Improved Modeling of Elevation Error with Geostatistics

GeoInformatica 2:3, 215±233 (1998)

# 1998 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

Improved Modeling of Elevation Errorwith Geostatistics

PETER FISHER

Department of Geography, University of Leicester, Leicester, LE1 7RH, United [email protected]

Received January 14, 1998; Revised and Accepted June 3, 1998

Abstract

The elevations recorded within digital models are known to be fraught with errors of sampling, measurement and

interpolation. Reporting of these errors according to spatial data standards makes several implicit and

unacceptable assumptions about the error: it has no spatial distribution, and it is statistically stationary across a

region, or even a nation. The approach explored in this paper employs actual elevations measured in ground and

aerial survey at higher precision than the elevations in the DEM and recorded on standard paper maps. These high

precision elevations are digitized and used to establish the real statistical and spatial distribution of the error.

Direct measurements could also have been taken in the ®eld by GPS or any other means of high precision data

collection. These high precision elevations are subtracted from values stored in the DEM for approximately the

same locations. The distribution of errors speci®c to the DEM can then be explored, and can be used in the

geostatistical method of conditional stochastic simulation to derive alternative realizations of the error modeled

and so of the DEM. Multiple versions of the derived products can also be determined. This paper compares the

results of using different methods of error modeling. The best method, which gives widely implementable and

defensible results, is that based on conditional stochastic simulation.

Keywords: Digital Elevation Models, error modeling, geostatistics, conditional stochastic simulation,

viewsheds

1. Introduction

Error modeling is a way of providing estimates of the consequences of the quality level in

data, and so enables informed judgements by users as to the suitability of the spatial data

for speci®c tasks. Error modeling has been applied to single data types, including raster-

based elevation models [6], [8], [22], [18], soil maps [7], [15], and vector shore lines [16],

and to multi-variable spatial models [25], [1]. The principal problem in error modeling,

however, is making the model re¯ect a realistic statistical and spatial distribution of the

error. A number of studies have shown that the rate of uncertainty in the derived spatial

information is dependent on the spatial autocorrelation of the error [6], [8], [11], [22], [18].

In this article an approach based on geostatistics is used to model the actual distribution of

errors in a DEM. The results of the error modeling on the derived spatial information

(viewsheds) are argued to better re¯ect the real distribution of the error than other

methods, and so to provide better uncertainty estimates.

This paper starts by presenting the need for error modeling, and then moves on to

examine the process of creating a DEM and the usual reporting of errors. Two alternative

error models which can be derived from these reports are then discussed and their

shortcomings outlined. The methods are applied to a study area in Scotland. An improved

estimation of error parameters is then discussed. Finally, a strategy for error modeling

using the revised estimation is presented.

2. Why error models?

Given that error occurs in spatial information (as in all other information), we know that

there is a risk that the outcome of any analysis of the information will be incorrect. It is

important to establish whether the chance of an incorrect outcome in any particular

situation is signi®cant for the application concerned. If it is, then the data may not be ®t-for-use in the particular context. Alternatively, it may be possible to generate analyses of

the outcome so as to predict the probability of the outcome being correct given the

database errors.

When some types of spatial data are combined within certain spatial operations, it may

be possible to predict the probability of the outcome being correct. Such predictions have,

for example, been determined for Union and Intersection overlay operations within GIS

where the data types input to the analysis are the same (e.g., two zonal coverages [21]).

For such predictive relationships between the data error and the probability of the accuracy

of the outcome, the data and the error have to be statistically well behaved, and the

analysis to proceed predictably within the GIS operation. This is only true of a few

operations. The so-called viewshed operation combines a point (the viewer or viewed

point) with a surface (the DEM) to yield a zone (the visible area), and so is not con®ned to

data of a single type. Furthermore, the data and the errors are not well understood, and the

GIS operation itself is not computationally predictable, but dependent on the data itself

(but see the recent paper by Huss and Pumar [19]).

Until such predictive equations are available for all operations, if we wish to establish

the effect of database error upon the results of an analysis, then we have to model the error

to yield alternative versions of the data, and so derive multiple versions of the outcome. In

this way, we can examine the effects of different values of error parameters, and different

sources of error on the probability of the analysis being incorrect [8], [18].

