errors within the inverse distance weighted (idw) interpolation procedure
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Errors within the Inverse DistanceWeighted (IDW) interpolationprocedureGeorgios Achilleos aa National Technical University of Athens, Laboratory ofGeometrical Representation of Space , Athens, GreecePublished online: 13 Oct 2008.
To cite this article: Georgios Achilleos (2008) Errors within the Inverse Distance Weighted (IDW)interpolation procedure, Geocarto International, 23:6, 429-449
To link to this article: http://dx.doi.org/10.1080/10106040801966704
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Errors within the Inverse Distance Weighted (IDW)
interpolation procedure
Georgios Achilleos*
National Technical University of Athens, Laboratory of Geometrical Representation of Space,Athens, Greece
(Received 18 December 2006; final version received 6 February 2008)
Interpolation procedure is broadly used in sciences that are concerned with spatial dataand continuous phenomena that can be depicted on a spatial surface. Interpolationmakes use of sampling data, which is accurate and qualitative, in order to produce acontinuous representation of the phenomenon in question. The data’s accuracy istransferred by the procedure to its results and should be known. This paper examineserror propagation by the interpolation procedure, using the Inverse Distance Weighted(IDW) method in the case of the Earth’s relief.
Keywords: DEM; elevation errors; interpolation; inverse distance weighted
1. Introduction
Interpolation is broadly used in almost all geosciences, given that it offers the possibility ofrepresenting a continuous spatial phenomenon and describing it using data sampled in thedata space of this phenomenon (Mikhail 1978, Tempfli and Makarovic 1978, Ebner andReiss 1984, Legates and Willmott 1986, McCullagh 1988, Carpenter 1991, Weibel andHeller 1991, Kumler 1994, Rahman 1994, Vozelinek 1994, Gao 1995, Tang 1976).
Such a phenomenon, which is approached by a continuous spatial surface, is theEarth’s relief. Other similar phenomena are: geomagnetic field, temperature, rainfall etc.
It is important to know the quality of the result of the interpolation procedure in orderto combine the various parameters that are involved in the procedure and to vary asrequired (Clerici and Kubik 1975, Frederiksen 1980, Tempfli 1980, Burrough 1986, Balce1987, Ostman 1987, Carter 1988, Li 1988, 1992, 1993a,b, Corte and Koolhoven 1990, Leeet al. 1992, Canters 1994). Knowing this can allow the estimation of the quality of theresults that are derived from further use of this result (Dutton 1992, Openshaw 1992).
Usually, in the case of the Earth’s relief, the control points of a Digital ElevationModel (DEM) created by interpolation do not exist, therefore, it is not possible to controlthis result in terms of its quality. In many cases the DEM is available but its source (data)and the elements and parameters of the procedure from which it was derived (‘metadata’)are unknown.
This paper aims to investigate, both in theory and practice, error propagation, and ingeneral, the propagation of data uncertainty from the result of an interpolation procedurefor creating a DEM. The Inverse Distance Weighted (IDW) interpolation method is used
*Email: [email protected]
Geocarto International
Vol. 23, No. 6, December 2008, 429–449
ISSN 1010-6049 print/ISSN 1752-0762 online
� 2008 Taylor & Francis
DOI: 10.1080/10106040801966704
http://www.informaworld.com
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to second power, which is simple and broadly used. Other interpolation methods can beaddressed in a respective approach.
2. Inverse Distance Weighted (IDW) interpolation method
2.1 About interpolation
Many definitions have been formulated with regard to the interpolation procedure(Burrough 1986, McCullagh 1988, Robinson 1994):
Interpolation is the procedure of estimating the value of properties at unsampled sites withinthe area covered by existing point observations/data. (Burrough 1986)
Producing secondary information is always based on a certain model governed byassumptions and conditions (Burrough 1986, McCullagh 1988). A basic assumption is thatthe information that appears on the sites of the phenomenon surface, where sampling isperformed, presents spatial dependence (auto-correlation) at a satisfactory level, and thatthe mathematical function being used approaches, to a satisfactory degree, the continuousphenomenon that is analysed (Robinson 1994).
In the case of the Earth’s relief, a linear mathematical function is often used in theinterpolation procedure. However, the Earth’s relief does not present in fact a linearchange. Given that there is no need for high accuracy in the result, this function is anaccepted approaching solution.
Because nearly all interpolation methods use the distance criterion in their application,the assumption of using ‘short’ distances is clear. Long distances are affected by theEarth’s curvature and cannot be used in the interpolation procedure. In the case of longdistances, other techniques are needed.
Interpolation as a subject-matter and a procedure has been for many years a matter ofinterest for the international scientific community. The way this matter is addressed differsfrom time to time, depending on the means and tools available, as well as on the value andusefulness of the result.
During the 1960s and 1970s, when computer science began to be a part of theapplications, at a limited level, the investigation of interpolation focused mainly on issuesregarding the improvement of processing speed, the quality of the result of themathematical functions used and the way it was presented. These matters have been thesubject of international conferences and workshops (International Society of Photo-grammetry and Remote Sensing 1972, 1976, 1980, International Congress: ‘MathematicalModels, Accuracy Aspects and Quality Control’, Otaniemi/Finland 1982, etc.).
Developments in the field of computer science gave the scientists the opportunity to usethe interpolation method or technique they wanted, the data quantity they wanted,without being concerned about the time and the processing power needed for theprocedure. Furthermore, research moved towards other factors that affect the interpola-tion procedure and which have not been examined enough so far, at primary level.Such factors are the quality and accuracy of the information used primarily ininterpolation, the distribution of this information and, in general, the capacity of theinformation used to describe accurately the surface of the phenomenon being examined(e.g. the Earth’s relief).
There is a great range of methods, models and techniques for the interpolationprocedure, which are based on parameters that are decisive for the quality of the result.Many of these methods and techniques have already been established and used to the
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maximum extent, because they provide acceptable results. At the same time, researchcontinues with the aim to improve them (Oswald and Raetzsch 1984, Gold 1989).
