VBA-code looks different now. Dim Output As Double Dim NumSegs As Long

ESRI 2003 – 18. European User Conference/10. Deutschsprachige Anwenderkonferenz Dim Dim Dim Dim

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aLoop1, aLoop2 As Long pArea As ISegmentCollection x1,y1,x2,y2 As Double dfx1, dfy1, dfx2, dfy2 As Double

Set pArea = [Shape] NumSegs = pArea.SegmentCount Output = 0 For aLoop1 = 0 To NumSegs – 1 select case aLoop1 case 0 x1 = pArea.Segment y1 = pArea.Segment x2 = pArea.Segment y2 = pArea.Segment case NumSegs – 1 x1 = pArea.Segment y1 = pArea.Segment x2 = pArea.Segment y2 = pArea.Segment case else x1 = pArea.Segment y1 = pArea.Segment x2 = pArea.Segment y2 = pArea.Segment end select dfx1 = x1 - x2 dfy1 = y2 - y1 For aLoop2 = 0 To NumSegs – 1 select case aLoop2 case 0 x1 = pArea.Segment y1 = pArea.Segment x2 = pArea.Segment y2 = pArea.Segment case NumSegs – 1 x1 = pArea.Segment y1 = pArea.Segment x2 = pArea.Segment y2 = pArea.Segment case else x1 = pArea.Segment y1 = pArea.Segment x2 = pArea.Segment y2 = pArea.Segment end select dfx2 = x1 - x2 dfy2 = y2 - y1

(NumSegs - 1).fromPoint.x (NumSegs - 1).fromPoint.y (1).fromPoint.x (1).fromPoint.y (aLoop1 - 1).fromPoint.x (aLoop1 - 1).fromPoint.y (0).fromPoint.x (0).fromPoint.y (aLoop1 (aLoop1 (aLoop1 (aLoop1

+ +

1).fromPoint.x 1).fromPoint.y 1).fromPoint.x 1).fromPoint.y

(NumSegs - 1).fromPoint.x (NumSegs - 1).fromPoint.y (1).fromPoint.x (1).fromPoint.y (aLoop2 - 1).fromPoint.x (aLoop2 - 1).fromPoint.y (0).fromPoint.x (0).fromPoint.y (aLoop2 (aLoop2 (aLoop2 (aLoop2

+ +

1).fromPoint.x 1).fromPoint.y 1).fromPoint.x 1).fromPoint.y

if aLoop1 = aLoop2 then Output = Output + (dfx1^2+dfy1^2) * 0.01 + 2*dfx1*dfy1*0.007 else Output = Output + (dfy1*dfy2 + dfx1*dfx2 + dfx1*dfy2 + dxy1*dfx2)*0.005 endif Next aLoop2 Next aLoop1 Output = Sqr ( Output ) / 2

The code consists of two loops applying formula (6). Both loops run from the first to the last point of the bounding polygon. The case-constructs provide the coordinates of the neighboring points. The if-then-else-branch splits the case of correlation between different points from the correlation of coordinates of the same point. The results are similar to the results from the previous examples. The maximum standard deviation dropped to 76m2. This is a result of the higher neighborhood accuracy modeled by the correlation. The general distribution, as shown in Figure 7, only changes very little. Table 2 shows a comparison between the two cases: Case 1 only has correlations between coordinates of a point whereas case 2 introduces correlations between coordinates of different points. Street parcels and agricultural parcels remain generally unaffected. Parcels for construction improve in precision of the area computation. It seems that correlations have a greater influence on small areas than they have on large ones. The strongest example for this is a rectangular area of 3,3m x 1,2m. The standard deviation in the case of uncorrelated coordinates was 0,26m2 and in the case of correlated points 0,33m2.

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Figure 7 Standard deviation of the areas with 10cm standard deviation of the coordinates, correlation of 0,7 between the coordinates of a point, and correlation of 0,5 between coordinates of different points

Mean relative precision

Type

Case 1

Case 2

2%

Construction Agricultural use

Max. relative precision

Percentage of Areas with less than 1% rel. precision

Case 1

Case 2

Case 1

Case 2

2%

9%

9%

27

27

0,8 %

0,5 %

7%

8%

77

92

0,6 %

0,6 %

5%

5%

85

85

Street

Table 2 Comparison between results of uncorrelated (case 1) and correlated (case 2) points

5.3 Distance Dependant Covariance Instead of a fixed covariance we can use a covariance function. Covariance functions specify the value for the covariance as a function of the distance between the points. There are different types of covariance functions like the Gaussian covariance function or the model of Hirvonen. Gauss:

C (d ) = C (0) ⋅ e − k

Hirvonen:

C (d ) =

2 2

d

C (0)

(1 + A d ) 2

2 p

(13) .