3. Database creation

Digital Elevation Models can be generated in a number of different ways [30]. The

principal methods all involve the direct measurement of elevations at speci®c locations

and the inference or interpolation of values at unvisited locations. In many grid-based

DEMs most values are inferred, because the elevations measured in the ®eld only rarely

coincide with grid-intersections unless the data were collected speci®cally for this type of

DEM either by direct ®eld measurement or by photogrammetry. Within gridded DEMs

elevation values can also be estimates of the average elevation (or some other summary

statistic) within the area of the grid-cell. National mapping agencies such as the US

216 FISHER

Geological Survey and the British Ordnance Survey (OS) have used a mixture of direct

photogrammetric measurement, and manual or automated digitizing of contours from

paper maps. In short, the detailed process by which the errors in a DEM are created

depends on the type of DEM and how it was created.

Whatever method is used the values recorded will incorporate an error (measurement

error, interpolation error, etc.). As in all digital spatial information, an error is the

deviation between a measurement which would be made on the ground at the exact

location speci®ed within the DEM and that recorded in the database. It has been widely

asserted that errors in a particular dataset can be estimated from related observations

(similar values) in an independent dataset of higher precision (larger scale [3]), and that is

the approach advocated here.

The British Ordnance Survey produces two different DEM products. For a number of

years a complete national coverage of the PanoramaTM DEM has been available with a

grid spacing of 50 m. The elevations were interpolated from contours digitized from the

national 1:10,000 map series. More recently the Ordnance Survey has also made available

the so-called Pro®leTM DEM which has a grid spacing of 10 m. 10 km tiles of this second

product are interpolated when ordered by a customer. The data in the two products have

been created in different operations, at different times, but from largely the same data. In

both datasets it is understood that the values are estimates of the actual elevation at the

intersection of the lattice. In this paper, the DEMs are referred to by their grid resolution,

although they are better known in the UK by their nominal scales or product names

(table 1).

Errors are reported for UK DEMs as a single value for the whole country. In the

documentation for OS 50 m DEM data it is stated that while the accuracy is not tested for

all DEM products, where it has been tested the Root Mean Squared Error (RMSE) is

between 2 and 3 m [26]. There has been no attempt (of which the author is aware) to

publish the results of those tests which apparently have been done, or to state their

geographic extent. Therefore a user must assume that the error rates are the same (2±3 m)

everywhere in the British Isles, from the highest mountain to the ¯attest coast.

In the US, various methods have been used for estimating the RMSE [4]. Most recently

it is supposed to be estimated by comparing the elevations at a minimum of 20 well located

sites in a survey of higher accuracy with the elevations recorded in the DEM [29].

The RMSE is the standard measure of error used by surveyors around the world. It is

based on the following formula:

Table 1. Error parameters derived from the comparison of ground survey locations and elevations recorded

within-grid square in DEMs.

DEM NameNominalScale

DEMResolution Mean Error

StandardDeviationof the Error

MaximumError

MinimumError

Panorama 1:50,000 50 m 2.1003 6.9565 28 ÿ 26

Pro®le 1:10,000 10 m 0.5089 2.6481 26 ÿ 11.5

IMPROVED MODELING OF ELEVATION ERROR 217

RMSE ��P�zÿ w�2

n

s�1�

where z is the elevation recorded in the DEM; w is the elevation measured at the higher

precision; and n is the number of locations tested. If we make the assumption that the

mean error is 0 (as is stated in the documentation of some accuracy speci®cations for

DEMs), then this formula is the same as the formula for the standard deviation of the

population, s:

s ��P��eÿ �zÿ w��2

n

s�2�

where �e is the mean of the errors.

3.1. Case study area

As part of a project on the visibility analysis of wind turbines in the British countryside,

contracted through the Macaulay Land Use Research Institute to the Countyside

Commission of Wales, DEMs at both 10 m and 50 m resolutions were made available

for a part of the Cairngorms in Scotland. The area was centred on an existing site of

multiple wind turbines (a windfarm; ®gure 1).