The accuracy that can be produced by an interpolation procedure in a DEM is a resultof the density and the distribution of the reference altitudes as well as of the selection ofthe interpolation procedure used (Schut 1976). Even the simplest interpolation methodmay be useful, if the density of the reference altitudes is significantly high and theirdistribution ideal.
All these methods, models and techniques derived from the scientists will determine theoptimum interpolation procedure, either for all cases or for groups of similar cases.However, strenuous research efforts in the last two decades resulted in the conclusion thatthis cannot happen.
McCullagh (1988) is assertive with regard to the interpolation procedure:
. an ideal interpolation method does not exist
. no general solution regarding interpolation has been found to be suitable for allapplications.
2.2 Inverse Distance Weighted (IDW) method
The Inverse Distance Weighted method is largely recognized and exists as the basicmethod in most systems that create and manage DEMs (Burrough 1986, Schut 1976).
The main characteristic of this method is that all the points of the Earth’s surface areconsidered to be dependent among them, on the basis of distance. Therefore, thecalculation of the altitude in this area depends on the altitudes of these points.
The relation presenting the altitude of the interpolated point with the altitudes of thepoints – data is inversely proportional to their distance, raised to a power. This power isusually square or cubic. A power higher than the cubic does not contribute to the qualityof the interpolation, since it nearly nullifies the effect and the participation of the mostdistant elevation points in the calculation of the altitude of the interpolated point. At thesame time, it produces a significant delay in performing the calculations of the procedure.
Generally, it is a fast enough method which, however, presents certain weakness as faras the result is concerned (DEM). The ‘bull’s eyes’ phenomenon appears often andsurrounds the elevation points. In other cases, it tends to smooth out the curved parts ofthe contour lines when DEM resolution is not so high (pixel size). On the opposite case,the method smoothes out the curved parts of the contour lines, creating at the same timesmall islands of contour lines at the points of the real contour lines. Moreover, anotherobservation about this method is its weakness on giving a correctly interpolated height inthe case of islands. This weakness is equally intense in other methods such as the Krigingmethod, the method of radial basis function, where in the case of islands they do notperform an interpolation and the whole island takes exactly the same elevation. This isknown as the ‘flat-problem’.
These phenomena may be reduced, even eliminated, if a smoothing procedure is usedfollowing the interpolation. In this way the result is improved, but at the same time it isaltered as to the interpolation function used. The use of smoothing procedures should bemade with attention and awareness so that the final surface that is created (DEM)describes the Earth’s surface in question, without great deviations (Burrough 1986, Lam1994, Murray 1994, Greville 1967).
The method provides satisfactory results when the elevation points in the study areaare many and uniformly distributed (ESRI 1992). Furthermore, because this method gives
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a weighted average of the points – data elevations as the interpolated elevation, the DEMcreated has elevation values at the elevation interval created by the reference points.
3. Error propagation by the Inverse Distance Weighted method
The interpolation function of the Inverse Distance Weighted method, which is used in theinvestigation is:
Hp ¼Pn
i¼1 hi�d2iPn
i¼1 1�d2i
� � ð1Þ
where:Hp is the calculated altitude of point P where the interpolation is effected; hi is theelevation used to measure the height at point P; di is the distances of the points from pointP; n is the number of points used in the interpolation procedure for measuring the heightat point P.
The points (i) used as data in the interpolation procedure appear in Figure 1. Thesepoints are the tops of the broken lines that shape the form of the contour lines, on the basisof which the interpolation is effected. Sometimes it is possible to select among these points,when for example they are more dense than needed in order to define the contour’sgeometry (contour generalization).
From the elements of equation (1), only the distances of the elevation points i (di) frompoint P present uncertainty. The heights (hi) do not present errors, because the elevationpoints are points of contour lines and contour lines do not present elevation errors. This istrue, provided that the value of the height of the contour line is correctly determined(without gross error).
Therefore, by applying the error propagation law to equation (1), and following manyand strenuous operations we have:
s2Hp ¼4
sumD2
Xni¼1
hi �Hp
d3i
!2
s2di
24
35þ 8
sumD2
Xni¼1
Xnj¼1
hi �Hp
� �hj �Hp
� �d3i d
3j
sdidj
" #ð2Þ
where sumD¼Pn
i ¼ 1 1d2i
:Distances di of the elevation points i to the interpolated point P present sdi variance
and sdidj covariance, as a result of the digitization of contour lines from the map and theapplication of a coordinate transformation on this data.
Figure 1. Points i and j as used in the interpolation procedure as elevation points.
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By applying the error propagation law in the distance equation, the following relationsare derived:
s2di ¼1
d2ixi � xp� �2s2xi þ yi � yp
� �2s2yi þ 2 xi � xp� �
yi � yp� �
sxiyih i
ð3Þ
sdidj ¼1
didj
xi � xp� �
xj � xp� �
sxixj þ xi � xp� �
yj � yp� �
sxiyjþ yi � yp� �
xj � xp� �
syixj þ yi � yp� �
yj � yp� �
syiyj
� �ð4Þ
Where (xi, yi) are the coordinates of the elevation point I, (xp, yp) are the coordinates of theinterpolated point P, sxiyi is the variance of coordinates (xi, yi) of the elevation point, sxixjis the covariance of abscissa xi, xj of two elevation points i, j (i 6¼ j).
Obviously, apart from the coordinates (xi, yi), (xj, yj) and (xp, yp) of the points takingpart in the interpolation procedure, it is necessary to know the uncertainty that governsthe coordinates of these points (i, j, P). Their uncertainty is described in total by theirvariances (sxi, syi), for each i, j and their covariances (sxiyi, sxiyj, sxjyi, sxjyj) for each i, j.
These parameters must be either known from the beginning so that the uncertainty canbe propagated by the interpolation procedure, or must be calculated by the preparationprocess of the file with the contour lines that are used in the interpolation.
This uncertainty is not known in most cases. However, it is necessary to have aprocedure for estimating the uncertainty by the preparation process of the file with thecontour lines.