(14)

The model of Hirvonen with the parameters A=1 and p=1 leads to the simple formula C (d ) =

C (0)

(1 + d ) . 2

This formula requires only small changes in the VBA-code from section 5.2. … Dim aDistSq As Double … if aLoop1 = aLoop2 then … else x1 = pArea.Segment (aLoop1).fromPoint.x y1 = pArea.Segment (aLoop1).fromPoint.y x2 = pArea.Segment (aLoop2).fromPoint.x

(15)

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y2 = pArea.Segment (aLoop2).fromPoint.y aDistSq = (x2-x1)^2 + (y2-y1)^2 Output = Output + (dfy1*dfy2 + dfx1*dfx2 + dfx1*dfy2 + dxy1*dfx2) * .01 / (1 + aDistSq) end if …

The form parameters A and p can be used to adopt the model to the data set. The influence of the correlation is greater than zero in this case. The influence will usually be smaller than the constant influence shown in the last section. Therefore, the values for the area precision will be between the values from section 5.1 and the values from section 5.2.

6. CONCLUSIONS We have seen that it is rather difficult to stipulate a value for the precision for areas. We cannot provide a unique value as we usually do for points. In general the standard deviation increased with the size of the area but poor shape influences the result heavily. Especially street parcels have worse precision than rectangular areas. The average value for the relative area precision was below 1%, except for the streets where some areas have with up to almost 10%. The assumption for this computation was a coordinate precision of 10cm standard deviation. Ground survey easily provides this precision. In photogrammetry 10cm are medium to high quality for large measurement areas. Satellite imagery however will not be able to provide this precision. Unfortunately the areas covered by GIS are often too large for ground survey and even with photogrammetry large numbers of images are necessary. In addition standard deviation has a statistical security of 68%. We must multiply the precision values by 3 to get a statistical security of 99% which may be necessary in some cases. This leads to an average relative precision of 3% with outliers between 20 and 30%. Introducing correlations between the coordinates provides more realistic precision measures. Unfortunately we usually do not know the correlations between coordinates if we only have a set of coordinates. Storing the original measurements (e.g., with Survey Analyst) might solve this problem. Michael Franz implementing a full measurement based system for the area of Ebergassing as his master thesis. This will then provide a starting point for further discussions.

ACKNOWLEDGEMENTS This work was supported by the project ReviGIS (Revision of the Uncertain Geographic Information) financed by the European Commission. The data was provided free of charge by the Austrian Federal Office of Metrology and Surveying. I appreciate the help of Gernot Tutsch on questions of ArcGIS and of Christian Gruber to improve the text.

REFERENCES (1994). Vermessungsverordnung. BGBl.Nr. 562/1994. Buyong, T. (1992). Measurement-based multi-purpose cadastral systems. NCGIA. Orono, University of Maine. ESRI (2003). Knowledge Base, ESRI. Ghilani, C. (2000). "Demystifying Area Uncertainty: More or Less." Surveying and Land Information Systems 60(3): 183 189. Joffe, B. (2003). Survey Analyst: A Dream Come True. ArcNews. 25. Kahmen, H. (1993). Vermessungskunde. Berlin, de Gruyter. Niemeier, W. (2002). Ausgleichungsrechnung: Eine Einführung für Studierenden und Praktiker des Vermessungs- und Geoinformationswesens. Berlin, de Gruyter. Reißmann, G. (1976). Die Ausgleichungsrechnung. Berlin, VEB Verlag für Bauwesen. Stoyan, D. (1993). Stochastik für Ingenieure und Naturwissenschaftler. Berlin, Akademie Verlag. Wolf, H. (1968). Ausgleichsrechnung nach der Methode der kleinsten Quadrate. Dümmlerbuch 7820 / Ferd. Dümmler's Verlag, Bonn.