For the purposes of the present paper, a particular version of the area from which the

windfarm could be seen is examined (the reverse viewshed [11]). The windfarm itself

includes 48 separate wind-turbines, each 50 m high. For experimentation here, the total

area in the landscape where a person approximately 2 m high can see the mid-height of any

of the turbine masts, 25 m above the ground, is shown in ®gure 2. This is the area shown

without any perturbation of the DEM to account for error; it is the binary viewshed.

4. A simple error model

The simplest error model can be de®ned by drawing random values from a normal

distribution which has mean� 0 and standard deviation�RMSE. From these we can

generate a ®eld of white noise which is similar in spatial extent to the DEM with this mean

and this standard deviation. To the error ®eld is added the elevations as they are recorded

in the DEM [6], [8], [11], [18]. The resulting DEM has the essential properties of both the

original DEM and the error which is known to occur within it. Furthermore, because the

DEM is known to be in error by the amount reported, it can be argued that such a derived

DEM is actually more realistic than the original DEM. Naturally, any number of versions

of the error ®eld can be generated and so multiple realizations of the DEM can be

generated each incorporating a slightly different version of the error ®eld.

218 FISHER

If a product can be derived from the DEM, such as the visible area, slope, etc. then by

repeated derivation of that product over alternative DEMs with different error ®elds but

the same mean and standard deviation, a set of possible versions of the product can be

derived. By collecting appropriate statistics from these possible versions, we can derive a

probable version of the product [8]±[10], [18]. In the case of the visible area, which is

usually presented as a binary area (coded as 0 for out-of-view and 1 for in-view), by adding

the different visible areas derived from the same view point across alternative DEMs with

different error ®elds the probability of any location being visible can be estimated. This is

called the Probable Viewshed [9], [10].

The ®rst case of error modeling for generating the probable viewsheds used the reported

value of DEM error in OS data (2 m RMSE which translates to a standard deviation of the

error� 2 m and a mean error� 0; ®gure 3). We can see that the locations immediately

Figure 1. The study area in the Cairngorms of Scotland. The area covers 25� 25 km, and the black areas

indicate the positions of the existing wind turbines. Higher ground is indicated in white, and lower in black.

Published with permission from the Ordnance Survey, Crown Copyright.


around the turbines have large probabilities �p� 1� of being able to see the turbines, but

elsewhere the probability is rarely over 0.5. This probable viewshed which is based only

on the way error is reported (no more information is used) does not match our real-world

experience, because such low probabilities are inconceivable and would point to this being

a very poor data product for the purpose. This observation is similar to that made

elsewhere [8], [10].

The problem is that the error model is too simple because reporting only the RMSE is

not enough. Simple observation of the DEM shows that the surface is relatively smooth.

The land surface as it is experienced in the real world is also relatively smooth; sudden

changes at cliffs and the like are very unusual. Therefore, the difference between the

elevations in the DEM and the actual surface (which equals the error surface) should also

be relatively smooth [14]. A smooth surface has a large positive spatial autocorrelation;

this ®rst error modeling method does not incorporate this spatial autocorrelation since it

uses white noise.

There are a number of ways that positive spatial autocorrelation can be introduced into

the error model. The simplest method is as follows [6], [11], [12]:

1. Generate a ®eld of white noise (approximately zero spatial autocorrelation) with

mean� 0 and standard deviation�RMSE;

2. Determine current value of the spatial autocorrelation (measured, in this case, by

Moran's I [13];

3. Swap two values chosen at random;

Figure 2. The binary viewshed from the test viewing location.

220 FISHER

4. Recalculate the spatial autocorrelation;

5. If the spatial autocorrelation has increased but is not greater than the target then retain

the swap otherwise reverse it;

6. Repeat steps 3±5 until the desired spatial autocorrelation is reached.

This approach yields error ®elds with appropriate mean and standard deviation and with

positive spatial autocorrelation. The probable viewshed shown in ®gure 4 was derived

using this error model, with the spatial autocorrelation measured by Moran's I� 0.9 [13].

Much of the area which has any probability of being visible in ®gure 4 has a high

probability of being seen; most locations we can either see or not see and only over a small

total area is there much doubt. This seems to ®t our real-world experience of the visible

area much better than the simple error model shown in ®gure 2. It also better re¯ects the

occurrence of the error since positive spatial autocorrelation is included.