This project developed such a procedure, which follows the flow chart below:
. Digitization of the contour lines from their analogue source (analogue map)
. Transformation of the coordinates of the file with the contour lines in order to gofrom the reference system of the digitizer (or the screen) to the coordinate referencesystem of the initial source (analogue map), and to the DEM under creation.
Through this diagram of the preparation of the file with the contour lines another file isalso created that includes the uncertainty in question, as defined. This uncertainty has twoparts; the first one is derived due to the digitization of the contour lines from the analoguesource, and the second one is the part of the uncertainty that is produced by thetransformation of the coordinates of the digitized contour lines.
With regard to the first part, the phase of the digitization of the contour lines fromtheir analogue source aggravates the digital data with an uncertainty percentage, which issummarized in the digitization error. The basic assumption in this research is that thiserror is represented by the vertical component of the deviation of each point, which isdigitized in the contour line (Figure 2). It is considered that the deviation’s parallelcomponent to the contour line (given also that digitization errors move within rational andrandom limits) does not produce a digitization error since the shape of the contour line isnot changed to create a position error.
In order to use the digitization error we must know in advance its size (vector length)and its direction. With regard to the size of the digitization error, it can be taken either as aunified estimated average of all data (contour lines), or if one has the real digitizationerrors in each point of the contours and uses them.
This vertical component of the digitization error is at the same time the main errorellipse at the point examined, as it appears on Figure 2.
The characteristics of the main error ellipse are: u ¼ sr, v ¼ 0, rxy ¼ 1, suv ¼ 0, y8 isthe angle of the perpendicular to the contour line with the X axis.
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The question is to find the error ellipse with regard to the sx, sy axes with thecharacteristics sx, sy, sxy.
On the basis of the equations of the error ellipses (Mikhail and Gracie 1981):
s2x sxysxy s2y
� �¼ R½ ��1 s2u suv
suv s2v
� �R0½ ��1 ð5Þ
where
R½ � ¼ cos y sin y�sin y cos y
� �
By doing the mathematical operations:
s2x sxysxy s2y
� �¼ s2r cos2y 1=2s
2r sin 2y
1=2s2r sin 2y s2r sin2y
" #ð6Þ
and
sx ¼ sr cos y
sy ¼ sr sin y
sxy ¼ 1=2s2r sin 2y
ð7Þ
And developing sxy, we have:
sxy ¼ 1=2srsr2 sin y cos y
sxy ¼ sr sin ysr cos y
sxy ¼ sxsy
ð8Þ
Namely, when sr and the angle y8 are known (digitization error and direction of theperpendicular in relation to the horizontal axis), one can calculate sx, sy and sxy.
In equation (4) the covariances sxixj, sxiyj, syixj, syiyj are also used. These covariancesmay be estimated by using the serial dependence of data (Davis 1973, Chatfield 1980,
Figure 2. Digitization error and main error ellipse.
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Cressie 1991). As indicated (Keefer et al. 1988), the coordinates X and Y of the points thatform the contour lines present auto-correlation and cross-correlation phenomena withregard to the horizontal position error of the abscissa X and ordinate Y (Lee et al. 1994).
Covariance sxixj and covariance syiyj are given in equation (9):
Ck ¼XN�kt¼1
xt � �xð Þ xt¼k � �xð Þ=N
8i 6¼ j
ð9Þ
These covariances are connected to their auto-correlations through the relation(10):
rk ¼ ck=c0 ð10Þ
where
c0 ¼ s2x
rk : auto-correlationRespectively, covariance sxiyj and covariance syixj are given by equation (11):
cxy kð Þ ¼
PN�kt¼1
xt � �xð Þ ytþk � �yð Þ
N 8k 2 0; 1; 2; 3; . . . ; N� 1ð Þ½ �
orPNt¼1�k
xt � �xð Þ ytþk � �yð Þ
N 8k 2 �1;�2;�3; . . . ;� N� 1ð Þ½ � ð11Þ
These covariances are connected to their cross-correlations through the relation(12):
rxy ¼ cxy kð Þ. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cxx 0ð Þcyy 0ð Þq
: ð12Þ
where
cxx 0ð Þ ¼ s2xcyy 0ð Þ ¼ s2y
rxy is the cross-relation. Therefore, with regard to the first phase of the data preparation, ifone knows the digitization error on each top of the broken line that defines the contourline, he can estimate the spatial uncertainty that exists in the file of the digitized contourlines, as defined.
Further on, the file for the digitized contour lines is subject to the transformationprocedure of the coordinates, in order to go to the coordinate reference system that is ofinterest when creating the DEM.
It is known that the transformation of the coordinates is a procedure for which theinternational scientific community has made great and important research efforts, in orderto find out all its parameters and conditions so that its users decide according to the casethey deal with which transformation method and parameters to use.
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This research effort focuses on the use of the affine transformation of coordinates,which presents many advantages and is also broadly used. Moreover, the procedurealgorithm is nearly integrated in each software for spatial information management.The affine transformation requires that the coordinate points be known, which arecommon to both reference systems (the digitizer’s reference system and the DEMreference system). The number of these common points cannot be determined, but it isin connection with many factors. Further on, transformation with simple linearequations results in the transfer of the data (file of contour lines) from one referencesystem to the other.
The equations used in the affine transformation are:
x ¼ a1 þ a2Xþ a3Y
y ¼ a4 þ a5Xþ a6Yð13Þ
where X, Y are the flat coordinates prior to the transformation, x, y are the flatcoordinates after the transformation, ai are the transformation coefficients (i ¼ 1,2 ,. . . , 6).
By applying the error propagation law, the result that determines the uncertainty in thetransformed file of the contour lines (sxi, syi, sxiyi) is given by equation (14) (Achilleos2002):
s2x ¼ a22s2X þ a23s
2Y þ s2a1 þ X2s2a2 þ Y2s2a3 þ 2a2a3sXY þ 2a2sXa1 þ 2a2XsXa2 þ 2a2YsXa3
þ 2a3sYa1 þ 2a3XsYa2 þ 2a3YsYa3 þ 2Xsa1a2 þ 2Ysa1a3 þ 2XYsa2a3
s2y ¼ a25s2X þ a26s
2Y þ s2a4 þ X2s2a5 þ Y2s2a6 þ 2a5a6sXY þ 2a5sXa4 þ 2a5XsXa5 þ 2a5YsXa6
þ 2a6sYa4 þ 2a6XsYa5 þ 2a6Y � sYa6 þ 2Xsa4a5 þ 2Ysa4a6 þ 2XYsa5a6
sxy ¼ a2a5s2X þ a3a6s2Y þ a2a6 þ a3a5ð ÞsXY þ a5sXa1 þ a5XsXa2 þ a5YsXa3 þ a2sXa4þ a2XsXa5 þ a2YsXa6 þ a6sYa1 þ a6XsYa2 þ a6YsYa3 þ a3sYa4 þ a3XsYa5 þ a3YsYa6þ sa1a4 þ Xsa1a5 þ Ysa1a6 þ Xsa2a4 þ X2sa2a5 þ XYsa2a6 þ Ysa3a4 þ X � Y � sa3a5þ Y2 � sa3a6 ð14Þ
These equations use, apart from the coordinates (X, Y) and the coefficients ai, whichare data either known or derived from the procedure, also their uncertainties, which areeither known from the previous phase, or are derived from this phase.
Given the principle of the verticality of the components of the digitization error vector,the correlation of sX and sY is a unit (- 1-) because its direction is given sr (vertical to thecontour line) (Keefer et al. 1988).
Therefore:
rXY ¼ 1 ¼ sXYsXsY
) sXY ¼ sXsY
sXai, sYai do not exist because the coordinates X, Y of the points in the previous depictionsystem for transformation existed prior to the adjustment of ai to the transformation
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model. Finally, under these circumstances and conditions, equations (14) may besimplified in the form:
s2x ¼ a22s2X þ a23s
2Y þ s2a1 þ X2s2a2 þ Y2s2a3 þ 2Xsa1a2 þ 2Ysa1a32XYsa2a3
s2y ¼ a25s2X þ a26s
2Y þ s2a4 þ X2s2a5 þ Y2s2a6 þ 2Xsa4a5 þ 2Ysa4a6 þ 2XYsa5a6
sxy ¼ a2a5s2X þ a3a6s2Y þ a2a6 þ a3a5ð Þ � sXY þ sa1a4 þ Xsa1a5 þ Ysa1a6 þ Xsa2a4þ X2sa2a5 þ XYsa2a6 þ Ysa3a4 þ XYsa3a5 þ Y2sa3a6 ð15Þ
Therefore, now all necessary information exists in order to apply the creationprocedure for a DEM and, at the same time, to create another file that includes theuncertainty of this DEM, which will be spatially distributed in the form of an elevationerror.
With regard to the interpolation of the transformed contour lines, the IDW method isused, as already stated. The selection of this particular function was based on itssimplicity, on its broad use and the facility in programming it in a computer. Otherinterpolation methods can also be addressed in a similar way and approach.
4. Investigating elevation errors due to interpolation
4.1 Investigation conditions
The errors and their propagation through the phases of creating a DEM are examinedfurther on, in order to see their effect on the final result.
There was the question of whether we will assume a unified digitization error value or ifwe will use the digitization errors of a particular file, provided that they exist. Given that toorder to estimate the elevation errors in a DEM by following the described procedure, thedigitization errors will not be known and the assumption of the unified value (sc) willnecessarily be followed, we investigate both cases in order to make a comparison and seehow well the assumption of a unified value (sc) approaches to a satisfactory degree thepossibility to have and use the real digitization errors.
In the investigation we use a file of the digitized contour lines, which is subject to theprocedure until the production of the DEM. This file is accompanied by a file with therespective digitization errors, as they are already defined. The digitization errors present asatisfactory average image with regard to their distribution, something that allows them tobe used as the initial research data.
Knowing the digitization errors in the file, the digitized contour lines can be affected byfollowing the procedure below. With a given digital file with the contour lines (original) ananalogue topographic map of 1:10 000 scale is prepared and printed on an ultra highresolution printer. Further on, this map is digitized by a research partner, who knows theprinciples and the specifications of the digitization procedure. The file created (copy) iscompared to the original file and the digitization errors are calculated. The person of thedigitization selects at the same time eight reference points that are used in the phase of thetransformation of the coordinates (change of the reference system) (Achilleos 2002).
The statistical elements of the distribution that present digitization errors in the file ofthe contour lines are: mean ¼ 2.6 m, sd ¼ 2.0 m, min ¼ 0 m, max ¼ 13.6 m.
In order to determine the unified value of the horizontal error sr (¼sc), which will bethe initial entry error in the transformation procedure, we use the following method
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described. We assume that the initial error of the analogue map is constant and equal tothe distinctive capacity (*0.25 mm) in the particular scale (1:10 000), namely sK ¼ 2.5 m.The error is analysed in two components in the X (sKX) and Y (sKY) axes, by assuming alsothat these two components are equal. Therefore:
sKX ¼ sKY ¼ffiffiffiffiffiffis2K2
r¼ �1:77m
Further on, the digitization error of the contour lines depends on: (a) the error made bythe user during the digitization (sd); and (b) the error made due to the resolution of thedigitizer (sm) (Warner and Carson 1991)). These errors are determined at this particularscale to be: sd ¼ +2.5 m and sm ¼ +1.0 m. Namely, the final error sc only due to thedigitization is:
sc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2d þ s2m
q¼ �2:69m
Therefore, from the error propagation theory the final errors sX and sY are derived, whichwill be considered initially for the transformation procedure:
sX ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2KX þ s2C
q¼ �3:22m
sY ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2KC þ s2C
q¼ �3:22m
And the resultant horizontal position error sr is: sr ¼ SQR ðs2X þ s2YÞ ¼ �4:55m.With regard to the transformation of the coordinates for changing from the digitizer’s
reference system to the reference system of the final DEM, the common points are used asrequired in both reference systems.