A number of problems with the model persist, however. First, there is an unjusti®ed

assumption that the mean error is zero [23], or unbiased. It is assumed that the RMSE over

the study area is constant or spatially stationary (indeed, in Britain it is assumed to be

constant nationally [26]). These are both unrealistic assumptions, and there is no possible

justi®cation for either, except simplicity. Finally, there is no information on the actual

Figure 3. The probable viewshed determined by adding together binary viewsheds through multiple

alternative realizations of the simplest error model plus the DEM. Standard deviation� 2 m for generation of the

noise ®eld, and the viewer was speci®ed to be as 2 m above the ground. The white areas have probability 1 and

the black have probability 0.


spatial autocorrelation; any arbitrary value of I could be used in the error model allowing

the generation of a smooth error ®eld.

5. Improving estimation of error parameters

To achieve an improved estimate of the error for any particular area, a set of measurements

made at a higher precision is required. In the United Kingdom, such measurements are

readily available. Spot heights from ground and aerial survey are available for the whole

country, and are shown on 10 m maps [24]. By digitizing these spot heights

it is possible to compare the values with the 50 m DEM values. These spot heights are

surveyed separately from the map information, being actual heights at a location, and so

are independent of the 1:10,000 contour maps on which both the 50 m and 10 m DEMs are

based since they are not used in the construction of those DEMs. The distribution of spot

heights for 100 square kilometres in the middle of the study area is shown in ®gure 5.

At every spot height location it is possible to subtract the actual values from the DEM

values to yield the error at the point. From this error information we can generate improved

location speci®c error statistics (table 1). The mean error (which could also be known as

the bias) records systematic error across the dataset. The standard deviation of the errors

(equivalent to the RMSE, but in this case the actual sample standard deviation without

Figure 4. The probable viewshed determined using error models based on stationary measures of spatial

autocorrelation (I� 0.9) and standard deviation (RMSE� 2).

222 FISHER

making any assumptions about the mean) shows the dispersion of the errors. The method

employed is approximate, and would be better on ¯atter ground where elevations within

a pixel varied less. Bilinear interpolation between the values at a spot height and the

surrounding DEM elevations would also have been possible, but it would not have been

possible with standard GIS functionality and without speci®c programming.

The 50 m DEM is seen to have a very real bias in the error (2.1 m), and a large standard

deviation (6.9 m). Furthermore, the histogram of the errors (®gure 6A) shows them to be

approximately normally distributed. These results are not necessarily representative of all

DEMs, but the mean and standard deviation are very different from the nationally reported

®gures of [26]. The errors found are therefore liable to be overestimates of the actual error

in the DEM. Using the same basic procedure, [24] found that the mean and standard

deviation of the errors in two areas elsewhere in Britain conformed more closely to the

values reported or implied by the OS (mean� 0 m, standard deviation� 2±3 m). The 10 m

Figure 5. Black dots indicate the locations of ground and aerial survey points in the vicinity of the test

viewpoint. Published with permission from the Ordnance Survey, Crown Copyright.


Figure 6. Histograms of the actual errors in A) the 50 m and B) the 10 m DEMs of the study area.

224 FISHER

DEM is more precise with a mean error of 0.5 m and a standard deviation of 2.6 m,

and again with a normal distribution (®gure 6B). This is a more precise representation

of the terrain, as might be hoped in a ®ner resolution, but again the results may be

overestimates.

For paper maps it has been established that the positional error of contours is related to

the slope of the ground, and this is embedded in the US National Map Accuracy Standards

[28]. Here, the point errors are compared with the slope of the land surface. For the 50 m

DEM there is a signi®cant correlation between the two (®gure 7A). On the other hand, the

errors in the 10 m DEM are not correlated with the slope (®gure 7B). There is a suggestion

here that the spot heights from the 10 m map may have been used in construction of the

10 m DEM, and so may not be suitable to test the accuracy of the DEM. Hence the methods

suggested here are applied only to the 50 m DEM.