The file with the used digitized contour lines, is accompanied by a satisfactory numberof reference points (eight), so that it is possible to apply the coordinate transformation. Inthe DEM created by the interpolation, the elevation errors are determined by applying theerror propagation law to the interpolation function. The size of the pixel used is of nointerest in this research, because elevation errors are controlled only at positions (X,Y)where the interpolation is effected, and can be anywhere. With regard to positions (X,Y)two cases are followed in the investigation:
(a) the interpolation points (X,Y) are the peaks of a raster, which correspond in thiscase to the size of a pixel 100 m
(b) the interpolation points (X,Y) are points of the contour lines of the initial originaldigital map that was used in this research.
In the first case, where the selection of the interpolation positions (X,Y) is made inraster peaks, we can compare the elevation errors of the DEM, which are derived by usingfirst the unified value of the digitization error and at the second time the real digitizationerrors of the file with the digitized contour lines.
The second case of the interpolation positions (X,Y) at the points of the contour lines,allows for the immediate determination of elevation errors that are created by theinterpolation. In this investigation, the initial original digital map is considered torepresent reality, it does not have a scale error neither a digitization error, because thismap was sent for digitization in an analogue form.
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4.2 Results of the first investigation: assumption of a unified digitization error value
In this investigation we assume that a constant digitization error is determined and isapplied throughout the range of the map being digitized. Their parameters anduncertainties, with regard to the transformation of the coordinates of the digitized mapfrom the digitizer’s reference system to the reference system of the original analogue map,are presented in Table 1. The horizontal errors prior to and after the transformation arepresented in Table 2.
In order to apply the error propagation equations through the interpolation procedure,we must know the covariances sxixj, sxiyj, syixj, syiyj. As already mentioned, these arecalculated by using the auto-correlations and cross-correlations of the horizontal errors inthe X and Y axes.
Table 1. Parameters and accuracy of the parameters of the coordinate transformation procedure.
a1 0.00040 sa1 1.98041a2 0.99090 sa2 0.00197a3 0.04750 sa3 0.00266a4 0.00010 sa4 1.98041a5 70.04940 sa5 0.00197a6 1.00550 sa6 0.00266sapost +3.96020 RMS 5.60143
RMS, root mean square.
Table 2. Horizontal errors due to digitization (prior to the transformation) and due totransformation (after).
Investigations
Case 1 Cases 2a, 3 Case 2b
DigitizationxMean 2.9024 3.0763 1.6571SD 0.7431 1.2741 1.1271Min 0.0000 0.0000 0.0000Max 4.5500 8.5479 2.6000
yMean 2.9120 2.9276 1.6628SD 0.8211 1.3276 1.2268Min 0.0000 0.0000 0.0000Max 4.5500 9.3265 2.6000
TransformationxMean 3.7054 3.5564 2.6429SD 1.2312 1.4011 1.4976Min 1.9922 2.7402 0.8983Max 4.9291 9.2282 8.5987
yMean 3.5636 3.3470 2.4129SD 1.1407 1.6574 1.7719Min 1.9804 2.3954 0.5401Max 4.9853 10.8723 10.5134
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Table 3 in column rk_x presents the auto-correlations of the horizontal errors in X (10lags) and in column rk_y the auto-correlations of the horizontal errors in Y (10 lags). Thistable (columns ck_x, ck_y) presents the covariances of the horizontal errors sxixj and syiyj.
The cross-correlations of the horizontal errors in X and Y (10 lags) are given inTable 4, columns rk_xy, rk_yx and the covariances sxiyj, syixj in Tables 4 and 5 in columnsck_xy and ck_yx, respectively.
When the DEM is created and the elevation errors are calculated, the basic statisticalcharacteristics of the total errors are calculated (Table 6).
The elevation errors estimated by this procedure are mapped in the form of circles, asdepicted in Figure 3.
Elevation errors are classified per elevation category (by 10 m). This classification ispresented in Table 7. On the same table are the elevation errors that are derived fromapplying the exact same procedure, this time by considering an initial horizontal error of10 m (digitization error at the map’s scale equal to 1 mm). It was considered necessary tomake this investigation at this point, in order to see what would happen in more adversecases of horizontal error from the digitization procedure.
The correlation coefficient of the two totals of the elevation errors of Table 7 isrxy ¼ 0.62. This means that the remaining 38% of the correlation index is due to the initialdigitization error, which was considered to be different in the two cases. In other words,the elevation errors of the DEMs depend significantly on the digitization errors.
Table 3. Covariances of the horizontal errors x and y from the 1st investigation.
rk_x rk_y sx sy sx2 sy2 sqr(sx2*sy2) ck_x ck_y
0.767 0.774 3.7 3.56 13.69 12.6736 13.172 10.50023 9.8093660.481 0.482 3.7 3.56 13.69 12.6736 13.172 6.58489 6.1086750.292 0.287 3.7 3.56 13.69 12.6736 13.172 3.99748 3.6373230.149 0.143 3.7 3.56 13.69 12.6736 13.172 2.03981 1.8123250.054 0.052 3.7 3.56 13.69 12.6736 13.172 0.73926 0.65902770.001 0.002 3.7 3.56 13.69 12.6736 13.172 70.01369 0.02534770.016 0.013 3.7 3.56 13.69 12.6736 13.172 70.21904 0.16475770.023 70.018 3.7 3.56 13.69 12.6736 13.172 70.31487 70.22812570.02 70.019 3.7 3.56 13.69 12.6736 13.172 70.2738 70.24079870.021 70.018 3.7 3.56 13.69 12.6736 13.172 70.28749 70.228125
Table 4. Covariances of the horizontal errors of the coordinates x and y (sxiyj) from the 1stinvestigation.
rk_xy rk_yx sx sy sx2 sy2 sqr(sx2*sy2) ck_xy
70.762 70.757 3.7 3.56 13.69 12.6736 13.172 710.0370670.486 70.47 3.7 3.56 13.69 12.6736 13.172 76.40159270.299 70.278 3.7 3.56 13.69 12.6736 13.172 73.93842870.157 70.133 3.7 3.56 13.69 12.6736 13.172 72.06800470.064 70.042 3.7 3.56 13.69 12.6736 13.172 70.84300870.006 0.006 3.7 3.56 13.69 12.6736 13.172 70.0790320.014 0.017 3.7 3.56 13.69 12.6736 13.172 0.1844080.023 0.02 3.7 3.56 13.69 12.6736 13.172 0.3029560.023 0.017 3.7 3.56 13.69 12.6736 13.172 0.3029560.023 0.017 3.7 3.56 13.69 12.6736 13.172 0.302956
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Table 5. Covariances of the horizontal errors of the coordinates y and x (syixj) from the 1stinvestigation.
rk_xy rk_yx sx sy sx2 sy2 sqr(sx2*sy2) ck_yx
70.762 70.757 3.7 3.56 13.69 12.6736 13.172 79.97120470.486 70.47 3.7 3.56 13.69 12.6736 13.172 76.1908470.299 70.278 3.7 3.56 13.69 12.6736 13.172 73.66181670.157 70.133 3.7 3.56 13.69 12.6736 13.172 71.75187670.064 70.042 3.7 3.56 13.69 12.6736 13.172 70.55322470.006 0.006 3.7 3.56 13.69 12.6736 13.172 0.0790320.014 0.017 3.7 3.56 13.69 12.6736 13.172 0.2239240.023 0.02 3.7 3.56 13.69 12.6736 13.172 0.263440.023 0.017 3.7 3.56 13.69 12.6736 13.172 0.2239240.023 0.017 3.7 3.56 13.69 12.6736 13.172 0.223924
Table 6. Statistical elements of elevation error in the created DEM.
Investigation
1 2a 2b 3
ElevationMean 0.4473 0.5224 0.3490 0.7628SD 0.5047 0.8677 0.6741 0.6994Min 0.0252 0.0272 0.0006 0.0011Max 3.1549 5.3033 4.9327 3.7700
Figure 3. Elevation errors from the first investigation (the size of the elevation error is representedby the size of the circle – the scale of the contour lines is not identical to the scale of the elevationerrors).
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4.3 Results of the second investigation: use of real digitization errors
In this investigation the same interpolation procedure is applied, with exactly the samedata, as in the first investigation, with the difference that the digitization errors are nolonger constant and uniform throughout the map, but they are real errors that areincluded in the file with the digitized contour lines and are differentiated in the area ofinterest.
The elements of the model of the transformation of the coordinates of the digitizedmap from the digitizer’s reference system to the reference system of the original digitalmap have been already presented in Table 1, in the first investigation. The horizontalerrors prior to and after the transformation are shown in Table 2.
Table 8, in columns rk_x, rk_y, presents the auto-correlations of the horizontal errorsin the X and Y axes, (10 lags). The same table in columns ck_x, ck_y, presents thecovariances of the horizontal errors X and Y.
The cross-correlations of the horizontal errors in the X and Y axes, for various delays,are shown in Tables 9 and 10, along with the covariances sxiyj, syixj. When the DEM iscreated and the elevation errors are calculated, the basic statistical characteristics of allthese errors are calculated (Table 6). The elevation errors that are estimated by thisprocedure are mapped in the form of circles, as depicted in Figure 4.
Table 7. Elements of the elevation error in connection with the altitude (categorization by 10 m ofelevation zone) for initial digitization error 4.55 m and 10 m.
Elevationzone
Mean elevationerrors for
initial digitizationerror 4.55 m
Mean elevationerrors for initialdigitization error
10.00 m
10 1.6787 1.394520 0.7110 1.450430 0.3457 0.555840 0.2871 0.290550 0.3859 0.515060 0.5910 1.228970 1.1186 4.341280 1.2568 4.9298
Table 8. Covariances of the horizontal errors x and y (sxixj, sxixj) from the 2nd investigation.
rk_x rk_y sx sy sx2 sy2 sqr(sx2*sy2) ck_x ck_y
0.567 0.58 3.55 3.34 12.6025 11.1556 11.857 7.145618 6.4702480.295 0.284 3.55 3.34 12.6025 11.1556 11.857 3.717738 3.168190.157 0.168 3.55 3.34 12.6025 11.1556 11.857 1.978593 1.8741410.09 0.148 3.55 3.34 12.6025 11.1556 11.857 1.134225 1.6510290.08 0.127 3.55 3.34 12.6025 11.1556 11.857 1.0082 1.4167610.038 0.053 3.55 3.34 12.6025 11.1556 11.857 0.478895 0.5912470.02 0.04 3.55 3.34 12.6025 11.1556 11.857 0.25205 0.4462240.004 0.023 3.55 3.34 12.6025 11.1556 11.857 0.05041 0.25657970.001 0.002 3.55 3.34 12.6025 11.1556 11.857 70.012603 0.0223110.012 70.019 3.55 3.34 12.6025 11.1556 11.857 0.15123 70.211956
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4.4 Comparing the results of the first and second investigations
When comparing the elements of Table 3, which presents the descriptive statisticalelements of the distributions of the elevation errors of the two investigations, it is obviousthat the two distributions are alike. A statistical control takes place with regard to theequality of the two mean values, of investigation 1 and investigation 2, in order to identifywhether these two distributions are finally identical (Hammond and McCullagh 1974,Kennedy and Neville 1976). After the control, it is concluded that the mean values of thedistributions of the two investigations of the mean elevation error are not equal.
The correlation coefficient among the elevation errors of the two investigations is:rxy ¼ 0.584, which shows that there is a significant similarity and correlation between thetwo distributions. In brief, the assumption of the unified value of the horizontaldigitization error, as determined, is a good enough estimation of the reality and can beapplied when the real digitization errors are not known. At this point it should bementioned that, when the real digitization errors are not known, simulation models andapproaches of the digitization errors can be used, which were developed in theinternational literature for this purpose. These models satisfactorily simulate thephenomenon and give better results than the assumption of the unified error value(Keefer et al. 1988, Marble et al. 1990, Achilleos 2002).