In addition to statistical summaries, the actual errors at speci®c geographical points are

available, and can be mapped (®gure 8). It can be seen that similar values do cluster,

exhibiting some spatial autocorrelation. It is also possible to use geostatistics to examine

the spatial distribution of the errors [17]. The variogram of the errors in the 50 m DEM is

shown in ®gure 9A, and illustrates the variance between pairs of values at different

spacings, and most importantly, the increasing variance with spacing of data pairs. This is

observable in the spatial distribution of the error in the 50 m data. To this empirical

variogram it is possible to ®t a theoretical variogram. A number of methods can be used,

but here the VARIOWIN [27] visual interface for variogram curve ®tting was used and the

best ®t model variogram found is shown in ®gure 9B. The intercept of the variogram curve

and the y axis of the graph is known as the nugget variance, and expressed the residual

variance in the observations. In the theoretical variogram (®gure 9B) this value is forced to

zero because of the assumption of smoothness of the error ®eld. This variogram records

the piecewise inclusion of three different spherical structures [27].

For the 10 m DEM the variogram is rather different, showing as it does no spatial

autocorrelation at the spacing of the points (there is no increase of variance with spacingÐ

®gure 10). It is assumed, however, that since the observations stored in the DEM are

denser than in the 50 m product (10 m cell size as opposed to 50 m) there is still positive

spatial autocorrelation between neighboring grid intersections, and the nugget should be

zero in any theoretical variogram ®tted.

6. Using the improved estimates

For the 50 m DEM a locally parameterized description of the error has been established in

the foregoing section including local estimates of the mean, standard deviation and spatial

autocorrelation. The description of the 10 m DEM is not so clear. While it is possible to

derive estimates for this DEM too, there remains some doubt as to whether the spot heights

are used in generating the DEM values. If the spot heights are indeed used for the creation

of the DEM then they are not an appropriate set of data with which to compare the

accuracy of that DEM, although they would show something of the process induced error.

To work further with the 10 m DEM a high precision ®eld survey would be necessary


Figure 7. Scatterplots of the amount of error and slope within the grid cells for A) the 50 m and B) the 10 m

DEMs.

226 FISHER

involving collection of 500±1,000 spot heights. This has not been done, and so further

analysis here focuses on the 50 m DEM only.

A number of different improvements in the error models presented above would be

possible from the parameters collected. First, the actual mean and standard deviations of

the error can be used in generating the white noise ®elds. Second, the locations of known

error can be included as ®xed points in the error ®elds. Third, it is possible to simulate error

with spatial autocorrelation matching the autocorrelation observed in the variogram.

Finally, it is possible to include all three improvements in one model and that is the

approach pursued here.

Conditional stochastic simulation, a process in the geostatistical toolkit, requires as

Figure 8. Error amounts at survey points in the study area. Symbols are proportional to the size of the error.

Grey circles are approximately zero error, white indicates positive error and black negative. Notice that the area

shown here is only a portion of the area of the total DEM available in the study. Published with permission from

the Ordnance Survey, Crown Copyright.


input a model variogram, such as that shown above. It is discussed at length in a number

of articles and texts [2], [5], [17], [20]. Alternatively, the method outlined above for

generating spatially autocorrelated error ®elds can be used, and the error at locations of

spot height observations never changed in the simulation process (conditioning of the

simulations by the known errors).

The probable viewshed when the error surfaces are derived by conditional simulation is

shown in ®gure 11. We can see that the distribution of probabilities is between those shown

in ®gures 2 and 4. A large area has a high probability of being visible, and a reasonably

large area has a smaller probability. This can be seen as an acceptable compromise

Figure 9. A) the empirical variogram and B) the best-®t model of the theoretical variogram for the actual error

values in the 50 m DEM.

228 FISHER

between the apparently over-pessimistic estimation of the probabilities of being visible in

the simple error model and the equally apparent over-estimation of large probabilities

when only the spatial autocorrelation is used. It should be emphasised that however the

Figure 11. The probable viewshed derived using the most complex error model which holds errors at known

locations (shown in ®gures 5 and 6) ®xed, and also builds in the degree of spatial autocorrelation observed in

that distribution.