Table 9. Covariances of the horizontal errors of the coordinates y to x (sxiyj) from the 2ndinvestigation.
rk_xy rk_yx sx sy sx2 sy2 sqr(sx2*sy2) ck_xy
0.174 0.127 3.55 3.34 12.6025 11.1556 11.857 2.0631180.127 0.046 3.55 3.34 12.6025 11.1556 11.857 1.5058390.108 0.011 3.55 3.34 12.6025 11.1556 11.857 1.2805560.117 0.015 3.55 3.34 12.6025 11.1556 11.857 1.3872690.098 0.01 3.55 3.34 12.6025 11.1556 11.857 1.1619860.078 70.03 3.55 3.34 12.6025 11.1556 11.857 0.9248460.059 70.053 3.55 3.34 12.6025 11.1556 11.857 0.6995630.018 70.036 3.55 3.34 12.6025 11.1556 11.857 0.2134260.034 70.039 3.55 3.34 12.6025 11.1556 11.857 0.4031380.04 70.054 3.55 3.34 12.6025 11.1556 11.857 0.47428
Table 10. Covariances of the horizontal errors of the coordinates y to x (syixj) from the 2ndinvestigation.
rk_xy rk_yx sx sy sx2 sy2 sqr(sx2*sy2) ck_yx
0.174 0.127 3.55 3.34 12.6025 11.1556 11.857 1.5058390.127 0.046 3.55 3.34 12.6025 11.1556 11.857 0.5454220.108 0.011 3.55 3.34 12.6025 11.1556 11.857 0.1304270.117 0.015 3.55 3.34 12.6025 11.1556 11.857 0.1778550.098 0.01 3.55 3.34 12.6025 11.1556 11.857 0.118570.078 70.03 3.55 3.34 12.6025 11.1556 11.857 70.355710.059 70.053 3.55 3.34 12.6025 11.1556 11.857 70.6284210.018 70.036 3.55 3.34 12.6025 11.1556 11.857 70.4268520.034 70.039 3.55 3.34 12.6025 11.1556 11.857 70.4624230.04 70.054 3.55 3.34 12.6025 11.1556 11.857 70.640278
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4.5 Results of the third investigation: interpolation in points of contour lines
In the third investigation an interpolation is effected in the digital data of the file of thecontour lines, at the particular points (X,Y) for which it is known from the original digitalmap are the points of the real and theoretically correct contour lines (in this investigationit is not considered that there is a map scale error). Therefore, on the basis of the digitaldata the heights (hp) at positions (X,Y) are determined by the interpolation procedure.(Following this, they are compared one-to-one to the real heights of the contour lines ofthe original digital map (h), at the same positions. The deviations observed (jhp – hj) are thereal elevation errors (sh) of the file of the contour lines that is derived from the proceduresof digitization, transformation and interpolation. The results from this investigation arepresented in Table 6 and in Figure 5 as circles.
In Figure 5 it is first observed that there are elevation errors in the areas of the mapwhere the curvature of the contour lines is intense, something that verifies the high degreeof dependency of these errors on the digitization errors. This result forms the fact thatdigitization errors present in their majority the same attitude, as indicated in the literature(Keefer et al. 1988, Achilleos 2002).
It is not possible to have a direct comparison of the results of this investigation with theresults from the previous two investigations, because the conditions under which theinterpolation was effected are differentiated. However, a relative comparison can be made.In Table 6 we may observe that the third investigation, which is a direct comparison toreality, gives the higher average of elevation errors, although these errors are free from themap scale error, which in the other two investigations was included and taken intoconsideration. Therefore, it seems that if the initial digital map in the third investigationwas considered to have a scale error, then the elevation errors presented in Table 6(Investigation 3) would have been bigger.
For the appearance of the image in this case, and taking into account that theconditions of the third investigation cannot be changed, because in an opposite case the
Figure 4. Elevation errors from the second investigation (the size of the elevation error isrepresented by the size of the circle – the scale of the contour lines is not identical to the scale of theelevation errors).
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advantage we have from knowing reality to which the comparison is made, would vanish,the second investigation was repeated; this time it was considered that there was no mapscale error. In this way, the two cases become partially comparable, although theinterpolation conditions are differentiated again among them, since in the secondinvestigation the interpolation was effected on the tops of a given raster, while in the thirdcase, the interpolation take place at specific points (X,Y) of the contour lines.
The results from repeating the second investigation, without taking the map scale errorinto account, are presented in Tables 2 and 6.
Table 11 shows the covariances of the horizontal errors X and Y, as well as the auto-correlations of the horizontal errors in the x and y axes. The cross-correlations of thehorizontal errors in X and Y, and the covariances sxiyj, syixj are presented in Tables 12and 13.
Table 11. Covariances of the horizontal errors x and y (sxixj, sxixj) in the repetition of the 2ndinvestigation without the use of the map scale error.
rk_x rk_y sx sy sx2 sy2 sqr(sx2*sy2) ck_x ck_y
0.481 0.58 2.6429 2.4129 6.98492 5.82208641 6.37705341 3.359747 3.376810.241 0.284 2.6429 2.4129 6.98492 5.82208641 6.37705341 1.683366 1.6534730.159 0.168 2.6429 2.4129 6.98492 5.82208641 6.37705341 1.110602 0.9781110.102 0.148 2.6429 2.4129 6.98492 5.82208641 6.37705341 0.712462 0.8616690.084 0.127 2.6429 2.4129 6.98492 5.82208641 6.37705341 0.586733 0.7394050.019 0.053 2.6429 2.4129 6.98492 5.82208641 6.37705341 0.132713 0.3085710.033 0.04 2.6429 2.4129 6.98492 5.82208641 6.37705341 0.230502 0.2328830.032 0.023 2.6429 2.4129 6.98492 5.82208641 6.37705341 0.223517 0.1339080.024 0.002 2.6429 2.4129 6.98492 5.82208641 6.37705341 0.167638 0.01164470.045 70.019 2.6429 2.4129 6.98492 5.82208641 6.37705341 70.314321 70.11062
Figure 5. Elevation errors of the third investigation (the size of the elevation error is represented bythe size of the circle – the scale of the contour lines is not identical to the scale of the elevationerrors).
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Figure 6 presents the elevation errors in the form of circles at the interpolation points.When comparing the elements of Table 6, it is observed that the reduction in the averageof elevation errors, when the horizontal error is not taken into account due to map scale, isof 33% in relation to the case in which the scale error is taken into account (or inverselyincreased by 50%). At the same time, the scattering of the elevation errors is reduced.When comparing the elements of Table 2 it is observed that the transformation increasesthe horizontal errors that exist until the digitization procedure, by 30% on average, whileat the same time the scattering of errors is also increased. This effect is of course importantbut lower in relation to the effect of the digitization in the horizontal errors. Finally, inTable 6 it is worth mentioning that even in the case in which the map scale error is nottaken into account, the elevation errors of the DEMs that are created due to the remainingprocedure are significant.
Between the two distributions shown on Table 6a deviation is observed; this deviationmay be due to the assumptions of the horizontal errors that took part in the equations, oreven to the different interpolation conditions between the two investigations, which do notmake them directly comparable. In any case the procedure proposed seems to be a goodestimation of reality.
Table 12. Covariances of the horizontal errors of the coordinates y to x (sxiyj) in the repetition ofthe 2nd investigation without the use of the map scale error.
rk_xy rk_yx sx sy sx2 sy2 sqr(sx2*sy2) ck_xy
0.152 0.127 2.6429 2.4129 6.98492 5.822086 6.37705341 0.9693120.119 0.046 2.6429 2.4129 6.98492 5.822086 6.37705341 0.7588690.114 0.011 2.6429 2.4129 6.98492 5.822086 6.37705341 0.7269840.121 0.015 2.6429 2.4129 6.98492 5.822086 6.37705341 0.7716230.115 0.01 2.6429 2.4129 6.98492 5.822086 6.37705341 0.7333610.093 70.03 2.6429 2.4129 6.98492 5.822086 6.37705341 0.5930660.055 70.053 2.6429 2.4129 6.98492 5.822086 6.37705341 0.3507380.008 70.036 2.6429 2.4129 6.98492 5.822086 6.37705341 0.0510160.014 70.039 2.6429 2.4129 6.98492 5.822086 6.37705341 0.0892790.017 70.054 2.6429 2.4129 6.98492 5.822086 6.37705341 0.10841
Table 13. Covariances of the horizontal errors of the coordinates y to x (syixj) in the repetition ofthe 2nd investigation without the use of the map scale error.
rk_xy rk_yx sx sy sx2 sy2 sqr(sx2*sy2) ck_yx
0.152 0.127 2.6429 2.4129 6.98492 5.822086 6.37705341 0.8098860.119 0.046 2.6429 2.4129 6.98492 5.822086 6.37705341 0.2933440.114 0.011 2.6429 2.4129 6.98492 5.822086 6.37705341 0.0701480.121 0.015 2.6429 2.4129 6.98492 5.822086 6.37705341 0.0956560.115 0.01 2.6429 2.4129 6.98492 5.822086 6.37705341 0.0637710.093 70.03 2.6429 2.4129 6.98492 5.822086 6.37705341 70.1913120.055 70.053 2.6429 2.4129 6.98492 5.822086 6.37705341 70.3379840.008 70.036 2.6429 2.4129 6.98492 5.822086 6.37705341 70.2295740.014 70.039 2.6429 2.4129 6.98492 5.822086 6.37705341 70.2487050.017 70.054 2.6429 2.4129 6.98492 5.822086 6.37705341 70.344361
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5. Comments, observations and conclusions
5.1 General conclusions
Following is a summary of the observations from this research:
. The horizontal errors that are created because of the digitization are very importantand, in size, can reach or even surpass the horizontal errors due to the scale error ofthe map being digitized.
. The transformation of the coordinates results in horizontal position errors, whichare much smaller than those created due to the digitization.
. The high values of the elevation errors appear in the map’s areas, which present asignificant curvature and complexity in the contour lines and high data density.
. The map scale error, which is used in the digitization, is a significant source ofelevation errors in DEMs. The impact of this factor on the quality of the final resultis estimated by research to be within 33–50%. Even if we assumed that this factordoes not exist, the elevation errors of DEMs would still be significant due to thedigitization and the remaining procedure of their creation.
. The elevation errors of the DEMs can be approached adequately by applying theprocedure presented in this paper.
5.2 Comments and observations
The procedure that is investigated and presented in this paper can approach the elevationerrors of DEMs, which derive in the phases of the creation of these DEMs; this is done inan objective way and by using the initial errors that exist in the data and the function onthe basis of which these errors are propagated.
The DEMs that are created and used many times, having as initial data the contourlines of an analogue topographic map, may still be in the market and used in applications.
Figure 6. Elevation errors from the second investigation without the use of the map scale error (thesize of the elevation error is represented by the size of the circle – the scale of the contour lines is notidentical to the scale of the elevation errors).
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Therefore, it would be a significant step to estimate the elevation errors included thereinand make them known to the users of these DEMs.
The approach that was made in this paper may be adapted in a similar way to othercoordinate transformation models and other interpolation models. Moreover, simulationmodels of the digitization error may be useful in this case, given that their ability toapproach these errors satisfactorily is proven.
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Burrough, P.A., 1986. Principles of geographical information systems for land resources assessment.Oxford: Oxford University Press.
Canters, F., 1994. Simulating error in triangulated irregular network models. EGIS, Available from:http://www.odyssey.maine.edu/gisweb/spatdb/egis/eg94019.html
Carpenter, J.D., 1991. Creating digital terrain models: procedure and problems. L. A. ComputerNews, ASLA Open Committee on Computers, 4 (1).
Carter, J.R., 1988. Relative errors identified in USGS gridded models. In: AUTOCARTO 9.Proceedings of Ninth International Symposium on Computer-Assisted Cartography. Baltimore,Maryland, 2–7 April, ASPRS/ACSM, 255–265.
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