Figure 10. The empirical variogram of the distribution of actual errors in the 10 m DEM.


new model compared with the other estimates (whether this version of the probable

viewshed had the smallest or largest area of large probabilities of being seen) is immaterial

to the decision that this is the best estimate. The model uses as much information as

possible about the statistical and spatial distribution of the local error, and so must be

better than any method relying on global estimates of aspatial parameters. The paramaters

for the model proposed here can be extracted for any sub-area of the DEM. The data in this

case are readily available from published paper maps, but could also be collected by direct

®eld survey. There is a considerable increase in the computation time over other methods,

but the improved error model is considered to be well worthwhile (®gure 11).

7. Error modeling and data quality

The method suggested here indicates an increasing gulf between what is acceptable in data

quality statements and what is necessary for error modeling. The data quality statement for

many DEMs requires no more than the RMSE, and is accommodated by the simplest error

model outlined here (®gure 3). The fact that it is the same for all OS-produced DEMs of

Britain could be seen as a triumph for simplicity. In terms of error modeling, the data

quality statement, being as it is aspatial and stationary over large areas, is nearly useless. It

fails to encapsulate what we can infer about the distribution of the error (large positive

spatial autocorrelation), and the inevitability of non-stationarity. The most complete (and

the only really justi®able) error model outlined, which is sensitive to all these faults,

requires development of a full empirical variogram (®gures 9 and 11), which will vary for

any particular area.

Data providers should therefore move towards delivering high precision spot heights as

a part of their DEM products. Those concerned with the consequences of error can use

those in a number of ways, from calculating local error values of RMSE and bias, to

determining the variogram to condition simulation models of the error. Many data

providers already hold those higher precision data, and merely need to make them

available, either as an addendum to the existing data ®les which are being distributed or as

a separate ®le.

The only time when this is inappropriate is when the spot heights are de®nitely used in

the generation of the DEM itself. This may happen with OS DEMs, but the author has not

been able to ascertain whether this is the case either for the tiles examined here or in

general. In such circumstances, collection of independent height values by high precision

survey must be the responsibility of the user. A clear de®nition of the data sources is also a

necessary part of the metadata.

8. Conclusion

The increase in complexity of the error models shown here has yielded an increasingly

plausible probable viewshed. The improved methods are the best currently available for

error modeling, yielding results which are completely defensible. In some situations, the

230 FISHER

second model (with simulated spatial autocorrelation) may suf®ce, but only if the local

error parameters for the study area conform to the description of error provided by the

data supplier.

To use error estimates in developing realistic models of error so that those errors can be

propagated into non-trivial spatial operations requires a complete description of the

statistical and spatial distribution of the errors. The best way to do that is not for the data

providers to give still more estimates of global parameters, but to provide well surveyed

data points of higher precision than the database itself.

The probable viewsheds which result in the conditional method used here are

considered the most completely defensible, although empirical proof is considered

impossible, because the probabilities derived relate to the database error, and so to nothing

measurable in the ®eld (the basis of empirical proof ). General checks might be made that

areas with high probability can indeed be seen from the locations described, but that is not

simple, and even then it is not entirely clear how to evaluate the results in terms of

probabilities. Estimation of the probabilities is, however, based on a spatially distributed

model of error using the best possible method of error simulation. The difference between

this and previous research is that the methods used are readily available in many

geostatistical packages already loosely coupled to GIS [5], and so they could be adopted

immediately to become part of the standard GIS toolkit.

Acknowledgment

The work reported here was funded by the Countryside Commission for Wales, as a

subcontract to the Macaulay Land Use Research Institute. I would like to particularly

thank David Miller (MLURI), and Peter Minto (CCW) for their separate contributions to

the work. I would like to thank Robert Jeansoulin for organizing the meeting in Paris

at which the work reported here was ®rst presented, and I would like to thank three

anonymous reviewers whose comments have immeasurably improved the manuscript.

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232 FISHER

Peter Fisher holds degrees from the Universities of Lancaster and Reading, and Kingston Polytechnic (now

University). He has lectured in Geography Departments at Kingston Polytechnic, Kent State University and is

now Professor of Geographical Information at the University of Leicester. He has published more than 50 papers

in refereed journals, and written more papers for conference proceedings. He is editor of the International Journalof Geographical Information Science, and is co-editor of the Research Monographs in Geographical InformationSystems.


Improved Modeling of Elevation Error with Geostatistics

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